title: Toy Dolls

text: The band released a best-of album, "Ten Years of Toys". 1993 saw the popular live song "I'm a Telly Addict" on the album "Absurd-Ditties". In 1997, the band released "One More Megabyte", which featured references to computers, mainly in the lyrics of its title track, and in tracks like the outro track. A popular cover of Ricky Martin's "Livin' La Vida Loca" made its way onto "Anniversary Anthems" in 2000. The band's 2004 album "Our Last Album?" fooled fans in the anticipation up to its release that it was the band's last album. The outro track on the album stated that it was not.


title: Toy Dolls

text: To celebrate the band's twenty-fifth anniversary in 2004, the official biography, "The Toy Dolls: From Fulwell to Fukuoka", was published by Ardra Press.


title: Toy Dolls

text: In 2006, The Toy Dolls contributed their cover version of "Toccata in Dm" to the album project "Artists for Charity – Guitarists 4 the Kids", produced by Slang Productions, to assist World Vision Canada in helping underprivileged kids in need.


title: Toy Dolls

text: For their thirtieth anniversary in October 2009, the band received congratulatory messages from a selection of musicians, including Baz Warne from The Stranglers, TV Smith from The Adverts Jake Burns from Stiff Little Fingers, and NOFX manager Kent Jamieson.


title: Toy Dolls

text: The Toy Dolls live shows consist of synchronized choreography, timed moves, dances, jumps, and spinning guitars.


title: Toy Dolls

text: "The Album After the Last One" was released in 2012. The band has been on a "The Tour After the Last One" and started a range of Toy Dolls-themed real ales under the label "The Beer After the Last One".


title: Toy Dolls

text: In 2018, the band played one show in Montebello, Quebec, for the city's annual Rockfest music festival, marking the first time in the band's 38-year history that they played a show in Canada.


title: Toy Dolls

text: As of 2007, the band has had fourteen different drummers and twelve different bassists. Olga is the only remaining original member of the band. After Olga, drummer Martin Yule had the longest tenure in the band (1987–2000). Yule currently owns a shop in Sunderland called "Hotrats." Former drummer Dave "the Nut" Nuttall owns Jalapeño Drums, a custom drum company based in Lancaster, Lancashire. The current lead singer/guitarist with The Stranglers, Baz Warne, also had a stint with the band. Olga has played bass on tours with the US punk band The Dickies, and also with The Adicts. Gary Dunn, former bassist of The Toy Dolls (1997–1999), teaches music at Harton Academy, and former drummer, Trevor "The Frog" Brewis, teaches music at Middlesbrough College, though Olga moved away from the northeast of England a number of years ago, and now lives in London.


title: Perron–Frobenius theorem

text: In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Hawkins–Simon condition);


title: Perron–Frobenius theorem

text: to social networks (DeGroot learning process); to Internet search engines (PageRank); and even to ranking of football


title: Perron–Frobenius theorem

text: teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau.


title: Perron–Frobenius theorem

text: Let positive and non-negative respectively describe matrices with exclusively positive real numbers as elements and matrices with exclusively non-negative real numbers as elements. The eigenvalues of a real square matrix "A" are complex numbers that make up the spectrum of the matrix. The exponential growth rate of the matrix powers "A" as "k" → ∞ is controlled by the eigenvalue of "A" with the largest absolute value (modulus). The Perron–Frobenius theorem describes the properties of the leading eigenvalue and of the corresponding eigenvectors when "A" is a non-negative real square matrix. Early results were due to and concerned positive matrices. Later, found their extension to certain classes of non-negative matrices.


title: Perron–Frobenius theorem

text: Let formula_1 be an formula_2 positive matrix: formula_3 for formula_4. Then the following statements hold.


title: Perron–Frobenius theorem

text: All of these properties extend beyond strictly positive matrices to primitive matrices (see below). Facts 1-7 can be found in Meyer chapter 8 claims 8.2.11–15 page 667 and exercises 8.2.5,7,9 pages 668–669.


title: Perron–Frobenius theorem

text: The left and right eigenvectors "w" and "v" are sometimes normalized so that the sum of their components is equal to 1; in this case, they are sometimes called stochastic eigenvectors. Often they are normalized so that the right eigenvector "v" sums to one, while formula_11.


title: Perron–Frobenius theorem

text: There is an extension to matrices with non-negative entries. Since any non-negative matrix can be obtained as a limit of positive matrices, one obtains the existence of an eigenvector with non-negative components; the corresponding eigenvalue will be non-negative and greater than "or equal", in absolute value, to all other eigenvalues. However, for the example formula_12, the maximum eigenvalue "r" = 1 has the same absolute value as the other eigenvalue −1; while for formula_13, the maximum eigenvalue is "r" = 0, which is not a simple root of the characteristic polynomial, and the corresponding eigenvector (1, 0) is not strictly positive.


title: Perron–Frobenius theorem

text: However, Frobenius found a special subclass of non-negative matrices — "irreducible" matrices — for which a non-trivial generalization is possible. For such a matrix, although the eigenvalues attaining the maximal absolute value might not be unique, their structure is under control: they have the form formula_14, where "r" is a real strictly positive eigenvalue, and formula_15 ranges over the complex "h"th roots of 1 for some positive integer "h" called the period of the matrix.


title: Perron–Frobenius theorem

text: The eigenvector corresponding to "r" has strictly positive components (in contrast with the general case of non-negative matrices, where components are only non-negative). Also all such eigenvalues are simple roots of the characteristic polynomial. Further properties are described below.


title: Perron–Frobenius theorem

text: Here a non-trivial coordinate subspace means a linear subspace spanned by any proper subset of standard basis vectors of "F". More explicitly, for any linear subspace spanned by standard basis vectors "e", ...,


title: Perron–Frobenius theorem

text: Definition 3: One can associate with a matrix "A" a certain directed graph "G". It has "n" vertices labeled 1...,"n", and there is an edge from vertex "i" to vertex "j" precisely when "a" ≠ 0. Then the matrix "A" is irreducible if and only if its associated graph "G" is strongly connected.


title: Perron–Frobenius theorem

text: Definition 4: The group representation of formula_17 on formula_18 or formula_19 on formula_20 given by formula_21 has no non-trivial invariant coordinate subspaces. (By comparison, this would be an irreducible representation if there were no non-trivial invariant subspaces at all, not only considering coordinate subspaces.)


title: Perron–Frobenius theorem

text: A real matrix "A" is primitive if it is non-negative and its "m"th power is positive for some natural number "m" (i.e. all entries of "A" are positive).


title: Perron–Frobenius theorem

text: Let "A" be real and non-negative. Fix an index "i" and define the period of index "i" to be the greatest common divisor of all natural numbers "m" such that ("A") > 0. When "A" is irreducible, the period of every index is the same and is called the period of "A". In fact, when "A" is irreducible, the period can be defined as the greatest common divisor of the lengths of the closed directed paths in "G" (see Kitchens page 16). The period is also called the index of imprimitivity (Meyer page 674) or the order of cyclicity. If the period is 1, "A" is aperiodic. It can be proved that primitive matrices are the same as irreducible aperiodic non-negative matrices.


title: Perron–Frobenius theorem

text: All statements of the Perron–Frobenius theorem for positive matrices remain true for primitive matrices. The same statements also hold for a non-negative irreducible matrix, except that it may possess several eigenvalues whose absolute value is equal to its spectral radius, so the statements need to be correspondingly modified. In fact the number of such eigenvalues is equal to the period.


title: Perron–Frobenius theorem

text: Let formula_22 be an irreducible non-negative formula_23 matrix with period formula_24 and spectral radius formula_25.


title: Perron–Frobenius theorem

text: The example formula_58 shows that the (square) zero-matrices along the diagonal may be of different sizes, the blocks "A" need not be square, and "h" need not divide "n".


title: Perron–Frobenius theorem

text: A matrix "A" is primitive provided it is non-negative and "A" is positive for some "m", and hence "A" is positive for all "k ≥ m". To check primitivity, one needs a bound on how large the minimal such "m" can be, depending on the size of "A":


title: Perron–Frobenius theorem

text: Numerous books have been written on the subject of non-negative matrices, and Perron–Frobenius theory is invariably a central feature. The following examples given below only scratch the surface of its vast application domain.


title: Perron–Frobenius theorem

text: The Perron–Frobenius theorem does not apply directly to non-negative matrices. Nevertheless, any reducible square matrix "A" may be written in upper-triangular block form (known as the normal form of a reducible matrix)


title: Perron–Frobenius theorem

text: where "P" is a permutation matrix and each "B" is a square matrix that is either irreducible or zero. Now if "A" is


title: Perron–Frobenius theorem

text: non-negative then so too is each block of "PAP", moreover the spectrum of "A" is just the union of the spectra of the


title: Perron–Frobenius theorem

text: The invertibility of "A" can also be studied. The inverse of "PAP" (if it exists) must have diagonal blocks of the form "B" so if any


title: Perron–Frobenius theorem

text: Conversely let "D" be the block-diagonal matrix corresponding to "PAP", in other words "PAP" with the


title: Perron–Frobenius theorem

text: identity plus a nilpotent matrix. But such a matrix is always invertible (if "N" = 0 the inverse of 1 − "N" is


title: Perron–Frobenius theorem

text: Therefore, many of the spectral properties of "A" may be deduced by applying the theorem to the irreducible "B". For example, the Perron root is the maximum of the ρ("B"). While there will still be eigenvectors with non-negative components it is quite possible


title: Perron–Frobenius theorem

text: A row (column) stochastic matrix is a square matrix each of whose rows (columns) consists of non-negative real numbers whose sum is unity. The theorem cannot be applied directly to such matrices because they need not be irreducible.


title: Perron–Frobenius theorem

text: If "A" is row-stochastic then the column vector with each entry 1 is an eigenvector corresponding to the eigenvalue 1, which is also ρ("A") by the remark above. It might not be the only eigenvalue on the unit circle: and the associated eigenspace can be multi-dimensional. If "A" is row-stochastic and irreducible then the Perron projection is also row-stochastic and all its rows are equal.


title: Perron–Frobenius theorem

text: The theorem has particular use in algebraic graph theory. The "underlying graph" of a nonnegative "n"-square matrix is the graph with vertices numbered 1, ..., "n" and arc "ij" if and only if "A" ≠ 0. If the underlying graph of such a matrix is strongly connected, then the matrix is irreducible, and thus the theorem applies. In particular, the adjacency matrix of a strongly connected graph is irreducible.


