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Question on an analytic number theory paper

My question is just a ``I don't understand what goes on in X of paper Y" so I don't know if I can post it; on the other hand it is research. I posted it in stackexchange but it received no ...
0
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0answers
17 views

Can factorization of very large numbers be aided by associating them with a series (described below) of quadratic polynomials?

My name is J. Calvin Smith. I graduated in 1979 with a Bachelor of Arts in Mathematics from Georgia College in Milledgeville, Georgia. My Federal career (1979-2012) in the US Department of Defense led ...
0
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0answers
11 views

Proving the number of coplanar subgraphs of a hypercube

Given an n-dimensional hypercube, join each vertex to obtain a complete graph. Then the number of subgraphs of four coplanar vertices is given by $6^n/8 -4^{n-1} +2^{n-3}$ Example: a 2D square has ...
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0answers
20 views

Notation for the regular and the adjoint representation of a finite group, in particular the symmetric group

The (left) regular representation of a finite group $G$ is theThe (left) regular representation of a finite group $G$ is the action on itself by left multiplication, $g\cdot h = gh$. The adjoint ...
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0answers
19 views

Homomorphisms and indecomposable decompositions of finite modules over polynomial rings

I am studying $\mathbb{N}^n$-graded, finitely generated modules $M$ over the $\mathbb{N}^n$-graded polynomial ring $K[X_1,X_2,X_3...,X_n]$. I know every such $M$ has an indecomposable decomposition $M\...
-1
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0answers
33 views

Differentiable values for x|x| [closed]

I was wondering if someone could tell me if the function x|x| where f(x)-x^2 if x<=0 and f(x)=x^2 for x>0 is differentiable at all values of x, particularly at x=0. if this is true, how would ...
1
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0answers
20 views

contracting homotopies of Koszul resolution of $\mathbb{C}[x_1, \ldots, x_n]$ and $\mathbb{C}_{q}[x_1, \ldots, x_n]$

Let $A : = \mathbb{C}[x_1, \ldots, x_n],$ $A_q : =\mathbb{C}_q[x_1, \ldots, x_n] = \mathbb{C} \langle x_1, \ldots, x_n \rangle / (x_ix_j = q x_jx_i)$. By Koszul resolution I mean $$\ldots \to A \...
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0answers
11 views

Tree-width of graphs in which any two induced cycles touch

Let $G$ be a graph s.t. any two cycles $C_1, C_2 \subseteq G$ either have a common vertex or $G$ has an edge joining a vertex in $C_1$ to a vertex of $C_2$. Equivalently: for every cycle $C$ the graph ...
2
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1answer
33 views

Auslander-Reiten sequence and projective covers

Let $R$ be an Artin algebra and let $0 \to A \to B \to C \to 0$ be an Auslander-Reiten sequence of finitely generated left $R$-modules. Is it always true that the projective cover of $B$ equals to the ...
1
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0answers
48 views

Generalized Thomas Ordowski conjecture at OEIS sequence A002326

OEIS is the online encyclopedia of integer sequences, Here is the link to the sequence $A002326$: https://oeis.org/A002326 For $n\geq 0$, the $n$th term in the sequence is defined as: $a(n)$ equals ...
8
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1answer
124 views

Definition of an n-category

What's the standard definition, if any, of an $n$-category as of 2020? The literature I can tap into is quite limited, but I'll try my best to list what I had so far. In [Lei2001], Leinster ...
1
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0answers
19 views

What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?

I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any. I'm looking at the description of a short-term position in ...
1
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1answer
51 views

Is having injection to hereditarily size sets equivalent with choice over ZF?

It's known that ZF proves that for every set $x$ there exists a set $H_x$ of all sets that are hereditarily strictly subnumerous to $x$. [see here]. Now is the following principle equivalent with ...
6
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1answer
108 views

From Topoi to Grothendieck categories

This question is mostly about a reference request. Let $\mathcal{E}$ be a Grothendieck topos. I am looking for a reference of the following two facts. I am aware that $(2) \Rightarrow (1)$ by Gabriel-...
1
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1answer
75 views

Given $f: X \to Y$, $g: X \to Z$, when does it exists $h: Y \to Z$ such that $hf \simeq g$?

I was wondering if the following general problem has a standard solution. Let $X, Y, Z$ be CW-complexes and $f: X \to Y, g: X \to Z$ be continuous functions. Is there a criterion to say when it does ...
3
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0answers
45 views

Dominant rational maps and compositions

According to many books (and also to Stack project, see https://stacks.math.columbia.edu/tag/01RI ), a morphism $f\colon X\to Y$ between schemes is said to be dominant if the image is dense in $Y$. ...
1
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0answers
38 views

Weakly homogenously Souslin sets and the measurability of $\omega_1$

I found this intriguing remark at the end of Woodin's Supercompact cardinals, sets of reals, and weakly homogeneous trees (1988): The assertion that every set of reals, in $L(\mathbb{R})$, is the ...
0
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0answers
22 views

Is a locally invertible weak limit of injective maps injective almost everywhere?

