2
votes
0answers
23 views

Combinatorial identity and Fuss-Catalan numbers

I would like to show that $$ \lim_{N\to\infty}\frac{1}{N^{np+1}}\frac1{p!}\sum_{j=0}^{p-1}(-1)^j\binom{p-1}{j} \left(\frac{\Gamma(N+p-j)}{\Gamma(N-j)}\right)^{n+1} =\frac1{np+1}\binom{(n+1)p}{p}, $$ ...
1
vote
0answers
38 views

Applications of Szemeredi's Theorem

Szemeredi's Theorem is a famous theorem in Additive Combinatorics, Ergodic Theory and maybe some other parts of Mathemtaitcs: (Szemeredi's Theorem) Let $\Lambda \in \mathbb{Z}$ be a subset of ...
0
votes
0answers
18 views

Refomulation of a theorem for special case

This question is about some theorems in the book "Analysis Now" from Pedersen. In particular Proposition $5.3.2$ and Theorem $5.3.3$. Here one has a essential $*$-isomorphism of some algebra $L(X)$ ...
0
votes
1answer
39 views

Dieudonné modules -reference request

I need a reference to start learning about Dieudonn\'e modules, and their application to the arithmetic of abelian varieities. I know that this is a copy of Reference for Dieudonné modules, ...
0
votes
0answers
13 views

Finite element method p2 2d

I'm preparing my graduate project and i really need some help to implement FEM 2D with quadratic element triangles, so i have done everything so far and i think the problem is in the assembling of the ...
7
votes
0answers
64 views

Consecutive numbers with mutually distinct exponents in their canonical prime factorization

Is it possible to find 23 consecutive positive integers each of which has mutually distinct exponents in its canonical prime factorization? Such numbers are sequence A130091 in OEIS. 24 such numbers ...
1
vote
0answers
18 views

Decomposing large symmetric banded sparse matrices

I'm investigating 3D image deblurring and one of the approaches I'm interested in is applying spatial regularisation. To do this I have generated a matrix $A$ which encodes the 6-connectivity of each ...
3
votes
0answers
31 views

Max min of functionals

I have an interesting question which I believe was probably already studied, but I could not find anything. Let $n, m \geq 1$ be fixed. Suppose that $|| \cdot ||$ is a norm in $\mathbb{R}^n$ and $f_1, ...
1
vote
1answer
19 views

Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$. When does a global joint probability $p(x,y,z)$ (possibly not unique) exist? The first compatibility condition to ...
0
votes
0answers
43 views

Embeddings of subshifts

Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a ...
0
votes
0answers
19 views

How can I calculate the global log canonical threshold?

Let $X$ be a normal variety and let $D$ be a $\mathbb{Q}$-divisor. The log canonical threshold of a pair $(X,D)$ is $lct(X,D)=sup\{c|(X,cD)$ is log canonical$\}$. If $X$ is $\mathbb{Q}$-Fano ...
3
votes
0answers
57 views

A strange condition on containment of special complex numbers in cyclotomic fields

In a recent theorem we have naturally come across this condition, that seems to be important, but rarely satisfied: $\sqrt{\frac 1 4 + a^m} \in \mathbb Q(\zeta_m, a)$ where $a\in\mathbb C^*$ and ...
2
votes
1answer
66 views

Is there a maximal connected Hausdorff space?

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected. Is there a maximal ...
1
vote
1answer
44 views

Maximal connected topologies

We call a space $(X,\tau)$ maximal connected, if it is connected, and for any topology $\sigma \supseteq \tau$ with $\sigma\neq \tau$, the space $(X,\sigma)$ is not connected. If $(X,\tau)$ is ...
0
votes
1answer
27 views

Defining a brownian bridge indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
-5
votes
0answers
19 views

LyX preamble to auto add space before and after math (CTRL+M) [migrated]

I tried everything, but nothing works. How can I make auto space bofore and after math formulas in the same line? Thank you.
2
votes
1answer
26 views

Null tetrad transformation

I have been going through the Chandrasekhar's "The Mathematical Theory of Black holes", in particular the chapter on Newman Penrose formalism. I have a question about what he calls a "class III ...
1
vote
0answers
59 views

