Okay, so it's time to stop. Let me introduce the today's speaker, which is the William DeMau of Charles University currently. He will talk about the homomorphism problem for Boolean structures. So I'll see all yours. Okay, thank you. Can you hear me okay? Yep. Okay, great. So today I'd like to talk about this problem we've been working on, homomorphism problem for Boolean structures. So this is a general homomorphism problem, not just for relational structures, but for structures that could have relations and operations. And I first want, before I get into this, I want to thank the organizer, Jakub. You've done a great deed organizing these lectures, because this is really amazing. It was my first Zoom lecture ever. I'm very excited. Thank you for inviting me to do this. And also I want to thank my colleagues, Libor Bardo and Antoine Moté, because they have contributed significantly to this work. And without them, I'm not sure I would be able to do it. And well, maybe I would, but it would take many times longer. So I really appreciate working with them. Okay, so let's get into this. What is this problem we're talking about? So we've managed to prove this new dichotomy result for general structures. Well, I'm not quite general, but for Boolean structures. So if you consider an arbitrary finite signature S, so S could have relation symbols, as well as operation symbols. And then if you take a Boolean structure B, the problem of deciding if you're given another S structure A, is there a homomorphism from A to B? This is either NP or NP complete. So maybe it seems like, well, that's already been done by Schaefer, right? This dichotomy has already been proved by Schaefer for Boolean structures as well, not only for Boolean relational structures, but what we've done is extended this to arbitrary structures with relations and operations. So let's talk about the background of motivation for this. So we're trying to study a generalization of the finite template CSP over an arbitrary finite structure. So where we allow both relation symbols and operation symbols in the signature. So this is generalization of Schaefer's dichotomeric theorem for relational Boolean structures from relational Boolean structures to arbitrary Boolean structures. So we've only done this so far for Boolean structures, that's why we're calling it a generalization of Schaefer's result. But still, there's no general theory for CSP over arbitrary structures, right? So what we've done so far is only for arbitrary structures, assuming that the template is Boolean. So the universe only has zero and one. Okay, so the main results that we have is a generalization of Schaefer's dichotomies to structures with relations and operations. And the theorem that we proved is that if B is a Boolean structure with finitely many relations and operations, then the CSP of B is NP or it's NP complete. And to say this more precisely what we actually proved, we'll need a definition which most we are probably familiar with. So we say that an n-ary function is essentially unary if there exists an I possibly constant function g such that this, we have this equation holds, right? So it just means that either f is constant or it depends on one variable of its arguments. Okay, so right, so then the theorem that we proved is that if B is a Boolean structure, then the CSP of B is NP if B as an operation that is not essentially unary, or there exists a polymorphism of the graph of B, which I'll explain what that is in a minute, that is constant or not, not essentially unary. And if all functions in B are essentially unary, then the CSP of B is equivalent to the CSP of the graph of B. So I'm not sure if everyone's familiar with what this graph of B is, but let me just try to explain what that is. So we have this structure B, whoops, okay, let me try that again. So B is the structure with some universe and some operations and relations maybe. And then the graph of B which I'm calling script G of B, that's the structure with where the universe is the same. But the operations, well, it only has relations. And the relations are the relations of B, as well as the relations you get from viewing the operations as relations, right? So if you take an operation F, let's say, and it goes from B to the N to B, then we will instead view that as a relation that is a relation in, so we'll view that, we'll view the relation of F as a relation in B to the N plus one, where you have N two pulls of B that you plug into F and then you get some output and that output is the N plus first element of the of the two pull that's in the relation. Does that make sense? I'm not sure I explained that very well, but in any case, what we're doing is we're, when you view an operation as a relation, that's what we're considering the graph, this graph of B. Okay, I hope that's clear. If it's not, please feel free to stop me. Okay, so let's move on. And so let me explain how this result is a bit stronger than what I stated. So what did I state? I said that we have this result that the CSP of B is either in P and B complete for these Boolean structures, but actually we've proved something a little bit stronger, which says that under the above conditions on B, which is that B is this Boolean structure, so it's the universe is zero and one, and it has finite, so finite, finitely many operations and finitely many relations, and these are finite parity. Then in that case, there's at most a polynomial number of homomorphisms from each instance of CSP B to B. Okay, so not only do we know that it's in PR, NPR and B complete, but also we can put a bound on the number of homomorphisms. And we can effectively enumerate these. In fact, that's how we prove that this is in NPR and B complete. So that gives us this polynomial algorithm for CSP of B. Okay, so what has been done already in this problem? Of course, a lot of work in CSP has been done for the constraint satisfaction problem for relational structures, but not much has been done for structures that have relations and operations. The only work we know of is this work of Federer and Madeline and Stewart, this paper from 2004, dichotomies for classes of homomorphism problems involving unary functions. And they show that purely algebraic CSP generalizes to the relational one in the following sense. Each CSP of a finite relational structure is equivalent to a CSP of a finite algebraic structure, module polynomial time during reductions. So in other words, they look at relational structures and they say, oh, this is actually equivalent to if you made this into an algebraic structure. But module polynomial time during reductions, not many to one reductions. So that's one caveat. But we will show that in general, there's no polynomial time reduction from CSP of the graph of B to the CSP of B. But I'm going to use this abbreviation for the duration of this talk. FMS refers to Federer, Madeline and Stewart, and they prove that they prove this reduction from the CSP of the graph of B to the CSP of B when all operations of B are essentially unary. Oops, sorry about that. Okay, so their theorem, which I'll call the FMS theorem, says that for a structure B whose operation symbols are at most unary, the CSP of B is equivalent to the CSP of the graph of B by a polynomial time menu one reductions. Okay, so for this one, they actually achieve this menu one reductions result. So this is a really nice result. But the serum doesn't extend to structures with operations of higher arity. So only when as I mentioned here, oops, sorry about this. Yeah, so for these structures where the operation symbols are at most unary, they have this result. But if you consider structures with operations that have higher arity, this result doesn't doesn't obtain. And we'll give an example where we have just a two element algebra with a single binary operation for which the result fails. Sorry, Graham, can I stop you for a minute? There is a question from Katarina. She's asking about the difference with the value setting. Is that an operation can output a Doppler rather than a single value? Is that correct? Can you repeat that question? The difference with the values, I think that is that an operation can output a topo rather than a single value? Okay. Should I ask a question? Can you can you tell me what is an operation exactly? Sure. Yeah, well, what's the difference with that with a function? This was my question. Okay. Yeah. So an operation, I mean, we're only considering if you have so an operation on, sorry, this is my first time using this, an operation on a set B is just a function from some power of B to B. All right. So, okay, okay. I got, okay. Not a general operate. It's not a general function. An operation has domain, some power of B, an operation on B has domain, some power of B, and codomain is B. Yeah, sure. Okay. Thank you. Okay. Thanks for the question. Okay. So shall we resume? Okay. All right. So I'm going to go through some background now. So the the border between P and NP complete in Schaefer's dichotomy theorem can be described in terms of polymorphisms. For a binary relational structure B, CSP of B is in P, if B has a polymorphism that is constant or not essentially unary. And I think did I already define essentially unary? I don't think I did. Sorry. I want to make sure that I did. Yes. Okay. Good. So we covered what essentially unary means. Okay. Okay. And otherwise it's NP complete. Okay. And the discovery of the role of polymorphisms for relational CSP as obviously had a big impact on CSP in general, as you can see in some results that I'm citing here. And these made possible to conjecture this borderline between P and NP complete CSPs over non-billion domains. And these were important for generalizations to infinite domains and promised CSPs, value CSPs. And we don't have analogs of any of these results for the non-relational setting. So as soon as you put in some operations and your structures, the theory is still wide open. So I think this opens up a lot of interesting questions for future work. Okay. So let's talk about some reductions. Here's a fact that a map from a structure A to B is a homomorphism, if not only if it's a homomorphism from the graph of A to the graph of B. This is a very easy to prove result. And therefore you get that the CSP of B is reducible to CSP of the graph of B, where the reduction is a map from an instance of the CSP of B to the instance of you just take the graph of the instance. Okay, that's sort of obvious. And so just to review some definitions that we're going to need in the lemmas that we use in our proof, quantifier-free primitive positive formula in a signature S is just a conjunction of atomic S formulas. And two S structures in B over the same domain are called term equivalent. If for every operation of the first structure, there exists a term, an S term, such that this operation is equal to that S term for every tuple. Okay. And conversely, for every operation of the latter structure, you can find a term such that this operation is equal to that term. Okay, so we're going to use this term equivalence in our lemmas that I'm about to go through. So the first one is that... Can I distribute for a second? Yes, sure. To the term equivalence of these structures, it does not say anything about the relations. You just, you forget about the relations and you just compare the algebra. Yes, it's basically just about the operations. Okay, thanks. Okay, so