0
votes
0answers
15 views

Probability that an integer contains no $1\bmod 4$ prime factor

What is the probability that and integer contains at most $r$ prime factors of form $1\bmod 4$? What is the probability that and integer contains at most $r_t$ prime factors of form $(2t+1)\bmod 2^k$ ...
0
votes
0answers
5 views

floating point representation via the perspective of TTE/computable analysis

Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not ...
-1
votes
0answers
34 views

Are polls good approximations

Let $X$ be a finite set and $A\subseteq X$ and $m$ be a natural number satisfying $m\le |X|$ and $\epsilon$ be a small positive number. I'm interested to know if one selects a random $Y\subseteq X$ ...
0
votes
0answers
11 views

Obtaining z-transform of a multivariate nonlinear difference equation

My research area is not Mathematics, but I am facing a conceptual mathematical issue, the answer to which I could not find in regular textbooks and other material that the internet fetched me and ...
-2
votes
0answers
13 views

segment intersecting a tetrahedron coordinates [on hold]

I am trying to write C++ code to find the intersection points of a segment intersecting a tetrahedron. I reduced the problem like this: For each face of the tetrahedron (a triangle), find the ...
-2
votes
0answers
34 views

Solution manual of real mathematical analysis PUGH [on hold]

Is there anybody know that where I can find the solution manual of real mathematical analysis PUGH ? Thanks a lot.
0
votes
0answers
122 views

A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...
4
votes
0answers
71 views

Has unconditional convergence ever been proved other than by deducing it from absolute convergence?

Nobody's answering this question so I'll try it here. This is really a reference request: Has a certain kind of proof ever been used? A series $\displaystyle\sum_n a_n$ converges absolutely if ...
6
votes
1answer
94 views

Sums of reciprocals involving divisor sums

This question was asked at MSE but never received an answer. Let $A\subset\mathbb{N}$ be a subset of the natural numbers, and let $\sigma(n)$ denote the sum of divisors of $n$. Recall that we have ...
1
vote
1answer
35 views

approximate diameter of polytopes in high dimensions

I just came across the following problem: Let us consider the unit corner of the n-cube $$ \Delta^n = \left\{(t_1,\cdots,t_n)\in\mathbb{R}^n\mid\sum_{i = 1}^{n}{t_i} \leq 1 \mbox{ and } t_i \ge 0 ...
2
votes
0answers
73 views

Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
1
vote
0answers
26 views

Simultaneous integral equation on $SU(n)$

Consider a curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ and a smooth real, bounded function $w: [0,T] \rightarrow ...
1
vote
2answers
73 views

Homotopy fibers and stratified fibrations

Suppose I have a map $f:X \to Y$ of topological spaces and a nice stratification of $X$ ( say such that the inclusion of each stratum is a Hurewicz cofibration) such that the restriction of $f$ to ...
2
votes
0answers
44 views

Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly. For example, take $X=\mathbb{P}^n$, ...
1
vote
0answers
57 views

Category-theoretic characterization of zero-dimensional spaces

Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
2
votes
0answers
50 views

Explanation that Twistor Space of $S^4$ is $\mathbb{C}P^3$?

I am attempting to read Atiyah's paper on self-duality in four-dimensional Riemannian geometry, and I came across the following basic example: Let $S_-$ be the $SU(2)$-bundle of anti-self dual ...
1
vote
1answer
78 views

Need an explanation of a deduction

When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the ...
1
vote
1answer
33 views

The Laplacian of an expression involving the Ricci tensor

While doing some computations on a compact Riemannian manifold I have reached the following expression: $$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$ where $\Delta_y$ is the ...
4
votes
1answer
114 views

Laplace-Beltrami and averaging

For a Riemannian manifold $M$ with metric $g$ and Laplace-Beltrami operator $-\Delta_{g}$, what conditions on $M$ guarantee that $-\Delta_{g} u(x)$ measures the difference between $u(x)$ and the ...
4
votes
1answer
96 views

Parametrized Atiyah-Singer index theorem

Let $M$ be any smooth manifold (could be unorientable - I think). Let $E,F \to M$ be two complex vector bundles. Let $S$ be any compact space, and let $D_s:E\to F,s\in S$ be a continuous family ...
0
votes
0answers
26 views

Conditions for Mellin inversion

Under which conditions is the function $$ g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R} $$ the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...
0
votes
0answers
16 views

Log-concave distributions: Weighted sum of pdfs

Assuming $f_n(\cdot)$ is a log concave function (e.g., pdf of Gaussian distribution) and $0\le q_n\le 1$ for all $n\le N$, I am trying to find conditions under which the following holds ...
1
vote
0answers
57 views

Integral homology of $S^{n-1}/\pi$, $H_*(S^{n-1}/\pi; \mathbb{Z}_p)$ [migrated]

Let $p$ be an odd prime number. Regard the cyclic group $\pi$ of order $p$ as the group of $p$th roots of unity contained in $S^1$. Regard $S^{2n-1}$ as the unit sphere in $\mathbb{C}^n$, $n \ge 2$. ...
3
votes
1answer
48 views

Integral domains equal to intersection of their height one localizations

Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$? It is a standard fact that Krull domains, and thus noetherian ...
0
votes
0answers
36 views

Is this permutation-sum problem NP-hard?