This is a cross-post. Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...
3
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1answer
88 views

Is every middle exact functor a derived functor?

Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to ...
4
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0answers
51 views

Slices for certain $C_p$-spectrum

By the work of Hill-Yarnall, for the group $G=C_p,$ all the slices for any spectrum, in particular, for $S^V \wedge H\underline{\mathbb{Z}}$, are classified. Here $V$ is a representation of $C_p.$ ...
2
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1answer
44 views

Injective choice function for finite Fano planes

Let $H=(V,E)$ be a hypergraph that is a finite Fano plane, that is, $V$ is a finite set and $E$ has the following properties: for $e_1\neq e_2\in E$ we have $|e_1|=|e_2|$, as well as $|e_1\cap e_2|=1$...
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0answers
25 views

Confusion about exponential versus inverse exponential distributed random variable

Note: I have asked this question on another forum as we well, but have not received a response. https://math.stackexchange.com/questions/3878100/confusion-about-exponential-versus-inverse-exponential-...
2
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0answers
62 views

Generalization Napoleon theorem, Bottema theorem and Brahmagupta theorem in one configuration

I am looking for a proof of Generalization Napoleon theorem, Bottema theorem and Brahmagupta theorem on one configuration as follows: Let four points $A, B, C, D$ in the plain, the perpendicular of $...
9
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4answers
223 views

Convexity and Lipschitz continuity

It is probably an easy question, but somehow I am stuck. Question Is the following statement true? If yes, how to prove it? Suppose that $f\in C^1(\mathbb{R}^n)$ is convex and $$ \langle\nabla f(x)-\...
1
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0answers
30 views

The map between a von neumann algebra which preserves support projections

Let $M$ be a von Neumann algebra. Suppose $T: S(M_{*})\cap M_{*}^+ \rightarrow S(M_{*})\cap M_{*}^+$ is a surjective isometry between normal state spaces of $M$ , then there exists a Jordan $*$-...
5
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2answers
75 views

When is a locally presentable category (locally) cartesian-closed?

Let $\kappa$ be a regular cardinal. A category $\mathscr C$ is locally $\kappa$-presentable iff it is the free completion of a small $\kappa$-cocomplete category under $\kappa$-filtered colimits. Is ...
-5
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0answers
43 views

See the description and prove that! [closed]

We know: AB = AC = BC Location of "D" in not specified, but we know it's on BC bow. Now prove that: AD = BD + CD ...
-4
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0answers
26 views

How do I prove (B/A) union (C/A) = (B union C) /A using: 1.) A containment proof and 2.) using set builder notation and logical equivalences? [closed]

What the title says. I need 2 answers using the methods listed above. Got an exam on monday, need to know how to do this
1
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0answers
87 views

Finding Isomorphism between $\mathbb{Z}^2\ltimes_{A,B(a,b,c,d)} \mathbb{Z}^4$ and $\mathbb{Z}^2\ltimes_{A,B(e,f,g,h)} \mathbb{Z}^4$

Let $A,B(a,b,c,d)\in\mathsf{GL}(4,\mathbb{Z})$ be given by $$A=\begin{pmatrix} I_2 & \begin{pmatrix} 0&0\\0&1 \end{pmatrix} \\0& -I_2\end{pmatrix},\quad B(a,b,c,d)=\begin{pmatrix} -2-b ...
7
votes
1answer
215 views

When do infinitary compactness numbers exist?

For a logic $\mathcal{L}$, let the compactness number of $\mathcal{L}$ (if it exists) be the least $\kappa$ such that every $(<\kappa)$-satisfiable $\mathcal{L}$-theory is satisfiable. Note that ...
9
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0answers
183 views

univariate integer version of Hilbert's 17th problem

Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
7
votes
1answer
152 views

Deformations of Hopf manifolds

Recall that a Hopf manifold is a quotient $\mathbb C^n\setminus 0$ by a free action of $\mathbb Z$ where the generator is acting by a holomorphic contraction. Question 1. Is it true that any ...
6
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0answers
119 views

Galois groups associated to matrices

When $A\in M_n(\mathbb{Q})$, we consider the pencil $A-xA^T$. Then $p_A(x)=\det(A-xA^T)$ is a self-reciprocal polynomial. $p_A$ can only be irreducible if $n=2p$ is even. Question: For every $p$, does ...
4
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2answers
81 views