if you have a signature and you have some structure in that signature and you have some finite set of identities in that signature, such that B models all over those identities, then for every instance of CSP of B, we can compute in polynomial time an instance, Y, such that Y models those identities. And the instance X has a homomorphism B if and only if it has a homomorphism, if and only if Y has a homomorphism to B. Okay, that's the first result we'll use. And the second is that if you have two structures that are term equivalent, and you suppose that every relation of A has a quantifier free primitive positive definition in B, then CSP of A has a polynomial time menu on reduction to CSP of B. Okay, so does this make sense? Okay, as a corollary, we retain this algebraic invariant for complexity of CSP of finite structures. So recall, a partial operation is a partial polymorphism. So what a partial a partial operation is, it's just this operation that's defined on some parts of the domain, but possibly not all of the domain. And it's a partial polymorphism if for all relations are and for all tuples in the relation. If you plug those tuples in to this function H, wherever it's defined, then it's in R. Right, so it doesn't have to be defined everywhere, but if it is defined, then it belongs to the relation. And of course this, I'm like expecting it's assumed that this evaluation of H on all of these tuples is evaluated component-wise. Right, so H is an n area operation or an n area function from R to B and R to the end of B. And then we've got these tuples in R. And when you plug them into R to H, you just take it component-wise. And the result, if it's defined, has to belong to R. Okay, there's this result of Romo from 1981 that says, if you've got two finite relational structures such that every partial polymorphism of B is also partial polymorphism of A, then the relations of A have a quantifier-free P definitions in B. Okay, so we're going to use this result. And the corollary of this is that for a finite structure B, the complexity of the CSP of B depends only on the clone generated by the basic operations of B and the set of partial polymorphisms of the relational reduct of B. So the relational reduct is just throw away the operations and just consider B with its relations. Okay, so in the Boolean case, the complexity of CSP depends only on the clone generated by the basic operations of B and the total polymorphisms of B. So we don't have to consider the partial polymorphisms in this case. Sorry, the total polymorphism of the graph of B. Okay, so this is invariant is weaker in the sense that it contains less information than the previous one. And indeed, every polymorphism of the graph of B is a polymorphism of the relational reduct of B. So in that sense, it's weaker. So it's unclear to us whether this weaker invariant is enough to separate the tractable and the NP-complete problems for structures with bigger domains than the Boolean case that we've so far considered. Okay, so let's get to the main theorem that we want to show here. And that is the dichotomy theorem for Boolean structures. So if you have a Boolean structure, then the CSP of B is in P. If B has an operation that is not essentially unary or the polymorphism, the polymorphism clone of the graph of B has a polymorphism that is constant or not essentially unary. Otherwise, it's the CSP is NP-complete. And the proof relies on this classic result of post, which is that if you have a clone of operations on 01, if this clone contains operations that an operation that is not essentially unary, then it contains one of the operations meet, join, majority, or minority. Okay, so let's go through the steps of the proof. They're fairly straightforward. So we won't get too technical. So I hope you'll bear with me. Okay, so if the Boolean structure has an operation that's not essentially unary, then by post theorem, the clone generated by the operations of B has either a semi-latest majority or minority operation. And by lemma two, which we already went through, we can assume that B has such an operation in its language. And we prove that the presence of such an operation puts the CSP of B in P. So it will be tractable. So in fact, we prove a stronger result in this case for each finite instance X, there will be at most a polynomial number of homomorphisms from X to B. Okay, so I'll go through each case. Maybe I'll go through the first couple of cases and you'll get the idea and we won't have to cover all of them. But there's only a few cases. So let's begin. So the first theorem is that if we have this Boolean structure, notice we've got this structure here. It's on the universe is zero one. And we have S is some operation. And then we might have other operations and relations. And in this case, we're going to assume that the operation is this meat operation, or maybe it's this join operation. If the structure has either of those, then the CSP of S structure is in P. And that's where we're going to show now. Okay, so if you fix a structure as an instance, then this structure X satisfies, you can find this structure X prime that satisfies all the equations that the original instance satisfies, such that if you have a homomorphism from X to B, then you also have a homomorphism from X prime to B and vice versa. So this is I probably let me let me try to say that again. So if you have an instance X, you can find an instance X prime that satisfies all the relation or all the identities that be satisfies, and such that there's a homomorphism from X to B, if not only if there's a homomorphism from X prime to B. Okay, and by lemma one, we can assume that that S, the S above, sorry the S, so let me make this clear, we can assume that the S interpreted in this structure X is a semi lattice operation because of this fact that you can find you can find a structure that satisfies the identities that B satisfies and has a homomorphism to B, if and only if the original instance has a homomorphism to B. Okay, so if we define this lesser equal, did you have a question? Okay, X lesser equal to Y, if and only if X is equal to S of XY, then for all H from A to B, the set, this inverse of one, the inverse under H of one is a principal filter in in this structure X lesser equal, right? So a principal filter is this upward closed set that's closed under S. So we can see that by saying that by seeing that if X is in the inverse image of H of one under H and X is less than Y, then H of Y is, well you can see these equations, right? So what we show is that each homomorphism, if you can find a homomorphism, it corresponds to one of these principal filters, and of course you have at most the number of elements in X, many principal filters, so there are most that many homomorphisms from X to B. Okay, so this is the idea is that you consider, if you have a homomorphism from A to B, and then you consider its inverse image, so it takes some, it takes some elements to one, and then you consider the inverse image of one, and then this is a principal filter in this structure X with this lesser equals, and so if you want to count the number of homomorphisms, all you have to do is consider the number of principal filters, and of course this is bounded by the number of elements in X in the domain. So this is how we know that we can find a bound on the number of homomorphisms from X to B. Can I have a question? So you are using this reduction here, I assume that this reduction doesn't change the number of homomorphisms significantly, going from X to X prime, is this correct? That's correct, so yes that's a good question, so we're assuming as I think this is Alex Kozda who's talking now. Yes. Yes, so that's a good question. So what we're doing now is going from an S structure S, X, so that's the input instance, and then we want to now work with the structure X prime that satisfies these identities, and Alex's good question is well if you start working with X prime, then now you have to know that you haven't changed the number of homomorphisms, and yeah that is correct. We have verified this and I'm sorry I can be a very clear answer why this is so, but maybe someone, one of my co-authors could come to my rescue at this point and tell me. The point is that X prime is obtained by factoring X by a congruence, and it's a congruence that satisfies that every homomorphism factors through this congruence, so that you don't change the number of homomorphisms at all. Like the kernel of every homomorphism from X to B would contain this congruence that we factor with, so that's the number, so that there is a bijection between the homomorphisms from X to B to the homomorphisms from X prime to B. Thank you. Thank you very much Anton. This is my co-author Anton. It's explaining that if we consider homomorphisms from X to B, and then we have this other X prime to B, and there won't be more homomorphisms from X prime to B than the real way from X to B, so yes thank you for clarifying that for us. Okay good. So right, so we do similar things in the majority case where instead of having this semi-ladis term, we have a majority term, which is some term that satisfies these identities that most of you are probably familiar with, and permeate these inputs, and we do similar things with this, and things seem to work out fine, right, so we fix a structure with a signature, and we have M that's a ternary operation, we're not sure what it is, but again we can assume that we have the input instance satisfies the same equations that the template satisfies, so we can assume that M is indeed a majority operation, and it satisfies these equations, and using these facts we're able to show that you have a bound on the number of homomorphisms, again by the fact that you have inverse images of these homomorphisms, must be principal filters in the lattice, and so there's only so many that you can have, right, because there are only so many principal filters that you can have, in fact that's bounded by the number of elements in the domain. Okay, so that's the idea with the majority case, and the affine case is similar, what I'm calling the affine case, if the operation is x plus y plus c, you got a similar result, so okay, so the proof of the main theorem is that if you've got b, a Boolean structure with finitely many relation and operation symbols, then CSP of b is in p or in p complete, and the proof is first, if you, the first case, if b has an operation that is not essentially unary, then CSP of b is in p by these, sorry lemmas, these were lemmas one, two and three that I just showed you, and then if b has only essentially unary operations, then b is term equivalent to b prime, whose operations are all unary, and the CSP of b and CSP of b prime are polynomial time equivalent by lemma two, and then the theorem of Federer and Stuart, and I'm forgetting the other author, Antoine, do you want to help me on this? All right, so the paper by Federer at all is showing that the CSP of b prime and the CSP of the graph of b prime are polynomial time equivalent, okay, so by Schaefer's dichotomy theorem, the CSP of the graph of b prime is in p if the polymorphism clone of g of b prime contains an operation that's essentially, not essentially unary, and otherwise it's np complete, okay, so finally the clone, the the polymorphism clone of the graph of b prime and the polymorphism clone of the graph of b are the same, and that yields the final