# All Questions

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### 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 ...
95 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 $x, y\in\mathbb{C}^n$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Weak topology on subsets of a normed space

I have few questions about the subsets of a normed space $X$ endowed with the weak topology. Let $E$ be such subset. When is the norm a continuous function on $E$? When is the metric induced by the ...
43 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 ...
38 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 ...
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### Relation between Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials. Pointwise Lagrange ...
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### A $C^{*}$ algebra associated to a group [on hold]

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar ...
171 views

### Average probability that a random cosine polynomial with bernoulli coefficients is small

Let $P_{n}(t)=\sum_{k=0}^{n}\varepsilon_{k}\cos(kt)$ where $\varepsilon_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any ...
849 views

### number theory which is close to analysis

I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. ...
184 views

### Functor of points of a tensor triangulated category

Is there is a functor of points approach to tensor triangulated categories parallel to Balmer's theory of prime spectra? Given a tensor triangulated category $\mathcal{T}$ an $R$-point can be ...
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### dual basis of cohomology algebra [migrated]

Let $H^*(M)$ be the cohomology algebra of oriented manifold $M$ with rational coefficients. Let $\{b_i\}$ be a basis of $H^*(M)$ as a vector space over $\mathbb{Q}$. Let the dual basis be ...
37 views

### shortest line segment that equally divide a triangle? [on hold]

Given the coordinates of all three vertexes of a triangle, what's the length of the shortest line segment that divide a triangle into two equal-area halves? This is a problem that a friend gave to ...
594 views

### Interplay between Loop Quantum Gravity and Mathematics

It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...
86 views

### Thales Style Level Sets

Encouraged by Joseph O'Rourke ( and inspired by the discussion at Thales' semicircle theorem in higher dimensions ), I ask about level sets in three dimensional space occuring from considering ...
129 views

### Probable direction of deviations from the expected value in binomial and hypergeometric cases

Suppose I have an urn with N marbles, with frequencies p and q for red and black marbles, and with p > 0,5. I take a sample of r marbles. It sounds intuitive to say that deviations from the mean ...
Let $G$, $H$ be topological groups and $f:G\rightarrow H$ a continuous group homomorphism which happens to be a homotopy equivalence of the underlying topological spaces. Let us assume that $G$, $H$ ...