Irreducibility of a general fibre

Let $A\subseteq B$ be an inclusion of affine domains over an algebraically closed field $k$ of characteristic $0$. Can someone give me a reference for the following fact? If $A$ is algebraically ...
3
votes
0answers
54 views

Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question. I can think of at least four topologies to put on $C_c(M)$: Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney ...
10
votes
2answers
268 views

(Non)existence of mirrors with more than two foci

Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...
-2
votes
0answers
36 views

How to estimate some combinatorial expression? [on hold]

How to estimate (or explicitly compute) the following sum $\sum_{j=1}^{k}\left|\binom{x}{j}\binom{k-1}{j-1}\right|$ from above? The most convenient estimation ...
0
votes
1answer
44 views

Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices. Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...
2
votes
1answer
36 views

Monoidal structure on simplicial sheaves

Let $\mathcal{C}$ be a site and let $sPh(\mathcal{C})_{proj}$ be the category of simplicial presheaves equipped with the projective model structure. This category is a closed monoidal model category ...
-2
votes
0answers
27 views

Deriving sequence of number added after sum of the number [on hold]

I just want to know , is there any mathematics operation by which we can achieve the below mentioned task: 1> I will sum up a sequence of number, it could be any number(For the task I am free to ...
2
votes
0answers
31 views

Hyperellptic curve defined by a set of rational points

If we fix a field $\mathbb{F}$ of positive characteristic, and a a genus $g$ , how many rational points are enough to build a unique hyperelliptic curve of genus $g$ over $\mathbb{F}$?. The thing is ...
1
vote
0answers
34 views

Weak topology on subsets of a Hilbert Space

I have few questions about the subsets of a (for example) Hilbert Space endowed with the weak topology. Let $E$ be such subset. When is the norm a continuous function on $E$? This happens for ...
1
vote
0answers
32 views

The number of $k$-free integers not exceeding $x$

We say that an integer $n\in\mathbb{N}$ is $k$-free if for each prime $p\mid n$, one has $p^k \nmid n$. Let $\mu_k(n)$ be the characteristic function of $k$-free numbers, where $k\ge 2$. Let ...
0
votes
0answers
22 views

Approximation property of Fréchet if range is restricted to an embedded Hilbert space

Let $W$ be a separable Fréchet space, and $H\subset W$ be a separable Hilbert space that is continuously embedded (equivalently, the topology of $H$ is stronger than the subspace topology generated by ...
2
votes
0answers
77 views

Constructing normal crossing varieties

Let $X_i$ be a smooth projective variety with a smooth divisor $D_i$ for $i=1,2$. Suppose that $D_1$ is isomorphic to $D_2$. Then does it make sense to construct a normal crossing variety $X=X_1 ...
3
votes
0answers
48 views

If $R$ is generated by idempotents, then $\text{Ann}(R)=0$?

Let $R$ be a ring (not necessarily commutative or unital) that is generated by idempotents. I'd like to know if $\text{Ann}(R)=0$ must hold. Here I use $\text{Ann}(R)$ to denote the set of all ...
6
votes
1answer
133 views

Notions of infinity in $\mathsf{ZF}$ without choice

Consider the following statements about a given set $X$ in in $\mathsf{ZF}$: (1) There is $x_0\in X$ such that there is a surjective map $\varphi: X\setminus\{x_0\}\to X$. (2) There is an injective ...
0
votes
0answers
53 views

Degree of compositeness of integer

Given $N\in\Bbb N$, call a number $(t,d,c)-$composite with respect to $N$ if there is there a $2t$ bit number $R$ such that number of distinct factor pairs $r_1,r_2$ of size $t\pm c$ bits such that ...
0
votes
0answers
25 views

How to prove Butler's inequality for the maximal slope of the kernel bundle?

In Butler' paper "Normal generation of vector bundles over a curve" (J.d.g.,1994). Proposition 1.4 said that $$prop^+(M_E)\leq \max\left\{-2,\frac{-prop^+(E)}{prop^+(E)-g}\right\}$$ where $E$ is a ...
3
votes
0answers
37 views

Concentration of measure for uniform distribution on Stiefel manifolds

This is my first post on MO, so I hope the question is suitable. I am looking at the uniform distribution on the Stiefel manifold, but more specifically, at the uniform distribution on the ...
2
votes
1answer
80 views

Classification of Hopf-Galois Extensions as Torsors

Faithfully flat Hopf-Galois extensions of rings: $A\to B$, with $H$ coacting on $B$ such that $B\otimes_AB\simeq B\otimes H$, are often thought of as being accessible substitutes for $G$-torsors in ...
3
votes
0answers
127 views

Have topographs been studied before?