result, okay, so I'll show an example now where this is the example that we were struggling with for a while to try to show that this result of, of, oh, Madeleine, Federer and Madeleine and Stuart doesn't doesn't apply in general if you if you take structures with operations that have higher arities than than one, and so this structure, we have we found a structure where the CSP of the graph of that structure does not reduce to the CSP of the structure, and it's very simple, but it took a while for us to prove, or at least for me, I think maybe my co-authors could have done it much quicker, but so if you take this structure that's just on zero one, and you have this binary operation where it's, I guess this is called the shepherd stroke, I don't know if that's what it's called, but it's it's very easy to describe this is the operation table, it looks like this, so zero times zero is one, and otherwise, oops, sorry, this is a mistake, okay, let me try that again, okay, so this operation is one only if both arguments are zero, otherwise, the the result is is zero, okay, so if you consider the relation induced by this operation, what do you have you have these triples zero, you have zero zero one, right, this is this first tuple in the relation, and then zero one zero, right, that's the second, so you construct the relation from this this operation, and you get this, and it's clear that that none of these operations, the the join meet majority or minority is a polymorphism of this of this structure, this relational structure, right, okay, and of course the constant map into zero one is also not an endomorphism B, so the CSP of the graph of B is NP hard, right, and on the other hand this operation, this operation, if you take this operation, x, y goes to x dot y dot x dot y, that's exactly the join, x join y, and okay, and so the the results we've gone through so far imply that this will be actually tractable, so the CSP of this structure is tractable, and yet the CSP of the graph is not tractable, the graph of the structure is not tractable, okay, so let me move on and in conclusion discuss some open questions and future work, of course this is brand new stuff and we haven't really even sketched the surface of CSP for general structures, I mean the the initial question that I was interested in because I'm I'm not really a specialist in relational structures, but I'm more a universal algebraist, so I deal with algebras with just universes and operations, and so for me it's very interesting to know when when do you know you have a homomorphism from one algebraic structure to another, and that was the initial question that got me interested in this, but then in working with Levor and Anton it's clear that when you throw in relations you have an even more general and more interesting question, when are there homomorphisms from these general relational structures to to another general relational structure, so this obviously we haven't we haven't made very much progress, all we've done is we've managed to to answer this question for Boolean structures, but of course there's a lot of open questions when you consider bigger domains than Boolean, and so let's discuss a couple of these, and so this raised a number of questions for for developing in general theory of CSP of arbitrary finite structures, not just relational structures, and the reduction from CSPs of relational structures to CSP of algebras in Federer, Madeline, and Stewart is a polynomial time-turing reduction, and we'd like to know uh does this generalize to or can we can we get a polynomial polytine many one reduction instead of just uh the stirring reduction, that's the first question, and then another question is that we showed for a finite structure uh be the complexity of CSP of B depends only on the clone generated by the operations of B and partial polymorphisms of the relational reductive B, and maybe the complexity of CSP of B is captured by the polymorphisms of the relational reductive B, and that's something that is unclear at this point, and finally for the relational for the Boolean dichotomy of above that we that we laid out, the main step is establishing that as long as B has an operation that's not essentially unary, then for all instances A there are only polynomial many more because it's from A to B, right, so it's generally unknown what other properties of finite algebras imply such polynomial bound, and that that would be an interesting question to resolve, okay, so there are also some more questions relating to um width and um relational width and and so forth, but I didn't discuss the results in our paper that have to do with that, so I think I'll just leave these on the slide and you can consider these if uh if that's what you're interested in, so okay, and I'd like to invite now my uh co-authors to give any comments or input that they would like to because I think they they might have noticed some areas that I that I was uh not very clear on or that I lacked uh sufficient um insight into that they would like to clear up, okay, thanks for the talk, thank you, there's some comments, questions, remarks, corrections, I'm sorry, what happens if you have a purely a operational structure, so a homomorphism for algebras? Right, so do we know the answer or anything? Yeah, that's a good question, um in fact the the in the federal medallion and stewart art uh paper they they proved that in fact this is more general than the relational problem, so you can represent this relational structure uh can be reduced to a csp of algebraic structures, so this is somehow more general or more difficult than the relational structure csp. Don't they also give a reduction in the other direction showing that the csp of an algebra reduces to the is equivalent to a csp of a relational structure up to turing reductions? Yes, so somehow it's the same pure relational structures and pure finite algebras, yes, but again their their paper is mainly only considering these structures with unary operations, right, so in this case they show these are equivalent. No, so sorry, do we have the economy or not? That's somehow a bit confused. A dichotomy for algebras? In this case where we're just talking about Boolean structures, yes. No, in general. No. What do we know in general? I'm confused as well. I'm confused as well because I thought there is no reduction, like it's not known whether every algebraic csp is equivalent to some relational csp. Yes, that's correct. Only for unary algebras is what you said. For unary algebras, yes, so every relational csp is equivalent to an algebraic csp where you only need like two unary operations. Yes. The framework of just pure algebras can be marginal. Okay, so it's open, is that what you say? It's open, yes. Okay, interesting. And what does the dichotomy look like for Boolean algebras? That should be somehow like for Boolean algebras. Yeah, for Boolean algebras it's simply if there is an essential operation, then it's easy. That's already proved. And if there is none, then it's also easy, right? Yes, it's always easy. Yes, yes, it's always easy. Okay, well, then yeah, it's a good question whether there is a reduction going the other way. Yes. Because I suppose this paper of Federer and others, it's from the times when... It's prehistory, it's medieval age. Yeah, essentially if you want to say you can forget about this problem, you can prove that csp dichotomy reduces to it, and that's that. This means you can forget about it, it's too difficult. So maybe they didn't even think about the other direction. Okay, interesting. Maybe let me just comment, because William told me, told us on this last slide, can you put the last slide please, William? Yeah, no, no, the last this one. Yes, there are some questions about width. So I just want to clarify why these questions are there perhaps. So for relational structures in general, we know that if it has bounded width, then it has relational width to three. All the algebraic things, pure algebraic things are solvable actually by local consistency. So they have bounded width. But the width is strange, and they do not have provably width to three. So it's also one thing which is in the paper. So there is no collapse of width, at least not that brutal as for relational csp. Yes, thank you for having that. Some more questions? I guess it would be interesting to have a specific example of an algebra for which the csp is open, the complexity of the csp is unknown.  It would be good. Yeah, I mean this is, yeah, renewed to me, this question is new. So we just managed to do it for just Boolean structures. And now from here we can consider other more general, hopefully more general results. Some more questions? So by the way, your deposition side of your dichotomy says that you always have a polynomially many insolutions. Is that it? Yeah, there are most polynomial many homomorphisms. Well, no, that's not correct. It's only in the case where the algebra has an essential operation. Because there is, I mean, the relational case is part of this, right, the purely relational case. And it's not true that you only have polynomially many homomorphisms. Well, not in general, but in this one case where it's tractable. Well, there are some tractable cases that come from the relation or purely relation. The thing is that you have polynomially many homomorphisms provided that your target algebra has an essential polynomial, an essential operation. Okay. So it's an example which has many solutions. I think there are some results in CSP about the structural restrictions, which classify when you have polynomially many solutions to something like this. So I have a question about this actually, because it kind of feels like when we are talking about Z2 modules or vector spaces, then we should have many homomorphisms. I'm not quite sure, but you're saying that it's polynomially this case too, when it's algebra with some Maltsev operation, let's say, and we are mapping one to the other. It's still polynomially many. Well, there is only like one Maltsev algebra here, right, which is Z2. And so the number of homomorphisms between vector spaces is very small, right? It's determined on the basis. Okay, this Z2 example is actually cute because Z2, like relational Z2 is very unbounded with, right? And the Z2 algebraic is bounded with. But it has polynomial many, but we don't know in general, for example, for Maltsev algebras. I don't think we know it. Yeah, we have, Kompatcher mentioned that from some results, it is actually true for, I don't know, nil-pult and super nil-pult and something like that, or a billion at least, but even Maltsev in general is open. So I begin to see why it's not exponential. It's because the input algebra is not given, like 0, 1, 2, 15th power. It's given by a list of elements. So I cannot go exponential. I guess that's a yes. Yes, that's correct. Yeah, yeah. But like even if you go 0, 1, 2, 15, 15th power, then there is only like 2, 2, 15 of them? Yeah, okay. Yeah, yeah. Yeah, which is linear. Yeah, sure. Okay. If there are more questions, I want to thank my co-authors for joining me in this project. I think it was really fun to do and they've made it a pleasure. It was really good working with them and they're both Antoine and Libor are great people to work with. So thank you very much. I'm sorry if I didn't do the work justice in this talk. Okay, I guess that's all with the questions. So thanks again.


