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https://mathoverflow.net/questions/118336 | 6 | Recall the [product lemma](http://books.google.com.au/books?id=WTAl997XDb4C&lpg=PA264&dq=easton%20%2Bjech%20class%20forcing&pg=PA234#v=onepage&q&f=false) from Easton's famous paper, which tells us something about when we have a forcing notion (which may be a proper class) that splits as a product with one factor $\lamb... | https://mathoverflow.net/users/4177 | From the product lemma to to a result about powersets | Let me handle just the set-forcing case, since I think the class forcing case is problematic.
Suppose that $\mathbb{P}$
is $\lambda^+$-c.c., and $\mathbb{Q}$ is $\leq\lambda$-closed. If
$G\times H\subset\mathbb{P}\times\mathbb{Q}$ is $V$-generic, then
the claim is that $P(\lambda)^{V[G][H]}=P(\lambda)^{V[G]}$.
Fir... | 8 | https://mathoverflow.net/users/1946 | 118361 | 66,802 |
https://mathoverflow.net/questions/83020 | 1 | We know in elliptic equation theory(or related area) that harmonic function has mean value property. Roughly speaking, harmonic function function at point x is equal to its average on the spherical surface(or ball) centered at x. Furthermore, a locally integrable function which satisfies mean value property is smooth a... | https://mathoverflow.net/users/17708 | X-harmonic and mean value property | In Oksendal's book, the transition probablity or generating semigroup is not fully discussed. Think of X-harmonicity and the mean value property (MVP) together in lemma 9.2.4 in Oksendal: the MVP is w.r.t to the measure on the boundary,$Q^{x}(dy)$ when $dy\in\partial D$. Therefore, to show that $f$ is continous in $x$ ... | 2 | https://mathoverflow.net/users/14390 | 118365 | 66,805 |
https://mathoverflow.net/questions/118277 | 2 | A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) = 0$, then $f(z) = zg(z)$ for some holomorphic function $g$. A crucial property of this type of factorization is that it preserves boundedne... | https://mathoverflow.net/users/30454 | Factorization of antisymmetric bounded holomorphic functions | No, $g$ is not necessarily bounded. Let $h(z,w)$ be (a branch of) $log (z+w-2)$ and let $f(z,w)$ be $(z-w)g(z,w)=(z-w)zwh(z,w)$.
Then on the one hand $zwh(z,w)$ is unbounded as $z$ and $w$ (in $\mathbb D$) both approach $1$, but on the other hand $f(z,w)$ remains bounded: the inequality $|z-w|^2+|z+w|^2=2|z|^2+2|w|^... | 2 | https://mathoverflow.net/users/6666 | 118369 | 66,808 |
https://mathoverflow.net/questions/118169 | 4 | Sorry for the long title. What I mean is that for two vectors (a\_1,...,a\_n) and (b\_1,...,b\_n) with the property $a\_i,b\_i \geq 0 $ and $ \sum a\_i =\sum b\_i =1$.
If $ -\sum a\_ilog(a\_i) > -\sum b\_ilog(b\_i) $ implies $\sum a\_i^2 < \sum b\_i^2$ or something similar?
| https://mathoverflow.net/users/19294 | Is there a relationship between Entropy of a fininte distrete probability distribution and the squre sum of the values of probability mass function of that distribution? | Of course, in general it is not true that inequality between entropies implies the same inequality between $\ell^2$ norms. Although this is true for $n=2$, it fails already for $n=3$ (as the entropy $H(p\_1,p\_2,p\_3)$ is obviously not constant on the level curve determined by conditions $\sum p\_i=1$ and $\sum p\_i^2=... | 6 | https://mathoverflow.net/users/8588 | 118373 | 66,809 |
https://mathoverflow.net/questions/118370 | 2 | on Williams Crossed product book,on page 198, it is mentioned that there is only one regular representation for C\_c(G), and that is the left regular representation.
I know that this representation is one of the regular representations, but why it is the only one.
Indeed I know in the process of constructing the reduce... | https://mathoverflow.net/users/30488 | crossed product | It looks like he's using the term "regular representation" to mean: a representation of $A \rtimes G$ on $L^2(G; H)$ constructed by combining a given representation of $A$ on $H$ with the left action of $G$ on $L^2(G;H)$. If you start with $A = {\bf C}$ then there is "only one" representation of $A$ to start with, so t... | 3 | https://mathoverflow.net/users/23141 | 118375 | 66,810 |
https://mathoverflow.net/questions/118381 | 0 | Let be R a commutative ring whit unit and let I a proper ideal of R. Let R' the completion of R respect to the ideal I (see Introduction to Commutative Algebra - M. F. Atiyah, I. G. MacDonald for the definition). It's true or false that:
>
> R' is an integral domain then R is an integral domain?
>
>
>
| https://mathoverflow.net/users/30492 | The completion of a ring R is a domain then the ring R is a domain? | No. $\mathbb{Z}/6$ completed at the prime $(3)$ is $\mathbb{Z}/3$. (I can think of true versions of this statment, but I'll hold off until it's clearer what you're looking for.)
| 6 | https://mathoverflow.net/users/297 | 118383 | 66,813 |
https://mathoverflow.net/questions/118377 | 1 | Good day to everyone!
My question concerns Hidden Markov Models and is pretty basic. In one of the books ("Introduction to Machine Learning" by Ethem Alpaydin, 2nd Edition, p.373), I get the following transition:
$P(O\_1, \ldots, O\_T \mid q\_t = S) = P(O\_1, \ldots, O\_t \mid q\_t = S) \cdot P(O\_{t+1}, \ldots, O\... | https://mathoverflow.net/users/30489 | Hidden Markov: representing joint probability for set of observations as a product of two subset probabilities | You are right that this equation amounts to conditional independence of the future and the past provided the present is fixed (i.e., the Markov property). However, one has to use the right chain. In this case the easiest is to consider the "compound" Markov chain on $S\times V$, where $S$ is the set of states of the or... | 2 | https://mathoverflow.net/users/8588 | 118391 | 66,816 |
https://mathoverflow.net/questions/118382 | 4 | Hi, I know that over $\mathbb{C}$ the dimension of an irreducible representation of a finite group $G$ must divide the order of the group. I've read somewhere that if $p$ does not divide the order of the group then the representations over $\tilde{F\_p}$ are "essentially the same" (Is this true? What does it mean other... | https://mathoverflow.net/users/30491 | Dimension of irreducible representations in characteristic p | I'm not sure where you are starting in terms of background and references, but the standard short book for such questions is Serre's *Linear Representations of Finite Groups* (Springer GTM 42, a good English translation by Len Scott of older lecture notes dating back in their first version to the 1960s). Part III mixes... | 7 | https://mathoverflow.net/users/4231 | 118392 | 66,817 |
https://mathoverflow.net/questions/118362 | 5 | Motivated by an answer to [this mathoverflow question](https://mathoverflow.net/questions/57656/standard-model-of-particle-physics-for-mathematicians) I've been making an effort to understand Baez and Huerta's article ["The algebra of grand unified theories"](http://www.ams.org/journals/bull/2010-47-03/S0273-0979-10-01... | https://mathoverflow.net/users/7631 | Meaning of a phrase from "The algebra of grand unified theories". | Our sentence indeed loses its point taken out of context. Condon and Cassen were not merely claiming that we can multiply a vector in $\mathbb{C}^2$ by an element of $\mathrm{SU}(2)$. They were claiming that when $\mathbb{C}^2$ is used as the Hilbert space for nucleons, this transformation is a *symmetry of the laws of... | 10 | https://mathoverflow.net/users/2893 | 118394 | 66,818 |
https://mathoverflow.net/questions/118372 | 1 | Hi to all,
Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n\_{\nu}$ ($\mathbb{R}^n\_{\nu}$ is the semi-Euclidean space) and $G\_1$ ,$G\_2$ two Lie subgroups of $G$. Let $G\_1 \times G\_2$ as a Lie subgroup of $G$ by identifying it with the inner product of $G\_1$ and $G\_2$ (suppose $G\_1$ , $G\_2$ have ... | https://mathoverflow.net/users/nan | Orbits of Product Lie Groups Action | If all actions are proper, orbits are closed embedded submanifolds.
If not, then they are only initial submanifolds (unique injectively immersed submanifolds with a possibly finer topology - think of a 1-param. subgroup of the torus with irrational slope;
see thm 6.4 in [(here)](http://www.mat.univie.ac.at/~michor/dgbo... | 1 | https://mathoverflow.net/users/26935 | 118403 | 66,822 |
https://mathoverflow.net/questions/118407 | 4 | Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, are interesting topological properties somehow coded in the resulting forcing extennsion. For example, would ${\mathbb S}... | https://mathoverflow.net/users/15819 | Forcing over the poset of nonempty open subsets of a nice topological space | The property that you are describing is called *coabsoluteness*. In other words, two regular spaces are said to be coabsolute if their regular open algebras are isomorphic. In the paper, A Characterization of Coabsoluteness for a Class of Metric Spaces by Catherine Gates, theorem 2.3 says that two locally compact metri... | 6 | https://mathoverflow.net/users/22277 | 118409 | 66,825 |
https://mathoverflow.net/questions/118405 | 6 | Is there a solution or progress of the following problem (maybe old conjecture): Is An immersed surface with constant mean curvature and with a circle as a boundary part of a sphere??. If we replace "immersed" by "embedded" I think the problem was solved by Alexandroff kind of long time ago. Is someone could enlighten ... | https://mathoverflow.net/users/30500 | Immersed surface with circle as a boundary | Check out [this paper of Rafael Lopez](http://www.ugr.es/~rcamino/publications/pdf/art14.pdf) and references therein.
| 4 | https://mathoverflow.net/users/11142 | 118412 | 66,826 |
https://mathoverflow.net/questions/118425 | 7 | Do there exist translations of Serre's early papers on spectral sequences? In particular, I am interested in the following ones:
Serre, Jean-Pierre Homologie singulière des espaces fibrés. Applications. Ann. of Math. (2) 54, (1951). 425–505.