Regarding unital positive operators

Let $\Omega$ be a domain in $\mathbb{C}^n$. Let $\mathbb{D}$ denote the open unit disc in $\mathbb{C}$. Let $C_b(\Omega)$ and $C_b(\mathbb{D})$ denote the space of all bounded continuous complex ...
0
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0answers
47 views

Total variation between Dirac mixtures

Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}^k$ and $0\leq a_i,b_i\leq 1$ be in the probability $n$-simplex. Define the finite measures $\mu=\sum_{i=1}^n a_i \delta_{x_i}$ and $\nu=\sum_{i=1}^n b_i ...
-3
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0answers
46 views

Extensions of Fields [closed]

Probably it's a simple question but I don't know how to prove that $\mathbb{Q}$($\sqrt{2}$) isn't isomorphic to $\mathbb{Q}$($\sqrt{7}$) like fields. I know that they are like $\mathbb{Q}$-Vector ...
28
votes
2answers
705 views

Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein?

If $V \hookrightarrow W$ and $W \hookrightarrow V$ are injective linear maps, then is there an isomorphism $V \cong W$? If we assume the axiom of choice, the answer is yes: use the fact that every ...
0
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0answers
15 views

Reference request: existence/uniqueness of solutions to convection diffusion equations

I am looking for a reference wherein existence and uniqueness results are proven for a system of PDEs of the form $$ \frac{\partial Q}{\partial t} + A \frac{\partial Q}{\partial x} = f(Q,x,t) + \frac{...
1
vote
0answers
26 views

Galois extensions of ring spectra and subextensions

In Galois extensions of structured ring spectra, Rognes introduces the notion of a faithful $G$-Galois extension of ring spectra. Let me recall what this means: We have a commutative ring spectrum $R$ ...
0
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0answers
43 views

Combinatorial optimization problem with interdependent constraints on points on a line segment

We are given a set $S$ of $n$ real numbers in $[0,1]$, with $0,1\in S$, and a value $\alpha\in(0,1)$. For each ordered triplet $(i,j,k)$ of values contained in $S$ (with $i\le j \le k$), we define its ...
-2
votes
0answers
30 views

Algorithm to evaluate game of go [closed]

could some one help me with an algorithm to evaluate a go board after a player has played his turn.the board is represented as a vector with a 1 in a dimension showing player 1's prices, and a 0.5 ...
-1
votes
0answers
12 views

Defining heavy-tailed/power law multivariate distribution with an explicit index

It is well known that univariate heavy tailed distribution can be defined by looking at its tail behavior, which is $x$ has heavy tail with index $\alpha$ if and only if $$f(x)\sim |x|^{-\alpha}$$ ...
2
votes
0answers
88 views

Bryant-Salamon $G_2$ manifold on the spinor bundle over $S^3$

I am trying to understand the spaces constructed in R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy. My first problem is, essentially, about ...
0
votes
1answer
37 views

Weak convergence to a “multi-Bernoulli” distribution

Let $(X_n)_{n\geq 1}$ be a sequence of random variables defined on the $d-$simplex ($d\geq 1$) : $\Sigma_d=\big\lbrace\boldsymbol{x}\in\mathbb{R}_+^d,\,\sum_{1\leq i\leq n} x_i=1\big\rbrace$. Assuming ...
-2
votes
0answers
25 views

Sequence expression for periodic and aperiodic sequence [closed]

Can you help me explain the following 2 sequences that can be periodic and aperiodic sequence look like? What does the cap arrow mean in sequence first item 1 and 3? Consider the following sequence ...
0
votes
1answer
57 views

the curvature wave equation

I was referred here from this question I asked on stackexchange. And now that I'm here, I see that this other question about geometric wave equations is very closely related to mine. But I have a ...
0
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0answers
19 views

Optimizing upper and lower bounds

Let $L_i:X\rightarrow [0,\infty)$ be continuous (objective) functions defined on a metric space $X$ and suppose that $$ L_1(x)\leq L_2(x)\leq L_3(x)\qquad (\forall x \in X). $$ Here, I imagine that $...
5
votes
0answers
130 views

If cohomology theory corresponds to intersection theory, valuation theory corresponds to -?

This is a meta question I asked myself. Cohomology theory is dual to an intersection theory. Is there anything valuation theory corresponds to in general? For instance, McMullen's polytope algebra is ...
4
votes
0answers
103 views

Normal singularities homeomorphic to a smooth space

I am looking for examples of normal complex spaces $X$ which locally around a singular point are homeomorphic to a smooth complex manifold. The only example I know is a curve with a cusp, but this is ...
2
votes
0answers
37 views

Is there a method to solve a non-linear quadratic matrix equation?

I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$ Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...

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