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https://mathoverflow.net/questions/255400 | 5 | Look at the expression
$$
f(x\_1,x\_2,x\_3) = x\_1^2+x\_2^2+x\_3^2+x\_1x\_2+x\_2x\_3+x\_3x\_1.
$$
The numbers $x\_1,x\_2,x\_3$ are non-negative, and I assume that $x\_1+x\_2+x\_3=3$. This is a sum of squares and "cyclic correlations" of consecutive variables. Then you can check that $f$ is minimized for the values $x... | https://mathoverflow.net/users/46852 | Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$ | The question is about the signature of a quadratic form
$$
\sum\_{i=1}^n x\_i^2 + \frac12\sum\_{1 \le \mathrm{dist}(i,j) \le p} x\_ix\_j
$$
(or about the spectrum of the corresponding linear operator). The matrix is a circulant matrix, see [Wikipedia](https://en.wikipedia.org/wiki/Circulant_matrix), and its eigenvector... | 15 | https://mathoverflow.net/users/98590 | 255418 | 115,462 |
https://mathoverflow.net/questions/255412 | 3 | Let $X$ be a space, $\ f:X\times X\rightarrow\mathbb{R}^+\cup\{0\}$ be a map satisfying the first two axioms for a metric (so $f(x,y)=0$ exactly when $x=y$, and $f$ is symmetric).
Now, consider the topology generated by the following sets (informally thought of as open balls), for $x\in X$ and $a\in \mathbb{R}$:
$$
U(... | https://mathoverflow.net/users/80352 | When is a real-valued function on a metric space a metric? | Following the example of my comment, with $X = \mathbb{R}^2$ and
$$f((x\_1, y\_1), (x\_2, y\_2)) = ((x\_1 - x\_2)^{1/2} + (y\_1 - y\_2)^{1/2})^2,$$
one sees that $f((0, 0), (0, 1/2)) = 1/2$ and $f((0, 1/2), (1/2, 1/2)) = 1/2$, whereas $f((0, 0), (1/2, 1/2)) = 2$, so the triangle inequality fails.
In general, i... | 3 | https://mathoverflow.net/users/2926 | 255421 | 115,463 |
https://mathoverflow.net/questions/254159 | 5 | Suppose that $\Gamma \curvearrowright (X,\mu)$ is a measure-preserving action of a discrete, finitely generated group $\Gamma = \left\langle S \right\rangle$ on a probability space $(X,\mu)$. Let $\pi: \Gamma \to L^{2}(X,\mu)$ be the associated Koopman representation $(\pi(\gamma)f)(x) := f(\gamma^{-1}x)$. I've come ac... | https://mathoverflow.net/users/24953 | Strong ergodicity and spectral gap | Yes, conditions 1 and 2 are equivalent (as you already noticed).
Let me start by replacing both definitions with their negations, and replace definition 2 with its $L^1$-version.
A and B below are the negations of 1 and 2 correspondingly, slightly rewritten:
A. There exists a sequence of measurable subsets $\emptys... | 5 | https://mathoverflow.net/users/89334 | 255426 | 115,465 |
https://mathoverflow.net/questions/255427 | 2 | In [A duality formalism in the spirit of Grothendieck and Verdier](https://arxiv.org/abs/1108.6020), Boyarchenko and Drinfeld consider a monoidal category $(\mathcal{M}, \otimes, \mathbf{1})$ together with an object $K \in \mathcal{M}$ such that there is an equivalence $D \colon \mathcal{M}^\mathrm{op} \to \mathcal{M}$... | https://mathoverflow.net/users/100993 | Associator of the "dual" monoidal structure of a Grothendieck--Verdier Category | Let $F : \mathcal{C} \to \mathcal{D}$ be an equivalence of categories and $(\mathcal{D},\otimes,1)$ be a monoidal structure on $\mathcal{D}$. Then we have an induced monoidal structure on $\mathcal{C}\,$: Choose a quasi-inverse $F^{-1} : \mathcal{D} \to \mathcal{C}$ and isomorphisms $\mathrm{id}\_{\mathcal{C}} \cong F^... | 2 | https://mathoverflow.net/users/98306 | 255434 | 115,466 |
https://mathoverflow.net/questions/255439 | 5 | Let $T$ be a small triangulated category. Under which conditions there exists a triangulated category $B$ closed with respect to (small) coproducts such that $T$ fully embedds into the subcategory of compact objects of $B$ (I don't need this embedding to be an equivalence; yet if some $B$ of this sort exists then one c... | https://mathoverflow.net/users/2191 | Which triangulated categories are subcategories of compact objects "somewhere"? | I do not know of an answer for a general triangulated category (non-topological triangulated categories are very unusual), but as soon as you ask for some more structure the thesis follows very quickly.
Let us suppose that $T$ is the homotopy category of some stable ∞-category $C$ (that is, that $T$ is a topological ... | 5 | https://mathoverflow.net/users/43054 | 255440 | 115,468 |
https://mathoverflow.net/questions/255446 | 11 | Could there be an undecidable statement $S$ in ${\sf ZFC}$ of which one will never be able to prove its undecidability for principal reasons (ie we will never know that $S$ is undecidable)?
If this is a silly question, I apologise; feel free to vote to close. I'll remove it quickly.
| https://mathoverflow.net/users/8628 | Meta-undecidability | One can never prove in ZFC itself that a given statement is independent of ZFC, because the assertion "*$S$ is independent of ZFC*" implies Con(ZFC), since every statement is settled by an inconsistent theory.
Thus, if a statement $S$ is independent of ZFC, then there is a model of ZFC inside of which $S$ is not ind... | 15 | https://mathoverflow.net/users/1946 | 255447 | 115,469 |
https://mathoverflow.net/questions/254293 | 12 | A commutative ring $R$ is reduced if $r^2=0 \Rightarrow r=0$ holds for all $r \in R$. Commutative rings are precisely the commutative algebra objects in the symmetric monoidal category $(\mathsf{Ab},\otimes)$. This leads to the following question: Has any notion of "reduced commutative algebra objects" been studied for... | https://mathoverflow.net/users/98306 | Is there a categorical notion of reduced commutative algebras? | One can use [idals](https://math.stackexchange.com/questions/1004616). The unit $1\_{\mathcal{C}} \in \mathcal{C}$ is called reduced if for all idals $e : I \to 1\_{\mathcal{C}}$ (i.e. morphisms $e$ satisfying $e \otimes I = I \otimes e$) with $e \otimes e = 0$ one has $e=0$. If $\mathcal{C}=\mathsf{Mod}(X)$ for some r... | 3 | https://mathoverflow.net/users/98306 | 255452 | 115,471 |
https://mathoverflow.net/questions/255416 | 7 | *(**Update**)*:
Courtesy of Myerson's and Elkies' answers, we find a second cyclic quintic for $\cos\frac{2\pi}{p}$ with $p=\text{1 mod 10}$ as,
$$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^2 - 625) z - 4 n^2 (n^4 - 25 n^2 - 125)$$
where $\beta=n^4 + 25 n^2 + 125.$ Its discriminant is,
$$D=2^{12}\,5^{20}... | https://mathoverflow.net/users/12905 | Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins | I think the depressed quintic in the question is what Emma Lehmer called the reduced quintic in her [paper](https://projecteuclid.org/euclid.dmj/1077476385), The quintic character of 2 and 3, Duke Math. J. Volume 18, Number 1 (1951), 11-18, MR0040338. In the proof of Theorem 4, she writes that the reduced quintic is $$... | 4 | https://mathoverflow.net/users/3684 | 255454 | 115,473 |
https://mathoverflow.net/questions/255338 | 5 | first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far.
In Isaac Chavel's book *Eigenvalues in Riemannian Geometry*, Chapter VI, pages 151-154, the heat kernel for compact manifolds is constructed.
I am hoping for someone that is fami... | https://mathoverflow.net/users/101486 | heat kernel on closed manifolds - error in Chavel's book? | Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that
$$L(H\_k \* F) = (LH\_k)\*F - F,$$
so if you define $F = \sum\_{l=1}^\infty (LH\_k)^{\*l}$ and $p= H\_k + H\_k \* F$, then
$$ L p = LH\_k + (L H\_k)\*F - L F = LH\_k + \sum\_{l=2}^\infty (LH\_k)^{\*l} - \sum\_{l=1}^\infty (LH\_k)^{\*l} = 0,$$
wh... | 4 | https://mathoverflow.net/users/16702 | 255455 | 115,474 |
https://mathoverflow.net/questions/255451 | 2 | If $f:X\rightarrow Y$ is a morphism of algebraic schemes with reduced and projective source $X$ and nonsingular target $Y$, then first order deformations of the map $f$ with both source and target fixed is in bijection with $H^{0}(X,f^{\*}T\_Y)$.
How do I understand the condition that $Y$ is nonsingular? Is there a ... | https://mathoverflow.net/users/16356 | Deformation of maps to singular target | The first order deformations are in general (say $Y$ has no embedded points in $f(X)$ and the image of every irreducible components of $X$ intersects the smooth locus of $Y$) parametrized by $\mathrm{Hom}\_X(f^\*\Omega\_Y,\mathscr O\_X)$. If $Y$ is smooth, then $\Omega\_Y$ is locally free, so this group is the same as ... | 3 | https://mathoverflow.net/users/10076 | 255456 | 115,475 |
https://mathoverflow.net/questions/255415 | 4 | Consider the usual sequential modifications of topologies (spaces) in the categories of topological spaces $\text{Top}$, topological vector spaces $\text{TVS}$ and locally convex spaces $\text{LCS}$ :
Let $X$ be a real vector space. All of the occurring topologies are assumed to be Hausdorff.
If $\tau$ is a topology,... | https://mathoverflow.net/users/58682 | Productivity of certain sequential subcategories of topological vector spaces | I think the result of [Ferrer, Morales, Sanchez-Ruiz] is simply wrong. Mazur announced in
*Sur la structure des fonctionelles linéaires dans certains espaces (L)*, Ann. Soc. Polon. Math. 19 (1946), 241
that every sequentially continuous linear functional on $\mathbb{R}^I$ is continuous iff $|I|$ is not a (two-value... | 2 | https://mathoverflow.net/users/99234 | 255469 | 115,480 |
https://mathoverflow.net/questions/255122 | 6 | The tensor square of the Leech lattice is an even unimodular lattice of dimension 576 which, unless I am very mistaken, has no roots. Its automorphism group contains a group of shape $2 \cdot \mathrm{Co}\_1^{\times 2} : 2$, but I expect it is larger than that. I would like to understand what I can about this group, e.g... | https://mathoverflow.net/users/78 | What is the automorphism group of the tensor square of the Leech lattice? | This post provides details for the answer outlined by Noam Elkies in the comments. It is Community Wiki, so anyone can edit it to improve it.
**Answer:** Let $\Lambda$ denote the Leech lattice. The full automorphism group of the lattice $\Lambda^{\otimes 2}$ is the group $2\cdot \mathrm{Co}\_1^{\times 2}:2 = (\mathrm... | 3 | https://mathoverflow.net/users/78 | 255470 | 115,481 |
https://mathoverflow.net/questions/255425 | 11 | First of all, I have to mention that I'm truly sorry if this question would seem inappropriate for this site for some people. Still, I think it is better to ask here rather on math.stackexchange.