Serre, Jean-Pierre Groupes d'homotopie et classes de groupes abéliens. Ann... | https://mathoverflow.net/users/30328 | Translations of Serre's early spectral sequences papers | You can look at this volume:
<http://www.worldscientific.com/worldscibooks/10.1142/8444>
ps: J.-P. Serre's papers are written in a very elegant style, his french is beautiful and nice to read :-)
| 6 | https://mathoverflow.net/users/27816 | 118426 | 66,834 |
https://mathoverflow.net/questions/118357 | 8 | I am interested into computational complexity of decision problem: Does a given 2-dimensional simplicial complex contain (any)triangulation of 2-sphere? This problem trivially lies into NP, because certificate is the subset of triangles. I think that it is NP-complete.
| https://mathoverflow.net/users/30484 | computational complexity | It is NP-complete. Here is a reduction from the graph 3-coloring problem.
For a given graph, consider a sphere with $n$ holes where $n$ is the number of vertices. Enumerate the holes by the vertices of the graph. Fill each hole by 3 discs (red, green and blue) by identifying each disc boundary with the hole boundary.... | 11 | https://mathoverflow.net/users/4354 | 118428 | 66,836 |
https://mathoverflow.net/questions/118429 | 6 | Why is the fundamental group of a compact manifold finitely presented?
| https://mathoverflow.net/users/23358 | Fundamental group of a compact manifold | Every compact manifold has the homotopy type of a finite CW-complex, and a finite CW-complex has finitely presented fundamental group by van Kampen's theorem.
| 19 | https://mathoverflow.net/users/4362 | 118435 | 66,839 |
https://mathoverflow.net/questions/118396 | 5 | Hi,
I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$.
Fix $T>0$. Let's calculate the eigenfunctions of the transformation $ \mathscr T\_kf=\intop\_{0}^T k(x,y)f(y)dy$:
$$ \lambda\psi(x)=\i... | https://mathoverflow.net/users/28037 | Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$ | What Branimir Ćaćić writes is correct. Another way to see that your $\lambda$'s where not right is as follows:
From $$\lambda\psi(x) = \int\_0^x y\psi(y) dy + x\int\_x^T \psi(y)dy$$ you get that $
$$\psi(0)=0.$$
Similarly, from $$\lambda\psi'(x) = \int\_x^T \psi(y)dy$$ you get $$\psi'(T)=0.$$
Hence, you have two... | 5 | https://mathoverflow.net/users/9652 | 118436 | 66,840 |
https://mathoverflow.net/questions/118444 | 13 | Let $H$ be a separable, infinite dimensional Hilbert Space and $Calk(H):=B(H)/K(H)$ denotes
the Calkin algebra. There is obvious surjection $\pi: B(H) \to Calk(H)$ but I'm interested
in somehow opposite question: is it possible to construct embedding $j\_1:Calk(H) \to B(H)$?
The same question for embedding $j\_2:B(H) \... | https://mathoverflow.net/users/24078 | Calkin Algebra and the embedding | (1) No. The Calkin algebra contains an uncountable family of mutually orthogonal nonzero projections.
(2) Yes. Embed $B(H)$ into $B(H \otimes H)$ by the map $A \mapsto A \otimes I$, then pass to $Q(H\otimes H) \cong Q(H)$.
(3) No. The Calkin algebra is simple.
| 26 | https://mathoverflow.net/users/23141 | 118446 | 66,842 |
https://mathoverflow.net/questions/118395 | 1 | Let's say we denote by $T^{(n,m)}M$ the vector-bundle of rank $(n,m)$ tensors on a manifold $M$ and by $\Lambda^pM$ the vector-bundle of $p$-forms on $M$. Is there a relationship (perhaps a diffeomorphism to some sort of direct sum expansion) between the various $\Lambda^p M$ (or the various $T^{(n,m)}M$) and iterated ... | https://mathoverflow.net/users/30496 | Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bundles on a manifold? | The answer is 'basically, no'. The tensor bundles that you list, such as $T^{(n,m)}M$ and its quotients (such as $\Lambda^p(TM)$, etc.), are *first order* prolongations of $\mathrm{Diff}(M)$, whereas $TTM$ is a *second order* prolongation of $\mathrm{Diff}(M)$. All of these are examples of functors from the category $\... | 7 | https://mathoverflow.net/users/13972 | 118457 | 66,848 |
https://mathoverflow.net/questions/118442 | 18 | I know (and am able to prove via Stone-Čech compactification) that the following is correct:
>
> **Theorem:** A metric space is compact if and only if its space of bounded, continuous, real-valued functions is separable in the uniform topology.
>
>
>
I use it in a paper for readers who are presumably not famil... | https://mathoverflow.net/users/15327 | reference for "X compact <=> C_b(X) separable" (X metric space) | The result does appear in Dunford/Schwartz, *Linear Operators Part I* (page 437), but is only stated as an exercise.
Edit after @JosephVanName' comment:
Conway's *Functional Analysis* has the result for completely regular spaces as Theorem 6.6 (page 140).
| 12 | https://mathoverflow.net/users/30516 | 118461 | 66,849 |
https://mathoverflow.net/questions/118469 | 14 | Hello,
for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I should have in order to understand this topic. Can you perhaps recommend a textbook or some other reference regarding Ho... | https://mathoverflow.net/users/30520 | Hopf Algebra for a physicist | Dominique Manchon's [lecture notes](http://arxiv.org/abs/math/0408405), which are very well-known amongst people working on Connes--Kreimer renormalisation, offer exactly the sort of detailed, accessible introduction to Hopf algebras and Connes--Kreimer renormalisation that you're looking for. However, you should first... | 12 | https://mathoverflow.net/users/6999 | 118470 | 66,852 |
https://mathoverflow.net/questions/118459 | 4 | Let $T = \{ x\_0,\ldots,x\_n \}$ be a set of $n+1$ different points in the real interval $[a,b]$. Let $X\_T$ be the associated *interpolation operator* on $C[a,b]$: it takes a function $f \in C[a,b]$ into the unique degree-$n$ polynomial $p$ such that $p(x\_i) = f(x\_i)$ for $i = 0,\ldots,n$. The *Lebesgue constant* $\... | https://mathoverflow.net/users/30512 | Lebesgue constant as condition number of polynomial interpolation | A direct consequence of the definition, yes. Note that $X\_T$ is in fact defined on $C(T)$, with the same norm $\Lambda \_ T$ w.r.to the infinity norm $\|u\|$ (viewing the vector $u:=(u\_0,\dots,u\_n)$ as an element of $C(T)$, taking $x \_ j$ to $ u \_ j$). Then $u=p{|\_ T}$ so $p=X\_T u$ by the uniqueness; and $\hat p... | 2 | https://mathoverflow.net/users/6101 | 118474 | 66,854 |
https://mathoverflow.net/questions/118479 | 9 | I am trying to understand what the obstructions are to orienting moduli spaces of pseudoholomorphic curves with totally real boundary condition.
I believe that Fukaya-Oh-Ohta-Ono have shown that if a Lagrangian is relatively spin, the moduli spaces of disks with boundary in it can be oriented.
My question has 3 r... | https://mathoverflow.net/users/477 | Orientations for pseudoholomorphic curves with totally real boundary condition | 1) The problem of orienting moduli spaces of pseudo-holomorphic discs with totally real boundary conditions is really a problem in index theory. It was solved Vin de Silva in his (unpublished) D. Phil. thesis, using Atiyah's Real K-theory, and independently by FOOO. There's an excellent account in Seidel's book (sectio... | 11 | https://mathoverflow.net/users/2356 | 118483 | 66,857 |
https://mathoverflow.net/questions/118486 | 1 | I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.
Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ generates two operators $pre: \wp(B) \to \wp(A)$ and $post: \wp(A) \to \wp(B)$ where $pre$ maps a set $X \subseteq A$ to its... | https://mathoverflow.net/users/13475 | Why the preimage rather than image in Stone-type dualities. | You should think of the preimage as taking the pullback of $\mathbb{F}\_2$-valued functions. For any topological space $X$, the space of continuous functions $X \to \mathbb{F}\_2$ may be identified with the Boolean algebra / ring of clopen subsets of $X$, the pullback of such a function is another such function, and ta... | 3 | https://mathoverflow.net/users/290 | 118487 | 66,859 |
https://mathoverflow.net/questions/118376 | 6 | By the modular group I mean either $SL(2,\mathbb{Z})$ or $PSL(2,\mathbb{Z})$.
Where can I find examples of these?
Another question: is there a good (ideally analytical, but possibly computer-aided) way to determine if a subgroup of the modular group generated by some given matrices has finite index, and possibly al... | https://mathoverflow.net/users/15242 | Are there noncongruence subgroups (of finite index) of the modular group generated only by 2 or 3 elements? | The answer to your Question 1. is that there are not rank 2 subgroups of $PSL\_2(\mathbb{Z})$, but there are rank 3 subgroups. This follows from a [result of Wohlfahrt](http://projecteuclid.org/euclid.ijm/1256059574). He shows in Theorem 5
that any non-congruence subgroup $\Gamma < PSL\_2(\mathbb{Z})$ has index $\geq 7... | 11 | https://mathoverflow.net/users/1345 | 118489 | 66,861 |
https://mathoverflow.net/questions/118504 | 0 | Let $(K, \nu)$ be a valued field and $x$ is transcendental over $K$. Is there exist a henselian extension of $(K, \nu)$ in between $(K, \nu)$ and $(K(x), \nu^{'})$ where $\nu^{'}$ is an extension (arbitary) of $\nu$.
| https://mathoverflow.net/users/30267 | Henselization of valued field | In general the answer is no: let $(K,v)$ be non-henselian and let $v\_x$ be the Gauss extension of $v$ to the rational function field $K(x)$:
$v\_x(a\_nx^n+\ldots +a\_0):=\min (va\_i : i=0,\ldots ,n)$.
Then $(L,v\_x)$ is non-henselian for every subfield $L\subseteq K(x)$.
Proof: by assumption there exists a monic... | 1 | https://mathoverflow.net/users/3556 | 118507 | 66,868 |
https://mathoverflow.net/questions/118455 | 4 | Let $G$ and $H$ be graphs, let $\vec H$ be a fixed orientation of $H$.
Denote by $D(G,\vec H)$ the number of orientations of $G$ that contain a copy of $\vec H$ and denote by $D'(G,H)$ the number of spanning subgraphs of $G$ that contain a copy of $H$.
Claim: $D'(G,H) \leq D(G, \vec H)$.
Note: a *spanning subgr... | https://mathoverflow.net/users/7368 | Alternative proof for counting problem in graphs | I think I have a proof. I take it kind of pedantically because I am a little suspicious that it looks both simple in ideas and convoluted in details, but see what you thnk . Following that are very preliminary comments on inductive proofs.
