# All Questions

**1**

vote

**1**answer

26 views

### Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...

**1**

vote

**0**answers

15 views

### A construction with homotopy colimits and homotopy pullbacks for descent

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...

**4**

votes

**1**answer

20 views

### Conservativity of multiplicative linear logic over intuitionistic multiplicative linear logic

It is well known that multiplicative linear logic (MLL) is conservative over intuitionistic multiplicative linear logic (IMLL). In other words, if an IMLL formula is provable in MLL then it is already ...

**-5**

votes

**0**answers

62 views

### Is my proof of the Schoenfeld's inequality correct? [on hold]

Full preprint here.
Theorem 4.1.
For any $x\ge 2$ we have
$$
\begin{equation}
\theta(x)-x<\frac{1}{8\pi}\sqrt x\log^2 x. \;\;\;\;\;\;\;\;\;\;\;(4.1)
\end{equation}
$$
Proof:
It's known that ...

**0**

votes

**0**answers

21 views

### Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function ...

**-3**

votes

**0**answers

34 views

### Quardic Equation [on hold]

We know that a quardic equation have two roots.After solving this equation:8x^2-33x-35=0 we get two roots.The first one is:5 and the second one is:-7/8.But the root -7/8 doesn't satisfy the given ...

**1**

vote

**0**answers

27 views

### Generalize Gauss-Bonnet Formula to non-simple closed curves

According to the Classical Gauss-Bonnet Formula, I think it should can be generalized to non-simple closed curves in the following sense:
For a domain $\Omega$ enclosed by an non-simple closed curve ...

**1**

vote

**1**answer

57 views

### A Poincare-Type Inequality and its generalization

Let $f(\theta)$ be a fixed positive $2\pi-$periodic $C^1$ function on $\mathbb{R}$ with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$
Does for any ...

**4**

votes

**0**answers

41 views

### What is the Turing degree of $\mathbb{C}_{exp}$?

Let $\mathbb{C}_{exp}$ be the theory of the complex numbers in the language of exponential rings. I am interested in the Turing degree of $\mathbb{C}_{exp}$. As the natural numbers are definable in ...

**-2**

votes

**0**answers

12 views

### Random sum of random variables, not in expectation [on hold]

If $N\geq 1$ is a finite random variable (in this case a binomial Bin(n,p) random variable conditioned to be $\geq 1$) then can we say the following?
$$\sum\limits_{i=1}^N \frac{1}{N^2} ...

**0**

votes

**0**answers

88 views

### Flatness and intersection of fibers

Let $f:X \to Y$ be a flat, proper, surjective morphism between noetherian schemes. Assume $Y$ is irreducible and smooth over $\mathbb{C}$. Suppose that $X$ is the union of two schemes $X_1$ and $X_2$ ...

**8**

votes

**0**answers

60 views

### Nilpotence of the stable Hopf map via framed cobordism

The Pontryagin-Thom construction shows that the stable homotopy groups of spheres are the same as the groups of stably framed manifolds up to cobordism. Specifically the Hopf map corresponds to the ...

**1**

vote

**0**answers

20 views

### Invariant Girsanov Theorem on a Riemannian manifold

This is somewhat a follow-up on this post.
Let $X_t$ be the stochastic process on a compact Riemannian manifold generated by the (possible time-dependent) second-order elliptic operator $L_t$. Let ...

**2**

votes

**1**answer

42 views

### “C^0 estimate for solutions to $\Delta(u)+e^{-u} \geq 0$”

Let $u: \mathbb{R}^2 \to \mathbb{R}.$ Suppose I have a solution to the equation
$$\Delta(u)+e^{-u} \geq 0$$ on $\mathbb{R}^2$. Let r be the radial coordinate on $\mathbb{R}^2$. Suppose that $$lim_{r ...

**1**

vote

**1**answer

78 views

### Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a complex submanifold (closed) of codimension(complex) r. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$.
1) we have the ...

**-1**

votes

**0**answers

17 views

### Roots of 3 polynomials in two variables in terms of the coefficients [on hold]

Let p,q,h\in\mathbb{C}[x,y]
If {p_{ij}}, {q_{ij}} and {h_{ij}} are the coefficients of p,q and h respectively.
(...

**2**

votes

**1**answer

459 views

### A stronger version of Fermat's last theorem

Motivated by Fermat's last theorem, one may wonder the following conjecture is true or not.
The equation $x_1^m+\cdots+x_n^m=1$ has nonzero rational solutions iff $n\geq m$.
Here a nonzero rational ...

**1**

vote

**1**answer

82 views

### Structure of $\text{Aut}_R(R[X])$

Let $R$ be a commutative ring with identity. I'd like to know how to determine the set $\text{Aut}_R(R[X])$ of all $R$-automorphisms of $R[X]$.
I've proved that all $\sigma\in\text{Aut}_R(R[X])$ ...