I recently grasped basic notions of Grothendieck universes for category theory. We most use the axioms of Zermelo and Fra... | https://mathoverflow.net/users/83143 | What is the use of Grothendieck universes in category theory? | Universes provide a convenient framework for stacking many levels of largeness. But a class in [NBG](https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory) cannot contain any other class. The most basic application which shows that universes are more convenient is that the category of ... | 5 | https://mathoverflow.net/users/98306 | 255473 | 115,483 |
https://mathoverflow.net/questions/255474 | 5 | *Note: I posted my question on [math.stackexchange](https://math.stackexchange.com/questions/2016550/integral-of-the-square-of-the-exponential-of-a-matrix?noredirect=1#comment4140463_2016550) but got no answer. That is why I am asking it here.*
Let $A$ be a $n\times n$ square matrix such that the real part of all eig... | https://mathoverflow.net/users/90045 | Integral of the entrywise square of the exponential of a matrix | Inspired strongly by Anthony's answer, here is a formula that works for arbitrary $A$. Let $M$ be the $n^2 \times n^2$ square matrix given by $$M= A \otimes I\_n + I\_n \otimes A\_n$$
i.e. in terms of indices
$$ M\_{ij,kl}=A\_{i,k} \delta\_{j,l} + \delta\_{i,k}A\_{j,l}$$
Then because $A \otimes I\_n$ and $I\_n \oti... | 10 | https://mathoverflow.net/users/18060 | 255484 | 115,488 |
https://mathoverflow.net/questions/255500 | 1 | I asked the following question in the mathstackexchange but I did not get an answer, probably the level of this question is not appropriate for mathoverflow, so I would like to apologize in advance if it is the case:
Let us consider the algebraic curve $$ f(z,w)=z(z-a^{-1})w^4-(z-a)^3=0 \ \ \ \ \ \ (\*)$$
I want to... | https://mathoverflow.net/users/101575 | Finding the genus of a certain algebraic curve | The genus $g$ of a Riemann surface is found from the Riemann-Hurwitz formula:
$$2g-2=\sum(n\_k-1)-2d,$$
where $d$ is the number of sheets, $n\_j$ are the orders of ramification points.
In your case, $d=4$, and there are $4$ ramification points, all of order $4$. So the genus is $3$.
The proof of the Riemann-Hurwitz f... | 3 | https://mathoverflow.net/users/25510 | 255505 | 115,490 |
https://mathoverflow.net/questions/255498 | 3 | It seems that Lan Nguyen proved [in a preprint on arxiv](https://arxiv.org/abs/1309.7675) of 2013 that the Tate-Shafarevich group of a rational elliptic curve is finite. However, I couldn't find any published version thereof. So is it now known that the Tate-Shafarevich group of a rational elliptic curve is finite?
M... | https://mathoverflow.net/users/13625 | Is the Tate-Shafarevich group of a rational elliptic curve finite? | MO is not the place to discuss the validity of preprints, but I think it is safe to say that the finitiness of the Tate-Shafarevich group for elliptic curves over $\mathbb{Q}$ is considered an open problem for rank $\geq 2$.
As far as I can tell the paper was never published.
| 18 | https://mathoverflow.net/users/43108 | 255506 | 115,491 |
https://mathoverflow.net/questions/255503 | 23 | The proof by Cauchy induction of the arithmetic/geometric-mean inequality is well known. I am looking for a further theorem whose proof is much neater by this method than otherwise.
| https://mathoverflow.net/users/7458 | Is Cauchy induction used for proofs other than for AM–GM? | A nice proof by Cauchy induction can be given for the identity
$$ \|A^n\|=\|A\|^n, $$
which holds for a bounded, self-adjoint operator $A:H\to H$ on a real Hilbert space $(H,\langle\cdot,\cdot\rangle)$. Here $\|\cdot\|$ denotes the operator norm.
Indeed, the inequality $\|A^n\|\le\|A\|^n$ is trivial by submultiplicat... | 42 | https://mathoverflow.net/users/36952 | 255513 | 115,493 |
https://mathoverflow.net/questions/255501 | 3 | Let $G$ be the wreath product $C\_m\wr S\_n$, where $C\_m$ is the cyclic group of order $m$ and $S\_n$ is the symmetric group on $n$ letters. I would like to understand the maximal subgroups of $G$ up to conjugacy. In particular, how many conjugacy classes are there, and how explicitly can they be described?
| https://mathoverflow.net/users/46987 | Maximal subgroups of a generalized symmetric group | The maximal subgroups that contain the base group $C\_m^n$ correspond exactly to the maximal subgroups of $S\_n$. There is a lot known about those (they subdivide into intransitive maximals, imprimitive amximals, and primitive maximals), and they are known explicitly up to $n$ a few thousand, but there is no hope of co... | 6 | https://mathoverflow.net/users/35840 | 255522 | 115,496 |
https://mathoverflow.net/questions/255511 | 27 | This question has been bugging me for a while and I can't seem to make sense of it on a clear conceptual level.
The theory of local rings is given by taking the theory of rings and adding the axioms
\begin{eqnarray}
(0=1) \vdash \bot \\
x + y = 1 \vdash \exists z : (xz = 1) \vee \exists z : (yz = 1)
\end{eqnarray}
Th... | https://mathoverflow.net/users/76299 | The philosophy behind local rings | I'm not sure if this constitutes a full answer, and a lot of it has already been said in some form by HeinrichD in the comments. Because your question is ultimately one of philosophy, I will focus mostly on history and philosophy (and a few applications); not so much the categorical or logical interpretations.
**Hist... | 20 | https://mathoverflow.net/users/82179 | 255528 | 115,498 |
https://mathoverflow.net/questions/255534 | 1 | Let $C$ be a locally presentable category. Is it true that the category of pointed objects $C^{\*/}$ in $C$ is also locally presentable?
| https://mathoverflow.net/users/84563 | Pointed objects in a Presentable Category | Yes. The category of pointed objects is the category of algebras of an accessible monad on $C$, namely the monad $c \mapsto \ast \sqcup c$ (that is induced by the unique monoid structure on $\ast$ with respect to the monoidal product $\sqcup$), and such a category of algebras is again locally presentable. See Adámek-Ro... | 4 | https://mathoverflow.net/users/2926 | 255535 | 115,500 |
https://mathoverflow.net/questions/255499 | 13 | In their seminal work on "representation stability" (<https://arxiv.org/abs/1008.1368>), Church and Farb deal with a stabilization procedure for representations (up to isomorphism) (over $\mathbf C$) of the symmetric group. It's a stabilization in the sense that it produces a representation of $\mathfrak S\_{n+1}$ out ... | https://mathoverflow.net/users/5239 | Stabilization of representation of the symmetric group | There are two conceptual approaches to what Church-Farb did: the theory of "central stability" that I defined in my paper
A. Putman,
Stability in the homology of congruence subgroups,
Invent. Math. 202 (2015), no. 3, 987-1027.
and Church-Ellenberg-Farb's notion of FI-modules from their paper
T. Church, J. Ellen... | 9 | https://mathoverflow.net/users/317 | 255537 | 115,501 |
https://mathoverflow.net/questions/255282 | 2 | Let $C$ be a category and let $F:C\rightarrow D$ be a functor with $D$ locally presentable and cartesian closed. When does the Yoneda extension $\widehat{F}=Lan\_{y} F:[C^{op},Set]\rightarrow D$ preserve finite products?
| https://mathoverflow.net/users/84563 | Yoneda extension preserving finite products? | (Update: the following answer applies only in the case that $C$ has finite products.)
The left Kan extension $\hat{F}$ preserves finite products just when $F$ does.
One direction is easy since $F \cong \hat{F} \circ Y$ and $Y$ preserves finite products.
The converse is an old result of Borceux and Day from their... | 5 | https://mathoverflow.net/users/8751 | 255541 | 115,502 |
https://mathoverflow.net/questions/255533 | 4 | Suppose that $f \in C^{\infty}\_c ( \mathbb{R}^2 )$, i.e. $f$ is a $C^{\infty}$ function with compact support defined on $\mathbb{R}^2$. The following link
[Approximation of smooth compactly supported functions on $\mathbb{R}^2$ using sums of products of one variable functions](https://mathoverflow.net/questions/1661... | https://mathoverflow.net/users/62049 | Approximation of a $C^{\infty}_c$ function by tensor products | Let $Q$ be a square with side $T$ which contains the support. Extend your function
to a doubly periodic one with periods $(0,T)$ and $(T,0)$. Then expand into the double Fourier series:
$$f(x,y)=\sum a\_{m,n}e^{2\pi inx/T}e^{2\pi im/T}.$$
This series can be differentiated term-by-term.
As $f\in C^\infty$ this converges... | 5 | https://mathoverflow.net/users/25510 | 255542 | 115,503 |
https://mathoverflow.net/questions/237937 | 7 | The Banach-Dieudonné theorem states that if $X$ is a metrizable locally convex Hausdorff space then the equicontinuous weak-\* topology $ew^\*$ on $X'$ coincides with the topology of precompact convergence and is therefore a locally convex topology. ($ew^\*$ is the final topology on $X'$ coinduced by the inclusions of ... | https://mathoverflow.net/users/58682 | Is the equicontinuous weak-star topology locally convex on the dual of an LF-space? | An interesting paper on spaces for which the $ew^\*$ topology coincides with the topology of precompact convergence is [S-spaces and the open mapping theorem](http://msp.org/pjm/1962/12-1/p23.xhtml) by Taqdir Husain. He calls such spaces S-spaces, a term that doesn't appear to have become standard.
Proposition 2 of t... | 6 | https://mathoverflow.net/users/99234 | 255549 | 115,507 |
https://mathoverflow.net/questions/255556 | 0 | Let $\text{Mat}(\mathbb{N},\{0,1\})$ be the set of all maps $A:\mathbb{N}\times\mathbb{N}\to \{0,1\}$. We define a matrix multiplication for $A, B\in \text{Mat}(\mathbb{N},\{0,1\}$) and $m,n\in\mathbb{N}$ by $$(A B)(m, n) = 0 \text{ if } \sum\_{i=1}^\infty A(m, i)B(i, n) < \infty \text{ and } (AB)(m, n) = 1 \text{ othe... | https://mathoverflow.net/users/8628 | Nilpotent infinite binary matrices | I'm assuming you mean entries are independent and 0 or 1 with probability .5 each.
I think the AA is identically 1, as $P(A\_{ij} = 1, A\_{jk} = 1) = \frac 14 $ as long as $ i \ne k$ and therefore happens i.o with probability 1.
| 2 | https://mathoverflow.net/users/nan | 255560 | 115,510 |
https://mathoverflow.net/questions/255550 | 8 | Studying curves in the Euclidean three dimensional space, one usually defines the curvature and the torsion of a curve. If I am not missunderstanding the thing, I guess that a curve has zero torision if and only if it is contained in a plane.