We think of graphs as sets of edges. I will change the notation slightly to h... | 1 | https://mathoverflow.net/users/8008 | 118509 | 66,869 |
https://mathoverflow.net/questions/118494 | 14 | The largest permutation group without 2-cycles is $A\_n$, which has size $n!/2$. I think the largest permutation group without 2-cycles or 3-cycles is much smaller, but I can't figure out if it should be polynomially smaller (eg. of size $n!/n^3$), or more dramatically smaller (eg. of size $(.5n)!$).
The largest gro... | https://mathoverflow.net/users/30530 | Largest permutation group without 2-cycles or 3-cycles | I can do a bit better! For even $n=2m$ there is a subgroup of order $2^{m-1}m!$ with no 2-cycles or 3-cycles. Let $W$ be the wreath product of a cyclic group of order 2 with $S\_m$. In other words, $W$ is the subgroup of $S\_n$ that preserves the partition $\{\{1,2\},
\{3,4\},\ldots,\{2m-1,2m\}\}$ of $\{1,2,\ldots,n\}$... | 16 | https://mathoverflow.net/users/35840 | 118513 | 66,871 |
https://mathoverflow.net/questions/118505 | 5 | (Yet another question in a series demonstrating my rather embarrassing ignorance of standard Lie theory... I hope this is not too basic for MO!)
To be a little more precise: let $G$ be a real connected Lie group, not necessarily simply connected, and let $R$ be its solvable radical.
Then (see the [EOM entry](http://... | https://mathoverflow.net/users/763 | Are maximal connected semisimple subgroups automatically closed? | Your impression is correct. Let $H$ be the universal cover of $SL\_2(\mathbb{R})$, it has infinite cyclic center, let $z$ be a generator of the center. Consider the product $H\times U(1)$ (where $U(1)$ is the group of complex numbers of modulus 1), and mod out by the (discrete, central) subgroup generated by $(z,e^{2\p... | 12 | https://mathoverflow.net/users/14497 | 118518 | 66,874 |
https://mathoverflow.net/questions/118511 | 4 | The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is a flat deformation of $S(\mathfrak{g})$, so these algebras should be similar in many ways. Does at least this general similarity hold?
| https://mathoverflow.net/users/30539 | Is the universal enveloping algebra of a finite-dimensional Lie algebra (left) noetherian? | Yes, if a filtered ring $R$ has the property that its associated graded ring is Noetherian, then $R$ is Noetherian. Universal enveloping algebras have a PBW filtration such that the associated graded algebra is $S(\mathfrak{g})$. This is proved in Noncommutative Noetherian Rings by McConnell, Robson, Small - see sectio... | 11 | https://mathoverflow.net/users/6481 | 118521 | 66,877 |
https://mathoverflow.net/questions/118477 | 17 | We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. This technique is used, for example, in the proof that GBC is a conservative extension of ZFC, by forcing to add a univers... | https://mathoverflow.net/users/10671 | Is every class that does not add sets necessarily added by forcing? | It is a fantastic question, Jonas! I've spent hours with it now, going back and forth several times about which way it might go.
But finally, I've got a negative answer, at least for some models $M$. My idea is
that some models of ZFC admit what are called *satisfaction
classes*, but these can never be added by class... | 16 | https://mathoverflow.net/users/1946 | 118525 | 66,879 |
https://mathoverflow.net/questions/118533 | 0 | I want the rodrigues like formula using sin and cos , not a matrix series expansion.
I've found some references for se(n) , n > 3 in :
<ftp://ftp.cis.upenn.edu/pub/papers/gallier/rodrig.pdf>
| https://mathoverflow.net/users/30544 | How to compute SE(2) group exponential and logarithm? | Just computing the exponential gives
$$
\exp\left(
\begin{matrix}
0 & 0 & 0 \cr
x & 0 & -t\cr
y & t & 0
\end{matrix}
\right)
= \left(
\begin{matrix}
1 & 0 & 0 \cr
x\frac{\sin t}{t}-y\frac{1{-}\cos t}{t} & \cos t & -\sin t\cr
x\frac{1{-}\cos t}{t}+y\frac{\sin t}{t} & \sin t & \cos t
\end{matrix}
\right),
$$
and invertin... | 2 | https://mathoverflow.net/users/13972 | 118538 | 66,884 |
https://mathoverflow.net/questions/118247 | 6 | Given an integer $k > 1$, define the sequences $X(k,n), Y(k,n)$ as follows:
$a=4k-2,$ $y\_0 = 1,$ $y\_1 = a + 1,y\_n = ay\_{n-1} - y\_{n-2}$
$b = 4k + 2,$ $ x\_0 = 1,$ $x\_1 = b - 1,$ $x\_n = bx\_{n-1} - x\_{n-2}$
For example, with $k = 2$ we get
$y\_j = 7, 41, 239, 1393, \ldots$
$x\_j =... | https://mathoverflow.net/users/30445 | Uniqueness of values in recurrence relations | You may consider the paper:
B. Ibrahimpasic, A parametric family of quartic Thue inequalities.
Bull. Malays. Math. Sci. Soc. (2) 34 (2011), no. 2, 215–230,
available at <http://www.emis.de/journals/BMMSS/vol34_2_2.html>
It seems that Theorem 3.1, with c=2k, answers your question.
| 2 | https://mathoverflow.net/users/21337 | 118542 | 66,885 |
https://mathoverflow.net/questions/118540 | 6 | Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$.
Assuming that $X$ is rational over $k$, is the quotient $X/G$ always rational?
If $k=\mathbb{C}$, we can use Castelnuovo's theorem and see that $X/G$ is unirational and hence rati... | https://mathoverflow.net/users/23758 | Quotients of rational surfaces | This is not always true, and cubic threefolds give a counterexample over the field $k=\mathbb{C}(t)$. Let $\mathcal{Y}$ be a smooth cubic hypersurface in $\mathbb{P}^4\_{\mathbb{C}}$. Let $L\subset \mathcal{Y}$ be a line. Denote by $\mathcal{X}$ the (locally closed) subvariety of $\mathcal{Y}\times L$ parameterizing pa... | 6 | https://mathoverflow.net/users/13265 | 118544 | 66,886 |
https://mathoverflow.net/questions/118536 | 14 | This is about Deligne's "La série exceptionnelle de groupes de Lie, C.R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 321–326".
When this came out, that was quite something! People were often talking about it, at lunch or coffee time - even making bets if I remember well.
I totally lost contact with all this, ... | https://mathoverflow.net/users/29333 | Deligne's 1996 note on exceptional Lie groups | My understanding is that the series corresponding to the first three rows of the magic square do not exist but the existence of the series for the last row, the original exceptional series, is still open. This follows from Dylan Thurston's (unpublished) computer calculation and from Pierre Vogel's (unpublished) papers.... | 7 | https://mathoverflow.net/users/3992 | 118547 | 66,887 |
https://mathoverflow.net/questions/118500 | 38 | Let $k$ be a field. There are two natural categories to consider:
* The category of simplicial commutative $k$-algebras.
* The category of connective $E\_\infty$ $k$-algebras (i.e., chain complexes of $k$-vector spaces in nonnegative dimensions with a coherently associative and commutative multiplication law).
The... | https://mathoverflow.net/users/344 | What is a simplicial commutative ring from the point of view of homotopy theory? | I don't know a really satisfying answer to this question, but here are a few observations.
1) The $\infty$-category of simplicial commutative $k$-algebras is monadic over the $\infty$-category of connective $k$-module spectra. The relevant monad is the nonabelian left derived functor of the "total symmetric power" on... | 37 | https://mathoverflow.net/users/7721 | 118552 | 66,889 |
https://mathoverflow.net/questions/118551 | 2 | **Background**:
In the 1700s, Euler solved the Basel Problem, which was to solve $\sum\_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing $\frac{\sin(x)}{x}$ in Taylor series form, and then writing it as the normalized product of the linear factors given ... | https://mathoverflow.net/users/1982 | Can infinite polynomials be expressed as a product of its linear factors? | Hello Gabriel,
I think you should indeed have a look at the theory of (entire) holomorphic/meromorphic functions, since $x\mapsto (x-1)\zeta(x)$ belongs to that class. You more particularly wish to learn about Hadamard or Weierstrass factorization theorems [see Wikipedia](http://en.wikipedia.org/wiki/Entire_function)... | 6 | https://mathoverflow.net/users/24309 | 118553 | 66,890 |
https://mathoverflow.net/questions/118481 | 39 | Let me give a reasonable model for the question in the title. In ${\rm Sym}\_n({\mathbb R})$, the positive definite matrices form a convex cone $S\_n^+$. The probability I have in mind is the ratio $p\_n=\theta\_n/\omega\_n$, where $\theta\_n$ is the solid angle of $\Lambda\_n$, and $\omega\_n$ is the solid angle of th... | https://mathoverflow.net/users/8799 | The probability for a symmetric matrix to be positive definite | **Edit:** According to [Dean and Majumdar](http://arxiv.org/abs/0801.1730), the precise value of $c$ in my answer below is $c=\frac{\log 3}{4}$ (and $c=\frac{\log 3}{2}$ for GUE random matrices). I did not read their argument, but I have been told that it can be considered as rigourous. I heard about this result throug... | 27 | https://mathoverflow.net/users/10265 | 118556 | 66,892 |
https://mathoverflow.net/questions/118503 | 6 | Suppose that $X$ is an algebraic variety over $\mathbb C$, *not necessarily smooth*. Is it still true that each $\mathcal D\_X$-module ($\mathcal D\_X$ is of course the sheaf of differential operators) that is coherent as an $\mathcal O\_X$-vodule must be locally free as an $\mathcal O\_X$-module?
Thank you in advanc... | https://mathoverflow.net/users/29992 | D-module that is coherent as O-module | [Edited to correct errors pointed out by David Ben-Zvi and to answer a query by serge\_I. These corrections reduce this ``answer'' to the status of a comment.]
