# All Questions

**1**

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6 views

### Calculation of the integral related to the gravitational shock wave

The following integral
$$\int\limits_0^\infty \frac{\cos{\left(\frac{1}{2}\sqrt{3}s\right)}}{\sqrt{\cosh{s}-\cos{\theta}}}\,ds$$
can be found in the paper
Tevian Dray and Gerard 't Hooft, The ...

**1**

vote

**0**answers

27 views

### Generic Smoothness Type of Results in Positive Characteristic

Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth.
We know that ...

**0**

votes

**0**answers

5 views

### The line graph of a complete graph

There exist a $\left\{P_{5},C_{4}\right\}$- decomposition of the graph $L(K_{9})$

**1**

vote

**0**answers

19 views

### Difference in the Four Color Theorem

How is proving that any planar graph with maximum degree of four has a four coloring, different from proving the four color theorem? If they are different, then how would one prove it?

**6**

votes

**2**answers

119 views

### Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step ...

**-2**

votes

**0**answers

34 views

### Adherent value of sin(sqrt(n)) [on hold]

I have been struggling against this question for several hours and I really need some help. It is from my undergraduate real analysis course. Your time and help is greatly appreciated.
I got struck ...

**0**

votes

**0**answers

25 views

### Isotopy class of closed 2-ball embedded in R^3

My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove?
It seems like it should be easy ...

**2**

votes

**1**answer

257 views

### Use of infinitude of primes in the Green-Tao theorem

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...

**2**

votes

**0**answers

31 views

### Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...

**7**

votes

**4**answers

228 views

### List of counting proofs instead of linear algebra method in combinatorics

I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...

**1**

vote

**1**answer

42 views

### Is there a generalization for the discrete fourier transform whereby eigenvalues are other roots of unity?

The eigenvalues of the discrete fourier transform are $\{1, -1, i, -i\}$ in approximately equal proportions.
https://en.wikipedia.org/wiki/Discrete_Fourier_transform#Eigenvalues_and_eigenvectors
Is ...

**-2**

votes

**0**answers

19 views

### Loxodrome loop on a surface of revolution

Prove that there can be a closed loxodrommic curve on a surface of revolution only when it is doubly connected.

**-2**

votes

**0**answers

28 views

### Finding topological properties under a metric on set of composition operators of L2 [on hold]

We define a new metric on all composition operators in L2: $$d{R}(A,B)=\sqrt{{\parallel P{R(A)}-
...

**-1**

votes

**0**answers

23 views

### A sequence of continuous functions that converge to the characteristic set of a G-Delta Set [on hold]

If $\mathcal{O} \subset I$ is an open subset of an interval then I know we can find a sequence of continuous functions $f_j \in C(I)$ such that $0 \le f_j(x) \nearrow \chi_{\mathcal{O}}(x)$ for all $x ...

**0**

votes

**0**answers

27 views

### Markov Chains and Simple Machine Learning

Suppose I have a large training set consisting of many strings of symbols.
$TS = \{Str_0, Str_1, ..., Str_n\}$
$Str_i = \{Sym_0 ... Sym_{len}\}$
These strings of symbols are each generated by the ...

**-1**

votes

**0**answers

21 views

### Correlation between probability of an event in the domains A and B and the event in an event AUB [on hold]

Given the event $E$ and finite sets $A$ and $B$ such that $P(E)=p_1$ in the domain $A$ and $P(E)=p_2$ in the domain $B$, then what can we say about $P(E)$ in the domain $A\cup B$?

**-6**

votes

**0**answers

39 views

### Need a help solving a rational integral [on hold]

Today I have spent all my day solving this integral, but no result yet. So I need your help. Will be very thankful.

**0**

votes

**1**answer

66 views

### Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let ...

**2**

votes

**1**answer

139 views

### A decreasing sequence involving the divisor function?

Define $N_k \geq 6$ to be the $k-th$ primorial number and let $\sigma(n)$ be the divisor function.
It seems that $u_k = \dfrac{\sigma(N_k)}{N_k \log\log N_k}$ is a decreasing function ?
By ...

**3**

votes

**2**answers

129 views

### Definition of the differential of the Cone of a morphism of complexes [on hold]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$.
The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...

**1**

vote

**0**answers

36 views

### End points of continua

Whyburn (1942) defined an end point x of a continuum X to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines end point locally.
...

**0**

votes

**0**answers

43 views

### canonical decomposition of a linear map

I have a linear map $A : {\mathbb F}^k \to {\mathbb F}^{k+m}$ where ${\mathbb F}$ is some field; (you can impose restrictions on ${\mathbb F}$ if it helps). I'd like to decompose $A$ as a product of ...

**2**

votes

**1**answer

83 views

### Looking for Severi varieties

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let
$$
\mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\},
...

**1**

vote

**0**answers

50 views

### Intuitive understanding of the mean curvature flow

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = ...

**2**

votes

**1**answer

89 views

### Symplectic group over integers and finite fields

For $H=\left( \begin{smallmatrix} 0 & I_n \\ -I_n & 0 \end{smallmatrix} \right)$ and a commutative ring $F$, the symplectic group $Sp(2n,F)$ is the set of all matrices $M\in F^{2n\times 2n}$ ...