This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...
-2
votes
0answers
103 views

About “covering” subgroups

Let $H$ be a subgroup of $S_n$ (the symmetric group with n elements). In the paper I read (cf. Thm 3.17 there), the authors define $H$ to be a covering if the following condition holds: for all ...
10
votes
1answer
209 views

What's an example of 2 elliptic curves with the same ground ring s.t. their associated cohomology theories detect different things?

My understanding is that a complex-oriented spectrum is a ring spectrum $E$ with a map $MU \to E$. Analogously, a ring with a formal group law is a ring $R$ with a map $L \to R$ (where $L$ is the ...
-7
votes
0answers
54 views

taylor-series condition [on hold]

I think it is interesting, if we have the formula $\frac{n^m - (n - 1)^ m }{m}$ ~ ${n ^ {m-1}}$ for example $\frac{100^3 - (100-1) ^ 3 }{3} $ = $9900$ ~ ${100 ^ 2}$, if the difference (I'll ...
10
votes
5answers
896 views

Proof that no differentiable space-filling curve exists

Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists? Or piecewise differentiable? Must every continuous space-filling curve be nowhere ...
1
vote
0answers
79 views

When does an algebraic space that is a torsor over a scheme have to be a scheme?

In Group actions on stacks and applications (Section 4 of part A), M.Romagny gives a definition of $G$-torsor over a scheme $S$ in which the total space need not be a scheme, just an algebraic space. ...
1
vote
0answers
56 views

Curvature of a principal bundle and the exterior covariant derivative

I am sorry if this is too elementary; I had posted it on math.stack but no one answered. Let $P\to M$ a principal fibre bundle with fibre $G$, and let $A\in \Omega^{1}(P)\otimes\mathfrak{g}$ be a ...
7
votes
1answer
121 views

Tauberian theorem with better error term

This is a fairly vague question. Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...
0
votes
0answers
27 views

Is the minimal solution of a Pell equation a positive integral power of the fundamental unit? [migrated]

Let $k=\mathbb{Q}(\sqrt{d})$ -- $d$ is a positive square-free integer -- be a real quadratic field, and let $\varepsilon_k$ be its fundamental unit. Let $(x,y)$ be the minimal solution to the Pell ...
7
votes
1answer
158 views

Example of a ring $R$ such that $\dim(R[[X]])<\dim(R[X])$

Dimension refers to the Krull dimension of a commutative ring. In the paper "Prime ideals in power series rings" J. Arnold gives an example of such a ring: Let $k$ be a field and $K=k(t)$ a ...
19
votes
0answers
200 views

Identities for power series like $\sum_n z^{n^3}$

Probably, one of the first power series that every mathematician encounter is the geometric series $$\sum_{n=0}^\infty z^n = \frac1{1-z}, \quad z \in \mathbb{C},\; |z| < 1 .$$ Also, a particular ...
0
votes
0answers
64 views

Are fibers at points of morphism of schemes closed subschemes? [on hold]

Is $X \rightarrow Y$ is a morphism of schemes and $y \in Y$, is the fiber of $y$ a closed subscheme of $X$? Is is true that the fibers of the projection $X \times_S Y \rightarrow Y$ (with $X,Y$ ...
-7
votes
0answers
52 views

Closed configuration of prime numbers and composed numbers [on hold]

Conjecture 1. Being P the product of the multiplication of several different prime numbers, and being Np ˂ P any prime number not being prime factor of P; being NL ˂ │P^(1/2)│ any prime number not ...
2
votes
1answer
65 views

Criteria for Compactness of a Closed in $L^2$ Spaces [on hold]

$(X, \mathcal{B}, \mu)$ is a measure space. Is there any well-known criteria for compactness of a closed set in $L^2(X, \mu)$? If the answer is negative what about $L^2(\mathbb{R}^n,\mu)$(in this ...
4
votes
1answer
140 views

Hausdorff measure of the graph

Is there any example of a real valued function on the real line whose domain has Lebesgue measure zero but the graph (in the plane) has positive one dimensional Hausdorff measure? Of course if such ...

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