Newest Questions
159,042 questions
6
votes
0
answers
235
views
A curious series for $L(2,(\frac{-3}{\cdot}))$
Let
$$K:=L\left(2,\left(\frac{-3}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac k3)}{k^2}=\sum_{j=0}^\infty\left(\frac1{(3j+1)^2}-\frac1{(3j+2)^2}\right),$$ where $(\frac k3)$ is the Legendre ...
- 15.6k
6
votes
1
answer
485
views
Three conjectural series for $\pi^2$ and related identities
Recently, I found the following three (conjectural) identities for $\pi^2$:
$$\sum_{k=1}^\infty\frac{145k^2-104k+18}{k^3(2k-1)\binom{2k}k\binom{3k}k^2}=\frac{\pi^2}3,\tag{1}$$
$$\sum_{k=1}^\infty\frac{...
- 15.6k
1
vote
0
answers
77
views
Definition of this formula for the $2p$ functions
I am reading this paper about constructive renormalization for fermions and I got a really basic question about it. There, the effective Lagrangian (with UV cutoff $\Lambda_{0}$ and IR cutoff $\Lambda$...
- 1,305
3
votes
0
answers
246
views
"weakly functorial resolution" of quasi-compact T_1 topological space by quasi-compact Hausdorff space
I have an arguably weird question: Let $X$ be a quasi-compact $T_1$ topological space, could there be a construction that takes such an $X$ as input and outputs a surjection
$$X' \to X$$
with the ...
- 619
2
votes
0
answers
60
views
Length of $\text{Tor}$ modules in complete intersection rings with characteristic $p$
Let $(R,m)$ be a complete intersection ring with characteristic $p$. If $M$ is a finitely generated $R$-module with finite length i.e. $l(M)<\infty$, it follows that $l(F(M))\geq p^nl(M)$, where $n=...
- 121
8
votes
1
answer
619
views
A question regarding symmetrizing the tensor product of vectors in two different ways
Let $V = \mathbb{C}^m$, endowed with the standard hermitian inner product which we will denote by $\langle \cdot, \cdot \rangle$, $n$ be a positive integer and $\Sigma_n$ denote the symmetric group on ...
- 5,215
6
votes
0
answers
299
views
A new series for $\sqrt3/\pi$?
Recently, I conjectured the following identity:
$$\sum_{k=0}^\infty\frac{(66k^2+37k+4)\binom{2k}k\binom{3k}k\binom{4k}{2k}}{(2k+1)729^k}=\frac{27\sqrt3}{2\pi}.\tag{1}$$
This can be easily checked ...
- 15.6k
2
votes
0
answers
147
views
Graph Laplacians, Riemannian manifolds, and object collisions
To preface this question, I am a part-time game developer and full-time optimization fiend.
I am working on object collisions at the moment and many resources I have found online are more-or-less just ...
- 21
6
votes
1
answer
315
views
In HoTT with LEM, are sets and pointed sets the same thing?
The operations of adding and removing a point (where removing is a consideration of a subset of elements x such that $(x = *) \to 0$) implements the equivalence of these 1-types, as far as I can see. ...
- 6,962
0
votes
2
answers
139
views
Graph vertices selection for paths sum minimalization
Let $G = (V, E)$ be a simple, finite, weighted, undirected, connected graph. Let $P = \{p_{ij}, p_{kl}, p_{il}, ... \}$ be a set of paths, where $p_{ij}$ is a path from $i$-th to $j$-th vertex. Is ...
0
votes
2
answers
118
views
Why is a plane graph Delaunay realizable if stellating a face makes the graph inscribable?
I have come across the following lemma in several papers (for instance see lemma 2.2 in this) (and some authors state this the proof immediately follows from basic properties of the stereographic ...
- 129
14
votes
1
answer
1k
views
What is decategorification?
A decategorification is, roughly, some procedure $\Phi$ which inputs some sort of $n$-categorical data and outputs some sort of $(n-1)$-categorical data. Whatever this means, categorification is ...
