**0**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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] } ...