The curvature is generalised by the second fundamental form of a submanifol... | https://mathoverflow.net/users/48866 | Torsion of submanifolds | The torsion of a curve $C$ *in* $\newcommand{\bR}{\mathbb{R}}$ $\bR^3$ is a a manifestation of two ``freak'' low dimensional accidents: the curve is $1$-dimensional and it lives in a $3$-dimensional Euclidean space.
Being $1$-dimensional allows us, after fixing an orientation, to choose an orthonormal basis of the ta... | 9 | https://mathoverflow.net/users/20302 | 255564 | 115,511 |
https://mathoverflow.net/questions/255573 | -1 | This is an additional question to [that question](https://mathoverflow.net/questions/255556/nilpotent-infinite-binary-matrices), inspired by Leonid Petrov.
Let $\text{Mat}(\mathbb{N},\{0,1\})$ be the set of all maps $A:\mathbb{N}\times\mathbb{N}\to \{0,1\}$. We define a matrix multiplication for $A, B\in \text{Mat}(\... | https://mathoverflow.net/users/8628 | Is this operation on infinite matrices associative? | No, even $(A^2)A\ne A(A^2)$ in general. We consider $A$ as an incidence matrix of an infinite graph (with possible loops, and directed, but in our example it is loop-less and undirected). Consider vertices 1, 2 and join them but infinitely many paths $1x\_{ij}y\_j2$, where $i,j=1,2,\dots$, and all vertices $y\_j,x\_{ij... | 1 | https://mathoverflow.net/users/4312 | 255576 | 115,513 |
https://mathoverflow.net/questions/255572 | 4 | Consider the integers modulo $m$, for composite $m$. Then an element $x$ only has a multiplicative inverse if it is relatively prime to $m$. Therefore, it is possible to have a set of more than one element, so that no element in the set can be written as a linear combination of the others; call such a set linearly inde... | https://mathoverflow.net/users/90005 | Maximum size of linearly independent subsets of $\mathbb{Z}/m\mathbb{Z}$ | A subset $S$, $|S|>1$, is linearly independent if and only for any $a\in S$ there exists a prime $p|m$ such that $\nu\_p(a)<\nu\_p(b)$ for all $b\in S\setminus \{a\}$. Here $\nu\_p(a)$ denotes the maximal $k$ such that $p^k$ divides both $a$ and $m$. Indeed, if it is the case, then $a$ is not a linear combination of ot... | 6 | https://mathoverflow.net/users/4312 | 255577 | 115,514 |
https://mathoverflow.net/questions/255538 | -1 | It is well-known (see Allouche monography for example) that if $f$ is an algebraic function over $K(X)$ then $f'$ is also algebraic. I wonder whether $f$ and $g$ are algebraically dependent, then $f'$ and $g'$ are also algebraically dependent. I think that it is false, but I do have no counterexample, neither a proof t... | https://mathoverflow.net/users/33128 | Preservation of algebraically dependence for derivative | The answer is no.
Example. Let $f=\Gamma(z)$, Euler's Gamma function, and $g(z)=\Gamma^2(z)$. Evidently they are
algebraicaly dependent. Then $g'=2ff'$.
Suppose that $f'$ and $g'$ are algebraically dependent, that is there is an equation $F(f',g')=0$ where $F$ is a polynomial with constant coefficients.
Then $F(f',2f... | 5 | https://mathoverflow.net/users/25510 | 255588 | 115,518 |
https://mathoverflow.net/questions/255587 | 6 | May not be a research-level problem for an expert, but non-trivial for a non-expert: why do we have
$$ \sum\_{k=0}^n \frac{(-1)^k}{2k+1} \binom{n}{k} = \frac12 \int\_0^\pi (\sin x)^{2n+1} dx $$
and what is the asymptotic / good lower bound for this as $n$ grows? Thanks!
| https://mathoverflow.net/users/47453 | A relation between a binomial sum and a trigonometric integral | You suspected right. It's easy. Convert $\sin^{2n+1}x=(1-\cos^2x)^n\sin x$ and integrate by substitution and apply binomial expansion
$$\int\_0^{\pi}\sin^{2n+1}x\,dx=2\int\_0^1(1-u^2)^ndu=2\sum\_{k=0}^n(-1)^k\binom{n}k\int\_0^1u^{2k}du=2\sum\_{k=0}^n\frac{(-1)^k}{2k+1}\binom{n}k.$$
There is more to this: the RHS has a ... | 11 | https://mathoverflow.net/users/66131 | 255595 | 115,523 |
https://mathoverflow.net/questions/255485 | 1 | Suppose we know $deg(m(x))=n-1=deg(f\_1(x))=deg(f\_2(x))$.
Suppose we know $C\_1(x),C\_2(x)$ where $deg(C\_i)=n$.
Then given $n$ values of $$C\_1(x)(x+1)m(x) +C\_1(x)(x+2)f\_1(x)\in\Bbb F\_q[x]$$ and $n$ values of $$C\_2(x)(x+2)m(x) +C\_2(x)(x+5)f\_2(x)\in\Bbb F\_q[x]$$ where $f\_1(x),f\_2(x)$ are unknown can we ob... | https://mathoverflow.net/users/nan | A polynomial recovery problem | No, you gain no information about $m(x)$ whatsoever.
In fact, suppose we know
$$ A\_{i,j} m(x\_{i,j}) + B\_{i,j} f\_i (x\_{i,j}) = y\_{i,j}$$
for $i=1,2$ and $j=1, \ldots, n.$ Suppose we know $x\_{i,j}$ and $y\_{i,j}$ as well as $A\_{i,j}$ and $B\_{i,j}$.
If $m(x)$ is any arbitrary function, we can interpolate degr... | 3 | https://mathoverflow.net/users/22512 | 255598 | 115,524 |
https://mathoverflow.net/questions/255597 | 2 | Hi everyone: Let $ \omega $ be a bounded open set in $ \mathbb{R}^{q} $, $ q\geq 2 $, and $ E $ a subset of the boundary $ \partial\omega $ that has harmonic measure zero in $ \omega $. Let $ V $ be the interior of the closure of $ \omega $. We know that some points of $ E $ can be inside $ V $. If I take an open ball ... | https://mathoverflow.net/users/100746 | Harmonic measure | No. Let $V$ be the disk $|z|<2$ in the complex plane, $E$ the arc $\{ e^{it}/2:|t|\leq1\}\subset V$.
Now draw a simple arc $\gamma(t),\; 0\leq <1$, $\gamma(0)=1$, $\gamma\backslash\gamma(0)\subset V\backslash E$, and $\gamma$ is spiraling around $E$,
so that the limit set of $\gamma(t)$ as $t\to 1$ equals $E$.
And let ... | 2 | https://mathoverflow.net/users/25510 | 255607 | 115,528 |
https://mathoverflow.net/questions/255570 | 1 | An operational quantity is a procedure which determines, for every pair $X,Y$ of infinite-dimensional Banach spaces, a map from $\mathcal{L}(X,Y)$ into the non-negative numbers.
Given two operational quantities $a$ and $b$, we will write $a\leq b$ if for any infinite-dimensional Banach spaces $X,Y$ and $T\in \mathcal... | https://mathoverflow.net/users/41619 | Operational quantities characterizing strictly singular operators and strictly cosingular operators | The quantities $\Delta$ and $\tau$ are better known in the literature as $sin$ and $sj$, respectively. In the paper "Note on Operational Quantities and Mil'man Isometry Spectrum" by Gonzalez/Martinon (Rev. Acad. Canar. Cienc. 3, 1991, pp103–111), it is shown that $sin$ and $sj$ fail to be equivalent---in particular, th... | 0 | https://mathoverflow.net/users/73784 | 255615 | 115,531 |
https://mathoverflow.net/questions/255602 | 0 | Chen Jing Run proved that every large enough even integer is either the sum of two primes or the sum of a prime and a semi-prime (that is, the product of two primes). Golbach's conjecture states that every even integer greater than 3 is the sum of two primes. Harald Helfgott (happy birthday to him) proved that every od... | https://mathoverflow.net/users/13625 | Upper bound for the number of even numbers sum of a prime and a semi-prime not fulfilling Goldbach's conjecture | If I understand correctly, $\overline{\mathcal{G}}(x)$ is the usual exceptional set $E(x)$ except for possibly finitely many exceptions in the range $(4\cdot 10^{18},e^{e^{36}})$, where the Goldbach conjecture has not been verified, and an effective version of Chen's theorem does not apply. With that in mind, a classic... | 5 | https://mathoverflow.net/users/43108 | 255619 | 115,533 |
https://mathoverflow.net/questions/255605 | 1 | So I've been fiddling around with the cauchy product of sequences lately, and am curious about a little identity I've found (which I'm sure is ubiquitous in finite differences, as I can't be the only one to have thought of it). I have been having trouble proving it, but the idea is rather straight forward. I think I mi... | https://mathoverflow.net/users/nan | Iterated sums--something like a differsum | Ok. Fixing the typo in the post, we can prove the claim without much trouble.
After fixing the typo, the question is really asking to prove the following for all nonnegative integers $n$ and all complex $s$ and $q$:
$$
\sum\_{k=0}^{n} {k+s-1 \choose k} {n-k + q-1 \choose n-k} = {n+q+s-1 \choose n},
$$
where ${z \choo... | 0 | https://mathoverflow.net/users/22512 | 255627 | 115,535 |
https://mathoverflow.net/questions/255639 | 2 | Please consider a tree graph. There is one unique path connecting any two vertices.
However, I wonder how to address the following question:
Starting from a generic tree, is there an algorithmic way to connect any two nodes of the tree with $M$ unique paths such that no two paths contain the same edge? How can the ... | https://mathoverflow.net/users/19724 | Modifying tree Graphs | Yes. See the paper [Minimum augmentation of a tree to a K-edge-connected graph](http://onlinelibrary.wiley.com/doi/10.1002/net.3230180104/abstract) by Ueno, Kajitani and Wada.
| 4 | https://mathoverflow.net/users/2233 | 255640 | 115,540 |
https://mathoverflow.net/questions/255646 | 3 | I am interested in partitions of an integer n which do not have 1 and 2 as its part. That is, I am looking to write
$$n=3^{n\_3}4^{n\_4}\cdots.$$
Now, I want to count this. And any other information about generating function or recursive formula will be helpful.
| https://mathoverflow.net/users/69977 | Restricted partitions without 1s and 2s | This is easy with generating functions. Let $a(n)$ be such a count. Then,
$$\sum\_{n=0}^{\infty}a(n)q^n=\prod\_{k=3}^{\infty}\frac1{1-x^k}.$$
For more information, you may like to [explore this](http://oeis.org/search?q=0%2C+0%2C+1%2C+1%2C+1%2C+2%2C+2%2C+3%2C+4%2C+5%2C+6%2C+9%2C+10%2C+13%2C+17%2C+21%2C+25&language=engl... | 6 | https://mathoverflow.net/users/66131 | 255648 | 115,543 |
https://mathoverflow.net/questions/255652 | 2 | Say I have $N = n\_1 + \ldots + n\_d$ balls, with $n\_1$ balls of color $c\_1$,... , $n\_d$ balls of color $c\_d$. How many ways can I arrange all $N$ balls in a sequence so that no two balls of the same color are adjacent? (Some choices of the numbers $n\_1$, ..., $n\_d$ are inadmissible.) I apologize if this is a bas... | https://mathoverflow.net/users/101646 | Sequences without repeated objects | Let $g\_i(x\_1,\dots,x\_d)$ be a sum of $x\_1^{n\_1}\dots x\_d^{n\_d}$ over all such sequences in which $c\_i\geqslant 1$ and the last ball has color $c\_i$. Then $g\_i=x\_i+x\_i \sum\_{j\ne i} g\_j$. The generating function for what you ask about is $g:=1+g\_1+\dots+g\_d$.