**1**

vote

**0**answers

17 views

### Volume under the intersection of scaled simplices

This is rather specific but I need to compute the volume under the intersection of rescaled simplexes, that is, the volume of the space:
$\left\{x \in \mathbb{R}^n|\sum_i c_{ki} |x_i| \leq1\; k = 1 ...

**0**

votes

**0**answers

11 views

### Fractal in discrete time series/discrete time sequence

Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series?
I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...

**0**

votes

**0**answers

15 views

### Laplace equation in polar coordinates [migrated]

Hy, I'm new in here but i have a question please help me. I want to solve a Laplace PDE in a polar coordinate system with finite difference method, but I have a problem with boundary conditions at r = ...

**0**

votes

**0**answers

21 views

### Is there a unique saddle value for a convex/concave optimization? [migrated]

I asked this question previously in math.stackexchange and couldnt receive any answer. The question is fundamental, though I couldn't find a reference or a proof somewhere.
Here is the question:
...

**5**

votes

**0**answers

112 views

### Can integers be distorted to make primes more regular?

Given a set $P$ of real numbers $\ge 1$, define the gap among different products in $P$ as
$$g(P) = \inf \big\{\prod_{i=1}^n p_i^{a_i} - \prod_{i=1}^n p_i^{b_i} \mid p_i\in P;\,\, p_i\ne p_j \,\text{ ...

**2**

votes

**2**answers

74 views

### Is the associated order of a minimal $T_0$ space always total?

Let's call a space $(X,\tau)$ minimal $T_0$ if it is $T_0$ and for every topology $\sigma\subseteq \tau$ with $\sigma \neq \tau$ we have that $(X,\sigma)$ is not $T_0$ any more.
We say $x\leq y$ in a ...

**0**

votes

**0**answers

16 views

### Uniform bound in Faedo-Galerkin method with time-dependent weight in inner product

Let $v_j$ be an orthonormal basis for $V=H^1(\Omega) \subset L^2(\Omega)$ which is orthogonal in $L^2(\Omega)$.
Let $w:[0,T]\times\Omega \to \mathbb{R}$ be a time-dependent weight which is smoooth ...

**-1**

votes

**0**answers

61 views

### Simple algebras [on hold]

I like to prove the following statement.
Proposition
Let $A$ and $B$ two $K$-algebras such that $ A \otimes_{K} B$ is simple. Then $A$ and $B$ are simple.
Proof
We only proof this proposition for ...

**1**

vote

**0**answers

31 views

### Infinite non-splittable graphs

Let $G=(V,E)$ be a graph. For $v\in V$ we set $N(v)=\{w\in V:\{v,w\}\in E\}$. We say that $G$ is splittable if there are $S,T\subseteq V$ with $S\cap T=\emptyset$ and $S\cup T = V$ such that
for all ...

**2**

votes

**0**answers

44 views

### What should one “do” to “strictify” a triangle of transformations coming from a lax commutative triangle of functors?

I would like to apologize for this rather stupid abstract nonsense question.
Let $h=f\circ g$ for composable functors $f,g$; assume that there exist left or right adjoints to $f$ and $g$. Then it ...

**4**

votes

**0**answers

79 views

### Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?

Assume that $V\neq HOD$ and let $\kappa = \min \{\alpha\in On \mid \mathcal{P}(\alpha) \not\subseteq HOD\}$.
Clearly, $\kappa$ is a cardinal.
Question: Is it consistent that $\kappa = ...

**2**

votes

**0**answers

34 views

### Multiplying three factorials with three binomials in polynomial identity

I have checked the following identity (1) below for $n\leq 40$ with a computer.
Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let
$Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let
...

**3**

votes

**1**answer

73 views

### When two Dedekind sums are equal

The (classical) Dedekind sum $s(h,k)$ is defined as
$$s(h,k)=\sum_{r=1}^{k-1}\frac{r}{k}\bigg(\frac{hr}{k}-\Big[\frac{hr}{k}\Big]-\frac{1}{2}\bigg)$$
for $\gcd(h,k)=1$.
A natural question is, when ...

**1**

vote

**1**answer

27 views

### What is the smallest partition lattice PART(M) containing the lattice P(N) of subsets of a finite set of N elements

It's been known for 35 years that every finite lattice can be embedded in a finite partition lattice ( Pudlak and Tuma algebra universalis 1980, Volume 10, Issue 1, pp 74-95). I don't follow the ...