- 64k
2
votes
0
answers
192
views
Completeness of the Infinity Category of A-infinity Categories
Is the infinity category of $A_{\infty}$-categories complete? By complete I mean do there exist arbitrary homotopy limits in the infinity category of $A_{\infty}$-categories?
I felt like this result ...
- 341
1
vote
1
answer
256
views
Intersection of two intermediate subalgebras
Suppose $B\subset A$ is an inclusion of simple $C^*$-algebras with a conditional expectation of (Watatani) index-finite type and $B^{\prime}\cap A=\mathbb{C}$. Then we know $B^{\prime}\cap A_1$ is ...
- 393
1
vote
0
answers
155
views
Lifting action of torus to torus bundle
Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it.
Let $\phi$ be a smooth action of $T^k$ on $X$.
The paper "Lifting compact group actions ...
- 205
4
votes
0
answers
218
views
Automorphism group of a Lorentzian lattice
Consider the even integral lattices $L_n:=Z\times Z\times Z^{n-2}$ (where $Z$ is the set of integers) with elements $x=(x_+,x_-,x_d)$ and inner product
$$(x,y):=x_+y_-+x_-y_++2x_d\cdot y_d.$$
Its ...
- 3,873
5
votes
1
answer
221
views
Arens regularity of $\mathrm{BV}(\mathbb{R})$
$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of ...
- 6,406
0
votes
1
answer
183
views
Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$
Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\...
- 245
0
votes
1
answer
195
views
Trying to solve for the remainder of $a^q$ modulo $q$
Let $q$ be a prime and $a$ be a number from $0$ to $q-1$ (not an equivalence class).
The elements $a^q$ are exactly the elements of order $q-1$ modulo $q^2$.
I'm trying to solve the equation:
$$a+2*\...
- 591
7
votes
2
answers
1k
views
Can a non integrable random variable satisfy a strong law of large numbers principle?
Given a random variable $X$, we denote by $X_1, X_2, \dots$ a sequence of iid copies of $X$.
Question: Does there exist a random variable $X$ with $\mathbb E[X^+] = \mathbb E[X^-] = +\infty$, but
$$\...
- 6,323
1
vote
2
answers
221
views
Beating the $1/\sqrt n$ rate of uniform-convergence over a linear function class
Let $P$ be a probability distribution on $\mathbb R^d \times \mathbb R$, and let $(x_1,y_1), \ldots, (x_n,y_n)$ be an iid sample of size $n$ from $P$. Fix $\epsilon,t\gt 0$. For any unit-vector $w \in ...
- 6,853
4
votes
0
answers
242
views
On the Dunford-Pettis property and multiplier algebras
I am not an expert in operator algebras, so if the answer to this question might be trivial, that might be one reason for that:
Let $\mathcal{A}$ be a $C^\ast$-algebra. Then $\mathcal{A}^{\ast \ast}$ ...
3
votes
1
answer
269
views
Proving that solutions to elliptic PDE is analytic using Cauchy–Kovalevskaya theorem?
It seems that Cauchy–Kovalevskaya is not commonly used in books on PDE theory. I am thinking about applying it somewhere interesting.
It is known that if $L$ is a uniformly elliptic operator, with ...
- 1,755
3
votes
1
answer
210
views
Exponential growth of shortest vector norm for successive lattices corresponding to powers of a matrix
Let $A\in M_{2\times 2}(\mathbb{Z}) $ be a two by two integer matrix such that $0,\pm 1$ are not eigenvalues of $A$ and $\left|\det(A)\right|>1$. I am interested in the growth of the norm shortest ...
0
votes
0
answers
129
views
Extend line bundle on regular curve to it's regular model
Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$.
Assume, $C$ admits a proper regular flat model $\...
- 6,048
1
vote
1
answer
177
views
Algebraic independence of polynomials when truncated imply algebraic independence of the entire polynomial?
Let $f_1,\ldots,f_m \in \mathbb{F}[x_1,\ldots,x_n]$ and suppose $\hat{f}_i = f_i$ $\bmod \langle x_1,\ldots,x_n\rangle^3$ (i.e. the linear and quadratic part of $f_i$). Then if $\hat{f}_1,\ldots,\hat{...
