-1
votes
0answers
16 views

Are there analogues for the following conjectures/theorems in 3-factor numbers, 4-factor numbers etc?

Prime numbers have 2 factors. There are numbers with 3 factors, 4 factors and so on. Are there analogues for the following conjectures/theorems in n-factor numbers: Prime Number Theorem Riemann's ...
1
vote
0answers
64 views

What are your favorite instructional counterexamples on sequences? [migrated]

In this article, I give counterexamples regarding real sequences. And in that one some others. In particular counterexamples answering questions like: "If for all $p \in \mathbb{N}$ $\lim\limits_{n ...
0
votes
0answers
32 views

On Shannon Capacity of graph

1. Is there a graph $G$ that is DEFINITELY KNOWN to definitively Shannon zero error capacity $\Theta(G)$ only at infinite strong power? All examples I come across achieve capacity at finite graph ...
2
votes
0answers
47 views

What are cohomology of Lie algebra with coefficients geometrically?

I want to find analog of following two statements. Let $G$ be a discrete group, $M$ is representation of $G$. Local systems on $BG$ are the same as $G$ representations (because $\pi_1 (BG) =G$). Let ...
2
votes
0answers
17 views

Directed Hypercube Minimal Cuts

If $[n]:=\{1,2,\ldots, n\}$ for some $n\in\mathbb{N}$, then the hypercube digraph of dimension $n$, denoted $H_n$, is the graph whose set of vertices is the power-set $\wp([n])$ where two vertices ...
-4
votes
0answers
26 views

am I evaluating this truth table correcly? [on hold]

Does (((p∨q)∧((q→r)⊕(p∧r)))↔(r∧q))→(p∨r) evaluate into the below truth table? ...
0
votes
0answers
11 views

Convergence in the Wasserstein metric and the square root function

Let $f$ be a smooth probability distribution on the unit square $S$ such that $f(x)>0$ on $S$. Let $\{g_i\}$ be a sequence of smooth probability distributions such that $g_i(x)>0$ on $S$ as ...
0
votes
1answer
70 views

integral or rational cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
0
votes
0answers
22 views

How to write an equation for a fixed range scale [on hold]

I apologize if this question isn't professional grade. I'm writing a report and I need to use an equation to express a relationship in the Methodology section. It's a two-part equation but it's the ...
-1
votes
1answer
59 views

Can a rational map be extended without using resolution of indeterminacies?

Suppose I have a finite morphism C --> D, where C is an open subvariety of some projective variety C' and D is open in D', also projective. Thus there is a rational map from C' to D'. Is there a way ...
0
votes
0answers
49 views

homology of configuration spaces of non-compact manifolds

Let $M$ be a manifold. Let $F(M,n)$ be the configuration space of $n$-tuples on $M$. Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered ...
-2
votes
0answers
59 views

Definition of infinity [on hold]

Recently I watched a video explaining how the infinite sum of the natural numbers is equal to -1/12.When seeing this there were some questions bubbling out from my head which I couldn't explain ...
0
votes
0answers
43 views

Decomposing matrices to lower ranks

Given real matrix $M\in\{0,1\}^{n\times n}$ of rank $r$. How many $M_i\in\{0,1\}^{n\times n}$ of rank $s$ does one need to write $M'=\sum_{i=1}^ta_iM_i$ for some $a_i\in\Bbb R$ where maximum absolute ...
0
votes
0answers
13 views

What is the AAN algorithm for computing the fast DCT, and does it work for arbitrary or even-sized input vectors?

I'm trying to implement a faster DCT algorithm for an image perceptual hashing library I maintain in Rust. I based my original implementation on listing2.c from ...
3
votes
2answers
190 views

SU(2) and differential forms

I am a physicist with some background in differential geometry and I apologize for any possible unprecise terminology. Consider the Lie group $SU(2)$ and its tangent space $su(2)$ forming a tangent ...
0
votes
0answers
55 views

Isomorphism in a derived category of chain complexes with rational coefficients

Let $C$ be the category of quasi-projective schemes over a base field $k$ (maybe I will need some assumptions on $k$). Let $Ab(C_{\tau})$ be the category of Abelian sheaves on a site $C_{\tau}$, where ...
-1
votes
1answer
55 views

p-groups and 2-generated abelian images

Let $p$ be a prime number. Is there a finite nonabelian $p$-group $G$ such that any finite epimorphic $2$-generated image of $G$ is abelian?
8
votes
1answer
217 views

Are all smooth functions composites of 0-, 1-, and 2-ary functions?