So in the midpoint of the 2016 season Camilo Anthony had enough he was leading the league in points per game He was trying to get his team to victories But the teammates random were just not good enough and it Camilo Anthony.  He was just scoring left right and center as Camilo Anthony does just non-stop scores He asked for a trade. He demanded a trade from the Denver Nuggets and there was plenty of teams that were interested in him He could have gone to Miami the Oklahoma City Thunder the Boston Celtics the Houston Rockets and joined James Harden and Dwight Howard He could have joined the Trailblazers and joined a young force and Damian Lillard He could have joined the New York Knicks and teamed up with his twin brother Camilo Anthony He could have joined the Lakers and teamed up with Kobe Bryant the Clippers who needed a small forward the Cavs who needed I'm no wonder, but he joined the Chicago Bulls. Okay, Melo went to Chicago, baby This is the team he was meant to go to two years ago. And now he's finally here Welcome to Chicago Camilo Anthony. He teams up with Derek Groz and 82 overall He teams up with Jimmy Butler and 88 overall and they still got the old man power to sell on the bench injured I Mean this team is nice. It's gonna be Derek Rose Jimmy Butler Camilo Anthony Power to sell a Joe Kim Noah who I would like to see Joe Kim trade. I'm not even gonna lie I don't really like Joe Kim Noah, but you know in this series. I like him, okay? Because you know, we gotta we gotta win games. We can't have no team chemistry at the moment I'm averaging about 30 points per game 10 rebounds six assists Denver's upset that they lost me, but they had to lose me because I was and look at this cheese I I'm always defensive over the year. I can't be going in defense play the year I got a win MVP not defensive over the year. Okay, but yeah, I'm leading the I'm leading the league in points per game I was bringing Denver Losses to be honest. I was bringing Denver a lot of scoring But the teammates around me were just not good enough so I requested a chain to the Chicago Bulls and Just like in real life when Camilo Anthony was frustrated with Denver He went to New York instead of going to New York. I go to Chicago And when Mello went to New York, he he got a little bit of an upgrade.  I think it was but anyway in this in this scenario. He cuts off his dreads and He's going into into Chicago with with no dreads, but he's gonna look like the Camilo Anthony that we know today and Yeah, it's gonna be exciting. I cannot wait to play with Derek Rose Jimmy Butler I got teammates that can actually do something man. I'm excited. I got Jimmy Butler. He can play defense He can score. He can pass the rock I got D Rose if I need somebody athletic in 2k Derek Rose is athletic by the way I shout out to Jimmy Butler. He got injured the other day, which is I was watching the game at the moment I don't even know how bad the injury is because I'm I'm recording this on the day that Jimmy Butler got injured But Derek Rose he went ham when Jimmy Butler got injured and I just hope the D Rose goes back to the MVP Derek Rose He was a beast. He was MVP I mean best part of the league at that time It was sad to see him go down with all those injuries and play like no games throughout You know the next couple of seasons But Derek Rose in this series.  All right, he's gonna be a freaking MVP I don't know why I'm pause on this screen, but I'll just leave it Comment down below What would you rate this face out of Camilo Anthony like because I didn't you know use the real Camilo Anthony in the game I had to make him so Comment down below. What do you think about how I made him give it? Give me a rating from 0 to 10. How well do you think Camilo Anthony looks in this series? He's gonna look a lot better with a headband because Yeah, he has a headband in real life and he looks a lot better trust me He looks a lot better in in 2k when he has a headband So I'm I actually searched up on my phone, you know pictures of Camilo Anthony and I was just matching up You know what he has in real life, you know the facial hair and the hair etc like that And just putting it in the game Because yeah, I want him to look pretty good. I want it to look like the real Camilo Anthony and Apparently also 2k is being sued For having like real NBA players tattoos in the game. So that's kind of weird. I I don't know what's happening with that I think they might have got it resolved already But that that sucks for 2k if they've got if they're getting sued for having, you know players tattoos in the game But yeah, also somebody said, you know, I should um, I should change Camilo Anthony's cheek size because he said it was too His cheeks were too fat So I'm actually gonna change it for the one for the one guy that commented that and make his cheeks a little bit You know less chubby, I guess is what I'm trying to say But this is pretty much what he looks like with the Chicago Bulls And if you guys want to see the first episode where I face up against the Brooklyn Nets With my new squad the Chicago Bulls leave a like subscribe if you're new Let's see if we can reach 700 like if we can reach 700 likes I'm gonna upload the next video of this series immediately Facing up against the Nets with my first game as a Chicago Bull I'm out believe See what I did there. Peace


In this controversial episode of Mind Pump, we talk about group exercise classes and why we're usually not fans of them. Ooh, put on your helmets. So we talk about the origins of group exercise classes and kind of what it's looked like throughout the years. We talk about the pull of these classes, why they attract so many people, and we talk about some of the positives. There are some positives to group exercise classes, but then we get negative, and we talk about all the bad parts of group exercise, why we've seen high rates of injury, why they're typically inappropriate classes for most people, why they attract the wrong type of person, and why we think they need to change their approach, and we give you some advice. If you're somebody that wants to take group exercise classes, we kind of help you out on picking the right one, and if you're a group exercise instructor, we also give you some advice on how to make your group class a better one for most people. Now this episode is brought to you by our sponsor, Keon. Now Keon is the company owned by Ben Greenfield, our good friend. He's like the biohacker extraordinaire. In fact, he's the only best one that exists. The rest of them are all fake, but Keon coffee is phenomenal. We've been drinking Keon coffee for a very, very long time and been trying to work out a partnership with them for a while. It's literally the best coffee that we've ever tasted. Now most coffees are covered in pesticides that tend to be oxidized and stale and rancid by the time they hit the shelves. Keon is not like that. It's 100% organic specialty grade coffee. Of course, it's fair trade certified, completely free from mold, pesticides, and other toxins. This coffee legitimately is super fresh. It tastes amazing. I'm not making this up. You're going to need to try this yourself. Again, we work very, very hard at working with Keon because when we drink their coffee, we always get blown away by the flavor. 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That's M A P S S P L I T .com and use the code split 50. That's S P L I T five zero no space for the discount. I was thinking the other day at him about some of the more controversial statements that we've made on the podcast. We are me. Well, you came up a couple times.  I backed you up on that. But you got a lot of heat. I was a lot of heat for it. And so I thought we should do an episode explaining kind of what we mean by that because that's kind of a strong statement. We're going to defend myself a little bit. I don't think you mean it should all die, right? But I think we should talk a little bit about... No, there's value in some group classes for sure. And I think the argument that I think is pretty valid that people make is that it got me off the couch or it got me motivated or it helped me feel better about coming in and exercising because I could bring a friend or I went with a friend. And so I get all the reasons why it could help somebody ease their way into fitness. And I can't argue that. I can't say that if you would have never got involved in it because you were too scared to do something all by yourself and so therefore going to a group class where you felt it was a little more comfortable for you to go in. Because I know that can be very intimidating, right? We know that with clients. That's one of the top reasons why people don't start, right? Is the intimidation factor or the lack of knowledge? I don't know what to do. Well, that's what we always highlight it to, like the group X class when you go into the gym and we kind of try and pitch that initially to new members because it's like, okay, well, maybe you don't really know specifically what to do in the gym, but go ahead and we have these classes for you. It's like sort of an introduction into fitness. So that's really all I see. Yeah. I mean, okay. So little back story, obviously, the three of us have run gyms and worked in gyms with group classes. Because that's always a big part of the big box gym is that you'll be running five or six classes every single day. Some of them is full of people, 30, 40 people at a time. And in our experience, and I can speak for all of us, I know we've talked about the soft air. I've seen good come from it, but I've seen as much bad come from it. It's almost like a one to one ratio. Oh, I don't think it's one to one. I think the reason why I made statements like I wish it would dies because I think a majority of it is primarily been bad.  And I think a lot of people get stuck, you know, I don't know if you guys did, but for much of my career, I was I trained a lot of group X instructors. Oh, as clients as clients. Yeah, I had a lot of many of them, you know, came to me to help them because they couldn't lose body fat, or they couldn't get their bodies to change yet they were teaching two to four classes of group X every single day or every and all the movements with them and sweating and everything. And that's what they were they why they were frustrated.  I'm doing all these classes and I can't seem to shred the this extra 10 pounds or 15 pounds or change my physique or build muscle. Like, and so I helped a lot of them out. And it was really, really challenging to help a group X instructor that's teaching multiple classes per day. Yeah, for me, it was injury. It was a lot of people. There were a lot of instructors that would come into me because they hurt because their hip or their knee or their shoulder hurt from doing certain repetitive movements. Well, this is what I saw when I worked with Orange Theory was, you know, people come in, they they love the workout. And, you know, six months, nine months a year later, you don't see a lot of those people, they start to, you know, fall off. And a lot of the reasons why most of them were falling off is because of chronic pain from the repetitive movements, the same movements over and over and not addressing what those people need. And what people don't realize is when you do these, especially like an example like Orange Theory, where you're running on a treadmill all out, right? They have you sprint on a treadmill. When you do that, and you've got really poor mechanics, and you do that on a two to three times a week basis, you're going to exaggerate the problem. 100%.  