Solve above linear system of $d$ equations... | 3 | https://mathoverflow.net/users/4312 | 255659 | 115,547 |
https://mathoverflow.net/questions/255643 | 3 | Let $A$ be an Abelian group, and let $G$ be a finite group which acts on a finite set $M$, such that a subgroup $H$ acts trivially on $M$ and $G/H$ acts freely on $M$. I can define a $G$-module $A^M$ (i.e. Abelian group with $G$-action), with the $G$-action inherited from $M$. By restricting onto $H$ I get a homomorphi... | https://mathoverflow.net/users/81457 | Group cohomology with coefficients in a permutation module | I'm not sure I understand your intuition, but the statement is true. It's a special case of [Shapiro's Lemma](https://en.wikipedia.org/wiki/Shapiro's_lemma): if $H$ is a subgroup of $G$, $A$ is a $\mathbb{Z}H$-module, and $A^G$ is the coinduced module, then
$$H^\ast(G,A^G)\cong H^\ast(H,A).$$
| 10 | https://mathoverflow.net/users/22989 | 255660 | 115,548 |
https://mathoverflow.net/questions/255673 | 27 | The title is admittedly noninformative but I could not figure out how to squeeze into it the description of the object I am interested in. Judge by yourself.
I have an alphabet on $d$ symbols. I want to build a rectangular array with $pd$ rows and $qd$ columns filled with them in such a way that each row contains $p$... | https://mathoverflow.net/users/41291 | What is the name of this combinatorial object and place to read about it? | What you're asking for is a code with two non-usual restrictions. It's a code over alphabet size $d$. You want each code word to have length $pd$ and you want to have $qd$ total code words.
The parameter you call $n$ is equal to $pd - \Delta$, where $\Delta$ is called the distance of your code. You want to minimize $... | 37 | https://mathoverflow.net/users/22512 | 255677 | 115,556 |
https://mathoverflow.net/questions/255664 | 3 | I am trying to compute the cohomology of the homotopy fibre $F$ of a continuous map $f \colon S^{2n-1} \to S^n$ ($n$ even and non-zero Hopf invariant). It is easy to see that through the Serre spectral sequence the non zero groups are (with integer coefficients)
$$
H^{n-1}(F) = \mathbb Z,\quad H^k(F) = \mathbb Z\_m
$$... | https://mathoverflow.net/users/101655 | The homotopy fibre of an map $f \colon S^{2n-1} \to S^n$ | In the following, any identity may actually be up to a sign. I don't want to keep track of that.
We are thinking about the fibration sequence
$$ \Omega S^n \to F \to S^{2n-1} \xrightarrow{f} S^n.$$
It may be little easier to think about the homology Serre spectral sequence of $\Omega S^n\to F\to S^{2n-1}$ rather than... | 5 | https://mathoverflow.net/users/437 | 255681 | 115,557 |
https://mathoverflow.net/questions/255680 | -1 | First, I apologize if the question doesn't fit this forum.
In a thread about Galois theory on a French math forum, I read "le sextique résolvent" and the spelling looks odd to me. I would have expected an ending in "ant" for this notion, as the derivate of a present participle.
I am therefore looking for references i... | https://mathoverflow.net/users/13625 | Resolvent in French | This should read "la résolvante sextique".
Resolvents were used before Galois - and are still used after him. For example the cubic resolvent that appears in the solution of the fourth degree equation is sometimes attributed to Lagrange. In fact Lagrange had already studied the permutations of roots of a polynomial e... | 3 | https://mathoverflow.net/users/10696 | 255682 | 115,558 |
https://mathoverflow.net/questions/255678 | 1 | Hi everyone: Let $ \Omega $ be a bounded open set of $ \mathbb{R}^{N} $, $ N\geq2 $, and $ F\subset \Omega $ with empty interior. Suppose there exists a superharmonic function $ u $ on $ \Omega\setminus F $ such that
$$ \lim\_{x\rightarrow y}u(x)=+\infty $$
for all $ y\in F $. Now, we define $ w(x) $ to be equal to $ ... | https://mathoverflow.net/users/100746 | Is this a superharmonic function? | First you have to assume that $F$ has measure zero, since a superharmonic function is locally integrable.
Under this assumption, setting $u=\infty$ on $F$ results in a superharmonic function. To see this, note that by the limit assumption
$$\lim\_{x\to y}u(x)=\infty$$
for all $y\in F$, the resulting function is stil... | 2 | https://mathoverflow.net/users/101287 | 255691 | 115,561 |
https://mathoverflow.net/questions/255411 | 3 | Let $R$ be an arbitrary ring with maximal ideal $I$, such that the quotient ring $\frac{R}{I}$ is finite. For each integer $i$, we define the quotient ring number as below:
$$N\_i(R,I)=\sum\_{a\in \frac{R}{I}}{a^i}.$$
It is easy to see that if $\frac{R}{I}=\mathbb{F}\_q$ be a finite field with $q$ elements, then we h... | https://mathoverflow.net/users/19885 | Quotient Ring number | The definition of $N\_i(R,I)$ depends only on $R/I$, which is a ring isomorphic to $M\_n(q)$ as HeinrichD has already observed.
Thus, the question is: what is the sum
$$
S = \sum\_{A\in M\_n(q)} A^i \quad?
$$
The answer that has been proposed is: either $0$ or $-I$.
This answer is correct. It is proved in
Brawley, ... | 6 | https://mathoverflow.net/users/75735 | 255693 | 115,563 |
https://mathoverflow.net/questions/224641 | 4 | Let's work in a set theory without assuming AC (for instance, but not necessarily, ZF). Fix a set $k$ satisfying $k\times k \simeq k$, and consider its powerset $X = 2^k$. I have a technical condition that is satisfied whenever $X$ is well-orderable, but I can't tell what other examples there might be. Excuse the pecul... | https://mathoverflow.net/users/4177 | How close to being well-orderable does this make my powerset? | Prompted by the discussion in comments above, Asaf wrote a blog post [Cofinality and the axiom of choice](http://karagila.org/2015/cofinality-and-the-axiom-of-choice/) that gave the following example of a model of ZF:
>
> It is consistent that every non well-orderable set has cofinality 2. This was shown by Monro (... | 3 | https://mathoverflow.net/users/4177 | 255710 | 115,571 |
https://mathoverflow.net/questions/255694 | 5 | Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$.
So here is my list of additional "axioms":
(1) $A$ is an AB5 category (and so, Grothendieck abelian).
(2) There is an exact conservat... | https://mathoverflow.net/users/2191 | On various relations between "additional axioms" for AB4 and Grothendieck abelian categories | I don't think (3) implies (1).
For example, the opposite category of the category of abelian groups satisfies (3), but is not AB5.
| 5 | https://mathoverflow.net/users/22989 | 255734 | 115,584 |
https://mathoverflow.net/questions/255726 | 3 | (Adapted from [Rockafellar](http://www.math.washington.edu/~rtr/papers/rtr066-MonoOpProxPoint.pdf))
>
> **Definition**: Let $H$ be a real Hilbert space with inner product $\langle \cdot
> ,\cdot \rangle$. A function $T: H \to H$ is said to be a monotone
> operator if \begin{equation} \langle z - z', Tz-Tz'\rangle... | https://mathoverflow.net/users/40747 | Partial results on composition of operators such that overall composition is monotone | For $0\le \theta\le\pi/2$, say that $T:H\to H$ is $\theta$-monotone (therefore monotone) iff for all $x$, $y$ in $H$, $(x-y,Tx-Ty)\ge \|x-y\|\,\|Tx-Ty\|\cos\theta$, that is, $x-y$ and $Tx-Ty$ make an angle not larger than $\theta$. Then, of course, if $T\_i$ is a $\theta\_i$-monotone operator for $i=0,\dots,1$ with $\t... | 1 | https://mathoverflow.net/users/6101 | 255745 | 115,587 |
https://mathoverflow.net/questions/255666 | 8 | Let $V$ be an $(N+1)$-dimensional vector space with an action of the symmetric group $S\_n$, such that $V$ is an irreducible $S\_n$ -module.
Let $\{p\_1,...,p\_h\}\in \mathbb{P}(V)$ be $h\geq N+2$ points such that $S\_n$ acts transitively on $\{p\_1,...,p\_h\}$.
Is it true then that there exist $N+2$ points $p\_{... | https://mathoverflow.net/users/nan | Irreducible $S_n$-modules and $S_n$-actions on projective spaces | **EDIT**: Actually, I made this way too difficult.
Points in projective space are in general linear position if the corresponding lines are the multiples of a basis of $V$. Thus, the question is just if the lines corresponding to $p\_1,\dots, p\_h$ span the vector space $V$ (since in this case, some subset will corr... | 5 | https://mathoverflow.net/users/66 | 255747 | 115,589 |
https://mathoverflow.net/questions/255748 | 3 | If I understand correctly, Stephen Simpson, in his book *Subsystems of Second Order Arithmetic*, deems second-order arithmetic as a two-sorted first-order theory. If this is correct, then it seems reasonable to infer that one could use forcing to add generic sets of integers ('reals') and form models of $SOA$ that cont... | https://mathoverflow.net/users/20597 | Forcing in Second-Order Arithmetic | Re: the sets of natural numbers forming a proper class, *they always do* in second-order arithmetic: there *is no sort* for sets of sets of numbers! So there's no distinction between collections of sets of naturals which are sets, and collections of sets of naturals which aren't sets. All there is to a model of $RCA\_0... | 7 | https://mathoverflow.net/users/8133 | 255754 | 115,590 |
https://mathoverflow.net/questions/254133 | 1 | In Koepke's paper, "Turing Computations On Ordinals", one has the following (well-known) result:
>
> A set $x$ is ordinal computable from a finite set of ordinal parameters if and only if it is an element of the constructible universe.
>
>
>
In Sacks' survey article, "E-Recursive Intuitions", one finds these (... | https://mathoverflow.net/users/20597 | How are Koepke's ordinal computability and E-recursion related? | I don't quite know what your main question is asking, or what (3) means, but (1) and (2) seem to have straightforward answers:
* Any nontrivial forcing makes $L$ non-E-r.e.: if $M\models ZFC$ and $M[G]$ is a nontrivial forcing extension of $V$, then $M[G]\models \neg\mathsf{(V=L)}$, and so (from the perspective of $M... | 2 | https://mathoverflow.net/users/8133 | 255760 | 115,592 |
https://mathoverflow.net/questions/255100 | 5 | I'm currently reading through Larson's book on the stationary tower and a point confused me.
