# All Questions

**-1**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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 ...