Group exercise was how people worked out way way back in the day. If you look at pictures of like some of the first gymnasiums, that's where the word gym comes from is from gymnasium. And you can actually find these pictures online, whether these old black and white photos, and you would see like rings hanging and you know, a couple kettlebells, like step ladders and everything. Exactly. But you but they would have these huge open floors. And the way that people would work out in there is they'd all go in there and they would do calisthenics calisthenics or group exercise classes. So they actually the origin of of gyms is a group exercise in then the group exercises we know it that started kicking off in the late 60s and 70s and really started taking off in the 80s when they had step aerobics and they started putting music to the classes. Some of the first classes that I remember seeing back when I started working out in gyms and I probably started working out in big gyms at maybe 1994 95 and I started working in them at 97 or so. They were mainly these kind of dance aerobic type classes. Then they started integrating steps into them. I don't know if you guys remember step aerobics. Yeah, those still get done, I think around the country. Body pump, step aerobics, jazz exercises was like the first version of all this kind of aerobic to music and doing all these like type of dance moves with exercise. So that was like one of the first and then you had like your Richard Simmons that came out and used that same kind of aerobic type of movement in a class but made it fun and all this to kind of shed weight and sweat, sweat to the oldies, all that stuff. It's group. It's basically just group movement. It's group movement. It's line dancing would be similar, right? Even though you don't consider line dancing to be a group exercise class, it's very similar in the sense that you're all following the same prescribed movements. You're following an instructor. You're kind of doing what they're telling you to do. And I can see the pull. I can see a lot of pull. Why they're so popular because some of the most popular things. I'll give you guys an example of how much pull group classes have. When I had a gym, when I was part owner of a gym down in Palm in the Palm Springs area, I had a competitor. There was this gym that was not too far from ours. And you know, I came from the aggressive marketing of 24 early days of 24 fitness. So I left 24 hour. I go have this gym with my partner and I still have that aggressive mentality. And you guys know what we used to do at 24 with competitor gyms, we would send our salespeople to go and we'd set put flyers in the parking lot or we'd offer some kind of a special discount to their members. Well, this pissed off the owner because he wasn't used to this kind of aggressive marketing. And he came and threatened us and did the whole thing, which obviously I took it to the next level because that's the worst possible thing you could do.  But if you want to kill the gym, you take their top group X instructors, it's one of the easiest things you could do.  And this is what I did. I took three of their top instructors, paid them more and inevitably, you know, what followed was 60 members and then people talking about where they went. And then we started taking a lot of members. There's a lot of popularity and pull. Part of it is the fun component. There's no doubt that working out and doing stuff with synergy in a group setting like it does like provide more energy to sort of feed off of other people and there's momentum behind it. And so people are really attracted to that because, you know, it's an experience is something that they look forward to. But like that's really the value of it is on the entertainment side of that. Right. And it is a shared experience. It's community. That's part of the, I mean, and we see the examples of that in all these other modalities. CrossFit's another example. CrossFit took the... It took all the competitiveness out of like all the different sports and just like threw it into an actual workout. And what CrossFit did is they took the strength of group exercise, applied it to resistance training and became very popular. And, you know, when you have a shared, here's the thing, exercise is hard. It's painful. It's difficult. It's challenging, especially if you're new to it. If you're new to it, because one of the things that happens when you work out on a consistent basis is you don't just build your strength, your physical strength and your physical endurance.  This is one of the things that you learn when you work out. In fact, when I would get a new client who never worked out, we would do an exercise and I know as a trainer, having trained many, many people, I know how many more reps this person would have. And you could see on their face, they couldn't tolerate the feeling of doing an exercise because they just, they weren't used to it. So it's a very difficult process. What makes that process easier is when you share the pain with other people. So it's like, we're doing this hard thing, but we're doing it together. You commiserate. Exactly. So I'm looking at to my left and oh, she's sweating over here and that guy looks like he's going to throw up and all right, we're going to keep going and pushing. Then when you leave, you have a, you feel a sense of accomplishment. Like I was able to get through that struggle and we were able to do it together. And that's one of the, that's one of the big pulls. It's also one of the big weaknesses though of group class because when you, when you learn to rely on that feeling of motivation, you are actually setting yourself up for failure because they never, it can't last. That feeling of motivation, that feeling of fun never lasts, which is here's one of the trademarks of somebody who does group exercise classes all the time. They, they have to change constantly to new and different classes and modalities. And every time they get bored, they stop, they'll stop for X amount of months.  Oh, this is a new kickboxing one or I'm going to try this boxing one. And you get this stop start thing, which is in my opinion, one of the worst things that I see with the group exercise, you know, dependence or whatever. It does another value of it. It's less expensive. So if you're, if you're trying to get into working out and you want some instruction, group exercise is way less than having a one-on-one instructor. Well, that's again, I think that's one of the main reasons why I will never die. I mean, this is like one of the things that even as a trainer is promoted a lot because it's, it's a way for you to be able to monetize your time a lot more effectively.  So it's beneficial for them, it's beneficial for the members coming in, but at the same time, is that the most effective strategy? You know, I would contest that. Right, right. And then the accountability factor. When you're going with friends or other people, I think people feel like, okay, this is going to make me more consistent because I know if I don't show up that, you know, Susie or John will ask me where I've been, why I didn't come to the class. I have a scheduled time that I have to be there, which kind of puts me on a schedule. It's a big sense of community. It does. Now here's the thing with accountability, the whole, like, it's a bit of a myth. Actually, if you look at the statistics on group exercise classes, it's basically the same as if you were going on your own in terms of consistency. It really doesn't. Now, there is that air of it's going to increase my accountability, but statistically speaking, when gyms add group exercise classes, it's not to increase member retention. Gyms add group exercise classes to gain new members. This is a common knowledge among gyms. The thing that keeps members is actually personal training. Personal training does a phenomenal because it actually provides results. Well, it will. That's a big one. Yeah. I also think it matters what modality that we're talking about too in regards to group training. For example, I think yoga is phenomenal and that's a group class. I think stretching as a group, although I would still make the case that, you know, targeted, targeted corrective stretching is far more beneficial than a generic yoga class. Of course. But yoga in general, I think, has got incredible benefits and taking a class that is in a group for something like that is less detrimental than taking a strength building class. Those are the worst. The strength building group classes are the worst because they're doing cardio with weights. Exactly. They're taking the strength element out and replacing it with cardiovascular. Yeah. I remember when the body pump classes first came out. Do you remember those? When those became popular and they're like, oh, I lift weights. I do body pump and I'd watch the class. I'm like, well, no, you're not. You're doing aerobics with weights. There are no rest periods. None. Typically.  The group wants to keep going. Everybody wants to keep this thing going and it's inevitable that there's not going to be any rest. Well, it's the entertainment factor that you get from the class and rest periods are not entertaining. Yeah. And as a trainer, when you're training someone one-on-one, the rest periods, good trainers know how to deal with rest periods just like they know how to deal with the workout itself. They're both very, very important. Oftentimes when someone's resting, they're going to ask you lots of, well, why don't I just keep going? I'm not doing anything right. I have to explain the benefit of it. Why would I have to rest and how we're building strength and why that's an important part of the workout and the programming. Try doing that in a group class. It's very difficult, but you mentioned yoga, Adam. The reason why yoga tends to be a better group class is because the traditional goal of yoga is not, because if you ever take a really good yoga class, you'll hear the instructor say this often. They'll say, this is your own practice. Listen to your body.  It's a slower pace. They can come and watch and see what you're doing. However, I've been to yoga classes that start to develop that competitive feel that you get from a lot of group classes.  I also think it's the risk versus reward thing that makes it different, right?  The detriments that come from that are far greater than, you know, doing a, you know, some sort of a stretch or pose that maybe you don't need very much or you're not doing very well. So that is also why I say things like, well, yoga to me is not a bad group class.  You don't sleep very much.  Meanwhile, you've kickstarted your new way of life because it's a new year. You're in a calorie restricted diet and you're doing that. The detriments of that are incredible. I mean, this is what for most of my career, I was stuck helping people that had already gone down this path. Very rarely did I get somebody that was a blank slate that came to me and said, Adam, I'm smart enough to know I need professional help and here's, you know, here's X amount of dollars, help me, you know, start off on the right track. I always got people that are most of the time, I should say, the people that I got had tried all these things first. And one of the hardest clients to help are the, are the chronic yo-yo dieters who have lost and gained, lost and gained. And they've done it through these high intensity type of classes and extreme diets to get there. And they've just not had any success for years. And now they finally broke down and are willing to pay any money to help them. Those, and I think that's where a lot of the passion of why I say things like, oh, I wish group training would die is because it's raw. Those are the hardest people to help super hard. And you mentioned cortisol junkie. So now I'm going to use insulin as an example of what we're talking about here. So we've heard of insulin resistance, right? Where your insulin, uh, because of your diet, your lifestyle starts to become less and less effective. So your body literally starts to become resistant to insulin. So your body has to produce more insulin to give you the same effect. And we know what that looks like downstream that turns into big problems. Now it's, it's a little bit more complex with cortisol, but to, to kind of understand what's happening here. If you're a cortisol junkie, if you have a high stress life, you're not getting good sleep. You're taking lots and lots of stimulants. You find yourself naturally, uh, being late to lots of appointments without realizing you're trying to give yourself a cortisol boost. Cortisol is a hormone that we need. It gives us energy. Um, it's naturally spikes in the morning. But once your body starts to, for lack of a better term, become resistant to cortisol, your body has to produce more and more of it for you to get the same feeling. So if you're feeling crappy and down, you're drinking all these stimulant drinks, you're not getting good sleep. You've got a lot of stress. Then you go take this really high intensity, crazy group exercise class, which is most of them. Many of them, the majority of group exercise classes, by the way, are revolve around intensity, revolve around lots of sweat, lots of action. Then you take this class and you get your body to squeeze out more cortisol. Then you finish and you're like, I feel good again. And so then when someone like, uh, you know, Adam or, you know, ourselves come up to a, to you and say, Hey, listen, this is too much intensity for you. You need to do something else. You have a high stress job. You're not getting good sleep. Diets poor. You're taking all these stimulants. You're drinking lots of coffee. And then they come back and say, but I feel good. I feel good after I do this really high intensity exercise class. It's like, yeah, temporarily it's because you, your body, you're not feeling the cortisol you already have. You have to squeeze out even more. And the way you do that is by doing these crazy intense classes.  