Let $\delta$ be Woodin and let $j:V\to M\subseteq V[G]$ be an elementary embedding associated with the (full) stationary tower $\mathbb P\_{<\delta}$. Then $V[G]\models {^{<\delta}}M\subseteq M$. This seems close to $\operato... | https://mathoverflow.net/users/38602 | The stationary tower and supercompactness | Apparently this has a name. To any large cardinal notion, there is a *virtual* variant, in which the elementary embeddings in question lie in some generic extension. Reformulating my scenario, I pointed out that a Woodin $\delta$ consistency-wise implies a virtually $\theta$-supercompact for every $\theta<\delta$. Usin... | 2 | https://mathoverflow.net/users/38602 | 255773 | 115,597 |
https://mathoverflow.net/questions/255592 | 7 | Let $q=p^f$ be a prime power such that $q \equiv 1 \pmod 5$. According to the list of irreducible (complex) character degrees of $SL(5, q)$ in Frank Luebeck's homepage ([here](http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/DegMult/tables/DegreesAndMultiplicitiesA4sc_1mod5.txt)), $SL(5, q)$ has 20 irreducible charac... | https://mathoverflow.net/users/99323 | On existence of a certain irreducible character of $SL(5, q)$ | [The author or this question made me aware of this thread, so I send the answer here.]
The description in the question is almost correct, except when it comes to the centralizer of the semisimple element $s$ in $PGL\_5(q)$. For $q \equiv 1 \pmod 5$ the characters of $SL\_5(q)$ of degree $1/5(q^2+1)(q^2+q+1)(q+1)^2(q−... | 10 | https://mathoverflow.net/users/61095 | 255790 | 115,602 |
https://mathoverflow.net/questions/255785 | 9 | The problem asks to prove that the Diophantine equation $x^{3}+y^{3} = (x+y)^{2}+(xy)^{2}$ does not have any solutions in natural numbers $x, y$.
I believe that this problem appeared in the section Задачи наших читателей of the Soviet magazine [Квант](http://kvant.mccme.ru/) somewhere between the first issue of 1980 ... | https://mathoverflow.net/users/1593 | On the exact reference of a cute Diophantine problem | It appeared in issue 8 of 1984 at the page 34.You can download this issue from here: <http://kvant.mccme.ru/oblozhka_djvu3.htm>
| 11 | https://mathoverflow.net/users/32389 | 255792 | 115,603 |
https://mathoverflow.net/questions/255737 | 13 | I've been reading some papers on Igusa zeta functions, and they seem to be implicitly using a "quantitative version" of Hensel's Lemma, which also asserts the number of lifts of a $\mathbb{Z}/p\mathbb{Z}$-point to a $\mathbb{Z}/p^k\mathbb{Z}$-point. I'm looking for something like the following:
>
> Let $X$ be a smo... | https://mathoverflow.net/users/5101 | A quantitative version of Hensel's Lemma | Let me try a very explicit proof. Consider the problem of taking a solution modulo $p^k$ and lifting to solutions modulo $p^{k+1}$. For this, we may assume that $X \subset \mathbb{A}^m$ is smooth and affine of dimension $n$, given by polynomials $f\_1, \dotsc, f\_r \in \mathbb{Z}\_p[x\_1, \dotsc, x\_m]$. Let $\mathbf{a... | 10 | https://mathoverflow.net/users/3753 | 255798 | 115,605 |
https://mathoverflow.net/questions/254239 | 3 | Problem:
Given:
A set of points (their coordinates) on a sphere's surface.
Goal:
Connect them forming triangles, such that:
a)The union of triangles cover the whole surface of the sphere.
b)The length of the edge AB,BC,CA in every triangle ABC corresponds to the geodesic distance between the appropriate point... | https://mathoverflow.net/users/28070 | Cover sphere's surface with triangles of geodesic distance edges | What you are looking for is a geodesic triangulation of the sphere with a given vertex set $v\_1, \ldots, v\_n$. The following three conditions are equivalent:
1. Such a triangulation exists.
2. The origin lies in the interior of the convex hull of $v\_1, \ldots, v\_n$.
3. Every open hemisphere contains at least one ... | 6 | https://mathoverflow.net/users/98590 | 255802 | 115,606 |
https://mathoverflow.net/questions/255801 | 3 | Is there an infinite cardinal $\kappa$ and a set $\frak{E}$ of subsets of $\kappa$ with the following properties:
1. $|e\cap f| \leq 1$ for $e,f\in {\frak E}$ with $e\neq f$, and
2. $|\frak{E}| > \kappa$
?
| https://mathoverflow.net/users/8628 | "Strongly" almost disjoint subsets | No. For each element $x \in \kappa$, let $g(x)$ be the set of elements in $\frak{E}$ that contain $x$. By assumption, for all $x \in \kappa$, we have $|g(x)| \leq \kappa$. We may clearly assume $\emptyset \in \frak{E}$. But now, $\frak{E}$=$\{\emptyset\} \cup \bigcup\_{x \in \kappa} g(x)$, and so $|\frak{E}|$$\leq \kap... | 5 | https://mathoverflow.net/users/2233 | 255805 | 115,608 |
https://mathoverflow.net/questions/255141 | 14 | Let $F(n, l, i, j)$ be the cardinality of the set
\begin{eqnarray\*}
\{(k\_1, \cdots, k\_n)\in\mathbb{Z}^{\oplus n}|0\leq k\_r\leq l-1\text{ for }1\leq r\leq n\text{, }k\_1+\cdots+k\_n=lj-i\}.
\end{eqnarray\*}
Define an $n\times n$ matrix $M(l, n)$ by
\begin{eqnarray\*}
M\_{ij}(l, n)=(-1)^{i+j}F(n, l, i, j).
\end{eqna... | https://mathoverflow.net/users/85722 | Eigenvalues of a matrix with entries involving combinatorics | One can prove this statement along the following lines.
1. Prove that ${\rm Trace}(M(l,n)) = 1+l +\cdots + l^{n-1}$.
2. Prove that $M(l,n)^p = M(l^p,n)$.
Clearly, these statements 1 and 2 together imply that the eigenvalues
are $1,l,\ldots, l^{n-1}$.
We will assume throughout that $l\geq 2$, otherwise the statem... | 10 | https://mathoverflow.net/users/38468 | 255809 | 115,609 |
https://mathoverflow.net/questions/255810 | 9 | Let $A$ be an $n\times n$ complex matrix, and write $A=X+iY$, where $X$ and $Y$ are real $n\times n$ matricies. Suppose that for every square submatrix $S$ of $A$, $|\mathrm{det}(S)|\leq 1$ (i.e., all minors of $A$ are complex numbers with modulus $\leq 1$). This includes the assumption that $|\mathrm{det}(A)|\leq 1$.
... | https://mathoverflow.net/users/27404 | Determinant of the "real part" of a matrix | Bound 1 does not hold already for $n=2$. Take a matrix $A=\pmatrix{e^{ia}&e^{ib}\\e^{ic}&e^{id}}$, it satisfies your conditions if $|a+d-b-c|\leqslant \pi/3$. On the other hand, $X=\pmatrix{\cos a&\cos b\\\cos c&\cos d}$ and $$\det X=\cos a\cos d-\cos b\cos c=\frac12\left(\cos(a+d)+\cos(a-d)-\cos(b+c)-\cos(b-c)\right)\... | 12 | https://mathoverflow.net/users/4312 | 255814 | 115,610 |
https://mathoverflow.net/questions/255820 | 53 | Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a homage to Mac Lane's classic. I'm in no way implying that set theory is not "real mathematics" (whatever that expression m... | https://mathoverflow.net/users/83143 | How should a "working mathematician" think about sets? (ZFC, category theory, urelements) | Set theory provides a foundation for mathematics in roughly the same way that Turing machines provide a foundation for computer science. A computer program written in Java or assembly language isn't actually a Turing machine, and there are lots of good reasons not to do real programming in Turing machines - real langua... | 72 | https://mathoverflow.net/users/8991 | 255824 | 115,612 |
https://mathoverflow.net/questions/255813 | 6 | I was wondering if the prime avoidance lemma is very useful or just a nice result. So far I know just only one application: let $R$ be a commutative noetherian ring and $I$ be a proper ideal of $R$. If $I$ consists only of zero divisors of $R$, then $I$ is contained in some associated prime ideal of $(0)$.
So my ques... | https://mathoverflow.net/users/97665 | Applications of the prime avoidance lemma | Prime avoidance can be used to show the following fundamental result on regular sequences:
>
> If $R$ is a noetherian ring, $\mathfrak{a}\subseteq R$ is an ideal, and $M$ is an $R$-module of finite type, then every maximal $M$-sequence in $\mathfrak{a}$ has length equal to the $\mathfrak{a}$-depth of $M$.
>
>
>
... | 4 | https://mathoverflow.net/users/11025 | 255838 | 115,621 |
https://mathoverflow.net/questions/255525 | 6 | Let $V \colon [0,1]\times \mathbb R^d \to \mathbb R^d$ be a Borel vector field which is globally bounded, $V \in L^\infty$.
I am looking for a reference for the following result (which I suppose it is true and already been proved somewhere).
>
> The set of integral curves of $V$, i.e.
>
>
> $\displaystyle \m... | https://mathoverflow.net/users/100976 | Set of integral curves of a vector field | Actually this follows plainly as a consequence of an important classical result in descriptive set theory, namely, the set of all Borel real-valued functions on a metric space is the smallest class containing the continuous functions, and closed under point-wise convergence, that is, the Baire class. (And, of course, n... | 5 | https://mathoverflow.net/users/6101 | 255843 | 115,623 |
https://mathoverflow.net/questions/255862 | -2 | I am wondering why the computable function is defined in the natural number set. Can people give me the answer or some resources that can solve my puzzle.
| https://mathoverflow.net/users/84326 | why do the Computability theory choose the natural number as the object of study? | The natural numbers are often used as a background for (ordinary) computability theory because they are a simple and widely known system of finitary objects. Alternative possibilities would be finite strings of symbols from a fixed finite alphabet (which were, if I remember correctly, used as the background in Shoenfie... | 1 | https://mathoverflow.net/users/6794 | 255863 | 115,630 |
https://mathoverflow.net/questions/255861 | 2 |
>
> Let $f : X \to Y$ be an [fpqc morphism](http://stacks.math.columbia.edu/tag/022B) of schemes, and let $\mathcal{G}$ be an $\mathcal{O}\_{Y}$-module (on the [small Zariski](http://stacks.math.columbia.edu/tag/020T) site) such that $f^{\ast}\mathcal{G}$ is quasi-coherent. Is $\mathcal{G}$ necessarily quasi-coherent... | https://mathoverflow.net/users/15505 | Is "quasi-coherent" an fpqc-local property of modules? | Yes, this is an application of descent theory (with a bit of care). Let $F = f^{\ast}(G)$ on $X$, a quasi-coherent $O\_X$-module by hypothesis. Then for the maps $p\_1, p\_2: X \times\_Y X \rightrightarrows X$ we have an evident composite isomorphism $$\theta: p\_1^{\ast}(F) \simeq (f \circ p\_1)^{\ast}(G) = (f \circ p... | 6 | https://mathoverflow.net/users/81332 | 255870 | 115,633 |
https://mathoverflow.net/questions/255823 | 2 | I am interested in the analog between etale and usual topology. The following question is one example. Any reference would be appreciate.
