This is Michael Vunch. I'm the plant pathologist at the Carrington Research Center. I'm here at the Oaks Irrigation Research Site, which is a research site just south of Oaks, North Dakota, on irrigated sand. We're in a production belt here that has quite a bit of irrigated production on sands, and white mold is a real problem on the soybeans here. When I started working on white mold and soybeans, I got a lot of complaints from growers and crop advisors that they could not get consistent, reliable control of white mold with fungicides on soybeans, even though they could on dry beans. Now the first thing that we looked at was we looked at fungicide efficacy, but from there we moved on to look at optimizing fungicide application timing. If you look at the previous, the other video that we are producing for this Oaks Field Day on fungicide application timing, you'll find the details of what we learned in those studies. When we completed the fungicide application timing studies, we moved on to looking at fungicide droplet size. Now, as you may know, the standard recommendation for applying a fungicide is that you apply the fungicide with fine to medium droplets, and the reason for this is that fungicides exhibit only local systemic movement. When they are deposited on a leaf, or a stem, or a petio, they move upward from that point of deposition a certain degree, and oftentimes from one side of the leaf to the other. But they only move upward a limited degree, and they never move downward. And so what happens here is that coverage becomes important. If you only get one droplet on the leaf, you're only protecting a part of that leaf.  And it's not like an herbicide where the product moves both up and down. And so coverage becomes important. And when you have finer droplets, you get a lot more of them, and you cover a lot more of the plant surface. Now, the problem that we run into for white mold and soybeans is that white mold is a disease that develops on the bottom of the canopy. And if you look at, if you refer to our results on fungicide application timing, you'll notice that unless the canopy closed earlier, you need to be applying your fungicide for white mold at the full R2 growth stage. At full R2, many times the canopy is at or near closure. And the canopy is often 24 to 27 inches tall. The white mold develops usually on the bottom third to half of that canopy. And you somehow have to get your product down into the bottom third to half of a canopy that's at or near closure, and it's over two feet tall. And find a medium droplets while they cover the top of the canopy very well. They do not have the velocity to get inside a canopy that's at or near closure. So we've been running these studies with a tractor mounted R&D sprayer that's equipped with a pulse width modulation system. For that pulse width system, we're able to change nozzles that differ in output while maintaining our overall spray volume the same. We just increase our pulse width as the output of the nozzle increases. And we run these all of the tractors so we can drive at a realistic driving speed. Most of these studies have been done at nine to ten and a half miles an hour. And we've been running them with T-jet tips primarily, but we most more recently initiated trials also with wheelchair tips that allow us to change dropper size without changing pulse width because they have a lineup of tips that changes dropper size without changing output. And what we have learned is the following. When the canopy is open, there's still ground showing between the rows. When the soybeans are at full R2 growth stage, we have optimized the performance of fungicides by applying with fine to medium droplets just like the standard recommendation. This is with T-jet tips. And when the canopy has been wide open, the optimal droplet size has been a droplet on that boundary of fine and medium. I.e. fine but on the coarse render find if you will, medium but on the fine render medium, right at the boundary of fine and medium. That's when the canopy closure has averaged across the footprint of the entire study less than 80%. When the canopy closure has averaged between 80% and 90%, we have optimized fungicide performance with a droplet size that is in the middle of the medium spectrum. When the canopy closure is more than 90% on average, on average, across the footprint of our study, we have optimized our fungicide performance with coarse droplets. And this is anywhere from average canopy closure of 92% to 99%. What happens then is that there is not much ground showing between the rows. You have a pretty tight canopy and then you need velocity. You need a heavier droplet size to get inside. But the important thing to remember is that what our results show to date is that you must choose your droplet size based on canopy characteristics. The more open the canopy is when you're applying the finer the droplet, the more close it is the coarser. And we're not talking ultra-coarse droplets. This is not dicamba droplets. We're talking coarse droplets in this case. And the reason why you don't want to apply a coarse droplet to an open canopy is the finer droplet can get inside an open canopy and it will give you better coverage. You want just as coarse a droplet as the minimum needed to get inside the canopy. This is the last year of these studies. And as you can see behind me, these are very large studies. We conduct these with anywhere from 12 to 16 experimental replicates because of inherent spatial variability of white mold and the need to differentiate small treatments. They're done very rigorously, again with a real-world sprayer at real-world driving speeds. You have any questions? Look at our results online. Google NDSU Carrington. Click on Plant Pathology and scroll down to the section on soybean white mold. Click on the presentation on the impact of spray droplet size and fungicide performance. Thank you very much.


First of all, I'd like to thank the organizers, Anton and Maxim, for organizing this fantastic event. And it's really my pleasure to be here, so my first thanks to the organizers. And when preparing this talk, I was thinking what would be the topic which kind of intersects with Samsung's scientific interests. I think I hope that I will show that in many places it touches Samsung's interests, which are very and very interesting. So I'll mention this. And finally, I will say more probably later, but let me mention that Samsung came to Leningrad in 1981, in 1981. And he was a very young fellow, but he had, and he has a very remarkable gift. He can make friends with many people, young and old. I'll tell more, but before I'll start this talk, let me show some of the Samsung friends whom he made in the 80s. Okay, so this was like this ends informal introduction. This is Trinity College Dublin. And now let's start. So the topic originated from local mirror symmetry. And one of the most important papers was by Anganajic, Daigraf, Klem, Marino and Waffen. They studied mirror symmetry for local Toric, Calabio three-folds. And the typical example they consider, and many other people consider, is when the Calabio manifold is a total space of canonical bundle over Toric del Petz's surface. So that's what's written over there. It's a canonical bundle over a surface S. And there are very simple examples of such Toric surfaces, like P2, the basic example. Another example is, here's the Bruch surfaces, which are also very simple. It's P2 blown at 0, 1, 2 or 3 points. Then there are non-smooth examples, which are weighted projected spaces. There are much more, but this would be sufficient. Even in this talk I'll consider only one example, which would be also interesting. But this is the general setup. So this is a non-compact Calabio three-fold, that's why it's end. Then the result is that corresponding mirror manifold is given by precise equation. For this, it can be described explicitly. It corresponds to Landau-Ginsburg model with superpotential W. The equation is written over there. It defines a manifold in a product of C2 times C star 2, because it's natural to consider E of PN, E of Q, where PN Q are just complex coordinates. Then in these examples I mentioned before, the mirror curve is just elliptic curve and can be given by a very simple equation. For instance, for the basic example, when S is the Hilsbrug surface F0, the superpotential has this very simple form. There are two complex parameters, or rather real, say, zeta positive plays the role of physical mass and kappa is the modulus of corresponding elliptic curve. These precise equations were discovered long ago by physicists. They were used by mathematicians like Arou, Katsarkov, Arlov. They were studying these examples very carefully because they are like playing ground where you can test many things. When your delpets surface is a weighted projective space, you have another equation which is also very simple. This was known long ago. But then surprisingly, well, it was said that the idea was in the air for many years, starting from that paper by Agonadzic and other people and her friends. But here, Mariniou had an idea that one need, well, which came from integrable systems and it was probably at the same time when Samson, together with Nikita, were studying quantum integrable systems, Baxter equation, their relation to supersymmetric quantum field theories. At the same time, roughly, Marcus Mariniou and his collaborators, Grassi, Hatsude and other, they had an idea that one needs to quantize this mirror curves basically by replacing classical coordinates P and Q by quantum mechanical operators capital P and capital Q, position and momentum operators. And as a result, so that's the idea, then instead of equation of a quantum curve, we have an eigenvalue problem for the corresponding quantum mechanical operator. And there are many interesting questions, both from physics, from mirror symmetry and from the spectral theorem. So in this talk, I will be concentrating on the spectral properties of these operators as operators in the hybrid space, so that the hybrid space structure will be very important. And so, but first it was discovered by Mariniou and his co-authors that these operators have several remarkable properties. These, well, properties mean conjectures, so they mean the following conjectures, that spectral properties of these quantum mechanical operators on the line is the simplest quantum mechanics with one degree of freedom. They encode in some way the enumerative geometry of the Toric-Colabiau three-fold X, the original manifold we started in the mirror symmetry. More precisely they're related to invariance like Gromov-Witton, Gopakumar-Waffer and other things in some rather complicated way. The second also that the spectral theory of these operators which we'll discuss provides a non-perturbative definition of the topological string theory on an initial manifold X. So that's what studying