title: Perron–Frobenius theorem

text: The theorem has a natural interpretation in the theory of finite Markov chains (where it is the matrix-theoretic equivalent of the convergence of an irreducible finite Markov chain to its stationary distribution, formulated in terms of the transition matrix of the chain; see, for example, the article on the subshift of finite type).


title: Perron–Frobenius theorem

text: More generally, it can be extended to the case of non-negative compact operators, which, in many ways, resemble finite-dimensional matrices. These are commonly studied in physics, under the name of transfer operators, or sometimes Ruelle–Perron–Frobenius operators (after David Ruelle). In this case, the leading eigenvalue corresponds to the thermodynamic equilibrium of a dynamical system, and the lesser eigenvalues to the decay modes of a system that is not in equilibrium. Thus, the theory offers a way of discovering the arrow of time in what would otherwise appear to be reversible, deterministic dynamical processes, when examined from the point of view of point-set topology.


title: Perron–Frobenius theorem

text: A common thread in many proofs is the Brouwer fixed point theorem. Another popular method is that of Wielandt (1950). He used the Collatz–Wielandt formula described above to extend and clarify Frobenius's work. Another proof is based on the spectral theory from which part of the arguments are borrowed.


title: Perron–Frobenius theorem

text: If "A" is a positive (or more generally primitive) matrix, then there exists a real positive eigenvalue "r" (Perron–Frobenius eigenvalue or Perron root), which is strictly greater in absolute value than all other eigenvalues, hence "r" is the spectral radius of "A".


title: Perron–Frobenius theorem

text: This statement does not hold for general non-negative irreducible matrices, which have "h" eigenvalues with the same absolute eigenvalue as "r", where "h" is the period of "A".


title: Perron–Frobenius theorem

text: Let "A" be a positive matrix, assume that its spectral radius ρ("A") = 1 (otherwise consider "A/ρ(A)"). Hence, there exists an eigenvalue λ on the unit circle, and all the other eigenvalues are less or equal 1 in absolute value. Suppose that another eigenvalue λ ≠ 1 also falls on the unit circle. Then there exists a positive integer "m" such that "A" is a positive matrix and the real part of λ"" is negative. Let ε be half the smallest diagonal entry of "A" and set "T" = "A" − "εI" which is yet another positive matrix. Moreover, if "Ax" = "λx" then "Ax" = "λx" thus "λ" − "ε" is an eigenvalue of "T". Because of the choice of "m" this point lies outside the unit disk consequently "ρ"("T") > 1. On the other hand, all the entries in "T" are positive and less than or equal to those in "A" so by Gelfand's formula "ρ"("T") ≤ "ρ"("A") ≤ "ρ"("A") = 1. This contradiction means that λ=1 and there can be no other eigenvalues on the unit circle.


title: Perron–Frobenius theorem

text: Absolutely the same arguments can be applied to the case of primitive matrices; we just need to mention the following simple lemma, which clarifies the properties of primitive matrices.


title: Perron–Frobenius theorem

text: Given a non-negative "A", assume there exists "m", such that "A" is positive, then "A", "A", "A"... are all positive.


title: Perron–Frobenius theorem

text: "A" = "AA", so it can have zero element only if some row of "A" is entirely zero, but in this case the same row of "A" will be zero.


title: Perron–Frobenius theorem

text: For a positive (or more generally irreducible non-negative) matrix "A" the dominant eigenvector is real and strictly positive (for non-negative "A" respectively non-negative.)


title: Perron–Frobenius theorem

text: This can be established using the power method, which states that for a sufficiently generic (in the sense below) matrix "A" the sequence of vectors "b" = "Ab" / | "Ab" | converges to the eigenvector with the maximum eigenvalue. (The initial vector "b" can be chosen arbitrarily except for some measure zero set). Starting with a non-negative vector "b" produces the sequence of non-negative vectors "b". Hence the limiting vector is also non-negative. By the power method this limiting vector is the dominant eigenvector for "A", proving the assertion. The corresponding eigenvalue is non-negative.


title: Perron–Frobenius theorem

text: The proof requires two additional arguments. First, the power method converges for matrices which do not have several eigenvalues of the same absolute value as the maximal one. The previous section's argument guarantees this.


title: Perron–Frobenius theorem

text: Second, to ensure strict positivity of all of the components of the eigenvector for the case of irreducible matrices. This follows from the following fact, which is of independent interest:


title: Perron–Frobenius theorem

text: Proof. One of the definitions of irreducibility for non-negative matrices is that for all indexes "i,j" there exists "m", such that ("A") is strictly positive. Given a non-negative eigenvector "v", and that at least one of its components say "j"-th is strictly positive, the corresponding eigenvalue is strictly positive, indeed, given "n" such that ("A") >0, hence: "r""v" =


title: Perron–Frobenius theorem

text: >0. Hence "r" is strictly positive. The eigenvector is strict positivity. Then given "m", such that ("A") >0, hence: "r""v" =


title: Perron–Frobenius theorem

text: This section proves that the Perron–Frobenius eigenvalue is a simple root of the characteristic polynomial of the matrix. Hence the eigenspace associated to Perron–Frobenius eigenvalue "r" is one-dimensional. The arguments here are close to those in Meyer.


title: Perron–Frobenius theorem

text: Given a strictly positive eigenvector "v" corresponding to "r" and another eigenvector "w" with the same eigenvalue. (The vectors "v" and "w" can be chosen to be real, because "A" and "r" are both real, so the null space of "A-r" has a basis consisting of real vectors.) Assuming at least one of the components of "w" is positive (otherwise multiply "w" by −1). Given maximal possible "α" such that "u=v- α w" is non-negative, then one of the components of "u" is zero, otherwise "α" is not maximum. Vector "u" is an eigenvector. It is non-negative, hence by the lemma described in the previous section non-negativity implies strict positivity for any eigenvector. On the other hand, as above at least one component of "u" is zero. The contradiction implies that "w" does not exist.


title: Perron–Frobenius theorem

text: Case: There are no Jordan cells corresponding to the Perron–Frobenius eigenvalue "r" and all other eigenvalues which have the same absolute value.


title: Perron–Frobenius theorem

text: Given "r" = 1, or "A/r". Letting "v" be a Perron–Frobenius strictly positive eigenvector, so "Av=v", then:


title: Perron–Frobenius theorem

text: So "A" is bounded for all "k". This gives another proof that there are no eigenvalues which have greater absolute value than Perron–Frobenius one. It also contradicts the existence of the Jordan cell for any eigenvalue which has absolute value equal to 1 (in particular for the Perron–Frobenius one), because existence of the Jordan cell implies that "A" is unbounded. For a two by two matrix:


title: Perron–Frobenius theorem

text: hence "J" = |"k" + "λ"| (for |"λ"| = 1), so it tends to infinity when "k" does so. Since "J" = "C" "A""C", then "A" ≥ "J"/ ("C" "C" ), so it also tends to infinity. The resulting contradiction implies that there are no Jordan cells for the corresponding eigenvalues.


title: Perron–Frobenius theorem

text: Combining the two claims above reveals that the Perron–Frobenius eigenvalue "r" is simple root of the characteristic polynomial. In the case of nonprimitive matrices, there exist other eigenvalues which have the same absolute value as "r". The same claim is true for them, but requires more work.


title: Perron–Frobenius theorem

text: Given positive (or more generally irreducible non-negative matrix) "A", the Perron–Frobenius eigenvector is the only (up to multiplication by constant) non-negative eigenvector for "A".


title: Perron–Frobenius theorem

text: Other eigenvectors must contain negative or complex components since eigenvectors for different eigenvalues are orthogonal in some sense, but two positive eigenvectors cannot be orthogonal, so they must correspond to the same eigenvalue, but the eigenspace for the Perron–Frobenius is one-dimensional.


title: Perron–Frobenius theorem

text: Assuming there exists an eigenpair ("λ", "y") for "A", such that vector "y" is positive, and given ("r", "x"), where "x" – is the left Perron–Frobenius eigenvector for "A" (i.e. eigenvector for "A"), then


title: Perron–Frobenius theorem

text: "rx""y" = ("x" "A") "y" = "x" ("Ay") = "λx""y", also "x" "y" > 0, so one has: "r" = "λ". Since the eigenspace for the Perron–Frobenius eigenvalue "r" is one-dimensional, non-negative eigenvector "y" is a multiple of the Perron–Frobenius one.


title: Perron–Frobenius theorem

text: the function "f" on the set of all non-negative non-zero vectors "x" such that "f(x)" is the minimum value of ["Ax"] / "x" taken over all those "i" such that "x" ≠ 0. Then "f" is a real-valued function, whose maximum is the Perron–Frobenius eigenvalue "r".


title: Perron–Frobenius theorem

text: For the proof we denote the maximum of "f" by the value "R". The proof requires to show " R = r". Inserting the Perron-Frobenius eigenvector "v" into "f", we obtain "f(v) = r" and conclude "r ≤ R". For the opposite inequality, we consider an arbitrary nonnegative vector "x" and let "ξ=f(x)". The definition of "f" gives "0 ≤ ξx ≤ Ax" (componentwise). Now, we use the positive right eigenvector "w" for "A" for the Perron-Frobenius eigenvalue "r", then " ξ w x = w ξx ≤ w (Ax) = (w A)x = r w x ". Hence "f(x) = ξ ≤ r", which implies


title: Perron–Frobenius theorem

text: Let "A" be a positive (or more generally, primitive) matrix, and let "r" be its Perron–Frobenius eigenvalue.


title: Perron–Frobenius theorem

text: Hence "P" is a spectral projection for the Perron–Frobenius eigenvalue "r", and is called the Perron projection. The above assertion is not true for general non-negative irreducible matrices.


title: Perron–Frobenius theorem

text: Actually the claims above (except claim 5) are valid for any matrix "M" such that there exists an eigenvalue "r" which is strictly greater than the other eigenvalues in absolute value and is the simple root of the characteristic polynomial. (These requirements hold for primitive matrices as above).


title: Perron–Frobenius theorem

text: Given that "M" is diagonalizable, "M" is conjugate to a diagonal matrix with eigenvalues "r", ... , "r" on the diagonal (denote "r" = "r"). The matrix "M"/"r" will be conjugate (1, ("r"/"r"), ... , ("r"/"r")), which tends to (1,0,0...,0), for "k → ∞", so the limit exists. The same method works for general "M" (without assuming that "M" is diagonalizable).