Let $k$ be an algebraically closed field and $X\_1$ and $X\_2$ be $k$-varieties. My question is as follows: For a given etale neighbourhoods $U,u\to X\_1\times\_k X\_2,(a\_1,a\_2)$... | https://mathoverflow.net/users/101744 | etale neighbourhood of a product | In fact, the example I gave in the comments shows that such $U\_i$ do not exist in general:
**Example.** Let $X\_1 = X\_2 = \mathbb A^1$, with base points $a\_1 = a\_2 = 0$. Let $U = D(xy-1)$ be the complement of the hyperbola $xy = 1$.
Suppose there exist étale neighbourhoods $U\_1$ and $U\_2$ of $0$ such that $U\... | 4 | https://mathoverflow.net/users/82179 | 255873 | 115,635 |
https://mathoverflow.net/questions/255842 | 2 | Let $M$ be a compact $m$ dimensional manifold with boundary $\partial M$.
Assume that $I\_{1}, I\_{2}$ are two linear functionals on $\Omega^{m}(M), \Omega^{m-1}(\partial M)$, respectively.
Assume that we have $I\_{1}(d\alpha )=I\_{2} ( \alpha )$ for every $m-1$ differential form $\alpha$ on $M$.
>
> Are $I\_{1}... | https://mathoverflow.net/users/36688 | An Stokes type theorem for some operations other than integral | If you assume that $M$ is oriented, then up to a multiple $I\_1$ and $I\_2$ are the usual integral. In this case $\partial M$ is oriented, and since this is a manifold without boundary, the integral induces a linear isomorphism $\Omega^{m-1}(\partial M)/d(\Omega^{m-2}(\partial M))\to\mathbb R$. Now for any $\beta\in\Om... | 5 | https://mathoverflow.net/users/64141 | 255878 | 115,637 |
https://mathoverflow.net/questions/255799 | 16 | Let $(M^n,g)$ be a smooth complete Riemannian manifold. Let $p\in M$ be a point. Recall that the cut locus of $p$ is the set of vectors $v$ in the tangent space $T\_pM$ such that $\exp(t v)$ is a minimizing geodesic for any $t\in [0,1]$, but not for $t\in [0,1+\varepsilon )$ for any $\varepsilon >0$.
**Question.** Do... | https://mathoverflow.net/users/16183 | Geometry of the cut locus | The key fact is that the cut time $t\_c : UM \to \mathbb{R}$, defined on the unit tangent bundle $UM$ of a complete, $n$-dimensional Riemannian manifold, is locally Lipschitz continuous around all $v \in UM$ such that $t\_c(v) < +\infty$. Hence the tangential cut locus at $p \in M$, that is
$$
\tilde{C}\_p = \{t\_c(v)... | 19 | https://mathoverflow.net/users/13915 | 255882 | 115,638 |
https://mathoverflow.net/questions/255869 | 7 | In the question [Eigenvalues of a matrix with entries involving combinatorics](https://mathoverflow.net/questions/255141/eigenvalues-of-a-matrix-with-entries-involving-combinatorics) No\_way asked about eigenvectors of $n\times n$ matrix $M$ with entries \begin{eqnarray\*}
M\_{ij}=(-1)^{i+j}F(n, l, i, j),
\end{eqnarra... | https://mathoverflow.net/users/5712 | Eigenvectors of a matrix with entries involving combinatorics | In fact for a fixed $n$, the matrices $M(l, n)$ for $l>0$ commute with each other and thus are simultaneously diagonalisable. For your second question, if $\{p\_j(y)\}$ is a sequence of polynomials satisfying
\begin{eqnarray}
\left(\frac{t}{\sinh t}\right)^y=\sum\_{j=0}^\infty p\_j(y)t^{2j}.
\end{eqnarray}
then the $i... | 5 | https://mathoverflow.net/users/85722 | 255886 | 115,640 |
https://mathoverflow.net/questions/255877 | 5 | Let $T$ be a $N\times N$ transition matrix for a markov chain with $N$ states. Thus $T\_{ij}$ is the probability of transition from state $i$ to state $j$ (and thus rows summing to one). Now consider the matrix $$T\_k=T\otimes T$$. Its easy to see that rows of $T\_k$ sum to one and each entry is non-negative. Thus, $T\... | https://mathoverflow.net/users/27249 | Intuition on Kronecker Product of a Transition Matrix | I think you just have a pair of independent chains. The probability of making a transition from (i,j) to (k,l) is $p\_{ik}p\_{jl}$, where the first component is the state of the first chain and the second of the second. The transitions out of the (i,j) state are found in the N(i-1) + j th row.
| 3 | https://mathoverflow.net/users/nan | 255890 | 115,643 |
https://mathoverflow.net/questions/255872 | 10 | I've seen several times people mentioning that the notion of an algorithm / a computation is taken as a primitive notion in L. E. J. Brouwer's intuitionism. For instance, in *Varieties of Constructive Mathematics* (1987, p.1), Bridges and Richman write that:
>
> In Bishop's constructive mathematics (BISH), and in B... | https://mathoverflow.net/users/84804 | Why is the notion of algorithm a primitive one in Brouwer's intuitionism? | Since the work of Church and Turing (say around 1936), the notion of algorithm is definitely not considered primitive in intuitionism. But Brouwer started intuitionistic mathematics more than 2 decades before (1907, 1912), and in the absence of a commonly accepted notion of algorithm, he formulated his idea (somewhat) ... | 4 | https://mathoverflow.net/users/101577 | 255892 | 115,644 |
https://mathoverflow.net/questions/254497 | 3 | Let $\mathcal{F}\_n$ be the set of all boolean functions of $n$ variables and let $\xi$ be a random variable with values in the set $\mathcal{F}\_n$ with the uniform distribution. We define a new random variable $$\eta = \operatorname{max}\{W\_{\alpha}(\xi): \alpha\in V\_n\},$$ where $W\_{\alpha}(\xi)$ is a Walsh-Hadam... | https://mathoverflow.net/users/85489 | The best linear approximation of a random function | Firstly, I think you probably meant to take an absolute value in your maximum, otherwise by complementing the functions you'd get zero expectation, no?
F. Rodier has shown the following in the paper [here](http://iml.univ-mrs.fr/editions/preprint2003/files/RodierFoncBool.pdf):
Let $\eta$ be the maximum of $|W\_{\al... | 2 | https://mathoverflow.net/users/17773 | 255893 | 115,645 |
https://mathoverflow.net/questions/255896 | 4 | I have the following question:
I have a function $f: \mathbb R \to \mathbb R$ which is differentiable everywhere.
I also have a set $G\subset\mathbb R$ which is dense in $\mathbb R$ and a $G\_\delta$-set.
I know that $f'(x)=0 \forall x\in G$.
Can I conclude that $f$ is constant?
The answer is yes if $f'$ is contin... | https://mathoverflow.net/users/101778 | Derivative is Zero on a dense G_delta set | Since the zero set is always a $G\_\delta$, the question is whether it being dense implies that the function is constant. This is false, and a counterexample is known as a [Pompeiu derivative](https://en.m.wikipedia.org/wiki/Pompeiu_derivative).
| 9 | https://mathoverflow.net/users/99234 | 255898 | 115,646 |
https://mathoverflow.net/questions/255884 | 3 | For any cardinal $\kappa$ we set $2^{<\kappa} = \big|\{A\subseteq \kappa: |A|<\kappa\}\big|$. Let $\kappa$ be an infinite cardinal. We say $\kappa$ is *exact* if $\kappa = 2^{<\kappa}$, and we say that $\kappa$ is *reachable* if there is $\lambda < \kappa$ such that $2^\lambda \geq \kappa$.
Note that $\aleph\_0$ is e... | https://mathoverflow.net/users/8628 | Exact and reachable cardinals | I'm not sure what kind of answer you're seeking, but here are some elementary things to say about your concepts.
What you would call the non-reachable cardinals are more widely known as the strong limit cardinals, and this is a fundamental cardinal concept in elementary set theory. (A cardinal $\kappa$ is a *strong ... | 3 | https://mathoverflow.net/users/1946 | 255900 | 115,647 |
https://mathoverflow.net/questions/255911 | 9 | Do there exist positive integers $a,b$ and a prime $p>\max(a,b)$ such that $p^3$ divides $(a+b)^p-a^p-b^p$?
The reader of Kvant magazine A. T. Kurgansky asked to prove that such $a,b,p$ do not exist, see [here](http://kvant.mccme.ru/1984/08/p34.htm "Kvant 1984").
But discussion here
[On the exact reference of a c... | https://mathoverflow.net/users/4312 | May $p^3$ divide $(a+b)^p-a^p-b^p$? | There are apparently lots of examples. The smallest is $a=1$, $b=2$, and $p=7$.
| 13 | https://mathoverflow.net/users/3199 | 255914 | 115,651 |
https://mathoverflow.net/questions/255919 | 1 | Let $V$ be a vector space. A reflection is a linear map $f: V \to V$ which has an eigenvalue $1$ with multiplicity $n-1$.
Let $S\_n$ be the symmetric group on $\{1,\ldots,n\}$. Then the reflections in $S\_n$ are transpositions.
Let $W$ be a Coxeter group. What are all reflections in $W$? Thank you very much.
| https://mathoverflow.net/users/11877 | References request: reflections in coxeter groups | As you find in "Humphreys, Reflection and Coxeter groups" ([link behind paywall](http://www.ams.org/mathscinet-getitem?mr=1066460)) in Section 5.7, the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$, this is, all elements in $W$ that are conjugate to the generators $S$... | 7 | https://mathoverflow.net/users/21291 | 255922 | 115,655 |
https://mathoverflow.net/questions/81840 | 2 | A point process $\Phi$ is said to be negatively associated if for any finitely many bounded Borel subsets $B\_1,B\_2,...,B\_n,$ we have that
$$\operatorname{Cov} \left( f\left(\Phi\left(B\_1\right),\ldots,\Phi\left(B\_l\right)\right)g\left(\Phi\left(B\_{l+1}\right),\ldots,\Phi\left(B\_{n}\right)\right)\right) \leq 0... | https://mathoverflow.net/users/11409 | Negatively associated point processes | I just came across this old question, while searching for whether it has been proved yet that spatial determinantal point processes are negatively associated. It turns out it has been proved, in [this paper by Subhro Ghosh](https://arxiv.org/abs/1211.2435).
| 4 | https://mathoverflow.net/users/1044 | 255928 | 115,658 |
https://mathoverflow.net/questions/255912 | 11 | Let $M(X)$ be the vector space (actually it's an algebra) of all equivalent classes of measurable functions $X\to \mathbb{C}$ (where $X$ is a measured space) modulo equality almost-everywhere.