quantum mechanics on the mirror manifold allows to define non-perturbatively the topological strings on the Colabiau three-fold. Obviously I won't discuss these properties, these things, these are two main problems and they're rather deep and kind of difficult. So I will discuss some very simple things which should be like basic quantum mechanics. Introduction to quantum mechanics, let's put it this way. Okay, so first very briefly let's remind the VAL operators. So our hybrid space will be L2 on the line, the very basic. Then we can see the Heisman operators P and Q, quantum mechanical operators. Here I put h bar equals to 1, it will reappear later in a different context. And then we define VAL operators. Usually in quantum mechanics we are taught that VAL operators are unitary operators. There is P and Q, capital P and capital Q are self-adjoint operators in a hybrid space. They define each of them. One parameter group of unitary operators and this unitary operators satisfy Hermann-Veyle commutation relations. And then the one parameter group means the parameter is real on the real line. But in this setup the parameter is pure imaginary. So instead of considering unitary operators bounded, we have self-adjoint unbounded operators. Namely in this setup we have operators U and V which are just exponents of self-adjoint operators. P and Q, they exist as a nice unbounded self-adjoint operators in a hybrid space. And B is a positive constant. It kind of plays the role of a Planck constant. You can absorb it in P and Q, obviously. And then they satisfy on their common domain this Hermann-Veyle-like relation. This is not a true relation everywhere because when you have unbounded operators in a hybrid space you cannot state any algebraic relation between them because domains may be different, you need to be very careful. So this relation is understood formally on a common domain, which is invariant under the action of U and V. And then they obviously satisfy this relation. Some say it's like pre-normal space. Exactly, exactly. For instance, this is the domain which is invariant. Unboundable dimension, dense space, yes. But by no means this is true in general that such relation holds. So this is rather formal. And of course example, using Schrodinger representation we can realize operator U as a shift in imaginary direction in L2 on the line and V as a multiplication by exponent. Obviously V is a nice self-adjoint operator, just multiplication by a function. But it may look weird that a pure imaginary shift is a self-adjoint operator. This is true and this is very simple. Like shift in a real direction is unitary, shift in pure imaginary is obviously self-adjoint. The easiest way to see it is to use a Fourier transform. And then you can describe the domains explicitly. This is very simple to describe domains of these self-adjoint operators. And I like to say that these are self-adjoint in a true mathematical sense. It's not a remission operator which means either symmetric or whatever. It's self-adjoint in precise definition that each operator with dense domain closable has an adjoint and they are equal. This is precisely what we mean by a self-adjoint. So these are the basic operators. And well, that's how we define the inverses. The V is positive, U is also similar to a positive operator. So that's the basic setup. These are just two formulas we need to kind of, we'll be using throughout this talk. And then the functional difference operator, for instance, in this setup, when S is a Hilsberg surface, then we have a family of operators, H of zeta, which is a very symmetric expression.  Each of the terms is a self-adjoint operator, but in general it's not true. That the sum of two unbounded self-adjoint operators is self-adjoint. There could be some problems. But in this case, one can prove that this is truly self-adjoint operator in a hybrid space. And why this operator is so interesting in kind of the basic, because if we can set zeta to be 1 like a mass, then its symbol has a remarkable symmetry. Its symbol is the sum of two hyperbolic cosines. And this is an analog of a usual quantum mechanical oscillator, whose symbol is psi square plus p square. So if you expand for small bit, you get a harmonic oscillator. Minus 4, I guess. If you subtract 4 and expand, it's a harmonic oscillator. So this operator should have some remarkable properties, as harmonic oscillator does. And this is the operator which just came from my records. It was never studied, neither by mathematicians, in the speckler theory of differential operators, because H formula is a pseudo-differential operator of infinite order. Because if you expand, it's pseudo-differential operator of infinite order. And remarkably, its properties are very close to the second-order differential operator, to the Schrodinger operator. So it came in this way. So there are two, and this is also another remark, kind of interesting, is that you can put, you can discretize Schrodinger operator, and usually people replace second derivative by a second difference. Then this operator with second difference has different properties, because it becomes unitary. But this is a better approximation, replacing by functional difference in imaginary direction. So that's the proper discretization of Schrodinger operator rather of second derivative. Sorry, but before we termed functional difference equations, when you have continuous, there's no difference replacing with those unitary, do you care about geneticity? No, this is, if you have, instead of imaginary shift, you have like x plus b, it will be unitary obviously, it will be unitary, right? It's a shift operator. And then it's kind of interesting that in fact, when the mass zeta is zero, this operator, we should call H of zero, which is u plus u inverse plus v, already appeared in our work with Ludwig Fadeev, when we were discussing quantum Liouville model on the lattice. This was the trace of local monodromy matrix, then this operator appeared in representation theory of quantum group SL2R and so on. And I remember that when around that time Dan Friedan was visiting Soviet Union, he was in Moscow and in Leningrad, and when he came to Steklov Institute, I think something was there, we discussed with Fadeev, with Dan Friedan, because he was interested obviously in Liouville theory at that time it was, and of course this is, that's where this operator came from. Then later it was studied extensively by Kashiev, because if one quantizes, considers kind of deformation, quantization of the Taishner space, considered as a symplectic manifold with vile Peterson's symplectic form, then this operator H plays the role of a dent twist operator, that's what Renat Kashiev was discussing. So this has a long history, but surprisingly the spectral theory of this operator H, which is very simple, have not been done, although many things were discussed. Now that's another set of operators for weighted projective spaces, they have very nice also property, it's also a self-adjoint operator. And then let me just mention results about this operator, and then I'll discuss them. So first of all when the operators H zeta, for positive zeta with massive case, and this operator H mn can be proved that they have a pure discrete spectrum, the inverse operators are of trace class, and you can compute the asymptotic for the eigenvalue counting function. You can prove the analog of Hermann-Veyle law, and instead of area, like for Schrodinger, you get log square with precise constants. So these are theorems, and where the constant c is of this form. So the story is that we, this is in our paper with mathematicians, Laptev and Schimmer, and who got really interested in this operator, the experts in functional analysis, and people there never seen operators of this type, of this form. And so let me spend maybe five minutes to comment on the proof of these results, which are rather general. For instance, let me just mention that these asymptotics for the eigenvalue counting function says that the eigenvalues are very sparse. For instance, n's eigenvalue grows like exponent of square root of n with some coefficient. So they are not like, but of course if you expand, then, yeah, so. So the main idea of the proof is very simple. It based on another physics method using coherent state transform, which maps L2 on the line into L2 on the plane. This is not, of course, this is a partial isometry, it cannot be isometry, defined by this form, and then there is some kind of remarkable identity which reflects integrability of this operator, and this single identity allows to prove all these results. Namely, if you compute quadratic form of operator H in terms of the coherent state transform, then quadratic form exhibits the symbol of the operator. The symbol of the operator appears in a quadratic form with some constants d1 and d2. That's the, and if you, and using this thing and the standard techniques estimating risk sums, risk means which are written over there, it's a sum function of lambda, which is the sum over all eigenvalues less than lambda, lambda minus lambda j summation over all j such that lambda j is less than lambda, and then you get the identity, and then one can estimate it from above and below by the same constant using Jensen's inequality, convexity, and this representation, using two different types of representation. So this is basically very simple, and the key is this formula for the quadratic form of the operator. And of course it's also non-difficult, rather easy to prove that these operators have a discrete spectrum. And then from this, from the Vale law, it follows that they are of the trace class, inverses of the trace class, and these are all positive operators. And then, so that's what was discussed before, and of course the main thing, which the conjection stated by Marini and co-authors is that in fact these operators are of trace class, inverses of these operators are of trace class, one can define Fredgol determinant of these operators as entire function, and then it's asymptotic expansion of the Fredgol determinant, or traces of the all positive powers of the inverse operator in code enumerative invariance of the Calabria of three-fold. And this conjection of Marini is very complicated because it uses Nikrasov-Shatashvili function, it uses some, a lot of ingredients, some Nikrasov-Shatashvili limit, some kind of quantum theta functions, it's really very involved. But these simple operators seem to carry this information. But before this, we study the massless operator. And there's also another interesting story where Samson was involved, because in 2012 we invited Fadeev to the Simon Center, and he came probably for a month or maybe for four weeks or whatever, and then he was, Fadeev was working with this operator for quite a while, and then, but somehow when we asked him what is the spectral properties, what is the resolvent of this operator, he kind of, they didn't study it. So I started to study this, and then we developed the full, it's a very simple operator, and we developed the scattering theory for this operator, like it's precisely like a Schrödinger operator with a potential that grows exponentially at one end and decays very fast at the other end. And this can be done very nicely, but there is some interesting feature of this analysis, which I like to outline, is the following.  