(1) (Over $\mathbb C$) Under strong assumptions on the singularities of $X$ (namely, that $X$ should be cuspidal, Ben-Zvi and Nevins, arXiv 0212094v3), if $\... | 4 | https://mathoverflow.net/users/8726 | 118558 | 66,894 |
https://mathoverflow.net/questions/118563 | 8 | If we take a knot $K$ in $S^3$, there are several ways to construct the associated Seifert surface. One way, which I am not familiar with, I just came across in a paper I am reading. It goes like this:
Consider the regular neighborhood $n\_K$ of $K$ in $S^3$, which is diffeomorphic to $K\times \mathbb{R}^2$. $S^3-n\_... | https://mathoverflow.net/users/1446 | Seifert surfaces via Alexander duality | This is an old argument that essentially predates much modern knot theory, and goes back to Serre.
The basic idea goes like this: let $C$ be the complement of a co-dimension two knot in $S^n$. Apply Poincare/Alexander duality to deduce that $C$ is a homology $S^1 \times D^{n-1}$. So $H^1 C \simeq \mathbb Z$, and $H^... | 11 | https://mathoverflow.net/users/1465 | 118568 | 66,901 |
https://mathoverflow.net/questions/118541 | 1 | Call two rational numbers $N$-*indistinguishable* if they have the same $p$-adic order for every prime $p$ less than $N$. Write $\sim\_N$ for the relation of being $N$-indistinguishable,
Say you are given two rational numbers $a$ and $b$ and are told that at least one of them is positive. For natural numbers $n$ and ... | https://mathoverflow.net/users/nan | Can positivity be detected by agreement of p-adic order?: A question on binomial coefficients | Unless I've misunderstood, the answer is "trivially no", because given any $N$ you just let $M$ be the product of the gazillionth powers of all the primes less than $N$, and then you're free to change $a$ to $a-M$.
More generally, knowing facts about valuations of numbers at some finite set of primes tells you preci... | 4 | https://mathoverflow.net/users/30035 | 118569 | 66,902 |
https://mathoverflow.net/questions/118571 | 2 | Good morning,
Let $H$ be a hypersurface in a complex projective space $\mathbb{CP}^N.$ Let $d$ be the distance de Fubini-Study on $\mathbb{CP}^N.$
1. Let $x = [x\_0: \ldots :x\_N]$ and $y=[y\_0:\ldots:y\_N]$ two points in $\mathbb{CP}^N.$ Is the following formula true $$d(x,y)^2 = \frac{\sum\_{i<j} |x\_i \bar{y\_j... | https://mathoverflow.net/users/11376 | on the density of hypersurfaces in complex projective spaces | Concerning part 2), you could take a hyperplane with multiplicity k and then find a nearby hypersurface, so you could make the quantity you want arbitrarily close to the distance from a point to a hyperplane (hence independent of the degree). If you're interested in hypersurfaces which fill out projective space, Donald... | 4 | https://mathoverflow.net/users/10839 | 118578 | 66,907 |
https://mathoverflow.net/questions/118472 | 4 | Hello,
One of the subgrouops of $SO(n)$ which acts transitively on the sphere $S^{n-1}$ is the (compact) symplectic group $Sp(n/4)$. The center of $Sp(m)$ is isomorphic to $\mathbb{Z}\_2$. Can we embed $Sp(n/4)/\mathbb{Z}\_2$ in $SO(n)$ (as a Lie subgroup)? If the answer is yes, what happens to its orbits? Precisely... | https://mathoverflow.net/users/nan | Transitive action on the sphere | (I add details to my comments.) The answer depends on $n=4r$. Write $G=Sp(r)/\mu\_2$. If $r=1$, then $G\simeq SO\_3$, so $G$ admits a faithful 4-dimensional representation into $SO\_4$. Similarly, if $r=2$, then $G\simeq SO\_5$, hence $G$ admits a faithful 8-dimensional representation into $SO\_8$. (Of course, in these... | 2 | https://mathoverflow.net/users/4149 | 118581 | 66,909 |
https://mathoverflow.net/questions/118579 | -2 | Hello.
I am currently doing some mathematics as an amateur, and recently I got to
publish an article at the Journal of Mathematics Research (JMR). The doi
for the article is 10.5539/jmr.v4n6p12 . And now, I have been trying to
get readers.
My real question, nevertheless, is whether it would be possible that my pap... | https://mathoverflow.net/users/30558 | Question about getting Review services | Many if not most reputable journals state up front whether their production is reviewed by Mathematical Reviews and/or Zentralblatt Math. This can work as a heuristic when deciding where to submit a paper you write, to ascertain that it gets the treatment you would expect for it.
MR and ZBM are not venues for peer-re... | 6 | https://mathoverflow.net/users/102 | 118582 | 66,910 |
https://mathoverflow.net/questions/118575 | 4 | The Kadison-Singer problem is considered in relation to the separable Hilbert space:
KS: Does every pure state on the diagonal (atomic) masa of $B(\ell\_2)$ has a unique extension to $B(\ell\_2)$?
What is the status of this problem for non-separable Hilbert spaces? Most of operator-algebraits are uninterested in (... | https://mathoverflow.net/users/29433 | Kadison-Singer problem in exotic Hilbert spaces | I think it's the same problem. If a pure state on $l^\infty$ had distinct extensions to states on $B(l^2)$, then you could embed $B(l^2)$ into $B(l^2(\kappa))$ for $\kappa > \aleph\_0$ and extend those extensions to $B(l^2(\kappa))$. Conversely, if every pure state on $l^\infty$ has a unique extension then every operat... | 8 | https://mathoverflow.net/users/23141 | 118590 | 66,914 |
https://mathoverflow.net/questions/118448 | 2 | We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a natural Lagrangian submanifold of it. A well known result is that any submanifold $X=\{(p,f(p)): p\in M\}$, where $f$ is a closed one form is Lagrangian. Denote by $[f]$ the de Rham cohomology class of $f$. Assume that we flow $X$ ... | https://mathoverflow.net/users/29480 | On the de Rham cohomology of 1-forms in cotangent bundle. | As has been noted in Peter Michor's answer, a Hamiltonian isotopy can certainly move the graph of a closed one-form to a Lagrangian submanifold that is not the graph of a one-form.
However, in the special case that the starting and ending submanifolds are both graphs of closed one-forms, the de Rham cohomology classe... | 3 | https://mathoverflow.net/users/424 | 118594 | 66,917 |
https://mathoverflow.net/questions/118574 | 3 | Conlay described in $\textit{Isolated Invariant Sets and the Morse Index (1976)}$ the bases of what would be known as Conley Index Theory.
For the sake of simplicity let's think of vector fields defined on manifolds (a more general situation is enough by just considering locally compact spaces). Im going to pose a f... | https://mathoverflow.net/users/30555 | Conley index for isolated invariant sets with no exit points | I maybe wrong but a way of seeing is the following: from the universal property of quotients you would like that any continuous map $f:N\to Y$ to a topological space $Y$ defines a continuous map of pointed topological spaces between $N$ collapsed with nothing and $(Y,y\_0)$ no matter what $y\_0$ is. Since you "collapse... | 3 | https://mathoverflow.net/users/6032 | 118610 | 66,925 |
https://mathoverflow.net/questions/118611 | 0 | Suppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|\_V$ such that $\|x\|\_V = \|x\|$ for all $x\in V$.
Is $(B,\|\cdot\|)$ a completion of $(V,\|\cdot\|\_V)$ with respect to the $\|\cdot\|\_V$ topology? I.e., can the spaces be considered the same/identic... | https://mathoverflow.net/users/25950 | Norms agreeing on dense subspace | This is standard, and the answer has been been indicated in the comments. Recall that in a normed space $X,$ the triangle inequality easily yields that $| \|x\| - \| y \| |\leq \|x-y\|$
for all $x,y \in X.$
Now let's turn to your dense subspace $V$ of the Banach space $B.$ Take an element $b \in B.$ There is a sequen... | 2 | https://mathoverflow.net/users/14450 | 118614 | 66,927 |
https://mathoverflow.net/questions/118584 | 2 | Let $V$ be a vector space of dimension $n$.
Let $S^k V$ be a representation of $GL(n)$.
I would like to know if there exists some characterization of finite dimensional $GL(n)$ modules $V\_1,V\_2$ such that exists $GL(n)$ mapping $T:S^k V \rightarrow V\_1\otimes V\_2$ such that for some $x$ matrix $T(x)$ is invertible... | https://mathoverflow.net/users/4246 | Representations of $GL(n)$ containing $S^kV$ | Regarding the question of pairs, here is the answer (hope, I'm not mistaken).
Let me first change the notation a little bit. Suppose, we are looking for irreps $V\_1=\Sigma^{\lambda}V$ and $V\_2=\Sigma^{\mu}V$ where $\lambda = (\lambda\_1,\ldots,\lambda\_n)$ and $\mu = (\mu\_1,\ldots,\mu\_n)$ resp., such that there e... | 5 | https://mathoverflow.net/users/10941 | 118618 | 66,929 |
https://mathoverflow.net/questions/118554 | 2 | Hi,
The complex simple algebraic group $Sp\_{m,\mathbb{C}}$ of $2m$-dimensional space $V$ has, for $m≥2$, an irreducible representation of dimension $m(2m−1)−1$ in a subspace of codimension $1$ of the space $\Lambda^2V$. Is it the irreducible representation of smallest dimension after $V$ itself?
Thank you.
| https://mathoverflow.net/users/nan | Non-trivial representation of second-smallest dimension | The irreducible complex representations of the simply connected simple group $G=Sp\_{r,{\mathbb C}}$ of type $C\_r$,
for $r>1$, of dimension $n<{\rm dim}\ G$
are listed in the [paper of Andreev, Vinberg, and Elashvili](http://link.springer.com/content/pdf/10.1007%2FBF01076005),
Table 1 (see also the [Russian version](h... | 2 | https://mathoverflow.net/users/4149 | 118621 | 66,931 |
https://mathoverflow.net/questions/95139 | 1 | I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in [Explicit formula for Cholesky factorization in a special case](https://mathoverflow.net/questions/95097/explicit-formula-for-cholesky-factorization-in-a-special-case), but the question might have... | https://mathoverflow.net/users/22051 | Sufficient conditions for inverse-positivity | Sorry for promoting my own results,
but I think the condition in my old paper
"A sufficient condition for the monotonicity
of a positive definite matrix" (Computational Mathematics and Mathematical
Physics vol. 41, No 9., pp. 1237-1238) may be of help.
(Unfortunately, I lost the file years ago).