title: Perron–Frobenius theorem

text: The projection and commutativity properties are elementary corollaries of the definition: "MM"/"r" = "M"/"r" "M" ; "P" = lim "M"/"r" = "P". The third fact is also elementary: "M"("Pu") = "M" lim "M"/"r" "u" = lim "rM"/"r""u", so taking the limit yields "M"("Pu") = "r"("Pu"), so image of "P" lies in the "r"-eigenspace for "M", which is one-dimensional by the assumptions.


title: Perron–Frobenius theorem

text: Denoting by "v", "r"-eigenvector for "M" (by "w" for "M"). Columns of "P" are multiples of "v", because the image of "P" is spanned by it. Respectively, rows of "w". So "P" takes a form "(a v w)", for some "a". Hence its trace equals to "(a w v)". Trace of projector equals the dimension of its image. It was proved before that it is not more than one-dimensional. From the definition one sees that "P" acts identically on the "r"-eigenvector for "M". So it is one-dimensional. So choosing ("w""v") = 1, implies "P" = "vw".


title: Perron–Frobenius theorem

text: This is not specific to non-negative matrices: for any matrix "A" with an eigenvalue formula_66 it is true


title: Perron–Frobenius theorem

text: Any matrix induced norm satisfies the inequality formula_68 for any eigenvalue formula_66 because, if formula_70 is a corresponding eigenvector, formula_71. The infinity norm of a matrix is the maximum of row sums: formula_72 Hence the desired inequality is exactly formula_73 applied to the non-negative matrix "A".


title: Perron–Frobenius theorem

text: This fact is specific to non-negative matrices; for general matrices there is nothing similar. Given that "A" is positive (not just non-negative), then there exists a positive eigenvector "w" such that "Aw" = "rw" and the smallest component of "w" (say "w") is 1. Then "r" = ("Aw") ≥ the sum of the numbers in row "i" of "A". Thus the minimum row sum gives a lower bound for "r" and this observation can be extended to all non-negative matrices by continuity.


title: Perron–Frobenius theorem

text: Another way to argue it is via the Collatz-Wielandt formula. One takes the vector "x" = (1, 1, ..., 1) and immediately obtains the inequality.


title: Perron–Frobenius theorem

text: The proof now proceeds using spectral decomposition. The trick here is to split the Perron root from the other eigenvalues. The spectral projection associated with the Perron root is called the Perron projection and it enjoys the following property:


title: Perron–Frobenius theorem

text: Perron's findings and also (1)–(5) of the theorem are corollaries of this result. The key point is that a positive projection always has rank one. This means that if "A" is an irreducible non-negative square matrix then the algebraic and geometric multiplicities of its Perron root are both one. Also if "P" is its Perron projection then "AP" = "PA" = ρ("A")"P" so every column of "P" is a positive right eigenvector of "A" and every row is a positive left eigenvector. Moreover, if "Ax" = λ"x" then "PAx" = λ"Px" = ρ("A")"Px" which means "Px" = 0 if λ ≠ ρ("A"). Thus the only positive eigenvectors are those associated with ρ("A"). If "A" is a primitive matrix with ρ("A") = 1 then it can be decomposed as "P" ⊕ (1 − "P")"A" so that "A" = "P" + (1 − "P")"A". As "n" increases the second of these terms decays to zero leaving "P" as the limit of "A" as "n" → ∞.


title: Perron–Frobenius theorem

text: The power method is a convenient way to compute the Perron projection of a primitive matrix. If "v" and "w" are the positive row and column vectors that it generates then the Perron projection is just "wv"/"vw". The spectral projections aren't neatly blocked as in the Jordan form. Here they are overlaid and each generally has complex entries extending to all four corners of the square matrix. Nevertheless, they retain their mutual orthogonality which is what facilitates the decomposition.


title: Perron–Frobenius theorem

text: The analysis when "A" is irreducible and non-negative is broadly similar. The Perron projection is still positive but there may now be other eigenvalues of modulus ρ("A") that negate use of the power method and prevent the powers of (1 − "P")"A" decaying as in the primitive case whenever ρ("A") = 1. So we consider the peripheral projection, which is the spectral projection of "A" corresponding to all the eigenvalues that have modulus "ρ"("A"). It may then be shown that the peripheral projection of an irreducible non-negative square matrix is a non-negative matrix with a positive diagonal.


title: Perron–Frobenius theorem

text: Suppose in addition that ρ("A") = 1 and "A" has "h" eigenvalues on the unit circle. If "P" is the peripheral projection then the matrix "R" = "AP" = "PA" is non-negative and irreducible, "R" = "P", and the cyclic group "P", "R", "R", ..., "R" represents the harmonics of "A". The spectral projection of "A" at the eigenvalue λ on the unit circle is given by the formula formula_75. All of these projections (including the Perron projection) have the same positive diagonal, moreover choosing any one of them and then taking the modulus of every entry invariably yields the Perron projection. Some donkey work is still needed in order to establish the cyclic properties (6)–(8) but it's essentially just a matter of turning the handle. The spectral decomposition of "A" is given by "A" = "R" ⊕ (1 − "P")"A" so the difference between "A" and "R" is "A" − "R" = (1 − "P")"A" representing the transients of "A" which eventually decay to zero. "P" may be computed as the limit of "A" as "n" → ∞.


title: Perron–Frobenius theorem

text: The matrices "L" = formula_76, "P" = formula_77, "T" = formula_78, "M" = formula_79 provide simple examples of what can go wrong if the necessary conditions are not met. It is easily seen that the Perron and peripheral projections of "L" are both equal to "P", thus when the original matrix is reducible the projections may lose non-negativity and there is no chance of expressing them as limits of its powers. The matrix "T" is an example of a primitive matrix with zero diagonal. If the diagonal of an irreducible non-negative square matrix is non-zero then the matrix must be primitive but this example demonstrates that the converse is false. "M" is an example of a matrix with several missing spectral teeth. If ω = e then ω = 1 and the eigenvalues of "M" are {1,ω,ω=-1,ω} with a dimension 2 eigenspace for -1 so ω and ω are both absent.


title: Perron–Frobenius theorem

text: A problem that causes confusion is a lack of standardisation in the definitions. For example, some authors use the terms "strictly positive" and "positive" to mean > 0 and ≥ 0 respectively. In this article "positive" means > 0 and "non-negative" means ≥ 0. Another vexed area concerns "decomposability" and "reducibility": "irreducible" is an overloaded term. For avoidance of doubt a non-zero non-negative square matrix "A" such that 1 + "A" is primitive is sometimes said to be "connected". Then irreducible non-negative square matrices and connected matrices are synonymous.


title: Perron–Frobenius theorem

text: The nonnegative eigenvector is often normalized so that the sum of its components is equal to unity; in this case, the eigenvector is the vector of a probability distribution and is sometimes called a "stochastic eigenvector".


title: Perron–Frobenius theorem

text: "Perron–Frobenius eigenvalue" and "dominant eigenvalue" are alternative names for the Perron root. Spectral projections are also known as "spectral projectors" and "spectral idempotents". The period is sometimes referred to as the "index of imprimitivity" or the "order of cyclicity".


title: Carry On (Supernatural)

text: "Carry On" is the series finale of the American television series "Supernatural". It serves as the 20th episode of the fifteenth season, and the 327th overall. The episode was originally broadcast on The CW on November 19, 2020, and was written by showrunner and executive producer Andrew Dabb and directed by co-showrunner Robert Singer. The show stars Jared Padalecki and Jensen Ackles as Sam and Dean Winchester, two brothers who hunt demons, ghosts, monsters, and other supernatural beings. The episode shows Sam and Dean six months after the events of the previous episode, as they begin their final hunt together which ends in tragedy.


title: Carry On (Supernatural)

text: In January 2019, the CW renewed the series for a fifteenth season. In March, various cast members revealed that the new season of the show would be its last, with the series finale scheduled to air on May 18, 2020. However, production was shut down due to the COVID-19 pandemic with two episodes left to film, which resumed in August, and concluded in September 2020. The finale, which was rescheduled to air on November 19, 2020, was preceded by a special titled "The Long Road Home". In its initial broadcast, "Carry On" received positive reviews from critics, who deemed the episode a satisfactory conclusion to the series.


title: Carry On (Supernatural)

text: Six months after the events of the previous episode, Sam (Jared Padalecki) and Dean Winchester (Jensen Ackles) resume their regular lives hunting monsters. Sam is still expressing sadness with his allies Castiel (Misha Collins) and Jack Kline (Alexander Calvert) gone but Dean reassures him that they now have a chance to live a "more normal life".


title: Carry On (Supernatural)

text: In Akron, Ohio, intruders break into a family's house. The father is killed with his body drained of blood, the mother has her tongue removed and the children are kidnapped. Sam and Dean investigate and use their father's journal to identify the intruders as vampires he had hunted in 1986, predicting that their next target is Canton, Ohio. Following them, they kill one of the intruders and force the other to reveal the location of the children, who are at a barn. Once there, Sam and Dean free the children but they fight the vampires, with Sam being knocked out.


title: Carry On (Supernatural)

text: One of the vampires is Jenny (Christine Chatelain), a woman that they had failed to save from being turned into a vampire 14 years prior. Sam wakes up and they kill the vampires but Dean is impaled in the back by a spike. Sam intends to leave to find medical supplies but Dean has him stay, as his wound appears to be fatal. Dean reassures him that this was always how it was going to end for him and thanks Sam for everything, telling him he loves him and is proud of him. After saying goodbye, Dean dies in Sam's arms. The next day, Sam burns Dean's body in a funeral pyre.


title: Carry On (Supernatural)

text: Dean finds himself in Heaven and reunites with Bobby Singer (Jim Beaver) outside of the Roadhouse bar. Bobby reveals that after becoming God, Jack and Castiel reshaped Heaven to give everyone anything they wanted and tore down all of the walls keeping the souls separated from each other. After being told that his friend Rufus Turner (Steven Williams), and his parents John Winchester (Jeffrey Dean Morgan) and Mary Winchester (Samantha Smith) live in the surrounding area, Dean takes the Chevrolet Impala for a ride through Heaven as "Carry On Wayward Son" plays on the radio.