You have trouble building muscle, uh, cold hands, cold feet, uh, you know, temperature intolerances, appetites, fluctuations. I either have cravings or I'm binging or I have no, no appetite whatsoever. Um, dark circles under the eyes, you know, and, and this is quite common. And the, the problem is group exercise classes attract this person quite a bit. I want to elaborate on you saying poor diet too, because I think when we say that the, the common person that hears that goes, Oh, I don't have a poor diet. I don't eat fast food. I don't drink sugary drinks. You know, I eat, uh, you know, chicken and salad and that's my, so when, when we say poor diet in relation to training this way, that could be, that could mean eating so low calorie and lean that you are, you're dramatically slowing down your metabolism. Uh, it's already not ideal for you to reduce calories that low, right? Say the 900 to 1500 calorie range, depending on the size of the person and the individual, there's a huge variance of course, but this is what's typical is you, you know, new way of life. You're going to sign up to this new class that you like and you're going to start your diet. It's one of the biggest mistakes that people make. They go from over consuming thousands of calories. They go into this calorie restrictive diet and then they go into this high intensity class and that extreme variance is may show them great results for the first couple of weeks. And this is where it's deceiving. They get great results for the first couple of weeks. They get that cortisol high like Sal was talking about and they feel good afterwards. The scale is going down.  You don't know what you're talking about. Problem with that is you body adapts to that and now where do you go? You're chasing that cortisol like crazy. You're already in some of the lowest calories you're used to consuming and you're pushing like crazy in your class and you're already probably going three to four to five days a week and now you're in this place where you're nowhere near your goal still but now the body is slowed down and not showing results. It's adapting to what you've told it to do and that's what this is where this, this is what happens to like 90% of the people that choose this way of training. Yeah. I also want to speak to one of my biggest pet peeves with group training and that is, you know, regression and progression and there really is no way to manage that, you know, in a group setting like that. No matter how much of an awesome coach you think you are, if you're doing this in where we're lifting weights and we're trying to like, you know, like create an environment for strength training specifically. It is such an individualized thing and I think that's why we're trying to stress the fact that trying to pull you outside of, you know, that group mentality where you can get swept into higher intensity, higher doses of intensity that your body like really will benefit from and so this is where we do see injuries or even if, you know, you don't know that appropriate amount, it kind of takes over all the natural signals of your body trying to tell you what is appropriate and what isn't. It's hard, it's very, very difficult to individualize in group exercise classes. Now with one-on-one resistance training, you can do that. You can start, now it doesn't always happen. By the way, you could do this with one-on-one. You do this on your own. You could fall into all these traps on your own, but the difference is you're not being encouraged to continue falling this trap. Now you will be when you do a group class. If I show up to a group class and I'm tired and, you know, my body's telling me you need to scale back a little bit. You need to do more correctional exercise or whatever, but then the music starts, everybody's going, go, you got to go push it. Come on, you can do it. Now you're pushing yourself and it just kicks the can down the road and it's very difficult to individualize. So what you'll see in a lot of group classes is people whose form is off, movement is off, and people who this type of training is inappropriate, but yet this is what the kind of people that it tends to attract. Now there are group classes that I have seen that are far better than others. And this is where the instructor slows things down, pays attention to each individual and gives options. I've seen classes like this where they say, here's the exercise, here's the regression, here's the progression. They explain intensity, they explain, you know, listening to your body. And, and that makes a big difference. I've also seen correctional exercise based group classes. Correctional exercise, I think is, although that is always, that's even, that's also better on an individual basis. I still, I think with group classes, you can definitely get away with correctional exercise because it's a slower pace and the focus tends to be a little bit different. Well, because again, the risk versus reward, the, the, there is no real risk. Like what, what is, what is harm in you doing a, you know, handcuff with rotation when you already have good shoulder mobility?  The other side of the coin would be somebody who doesn't need it and they're doing it. What is the harm in that? There's no harm in it whatsoever. You're just moving. There's not huge benefit if you're working on something that you don't need to work on, but there's not detriments like there are if somebody is doing a high-intensity class in a calorie-restricted diet with poor mechanics, there's huge risk involved in that. So that's the reason why that's okay.  There was a, there was a period there where bootcamp classes were getting really popular. And, you know, like we said earlier, the cost of training in a group is lower than it is one-on-one. So I was a, I had a gym, personal training studio, and I thought, okay, this is a great way business-wise to gain new potential members because it's a lower barrier to enter rather than spending the, you know, I don't know what the range is now in the country between 50 to 100-something dollars per session for personal training. I can charge $100 a month and the people can take a class and then from there, then maybe we'll get more personal training clients. But I eventually had to stop doing it. And the reason why I had to stop doing it is when I had, you know, 10 people in my, I'm not even a big class. If I had 10 people in there, all interested in fitness, even if all of their goals are the same, everybody could have the same goal. We all want to lose weight.  So I'd set up their circuits or I'd set up the exercises. I'd watch them do it. And I'd have to stop. Wait, hold on a second. Okay. I need you to do it this way. And you need to stop doing that. And dude, I did the same thing. I had the same issues in like, it just got away from me so quickly.  Can she just jump in our workout? And I'm always like, Oh my God, here we go. You know, because it's a brand new set of variables for me to pay attention to, you know, in conjunction to the other person that's right there that I already know what I've been working with for years. And it's very specific. I still have to look for those little nuance things that's going on with their body. Now I'm over here like totally distracted, took away from her workout. What do you guys say to them to somebody who tells you that, you know, well, it's, is it not better than nothing? And what if I would never get in an exercise unless I was taking my class? What do you say to somebody that has that mentality? Well, look, as long as the class is appropriate and you move well, and you're listening to your body, is it better than, than nothing? Yes. Now the problem is that's not common. What's more common is it's inappropriate. The person doesn't move well. And they train themselves with too much intensity. Um, group environments. Okay, here's the deal. Remember when you were a kid and you were in school and the teacher is going through something and you don't know what he or she is talking about, but in front of the whole class, you're not going to raise your hand and say, Hey, hold on a second, stop right there. You were embarrassed to see like, I'm just going to keep going. This is what happens in group classes and group classes. You have 30 people in there and the person's not feeling right or I'm just going to push through because they don't want to stop. Ask the instructor, hold on a second. This bothers my knee. Hold on a second. I don't feel good. Should I need, do I need to stop? Now there are a lot of people who have the confidence to stop or walk out of a class or to just stop exercising altogether and not do that particular exercise. But if you're really honest with yourself, okay? If you're, you got to be real honest. I know some people like, Oh, I would just stop really. Most people don't. Most people just push themselves all the way through and end up doing something that's very inappropriate. And what that'll teach you is one of the worst lessons you could possibly learn with exercise. It's the thing that we have to continuously as personal trainers, reverse every single time, which is this, do not ignore your body signals and group classes can definitely be a breeding ground of training people to ignore how their body feels. Just keep going. Five more. Don't stop now. These are all things you hear in group, group exercise classes. This was one of the main reasons why I had such a big issue with CrossFit when it first started with resistance training, because it was that same thing. We got time. Five more. Who could do the most? Keep going. With your exercises and with even risky, extremely risky exercises. You're not just dealing with your body. You're now dealing with barbells and dumbbells and very close exercises. You know that. Orange Therian F45 is a direct response to that with CrossFit. It's a lesser evil. They try to address the dangers and risk of Olympic lifts with people for time and speed and count. And they try to go, okay, let's eliminate these Olympic lifts, but still take this community concept and competitiveness and circuit-based type training, but let's eliminate the ones that are really, really risky.  