And then, like we know from elementary quantum mechanics or functional analysis, that the Green's function for the second derivative operator is continuous on a diagonal, and its first derivative is discontinuous. And therefore, the delta function, which appears in the equation for the Green's function, comes from Heavisite formula. The derivative of a jump is a delta function at this point. Here the resolvent, let's consider the unperturbed, the free operator H0, which is u plus u inverse. It's an integral operator. It can easily be solved by Fourier transform, then integrating back its integral operator of this form. Let me, of this form, integral operator where this is a nice, smooth function everywhere is smooth at x equals zero. And then how come that this resolvent satisfies the equation H of r equals lambda times r plus delta function? What is the mechanism how the delta function appears? And it turns out that it's kind of interesting. It appears because of Sochotsky-Plemel formula, so which one can think of as a smooth version of Heavisite formula. Namely, it's easy to prove, well, it's elementary, that this property follows from Sochotsky-Plemel formula, where this you can consider as a smooth version of a Heavisite function. Also, it may be important, maybe not, but Polikov observed that this expression can be interpreted as one partition function for both the Einstein statistics. This is, of course, trivial, because you expand denominator into geometric series and you get both the Einstein partitions function. But maybe it's important, maybe not. But anyway, this is the Sochotsky-Plemel formula from which the whole thing follows. And then we have a free resolvent. Then one can study the solution of the Eigenvalue problem, scattering solution, where lambda is parameterized by two hyperbolic cosine of 2 pi bk, sorry. And then, actually, in this case, one can think of this problem as a Schrodinger operator on a half line from 0 to infinity. Then it is known that it has Yoast solutions and a scattering solution, which is 0 at the origin. And then the scattering solution here can be written nicely as Fourier transform of a product of two Fadeev's quantum dialogarin functions. And Yoast solutions can be also defined. And then we have precisely the same formula as for the Schrodinger equation, that the scattering solution is the linear combination of the Yoast solution, where this will define the plancherial measure. This will define the spectral function. The modulus square inverse defines the spectral function for this operator. This is very simple, but this is one can think of this operator as a Q-analog of the Schrodinger operator with potential e to the x, whose solutions are modified Bessel functions of the second kind. And then, orthogonality and completeness relation, spectral decomposition for this operator is classic result in theory of special functions. It's called Kontarovich-Lebedev transform. So this can be considered as a Q-analog of Kontarovich-Lebedev transform. Then the resultant for the perturbed operator has a very simple formula. It's the same type, like for the Schrodinger operator, and then starting. And then the last operator, U, which is a unitary operator from L2 on the line to L2 on the half line with the spectral measure, defines the spectral decomposition because operator UH, U inverse, is a multiplication operator. So this finishes the spectral theory for this operator, for operator H. This is very simple, so I'll just mention it. Now what happens with the operator, with our operators when the mass is non-zero? So, and then there is a whole story, which is rather involved because there is a conjecture. Starting from Nekrasov-Shetashvili paper, we know that total lattice, the periodic total lattice is integrable and one can find some exact quantization conditions using Nekrasov-Shetashvili function, which is a young, young potential for this theory. And this Nekrasov-Shetashvili function is obtained as a super potential for some super symmetric gauge theory in a rather complicated fashion. So that's the story. In particular, for the simplest Schrodinger operator, negative second derivative plus hyperbolic cosine, there is some quantization condition, which is impossible to check because it uses Nekrasov-Shetashvili function. So there is some interesting story going on, even with the classical, with modified Mathieu equation, when you replace ordinary cosine potential by a hyperbolic cosine. This gives you results from Nekrasov-Shetashvili theory. But then the claim is made by Marini and Hatsuda and other people, it's a conjecture that for all these integrable operators from mirror curves, there is a certain quantization condition, exact, which uses two ingredients. One ingredient is Nekrasov-Shetashvili function, which should be used for the operators and the dual operators. I forget to mention that we have a pair, u and v, and then there is a dual pair, u tilde say, and v tilde, which formally commute with u and v, only formally, they don't commute as self-adjoint, but formally they commute, and they have corresponding q tilde. It's like b goes to 1 over b of this sort. And then for one q you construct one, Nekrasov-Shetashvili function, for another q tilde you construct function for q tilde, and then the quantization condition is for the sum of these two functions f plus f tilde, and then this is a quantization condition for the eigenvalues of the discrete spectrum, and this is remarkable that usually when in defining young-young superpotential from supersymmetric gauge theories, you don't usually pay attention whether your solution is, is it in L2 or in some other space, but in this case this quantization condition defines a true L2 function, which is solution of the eigenvalue problem, which is kind of mysterious, and it's really, if it really holds, it's a miracle. But there is a second part of the story, that this quantization condition doesn't, the energy, the eigenvalues are not directly involved in this quantization condition, it's rather a certain function of these energies, and this certain function is obtained as a quantum A period of the mirror curve, and this is the object which nobody knows how to compute, the quantum A period of a mirror curve. Mirror curve is elliptic curve, it has A and B periods which can be computed, and they are used to obtain semi-classical asymptotics of the eigenvalues, but the quantum A period, whatever it means, there is some conjecture by Waffer's digraph, but it's not clear what it really means. This quantum period is allowed to determine how the lambda eigenvalue depends on the parameter T in Nekrasov-Shatashvili function. So there are two conditions involved. One is a pure quantization condition and another is change of variable. And so whether it's very useful or not, it's not clear because for potential Kosh X, for Schrodinger operator, there is also good swiller quantization condition, which is also exact. Probably good swiller condition in this case is a special case of Nekrasov-Shatashvili, but this would be interesting to check. I think maybe Tashner checked it, maybe not. It's not clear. Yeah, but either it's exactly that yo is good swiller condition. Exactly. Yeah, but good swiller in Schrodinger's work, because good swiller was the first who proved that the quantum, periodic quantum total lattice is completely integrable and he gave exact quantization condition. Later it was explained by separation of values. There are some formulas that if you take same function W and differentiate with respect to zeta, it centers as a parameter, then that would give the energy, evaluating on the solutions of discrete spectrum. Right. And bad things that protect Nekrasov. Right, but it would be interesting to check correspondence with good swiller. Okay, so and then the idea was that since this exact quantization conditions are very difficult in the conjecture that there is another way to treat the eigenvalues. It's considered the trace identity. Namely, you have Friedholm determinant, which is entire function of a parameter lambda. Its zeros are precisely the eigenvalues. Then one can study either its behavior when parameter lambda is zero, just Taylor expansion, but then one can consider its asymptotics for large lambda. And asymptotics for large lambda contain a lot of information about eigenvalues. They kind of reflect some of eigenvalues, some of squares of the eigenvalues regularized. And this is the thing which one can do. So I'll finish by giving some ideas how the trace identities work. Because that's the approach that kind of related to the Schrodinger operator and which probably can be done nicely in this case. So let me briefly review the history of the trace identity for Schrodinger and this type of operator. So that's what I said, that you have a regularized determinant of the operator. I write it in this form. It should be regularized at the Friedholm determinant. And then you consider the asymptotics of this operator for large values of parameter lambda. And express it in terms of the coefficients. So first, let's start with the basic example. It's the Schrodinger operator with a potential v of x, either on the interval with some boundary conditions or on the whole line, half line, anywhere, with appropriate boundary conditions. First, let's consider the Stroum-Lewin operator on zero pi, say, with smooth potential v of x. This is, of course, celebrated Gal-Font-Leviton-Dickey trace identity. That's a beautiful formula of Gal-Font-Leviton where you can see the Stroum-Lewin operator on the interval, say, with zero Dirichlet boundary conditions. Then some of the eigenvalues obviously diverge. You regularize it. There is a canonical way. Then it equals to the integral of a potential. So that's Gal-Font-Leviton, 1953 or something like that. And then, of course, this was developed further by Gal-Font-Leviton-Dickey and there is the whole beautiful machinery of trace identities. Schrodinger operator on a half line. Then on the interval, there is no condition on a potential. Just smoothness. It's sufficient. On a half line, then, of course, potential should be rapidly decreasing. Then there is what is called Busleif-Fadier trace identities, which also express regularized sum of eigenvalues plus a contribution from a continuous spectrum in terms of the potential. It's some integrals of potential, v of x, its derivatives, values of v at x equals zero, its derivatives, and