**2**

votes

**1**answer

46 views

### If $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$, then is $u \in L^\infty(0,T;X_1)$?

Let $X_0 \subset X_1 \subset X_2$ be continuous embeddings, with $X_0 \subset X_1$ compact.
Suppose $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$.
Is then $u \in L^\infty(0,T;X_1)$?
To apply ...

**2**

votes

**0**answers

36 views

### Spectra of certain totally positive matrices

Let $S$ be the set of $3 \times 3$ matrices $A$ satisfying the following conditions:
All minors are $>0$ (i.e., $A$ is a strictly totally positive matrix);
all principal minors are $>1$, ...

**-1**

votes

**0**answers

33 views

### Metrics and Measures on a Category of Cats : a cauchy complete category of categories

Is there a suitable way to restrict the functors between objects (and I suppose the objects themselves) of a category of categories such that we have a cauchy complete category. Can we find a complete ...

**8**

votes

**0**answers

67 views

### Is the restriction map for embeddings of manifolds with boundary a fibration?

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...

**11**

votes

**0**answers

188 views

### Have Grothendieck's notes in Montpellier already been investigated?

Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...

**3**

votes

**0**answers

25 views

### Carving a rectilinear polygon

In this question, carving a polygon $P$ means removing an axis-parallel rectangle adjacent to the boundary of $P$. Carving $P$ might break it into two or more polygons.
You are given a square $P$. ...

**1**

vote

**0**answers

30 views

### Fundamental system of neighborhoods. Self similar set

I have been reading the text Analysis on Fractals of Jun Kigami. There is a theorem about the fundamental system of neighborhoods of a point in a self similar set. It is stated as follows
Let ...

**-2**

votes

**0**answers

46 views

### Does continuity in one variable and locally Lipschitz in another imply uniformity in the first? [migrated]

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function.
Suppose $ f(t,x):D ...

**-3**

votes

**0**answers

18 views

### Square wave in the limit of infinite frequency [on hold]

What is the new function obtained when one takes the limit of the square wave function when the frequency is taken to infinity?
Does it depend on how the function is written down (e.g. defined as ...

**4**

votes

**1**answer

74 views

### A non locally compact group of finite topological dimension?

Is there a topological group which is Hausdorff, first countable,
locally connected and has finite topological dimension, yet fails
to be locally compact?

**2**

votes

**1**answer

103 views

### Opposite of an E2-algebra

Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of ...

**0**

votes

**0**answers

31 views

### CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...

**0**

votes

**0**answers

120 views

### Has the attempt of proof of the Frankl conjecture by Vladimir Blinovsky been checked? [on hold]

I found his article in arxiv: http://arxiv.org/pdf/1507.01270.pdf. But i didn't find any response to the article and as I'm an undergraduate I have no knowledge to judge if this approach is promising.
...

**5**

votes

**1**answer

222 views

### Hahn-Banach theorem for arbitrary locally compact fields?

Does anyone know if the Hahn-Banach theorem is true for every locally
compact field? Specifically, let $F$ be a finite algebraic extension of
either $Q_p$, the $p$-adic completion of $Q$, or of
...

**0**

votes

**0**answers

50 views

### smoothness of boundary under Riemann mapping

Suppose there is a smooth Jordan curve separating the complex plane. For complicity, assume the curve is given by a graph $(x, \phi(x))$, where $\phi(x)$ is smooth, bounded, and derivatives are ...

**0**

votes

**0**answers

45 views

### an example for fundamental group of graph of groups [on hold]

suppose we have a graph $X$ with the vertex set $\left\lbrace v_1,v_2,v_3 \right\rbrace $ and the edge set $\left\lbrace e_1,e_2,e_3 \right\rbrace $ like a triangle. let $(\Gamma,X)$ be a graph of ...

**6**

votes

**0**answers

76 views

### Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...

**-4**

votes

**0**answers

22 views

### Integral of cos(x) wrt t when integral of sin(x) wrt t is known [on hold]

If $\int_0^T sin(\theta) dt = A$, where $\theta$ is a variable, A is constant.
Then can we find out $\int_0^T cos(\theta) dt$ = ?

**3**

votes

**2**answers

47 views

### $V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 :
Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von
Neumann ...

**1**

vote

**0**answers

18 views

### Is support function of a convex curve in $\mathbb{R}^2$ absolutely continuous? [on hold]

There is an example of a convex function of two variables that is convex but not absolutely continuous. The level set of this function is a convex curve.
To construct one example of such a function ...

**0**

votes

**0**answers

29 views

### HNN extension group with finitely generated base

Let $B$ be a group and let $A_1$ and $A_2$ subgroups of $B$ with $\phi :A_1\rightarrow A_2$ an isomorphism. Let $\left<t\right>$ be the infinite cyclic group, generated by a new element $t$. The ...

**1**

vote

**0**answers

41 views

### Solvable Lie algebra whose nilradical is not characteristic

It is well known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic p > 0 need not be characteristic (that is, invariant under all derivations of the algebra), but ...

**4**

votes

**1**answer

77 views

### Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...

**-2**

votes

**0**answers

27 views

### Algorithm to rate board of tic-tac-toe [on hold]

At start I want to say that im programmer and I don't want anyone to write me code, just to help me what I can use.
Is there any algorith which I can use to rate a board of tic-tac-toe ?
What I want ...

**3**

votes

**0**answers

30 views

### Can approximately periodic functions be perturbed to periodic functions on a locally compact group?

Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...