title: Carry On (Supernatural)

text: Sam continues with his life, getting married and having a child, whom he names after Dean. As he grows older and his health deteriorates, he is visited by Dean Jr. who is shown to have become a hunter as well before Sam peacefully dies of natural causes. He then reunites with Dean in Heaven, on a version of the bridge first seen in the pilot episode.


title: Carry On (Supernatural)

text: In January 2019, The CW renewed "Supernatural" for a fifteenth season. In March 2019, cast members Jared Padalecki, Jensen Ackles, and Misha Collins confirmed that the fifteenth season would be the last. Series creator Eric Kripke commented that "In a show about family, it is amazing, and it is the pride of his life, that it became family. So thank you guys for that." With 15 seasons and 327 episodes, the show holds the title as the longest-running show in The CW's history.


title: Carry On (Supernatural)

text: The series finale was originally set to air on May 18, 2020. However, in March 2020, Warner Bros. Television shut down production on the series due to the COVID-19 pandemic with two episodes, including the finale, left to go. Showrunner and executive producer Andrew Dabb revealed that the season would go on hiatus after episode 13 of the season, broadcast on March 23. Dabb clarified that the series had completed production on 18 of the 20 episodes for the season, but the post-production process could not be completed on the episodes because of the shutdown in production. Dabb also assured that the series' cast and crew, The CW, and Warner Bros. were fully committed to filming and airing the unproduced episodes with its proper finale.


title: Carry On (Supernatural)

text: In August 2020, The CW announced that the season would resume airing on October 8, 2020, and that the series finale would air on November 19, which was preceded by a special titled "The Long Road Home". Filming on the last two episodes began in Vancouver on August 18, and concluded on September 10, 2020.


title: Carry On (Supernatural)

text: According to an interview with "Entertainment Weekly", the original plans for the series' final moments would have had Dean going to Heaven and finding a party being held in a version of the Roadhouse Bar. In attendance would have been various characters from throughout the series along with Kansas themselves performing the titular song. The pandemic forced the studio to cancel this idea and the ending was rewritten.


title: Carry On (Supernatural)

text: The episode received 1.4 million viewers and was the most-watched "Supernatural" episode since April 2019.


title: Carry On (Supernatural)

text: "Carry On" received mainly positive reviews. Emily Tannenbaum of "IGN" gave the episode an "amazing" 9 out of 10 and wrote in her verdict that "the "Supernatural" series finale took its time, gracefully balancing reference and nostalgia with a hunt and endgame worthy of its legacy." Alex McLevy of "The A.V. Club" gave the episode a B+ and praised the episode for its story, conclusion of major character arcs, and its ending that gave him an "emotional gut punch" and ended the series as "a damn good one." While Saim Cheeda of "Screen Rant" lauded the episode for focusing on the Winchesters, Maryann Sleasman of "TV Guide" wrote, "I may not be thrilled by where we ended up, but the trip has been incredible."


The European Bank for Reconstruction and Development (EBRD) is an international financial institution founded in 1991. As a multilateral developmental investment bank, the EBRD uses investment as a tool to build market economies. Initially focused on the countries of the former Eastern Bloc it expanded to support development in more than 30 countries from Central Europe to Central Asia. Similar to other multilateral development banks, the EBRD has members from all over the world (North America, Africa, Asia and Australia, see below), with the biggest single shareholder being the United States, but only lends regionally in its countries of operations. Headquartered in London, the EBRD is owned by 71 countries and two European Union institutions, the newest shareholder being Algeria since October 2021. Despite its public sector shareholders, it invests in private enterprises, together with commercial partners.


title: European Bank for Reconstruction and Development

text: The EBRD is not to be confused with the European Investment Bank (EIB), which is owned by EU member states and is used to support EU policy. EBRD is also distinct from the Council of Europe Development Bank (CEB).


text: Located in London, the EBRD was intended at its beginning to focus on the switch from centrally planned economy to market based economy in the Central and Eastern European countries by investing in the private sector, mainly banks, industries and businesses, and by promoting policies that would favor them. It was created as a result of the end of the Cold War because of particular issues regarding the international system in place, the historical and political context at the time. Furthermore, its recipient countries had to insure that they would work towards "multi-party democracy and political pluralism" in which was considered as one of the main particularity of the EBRD distinguishing the latter from other financial institutions. Firstly, notably advocated by European political actors such as the French President Mitterrand in 1989, the idea of creating this financial institution involving the European countries started before the fall of the Berlin Wall. The reception of this proposition was mitigated depending on the country, for instance it received hostility from the British whereas the French were important proponents of the project and influenced the process. In January 1990, negotiations about the EBRD objectives, its lending policy regarding the public and private sectors, started and involved all the EU countries at the time, as well as various other European countries, non-European countries and representatives from the EU and the European Investment Bank. The Soviet Union was represented by its own delegation in the process and thus participated in the negotiations as well as the USA. Later on, because of dissensions on the political and economic objectives of the institution, the inclusion of the Soviet Union was threatened. Considered as short negotiations, the agreement (for reasons relating to the situation in Eastern Europe and to a wave of privatization across Europe) leading to its creation was signed on 29 May 1990 by actors such as 40 states as well as two European organizations and by March 1991 took its first effects as the bank was inaugurated. The creation of the EBRD has led to the financing of various projects in the years following its first operations, for instance there were twenty projects in 1992 in which the EBRD provided financing worth ECU 2.1 billion.


text: Divided between its headquarters located in London, and field offices located elsewhere, the European Bank for reconstruction and development is made of a three-tier structure composed of firstly, the president and the staff, secondly the Board of Governors and finally the Board of Directors. The bank’s structure has changed over time, because of concerns about competition within the institution. In fact, in the beginning of the 90s, there were two different banking divisions (on one side, the merchant banking related to the private sector and on the other side the development banking side working mostly with the public sector). Consequently, the two divisions came to be merged into a single one, substituting this structure with another one instead dividing the bank with seven policy and country regional sub-divisions.


text: The Board of Directors consists of 23 members elected by the Board of Governors. These members can not be on the Board of Governors. Its function is to direct the general operations of the bank, notably:


text: The EBRD is composed of many members with voting powers, from European and non-European states to the membership of other institutions such as the European Investment Bank, however depending on the geographical location of each member, voting rights differ. More precisely, European and other creditor members hold a majority voting power. At its beginning, the EBRD was owned by more than 40 members, in 2015 the number of countries owning it was 61. and as of March 2022, there were 71 countries as owners.


text: To become a member of the bank, a country needs to fit the EBRD's Articles of Agreement. These Articles stipulate that a country can only become a Member State if it is a state in Europe or a non-European member of the IMF. Institutions like the European Union and the EIB also participate in the bank operations. Since its fondation the EBRD has almost doubled in membership size. The latest accessions to the bank are countries located in Africa.


text: The EBRD is a public bank, meaning that public institutions such as states are shareholders in those institutions. More precisely, public financial institutions are controlled major predominantly by public authorities (more than 50%) whereas in institutions with public participations, public authorities aren’t majority shareholders. Public banks were particularly involved in the economic transition of the former Soviet Republics and Central and Eastern European countries as private banks and other sources of financing didn’t want to invest for reasons such as those countries’ macroeconomic difficulties as well as political reasons regarding the country’s stability for instance.


text: In the context of the EU’s investment banking, the EBRD and the EIB have been involved in a rivalry regarding the status of the “EU’s premier development bank”. Major criticism about the EBRD in this situation is related to the fact that non-EU countries are also important shareholders whereas the EIB is completely owned by the EU. More recently, another source of rivalry was about becoming the new “European Climate and Sustainable Development Bank (ECSDB).


text: More recently, since the start of the Covid-19 pandemic, public banks have had their influence and role increased as they have become important actors of the economic recovery response. As for the European Bank for Reconstruction and Development, it has committed €21 billion between 2020 and 2021 of activities. More precisely, its own package was established within the scope of the Resilience Framework, the Trade Facilitation Programme and the Vital Infrastructure Support Programme etc. In 2020 alone, it has invested €11 billion as part of 411 projects which corresponds to a 10% raison compared to 2019. 72% of its investments were received by the private sector. Moreover, because of this specific context, the EBRD focused some of its operations in sectors other than green investment meaning that it only focused 29% of its investment in this domain (compared to 46% in 2019).


text: According to the EBRD 2020 Sustainability Report, donors provided €589 million to support EBRD activities, taking the form of grants, loans and risk-mitigating instruments.


text: The EBRD funds come mainly from bilateral donors, from the Climate Investment Funds (CIF), the European Union (EU), the Global Environment Facility (GEF), and the Green Climate Fund (GCF).


text: According to the EBRD, the bank provides different direct financial instruments such as loans, equity investments and guarantees. It also works on the assistance of SMEs through the intermediary of other banks or investment and venture capital funds which are linked to the EBRD’s investments.


text: Each project is analysed, first by the EBRD management with the negotiations and the subsequent signing of a mandate letter drafting the important information about the project plan, the expected expenses as well as the responsibilities. Secondly, the project is to be finally reviewed by the management. Later on, it is submitted to the Board of Directors for approval. After being signed for both parties, it becomes legally binding. It leads to the transfer of funds from the EBRD to the client which will in the future start to repay the Bank’s loan.


text: The EBRD is unique among development banks for two reasons. First, it was the first multilateral development bank to have an explicit environmental mandate in its charter (since 1995), and second, in that it will not finance thermal coal mining and coal-fired electricity generation due to their environmental impact.


text: The following table shows the development of investment volume into the Green Economy Transition (GET) approach, in support of the Paris climate goals.


text: The EBRD had pledged, prior to 2015 Paris Agreement, to dedicate above 40 per cent of its financing to green investment by 2020. This goal was accomplished for the first time in 2017. With Russia actually being the biggest donor to an NDEP Support Fund for the environments inside the bank, with total contributions amounting to €60 million. €5 million was donated by Russia in the same year of 2015.


text: Not only involving the European institutions, the EBRD financed projects also implicate private actors such as firms as well as foreign investors in the process. As part of its lending policies, the EBRD mostly lends capital to the private sector as lending to the public sector is capped at 40% maximum.


text: As a neoliberal institution, the EBRD participates in policy debates by publishing macroeconomic analysis as well as economic forecasts that are used by both governments and international financial institutions.