so on. There is a complete characterization. Then when you consider Schrodinger operator on a line with rapidly decreasing potential, then you can generalize Busleif-Fadier. It will be Fadier-Zaharov trace identities, which everybody knows because you use them for complete integrability of the KDV equation, which express precisely that certain spectral data in terms of integrals of potential, of polynomials of potential v of x and its derivatives, which trace identities represent all local commuting integrals for the KDV on a real line with rapidly decreasing case. There are also integrals in periodic boundary condition, but that's a different story I won't discuss it. So, it's periodic boundary condition also can be considered. Lambda should go to negative infinity. Thank you. Negative. Yeah, yeah. It's still in the formula. Right. Exactly. Exactly. Right. So, now, this is just what I said before. We can see the pure discrete spectrum. What is determinant of H is difficult to define because it needs a family, but if you have a family, you can define and normalize. Determinant by this, either by Adamar product, if the operator H inverse is of trace class, then the sum of the eigenvalues or rather one of eigenvalues converges. This is Adamar product. Or you define if your operator is not, if H inverse is not of trace class, but its power is of trace class, you define it somewhat differently. If H inverse minus epsilon for some small epsilon, you define it like that. So, basically, that's the object. Our operators from mirror curves, they belong to the first class. They belong to this class. And this is basically Fredholm determinant. Or you have to replace lambda by one over lambda. It doesn't matter. This is Fredholm determinant. Just ordinary determinants. Characteristic determinant of an operator. So, we have these properties. And then, well, there is another simple... Now, how to obtain asymptotics of this determinant when lambda goes to minus infinity? And how are they related to the distribution of the eigenvalues? Well, we define the operator zeta function to H, operate H. We define the operator zeta function, which is given by this series for real S large enough, you know, okay, greater than one, even one-half depends. And then, if, since the operator zeta function admits a miramorphic continuation to the whole S plane, then we can just by milline transform, you can express log of this determinant as an integral of the operator zeta function with these factors. Where sigma, you integrate along the vertical line in a complex plane. And sigma is large one. Large, greater than zero. And then, to get asymptotics of this determinant when lambda goes to minus infinity is very simple. You shift the contour of integration to minus infinity. And how far you shift? You get asymptotic, you get terms of your expansion plus an error term. And if you shift it to infinity, you get asymptotic series. Otherwise, you get exact formula with a remainder term. So, that's standard representation, like everywhere. And so, that's one way of obtaining the asymptotics. So, and then, of course, here we see, like, naively that since zeta function admits another continuation, poles are at, when S is an integer, say, when S is a positive integer, well, you don't go to the right. You go to the left. When S is, so S is a negative integer, then you get inverse powers of lambda. Inverse powers of lambda. And values of zeta and zeta of S at S equals to negative integer. So, you get regularized. You get regularized sum of eigenvalue squares and so on by zeta function, and you get this expansion. So, that's the standard part. Now, the non-trivial part is to obtain the same expansion using the potential v of x to get a different representation for this asymptotic series which is obtained from this formula. What I said is just a way of defining the sum of the eigenvalue, sum of square, how to regularize this. And then, there is a way which I put it here in rather cryptic form, but I can explain. How can one do it for Schrodinger? Now, we are discussing Schrodinger equation with growing potential. Growing potential of such that the spectrum is discrete. How to get a trace formula for this class? This was not really considered. But then, there is a way of, if operator is positive, you can see the asymptotics when lambda goes to negative infinity. It's like doing WK beam method without turning points. And this was known long before WKB. It's called Louisville Green method when there is no turning points. And in some cases, which one should be very careful, Louisville Green method gives uniform asymptotics for large negative lambda, its large negative energies, and uniformly in X. Then, doing this, so you can see the solution of this equation for any lambda negative. So it cannot be an eigenvalue. And then, you can see the solution which decays at one end, say at plus infinity. And then, you can see these asymptotics by this Louisville Green method when lambda goes to negative infinity. And these asymptotics should be uniform in X. So that's the leading term. When then subtracting these asymptotics, you can reduce your Schrodinger equation to Riccati equation. But you need to know what to subtract. This Louisville Green says precisely to Riccati equation where instead of a potential, will be some combination of potential in terms of this Louisville Green formula. So it depends how fast potential grows. To infinity involves certain integration. And this can be done explicitly. And then you can get Riccati equation and you get coefficients in the asymptotic expansion. Here, I didn't write it because here you expand this in inverse powers of lambda with coefficients. And then you can get these coefficients recursively by using Riccati equation.  Let's consider a couple of examples. This brings us to the 19th century actually. So first, the trivial example. The harmonic oscillator. You will all know that it's kind of discrete spectrum, but in fact all solutions are important. They give special functions, parabolic cylinder functions called Weber functions. Then this is the determinant. It's inverse gamma function with k-efficient. And then trace identities obtained by Louisville Green method gives you the sterling formula for gamma function. So trace identities are very meaningful because this is a... they combine information from two different points. So this gives sterling formula for zeta function, for gamma function. So there's one example where some... you cannot make a simple example. Then the second case, as I mentioned before, essentially is e to the 2x on the half line with directly boundary conditions at the origin. Solutions are modified basically function of the second kind. Then determinant is just the... this k is a McDonald's function, k function, evaluated at point 1, because log 1 is 0. And then it's function k of i square root of lambda. So again, you get using asymptotic expansion of this at lambda goes to infinity. There is some... by Green-Lewville method. You obtain formulas for the sum of the zeros of Bessel function. There are squares, regularized sums. Actually, just as a side remark, this is a toy operator which Poyer was using when he discussed approximation, some naive approximation to remain zeta function. It's not important. Now, the last case. The last case is very... excuse me. The last case is rather involved, but there is similarity. Since operator, although H is now pseudo-differential operator of infinite order, it behaves like a second-order differential operator. And one can also develop the analog of Louisville-Green method and obtain the corresponding trace identities. I hope to tell about it maybe next time at Samsung's birthday, maybe next year. But some details are rather involved. But this can be considered as a constructive way of obtaining some invariance like Gopakumar-Waffer and others from this trace identities. Because this potential is just hyperbolic cosine.  And then one can compare this bunch of numbers with exact invariance like Marini and his co-authors were doing. It's not done completely, but that's a way to avoid these quantization conditions and quantum A periods, because this is not really because it's very difficult. Not doable. Not doable, right. And get some information. And the key, of course, is that this remarkable operator is as nice as harmonic oscillator. So, of course, we cannot get a formula for the eigenvalues, but we can completely characterize its eigenfunctions or maybe the general solution like parabolic cylinder functions in terms of some integral representation. What we mean? We have special functions. Meaning that we have a function like Bessel function. We have some serious expansion which doesn't help much, but we have an integral representation which really is the thing which allows us to connect behavior at large and at small values of parameters. Same should be here because this is a remarkable operator which should be even better than harmonic oscillator. So probably this is like the bottom line of this talk. That one should look at this operator very carefully. And I think, yeah, it's time to stop. So, that's... Happy birthday Samson. So, you gave some examples of those kind of planning operators with exponential potential and with the shifts which we have as second-order operators. Is it clear what is the class kind of right? There were many examples, but is there some more general class of operators which became similar to second-order differential operators? Well, I think operator U plus U inverse, you can add anything. You can add capital V, you can add small V of X would be of the same class.  That's the true analog of a second... And then an arbitrary potential. Or arbitrary reasonable... Right, right, exactly. Yeah, there is also the harper operator. Can you use it to get some information? Yes, exactly. That's a very good point. When we consider the case when U and V are self-adjoint. If U and V are unitary, that's the famous half-stadter harper model. But this is a totally different harper operator. It has a periodic spectrum. It's like Mathieu's equation with cosine X and hyperbolic cosine. Spectral theory for one doesn't correspond to spectra. They're kind of like some analytic continuation. Yes, so yeah. Do you drink? Yes. Does the data function verify functional equation? I don't know. I'm asking this because for the harmonic oscillator the spectrum is integer so you get the classical... Right, right, right. You know, there is some weird equation Bender, you know, road, yeah but it's not true. You know Bender. Do you think that we try to consider elliptic difference analog of these equations because it's like few difference operators when the front can try... More complicated. Locals we can want to... Not to worry, but need itself. You don't know. No, I don't know. It's like the baby example. Right, but it's very... Let's thank the speaker again.