One can define a notion of sequential convergence on $M(X)$. A sequence $(f\_n)$ of $M(X)$ converges to $f$ iff (by definiti... | https://mathoverflow.net/users/99246 | What is the structure associated to almost-everywhere convergence? | Yes, this defines a "convergence vector space". In fact, it's probably the original motivating example for the generalization. In Fréchet's thesis he discussed L-spaces, which are essentially sequential convergence spaces: a set equipped with families of convergent sequences at each point satisfying the axioms that the... | 17 | https://mathoverflow.net/users/99234 | 255930 | 115,659 |
https://mathoverflow.net/questions/255897 | 24 | [Lagrange's four square\_theorem](https://en.wikipedia.org/wiki/Lagrange's_four-square_theorem) states that every positive integer $N$ can be written as a sum of four squares of integers. At present, let's focus only on positive integer summands; that is, $N=a\_1^2+a\_2^2+a\_3^2+a\_4^2$ with $a\_i\in\mathbb{N}$.
If n... | https://mathoverflow.net/users/66131 | two's and three's survive in gcd of Lagrange | The claim is certainly true for $n$ sufficiently large, and "sufficiently large" could be specified explicitly with more care.
We follow the suggestion of Fedor Petrov, and rely on the results of Brüdern & Fouvry (J. reine angew. Math. 454 (1994), 59-96) and of Heath-Brown & Tolev (J. reine angew. Math. 558 (2003), ... | 21 | https://mathoverflow.net/users/11919 | 255936 | 115,662 |
https://mathoverflow.net/questions/255812 | 0 | We adopt common notations in the study of Szemerédi's regularity lemma and only focus on simple graph $G(V,E)$. For any two disjoint vertex sets $A,B\subset V$, we say the pair $(A,B)$ is **$\varepsilon-$regular** if for every $X\subset A$, $Y\subset B$ satisfying $|X|>\varepsilon|A|$ and $|Y|>\varepsilon|B|$, we have ... | https://mathoverflow.net/users/81507 | Intersection property of Szemerédi's regularity condition | The regularity lemma has the unfortunate feature that, whilst it is conceptually extremely simple, there doesn't seem to be a terribly compact way of specifying regularity arguments precisely. This leads to the situation where lots of things are obvious to experts, and have obvious (to experts) proofs, but nobody reall... | 2 | https://mathoverflow.net/users/25485 | 255944 | 115,667 |
https://mathoverflow.net/questions/255722 | 40 | I've been learning Arakelov geometry on surfaces for a while. Formally I've understood how things work, but I'm still missing a big picture.
---
**Summary:**
Let $X$ be an arithmetic surface over $\operatorname{Spec } O\_K$ where $K$ is a number field (we put on $X$ the good properties: regular, projective,...)... | https://mathoverflow.net/users/47136 | Why are Green functions involved in intersection theory? | The Green's function is used not to measure distances in the surface but to measure distances in the line bundle. A Green's function on $X\_{\mathbb C}$ that blows up at $D$ can be used to measure sections of $\mathcal O(D)$. Indeed, if we represent a section of $\mathcal O(D)$ as a holomorphic function $f$ on $X\_{\ma... | 24 | https://mathoverflow.net/users/18060 | 255949 | 115,668 |
https://mathoverflow.net/questions/255360 | 11 | Let $X$ be $\mathbb{P}^1\_{\mathbb{F}\_q}\smallsetminus \{a\_1,...,a\_r\}$, where $a\_1,...,a\_r$ are some $\mathbb{F}\_q$-rational points. Let $\bar X :=X\_{\bar{\mathbb{F}}\_q}$. There is a short exact sequence $$1\rightarrow \pi\_1(\bar X)\rightarrow \pi\_1(X)\rightarrow \operatorname{Gal}(\mathbb{F}\_q)\rightarrow ... | https://mathoverflow.net/users/98901 | Is there a presentation to the kernel of the prime-to-$p$ fundamental short exact sequence of curves over finite fields? | Well, $N$ is generated by the commutators of elements of $\pi\_1'(\overline{X})$ with elements of the maximal pro-$p$ subgroup of $\operatorname{Gal}(\mathbb F\_q)$. This is just because any such element of $\operatorname{Gal}(\mathbb F\_q)$ lies in the kernel of the map to the maximal prime-to-$p$ quotient of $\pi\_1(... | 7 | https://mathoverflow.net/users/18060 | 255953 | 115,669 |
https://mathoverflow.net/questions/255948 | 1 | As the title ask, what is the best lower and upper bounds for the product below :
$$ \prod \limits\_{x < p \leq y} \frac{p+1}{p}$$
such that $p$ denote the prime numbers in which fullfil the conditions under the product ?
| https://mathoverflow.net/users/95470 | Best known bounds for a product over primes in an interval | Your product is the following $$\prod\_{y<p\leq x} \left(1+\frac{1}{p}\right).$$
You can use the fact that $$\prod\_{y<p\leq x} \left(1+\frac{1}{p}\right)\leq \prod\_{y<p\leq x} \left(1-\frac{1}{p}\right)^{-1} \leq \frac{\pi^2}{6} \prod\_{y<p\leq x} \left(1+\frac{1}{p}\right).$$ Note that $\frac{\pi^2}{6}=\frac{1}{\zet... | 5 | https://mathoverflow.net/users/76102 | 255955 | 115,671 |
https://mathoverflow.net/questions/224757 | 12 | If we consider $G$ a compact Lie group, there is a left invariant Riemannian metric whose the sectional curvature is nonnegative (see Milnors' paper). When can we find a left invariant metric that has positive sectional curvature?
The unique simply connected Lie group with a left-invariant metric that is positively c... | https://mathoverflow.net/users/48865 | What is known about Lie groups with (strictly) positive curvature? | The following result is, for example, exercise 3 on pg. 104 of Do Carmo's Riemannian Geometry book.
>
> Suppose $X$ is a Killing field on a compact even dimensional Riemannian manifold of positive curvature. Then $X$ has a zero.
>
>
>
Using this result, it's very easy to prove the following generalization of W... | 10 | https://mathoverflow.net/users/1708 | 255978 | 115,679 |
https://mathoverflow.net/questions/255976 | 1 | Let $G$ be a compact non-connected nilpotent Lie subgroup of $O(n)$. We know that $G\_0$, its identity component, is always a torus. Is it true that $G\_0$ is always central in $G$?
What about general $G$ (which may not be subgroup of $O(n)$)?
| https://mathoverflow.net/users/98833 | Compact non-connected nilpotent Lie subgroup of $O(n)$? | If this is your question, yes it's true that for every nilpotent compact Lie group $G$, $G\_0$ is central.
Indeed as you already noticed, $G\_0$ is abelian, so the action by conjugation on $G\_0$ factors through the finite group $F=G/G\_0$. Let $V$ be the universal covering of $G\_0$: this is a vector group (= finit... | 3 | https://mathoverflow.net/users/14094 | 255983 | 115,680 |
https://mathoverflow.net/questions/255981 | 5 | This question is a follow-up to [this](https://math.stackexchange.com/questions/2033992/what-conditions-on-a-map-of-schemes-guarantee-that-pullback-of-global-sections-i?noredirect=1#comment4181112_2033992) question which I asked on MSE.
Let $f: X \rightarrow Y$ be a surjective morphism of schemes, and $\mathscr{F}$ ... | https://mathoverflow.net/users/56878 | When do surjective morphisms induce injective maps on global sections of coherent sheaves? | Begin with $Y$ equal to the affine plane, $\text{Spec}\ R$, for $R=k[s,t]$. Let $\overline{f}:\overline{X}\to Y$ be the blowing up of $Y$ at the ideal $\mathfrak{m} = \langle s,t \rangle$. Define $I\subset \mathfrak{m}$ to be the ideal $\langle s,t^2 \rangle$. Define $\mathcal{G}$ to be $\widetilde{R/\mathfrak{m}}$, de... | 6 | https://mathoverflow.net/users/13265 | 255997 | 115,685 |
https://mathoverflow.net/questions/255967 | 12 | Suppose I have a positive number $d \in \mathbb{R}$ and a sequence of numbers $a\_n \in [0,d]$ for $n \in \mathbb{N}$ with the following properties
$$
\sum\_{i=1}^{\infty} a\_i^r \in \mathbb{Z}
$$
for all $r \in \mathbb{N}$ and
$$
\sum\_{i=1}^{\infty} a\_i \leq d \ .
$$
>
>
> >
> > Does it follow that only fini... | https://mathoverflow.net/users/3995 | Integer-valued power sums | The function
$$
f : z \in \mathbb{C} \longmapsto \sum\_{i} \frac{a\_i}{1-a\_iz}
$$
is meromorphic on $\mathbb{C}$ and has integral Taylor coefficients. It follows from a theorem of Borel that such a function must be in $\mathbb{Q}(z)$; see for example Richard Stanley's answer [here](https://mathoverflow.net/questions/... | 13 | https://mathoverflow.net/users/21724 | 255998 | 115,686 |
https://mathoverflow.net/questions/255979 | 7 | A grad student asked me this question during office hours, and I couldn't for the life of me come up with a proof or counterexample:
For a given $F:\mathbb{R}^3 \to \mathbb{R}^3$, does $(\nabla \times F) \cdot F= (\nabla \times F) \cdot \nabla f$ in $R^3$ always have a solution in $f$?
| https://mathoverflow.net/users/101819 | Does $(\nabla \times F) \cdot F= (\nabla \times F) \cdot \nabla f$ have a solution? | There exist $F$ for which there is no global solution $f$ to the above equation. Here is how you can construct an example:
First, regard $F$ as a vector field on $\mathbb{R}^3$ and consider its dual $1$-form $\phi$. The left hand side of your equation can then be written as the Hodge dual of $\phi\wedge\mathrm{d}\phi... | 28 | https://mathoverflow.net/users/13972 | 256002 | 115,688 |
https://mathoverflow.net/questions/256000 | 3 | Let $\sigma$ be a permutation. If two positive braids represent $\sigma$ and are of minimal length among the braids representing $\sigma$, then they are equal. From what I could gather, this result is Theorem 9.2.5 in:
Epstein, David, et al. Word processing in groups. AK Peters, Ltd., 1992.
This book refers to a ... | https://mathoverflow.net/users/68468 | Injection from Artin monoids to Coxeter groups | This is a general fact on Artin-Tits groups attached to Coxeter groups. Let $B(W)^+$ be the Artin-Tits monoid of the Coxeter group $W$. Then there is a canonical surjection $B(W)^+\rightarrow W$ which you mentioned.
Now starting from any reduced expression $s\_1 s\_2 \cdots s\_k$ of an element $w$ in $W$ where $s\_i... | 6 | https://mathoverflow.net/users/26751 | 256005 | 115,689 |
https://mathoverflow.net/questions/255791 | 3 | I posted the following question on stackexchange but didn't get any replies; I'm hoping perhaps someone can help me here.
