# All Questions

**-6**

votes

**0**answers

80 views

### smoothness of free boundary [closed]

For in $\mathbb{R}^n$, there is a an positive obstacle it has the expression $\phi(x) = \chi_{BR(0)} \cdot \max\{0,a(x)\}$, where $a(x)$ is an analytic function or smooth function on the whole of ...

**-2**

votes

**0**answers

28 views

### Some regularity results of free boundary quesions for a special case [on hold]

In $R^n$, there is an obstacle $\phi \in C_0(R^n)$, and $\phi$ is analytic on its support and has analytic continuation in an open set containing its support. Then I just want to solve the following ...

**0**

votes

**0**answers

14 views

### Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T

My question is given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ isntants:
$$\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T$$
...

**4**

votes

**1**answer

174 views

### Panning for gold nuggets: a type of isoperimetric problem

Let $C$ be a unit-radius circle in the plane.
Suppose you have a total length $L$ of string available, and
your task is to connect chords of $C$ using no more
than $L$ of string to minimize the ...

**3**

votes

**1**answer

108 views

### Gaussian Curvature of Exponentiated 2-Planes

Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map ...

**3**

votes

**0**answers

152 views

### A question on the size of an admissible ordinal

Let $\mathbf{L}_{\varsigma}$ be the level of ordinal $\varsigma$ of Gödel's constructible universe $\mathbf{L}$. Let $\Sigma_{3}$-KP be Kripke-Platek set theory with infinity and ...

**-4**

votes

**0**answers

52 views

### Partial differential equation [closed]

I have been trying to solve this equation for a while. Is there anyone who can help me to solve this? Any comments appreciated.
$$\frac1{r}\frac\partial{\partial r}\left(r \frac{\partial E}{\partial ...

**-3**

votes

**0**answers

30 views

### Necessary Condition For Ellipticity [closed]

What is the ellipticity condition for a higher order linear differential equations in two complex variables?

**1**

vote

**0**answers

91 views

### Infinite matrices with a finite number of non-zero values on each row

The little bit of literature on infinite matrices I've been able to find studies a general setting in which the theory is hindered by constantly having to worry about whether or not various infinite ...

**9**

votes

**1**answer

303 views

+50

### Between compact and locally uniform: What is the name of this convergence?

Let $X$ be a topological space, $(Y,d)$ a metric space, $f\in Y^X$, and $(f_n)$ a sequence in $Y^X$ with the following property:
For every $x_0\in X$ and every $\varepsilon>0$, there exist a ...

**-6**

votes

**0**answers

64 views

### How can I prove that [closed]

Let we have the space C[a,b] ( the space of all functions that are continuous on closed interval [a,b] ) And we have two norms on this space
||X||1=max|x(t)| such that t belongs to [a,b]
...

**-3**

votes

**0**answers

92 views

### Decidability of a language [closed]

L = {M,w | Turing Machine M on input w revisits
the left-most tape cell at least once after initially leaving it }. Is L decidable?

**-1**

votes

**1**answer

224 views

### Does anyone know of theorems/lemmas that are named after poets/authors/musicians? [closed]

This is a soft question, and if anybody knows of lemmas/theorems, that are used in mathematics, but are named in an unusual way. The lemmas/theorems could be named after poets/philosophers/musicians. ...

**8**

votes

**2**answers

255 views

### What is the BRST-anti-BRST formalism?

What is the BRST-anti-BRST formalism?
Is the Sp(2) doublet the ghost, antighost pair?
Introductory accounts of this subject seem to be hard to find. I would appreciate a reference for someone who ...

**1**

vote

**2**answers

263 views

### How to prove that a kernel is positive definite?

For example, how to prove
$\forall(x,y)\in R^N\times R^N,K(x,y) = \displaystyle\frac{1}{1+\frac{||x - y||^2}{{\sigma}^2}}\\$
where $\sigma > 0$ is a parameter, is positive definite? I have tried to ...