text: More recently, other than its original lending role, the EBRD has played a mediating role in Europe, for instance in Moldova in 2013 and even in the Central Asian region (most notably in countries such as the Kyrgyz Republic and Tajikistan for instance) by intervening as a third party in commercial disputes which has become one of its major areas of activities. As a mediating actor, the EBRD has managed to involve other actors such as businesses and Chambers of Commerce among others. Moreover, a regional forum on commercial mediation was established by the bank as part of its work, which has managed to involve various representatives from countries in Europe and Asia, such as judges and experts for instance.


text: It is notable that one of the main objectives of the European Bank for Reconstruction and Development was to become a link between the European Union, its institutions and the countries it would focus on. As the recipients were previously centrally planned economies, the ERBD centered through its operations on working with the private sector, which was the principal target of central planning. Focusing on the integration and modernisation process, as part of its projects, the EBRD has notably invested in important sectors such as telecommunications in the Central and Eastern European countries.


text: Since the 90s, recipients of EBRD financing have diversified, in fact, from mainly targeting EEC countries, the bank has moved to the Asian continent and increasingly focused on other countries such as China. More generally, Suma Chakrabarti, the EBRD president from 2012 to 2020, has pushed in the beginning of the 2010s for modernizing the Bank. Reforms encompassed the Bank's structure in itself, for instance by changing the management team, and its processes (for instance about policy dialogue or methodology regarding the concept of transition) in order for them to be more efficient.


text: Since the 2000s, the focus has shifted towards the issue over sustainable market economies, notably in the context of the financial crisis of 2007 which harshly impacted population and their trust in financial institutions, democracy and free market. As a result, a rethinking of the concept of transition has taken place within the EBRD and has impacted the activities of the Bank. Furthermore, the concept of transition was reviewed in 2015. In fact, it led to the idea that the ideal market economy is made of a greatly competitive, resilient and integrated market and is complemented by a good governance system conducted by a well-functioning state which has to be environmental friendly and inclusive.


text: In Armenia, EBRD has been operating since 1986, investing over €2 billion in over 200 projects. More than 90% of these investments have been made in private enterprises. The bank's key areas of investment include the financial sector, SMEs, and sustainable infrastructure. They invest, engage in policy-making, and offer technical assistance in order to encourage these areas locally. The EBRD invests around €150 million in Armenia's economy every year. Its partner financial institutions lend approximately €100 million to local SMEs through a variety of programs that promote trade, green energy, and financial access.


text: Gradually, climate change and sustainable development became the focus point of the European Union, then consequently found its importance as one of the main challenges that the EBRD has to tackle. It became linked to the objective of forming sustainable market economies. In 2021, Anastassia Obydenkova research has indicated that positive evolutions have been made, for instance that carbon emission from some of the states benefiting from EBRD loans have decreased.


text: The EBRD agreed a new green economy approach for 2021-2025 and adopted a target of becoming a majority green bank by 2025. It invested €3.2 billion in climate mitigation, climate adaptation and other activities related to environmental issues in 2020. Moreover, the EBRD signed up to the Sustainable Blue Economy Finance Principles which represent the world’s first global guiding framework for banks, insurers, and investors to invest in the ocean economy in a sustainable way.


text: The EBRD also aim to support the corporate transition to decarbonisation, for instance in Poland (EV battery performance), in Turkey (elimination of plastics in single-use hygiene products), and in Ukraine (green supermarkets); and support green infrastructures.


text: Linked to its objective of contributing to political transformation in the EBRD's country targets in which projects are financed, this institution contributes to political transformation in these same states. From the beginning of the negotiations that led to the creation of the EBRD, the question of democracy and human rights were included in the matter discussed. Those issues are clearly stated in the agreement.


text: The EBRD is devoted to strengthen its approach to human-rights and also to gender-based issues. For instance, it posted a 14.8 percent year-on-year increase in the number of projects it supports that promote gender inclusion. Moreover, the EBRD established a cross-departmental Human Rights Working Group and has developed internal human rights guidance.


text: The EBRD provided €802 million of support under the Vital Infrastructure Support Programme launched in April 2020, an emergency support programme for infrastructure providers which is part of its overall Solidarity Package response to Covid-19 adopted on 13 March 2020. The package delivered emergency liquidity and the EBRD invested more than €10 billion, both in 2019 and 2020.


text: In the 90s, the start of the EBRD actions have been considered as slow as the amount of money loaned was rather modest compared to what was available.


text: There have been criticism regarding potential conflict of interest as it appeared that instead of focusing on public interests, some bankers' operations were influenced by their clients' needs, contrary to the objective of the EBRD.


text: According to an economist named Branko Milanović, the analysis of global distribution of incomes has shown that the positive globalisation impact has mainly favoured the super-rich part of the population, thus deepening the inequalities of wealth concentration within the population. Furthermore, this uneven concentration is notably visible in the countries benefiting from the EBRD activities. This has led to a reflection over the idea of winners and loosers regarding globalisation.


text: In contrast to the highlighted inclusion objective, concerns regarding inclusion, which notably involves gender equality, have been rather absent from the Bank's attributions until recently. For instance, a Strategic Gender Initiative was validated by the EBRD President in 2013 as well as various projects involving gender equality related objectives.


text: NGOs have criticised the EBRD on the lack of progress the EBRD makes in its main mission, the “transition towards open and democratic market economies.”


text: Inequality of the lending : most projects were related to the Visegrad countries that were the ones with the most developed private sectors meaning that the ERBD was underinvesting in the countries that were the most supposed to receive its participation. For instance, those countries (Cezch Republic, Hungary, Poland) were involved in 46% of the projects signed during the 1991-1993 period.


text: Some NGOs have criticized the EBRD for financing projects they consider to be environmentally and socially harmful. Although it has increased its investments into energy efficiency and sustainable energy in recent years, these NGOs consider the bank continues to diminish the impacts of green investments by financing carbon-intensive development such as coal, oil and gas production, transportation and generation, motorways, and airports. Among the contested projects are the Ombla power plant in Croatia, the Kumtor Gold Mine in Kyrgyzstan, and the Šoštanj lignite power plant in Slovenia.


text: The EBRD's activities in the Balkans have attracted particular controversy and criticism, especially when they have centered on national parks or free-flowing rivers. This has often involved the completed or proposed construction of hydroelectric dams and road infrastructure. Indeed, a 2017 report alleged deficiencies in monitoring and mitigation measures that had been designed to lessen the environmental impact of dam projects financed by the EBRD, while, in March 2018, outdoor clothing label Patagonia helped launch "The Dam Truth" campaign, which directly requests international banks including the EBRD to "stop investing in the destruction of Europe's last wild rivers".


text: In 2011, the EBRD approved a €65 million loan to ELEM, the Macedonian electricity utility, for a dam at Boskov Most. The Standing Committee of the Bern Convention requested immediate suspension of the project, with reference to the high biodiversity of the area and its importance as a core reproductive area for the Balkan lynx, one of the most endangered mammals on the planet. In January 2017, the bank cancelled the loan saying the "conditions for disbursement were not met."


text: Again in North Macedonia, the EBRD was criticised by environmentalists after plans were announced to bisect National Park Galičica in the UNESCO Ohrid-Prespa Transboundary Biosphere Reserve with an A3 express road, which would have required certain zones of protection in the national park to be downgraded. Scientists from North Macedonia and across the world signed a declaration in opposition to this and other projects proposed for the Ohrid-Prespa region, a message that was reinforced by a Joint Reactive Monitoring Mission from the World Heritage Centre, ICOMOS and the IUCN, which requested total cancellation of proposed A3 road sections. This recommendation was underlined by the World Heritage Committee at its 41st session in Kraków. Eventually, in February 2018, the Republic of North Macedonia abandoned plans for the road, redirecting the EBRD's funds to other infrastructure projects.


text: Even though, its major objectives is to push for democracy and respect of human rights, the EBRD has been accused of not working towards those goals by not applying enough pressure and even "holding its annual meetings in dictator-run countries". Furthermore, reports have highlighted the fact that an important part of the loans provided by the EBRD go to countries with authoritarian leaders as of the beginning of 2022, including states such as Belarus or Egypt among others, and this despite its major objective of promoting democracy.


text: The EBRD announced on 23 July 2014 that it would suspend new investment projects in Russia, following an earlier declaration by the European Council. The European Council declaration was made in the context of the 2014 pro-Russian unrest in Ukraine. As of 2014 Russia has been the biggest funding recipient of all countries. In 2013, the Russian Federation received €1.8 billion for investments from the EBRD and 1 billion € from the EIB. Russia employed the funds to finance a variety of projects like pipeline valves, property acquisitions, and a loan to a hypermarket chain. Two Russian projects were awaiting funding from the EBRD: a €300 million plan for promoting energy efficiency, and a $180 million loan to lease agricultural and forestry equipment. The bank stated that it will continue to manage on-going projects in Russia. Despite denying Russia new funds the EBRD continues to insist on its 6.1% of ownership in the Moscow Stock Exchange, seeking profits from Russia with the Privatisation of the Soviet economy.


text: Building Southern Gas Corridor, European countries and companies support gas export from Azerbaijan and contribute to finance Azerbaijan's government. The Aliyev regime is considered by many NGOs and watchdogs organizations as repressive and activists and journalists are regularly arrested on false charges and imprisoned. The International Federation for Human Rights (FIDH) wrote in 2015, that "for more than a decade Azerbaijan has made shameless use of caviar diplomacy to charm European governments, its most important oil and gas clients". The EEC BankWatch warns that "Developing Shah Deniz stage 2 and the Southern Gas Corridor is likely to cement further the oppressive structures of the Aliyev government".


text: The EBRD has invested over 3 billion euros ($3.5 billion) in 177 projects in Azerbaijan. The EBRD approved three loans for the Shah Deniz gas field (US$200 million, US$250 million and US$100 million) as well as a US$500 million loan for the Trans-Anatolian gas pipeline (TANAP), and stated its financial support to the Trans Adriatic Pipeline (TAP) could amount to EUR 1.2 billion.


title: Replenishment oiler

text: A replenishment oiler or replenishment tanker is a naval auxiliary ship with fuel tanks and dry cargo holds which can supply both fuel and dry stores during underway replenishment (UNREP) at sea. Many countries have used replenishment oilers.