I don't know wheth... | 1 | https://mathoverflow.net/users/9833 | 118625 | 66,934 |
https://mathoverflow.net/questions/118626 | 43 | Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?
| https://mathoverflow.net/users/30376 | real symmetric matrix has real eigenvalues - elementary proof | If "elementary" means not using complex numbers, consider this.
1. First minimize the Rayleigh ratio $R(x)=(x^TAx)/(x^Tx).$ The minimum exists and is real.
This is your first eigenvalue.
2. Then you repeat the usual proof by induction in dimension of the space.
3. Alternatively you can consider the minimax or maximin... | 55 | https://mathoverflow.net/users/25510 | 118627 | 66,935 |
https://mathoverflow.net/questions/118636 | 6 | [KET](http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem) is often used to construct stochastic processes in continuous time when the state space is $\Bbb R^d$. As far as I am familiar with its proof, it uses standard monotonic class-like arguments together with Caratheodory Extension Theorem. Neither of the two... | https://mathoverflow.net/users/11768 | Topological conditions of Kolmogorov Extension Theorem | The KET fails for general measurable spaces, the classical example can be found in a [paper](http://gymarkiv.sdu.dk/MFM/kdvs/mfm%2020-29/mfm-25-4.pdf) by Andersen and Jessen. Topological assumptions are necessary so that the resulting measure is not only finitely additive but countably additive. There exists a quasi-to... | 11 | https://mathoverflow.net/users/35357 | 118638 | 66,937 |
https://mathoverflow.net/questions/118633 | 1 | Let $a\_n>0$ and $b\_n>0$ be two strictly declining sequences such that the series
$$\sum\_{n=1}^\infty \frac{a\_n}{b\_n}$$ is convergent. For $\sigma>0$ define $$f^N(\sigma) = \sum\_{n=1}^N \frac{a\_n}{b\_n + \sigma/N}$$ Is it generally true that $\lim\_{N \to \infty} f^N(\sigma)$ is independent of $\sigma$ or are the... | https://mathoverflow.net/users/30573 | Series of quotients with perturbed denominator | Since $\sum\_ {n=1}^\infty \frac{a\_n}{b\_n } < \infty$ and $0 \le \frac{a\_n}{b\_n + \sigma/N}\le \frac{a\_n}{b\_n} $, we have that $\sum\_{n=1}^N \frac{a\_n}{b\_n + \sigma/N} \to \sum\_ {n=1}^\infty \frac{a\_n}{b\_n }$ as $N\to\infty$, just by dominated convergence.
| 1 | https://mathoverflow.net/users/6101 | 118644 | 66,940 |
https://mathoverflow.net/questions/117006 | 31 | [Graham's number](http://en.wikipedia.org/wiki/Graham%27s_number) achieved a kind of cult status, thanks to Martin Gardner, as the largest finite number appearing in a mathematical proof. (It may no longer hold that record, but that is not my concern here.) I was surprised to learn relatively recently that it is not ac... | https://mathoverflow.net/users/3106 | Reconstructing the argument that yields Graham's number | I talked to Ronald Graham last night at the Joint Mathematics Meeting in San Diego and asked him the question here. He said he'd made up Graham's number when talking to Martin Gardner because 1) it was simpler to explain than his actual upper bound, the one that appears in his paper with Rothschild, and 2) it's bigger,... | 24 | https://mathoverflow.net/users/2893 | 118650 | 66,943 |
https://mathoverflow.net/questions/118646 | 0 | We know PGL(2, $q$) has elements of order $q+1$ or $q-1$. Suppose $k\neq 1$, $2$ divide $q+1$ or $q-1$. It is clear that PGL(2, $q$) has an elements of order $k$. I would like to know what is the number of the elements of order $k$ and how we can get it?
| https://mathoverflow.net/users/29634 | The number of elements of order k in PGL(2, q) | By the sizes of conjugacy calasses of PGL(2, q), if $k$ divides $q+1$, then the number of elements of order $k$ is $\phi (k)q(q-1)/2$ and if $k$ divides $q-1$, then the number of elements of order $k$ is $\phi (k)q(q+1)/2$.
| 0 | https://mathoverflow.net/users/30203 | 118653 | 66,945 |
https://mathoverflow.net/questions/118523 | 30 | Are there integers $a, b > 1$ such that $\pi = \log\_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - otherwise the Gelfond-Schneider theorem would tell the answer.
When I asked this question about 10 years ago, I got an answer... | https://mathoverflow.net/users/28104 | Is pi = log_a(b) for some integers a, b > 1? | There is a reason why one can say a little bit more about this question in the case of $\pi$. Because $\pi$ is (essentially) the natural logarithm of a rational number, questions like this are easily derived from Schanuel's conjecture, which states:
>
> If $\alpha\_1, \alpha\_2, \ldots, \alpha\_n$ are complex numbe... | 34 | https://mathoverflow.net/users/3106 | 118654 | 66,946 |
https://mathoverflow.net/questions/118034 | 16 | This is motivated by this MO [question](https://mathoverflow.net/questions/117567).
If $A\in{\bf M}\_n({\mathbb R})$ is row-stochastic (entrywise non-negative, and $\sum\_j a\_{ij}=1$ for all $i$), then $M:=A+A^T$ is
* symmetric,
* entrywise non-negative.
One finds easily the
* additional property that
$$\sum\... | https://mathoverflow.net/users/8799 | What is $A+A^T$ when $A$ is row-stochastic ? | My answer builds on Brendan McKay's idea. We will show the vertices have the form he describes..
It's obvious that vertices in the polytope of row-stochastic matrices have this form, because each row is independent and the equations for each row form a simplex.
So it's enough to show that vertices in the polytope o... | 3 | https://mathoverflow.net/users/18060 | 118660 | 66,948 |
https://mathoverflow.net/questions/118649 | 1 | $P$ is a system of polynomials in $n$ variables over $\mathbb{Q}$. $Q$ is a singe such polynomial. Let $V$ be the zeros of $Q$. I know from some symmetry argument that for every $y \in [0,1]^n \setminus V$ there are at least $k$ elements in $\{ x:\frac{P(x)}{Q(x)}=\frac{P(y)}{Q(y)} \}$. I want to prove that, for $y$ in... | https://mathoverflow.net/users/30574 | System of polynomial equations: P(x)=P(y) rather than P(x)=0 | Yes, you can formalize this intuition. You can look for solutions over the algebraic closure of the field $\mathbb Q(x\_0)$, setting $y=x\_0$. If there are just $k$ solutions, then at every point except for a positive codimension closed set there will just be $k$ solutions of the original problem.
The reason this wor... | 0 | https://mathoverflow.net/users/18060 | 118666 | 66,950 |
https://mathoverflow.net/questions/118475 | 3 | Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring and $M$ a finitely generated $R$-module. Let $x\_1,...,x\_t$ be an $M$-regular sequence and $I = (x\_1,...,x\_t)$. Is it true that
$$\mathrm{Tor}\_1^R(R/I^n, M) = 0$$
for all $n \geq 1$?
| https://mathoverflow.net/users/17901 | Vanishing of Tor | I've posted a proof [here](https://math.stackexchange.com/questions/268542/ideal-generated-by-a-regular-sequence-and-the-vanishing-of-tor) for the special case when $M$ is cyclic. Furthermore, I've mentioned that the result holds for finitely generated modules when the sequence is $R$-regular and $M$-regular.
| 2 | https://mathoverflow.net/users/23950 | 118668 | 66,951 |
https://mathoverflow.net/questions/118661 | 6 | I am interested in a recommendation for a good book which discuses representation theory of GL(n)(say over field of complex numbers).
I know only a basic representation theory.
The question I am interested in are how looks decomposition of $GL(n)$ module $V\otimes W$, where $V$,$W$ irreps.
I am interested in book or ... | https://mathoverflow.net/users/4246 | Good book on representation theory of GL(n) | As previous questions about books on representation theory and Lie theory indicate, there are a lot of them out there aimed at different parts of the subject. (So maybe community-wiki is indicated?) It's good to be clear at the outset that the problem of finite dimensional tensor product decomposition over $\mathbb{C}$... | 5 | https://mathoverflow.net/users/4231 | 118669 | 66,952 |
https://mathoverflow.net/questions/118290 | 5 | Suppose I define a multicategory $M=(Ob(M),Hom\_M)$ to be *simply closed* if
* for every sequence $S=(b\_1,\ldots,b\_n;x)$ of $n+1$ objects in $M$, we provide an object $Exp(S)\in Ob(M)$, and
* for every sequence $A=(a\_1,\ldots,a\_m)$ of $m$ objects in $M$ we provide a bijection
$$\phi\_S^A\colon Hom\_M(a\_1,\ldots... | https://mathoverflow.net/users/2811 | A (too?) simple notion of "closed multicategory" | In fact, if you include naturality, then your second condition becomes automatic, and moreover it suffices to consider $n=1$. See e.g. section 3 of [this paper](http://arxiv.org/abs/0904.3137).
| 4 | https://mathoverflow.net/users/49 | 118675 | 66,956 |
https://mathoverflow.net/questions/118693 | 3 | Let $n\in\mathbb{N}$
Is the volume functional continuous on the set of isometry classes of compact riemannian $n$-manifolds with volume $\geq \varepsilon$\
(with respect to Gromov--Hausdorff distance)?
Without the volume bound,
a collapsing torus gives a counterexample.
But it seems that this is the only singularity... | https://mathoverflow.net/users/20557 | Is the volume functional contiunuous for compact manifolds with lower bounds on volume? | I assume you talk about continuity in Gromov-Hausdorff topology.
In general, volume is not continuous, as easy examples show (see e.g. Colding's paper "Large manifolds with positive Ricci curvature", examples 1-2 which have a lower volume bound).
On the other hand, for manifolds with a lower Ricci curvature bound,... | 5 | https://mathoverflow.net/users/1573 | 118694 | 66,963 |
https://mathoverflow.net/questions/118686 | 7 | Given $n$, the number of vertices, what is the number of triangle-free simple graphs on $n$ vertices (or asymptotically)?
A more difficult problem is, given $n$, $m$, what is the number of triangle-free simple graphs on $n$ vertices with $\le m$ edges (or asymptotically)?
| https://mathoverflow.net/users/26659 | counting triangle free graphs | Next terms are 581460254001, 31720840164950 (sent to OEIS). All these numbers were found by exhaustive enumeration. As far as I know, the theoretic enumeration problem is unsolved, even for labelled graphs.