I will formalize my question in terms of algebraic theories. Background: Recall that an algebraic theory (in the sense of Lawvere) is a category $\mathcal{C}$ which is closed under taking finite ...
-4
votes
0answers
60 views

Formula to sum 1/sqrt(i) [on hold]

Is there a formula to calculate the sum of 1/$\sqrt i$ for n numbers? My application repeatedly calculates $\sum\limits_{i=k}^{k+m} \frac{1}{\sqrt i}$ , for different values of k and m. It spends ...
0
votes
0answers
67 views

knots complements and geometry [on hold]

Let $K$ be a knot in $S^{3}$. If I understand correctly the complement $S^{3}-K$ is an Eilenberg Maclane space. Is $S^{3}-K$ always a hyperbolic 3-manifold ?
4
votes
2answers
105 views

Geometric dominating set: NP-complete?

Let $G=(V,E)$ be a geometric graph, a graph embedded in the plane whose edge lengths are the Euclidean distance between its endpoint vertices. Say that a set of vertices $D \subseteq V$ is a geometric ...
1
vote
1answer
67 views

Lyndon–Hochschild–Serre spectral sequence for not normal subgroup

Is there analog of Lyndon–Hochschild–Serre spectral sequence for not normal subgroup? What can you say about it? Can you describe $E^{p, q}_1$ ? What is about $E^{p, q}_2$? What is the best technique ...
1
vote
0answers
53 views

A dual version of the generalization a theorem of Øystein Ore in group theory

This post is a dual version for the Generalization of a theorem of Øystein Ore in which we have proved: Theorem: Let $(H \subset G)$ be an inclusion of finite groups such that the lattice ...
0
votes
1answer
45 views

Series estimate

Let $\theta\in(0,1)$ be given. I define for $a>0$ and $\lambda \ge 1$, $ S(\lambda,a )=\sum_{k\ge 1} k^{\frac12-\theta}e^{-a\vert k-\lambda\vert}. $ I want to prove that $$ ...
1
vote
0answers
104 views

Consequences failure of $\tau$ conjecture

A question Bounds for constructing $n!$ with additions, subtractions, and multiplications starting from $1$ was asked on constructing $ak!$ with ring operations. $\tau$ conjecture states if $\exists$ ...
0
votes
0answers
62 views

Ring of integers in Artin-Schreier extension

Question put in mathstackexchange but received no answer. It is well-know( see Goldschmidt book: Algebraic Functions and Projective Curves) that a for $q$ a power of $2$ a quadratic separable ...
4
votes
1answer
65 views

Some questions about “inspecting” the boundary of a closed ball in Hilbert space

Let H be a separable Hilbert space and suppose that H is infinite dimensional. Let B be a closed ball of H-which has a positive radius-and let S be the boundary of B. A non-empty subset C of H is an ...
-2
votes
0answers
27 views

Find steady-state solution [on hold]

Governing equation: ∂Ω/∂t = (∂^2)Ω/(∂x^2)-DH(∂Ω/∂x) Find the steady-state solution Ω(x) from the governing equation with boundary conditions Ω(0) = 1 and Ω(1) = 0
-2
votes
0answers
82 views

Mathematical Expositions on Motivation [on hold]

I wanted to ask this question, However I hope that it is not too soft for this site. What I want to ask is if writing an exposition on motivation for a topic of mathematics would be relevant? I.e., I ...
0
votes
0answers
84 views

Solution to a PDE with constant data - what is the fault in my proof?

Let $C=\Omega \times (0,\infty)$. We want to find a solution $v \in H^1(C)$ such that given $u \in H^{\frac 12}(\Omega)$, $$\int_0^\infty\int_\Omega \nabla v \nabla \varphi + v_y\varphi_y = ...
-1
votes
0answers
53 views

Number of solutions of the congruence, $x-y \equiv z \pmod{n}$,$x,y\in U_{n}$?

I known number of solution of the congruence, $x+y \equiv z \pmod{n}$,$x,y\in U_{n}$ is $N(z)=n\prod_{ p\backslash n}\left(1-\frac{\varepsilon(p)}{p}\right)$,where $\varepsilon(p)$=$\left\{ ...
36
votes
0answers
496 views

What is a Frobenioid?

Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title. Recently, there has been a flurry of new discussion ...
0
votes
0answers
54 views

projective map from $\overline{\mathcal{M}}_{0,n}$

Suppose I have a morphism $f:\overline{\mathcal{M}}_{0,n} \to \mathbb{P}^N$ birational onto its image, and I know exactly what $F$-curves are contracted (or "dually", what divisors are contracted). Is ...
-6
votes
0answers
69 views

Please reply fast. [on hold]

Is there any formula for product of sines like sin a sin b sin c sin d. Please tell me the formula for this , if any for both product upto a specific number of terms and that to infinite number of ...
8
votes
1answer
189 views

Reference request: Topology on the space of smooth compact submanifolds

In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...
2
votes
1answer
78 views

How many expressions can be formed with $k$ commutative and associative functions?