I am happy to introduce Jason Parker from Bandon University who is going to talk on the Covariant Isotropy of Gothic Toposis. So please go ahead. Great, so thank you for the opportunity. So I'm Jason Parker and just FYI, I do sometimes stutter or stammer, so you hear me stuttering or stammering, that's totally normal. So yeah, so I'm talking about Covariant Isotropy of Groten Dika Toposis. So I'll just start with some kind of background. Whoops, I have to get that going. Okay, yeah, so basically Covariant Isotropy is a somewhat recent categorical construction that basically gives an abstract notion of conjugation or inner automorphism for an arbitrary category. Oh, I think I have a, okay. So basically in prior work with Peter Prostra and Phil Scott, we use techniques from categorical logic to characterize the Covariant Isotropy group of any locally presentable category C and in particular of any pre-sheaf category. So in this talk, we'll first give an overview of Covariant Isotropy and then show that its characterization for any pre-sheaf category basically extends to any Groten Dika Toposis. And this is based on my recent pre-print with the same title as this talk that's currently on the archive. Okay, so just to first motivate this topic. So basically George Bergman proved in a certain paper from I think 2012 that the inner automorphisms groups can be characterized purely categorically as the group automorphisms that extend naturally along any group morphism out of the domain. So see this, first observe that if I have say an inner automorphism alpha of some group G induced by some element S in G then for any group morphism F from G to some further group H, we can push forward alpha along F to define a further inner automorphism alpha sub F of the codomain H simply by conjugation with F of S and H. So therefore in particular, if I push alpha along the identity on G, we just have alpha. Now this family of automorphisms that I obtained from the starting inner automorphism alpha is coherent in the sense that if I have any group of homomorphisms F from G to G prime and then F prime from G prime to G prime prime then the following square will always commute. So we have alpha sub F on the top, alpha sub F prime after F on the bottom and F prime on either side. So basically if I push forward these or the inner automorphism alpha along F and then along F prime after F then those kind of extended or those pushed forward inner automorphisms will be coherent in the sense. Okay, so for group G, that's called an arbitrary family of automorphisms with above naturality property an extended inner automorphism of G. So concretely, this is a family of group automorphisms, one for each group morphism out of G on the codomain of that morphism. And these are all coherent with each other. And now I don't assume a priori that these are induced by a starting inner automorphism of G. So this is just an arbitrary family of automorphisms on the codomains of all the group morphisms out of G that are coherent in the sense on the previous slide. So this is just a natural automorphism of the projection functor from the group G slice group to group. So Bourbon proved that if I have any automorphism alpha of a group G then alpha is an inner automorphism. So it's induced by conjugation with some elements. If and only if there is an extended inner automorphism in this above sense that induces alpha. So who is component at the identity on G is just alpha itself. So we just saw that if alpha is inner then it induces any such extended inner automorphism but the converse is also true that if I give you any arbitrary extended inner automorphism of a group G then it must come from an inner automorphism in the conjugation theoretic sense of the starting group G. So basically this gives a completely a categorical characterization of inner automorphisms of groups. They're exactly those group automorphisms that can be coherently or functorily extended along group homomorphisms out of the domain. Okay, so how does this relate to your kind of inspire covariant acidity? So we have a functor Z from a group to group that sends any group G to its group of extended inner automorphisms. And these do form a group. So we refer to Z as the covariant isotropy group functor of the category group. And actually Bergman's theorem shows that this is in fact isomorphic to the identity functor on the category group. So in fact, any arbitrary category C has its own covariant isotropy group functor ZC that sends each object C to its own group of extended inner automorphisms. So these are families of automorphisms, one automorphism on the codomain of each arrow out of the object that are coherent with each other in the sense as described on a prior slide. So these are just natural automorphisms of the projection functor from the slice category under the object to the category. So basically again, these are just families of automorphisms, one automorphism on the codomain of each arrow out of the object that are natural or coherent. So we can turn Bergman's characterization of inner automorphisms in group into a definition of the notion of inner automorphism in an arbitrary category C. So if we have an object C and an automorphism alpha, of C, then we can define alpha to be inner just if it's induced by an extended inner automorphism. So if there is an extended inner automorphism whose component at the identity on C is alpha. So basically this is saying that alpha is inner just if it can be coherently extended along any arrow out of the object. So group is the category of models of an algebraic or an equational theory so just a set of equational axioms between terms in the sense of categorical logic. So in prior work, we basically generalized ideas from the proof of this result of Bergman to give a logical characterization of the inner and extended inner automorphisms in this more categorical sense of the category of models of any finite area quasi-equational theory T. Okay. So, yes, okay. Oops, there we go. Okay, so just I'll just get some background on those quickly. So a finite area quasi-equational theory T in the sense of the paper by Palmgren and Vickers called, I think, partial horn logic and Cartesian categories. So it's over a multi-sorted finite area equational signature. So it has just sorts and operation symbols. So it's a set of implications called the axioms which are between funnitary horn formulas which are just conjunctions of equations between the terms. Now, the kind of relevant feature of these theories and these are also known as say Cartesian theories or essentially algebraic theories is that the operations symbols may only be partially defined. So we have a term T, we'll write T down arrow as an abbreviation for T equals T meaning that T is defined. So this is not always a theorem of the given underlying logic. So now, if we have any regular cardinal lambda then one can define an extension of these theories to lambda. So we can define say a lambda area quasi-equational as the theory T in the same way but now allowing for operations with possibly lambda arity and then also lambda area conjunctions. Okay, so just to give some examples of these. So any algebraic theory is a funnitary quasi-equational theory as are the theories of categories, group points, stripped monoidal categories and any pre-sheaf category. Now, for this talk, so if we have a small site, then the Rodin de Topos on that site is in fact the category of models for a lambda area quasi-equational theory where lambda is the smallest regular cardinal larger than any of the sips. So the sorts of the objects of C for any arrow F I have a unary operation symbol going in the opposite direction and for any covering sieve J we have a lambda area operations symbol sigma J of that typing. And then you have axioms expressing the contravariant functoriality and the fact that any matching family has a unique amalgamation. So therefore the models of this theory are exactly the sheaves on the site. Okay, so now I will just kind of briefly give an overview of the main results for covariant isotropy of Cartesian or quasi-equational theories. So it's fixed a lambda area quasi-equational theory over a lambda area signature and we have its category of models T mod. So again, I'll just review the characterization of its covariant isotropy group which was achieved for funnitary theories in prior work and extended to lambda area in my recent pre-print. So basically using the quasi-equational syntax of T one can define a notion of definable automorphism for a model M of T and these then form a group, a depth in of M. So if T is a single sorted for simplicity then given any model M one can form the T model M bracket X obtained from M by freely adjoining an indeterminate element X. So the elements of this model can be seen as congruence classes of terms involving X regarded as a new constant and also constants from the model M and here two terms are congruent if they are provably equal in the diagram theory of the model extended by this axiom saying that this new constant is defined. So the diagram theory of M extends T by basically adding constants from the model M and axioms expressing the relations that hold in M. Okay, so if we have some element bracket T and M bracket X we say it's substitutionally invertible if there are some possibly other elements bracket S in this model that is basically the substitutional inverse of T with respect to this diagram theory plus X. So we substitute S for X in T we get just X in the diagram theory and conversely. If we have say an operation symbol of Sigma then a given element bracket T and M bracket X commutes generically with F if T of M with X1 through XN proves the sequence. So first of all, this model is just the diagram theory extended by N new pairwise distinct constants which are all defined. And then the sequence basically is expressing that if F applied to this N tuple of new constants is defined then if I substitute it for X in T that is equal to F applied to these sort of N variants of T. So basically this is supposed