They really owe a lot of that to CrossFit. CrossFit did it really, really well first. And they took, like you said, Sal, they took all the elements of group training like from Jazzercise days. They took strength training, which we know is the best thing for everybody. And they've kind of meshed them together to create this community fill of crazy circuit-based type of training. And so it exploded. And of course, the backlash on CrossFit, the injury rate and the dropout rate and the plateaus that a lot of the average people end up hitting. Now here comes Orange Theory F45 in direct response to that. Okay, here's, this is a safer option. I mean, shit, I remember when the Orange Theory was first opening up, that was part of the pitch. It's like, if you liked CrossFit, but you've been hurt from doing it, or you have a friend that's been hurt and you're scared, we used to get that all the time. I'd get somebody that would sign up there. And they really wanted to do CrossFit with their friend, but she got hurt. And so they are scared to do it. So I heard this is like this, or this gives you this, and it's safer. Yeah, safer. And so, you know, you're using TRX, a lot of body weight stuff, the dumbbells don't go higher than 20 pounds. Now you taught Orange Theory classes for a little while, but I know you taught them different because you had, yeah, I broke all the rules. Yeah. So what did you do different? Like, how did you let me let me do that? Because I have seen group exercise classes. I don't want to say they're all bad. Okay, I'm just talking about a lot of them is that I've seen. But there is a way you can do it that'll make it better. And I know you taught some of these classes. How did you do it? How did you change it so that it was more? Well, there was a lot of things that I did. And a lot of that is I owe that to the respect and flexibility that Brendan gave me. I mean, that was something that I, it's the only thing that kept me there, right? It was the ability for him to let me kind of run the class, how I saw fit, as long as I kind of stayed within the parameters, right? I wasn't completely changing everything. Although I did modify and change a lot of things with the the programs that they were putting out. But the main problem that I saw was this, I looked at a treadmill. So the way if those that are unfamiliar with like how that works, there's a set of rowers, there's a set of treadmills, and then there's a station where there's TRX and weights. And the idea is that you have two groups in the class, two groups of about 13 to 15 people. And at all times you have somebody in the weight area, and then you have in the cardio area, the cardio area would be considered the treadmills and the rowers. And then there's just this rotation that they either break up in like five minute blocks, where they're moving around, or half the class you're spent over in the cardio section, half the class you're spent over in weight. So in theory, the programming behind it is kind of smart. It's decent. And that's the reason why it's exploded so well. The hard part to teach to it, again, is I've got these 15 people that I'm looking at. And I remember I'd be walking down the treadmill and I see them and they have what they, they, it's called base pace, push and all out, which base is like your, you know, they're telling you basically keeping your heart rate kind of low, like around the target heart rate zone, like lower level, lower level, you're still jogging for most people or power walking uphill is kind of base and then your push kind of getting after a little bit and then all out is everything you got, right. And what I had a problem with was these push and all out paces. And when I looked at the mechanics on these people running, and by no means am I a running expert, I don't probably don't have great mechanics, which is probably why you don't see me run a lot. And I'm looking at the way these people are running.  And what I know is if that was a client, a one on one client of mine, I would address it, I would say, well, first of all, let's stop running. Let's figure out what where the breakdown is in your movement. And let's start addressing that. So what ended up happening was I started telling a lot of people to walk, you know, ignore the push and all out, whether I say it over the loudspeaker or not, you shouldn't be doing it. And before long, you would start to look into my classes and everybody's walking everyone's walking on the treadmill. And then the other thing that I had to do because the class is geared around, you know, staying in the orange zone is they get points for how much they're in the orange and orange is like high level target heart rate or even above. It's like whether they're they're burning max calories, right? So when you're in the orange, you're getting all these points. So it turns this into a little bit of a competitive thing. Everybody's watching each other because they can see everybody else's and they're trying to, you know, get as many points as you can. And you get the most points, obviously, based off of heart rate when you're doing cardio, because you're running, right? Well, that mentality carries over when they get into the weights, because you could if you push on weights, you can get your heart rate up there and get those extra points start turning it into cardio. Yes, you start turning it into cardio. And I had this is something I had to constantly do stop my class, turn the music down and go, stop, if you're in the weight room area, rest between your sets. If you can go with no breaks round and round and round on the weight, you're choosing too light of a weight, you choose a heavier weight and slow down, slow down what you're doing and rest. I'd like to look over here and see some of you sitting on the bench for 15 to 20 seconds to gather yourself before you do the next set. If you can go from set to set to set to set with no rest, then all you're doing is cardio with weights. Now you may as well just stayed over in the treadmill area, we're not getting the max benefits from the weight training. And this was something I was constantly having to drill down. And no matter how much I drilled it down, still 80% of the class didn't listen. I did have, you know, there was a percentage, a smaller percentage of people that really respected what I had to say and would go like, okay, I get what he's trying to explain to us and they would take it to heart and they would really listen. Now I saw that you would also, because this when we first started Mind Pump, you were doing this. And I saw that you would also incorporate a correctional exercise component, which wasn't a part. Not at all. In fact, I take credit for what they do now because it was non-existent before. That was a big problem I had too. So the last 10 minutes of the class was geared around, you know, kind of giving people highlights of what's going on at Orange Theory. And we've got this and we've got that going on and check basically like a news bulletin, right?  And instead, I chose to teach like one correctional exercise per class. And I only did one because I only had about five to 10 minutes to do this. So I couldn't instead of taking everybody through this generic cross your legs, you know, windmill, like doing 10 different like cool down type of movements that's generic is I would tell people, listen, if you're somebody who's dealing with knee pain, part of that can be from the running that's going on the treadmill. And your IT is all tight and you're feeling it. And so I'd explain what's going on. And then I'd say, here's a stretch to help alleviate that. And I would make them all do it with me.  So I try to add as much of my personalized flavor to it as I could. And I believe I can't, I'm not the only coach that teach that has taught group classes that probably tries to make it more individualized and better. Well, the reason why I wanted to say that is because if there's group group instructors listening, because here's the other part of this, this is the other part that I didn't ask you how popular were your classes? Yeah, I was I was booked out for months. Right. So and the reason why so the reason why I wanted to communicate that is because I think sometimes and I used to feel this as a personal trainer sometimes it's even more difficult as a group instructor. Sometimes as a personal trainer, you would feel the client you feel pressure from the client telling you how to train them. They try and dictate what was going to happen. Yeah, no, no, no, this is what I want. No, no, no, no, I want to feel this. No, I want to be real sore. I want to feel sore. I want to be real sweaty. And when I became it when I was a trainer early on, I would actually give into that because this person's paying me. I want to keep this client as I became more confident. I started to educate them and say, no, actually, this is how we're going to do it. This is what we want to do. And initially the fear is I'm going to lose clients. The reality is you gain more clients. You don't lose clients. So the reality is by you doing that, but by and I'm sure you had to fight people, I guarantee you did. Oh, yeah. Well, and I had and I probably had to argue with people in your class. I did and I had it and I also had to support what I was saying, right? I couldn't just say, Hey, we're gonna do this one stretch and then just make everybody do it, change the rules. I had to sell it. Yeah, I had to sell the idea of why I was, you know, breaking the, you know, the layout of how they had things and why I think this is better for them. And yeah, initially it took a little bit of time for me to sell them on that. It definitely helped that at that time too, we were starting mind pumps. So I would, it was great. We were talking about things on the podcast and I'd reference, you know, Hey, if you, if you're not comprehending what I'm talking about right now, we just did this episode where we talk all about this in detail. Because I know I don't have more than a couple of minutes. That was what the challenge for me was I only had minutes to like explain this because the class, like they come in, they get, they get right on the rowers and treadmills. They start warming up. It's go, it's good. And it's, and it's busting people in and out.   I see this person over here limping on one side. I see this person over here with horrible squat mechanics. You know, I see this person over here with terrible rowing mechanics. I see this person going way too fast.  It's the fast food of fitness. It's very similar parallel to that. You know, like it's, there is people and inherently want to eat, you know, correctly for themselves to get the right kind of nutrition to feel better to, they want the end result of like going on that path, but they're totally susceptible to just going in the drive through and getting whatever's quicker. And there, and there's, there's people that, and I want to make this clear too, because I don't want to come off like I'm just bashing like orange theory. It's very similar to how I feel about CrossFit. Like, I mean, I feel like I could take the class. I could take the class and have a great results from it. I would know, I would know where to, I would know where to push myself. I would know the importance of slowing down my tempo when I'm over in the weight area. I would choose to walk when I know that's what's best for me. If I didn't nutritionally support what I need calorie wise, like, so all that stuff would be happening. And I think there's the same amount of people that can do CrossFit and can be totally fine and get great results from it. What I know is being a, from being a trainer for as long as I have, is that it doesn't attract that person as much. I mean, they're, they're a, they're maybe, they, they're maybe 5% of the actual people that are busting through this. And I'm not talking to them.   