- 343
6
votes
1
answer
225
views
Does $U_q (\mathfrak{sl}_2)$ have a universal $R$-matrix?
Consider the standard quantum group $U_q (\mathfrak{sl}_2)$ over the field $\mathbb{C}(q)$ of rational functions (or over $\mathbb{C}$ if $q \in \mathbb{C}$ is not a root of unity), with the usual ...
- 601
1
vote
1
answer
184
views
Is multiplication by $d$ on the Jacobian of a nodal curve étale?
Let $k$ be an alg.closed of char$k=0$ and let $A$ be an abelian variety over $k$. This
Lemma on stacks project states that $[d]\colon A\to A$ is étale. In particular, when $A$ is the Jacobian of a ...
- 123
1
vote
0
answers
64
views
Is $\sf MK$ bi-interpretable with this modification of it posing an Ur-proper class for every set?
Let's take $\sf MK$ set theory.
Adopt the notation of upper case ranging over all objects, lower case only range over sets (i.e.; elements of classes), and $\frak A,B,C,..$ to range only over proper ...
- 11.3k
8
votes
0
answers
359
views
Bounding a sum of reciprocals of square-free integers
(Cross-posted from MSE, as the question did not get any clear answer)
Fix positive integers $k$ and $n$. Let $N_1,\dots,N_r$ be all the integers less than or equal to $n$ that are squarefree and have ...
- 189
1
vote
1
answer
63
views
Covering convex regions with disks optimizing on area and perimeter
Question: Are there planar convex regions $R$ and integers $n$ with the property: if $R$ is covered by $n$ disks of possibly different sizes such that (1) the total area of the covering disks is ...
- 5,979
9
votes
3
answers
427
views
Exponentiation of Dedekind cardinals
Question: Let $\mathfrak n$ be an infinite cardinal. In ZF (set theory without the axiom of choice) can either of the implications
$$\mathfrak n=\mathfrak n+1\implies2^\mathfrak n=2^{\mathfrak n+1}\...
- 13.4k
2
votes
0
answers
231
views
A question from Leon Simon's "Lectures on Geometric Measure Theory"
In a book I am reading (Leon Simon, Lectures on Geometric Measure Theory) at some point the author claims that if a certain property $(P)$ holds for almost every $n$-plane $\pi\subset \mathbb{R}^{n+k}$...
- 1,149
8
votes
1
answer
779
views
Connеcted components of irreducible algebraic varieties
I am wondering what is the possible (or maximum) number of connected components for an irreducible algebraic variety in $\mathbb R^n$ defined by a degree $d$ polynomial (i.e. hypersurface) in $\mathbb ...
- 185
-5
votes
1
answer
270
views
Calculus based on pdf [closed]
Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
- 201
1
vote
0
answers
197
views
Intersection of three quadrics: associating something geometric to these analogously to intersection of two quadrics
Consider the smooth intersection of two $4$-dimensional quadrics $Y = Q \cap Q' \subset P^5$. To the Fano threefold $Y$ we can associate a genus $2$ curve as follows. Take the pencil of quadrics $\{ ...
- 123
21
votes
3
answers
2k
views
Is every linear functional on a smooth finite dimensional vector space automatically smooth?
By a smooth finite dimensional vector space I mean a smooth manifold $M$ together with smooth operations $+ : M \times M \rightarrow M$ and $\cdot : \mathbb{R} \times M \rightarrow M$ turning $M$ into ...
- 1,307
4
votes
0
answers
102
views
Existence of a rational curve in the center of a birational contraction for symplectic singularities
Let $M$ be a holomorphically symplectic
complex manifold, and $f: M \to X$
a holomorphic, birational contraction to a Stein
variety $X$, contracting a subvariety $E$
to a point, and bijective outside ...