to express that T is commuting or that T commutes in a sort of generic sense with this operation symbol F. And likewise, this congruence class bracket T will reflect the finalness of F if the same theory proves this sequence which basically says that if T with F of this tuple for X is defined and so is F of this tuple. So this is again expressing that this congruence class bracket T reflects defining this of F in this generic sense. Okay, so we can then define a group. So def N of M to be the group of all of these elements bracket T in M bracket X that are substitutionally invertible and commute generically with and reflect defining this of every operation symbol of the signature sigma. So basically this is the group of definable automorphisms of M where we regard an element bracket T in this in this T model as inducing a definable automorphism if it satisfies these properties. Okay, and if T is multi sorted one can extend the above the definitions appropriately. So basically you just have to of course account for the possibly many sorts which is not too hard to do. And that is shown in certain of the references. Okay, so then we proved in prior work that if we have a lambda area quasi-equational theory then for any model M of this theory it's covariant isotope group. So it's group of extended inner automorphisms is actually isomorphic to its group of definable inner automorphisms. So, and then in prior work namely in this initial paper isotope of algebraic theories we use this result to show that the sort of inner automorphisms categorically defined in many categories of algebraic structures like monoids, groups and abelian groups non-commutative rings, et cetera are exactly the sort of conjugation theoretic inner automorphisms that you would expect. So in this sense, covariant isotope does actually provide kind of a generalised notion of inner automorphism for a category. Checking the time, okay. So for pre-sheet categories so in a recent paper we also characterized the covariant isotope group of any pre-sheet category for of course a small category C. So we have a pre-sheet F and we show that the group of definable automorphisms of F consists up to isomorphism of exactly the natural automorphisms alpha of F induced by some element psi in the automorphism group of the identity functor on C. So here psi is a natural automorphism of the identity functor on C. And to say that alpha is induced by psi means that the component of alpha at any object of C is just the functor F applied to the component of psi at C. So basically the, this shows that the only inner automorphisms of pre-sheets are those induced by natural automorphism of the identity functor on the base category. So then follows that the covariant isotope group functor of any pre-sheet category is actually a constant on the automorphism group of the identity functor of the index category. Okay, so now in my recent preprint I wanted to see if this would extend to arbitrary growth and deptopuses or sheaf categories. So to start, for certain convenient technical reasons I first just examined sub-canonical sites where no object is covered by the empty sieve. So basically for any object C, the empty sieve is not a covering sieve of C. Okay, so then if F is any sheaf over such a site, I show that its group of definable inner automorphisms consists up to isomorphism of exactly those natural automorphisms alpha of F induced by some natural automorphism psi of the identity functor on C in the sense I described on the prior slide. So now this proof is quite technical and it's the most non-trivial aspect of the overall results in that preprint. So although this result is the same as for pre-sheafs, so far at least, its proof is still not obvious or non-trivial and so, yeah. Yeah, so as for pre-sheaf categories, if CJ is any small sub-canonical site in which no object is covered by the empty sieve, then the covariant STOB group of that sheaf category is constant on the automorphism group of the identity functor of the base category. So now we want to hopefully remove the assumptions of sub-canonicity and no object of being covered by the empty sieve. So the second property is easier to dispense with. So if we have any small sub-canonical site which doesn't necessarily have this property, one can find pretty easily another sub-canonical site which does have this property, so in which no object is covered by the empty sieve and where the sheaf categories are equivalent and the automorphism groups of the respective identity functors are isomorphic. So therefore, if CJ is any small sub-canonical site where objects may or may not be covered by the empty sieve, then the covariant isotropy group of the sheaf category is still constant on the automorphism group of the identity functor on the base category. So now we want to consider any small site not necessarily sub-canonical. So first, if we have any locally small category E with a small full dense sub-category C then I just proved this lemma saying that the automorphism groups of the identity functors on those two categories are isomorphic. So, and I use that to prove the following results. Well, actually, okay. So if we have any small site CJ, not necessarily sub-canonical, then it's well known that there is a sub-canonical topology K on the small full dense sub-category AYCs. That's just the full sub-category of the sheafifications of the representables for which the original topos is equivalent to the topos of sheaves on this site. So then the covariant isotropy group of the original sheaf topos is constant on the automorphism group of the identity functor on this full sub-category AYC. And by the lemma in the first point that's in turn isomorphic to the automorphism group of the identity functor on the actual topos. So basically, if we have any small site, not necessarily sub-canonical or and where objects could be covered by the empty sieve, then the covariant isotropy group of this topos is constant on the automorphism group of the identity functor of the topos, which may also be called the center of the topos. Okay, so in particular, since of course any pre-sheaf category is known to be the category of sheaves for the trivial and sub-canonical topology T on C where only maximal sieves cover, we recover our early result for pre-sheaf toposes. And now if the site is not sub-canonical, there is in general no relation between the automorphism groups of idC and the identity functor of the topos. So for example, if the automorphism group of idC is non-trivial, say for example, if C is the one object category corresponding to an abelian group, which is non-trivial, then, and J is the topology where every sieve covers, then it's easy to show that J is not sub-canonical and that the only sheaf is the terminal pre-sheaf, so the sheaf category is trivial, so that's the automorphism group of the topos is trivial even though the automorphism group of idC is non-trivial by assumption. And yeah, so our results shows a major difference between covariant and contravariant isotropy of Groedendie toposes, so the latter was studied in this paper called isotropy and crosotobuses by Fung, Kostra, and Steinberg, and they show that so contravariant isotropy is always representable by a sheaf of groups, while we have now shown that a covariant isotropy is always constant, and in fact, on the group of global sections of this sheaf of groups. Okay, so to start wrapping up, so via George Bergman's purely categorical characterization of the inner automorphisms of groups, covariant isotropy can be seen as giving a notion of conjugation or inner automorphism for arbitrary categories. So basically you can define an automorphism in an arbitrary category to be inner if there is an extended inner automorphism of the object that induces that initial automorphism or in other words, the automorphism is inner if it can be coherently extended along any map out of the object because this is the result that holds for, that characterizes the inner automorphisms of groups defined in terms of conjugation. So we've characterized the covariant isotropy group functor of T-mod for any lambda, airy quasi-equational theory T. So the covariant isotropy group of any model M or its group of extended inner automorphisms is isomorphic to the group of definable automorphisms of M. And this, I think that this characterization is most useful because it can provide a more kind of concrete or perhaps computational way to characterize the extended inner automorphisms of a given model because those, I mean, the definition of extended inner automorphism does not kind of in itself allow for any obvious way to characterize them. Whereas if you work with the syntax of a theory for which this category is the category of models then via this isomorphism that kind of provides a more perhaps down-to-earth computational concrete way to hopefully characterize the covariant isotropy of the category. Yeah, so using this results, we've shown that the characterization of covariant isotropy for pre-sheet topuses that we achieved earlier, in fact, essentially extends to all Rotendieke topuses. So for any small and sub-canonical site, CJ, the covariant isotropy group of the Rotendieke topos is constant on the identity functor on C. And as we saw before, this basically amounts to saying that the only inner automorphisms of a sheaf on any such site are the ones induced by a natural automorphism of itsy. So although this result shows that covariant isotropy as opposed to contravariant for Rotendieke topuses is in some sense degenerates or at least constant, the proof of this is still non-trivial. And we also intend to build on this result to characterize the covariant isotropy of categories of sheaves of algebraic structures, not just sets, which will be non-constant in general. Since in my thesis, I showed that the covariant isotropy of categories of pre-sheaves of algebraic structures is non-constant in general. Okay, so that concludes my talk. Thanks very much. Oh, and have some references here as well. Thank you very much for your talk, Jan.