A new, tighter tardiness bound has been found for global Earliest-Deadline-First scheduling of jobs on symmetric multiprocessors. But this bound seems to be particularly hard to compute. In ...
2
votes
1answer
40 views

Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...
3
votes
2answers
124 views

Relationship between $H_*(X, A)$ and $H_*(Y \cup_f X, Y)$? $\pi_*(X, A)$ and $\pi_*(Y \cup_f X, Y)$?

Let $A$ be a subcomplex of a CW complex $X$, let $Y$ be a CW complex, and let $f: A \to Y$ be a cellular map. What is the relationship between $H_*(X, A)$ and $H_*(Y \cup_f X, Y)$? Is there a similar ...
2
votes
1answer
174 views

Is there a nonabelian free group inside a group of positive rank gradient?

Let $G$ be a finitely generated residually finite group with positive rank gradient, and let $F_2$ be the free group on $2$ elements. Must there be an embedding $i \colon F_2 \to G$ ? A group ...
-1
votes
0answers
37 views

A question on spectrum [migrated]

Let $A,B \in {C^{n \times n}}$ and ${\sigma (A + B)}$ is spectrom of $(A+B)$. Suppose $M = \left\{ {\lambda \in C:\lambda \in \sigma (A + B),\left\| B \right\| \le \varepsilon } \right\}$ $F(A) = ...
3
votes
0answers
33 views

Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
0
votes
1answer
54 views

Are constructive characterisations of k-regular (simple) graphs known?

By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some ...
4
votes
1answer
89 views

Kan extensions of pseudofunctors

Can anyone suggest a reference for (left) Kan extensions of pseudofunctors? In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors $\mathscr A \xrightarrow{G} \mathscr ...
3
votes
1answer
27 views

convert a special case of nonlinear fractional programming into a convex problem

Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this ...
-3
votes
1answer
32 views

upper bound for a convex fractional function [on hold]

Consider the following convex fractional function $$f\left( {\bf{x}} \right) = \frac{1}{{1- {\bf{x}}}}$$ where ${1- {\bf{x}}} > 0$. Is it possible to obtain a linear or quadratic upper bound ...
-1
votes
0answers
100 views

Is $(X_G, d_G)$ , compact manifold?

Let compact topological group $G$ acts on $(X,d)$ . We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It is clear that ...
12
votes
2answers
258 views

Are there open problems for primes which are known for probable primes?

Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$. Probable primes are the union of the primes and base two pseudoprimes. This definition is much ...
7
votes
1answer
339 views

Sums of unique squares

Let $\mathbb{N}$ denote the positive integers and let $S = \{n^2: n\in \mathbb{N}\}$. For any positive integer $k$ we define $$\text{sq}(k) = |\{F\subseteq S: F\neq \emptyset, F\text{ is finite and } ...
-1
votes
0answers
31 views

Why do there is a unique continuous homomorphism? [migrated]

Is this a right place to ask help for an exercise? Let $n\geq 2$ be an integer and $D=\mathbb Z[1/n]$. Let $A$ be a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ...
6
votes
0answers
125 views

On an unpublished result of Magidor

In 1970th, Magidor proved the following important results: (1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and ...
1
vote
1answer
100 views

Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$

Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that $$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$
2
votes
2answers
96 views

Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form? \begin{bmatrix} a^2 ...
1
vote
0answers
19 views

Strong solution to an SDE with a discontinuous diffusion term

I am having an SDE for which I would be in trouble if there were no strong solution. The SDE is - $ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$ where $b_1$ and $b_2$ are two ...
-3
votes
1answer
97 views

Encyclopedia of Mathematics?(non-Alphabetical) [on hold]

Do you know any Encyclopedia of Mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level. And what's the difference between say, ...
2
votes
0answers
47 views

Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
20
votes
2answers
755 views

What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is $\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...
6
votes
2answers
153 views

Repeats of all binary strings of length k

The question seems like it should be known, but I was not able to find it anywhere. How many binary strings of length $n$ are required so that for every $k$ positions in these strings, all $2^k$ ...
2
votes
0answers
80 views

An equality of discriminant and resultant divisors

Let $\Phi$ be the root system of a split group $G$ over a field $k$. The differentials $d\alpha$ of the roots define a polynomial called the discriminant $$\prod_{\alpha\in\Phi}d\alpha$$ on $\mathfrak ...
8
votes
0answers
54 views

What are some useful invariants for distinguishing between random graph models?

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as: The Erdos-Renyi model The Stochastic Block model The Watts-Strogatz model The Barabasi-Alber model ...
1
vote
0answers
61 views

Closed form answer to a naive integral [on hold]

Let a and b be positive real numbers. How to find a closed form answer to the integral $$\int_0^t \left(-a t + \big(1+ \dfrac{2bt}{3}\big)^{-3/2}\right)^{5/3} dt$$ If it is not possible to find a ...
-4
votes
0answers
36 views

Linear Independence for functions defined by integration [on hold]

Given that the set of functions $$f_i(x,y), \quad i=1,\dots,n$$ are linearly independent for $(x,y) \in [0,1]^2$. Is the set of functions, $g_1,\ldots,g_n$, defined by $$ g_i(x) = \int_{y\in [0,1] } ...

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