0
votes
0answers
2 views

When are isotrivial families split by a finite base-change?

A well-known theorem of Grauert and Fischer states that a smooth proper family of complex manifolds is a locally trivial fibration as soon as all the fibers are isomorphic. It is also easy to obtain ...
0
votes
0answers
12 views

How do eigenvalues change if we duplicate a row and column of a symmetric matrix

Let ${\bf A}$ be a size $n \times n$ symmetric positive semidefinite matrix with the first column being ${\bf a}_1$. If we define a new matrix, \begin{align} {\bf B} = \left[\begin{array}{cc} a_{11} ...
1
vote
0answers
12 views

$RO(G)$-Graded Cohomology Theories

Let $G$ be a compact Lie group with real representation ring $RO(G)$. Recently, I have been learning about some aspects of $RO(G)$-graded cohomology theories (for a precise definition, see Chapter ...
0
votes
1answer
53 views

Example of a specific manifold

I want to find a example of a manifold that has positive scalar curvature but is not half conformally flat. Does there exists such manifolds? Thanks.
0
votes
0answers
11 views

Proximal operator of modified L1 matrix norm

In literature proximal operator $prox_{\lambda f} : R^n \rightarrow R^n$ of $f$ is defined as: $prox_{\lambda f}(V) = argmin(X) (f(X) + (1/2 \lambda)||X-V||^2_2)$ Consider now $g(X) = ...
4
votes
0answers
66 views

The definition of < in Robinson's Q

I recently had to explain how the basic axioms in Simpson's Subsystems of Second Order Arithmetic were interpretable in Robinson's Q. Most of the axioms are actually the same, except that Simpson ...
0
votes
0answers
18 views

Entropy of matrix vector product

Consider a random $n$ by $n$ circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$. Let $M'$ be the $m$ by $n$ matrix which is formed by taking the first $m$ rows of ...
1
vote
0answers
28 views

Question about the specialization map for Etale Fundamental Groups

Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...
2
votes
0answers
51 views

Both NP-hard but different

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?
3
votes
0answers
46 views

The definition of Reedy category

The common definition of Reedy category seems to be this one that a Reedy category is a small category $R$ with two wide subcategories $R_+$ and $R_-$ and an ordinal-valued degree function on its ...
-7
votes
0answers
54 views

The Pythagorean Triples and the primes [on hold]

for many primes $\zeta>43$, there exists infinitely many Pythagorean triples $x,y,z$ such that $$\zeta=\delta\left(\cfrac{1}{x}+\cfrac{1}{y}+\cfrac{1}{z}\right)$$ where $\delta$ denote the ...
6
votes
0answers
28 views

Max flow, min cut on manifolds

If a graph has some half edges marked "input" and some half edges marked "output", it is well known that the smallest number of edges which must be cut to disconnect input from output is equal to the ...
3
votes
0answers
24 views

Set system with prescribed intersection sizes

Questions: What is the asymptotic maximal size of a $4$-uniform (every set has 4 elements) set system $\mathcal{A}$ of subsets of $[n]$ such that, no two sets have size of their intersection $2$? In ...
6
votes
3answers
212 views

Examples of famous 'workhorse' theorems

I use the term 'workhorse' to describe a theorem which is technically challenging to prove, perhaps very deep, but the statement is either uninteresting at first glance or too imposing to be ...
0
votes
1answer
99 views

is cross product of 2 or 3 vectors in 4 D is possible ? why ? i'm an engineer and please kelp me in explaining in a way i can understand? [on hold]

Is cross product of 2 or 3 vectors in 4 D is possible ? why ? i'm an engineer and please kelp me in explaining in a way i can understand ? It is also not clear that why vectors depend on the dimension ...
0
votes
0answers
21 views

A question on Gorenstein Liaisons of space curves

Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and ...
0
votes
0answers
146 views

Grothendieck's letter to serre

Is it the letter dated in 27/08/1965 of Grothendieck where he presents to Serre the Standard conjectures on algebraic cycles?
-6
votes
0answers
92 views

A new set of Primes [on hold]

Considering the following do there exists a fatal error? For infinity number of prime $\beta$, there exists $\zeta$, $\alpha$ and $\gamma$ $\mathfrak{Primes}$ such that ...
0
votes
0answers
29 views

edge graph reconstruction conjecture : set vs multi set

Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted ...
0
votes
0answers
50 views

Tarski's undefinability theorem and IF logic

The Tarski undefinability theorem is valid when there are several constrains on the logic on which it is applied. The most important is that the logic mus to be closed under contradictory negation. ...
1
vote
0answers
48 views

A topological criterion for connectedness of a semi-ample divisor

I have a half page long proof of the following statement, and I would like to know if this is (a corollary of) a well known statement. Maybe there is a reference or a three lines proof? Statement. ...
1
vote
0answers
27 views

Parametrized cancelations in stable Morse theory

Let $B$ be a closed manifold. Let $\pi : M\to B$ be a submersion such that each fiber is a manifold without boundary. Let $f : M \to \mathbb{R}$ be a function such that the restrictions $f_x$ to each ...
1
vote
1answer
44 views

Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...
1
vote
1answer
108 views

Localization of symmetric monoidal category

Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T ...
2
votes
0answers
70 views

Does the set of automorphisms of a cyclic group exhibit some sense of randomness?