As for asymptotics, it is an old result of Erdős, Kleitman and Rothschild that almost all triangle-free graphs... | 12 | https://mathoverflow.net/users/9025 | 118716 | 66,974 |
https://mathoverflow.net/questions/88929 | 11 | I have a question concering the family of Pellian equations
$$x^2 - (k^2+1)y^2 = k^2. \qquad (\*)$$
For an integer $k\geq 2$, the equation (\*) has at least three classes of solutions
in integers, corresponding to the fundamental solutions
$(x\_0,y\_0) = (k, 0)$, $(k^2-k+1, k-1)$, $(k^2-k+1, -(k-1))$.
Each fundamenta... | https://mathoverflow.net/users/21337 | a family of Pellian equations | Oh, I think the answer is definitely yes!
Let $\{k \to x,y\}$ be any solution of $x^2 - (k^2+1)y^2 = k^2$, and let $K$ be the set of $k$ for which a solution has $0 < k < y-1$. In a paper recently
submitted to *Glasnik Matematicki* we call these solutions
*exceptional* solutions. Andrej's conjecture is that f... | 3 | https://mathoverflow.net/users/30445 | 118718 | 66,976 |
https://mathoverflow.net/questions/118687 | 0 | Hi,
During my research I found an interesting fact, and I'd like to know if it's interesting for others as well.
Find a function $g(x,t):[0,T]\times[0,T]\rightarrow[0,T]$ such that for any twice differentiable $f(x):[0,T]\rightarrow[0,T]$ such that $f(0)=f'(0)=0$, the equality
$$ f(x)=\intop\_0^Tf''(t)g(x,t)dt$$
hold... | https://mathoverflow.net/users/28037 | RFC for definite integral connection to second derivative | this looks like a simple consequence, upon twice partial integration, of $f(x)=\int\_0^T f(t)\delta(t-x)dt$, so your $g(x,t)=(x-t)\theta(x-t)$
$\delta(x)=d\theta(x)/dx$ relates Dirac delta function and Heaviside step function.
| 2 | https://mathoverflow.net/users/11260 | 118720 | 66,977 |
https://mathoverflow.net/questions/118715 | 2 | You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar products. Suppose you are given a complex symmetric bilinear form $B: H\times H \to \Bbb C$, with norm $c$ over $<,>''$ i.e.
$$B(x,x) \leq c(|x|^2 + |x|'^2) ... | https://mathoverflow.net/users/17965 | Decomposing bilinear forms in Hilbert spaces | I think you mean $|B(x,x)| \leq c(|x|^2 + |x|'^2)$, etc. As the problem is stated, you can take $A(x,y) = \langle x,y\rangle$ and $C(x,y) = B(x,y) - \langle x,y\rangle$.
Then the answer is no, there's a counterexample where $H$ is two-dimensional. First, let us not demand that $\langle\cdot,\cdot\rangle$ and $\langle... | 5 | https://mathoverflow.net/users/23141 | 118724 | 66,981 |
https://mathoverflow.net/questions/118607 | 5 | We order $\mathbb{N}^{<\mathbb{N}}$ as following: if $|\sigma| < |\tau|$ then $\sigma < \tau$; if they are of same length then they are ordered lexicographically. It is provable over $\operatorname{RCA}\_0 + I\Sigma\_2$, that this is a well ordering (of type $\omega^\omega$).
It is said that this linear ordering coul... | https://mathoverflow.net/users/15424 | About the well ordering of finite sequences of numbers | The statement that the indicated order on $\mathbb{N}^{<\mathbb{N}}$ is well-founded is incomparable with $B\Sigma\_2$.
I will denote the order by $\prec$.
The statement that $\prec$ is well-founded can written as the statement that each non-empty set $X$ contains a $\prec$-minimal element, i.e.
(1) $\forall X \lef... | 3 | https://mathoverflow.net/users/3365 | 118728 | 66,983 |
https://mathoverflow.net/questions/118688 | 15 | I'm still studying maths at undergraduate level, but intend to continue exploring topics in pure maths after I have graduated, so am thinking already about what directions I'd like to persue now, (as I like to be several steps ahead of myself!)
One area which seems particularly interesting is 'reverse mathematics'. I... | https://mathoverflow.net/users/25997 | Understanding the nature and structure of proofs; Reverse Mathematics and Proof Theory. Prerequisites? Good introductory texts? | You probably already know this, from the references you mentioned, but since you didn't explicitly list it, I'll say it here. The "bible" of reverse mathematics is Steve Simpson's book "Systems of Second-Order Arithmetic".
As for the relevant areas of logic, I'd say that recursion theory is (currently) more closely c... | 10 | https://mathoverflow.net/users/6794 | 118733 | 66,987 |
https://mathoverflow.net/questions/118676 | 0 | Does anybody know a closed-form solution for the following expression (N>=1)? I don't even know where to begin with the i+n denominator.
Sum of i=0 to n Combin(n,i) \* (2i/(i+n)) / (2^n)
| https://mathoverflow.net/users/30581 | sum of a binomial-weighted entity | $\sum\_{i=0}^{n}2^{-n} {n\choose i}\frac{2i}{i+n}=2^{1-n}\frac{n}{1+n}F(1-n,1+n;2+n;-1)$
with $F$ the [hypergeometric function](http://en.wikipedia.org/wiki/Hypergeometric_function).
| 1 | https://mathoverflow.net/users/11260 | 118734 | 66,988 |
https://mathoverflow.net/questions/93634 | 4 | Let $M$ be a finite set, and
$S(M) = \{(f\_0, f\_1) | f\_0, f\_1: M → M\}$.
Each element of $S(M)$ can be considered as a finite directed graph with the set of nodes $M$, which has exactly two arrows from each node, the arrows being labeled $0$ and $1$. Let us take a look at the simplest operations on graphs of thi... | https://mathoverflow.net/users/22795 | Algebraic structure generated by primitive graph operations | Perhaps this paper might be tangentially related to the ideas you are developing?
[Bilal Khan, Kiran Bhutani, Delaram Kahrobaei. A Graphic Generalization of Arithmetic, Electronic Journal of Combinatorial Number Theory, 2007.](http://www.integers-ejcnt.org/vol7.html)
| 1 | https://mathoverflow.net/users/30593 | 118736 | 66,989 |
https://mathoverflow.net/questions/118532 | 5 | I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in [Euler's form](https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf), and I still don't quite understand how he got from one equation to the other. Lambert's solution is probably somewhe... | https://mathoverflow.net/users/28016 | What was Lambert's solution to $x^m+x=q$? | I understand your question as a historical one: how did Lambert solve the trinomial equation? Let me try to walk you through his derivation in his 1758 paper "*Observationes variae in mathesin puram*".
In paragraph 35 (see [here](http://ilorentz.org/beenakker/MO/Lambert35.png)) Lambert first explains his method of s... | 6 | https://mathoverflow.net/users/11260 | 118762 | 67,003 |
https://mathoverflow.net/questions/118747 | 2 | Question 1: In $\mathbb{R}^2$, let $l\_1$,$l\_2$ be two parallel lines and $l\_3$ another line which is not parallel to $l\_1$. Given two measurable sets $E\_1$ and $E\_2$ in $l\_1$ and $l\_2$ respectively, both of which have positive (1-dimension) Lebesgue measure. Denote by $l\_{u,v}$ the line passing through two poi... | https://mathoverflow.net/users/30259 | Operation on measurable sets in lines, containing an interval? | We will need the following result from the paper that I mentioned in the comments "An elementary proof and an extension of a theorem of Steinhaus" by M. Kuczma and M. Kuczma that generalizes the Steinhaus theorem.
Thm:(Kuczma and Kuczma) Let $A,B$ be sets with positive inner Lebesgue measure and let $f(x,y)$ be a rea... | 2 | https://mathoverflow.net/users/22277 | 118765 | 67,005 |
https://mathoverflow.net/questions/118476 | 4 | I am reading a paper and stuck with an inequality used in that paper.
$\varepsilon^n=(\varepsilon\_1^n, \varepsilon\_2^n,\ldots,\varepsilon\_n^n)^T$ is a vector of i.i.d. random variables with mean 0 and variance $\sigma^2$. Assume that $\varepsilon\_i^n$ have finite $2k$'th moment $E(\varepsilon\_i^n)^{2k}<\infty$ ... | https://mathoverflow.net/users/30524 | Prove an inequality related to moments | As observed in a (now deleted) previous comment, the exponent of $\|\alpha\|\_2$ should be $2k$ instead of $2$ for homogeneity reasons.
If the $\varepsilon\_i$ are symmetric, then this can be proven by a variant of the exponential moment generating function method used to prove Khintchine's inequality. Indeed, if we ... | 5 | https://mathoverflow.net/users/766 | 118769 | 67,009 |
https://mathoverflow.net/questions/118772 | 4 | Is there a name for regular bands that satisfy $xyx=yx$ for all $x$,$y$?
| https://mathoverflow.net/users/nan | Name for a regular band | A right regular band. The dual notion of left regular bands has been used to study random walks on hyperplane arrangements and oriented matroids. The paper of Ken Brown on Semigroups, rings and Markov chains is a good place to start. My paper with Margolis and Saliola gives more details on the representation theory as ... | 4 | https://mathoverflow.net/users/15934 | 118773 | 67,011 |
https://mathoverflow.net/questions/118763 | 9 | Hello,
Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones.
I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = \mathbf{1}$$
$X$ and $W$ are all assumed to have real-valued entries, and $X'$ denotes the transpose of $X$.
I don't, yet, ha... | https://mathoverflow.net/users/30601 | Rescaling positive definite matrices to force a unit eigenvector | Consider the simplex of nonzero diagonal matrices W with nonnegative entries up to scaling, and the simplex of nonzero vectors V with nonnegative entries up to scaling.