- 9,237
3
votes
0
answers
94
views
Strong uniqueness implies weak uniqueness for mean field FBSDE
Assume the standard probability setting and $B_t \in \mathbb{R}^d$ be the Brownian motion. Let $\xi$ be some random variable and $(X_{\xi}(t),Y_{\xi}(t),Z_{\xi}(t))$ be the solution to the following ...
- 54
1
vote
0
answers
48
views
Verdier (w) condition implies the $w_f$ condition when the restriction of $f$ in each stratum is a submersion?
Let $X\subset\mathbb{R}^n$ be and let $\Theta=(X_\beta)_{\beta\in I}$ a Verdier stratification for X. Let $f:X\rightarrow\mathbb{R}$ be a polynomial function, such that $f_{|_{X_\beta}}$ is submersion ...
11
votes
1
answer
605
views
If the universal cover has three boundary components, does it have infinitely many?
Suppose that $M$ is a compact, connected three-manifold with boundary. Suppose that $\pi_1(M)$ is infinite. Suppose that $\tilde{M}$, the universal cover of $M$, has at least three boundary components....
- 28.2k
0
votes
1
answer
142
views
Integral inner product with exponential function
Suppose on some unknown interval $[0, I]$ we have non-negative functions $f, g : [0, I] \rightarrow \mathbb{R}^{\geq 0}$.
If we know that
\begin{aligned}
\int_0^I f & = c \\
\int_0^I e^f & = e^...
- 129
1
vote
0
answers
118
views
Degrees of trigonometric numbers
For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers.
What is its degree?
That is, what is the minimal degree of ...
- 8,086
13
votes
1
answer
596
views
A symmetric function related to sums of square roots
Let $x_1,x_2,\dots,x_n$ be indeterminates (say over $\mathbb{Q}$). For
every sequence $\epsilon=(\epsilon_1, \dots,\epsilon_n)\in\{-1,1\}^n$
define $$ y_\epsilon = \sum_i \epsilon_i \sqrt{x_i}. $$ Let ...
- 50.8k
4
votes
1
answer
217
views
Is there a Bailey–Borwein–Plouffe (BBP) formula for $\gamma$ (euler-mascheroni constant)?
I was reading about BBP type formulas and there was a lot about $\pi$ and some $\log$'s. I started searching for some other constants and could find $2$ formulas for the catalan constant and learned ...
- 521
4
votes
1
answer
174
views
Maximal intersecting families on $\omega$ that are not ultrafilters
A family ${\cal S}\subseteq{\cal P}(\omega)$ is intersecting if any two members of ${\cal S}$ have non-empty intersection. Zorn's Lemma implies that every intersecting family is contained in a maximal ...
- 48.9k
8
votes
1
answer
319
views
Why does this combinatorial sum vanish?
We define the coefficients $c_{k,k-i}$ of ${n \choose k}$ by the following easy expansion:
\begin{align*}
& {n \choose k} = \frac{1}{k!} n(n-1) \dots (n-k+1) = \frac{1}{k!} \prod\limits_{t=...
- 1,295
1
vote
0
answers
36
views
Weighted least squares with matrices as unknowns
Let $M \in \mathbb{R}^{m \times n}$. Let $S \in \mathbb{R}^{m \times N_t}, U \in \mathbb{R}^{n \times N_t}$, with $ N_t \gg m,n$.
Moreover, $\epsilon = S - M U$, with $\epsilon$ zero mean white noise ...
- 123
1
vote
1
answer
362
views
Tensor product and semisimplicity of perverse sheaves
Let $X/\mathbb{C}$ be a smooth algebraic variety. Let $D_c^b(X,\bar{\mathbb{Q}}_{\ell})$ be the category defined in 2.2.18, p.74 of "Faisceaux pervers" (by Beilinson, Bernstein and Deligne). ...
- 615
1
vote
0
answers
57
views
CP maps obeying an equality
Start with a completely positive unital map $\psi:A\to B$ between $C^*$ algebras with identity. The equality $\psi(a^*a)=\psi(a)^*\psi(a)$ is true for all $a\in A$ in the case where $\psi$ is a star ...
- 1,143