Hello everyone, this is Amit Dan of AmitDan.com. Today I'll have another episode of unboxing and review video I'll be opening a portable pocket radio prunas J166 so lately I have been getting interested in radio and this is a piece of radio that I Purchased after many many years as a matter of fact, I even don't remember when I Used to radio last time so it has to be probably a few decades ago so all of a sudden, you know, I Grew some interest in radio and I was just like you know doing reading some reviews and looking at some radios So this is a very inexpensive radio. You cannot go wrong It's on sale and I purchased with my own money. So I spent about $13 Canadian, which is totally, you know Unbelievable price for a pocket radio So I'm going to see that, you know, whether this works and what kind of stations we can capture from here in downtown, Toronto so this is Only 119 gram it runs on two double a batteries. There's no way you can, you know recharge or You can put in our plug-in an adapter So you have to just, you know, keep buying or using batteries and I couldn't find how much the battery would last anywhere So but my guess would be that, you know, it would last a long time and it's a very lightweight only 119 gram without batteries and in terms of size, it's 2.64 by 1.1 by 5.0 3 inches So it's kind of even smaller than a cell phone. So it's a very, you know, small size So if you are traveling then it should be very like, you know a negligible way to carry with you and I'm hoping to travel with you and try the radio from different places on earth to see what kind of stations I can get and it has AM, FM and WB Frequency range band. So what it's that, you know, it can get the weather Stations, NOAA, NOAA for the US and possibly with our radio in Canada and What I'm going to do, let's open this up and take a look at it and then once, you know, I use it a few Days then I'm going to, you know, come back and touch the second part of the review with today's review So let's start with the unboxing So we have some specs right here and this is how the packaging looks. It's nothing fancy, nothing bad either. And the name is I believe you spell it pronus pronus and J166 is the model number and here we have some specs again so FM87 to 108 MHz and AM520 to 1710 and weatherband 162 to 162.55 and you need to double a batteries and then yes, you can use a earphone output and the speaker is supposed to be, you know, very Low volume or that's what I'm expecting. So I'm going to see, you know, once they open it how does sound quality So let's start the unboxing So the box is empty right now and I'm going to just, you know, leave it here So this is what we get So there is even this manual Our brochure that touch and you just just see about the, you know, various Buttons and how they work. So I'm not going to go through now Uh, but I might go through it later on. So let's set this inside for now And then we have the radio here so the first thing I can tell you that this is extremely lightweight.  So we have the FM WB and the AM right here And here's a knob to tune And then you have the volume button. Then you have the band selector which is right here And there's actually a telescopic antenna, which is right here And there is the 3.5 audio jack output. So you can use it with your headset or headphones So let me just try to Unfold the antenna and As you can see It's quite a long antenna. So it should be able to capture some Good Stations and let me just put it this way So it doesn't look or feel bad at all at this point. I'm going to start, you know Trying the radio and then, you know, I'll come back to the second part of the review Hello everyone This is the second part of the video. So I have had a few weeks to Try and test it out the prunas j166 radio And I'm just going to go over a few things today So the first thing, uh, it runs on two double batteries, which you can see Right here and I put this on about three weeks ago and I'm listening a little bit every day It looks like, you know, the battery life is very good There's no problem so far and then there's the telescopic antenna, which is right here So if you extend it, you can actually kind of stretch long and Now let's start with the the band. So Here's the off button right here and when you first Turn it on then the first band you get is the weather band and then here's the FM And then here's the aim and this is the volume slider and this is the tuning slider The tuning slider is slightly tight, but you know, you have to put a little bit pressure But it's it, you know, it has both both good side and bad side and Let's try out a few bands. So I'm going to just turn on the weather band, which is the first one And we have the volume here and On the weather band actually I only get two different bands one us and one canadian so the canadian one has very clear reception and The us one seems to be coming from buffalo and sometimes, you know, I get that and sometimes I don't So let me just try it out if I can get the us one today So this is where usually I get this But right now I see I'm not getting anything So I'm just going to switch to the canadian one Okay So this is the canadian one and it's 24 hours and I couldn't figure out where it's coming from Like if this is a central band for canadian weather could use the weather band But you know, this has very good reception. So I'm getting this all the time In the afternoon Let's go back to fm So fm I get a lot of stations and You know, some of them come very clear So the only thing is that you have to tune manually. There's no digital tuning But I wouldn't worry about that too much because of the price And then let's move on to the aim band So this band selection Can be a very little bit tight because you know, it's very tight and sometimes it's hard to select Let's say you're trying to put in fm, but you're going to slide it to a weather band. So it's not that like, you know Easy to maneuver, but it's not a big deal uh, so When I chose m band I noticed that I get a lot of station in chinese and indian languages and before I even didn't know that, you know, there are so many stations. They are playing on fm that i'm on m band in india and chinese languages And other languages So this is a french station English And lots of news channel So right now actually I don't get that many fm. I'm sorry m bands in the daytime But if you tune in at night, then you're gonna You're gonna get a lot more m Bands at night than during the day So this looks like this is another Indian channel So at night you're gonna actually get a many more chinese and indian channel So that's something I was really surprised Okay, so let me just turn this off right here Okay, so I turned it off and Here we have uh Our radio and the build quality it actually not it's not too bad, you know, it doesn't feel like, you know, it's a only so Inexpensive radio and I'm surprised with you know everything overall So when I purchased this radio the cost was about like, you know, $12 canadian I didn't expect anything and even I was thinking maybe I was not going to get any Stations or anything because it's so, you know, in expressive But I was surprised at the audio quality and I was surprised at you know, what I'm getting So definitely I cannot complain about anything based on, you know, the cost and what you get So another thing is that this is really lightweight and portable so you can take it anywhere when you're traveling the only Thing you have to keep in mind that, you know, everything is manual So you cannot give in or anything digitally and there's no memory. No nothing but the bad battery Is very low It doesn't take that much battery because I put this like in a few weeks ago and still they're running So I don't think that you know, that's going to be an issue even if you're listening to longer hours So overall I'm happy with the product and again This is not a paid or affiliated review and all my reviews are non-affiliated, unbiased and honest So thank you for watching this video. There's another radio review coming up I purchased another radio, which I'm going to be, you know, putting shortly and This is my first radio. I haven't had a radio for many decades and this is something I'm, you know, getting into recently And definitely I plan to, you know, do more Reviews and videos and radios in the future So thank you for watching. There will be more videos coming up. So I'll see you shortly. Thank you


Jerry, of the circus! Like he's half asleep all the time. Oh, Ham! Yeah? What you doing, Ham? I'm doing right now. I'm just thinking. Well, what you thinking about? Well, I was just thinking about what I'm supposed to do. I was supposed to do something, but I don't remember what it was, so I'm just thinking to see if I can find out. You sure forget quick. That's because I had my mind on your friends and rutions that they put on the riding show yesterday. Yeah, they sure are good, them rutions. He means the Rousseau troops. Yeah, I know. You liked them, huh, Ham? Oh, they were really something. Say, they can ride them horses better than anybody I ever did see. When they were riding, standing up, I really got scared. I never saw anything like that before. I guess that's because I never been what they were before, and that's why I didn't see them. They're just about the best riders in the world, I guess. The best trick riders, anyway. Well, you boys going to now. You going someplace, I'll bet you. Well, we were heading down the line to see my dad. Oh, he's out there on the track with Blaze, see him? Oh, sure, there he is, Tom, and he's got Blaze all saddled. I guess he's waiting for Warren to come and give Blaze his last workout. Come on, let's get over there. Can I come along and watch? Don't you think maybe I shouldn't order? Sure, come along, Ham. But hurry. You want me? You mean me? I should hurry? Oh, me. Well, Mr. Rogers. He sees us, Jerry. Come on, Ham. I'm coming, yes, sir. Poor dad. He sure is worried about that raceless afternoon. I don't blame him. It sure would be awful if he didn't win. Hello, boys. Hiya, dad. Hello, Mr. Rogers. Who's your friend there in back of you? Oh, Ham. He wanted to watch Blaze do his workout. Yeah, the