This is a generalization of a question I posted to Math.SE here. Suppose we have $k$ binary functions $f_1, f_2, \ldots, f_k$, all of which are commutative and associative, and $n$ indeterminates ...
-1
votes
0answers
188 views

A Small Discussion I want to have with Professional Mathematicians [on hold]

My professor is a really learned man, having written and published more than 18 research papers within 20 years. Hence I greatly admire him and value his opinion. He is, of course, a PhD in ...
1
vote
0answers
21 views

minimal entropy approximation of a discrete random variable

Let $X$ be a $\mathbb{N}$-valued random variable. Define $$ H^\epsilon_n(X) = \inf_f H(f(X)) $$ where $f$ runs over all functions $\mathbb{N} \to \mathbb{N}$ such that $\Pr(f(X)\neq X)<\epsilon$ ...
0
votes
0answers
28 views

How to show validity in classical logic? [on hold]

Firstly, I would like to know what does it mean to be a valid expression in classical logic. Secondly, How do we show validity of a formula (in sequent calculus) such as: (∀x A → ∃B) → ∃x(A → B) As ...
13
votes
1answer
227 views

A congruence involving binomial coefficients

The following open problem was shown to me by Maxim Kontsevich. I state it in a different but equivalent form. Let $a(n)$ be the sequence at http://oeis.org/A131868, that is, $$ a(n) ...
3
votes
1answer
186 views

book about string theory a la Von Neumann [duplicate]

Can we summarize string theory (in its actual state) in some principles and fundamental equations like electromagnetism, general relativity, quantum mechanics and classical mechanics ? I am looking ...
2
votes
0answers
24 views

Signs associated to self-dual simple objects in a fusion category

Every self-dual simple object $X$ in a fusion category can canonically be assigned a number $a$, from its "snake" associator element: The square of $a$ equals Muger's "squared dimension" of $X$, an ...
-4
votes
0answers
75 views

Algebraic Curves: Exercise 2.17 (William Fulton) [on hold]

Let $V=V(Y^2-X^2(X+1))\subset\mathbb{A}^2$, e $\overline{X}, \overline {Y}$ the residues of $X,Y$ in $A(V)$ its coordinate ring; let $z= \dfrac{\overline{Y}}{\overline{X}}\in K(V)$. Find the pole sets ...
0
votes
0answers
105 views

Numbers summing to distinct integers

We want to find $r$ positive integers $\{a_i\}_{i=1}^r$ such that of atmost $(2s+1)^r$ values obtained from $$\sum_{i=1}s_ia_i$$ where $s_i\in\{-s,-s+1,\dots,0,\dots,s-1,s\}$ with $s\leq r$, we insist ...
0
votes
0answers
15 views

Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$. Suppose we have diagonalized using $LMR=D$. I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of ...
-2
votes
1answer
124 views

Direct image of structural sheaf [on hold]

I am sorry if my question is not of high level!! Let $\pi:X\rightarrow Y$ be a double cover where $X$ and $Y$ are projective smooth curves. Is it true that $R^1\pi_*\mathcal O_X=0$ ? Why ? Thanks ...
4
votes
1answer
218 views

Formal group law over $\mathbb{F}_p$

Let $p$ be a prime. For each $n > 0$ there is a unique 1-dimensional commutative formal group law $F$ over $\mathbf{Z}$, $F(X, Y) = X + Y + \dots \in \mathbf{Z}[[X, Y]]$, whose logarithm function ...
1
vote
0answers
29 views

Enumerating matrices function of ranks

Is there an expression/approximate expression for number of real matrices $M\in\{0,1\}^{n\times n}$ of rank $r\leq n$?
4
votes
2answers
235 views

how to calculate the sum of remainders of N?

I'm trying to sum the remainders when dividing N by numbers from 1 up to N $$\sum_{i = 1}^{N} N \bmod i$$ It's easy to write a program to evaluate the sum if N is small in O(N) but what if N is large ...
0
votes
0answers
94 views

Weyl group representation

Let $G$ be a reductive p-adic group. Let $W$ be a weyl group. if $w$, and $w_o \in W$. I want to know in which case we have $w w_o w^{-1}= w_o$ ? in case if $w_o(\theta)=\theta $ where $\theta$ is a ...

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