title: Replenishment oiler

text: The United States Navy's hull classification symbol for this type of ship was AOR. Replenishment oilers are slower and carry fewer dry stores than the U.S. Navy's modern fast combat support ships, which carry the classification AOE.


title: Replenishment oiler

text: The development of the "oiler" paralleled the change from coal- to oil-fired boilers in warships. Prior to the adoption of oil fired machinery, navies could extend the range of their ships either by maintaining coaling stations or for warships to raft together with colliers and for coal to be manhandled aboard. Though arguments related to fuel security were made against such a change, the ease with which liquid fuel could be transferred led in part to its adoption by navies worldwide.


title: Replenishment oiler

text: One of the first generation of "blue-water" navy oiler support vessels was the British RFA "Kharki", active 1911 in the run-up to the First World War. Such vessels heralded the transition from coal to oil as the fuel of warships and removed the need to rely on, and operate within range of coaling stations. During the Second World War, the United States Navy's dramatically enlarged fleets, especially those in the Pacific Theater, required massive quantities of black oil, diesel oil, avgas, and other fuels and lubricants to support American land, sea, and air operations against remote, widely dispersed Japanese forces. Those supply demands resulted in U.S. Navy personnel refining many established practices for oilers and creating new procedures for replenishing warships while underway and for transporting highly combustible materials with increased effectiveness through hostile waters and over vast ocean distances.


title: Replenishment oiler

text: Modern examples of the fast combat support ship include the large British Fort class, displacing and measuring in length and the United States' , which displaces and has an overall length of .


title: Replenishment oiler

text: For all but the largest navies, replenishment oilers are typically one of the largest ships in the fleet. Such ships are designed to carry large amounts of fuel and dry stores for the support of naval operations far away from port. Replenishment oilers are also equipped with more extensive medical and dental facilities than smaller ships can provide.


title: Replenishment oiler

text: Such ships are equipped with multiple refueling gantries to refuel and resupply multiple ships at a time. The process of refueling and supplying ships at sea is called underway replenishment. Furthermore, such ships often are designed with helicopter decks and hangars. This allows the operation of rotary-wing aircraft, which allows the resupply of ships by helicopter. This process is called vertical replenishment. These ships, when operating in concert with surface groups, can act as an aviation maintenance platform where helicopters receive more extensive maintenance than can be provided by the smaller hangars of the escorting ships.


title: Replenishment oiler

text: Their size, additional facilities, and ability to support the operation of other vessels, means that replenishment oilers have been used as command ships, with some ships, such as the French , this capability being built into the vessels from the start.


title: Replenishment oiler

text: Because the replenishment oiler is not a combat unit, but rather a support vessel, such ships are often lightly armed, usually with self-defense systems (such as the Phalanx CIWS close-in weapons systems), small arms, machine guns and/or light automatic cannons. They may also carry man-portable air-defense systems for additional air defense capability.


title: Replenishment oiler

text: In the United States Navy, an Oiler is a Combat Logistics ship that replenishes other ships with fuel and in some cases food, mail, ammunition and other necessities while at sea, in a process called Underway Replenishment or UNREP. Up through the Second World War Navy oilers used commercial tanker hulls, with the addition of UNREP gear, defensive guns, and military electronic and damage-control equipment; since the 1950s however they have been built from the keel up as specialized naval auxiliaries. They were previously classified as Fleet Oilers in the 20th Century; under the current MSC operation their full classification is listed as Fleet Replenishment Oilers. Since the 1960s the classification Transport Oiler (AOT) has applied to tankers which ship petroleum products to depots around the world, but do not engage in UNREP.


title: Replenishment oiler

text: The first fleet oilers were identified by the hull designation AO, which is still in use. Large, fast multifunction oilers which also provide ammunition and dry stores are identified as Fast Combat Support Ships (AOE), and mid-size ones Replenishment Oilers (AOR). The AOR designation is no longer in use. All of these oilers provide the combined services of the AO, AE, AFS and AK.


title: Replenishment oiler

text: The style "USNS" and prefix "T" identify a ship as being operated by a civilian crew under the Military Sealift Command (known as the Military Sea Transportation Service until 1970).


title: 1984 Canadian federal election

text: The 1984 Canadian federal election was held on September 4, 1984, to elect members to the House of Commons of the 33rd Parliament of Canada.


title: 1984 Canadian federal election

text: In one of the largest landslide victories in Canadian political history, the Progressive Conservative Party (PC Party), led by Brian Mulroney, defeated the incumbent governing Liberal Party led by Prime Minister John Turner. This was the first election since 1958 in which the PC Party won a majority government.


title: 1984 Canadian federal election

text: Mulroney's victory came as a result of his building of a 'grand coalition' that comprised social conservatives from the West, Red Tories from the East, Quebec nationalists, and fiscal conservatives. Mulroney's PCs won the largest number of seats in Canadian history (at 211) and his party also won the second-largest percentage of seats in Canadian history (at 74.8%), only ranking behind Progressive Conservative Prime Minister John Diefenbaker's triumph in the 1958 federal election (at 78.5%). This was the last time that the winning party won every province and territory and the last time that the winning party received over 50% of the national popular vote. The Liberals suffered what at that time was the worst defeat for a governing party at the federal level (in terms of percentage of seats). The New Democratic Party (NDP) held on to nearly all of their seats.


title: 1984 Canadian federal election

text: The election marked the end of the Liberals' long dominance of federal politics in Quebec, a province which had been the bedrock of Liberal support for almost a century; they did not win a majority of Quebec seats again until three decades later in 2015.


title: 1984 Canadian federal election

text: The election was fought almost entirely on the record of the Liberals, who had been in power for all but 1 year out of 21, since 1963.


title: 1984 Canadian federal election

text: Pierre Trudeau, who had been Prime Minister from 1968 to 1979 and since 1980, retired from politics in early 1984. He was succeeded by John Turner, a former Cabinet minister under both Trudeau and Lester B. Pearson.


title: 1984 Canadian federal election

text: Turner had been out of politics for nine years. Upon assuming the leadership, he made immediate changes in an attempt to rebuild the Liberals' struggling reputation. For example, he announced that he would not run in a by-election to return to the House of Commons, but would instead run in the next general election as the Liberal candidate in the British Columbia riding of Vancouver Quadra. This was a sharp departure from usual practice, in which the incumbent in a safe seat resigns to allow a newly elected party leader a chance to get into Parliament. But the Liberal Party had lost favour with western Canadians, and policies such as the National Energy Program only aggravated this sentiment. Turner's plan to run in a western Canada riding was, in part, an attempt to rebuild support in that region. Going into the election, the Liberals held only one seat west of Ontario—that of Lloyd Axworthy, from Winnipeg—Fort Garry, Manitoba.


title: 1984 Canadian federal election

text: More seriously, there was great disaffection in Quebec with the Liberal government, despite their traditional support for the party. Conflict between the provincial and federal parties, a series of scandals, and the 1982 patriation of the Canadian constitution without the approval of the Quebec provincial government had damaged the Liberals' brand in the province. Hoping for success in Quebec, leader Joe Clark began actively courting soft nationalist voters in the province, and was one of the main reasons businessman Brian Mulroney, a fluently bilingual native of Quebec, was chosen as Clark's replacement.


title: 1984 Canadian federal election

text: Although Turner was not required to call an election until 1985, internal data initially showed that the Liberals had regained the lead in opinion polls. Turner and his advisers were also mindful of the fact that Trudeau had seemingly missed an opportunity to take advantage of favourable opinion polls in the latter half of the 1970s, when he waited the full five years to call an election only to go down to an (albeit temporary) defeat. Another factor was that the majority that the Liberals had won at the previous election had slowly been eroded in the years that followed, as a result of several by-election losses and floor-crossings. Turner's minority government left it in serious danger of being toppled by a motion of no confidence. With this in mind, the new Prime Minister requested that Queen Elizabeth II delay her tour of Canada, and asked Governor-General Jeanne Sauvé to dissolve Parliament on July 9. In accordance with Canadian constitutional practice, Sauvé granted the request and set the election for September 4.


title: 1984 Canadian federal election

text: The Liberals' early lead began to slip as Turner made several prominent gaffes. In particular, he spoke of creating new "make work programs", a concept from earlier decades that had been replaced by the less patronizing-sounding "job creation programs". He also was caught on camera patting Liberal Party president Iona Campagnolo on her posterior. Turner defended this action as being a friendly gesture, but it was seen by many as condescending.


title: 1984 Canadian federal election

text: Other voters turned against the Liberals due to their mounting legacy of patronage and corruption. An especially important issue was Trudeau's recommendation that Sauvé appoint over 200 Liberals to patronage posts just before he left office. This action enraged Canadians on all sides. Although Turner had the right to advise that the appointments be withdrawn (something that Sauvé would have had to do according to constitutional convention), he didn't do so, and in fact, appointed more than 70 Liberals to patronage posts himself despite a promise to bring a new way of politics to Ottawa. He cited a written agreement with Trudeau, claiming that if Trudeau had made the appointments, the Liberals would have almost certainly lost the election. However, the fact that Turner dropped the writ a year early hurt his argument.


title: 1984 Canadian federal election

text: Turner found out that Mulroney was allegedly setting up a patronage machine in anticipation of victory. At the English-language televised debate between Mulroney, Turner and New Democratic Party leader Ed Broadbent, Turner started to attack Mulroney on his patronage plans, comparing them to the patronage machine run by old Union Nationale in Quebec. However, Mulroney turned the tables by pointing to the raft of patronage appointments made on the advice of Trudeau and Turner. Claiming that he'd gone so far as to apologize for making light of "these horrible appointments," Mulroney demanded that Turner apologize to the country for not cancelling the appointments advised by Trudeau and for recommending his own appointments. Turner was visibly surprised, and could only reply that "I had no option" except to let the appointments stand. Mulroney famously responded:


title: 1984 Canadian federal election

text: You had an option, sir. You could have said, 'I am not going to do it. This is wrong for Canada, and I am not going to ask Canadians to pay the price.' You had an option, sir—to say 'no'—and you chose to say 'yes' to the old attitudes and the old stories of the Liberal Party. That sir, if I may say respectfully, that is not good enough for Canadians.


title: 1984 Canadian federal election

text: Turner, clearly flustered by this withering riposte from Mulroney, could only repeat "I had no option." A visibly angry Mulroney called this "an avowal of failure" and "a confession of non-leadership." He told Turner, "You had an option, sir. You could have done better." Mulroney's counterattack led most of the papers the next day; it was often paraphrased as "You had an option, sir; you could have said 'no'." Many observers saw this as the end of any realistic chance for Turner to stay in power.