There is a map, $V=\max(WX′XW\mathbf 1,0)$, from the first simplex to the second, with $\max(a,0)$ interpreted entrywise. This is well-defined becaus... | 4 | https://mathoverflow.net/users/18060 | 118782 | 67,016 |
https://mathoverflow.net/questions/118738 | 17 | As probably many other people here, I learned integration, as an undergrad, from Rudin's books. I recently realized, however, that I don't quite use Lebesgue integration in my work, or at least I use it less and less. Nor in fact do I think any longer of Riemann integration as *the intuitive* definition of the integral... | https://mathoverflow.net/users/29333 | Monte Carlo integration | this is perhaps more a comment than an answer, but here is one [expert opinion](http://terrytao.files.wordpress.com/2011/01/measure-book1.pdf) on whether one can base the foundations of integration theory on Monte Carlo integration:
>
> Another way to obtain continuous
> measure as the limit of discrete
> measure... | 9 | https://mathoverflow.net/users/11260 | 118796 | 67,022 |
https://mathoverflow.net/questions/60223 | 5 | Let $p$ be a fixed prime number. Roughly speaking, I am interested in the following ratio
$$
\frac{ |\{\text{ all CM number fields of degree }2g}|}
{ |\{\text{CM fields of degree 2g, such that p splits completely in K}\}|}
$$
A possible definition could be the following:
let $d\_{K}$ be the discriminant of K,
then ... | https://mathoverflow.net/users/4856 | Density of fields with a decomposition condition | Let $K$ be a CM field with maximal real subfield $k$. Most of the time the normal closure
of $k$ will have the symmetric group $S\_g$ as Galois group; thus the number of fields
$k$ in which $p$ splits completely has density $1/g!$ among all of them by density theorems due to Kronecker, Frobenius and/or Chebotarev. In a... | 3 | https://mathoverflow.net/users/3503 | 118797 | 67,023 |
https://mathoverflow.net/questions/118793 | 7 | In [THIS PROBLEM](https://mathoverflow.net/questions/118258/maximal-ideals-in-rx/118311) it is proved that for any non-zero ring $R$ with identity, $R[x]$ has an infinite number of maximal left ideals. Is it possible for an uncountable non-zero ring $R$ with identity, $R[x]$ has only a countable number of maximal left ... | https://mathoverflow.net/users/nan | Number of Maximal Left Ideals | Generalizing the answer by wccanard: Let $R$ be a commutative ring. Then the kernel of $R[x] \twoheadrightarrow R\_{red}[x]$ consists of nilpotent elements, hence this map induces a *homeomorphism* $\mathrm{Spec}(R\_{red}[x]) \cong \mathrm{Spec}(R[x])$. It restricts to a homeomorphism between the subspaces of closed po... | 4 | https://mathoverflow.net/users/2841 | 118804 | 67,027 |
https://mathoverflow.net/questions/118755 | 2 | Let $x\in M$, $M$ - finite dimensional smooth manifold. Is there an example of a finite dimensional Lie group action on $M$ with no slice at $x$?
| https://mathoverflow.net/users/30598 | Lie group action with no slice | Recall that if a free action of G on M has a slice S at a point x then the natural map of G x S into M given by (g,s) maps to gs would be a diffeomorphism onto a tubular neighborhood of the orbit Gx. So for a counterexample take the action of the real line on the 2-torus given by a 1-parameter subgroup with irrational ... | 8 | https://mathoverflow.net/users/7311 | 118813 | 67,032 |
https://mathoverflow.net/questions/118785 | 1 | Consider the following exponential kernel:
$k(x\_1, x\_2) = \exp\left(\frac{|x\_1 - x\_2|}{L}\right)$,
which is symmetric and non-negative definite. By virtue of Mercer's theorem, we have
$k(x\_1, x\_2) = \sum\_{i = 1}^\infty \lambda\_i \phi\_i(x\_1) \phi\_i(x\_2)$
where $\lambda\_i$ and $\phi\_i$ are the eigen... | https://mathoverflow.net/users/21503 | Eigenfunctions and eigenvalues of the product of two exponential kernels | You have two independent sets of variables, so it is a tensor product, not a Hadamard product, no? I am not sure I understand Suvrit's comment...
I think you just get
$K((x\_1,y\_1),(x\_2,y\_2)) = k(x\_1,x\_2)k(y\_1,y\_2) = \sum\_{i=1}^\infty \lambda\_i \phi\_i(x\_1)\phi\_i(x\_2)\sum\_{j=1}^\infty \lambda\_j \phi\_... | 2 | https://mathoverflow.net/users/30364 | 118817 | 67,033 |
https://mathoverflow.net/questions/118811 | 4 | As everyone knows, the standard middle-thirds Cantor set is constructed by dividing the interval into three equal parts, removing the middle one, then applying the same procedure to the remaining two intervals, etc.
The resulting set has Hausdorff dimension $s=\log 2/\log 3$, in view of Hutchinson's formula:
$$
\sum... | https://mathoverflow.net/users/8131 | Hutchinson's formula for asymptotically homogeneous Cantor sets | Since you remove "more", it should be more or less clear that $\text{HD}\; C\_\delta \le \text{HD}\; C$ (where $C$ is the standard Cantor set and $C\_\delta$ is the "perturbation" you describe). For proving the opposite inequality it is enough to exhibit a measure on $C\_\delta$ whose Hausdorff dimension is equal to $\... | 3 | https://mathoverflow.net/users/8588 | 118818 | 67,034 |
https://mathoverflow.net/questions/118816 | 2 | Hello!
It may be a stupid question, i'm trying to find a closed form for an integral similar to a Fourier transform on $S^{n}$ but i'm stuck...
Let $\alpha>0$, the integral i can't solve is
`$$I(p,\alpha)=\int_{S^{n}}e^{i\alpha\left<p,q\right>}d\mu_{S^{n}}(q)$$`
where $p,q\in S^{n}$, $\left< \cdot,\cdot \right>$ ... | https://mathoverflow.net/users/4971 | Closed form for Fourier transform-like Integral on $S^{n}$ | The integral obviously doesn't depend on $p$, since it must be rotationally invariant, and, using polar coordinates centered on $p$, one obtains
$$
I(p,\alpha) = \mathrm{vol}(S^{n-1})\int\_{-1}^1 e^{i\alpha t}(1{-}t^2)^{(n-2)/2}dt
=\frac{2\ \pi^{n/2}}{\Gamma(n/2)}\int\_{-1}^1 \cos(\alpha t)(1{-}t^2)^{(n-2)/2}dt.
$$
Thi... | 3 | https://mathoverflow.net/users/13972 | 118821 | 67,036 |
https://mathoverflow.net/questions/118801 | 2 | N.Hitchen in his paper about geometry of three forms wrote that "for a Real vector space $V$ of dimension six, the group $GL(6,V)$ has an open orbit and he referenced it to a thesis which was written in 1907 , how can we prove this fact, is there any open access reference?
| https://mathoverflow.net/users/nan | why the group $GL(6,V)$ has an open orbit? | In any case, the proof is very simple. Consider the $3$-form
$$
\phi\_0 = dx^1\wedge dx^2\wedge dx^3 + dx^4\wedge dx^5\wedge dx^6.
$$
I claim that the subgroup $G\subset\mathrm{GL}(6,\mathbb{R})$ that stabilizes $\phi\_0$ consists of the obvious subgroup $G\_0=\mathrm{SL}(3,\mathbb{R})\times\mathrm{SL}(3,\mathbb{R})$ t... | 11 | https://mathoverflow.net/users/13972 | 118824 | 67,039 |
https://mathoverflow.net/questions/118593 | 0 |
>
> Does there exist a lattice-ordered, not totally ordered, group of rational rank $1$?
>
>
>
Rational rank 1 means isomorphic to a nonzero subgroup of $\mathbb{Q}$. There exist totally ordered groups of rational rank 1.
| https://mathoverflow.net/users/30267 | Lattice-ordered group of rational rank 1 | If "rational rank 1" simply means that the underlying group is isomorphic to $\mathbb{Q}$, then every such lattice-ordered group structure $(\mathbb{Q}, \preceq)$ is totally ordered, in fact isomorphic to $\mathbb{Q}$ with its usual order $\leq$.
By a classical result (see the first sentence [here](http://www.jstor.... | 3 | https://mathoverflow.net/users/2926 | 118825 | 67,040 |
https://mathoverflow.net/questions/117536 | 1 | Let $G$ be an abelian (not elementary) finite $p$-group. In what conditions the automorphism group of $G$ is solvable?
| https://mathoverflow.net/users/27831 | finite abelian p-groups with solvable automorphism group | I believe that for $p\ge5$, these are the direct products of cyclic groups of pairwise distinct orders, for if $C\_{p^e}\times C\_{p^e}$ is a direct factor of $G$, then $GL(2,p)$ is a homomorphic image of a subgroup of $A=\text{Aut}(G)$.
Similarly for $p=2$ or $3$, I expect the groups $G$ you are looking for are thos... | 2 | https://mathoverflow.net/users/18739 | 118828 | 67,042 |
https://mathoverflow.net/questions/118831 | 11 | This is a question I posted on SE, and I have been advised to post it here.
<https://math.stackexchange.com/questions/146427/subspaces-of-l-p-and-banach-mazur-distance>
It is well-known that every subspace of $l\_2$ is *isometric* to $l\_2$. When $p\neq 2$, $l\_p$ has subspaces that are not even isomorphic, let al... | https://mathoverflow.net/users/30624 | Subspaces of $l_p$ and Banach-Mazur distance | (1) is correct. It follows from the fact that there is a sequence $(E\_n)$ of finite dimensional subspaces of $\ell\_p$ s.t. $\gamma\_p(E\_n) \to \infty$. Here $\gamma\_p(X)$ is the factorization constant of the identity on $X$ through an $L\_p$ space; that is, $\gamma\_p(X)= \inf \|T\|\cdot \|S\|$, where the infimum i... | 8 | https://mathoverflow.net/users/2554 | 118835 | 67,043 |
https://mathoverflow.net/questions/118742 | 1 | If there is a finite group $G$ with a cyclic normal subgroup $C\_n$, one can describe the indecomposable representations of $G$ through induction. How does $Ind\_{C\_n}^G$ decompose? For representations over fields, I know that Clifford's theory whould help. But what happens if the representations should be over a ring... | https://mathoverflow.net/users/30594 | Decomposition of an induced representation | Not really an answer, but this is already difficult in the complex case, when $C$ is a central
subgroup. For example, if $C = Z(G),$ and we induce a faithful irreducible $C$-module to $G,$ the number of distinct irreducible constituents is bounded above by the number of conjugacy classes of $G/C,$ but I don't know man... | 2 | https://mathoverflow.net/users/14450 | 118839 | 67,046 |
https://mathoverflow.net/questions/118647 | 1 | Let $X$ be a smooth variety and consider the diagonal $\Delta \subseteq X \times X$. It seems to be well-known that the exceptional divisor in the blow-up of $X \times X$ along $\Delta$ is isomorphic to the projectivized tangent bundle $\mathbb P(\mathcal T\_X)$ of $X$ but I can't find a reference or a proof; where mig... | https://mathoverflow.net/users/1528 | On the blow-up along the diagonal in a product | Chuck, I think you can find references for both of the statements 1,2 that Allen gives in Fulton's Intersection theory, Appendix B, Section 5, and maybe also section 7. There, I think (1) is proved more generally for a regularly embedded subscheme. I asked a question about (2) some time ago. You might find what you are... | 1 | https://mathoverflow.net/users/25854 | 118840 | 67,047 |
https://mathoverflow.net/questions/118842 | 9 | This might be ridiculously obvious, but...