I'm talking to the other 95% of the population that hurt, that sit on the, that we're sitting on a couch one day and their sister-in-law or a friend was talking to them and said, oh my God, Oran Steere or CrossFit or F45 changed my life. It's amazing. And that is what motivated you to come in. And that was your first decision of, oh, I'm going to start training and this is where I go. And you have no clue about any things we're talking about. The only thing personally I've seen in terms of like a, you know, quality group training session was when I was in sports and when there was like dedicated education involved, like the entire way through, like where we all as a team had to rest after each station a certain amount of time to get the maximal amount of, you know, strength output in this situation. And I think that, you know, it's unfortunate the general public just doesn't have that kind of access to, you know, a quality type of a class where you're actually like going through a weight training class, like you're learning, you know, how to actually go through these mechanics properly and technique and is very much stressed and highlighted. And, you know, everything is like, it is our own sort of initiative that we're catering towards this. Like I want what experience is going to make me feel good. I want, you know, to be motivated by loud music. It's like all these external factors that are entertaining that sound great and awesome, but really at the root of it, it needs to be, you know, the actual educational piece. Well, the best group classes that I've ever seen were taught by instructors who also had one-on-one training experience. And the reason why I'm saying that is because when you train people one-on-one, you have to really focus on that individual, their movement. You have to really look at what's working and what's not working. So then they take that to the group setting. Also, I've seen group classes where they're very appropriately labeled beginner or level one, level two, level three. And then the instructor in that class kicks people out when the class is not appropriate. Very few instructors do this. You don't want to do this as an instructor. You don't want to kick people out of your class. But I've seen really good instructors do this where it's a level three, whatever class, they start the class, they're doing it, and then, you know, they walk over to somebody and say, hey, listen, this is not appropriate for you. I want you to do something different. This class is too advanced for you. That's when I see group classes do well. Unfortunately, you don't see a whole lot of those. And part of it is us as consumers. We don't want to learn anything. We want to go in and just go. Just tell me what to do. Yeah, tell me what to do. I just want to sweat. Let's make this happen. And it kind of feeds into that. And so often times, group classes become a part of the problem rather than the actual solution. But it can be done right. I don't want to say broadly that they all disappear. It can be done right. And there are a percentage of people that it makes sense for. I think the problem is that a lot of people think they fall in that percentage and you don't say sorry. You're not that person that has the great mechanics that really understands what you should be doing nutritionally. The majority just aren't. Yes. And don't look at instruction like when you buy toilet paper. Don't look and say, oh, I can take this many classes for $100 versus two sessions with a personal trainer. So the value is in this one thing over here. I got 30 classes. It doesn't work that way.  That's just the truth. It's typically a much better investment. But again, there are people who are benefit from. The irony of us even discussing this right now. I was, yesterday in my DMs, I get a message from an old Orange Theory client that I knew. And I also knew that she would sporadically listen to Mind Pump and she reached out to me yesterday and was like, I should have listened to you three years ago when you were telling me I should drop the classes down to just once or twice a week and focus on it. And she goes, because now I have bursitis in my hips. And what's tough is that you have someone like that listening to the show. I'm constantly telling them and they still did, which is what's going to happen when we do this episode. There's going to be people that are going to deaf ear it because they're in love with it. I know that because I dealt with this my whole career is it's really tough to get somebody who is enjoying that community feel, they feel good after the class and get that cortisol that spike like we're talking about.  And that's all that we're trying to help prevent is somebody setting themselves up for either injury, chronic pain, or slowing their metabolism down dramatically or just becoming so disenfranchised, they stop, right?  And then, you know, maybe they come back, maybe they don't. One of the big problems with the fitness industry, the gym industry, the class and instructor industry is that they do a phenomenal job, an excellent job of attracting people, terrible job at setting people up. Yeah, long term success. It's a terrible job. Most people that you know, who don't work out at one point started and stopped. So we don't have a like start working out problem. That's not the issue getting people going. The issue is how do we set people up? Long term for consistent long term success. And unfortunately, group classes tend to feed the, you know, attract and then stop type of mentality and problem that we encounter quite a bit. So so that's why we think group classes, many of them are not that good. And that's our opinion. And with that, go to mind pump free.com and download our guides, resources and books. They're all totally free. You can also find the three of us on Instagram. You can find Justin at mind pump Justin, me at mind pump, Sal and Adam at mind pump, Adam.


Functional camp was this really engaging, inspiring sort of place. There are a lot of smart people at this conference who care a lot about writing code and doing it well. We'll see people that are just starting to put a toe in the water of functional programming and don't really know how to think in terms of functions and are still trying to make a transition from object-oriented. We knew about functional programming. We knew the word, all those things out there. But it's a kind of, everything is new. We are entering into a new world here. They have three tracks, right? So, introduction to functional programming, applying functional programming and mastering functional programming. It feels to me like just a team of people who are collaborating to help each other lift their knowledge and skills wherever they may be in that journey. I really enjoy this sort of mix that comes to this place.  It's just a nice place where everyone is just learning and figuring things out. There's a lot of cool people here and helpful people and they should come. Everybody was there with an open mind hoping to learn new things and push the boundaries of what was possible and really trying to just enjoy programming again. Come here, talk to people, have fun. It's a lot of fun.


Guys subscribe and turn notifications on leave a like and drop a comment down below. Get here first and win giveaway calls I will pin as first comment on some of my videos. Be fast and good luck. What is going on guys welcome back to the channel and I appreciate you being here today.  But before we go any further guys if you do enjoy the video and would like to show you support you can by hitting that like button and if you are new around here and enjoy daily Destiny 2 videos be sure to subscribe. Okay so it's located within the end part of the rear silver lost sector pinpointed on screen now. Follow the route I take on screen to get there just make sure you head out in the right direction first up and this is from spawn point by the way. So when you get to this lost sector use your team for your hat and travel straight to the end of this lost sector there will be two hub goblins located. The teleporter tower to this Ascendant Challenge is up there. Jump straight in just be careful as when I jumped in I was greeted by a few knights who wanted my blood. This Ascendant Challenges is literally a straight run to the end running and jumping nothing real hard about it.  When you get to the end you will have to kill a solo knight just be careful as many adds do spawn in with this knight as well as shriekers who murder your health. But it shouldn't be too hard people so take your time once you are here and just take out that knife with all means possible. Then guys you are done grab that chest and away you go. Well people this is the location of the Ascendant Challenge for week 6. If you guys enjoyed the video leave a like really does help out. If you are new around here and enjoy daily Destiny 2 videos be sure to subscribe and guys hopefully I will see you on that next one.


Hi everyone, Joe for jaspeyscasebreaks.com, coming at you with 2019-2020 Panini Prism Basketball, one sell-o-pack break number 36. We originally had eight mixer spots given away, but as I mentioned a number of times before I started this video, we miscalculated, sorry. We only have six mixer spots left. So when we do the re-randomizing at the end of this video, top six, spots one through six, will still get mixer spots in that mixer right there. And spot seven and eight will get the cost of the mixer in break credit, $95 of break credit will be sent to you. I apologize about that. I know you just wanted to get into the mixer, but we'll give you the money instead. Apologies again to these folks here. Although no one objected, no one wanted refunds, which I offered, so I guess everyone's okay with it. Thank you for your understanding. There are the teams right there. So let's do the break first, very quick break. Three and a two, five times for each list. One, two, three, four. Five times we got Michael Hughes down to Scotty, Scot E. Three and a two, five times for the teams, two. After five times we got the Hawks down to the Pelicans. All right, so Michael Hughes with the Hawks, Frank with the Warriors, David with the Suns, Scott with the Cavs, Asa with the Spurs, even Kelly with the Nuggets, Anthony with the Clippers, even Kelly with the Knicks, Mike Tower with the Wizards, Derek with the Thunder, Kenneth with the Timberwolves, Nicky got my Lakers, Ryan Redman with the Magics, Sean with the Jazz, Joe with the Kings, Sean with the Pacers, Mike with the Rockets, Maverick with the Hornets, Frank with the Trail Blazers. I think in the Top Gun II, Maverick, are they gonna be flying Hornets? F-18 Hornets? Frank with the Blazers, Jeff with the Heat, and Stephen Kelly with the Grizz, that movie got pushed back. Frank with the Bucks, very disappointed. Asa with the Pistons, Patrick with the Bulls, Mark with the Nets, I think it got pushed back. Andy with the Mavs, Frank with the Raptors, Chad with the Sixers, Josh with the Celtics, and Scott with the Pelicans. Let's sort by column B. I don't think anyone's gonna trade, right? I think we're mostly interested in seeing who wins those spots or the break credit. All right, TWC, trade window closed, let's print, let's rip. There is a little slim chance, but there is a chance that there could be like a Jaumaran or Zion Silver or something crazy like that, or in the red, white, and blue pack. Could be a possibility. There's Alfred Payton Green, and then we got Thomas Bryant Silver. There's a Tybal on the back, it's a rookie. Otto Porter, Joel Embiid, and a Matisse Tybal. It's not too shabby for Chad Wright and the 76ers. All right, nice. All right, thanks everybody.  Top six get mixer spots, seven and eight. New dice roll as well. Seven and eight will get that break credit, which I will send immediately after this video. Check your emails. If you don't see it, check your spam folder. Let's roll it, let's randomize it. Four and a six, 10 times for the list. One, two, three, four, five, six, seven, eight, nine, and 10th and final time after 10. Top eight is gonna be, so the break credit for spot seven and eight is gonna go to Derek and Stephen Kelly, and the rest of you here will get spots in the break. Josh, Sean, David, Sean, Jeff, and Asa. There you go, congrats to all of the winners right here. Thank you very much for getting into the action. Appreciate it. Coming up next will be the basketball mixer. I'm Joe. I'll see you next time, jaspyscasebreaks.com.