title: 1984 Canadian federal election

text: The last days of the campaign saw multiple Liberal blunders pile together. Turner continued to refer to "make work projects" and made other gaffes that caused voters to see him as a relic from the past. Turner even rehired much of Trudeau's staff during the final weeks in an attempt to turn the tide, but this did nothing to reverse sliding poll numbers. Even Trudeau himself did not campaign for Turner, instead only making appearances to support Liberal candidates.


title: 1984 Canadian federal election

text: Besides the Tories, the NDP also benefited from the slip in Liberal support. Under Broadbent, the party had seen greater support in opinion polling than ever before, and had actually replaced the Liberals as the second party in much of western Canada.


title: 1984 Canadian federal election

text: The Revolutionary Workers League fielded five candidates: Michel Dugré, Katy Le Rougetel, Larry Johnston, Bonnie Geddes and Bill Burgess. All appeared on the ballot as independent or non-affiliated candidates, as the party was unregistered.


title: 1984 Canadian federal election

text: Turner's inability to overcome the alleged resentment against Trudeau, combined with his own mistakes, resulted in a debacle for the Liberals. They lost over a third of their popular vote from 1980, falling from 44 percent to 28 percent. Their seat count fell from 135 at dissolution to 40, a loss of 95 seats—the second worst defeat of a sitting government in Canadian history at the time (after the 1935 election), and among the worst defeats ever suffered by a governing party in a Westminster system. It was the worst performance in their long history at the time; the 40 seats would be their smallest seat count until they won only 34 seats in 2011, when they also fell to third place in the seat count for the first time at the national level. Eleven members of Turner's cabinet were defeated.


title: 1984 Canadian federal election

text: Despite their hopes of winning more support in the West, the Liberals won only two seats west of Ontario. One of those belonged to Turner, who defeated Tory incumbent Bill Clarke in Vancouver Quadra by a fairly solid 3,200-vote margin. The other belonged to Lloyd Axworthy, who was re-elected in Winnipeg—Fort Garry by 2,300 votes.


title: 1984 Canadian federal election

text: Particularly shocking was the decimation of the Liberals in Quebec. They won only 17 seats, all but four in and around Montreal. The province had been the bedrock of Liberal support for almost a century; the 1958 Tory landslide was the only time since the 1896 election that the Liberals had not won the most seats in Quebec. They would not win the most seats in the province again until the 2015 election (albeit they won the province's popular vote in 2000). In Ontario, the Liberals won only 14 seats, nearly all of them in Metro Toronto.


title: 1984 Canadian federal election

text: Early in the election, Mulroney focused on adding Quebec nationalists to the traditional Tory coalition of western populist conservatives and fiscal conservatives from Ontario and the Atlantic provinces.


title: 1984 Canadian federal election

text: This strategy, as well as denouncing alleged corruption in the Liberal government, resulted in a major windfall for the Tories. They won 211 seats, three more than their previous record of 208 in 1958. They won both a majority of seats and at least a plurality of the popular vote in every province and territory, the only time in Canadian history a party has achieved this (the nearest previous occasion being in 1949, when only Alberta kept the Liberals from a clean sweep). They also won just over half the popular vote, the last time to date that a Canadian party has won a majority of the popular vote.


title: 1984 Canadian federal election

text: The Tories had a major breakthrough in Quebec, a province where they had been virtually unelectable for almost a century. However, Mulroney's promise of a new deal for Quebec caused the province to swing dramatically to support him. After winning only one seat out of 75 in 1980, the Tories won 58 seats in 1984, more than they had ever won in Quebec before. In many cases, ridings where few living residents had ever been represented by a Tory elected them by margins similar to those the Liberals had scored for years.


title: 1984 Canadian federal election

text: The NDP lost only one seat, which was far better than expected considering the size of the PC tidal wave. Historically, third parties do not do well in landslides. More importantly, their 30 seats were only ten behind the Liberals.


title: 1984 Canadian federal election

text: Despite losing less than a percentage point in terms of total vote share, the NDP still finished with less than two-thirds of the Liberals' popular vote. However, the NDP vote proved itself to be much more efficient since it was not as evenly distributed across the country. While they were a distant third everywhere east of Ontario, the NDP won only one seat fewer than the Liberals in Ontario despite finishing a distant third there as well. In Western Canada, the NDP were by far the second-largest party behind the Tories. Some NDP incumbents essentially held on to their own share of the vote, and simply had to survive a swing from Liberal to PC which in many cases proved insufficient to unseat the sitting New Democratic MP.


title: 1984 Canadian federal election

text: Although the NDP had long since established itself as the third major party in Canada, this was closer than any party had gotten to the Grits or Tories since 1921, when the Progressive Party briefly surpassed the Tories. This led to speculation that Canada was headed for a UK-style Labour–Conservative division, with the NDP knocking the Liberals down to third-party status. It would be as close as the NDP would get to becoming the Official Opposition until 2011, when the party gained the second-most seats in the House of Commons and the majority of seats in Quebec.


title: 1984 Canadian federal election

text: The Social Credit Party, which for a long time had been the country's fourth-largest (and occasionally even third-largest) party, suffered a massive drop-off in support from the previous election, in which they had already lost a major share of the vote and all their remaining MPs. Having performed poorly in various by-elections in the years that followed, the party suffered a blow to its image in June 1983, when the party executive voted to re-admit a faction led by Holocaust denier James Keegstra. Party leader Martin Hattersley resigned in protest.


title: 1984 Canadian federal election

text: In Quebec, most of the support which had helped keep the party viable in its final years turned to the Progressive Conservatives. The party essentially disappeared there—even among ridings the party had won as recently as 1979, it could field only one candidate (who finished in last place). What remained of Social Credit essentially reverted to being a western-based party, which was only able to run 52 candidates in 51 ridings (with two Socreds standing in a British Columbia seat). It was its second-smallest slate since first running candidates east of Manitoba four decades earlier, and barely enough to allow Social Credit to retain its party registration. The party lost 92 percent of its vote from 1980 and dropped from fourth place to ninth in the popular vote. For all intents and purposes, this was the end of Social Credit as a viable national party. It would make a desultory final appearance in 1988 before collapsing altogether in 1993.


title: 1984 Canadian federal election

text: The satirical Rhinoceros Party, despite a slight drop in their popular vote tally from the previous election, recorded its highest-ever finish at a general election, finishing as the fourth-largest party. Of the minor parties, only the Parti nationaliste du Québec and the Confederation of Regions Party of Canada managed to record more votes per candidate than the Rhinos, and even then only by small margins.


title: 1984 Canadian federal election

text: The Parti nationaliste du Québec, a successor to the previous Quebec-nationalist Union populaire party, ran for the first (and, ultimately, only) time in this election. Despite getting nearly six times the votes that their predecessors did in 1980, and finishing fifth in the popular vote, like the Socreds they proved unable to compete with the Progressive Conservatives, and failed to win any seats. The party would eventually collapse in 1987, though several of its members, along with dissident Tories and Liberals, would go on to found the more successful Bloc Québécois.


title: 1984 Canadian federal election

text: The Confederation of Regions Party of Canada, formed mostly by disaffected former Socreds, were another party who debuted in this election. While they placed sixth in the popular vote and attracted a little over quadruple the vote of their forerunners, they still failed to seriously challenge for any seats. Much like the Socreds, they too disappeared from the national scene after 1988, though they continued on a regional level for several years afterwards.


title: Logic bomb

text: A logic bomb is a piece of code intentionally inserted into a software system that will set off a malicious function when specified conditions are met. For example, a programmer may hide a piece of code that starts deleting files (such as a salary database trigger), should they ever be terminated from the company.


title: Logic bomb

text: Software that is inherently malicious, such as viruses and worms, often contain logic bombs that execute a certain payload at a pre-defined time or when some other condition is met. This technique can be used by a virus or worm to gain momentum and spread before being noticed. Some viruses attack their host systems on specific dates, such as Friday the 13th or April Fools' Day. Trojans and other computer viruses that activate on certain dates are often called "time bombs".


title: Logic bomb

text: To be considered a logic bomb, the payload should be unwanted and unknown to the user of the software. As an example, trial programs with code that disables certain functionality after a set time are not normally regarded as logic bombs.


title: Logic bomb

text: It has been reported that in 1982, the Trans-Siberian Pipeline incident occurred because of a logic bomb. (It was later reported that this story may be a hoax.) A KGB operative was reported to have stolen the plans for a sophisticated control system and its software from a Canadian firm, for use on its Siberian pipeline. The Central Intelligence Agency (CIA) was supposedly tipped off by documents in the Farewell Dossier and had the company insert a logic bomb in the program for sabotage purposes.


title: Overtoun Bridge

text: Overtoun Bridge is a category B-listed structure over the Overtoun Burn on the approaching road to Overtoun House, near Dumbarton in West Dunbartonshire, Scotland. It was completed in 1895, based on a design by the landscape architect H. E. Milner.


title: Overtoun Bridge

text: Since the 1950s, numerous reports of dogs either falling or jumping from the bridge have been reported. With the incidents often resulting in serious injury or death upon landing on the rocks some below, the bridge has been nicknamed the "Dog Suicide Bridge". Various explanations for these deaths have been proposed, ranging from natural accidents to paranormal activity.


title: Overtoun Bridge

text: In 1859, the Overtoun Farm was acquired by Scottish industrialist James White, who had just started in the business of chemical manufacturing. He built the Overtoun House three years later in 1862. When White died in 1884, his son, John Campbell White, inherited the house and its estate and started planning to extend the driveway of the house across a deep ravine in order to provide easier access. He hired landscape architect and civil engineer Henry Milner to design a bridge.


title: Overtoun Bridge

text: The bridge was constructed using rough-faced ashlar and was completed in June 1895. It comprises three arches: a large central arch spanning a deep valley at the bottom of which flows the Overtoun Burn, flanked on each side by lower, smaller pedestrian arches.