For each $n \in \mathbb{N}$, let $M\_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant function $d\_n:M\_n \to \mathbb{R}$ is a Morse function if and only if $n = 2$ since the zero matrix is a degenerate ... | https://mathoverflow.net/users/18263 | When is the determinant a Morse function? | I'm not sure exactly what a characterization would be. Is there a characterization of the submanifolds of $\mathbb R^n$ for which $x\_1$ is a Morse function?
Critical points of the submanifold will come in two forms - critical points of $M\_n$ that lie on the submanifold, and noncritical points of $M\_n$ that become ... | 7 | https://mathoverflow.net/users/18060 | 118843 | 67,049 |
https://mathoverflow.net/questions/118845 | 1 | In answering [this MO question](https://mathoverflow.net/questions/25588/free-monad-or-monad-defined-from-an-adjunction/118756#118756), the issue was raised of characterizing when a given endofunctor $R:C\to C$ has the form $U\circ F$ where $F:C\to D$ is left adjoint to $U:D\to C$, i.e. which admit a monad structure. I... | https://mathoverflow.net/users/11540 | Can we characterize endofunctors which admit a monad structure? | Characterizing endofunctors on various categories which admit a monad structure is the same as characterizing objects in various monoidal categories which have a monoid structure: "=>" If $(C,\otimes,\dotsc)$ is a monoidal category and $X \in C$, then $- \otimes X : C \to C$ has a monad structure iff $X$ has a monoid s... | 6 | https://mathoverflow.net/users/2841 | 118851 | 67,053 |
https://mathoverflow.net/questions/118868 | 1 | Let $L$ be an atomistic ortholattice (i.e. every element can be written as a join of atoms) with top and bottom elements 0 and 1, and let $M$ be a distributive atomic sub-ortholattice of $L$.
Is $M$ generated by its atoms, in the sense that every element in $M$ can be written as a join of the atoms in $M$?
| https://mathoverflow.net/users/6985 | distributive sublattices of atomistic ortholattices | No. Let $L$ be the power set of ${\bf N}$ ordered by inclusion. It is an atomic Boolean algebra. Let a subset of ${\bf N}$ belong to $M$ if its characteristic function is periodic. Then $M$ is a nonatomic Boolean subalgebra of $L$.
| 3 | https://mathoverflow.net/users/23141 | 118869 | 67,059 |
https://mathoverflow.net/questions/118858 | 1 | **A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$.**
Could someone post some left-separated space to help me understand such definition?
| https://mathoverflow.net/users/18465 | Understanding the left-separated spaces | This is just filling in some details of Ramiro de la Vega's comment: Given any space $X$, form a (possibly transfinite) sequence of points by the following induction. As long as (the range of) the sequence you've built so far isn't dense in $X$, choose arbitrarily a point not in its closure, and append the chosen point... | 5 | https://mathoverflow.net/users/6794 | 118876 | 67,061 |
https://mathoverflow.net/questions/118873 | 3 | I understand that the Mellin transform of a modular form is expected to satisfy RH when it is an eigenform of all Hecke operators, in which case it has an Euler product. Now about when the form is not an eigenform: Is it known a case where the zeros are all in the critical strip?
| https://mathoverflow.net/users/30637 | On the location of zeros of L functions from modular forms | When it is not a Hecke-Eigen form, the Hecke L-series connected with the modular form does not have an Euler product. However it can still be written as a linear combination of Hecke L-series that have Euler-products. Thus the situation will resemble the case of linear combinations of Dirichlet L-series. In particular ... | 3 | https://mathoverflow.net/users/10811 | 118883 | 67,063 |
https://mathoverflow.net/questions/118875 | 1 | Let $f: \mathbb{C} \rightarrow \mathbb{C} $ be a function of the form
$$ f(z) = z^n + z^{n+ 1} g(z) $$
where $g$ is a $\textbf{smooth}$ function (not necessarily holomorphic).
Is it true that the number of solutions counted with a sign ``near'' the origin
for the equation
$$ f(z)-\nu =0 $$
is $n$, where $\nu$ i... | https://mathoverflow.net/users/4463 | If you perturb a polynomial by a smooth function, then is the signed number of small zeros of the perturbed equation the same as the lowest non zero derivative? | The answer is yes. Actualy Rouche's theorem is valid for smooth functions if instead of
the number of zeros you count the signed number. (I suppose that this is what you want to count).
That is the zero with positive Jacobian is counted
with + and zero with negative Jacobian is counted with -. This is called the topolo... | 0 | https://mathoverflow.net/users/25510 | 118885 | 67,065 |
https://mathoverflow.net/questions/118880 | 1 | Suppose $X\_i, \; i=1,2,3...,n$ are each Gaussian, then it is not in general true that the set is jointly Gaussian (a multivariate Gaussian).
Does a similar statement hold if the variates are *pairwise* Gaussian? i.e. if we have that $X\_i, X\_j$ are a bivariate Gaussian for all $i, j$, then it is not in general true... | https://mathoverflow.net/users/8916 | Pairwise Gaussian vs Jointly Gaussian (k-wise Gaussian vs n-wise Gaussian) | Let $X\_i$, $i=1\ldots3$, be iid standard normal random variables. Let $Y = |X\_3|$ if
$X\_1 X\_2 > 0$ and $-|X\_3|$ if $X\_1 X\_2 \le 0$. Then $X\_1, X\_2, Y$ are normal and pairwise
independent, but they are not jointly normal since $X\_1 X\_2 Y \ge 0$.
| 4 | https://mathoverflow.net/users/13650 | 118891 | 67,068 |
https://mathoverflow.net/questions/118857 | 17 | I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? Presumably we do homotopy type theory in some $(\infty,1)$-topos, so we can axiomatize the notions accordingly? Does anyone ... | https://mathoverflow.net/users/11546 | Forcing in Homotopy Type Theory | In as far as we regard [forcing](http://ncatlab.org/nlab/show/forcing) as forming [internal sheaves](http://ncatlab.org/nlab/show/internal%20sheaf), the question is asking how to say "internal category of sheaves" in [homotopy type theory](http://ncatlab.org/nlab/show/homotopy%20type%20theory).
It is expected that t... | 25 | https://mathoverflow.net/users/381 | 118894 | 67,070 |
https://mathoverflow.net/questions/118786 | 0 | This question arose from sums over zeros of Dedekind zeta function.
It is known that complex zeros of Dedekind zeta function are in pairs $\rho, 1 - \rho$.
>
> Is it true that potential complex zeros not on the critical line of Dedekind zeta function must be in quadruples
> $\rho, 1 - \rho, \overline{\rho}, \ov... | https://mathoverflow.net/users/12481 | Are potential complex zeros not on the critical line of Dedekind zeta function in quadruples? | The functional equation tells you that if $\rho$ is a zero, so is $1-\rho$. On the other hand, since the Dirichlet series coefficients are real, we have that for $\text{Re}(s)>1$, $\zeta(\bar{s})=\overline{\zeta(s)}$. By analytic continuation this holds for all $s\ne 1$. So if $\rho$ is a complex zero off the critical ... | 2 | https://mathoverflow.net/users/6756 | 118895 | 67,071 |
https://mathoverflow.net/questions/118887 | 8 | **Definition 1**: Given a group $G$, a subset $X \subseteq G$, and a natural number $k$,
we say that $X$ is (left) $k$-generic in $G$ if there are $k$ many left translates of $X$ that cover $G$.
That is, if there exist $a\_1, \dots, a\_k \in G$ such that $G = \bigcup\_i a\_i \cdot X$ (where $a\_i \cdot X = \{ a\_i \cdo... | https://mathoverflow.net/users/30642 | What is the probability that a random subset of a finite group is generic? | I think your conjecture is right. I hadn't heard of this fact before, so can't provide a reference. Here is my attempt at a proof.
The idea is that I want to reverse things: Fix a set $A=\{a\_i\}$ with $|A|=k$ and show that the probability that a set $X$ is $A$-generic (i.e. $AX=G$) is exponentially small in $n$ (uni... | 7 | https://mathoverflow.net/users/11054 | 118901 | 67,073 |
https://mathoverflow.net/questions/118903 | 33 | I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary field when possible. I always introduce at least $\mathbb{F}\_2$ as an example of a finite field. To help motivate this level... | https://mathoverflow.net/users/1044 | Elementary applications of linear algebra over finite fields | How about binary linear codes? You can "see" the Hamming distance between codewords, and use linear transformations to encode/decode
| 26 | https://mathoverflow.net/users/11054 | 118904 | 67,075 |
https://mathoverflow.net/questions/118899 | 4 | [This question](https://mathoverflow.net/questions/43514/how-do-eigenvectors-and-eigenvalues-change-when-we-remove-a-row-column-pair-of-a) asks about the changes in eigenvalues when a row and a column is added to a symmetric matrix. [Cauchy's interlacing theorem](http://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_inte... | https://mathoverflow.net/users/42371 | How does appending a row and a column change eigenvalues of a non-symmetric matrix? | You need more than just $x$, $y$, $z$ and the eigenvalues of $A$ to determine or bound the eigenvalues of $B$. For example, consider
$$ B = \pmatrix{1 & 0 & | & 1\cr t & 0 & | & 0\cr \hline 0 & 1 & | & 0\cr}$$
where $A$ has eigenvalues $1$ and $0$, but any real number $\lambda$ will be an eigenvalue of $B$ for suitable... | 4 | https://mathoverflow.net/users/13650 | 118911 | 67,080 |
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