I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome. ...
2
votes
1answer
58 views

Projection onto $\ell^{2,1}$ ball

Does anyone have an idea how to project onto the $\ell^{2,1}$ ball efficiently, i.e. how to solve $$ u = \arg \min_u \|u-f \|^2 \text{such that } \left(\sum_i \big(\sum_j |u_{i,j}|\big)^2 ...
2
votes
1answer
102 views

Number of Matrices with bounded determinant

Here's my question: Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...
0
votes
0answers
15 views

Existence and Uniqueness of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
3
votes
0answers
54 views

What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...
-3
votes
0answers
48 views

What are the odds in the card game war that the game will start and finish on the first battle? [on hold]

What are the odds? My 6 year old daughter and I played war. On the first card we both played sevens and the battle bagan. We played three face down and one up and they matched again. Again, we ...
0
votes
0answers
28 views

Embedding dimension: local finiteness & intuition for more general spaces

Can every complex analytic space be covered by Stein spaces of finite embedding dimension? I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is $$ ...
1
vote
1answer
117 views

Examples of nontrivial local systems in Decomposition Theorem

There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that $$Rf∗IC_X \cong \oplus_a ...
2
votes
3answers
375 views

Reference request: SGA7

I want to read SGA7. Without considering the others SGA and EGA, Which are the textbooks for monodromy theory?
2
votes
1answer
46 views

Double quotients of Coxeter groups have the chain property?

Let $(W,S)$ be a Coxeter group with Bruhat order $\leq$ and length function $\ell(w)$. Definition: a subset $X \subset W$ has the chain property if whenever $x,y \in X$ with $x < y$, there ...
3
votes
2answers
118 views

A Lie algebra identity

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix a basis $e_1,\dots,e_n$ of $\mathfrak{g}$ and let $e^1,\dots,e^n$ be its dual basis. We also use $e^i$ to denote the left-invariant ...
-5
votes
0answers
33 views

Precalculus Help? [on hold]

The lengths of the sides in a right triangle form three consecutive terms of a geometric sequence. Find the common ratio of the sequence. (There are two distinct answers. Enter your answers as a ...
2
votes
0answers
56 views

Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background: There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...
2
votes
1answer
71 views

References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat ...
3
votes
1answer
55 views

$C^\infty$-vectors in general representations of Lie groups on locally convex spaces

This question is related to this one. Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything like that). In ...
6
votes
0answers
70 views

Newton polygons of modular polynomials

This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that ...
-1
votes
0answers
40 views

Question about lim and sup of a function [on hold]

Let f(t,d) be a continuous function in t. Prove that: $\lim\limits_{m\to \infty}\sup\limits_{t\in[t_1,t_2]}f(t,d_m)=\sup\limits_{t\in[t_1,t_2]}\lim\limits_{m\to \infty}f(t,d_m)$ Does anyone have any ...
3
votes
1answer
114 views

Topology on the dual of a Frechet space

If $F$ is a Frechet space, is there any locally convex space topology on the dual $F'$, such that for each local diffeomorphism $f$ from an open subset $U$ of $F$ to $F$, the map $U \times F' ...
2
votes
1answer
96 views

Interpretation of Hochschild Homology groups

In all the literature I've come across there are many concrete interpretations of the first few Hochschild Cohomology groups. For example $HH^1(A,M)\cong Derivation/Inner Derivations$ etc.... In ...
6
votes
0answers
49 views

Multiplicative domains and conditional expectations

Let $M$ be an injective von Neumann subalgebra of $B(H)$. For a completely positive map $\phi:B(H)\to B(H)$, let $Mult(\phi)$ denote be the multiplicative domain of $\phi$. For any conditional ...
0
votes
1answer
70 views

Maximize combinatorial sum for boolean function

I am trying to maximize the function $$ S(f)=\sum_{j=0}^{n-\frac{n-1}{t}}(-1)^j{n-\frac{n-1}{t}\choose{j}}\sum_{i=0}^{\frac{n-1}{t}}(-1)^{f(i-j)}(t-1)^i{\frac{n-1}{t}\choose{i}} $$ for a function ...
-6
votes
0answers
40 views

solve for three unknowns. [on hold]

Apple cost 97 dollars. Orange cost 56 and lemon cost 3. The total amount spent is 16047 dollars and total fruits bought is 240 and each one is bought atleast one. Calculate how many of each have been ...
0
votes
0answers
63 views

Rationally building bridges from Jacquet-Langlands to Langlands functoriality conjectures

For now I mainly worked on very classical proofs (viz. Bolte & Johansson, Bergeron) of the Jacquet-Langlands correspondence, but I hope to be able to understand in what this special case lead ...
4
votes
0answers
74 views

The (global) theory of Borel equivalence relations

What do we know about the complexity of the theory of Borel equivalence relations, with the Borel reducibility order $\leq_B$? That is, let $\mathcal{B}$ be the set of all Borel equivalence ...
6
votes
1answer
168 views

If $G$ is compact, $H \leq G$ open, $V$ an irreducible $H$-rep, is $\text{Ind}_H^G$ semisimple?

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a finite-dimensional topological ...
4
votes
1answer
93 views

Graphs where each edge belongs to the same number of 1-factors

Let $G$ be a simple connected graph that has at least one 1-factor. We'll define: $G$ has property A iff it is edge-transitive. $G$ has property B iff each edge belongs to the same number of ...

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