**1**

vote

**0**answers

52 views

### rational cohomology of finite dimensional real grassmannian

Let $G_k(R^n)$, $n>k$, be the finite dimensional real grassmannian. What is the rational cohomology algebra $H^*(G_k(R^n);Q)$? I have searched out that $H^*(BO_k;Q)=Q[p_1,p_2,...,p_[k/2]]$ is the ...

**-5**

votes

**0**answers

27 views

### Prize allocation of scratch codes to ensure correct number of prizes to give away [on hold]

We would like to give away 100 prizes.
We have 13.5 million codes, divided into 20 categories. To win a person must collect one code in each category. If 15 of those categories have 877,195 codes ...

**0**

votes

**1**answer

71 views

### Finiteness of geometric valuations

I feel the following fact has been used in many argument in algebraic geometry, but I was not be able to prove it or find the precise reference:
Let $X$ be a $\mathbb{Q}$-factorial variety with log ...

**21**

votes

**1**answer

1k views

### Mathematicians wearing hats on arbitrary total orders

I've been pondering the following generalisation of a famous problem (the special case where $T = \mathbb{N})$:
Question: We have some totally-ordered set $T$ of mathematicians, each wearing a hat ...

**1**

vote

**2**answers

85 views

### How can i change 8_19 to (3,4)-torus knot K(3,4)?

In the knot table, it is well-known that 8_19 is (3,4)-torus knot. But, it is not clear to me. How can i change 8_19 to (3,4)-torus knot K(3,4)? Moreover, it is well-known that braids of two ...

**-2**

votes

**1**answer

58 views

### Show that among any 6 non-negative integers one can find 2 integers so that their difference is divisible by 5 [on hold]

I have the following question in homework I have been assigned in Discrete Mathematics relating the pigeonhole principle:
"Show that among any 6 non-negative integers one can find 2 integers so that ...

**1**

vote

**1**answer

67 views

### A certain matrix associated to graphs

I am not very familiar with graph theory, but I need some results for my work. Thus, the question is, whether the following has already been studied and where I can find it. Let $G=(V,E)$ be an graph ...

**1**

vote

**2**answers

77 views

### Partial Sum of the Binomial Theorem [duplicate]

The binomial theorem states $\sum_{k=0}^nC_n^kr^k=(1+r)^n$. I am interested in the function \begin{equation}
\sum_{k=0}^mC_n^kr^k, \quad m<n
\end{equation}
for fixed $n$ and $r$, and both $m$ and ...

**1**

vote

**0**answers

28 views

### Classes of knots that have known Bridge spectra

Bridge spectra is a knot invariant first defined by Doll, who established some basic properties. Tomova has shown that high distance knots have bridge spectra $(n,n-1,\ldots,2,1,0)$. Zupan has ...

**3**

votes

**0**answers

40 views

### Matroid rank decay

Consider a uniform vector matroid $M(0)=U_{m,n}$ of rank $m$ with $n$ points, $n>m>2$ (you can think of it as a set of $n$ points in general position in vector space $F^m$ for some large field ...

**2**

votes

**0**answers

32 views

### Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$.
For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...

**0**

votes

**0**answers

66 views

### Behaviour of first $l^2$-Betti number under quotienting

Let $G$ be a finitely generated group, and let $H = G / N$ be a quotient of it. We have two observations:
1) In general, it is $\textbf{not}$ true that $\beta_1^{(2)}(G) = 0 \Rightarrow ...

**0**

votes

**0**answers

85 views

### Is it a correct description of the bounded above derived category of coherent sheaves?

Let $X$ be a (Noetherian) scheme. Let $D^{-}_{\text{coh}}(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with bounded above and coherent cohomologies.
Do we have the following ...

**2**

votes

**1**answer

50 views

### $t$-analogue of the symmetric power of an additive character over $\Bbb{F}_q^*$

Let $G$ be a finite group and let $f: G \longrightarrow \Bbb{C}$ be any complex-valued function. For integers $k, n \geq 0$, an indeterminant $t$, and $x \in G$ let $f_k(x) := f \big( x^k \big)$ and ...

**6**

votes

**0**answers

139 views

### On a permutation module for GL(n,q)

Let $G=GL(n,q)$ be the general linear group of degree $n$ over the $q$ element field. Let $X$ be the set of full rank $n\times r$ matrices where $1\leq r\leq n$. Then $G$ acts transitively on the ...

**8**

votes

**3**answers

627 views

### “Epicycles” (Ptolemy style) in math theory?

By analogy:
The epicycles of Ptolemy explained the known facts in the sun system and in this sense were not "wrong". But they distracted from a better insight. From another viewpoint, everything fell ...

**5**

votes

**1**answer

230 views

### When is the category of small (pre)sheaves a(n elementary) topos?

When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos.
When $C$ is ...

**3**

votes

**1**answer

41 views

### Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...