I understand that for many iterative methods, convergence rates can be shown to depend on the condition number of the coefficient matrix $A$ in the linear equation
$$Ax=y.$$
Therefore, if a preco... | https://mathoverflow.net/users/170703 | Question about preconditioning | The Frobenius norm remains the same if we do the orthogonal transform, so
$$
\min\_{P,Q} \left\| AP^{-1} - Q \right\|\_F =
\min\_{P,Q} \left\| Q^{-1}AP^{-1} - I \right\|\_F
$$
which is essentially nothing more than a two-sided preconditioning. While for some reason you consider only right-sided in the beginning (usual... | 1 | https://mathoverflow.net/users/97620 | 256030 | 115,699 |
https://mathoverflow.net/questions/254296 | 2 | i have got a problem with some assumptions to solve a parabolic variational inequality. My Problem is: Find a function $u$ with
\begin{align}
u\in L^2(0,T;V),~ u' \in L^2(0,T;V') \\
(u'(t),v-u(t)) + a(u(t),v-u(t)) + \psi(v) -\psi(u(t)) \geq (f(t),v-u(t)) \\
u(0) = u\_0.
\end{align}
We want to approximate $\psi$ wit... | https://mathoverflow.net/users/100894 | Derivative of Yosida-Approximation | Since no one else has written an answer, I'm copying my comment:
If $\psi$ is convex and bounded from below (which I'm assuming since you didn't state anything), the Moreau envelope $\psi\_j$ admits a minimizer (in fact, the same as $\psi$) for any $j$. Since the Moreau envelope is differentiable, this minimizer -- c... | 2 | https://mathoverflow.net/users/30516 | 256038 | 115,701 |
https://mathoverflow.net/questions/255920 | 2 | Consider $[0,1]$ and thereon $D\_n = \{ [k2^{-n}, (k{+}1) 2^{-n} ) : 0\le k \le 2^n-1\}$ and $\mathcal F\_n$ the generated sigma-algebras by $D\_n$. We do know that uniformly bounded (in $L\_p$-norm) martingales converge.
In this particular situation, $X\_n$ is a linear combination of dyadic indicator functions, and... | https://mathoverflow.net/users/101787 | martingale convergences wrt dyadic intervals | Yes, the Martingale convergence theorem can be proven by a weak compactness argument. Let me restrict to the case of a Hilbert space $H$ for simplicity. Any convergence in $H$ that is proven using completeness can be obtained by a weak compactness argument.
The $L^2$ Martingale convergence theorem is deduced from the... | 1 | https://mathoverflow.net/users/6129 | 256040 | 115,702 |
https://mathoverflow.net/questions/255982 | 2 | I have a convex optimization problem as follows:
\begin{align\*}
maximize\_{x\in R^n} &\sum\_{i=1}^n a\_i \log(x\_i)\\
st\quad & \sum\_{i=1}^n p\_{ij} (x\_i-1) = 0 \quad \forall j\\
& x\_i > 0
\end{align\*}
where, $a\_i \geq 0$ and $P$ is an $n\times m$ probability matrix with rows sum up to 1.
( I am assuming ... | https://mathoverflow.net/users/99932 | A convex optimization problem | Your problem is of the form
$$
\min F(x)\quad \text{s.t.}\quad Ax=b
$$
or
$$
\min F(x) + G(Ax)
$$
with $G$ being the indicator function of the single point $b$. Hence, you can try basically all methods from the slides ["Douglas-Rachford method and ADMM"](http://www.seas.ucla.edu/~vandenbe/236C/lectures/dr.pdf) by Lieve... | 3 | https://mathoverflow.net/users/9652 | 256043 | 115,703 |
https://mathoverflow.net/questions/255990 | 9 | I want to think of ZFC as not fully determining the powerset of the naturals, because you can add subsets with forcing and otherwise have a lot of control over the cardinality of the powerset of the naturals. But that suggests the question: what subsets does ZFC determine? There is also the related question: how compli... | https://mathoverflow.net/users/35714 | What is the descriptive complexity of a set added by Cohen forcing? | Since Cohen forcing is weakly homogeneous, all hereditarily ordinal definable sets in the Cohen extension are already in the ground model. That applies in particular to any ordinal definable real, and (in even more particular) to any real with a lightface $\Sigma^m\_n$ definition.
| 12 | https://mathoverflow.net/users/6794 | 256051 | 115,707 |
https://mathoverflow.net/questions/256042 | 0 | If there is a system of nonlinear equations and all variables are unbounded real numbers and the functions are continuous, is there a general result on the complexity of solving it? More specifically, can a statement be made about how the computational time to numerically solve the system increases with the number of v... | https://mathoverflow.net/users/101852 | Solving a nonlinear equation system: is there a general result on complexity? | There is not even an algorithm to test whether a solution exists. See e.g. [Richardson's theorem](https://en.wikipedia.org/wiki/Richardson's_theorem).
To have something sensible, you need at least to be able to restrict the domain to a bounded region.
EDIT: ... and even then, depending on the precise formulation of... | 3 | https://mathoverflow.net/users/13650 | 256052 | 115,708 |
https://mathoverflow.net/questions/255974 | 6 | Let $E/F$ be a finite field extension. Let $G\_E$ be a reductive linear algebraic group defined over $E$ and let $G=\mathrm{R}\_{E/F}G\_E$ be the Weil restriction of scalars. Then $G$ is a linear algebraic group defined over $F$ and $\mathrm{R}\_{E/F}$ is a functor from the category of linear algebraic groups defined o... | https://mathoverflow.net/users/37777 | Automorphism of restriction of scalars | You surely meant to assume the finite extension of fields $E/F$ is separable (otherwise ${\rm{R}}\_{E/F}(H)$ is *never* reductive for a smooth connected affine $E$-group $H \ne 1$). Also, this is one of those cases where more generality clarifies the situation. The reason I say this is that by limiting yourself only to... | 10 | https://mathoverflow.net/users/81332 | 256057 | 115,712 |
https://mathoverflow.net/questions/256019 | 4 | Recall that a dyck $n$-path is a lattice path of length $2n$ with steps $U$ (ups corresponding to $(1,1)$) and $D$ (downs corresponding to $(1,-1)$) such that it starts at $(0,0)$ and never goes below the $x$-axis.
[This sequence on OEIS](http://oeis.org/search?q=1%2C5%2C21%2C81%2C302&sort=&language=english&go=Search... | https://mathoverflow.net/users/61949 | Number of certain Dyck paths | First I will show one way to derive the generating function for Dyck paths, and then I will adapt it to count Dyck paths that avoid $U^k$ for any $k$.
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Define $F(x,u)$ to be the generating function that counts partial Dyck paths using $x$ to mark the number of steps and $u$ to mark the ending height. Here, "pa... | 7 | https://mathoverflow.net/users/36497 | 256070 | 115,713 |
https://mathoverflow.net/questions/256062 | 5 | Can an AW\*-algebra be recovered (up to Jordan isomorphism) from its lattice of projections? This is possible in the commutative/Boolean case.
| https://mathoverflow.net/users/99234 | Can an AW*-algebra be recovered from its lattice of projections? | Yes, this follows from Theorem 4.2 in Dye's Theorem and Gleason's Theorem for AW\*-algebras by Jan Hamhalter.
Link: <https://arxiv.org/abs/1408.4597>
Here the statement is: given an AW\*-algebra $A$ without type I$\_2$ summands, and an AW\*-algebra $B$, then every orthocomplement-preserving order morphism $\varphi:P(... | 7 | https://mathoverflow.net/users/100607 | 256071 | 115,714 |
https://mathoverflow.net/questions/255559 | 0 | A Cayley graph (resp. digraph) $Cay(G,S)$ is called a $CI$-graph (resp. $DCI$-graph) of $G$ if, for any Cayley graph (resp. digraph) $Cay(G, T)$, whenever $Cay(G,S) \cong Cay(G, T)$ we have $S = T^\sigma$ for some $\sigma \in Aut(G)$. A group $G$ is called a $CI$-group (resp. $DCI$-group) if all Cayley graphs (resp. di... | https://mathoverflow.net/users/75264 | DCI-properties of Cayley graphs | Corollary 4.5 of Babai's paper states that $D\_{2p}$ is CI for the category of "colour-graphs". Looking at the definition on page 330, you can see that Babai does not require "colour-graphs" to be undirected. So his result applies to digraphs, and therefore tells us that $D\_{2p}$ is a DCI-group.
| 1 | https://mathoverflow.net/users/68305 | 256075 | 115,715 |
https://mathoverflow.net/questions/256006 | 5 | Suppose that $M$ is a closed infinite dimensional subspace of $L\_4(0,1)$ which is also a closed subspace of $L\_1(0,1)$. Hence $M$ is isomorphic to $\ell\_2$ as a subspace of $L\_p(0,1)$ for $1\leq p\leq 4$. Note also that $L\_4(0,1)\subset L\_1(0,1)$ and $\|f\|\_1\leq \|f\|\_4$ for each $f\in L\_4(0,1)$.
Is it pos... | https://mathoverflow.net/users/39421 | Simultaneous near-best approximation with respect to two norms | It seems that the answer is yes.
Denote by $M\_q$ the space $M$ equipped with the $L\_q$-norm; by the closednedd condition, it is a Banach space for both $q=1$ and $q=4$. The identical mapping $M\_4\to M\_1$ is bounded; by the bounded inverse theorem, so is its inverse. Thus there exists $\mu$ such that $\|g\|\_4\leq... | 3 | https://mathoverflow.net/users/17581 | 256087 | 115,719 |
https://mathoverflow.net/questions/256047 | 1 | Let $S$ be a smooth closed submanifold of $M$. Let $U$ be a tubular neighborhood such that for any $x\in U\setminus S$ there is a unique minimizing geodesics. We now consider the distance squared to $S$, $d\_S^2(x):=dist^2(x,S)$, $x\in U$. Can we write down the taylor expansion of $d\_S^2(\exp\_p(t\nu))$, for any $t$ a... | https://mathoverflow.net/users/101796 | Squared distance function function from a submanifold | Have a look at (where your $S$ is a mostly curve):
* MR2024928 Reviewed Gray, Alfred Tubes. Second edition. With a preface by Vicente Miquel. Progress in Mathematics, 221. Birkhäuser Verlag, Basel, 2004. xiv+280 pp.
A remark: If $S$ is a point, the second derivative (with respect to Riemannian normal coordinates) ... | 3 | https://mathoverflow.net/users/26935 | 256089 | 115,721 |
https://mathoverflow.net/questions/256072 | 2 | Let $A$ be a finite dimensional algebra over a field K. Let $M$ be an $A$-module and $0 \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots$ be a minimal injective resolution of $M$. The dominant dimension of $M$, denoted by domdim$(M)$, is the largest number $t$ or $\infty$ such that $I^0, I^1, \ldots, I^{t-1}$ are pro... | https://mathoverflow.net/users/83554 | Questions about dominant dimension | 1.There is no such relation in general. For specific algebras, such as non-semisimple higher auslander algebras, the dominant dimension of noninjective modules is always bounded by domdim(A).
2. Let $A$ be not selfinjective. Then there is a injective module which is not projective. this module has always dominant dim... | 2 | https://mathoverflow.net/users/61949 | 256098 | 115,725 |
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