title: Overtoun Bridge

text: During the 1950s, locals started referring to the bridge as the "Bridge of Death" or the "Dog Suicide Bridge", as it was reported that dogs were leaping from the bridge into the ravine below. The story gained more prominence during the late 2000s and early 2010s. Since the original incidents were reported,


title: Overtoun Bridge

text: In 2004, Kenneth Meikle was walking with his family and Golden Retriever, when the dog suddenly bolted and jumped off the bridge. It survived, but was traumatized by the experience. Going into 2005, at least five other dogs also jumped over the course of six months. In 2014, Alice Trevorrow, who was walking with her Springer Spaniel named Cassie, reported a strange experience on Overtoun Bridge. "I had parked up and as she is so obedient I didn't put her lead on... Me and my son walked toward Cassie, who was staring at something above the bridge... she definitely saw something that made her jump. There is something sinister going on. It was so out of character for her."


title: Overtoun Bridge

text: A number of theories have been proposed as to the behavior of dogs on the bridge. In 2014, canine psychologist David Sands proposed that the surrounding foliage – giving the in-reality extremely steep drop off the side of the bridge the appearance of even ground – combined with the residual odor from male mink urine in the area could be culprit for luring dogs to jump off the bridge. This theory was protested by a local hunter and resident of 50 years, John Joyce, who stated that there were "no mink [in the area]." However, in an investigation by the Royal Society for the Protection of Birds, Officer David Sexton found that one end of the bridge reportedly favored by dogs contained "nests of mice, squirrels, and minks." Furthermore, in an experiment in which ten dogs were exposed to canisters filled with mouse, squirrel and mink scent, seven of the dogs "all went straight for the mink scent, many of them quite dramatically."


title: Overtoun Bridge

text: The Scottish Society for the Prevention of Cruelty to Animals has investigated the bridge and surrounding area, but none of their findings proved conclusive.


title: Overtoun Bridge

text: In 2019, the owners of Overtoun House, Bob and Melissa Hill, said that in 17 years of residing at the House, they had witnessed a number of dogs become agitated, jump up, and fall from the bridge. Bob Hill, originally a pastor from Texas, stated that the scent of mink, pine martens, and other animals agitated the dogs, resulting in their jump onto the bridge wall: "The dogs catch the scent of mink, pine martens or some other mammal and then they will jump up on the wall of the bridge. And because it’s tapered, they will just topple over." However, Hill also stated his belief that the grounds of the House held some sort of spiritual quality.


title: Overtoun Bridge

text: Local teacher Paul Owens argues that the bridge and nearby Overtoun House are haunted by supernatural activity. He claims that dogs and other animals are sensitive to such supernatural activity, so he proposes that dark spirits are responsible for luring dogs to their deaths.


title: Overtoun Bridge

text: In October 1994, paranoid schizophrenic Kevin Moy threw his two-week-old son Eoghan to his death from the bridge because he believed that his son was an incarnation of the Devil due to a birthmark. He chose the location due to its association with dark spirits going back to the druidic days. Following his act of murder, Moy then attempted to commit suicide several times by jumping off the bridge and slashing his wrists, but he was detained and placed in a mental health hospital.


title: Overtoun Bridge

text: In October 2022, Heel Films used Overtoun House, the bridge and the alleged paranormal activities as a basis for their 2023 short film "The Bridge" written and directed by Scott McMillan; the short film stars Christopher Wallace and Susan Sims


title: Azoospermia

text: Azoospermia is the medical condition of a man whose semen contains no sperm. It is associated with male infertility, but many forms are amenable to medical treatment. In humans, azoospermia affects about 1% of the male population and may be seen in up to 20% of male infertility situations in Canada.


title: Azoospermia

text: Azoospermia can be classified into three major types as listed. Many conditions listed may also cause various degrees of oligospermia rather than azoospermia. Pretesticular and testicular azoospermia are known as non-obstructive azoospermia, whereas post-testicular azoospermia is considered obstructive.


title: Azoospermia

text: Pretesticular azoospermia is characterized by inadequate stimulation of otherwise normal testicles and genital tract. Typically, follicle-stimulating hormone (FSH) levels are low (hypogonadotropic) commensurate with inadequate stimulation of the testes to produce sperm. Examples include hypopituitarism (for various causes), hyperprolactinemia, and exogenous FSH suppression by testosterone. Chemotherapy may suppress spermatogenesis. Pretesticular azoospermia is seen in about 2% of azoospermia. Pretesticular azoospermia is a kind of non-obstructive azoospermia.


title: Azoospermia

text: Testicular azoospermia means the testes are abnormal, atrophic, or absent, and sperm production severely disturbed to absent. FSH levels tend to be elevated (hypergonadotropic) as the feedback loop is interrupted (lack of feedback inhibition on FSH). The condition is seen in 49–93% of men with azoospermia. Testicular failure includes absence of failure production and low production and maturation arrest during the process of spermatogenesis.


title: Azoospermia

text: Causes for testicular failure include congenital issues such as in certain genetic conditions (e.g. Klinefelter syndrome), some cases of cryptorchidism or Sertoli cell-only syndrome as well as acquired conditions by infection (orchitis), surgery (trauma, cancer), radiation, or other causes. Mast cells releasing inflammatory mediators appear to directly suppress sperm motility in a potentially reversible manner, and may be a common pathophysiological mechanism for many causes leading to inflammation. Testicular azoospermia is a kind of non-obstructive azoospermia.


title: Azoospermia

text: Generally, men with unexplained hypergonadotropic azoospermia need to undergo a chromosomal evaluation.


title: Azoospermia

text: In post-testicular azoospermia, sperm are produced but not ejaculated, a condition that affects 7–51% of azoospermic men. The main cause is a physical obstruction (obstructive azoospermia) of the post-testicular genital tracts. The most common reason is a vasectomy done to induce contraceptive sterility. Other obstructions can be congenital (for example, agenesis of the vas deferens as seen in certain cases of cystic fibrosis) or acquired, such as ejaculatory duct obstruction for instance by infection.


title: Azoospermia

text: Ejaculatory disorders include retrograde ejaculation and anejaculation; in these conditions sperm are produced but not expelled.


title: Azoospermia

text: Idiopathic azoospermia is where there is no known cause of the condition. It may be a result of multiple risk factors, such as age and weight. For example, a review in 2013 came to the result that oligospermia and azoospermia are significantly associated with being overweight (odds ratio 1.1), obese (odds ratio 1.3) and morbidly obese (odds ratio 2.0), but the cause of this is unknown. The review found no significant relation between oligospermia and being underweight.


title: Azoospermia

text: Genetic factors can cause pretesticular, testicular, and post-testicular azoospermia (or oligospermia) and include the following situations: The frequency of chromosomal abnormalities is inversely proportional to the semen count, thus males with azoospermia are at risk to have a 10–15% (other sources citing 15–20% incidence) abnormalities on karyotyping versus about <1 % in the fertile male population.


title: Azoospermia

text: Pretesticular azoospermia may be caused by congential hypopituitarism, Kallmann syndrome, Prader-Willi syndrome and other genetic conditions that lead to GnRH or gonadotropin deficiency.


title: Azoospermia

text: Testicular azoospermia is seen in Klinefelter syndrome (XXY) and the XX male syndrome. In addition, 13% of men with azoospermia have a defective spermatogenesis that is linked to defects of the Y chromosome. Such defects tend to be de novo micro-deletions and affect usually the long arm of the chromosome. A section of the long arm of the Y chromosome has been termed Azoospermia Factor (AZF) at Yq11 and subdivided into AZFa, AZFb, AZFc and possibly more subsections. Defects in this area can lead to oligospermia or azoospermia, however, a tight genotype-phenotype correlation has not been achieved.


title: Azoospermia

text: Post-testicular azoospermia can be seen with certain point mutations in the cystic fibrosis transmembrane conductance regulator (CFTR) gene commonly associated with congenital vas deferens abnormalities.


title: Azoospermia

text: Genetic counselling is indicated for men with genetic causes of azoospermia. In terms of reproduction, it needs to be considered if the genetic defect could be transmitted to the offspring.


title: Azoospermia

text: BRD7, a transcription regulatory protein, is normally highly expressed in the testis. Absent or reduced expression of BRD7 protein was observed in the testes of azoospermia patients exhibiting spermatogenesis arrest. Homozygous knockout mice ["BRD7"(-/-)] are infertile and have higher levels of apoptosis and DNA damage in their germline cells.


title: Azoospermia

text: The human breast cancer susceptibility gene 2 ("BRCA2") is employed in DNA repair. A common single nucleotide polymorphism in "BRCA2" is associated with idiopathic male infertility with azoospermia.


title: Azoospermia

text: Four genes involved in DNA double-strand break repair and chromosome synapsis ("TEX11", "TEX15", "MLH1" and "MLH3") have key roles in genomic integrity, meiotic recombination and gametogenesis. Polymorphisms in these genes were tested for associations with male infertility. Single nucleotide polymorphisms in two of these genes ("TEX11" and "MLH3") were found to be associated with male infertility involving azoospermy or oligospermia.


title: Azoospermia

text: Azoospermia is usually detected in the course of an infertility investigation. It is established on the basis of two semen analysis evaluations done at separate occasions (when the seminal specimen after centrifugation shows no sperm under the microscope) and requires a further work-up.


title: Azoospermia

text: The investigation includes a history, a physical examination including a thorough evaluation of the scrotum and testes, laboratory tests, and possibly imaging. History includes the general health, sexual health, past fertility, libido, and sexual activity. Past exposure to a number of agents needs to be queried including medical agents like hormone/steroid therapy, antibiotics, 5-ASA inhibitors (sulfasalazine), alpha-blockers, 5 alpha-reductase inhibitors, chemotherapeutic agents, pesticides, recreational drugs (marijuana, excessive alcohol), and heat exposure of the testes. A history of surgical procedures of the genital system needs to be elicited. The family history needs to be assessed to look for genetic abnormalities.


title: Azoospermia

text: Congenital absence of the vas deferens may be detectable on physical examination and can be confirmed by a transrectal ultrasound (TRUS). If confirmed, genetic testing for cystic fibrosis is in order. Transrectal ultrasound can also assess azoospermia caused by obstruction, or anomalies related to obstruction of the ejaculatory duct, such as abnormalities within the duct itself, a median cyst of the prostate (indicating a need for cyst aspiration), or an impairment of the seminal vesicles to become enlarged or emptied.


title: Azoospermia

text: Retrograde ejaculation is diagnosed by examining a post-ejaculatory urine for presence of sperm after making it alkaline and centrifuging it.