Mr. Rogers, if you don't mind, sir. Oh, glad I have you, Ham. Say, Tom, did you happen to see Warren over at his table? No, sir, I didn't. I wish that boy would be a little more prompt. He's 10 minutes late already. I saw him, Mr. Rogers, sir. You did? Where? He was down at the other end of the line talking with Mr. Elston just a while ago, sir. Talking with Elston, you say? Yeah, sir. Who's that, Mr. Rogers? Elston owns the two fastest horses in the state. Well, which ones are they? He's the owner of Morning Sun and Silver Streak. Really? Yep. But lucky for the rest of us, he's only entering one. He's just running Morning Sun under his colors. See, I heard someone talking about those two horses yesterday. Oh, that Morning Sun is the one big threat of the race this afternoon. He can win it all right. My boss says his loss is going to win the race, and he knows what he's talking about. And that's what he told me. He said that to me. Well, I'm everybody that enters a horse hopes that his entry will win. But there's just going to be one winner when the race is over. Look at Dad. Here comes Warren now. Well, about time, I'd say. Come on, Warren. He sure has taken his time. I'm waiting for you, Warren. I've had Blaze out here on the track for 10 minutes already. I got something to tell you, Mr. Rogers. Bad news, I'm afraid. What's the matter? I'm not riding, Blaze. What are you talking about? You heard me. I'm not riding for you today. But you've got to. You can't let me down just hours before the race. There isn't another drucky available. I'm sorry, but Mr. Elson's going to enter both of his horses in the race, and I'm going to ride Silver Streak. Oh, how can you do a thing like this to me? I'm sorry. Listen, Warren, if you ride Blaze and Blaze wins, I'll give you... Well, I'll make it worth your while. Between you and me, Mr. Rogers, Blaze hasn't got a chance. Silver Streak is going to win the race, and I'll be riding him. Oh, now listen to me, Warren. I'm sorry, Mr. Rogers. I'm in a hurry now. Too many crickets. That's awful. Oh, Dad, what are you going to do? I don't know, son. I guess I'm through. I know. What? Why don't you ride Blaze, Tom? Say, that's an idea. Excuse me, sir, but Mr. Tom was just about the same size as Warren. Yes, but he's not a jockey. Why, you couldn't ride in a big race like the handicapped Tom? Sure I could, Dad. Yes, he could, Mr. Rogers. He's a good rider. What? What's this, you say? He rode Splendor and did jumps with him. What? He says I'm a natural born jockey. Honestly, he did, Dad. Whitey? Who's Whitey? He's a man who has charge of all the horses with the circus. Really, Mr. Rogers? Tom will surprise you. He can ride kings. Now, why don't you let him try to ride the horse now? And if he can ride, well... Well, I don't know what else to do, but... Well, you let him try. Yeah, Dad, let me try. It won't hurt to see if I can ride him. Well, it certainly would be a life saver if you could, Tom. Can I try? Well, all right. Go ahead. Hey, wait a minute, Dad. But before you get on that horse, I got something I wanted to give you. It was going to turn the trick. What are you talking about, Ham? Here, Mr. Tom, you take this lucky rabbit's foot. What are you rubbing it on your head for? That's going to make sure it's lucky. You can't go wrong now. Dude, thanks, Ham. Help me up, Dad. Okay. Up you go. Now, take it easy until you get the feel. Are you all right? Sure. Come on, please. You'll see how good he can ride. Well, he sure seems to know what he's doing. It's the best thing that could happen.  Now, Mr. Tom's sure going to win for you. He's going to try hard and with that rabbit's foot, he can't miss, no, sir. I hope you're right, Ham. Ham is never almost at a time, only sometime. Just look how nice he rides. Bring him back here. Oh, but, Jerry, I'm really afraid of let Tom ride him. Don't be afraid.  I wish I was sure of that. Well, just like Tom told you, why do you say he was a natural born jockey? Should I try him out now? All right, son, but be careful. Time is now. All right.  Get out there, please. Look at him go. Well, that's faster than he took off when Warren was riding him. Say, I believe you're right, Jerry. God, do it clear. That boy is all right. Hey, it might work. What, Mr. Rogers? I mean, I think Tom might be able to ride him this afternoon. Jerry can. Wait, wait now. Let me clock him at the quarter turn. There. What did he do then? Let's see. Well, say... How long did it take him? Only 23 seconds. Say, that's the best time ever. I told you Tom could ride. Oh, looks like you're right, Jerry. Ride, say, that boy is good. He's flying with that horse. You know he'll try hard to win. Sure he will. He'll try hard. That's what he'll do. Say, he'll try harder than Warren would have. What looked like bad luck has turned to good luck. Excuse me, sir. Don't forget to get his time at the half mile, sir. Oh, thanks, Ham. I nearly did forget it. There he goes now. How long did it take him? Hmm, 27 seconds. That's not so good. He seems to slow down. He's tiring, Jerry. Maybe this afternoon when the race is on and all the folks are yelling and hollering, well, we'll keep him going. Let's hope so, Jerry. But to be truthful with you... Yeah? Well, Jerry, I've given up hope. I'm afraid of Morning Sun and Silver Streak, Mr. Elston's horses. They're fast, Jerry. Real fast. How about Flashaway, my boss's horse? Oh, yes, Ham. Flashaway is fast, too. No, Jerry. I don't think we have much of a chance. And that boy you can ride him all right, though, Mr. Rogers. I'll say can. And it might make a difference, too. Oh, it's not the rider this time, Jerry. It's the folks that have all been kidding me about Blazer right. He just hasn't got the speed. He's all right for about a half mile, but then he slows down. Oh, there may be a way to overcome that. Well, can you think of a way, Jerry? Well, I was thinking that if Tom would hold him back and not let him get away so fast, he might save some of his speed that way. Oh, but then all the other horses would get so far ahead of him he wouldn't have a chance. Here he comes. Here he comes, Mr. Rogers, sir. You did all right, son. Exactly 50 seconds. Oh, where's the... Here it is. Well, it's not so good when you figure you did the quarter mile in 23 seconds, and then it took 27 to do the second quarter. That only means one thing, Tom. Blazer slows down. He tires. The race is too long for him. Here, Mr. Tom, I'll help you down. But I was holding him back. He wanted to go faster. Holding him back? Tom, why did you do that? You knew I was timing him. Well, I thought it would be better if I held him back. But why? Well, I didn't want him to get used to getting his reign so soon. Now, well, then he'd think I was going to let him out in the same place when the race was on this afternoon. Good boy. See, Mr. Rogers, that's just what I said. Sure. I wanted him to save all his speed for later. Well, I don't know, Tom. You do as you please. You seem to know how to ride him all right. Don't be so discouraged. No, Dad. Please don't. I'm going in there this afternoon to win, and I'm going to win. Hooray for you.  Well, you've got the right spirit, son. But I'm afraid spirit alone can't win the race. You'd better take Blaze back to a stable now and cover him up. I've got to go to the officials and tell them I've changed jockeys. I'll be back, though, in a little while. Okay, Dad. Come on, Jerry. Right with you. Come on, Ham. Hey, Yazi. Yeah, I'm right here. Your dad is sure discouraged, all right. Well, I think we've got a chance to win. I know for sure Blaze could go faster than he did. You sure enough going to give that rabbit food I gave him.  Hey, wait a second. I'll open the gate. We'll do it. Won't we, Blaze? Hey, Jerry. Yeah? Yesterday when Blaze ran after the fire department, you said you had an idea. I have. Well, what is it? Well, I can't tell you now, but you'll find out after the race is over this afternoon. Come on, boys. Let's put the horse up and get his blanket on him. In you go, Blaze. Here's his blanket. Let's take a saddle off first. Okay. I'll get his bridal off for y'all. There we are. Now, hand me that blanket. Yep, here it is. There. Now, you just take it easy, Blaze. Take a good rest so you can do your best this afternoon. We just got to win that race. Don't you think you'd best feed him some oats now so he'd be good and strong? Well, I don't know. Maybe Dad fed him already. He didn't yesterday before he tried out. Remember? He said he never fed Blaze until after he ran. Yeah, I guess that's right. Well, go ahead and pour some oats in his bin, hand. Yeah, that's what I'm going to do. That's just what I'm going to do. I think you'd better take a rest, too, Tommy. Not going to be so easy riding Blaze with all those other horses in the race. Well, I guess I'll have to keep my eyes open all right. I have to do a lot of steering. I'll say you will. Just look at that horsey. He was sure enough hungry all right. My gunners looked like he didn't have none to eat for a long time. Here comes your Dad back again. So soon? Uh-huh.  Yeah, here we are, Dad. Well, you're OK with the officials, Tom. I'll get you your riding outfit later. Hey, what's going on here? What's the matter? Who gave Blaze all those oats? Well, golly, we thought that... Is something wrong, sir? Wrong. Wrong. I'll say there is. We're as good as finished now. Now, all the idea of feeding a racehorse just before the race. Oh, we didn't think you'd fed him yet. Oh, I thought you boys knew better than that. Well, did you feed him already? Oh, of course I fed him. I fed him real early this morning, and I just gave him a little bit. Oh, that horse hasn't got a chance now. He won't be able to run at all after eating all those oats. He only had a little bit, sir, just a little bit. Yes, just a little bit. That little bit is just enough to make him so slow he won't even be able to get around the track and finish the race, let alone win it. Golly, we're awful sorry, Mr. Rogers. Oh, it's all right, boys. Oh, I suppose I should have known better than to enter Blaze and the race in the first place. Oh, cheer up, Mr. Rogers. You might be surprised. You never can tell about a race, you know.