**-1**

votes

**0**answers

33 views

### Does $\sum_{n=1}^{\infty} f(z^n)$ converges locally uniformly on unit disk [migrated]

If $f(z)$ is analytic in the unit disk and $f(0)=0$ , show that $$f(z)+f(z^2)+\cdots f(z^n)\cdots $$ converges locally uniformly to an analytic function in the unit disk.
I am thinking to apply ...

**-5**

votes

**1**answer

140 views

### An axiomatic system with a set of constants that form a complete ordered field [closed]

I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...

**1**

vote

**0**answers

25 views

### Breaking down the co power of a topological space

Consider a compact, Hausdorff topological space which is homeomorphic to its own co-power over an index set $I$, so $X \cong \prod_{i \in I } X$. Is there necessarily another topological space, which ...

**2**

votes

**1**answer

203 views

### Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$.
As one could find in G.M. Troianello "Elliptic Differential Equations and ...

**0**

votes

**1**answer

171 views

### The number of solutions of a Diophantine equation [on hold]

Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...

**7**

votes

**2**answers

219 views

### Peano arithmetic vs. fast-growing hierarchy with pathological fundamental sequences

Fundamental sequence for a countable limit ordinal $\alpha$ is an increasing sequence $\{\alpha[i]\}$ of ordinals of length $\omega$ such that $\lim_{i\rightarrow\omega}\alpha[i]=\alpha$. There are ...

**6**

votes

**0**answers

133 views

### Does every commutative $*$-algebra of operators on a prehilbert space have a character?

My question can be equivalently stated as follows:
Let $A$ be a complex commutative algebra with involution, and assume there exists a non-zero homomorphism $\pi:A\to L(V)$ to the algebra of ...

**5**

votes

**0**answers

152 views

### Gompf's invariant of $2$-plane fields

I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...

**8**

votes

**2**answers

355 views

### Infected square

I saw the following problem in Mathematical Puzzles from Peter Winkler (very good book, by the way): imagine you infect k cases of a chessboard nxn and the infection spreads to a case if it has at ...

**3**

votes

**1**answer

117 views

### Immersed versus embedded surfaces representing a same homology class

I am working on the Gromov norm of submanifolds in the total space E of surface bundles over surfaces. So I am interested in knowing the minimal genus of a surface representing a given homology class ...

**2**

votes

**0**answers

154 views

### Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...

**0**

votes

**0**answers

42 views

### Hyperquotient singularities and Newton polyhedra

Following the same notation as M. Reid "Young person's guide to Canonical Singularities":
Suppose that $Y\subseteq \mathbb{A}^{n+1}_\mathbb{C}$ is a smooth affine hypersurface defined by ...

**-1**

votes

**0**answers

49 views

### Probability: short circuiting amongst recycled batteries [closed]

I am a student at University of Illinois with some batteries to recycle. The student union collects batteries, but requires the terminals to be taped. I think this makes a lot of sense for 9 volt ...

**11**

votes

**2**answers

643 views

### from a circle to higher spheres

Question: Is there a group $G$ and a CW-complex $X$ such that
1) $X$ is homotopy equivalent to the circle $S^{1}$.
2) $G$ acts on $X$
3) the space of fixed points $X^{G}$ is weakly equivalent to ...

**-3**

votes

**3**answers

257 views

### Determinant of matrix from set {-1, 1}

Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$?
I have ...

**3**

votes

**0**answers

42 views

### Topological pressure for subshifts on a countable alphabet

Apologies for asking two similar questions within a week of each other, I had hoped that asking a finite alphabet version of this question would lead to enlightenment but unfortunately it didn't.
...

**-1**

votes

**2**answers

107 views

### Converting p-adic to decimal [closed]

Is it possible to convert irrational p-adic numbers to a standard number? Rationals and negative rationals are relatively straightforward, but is there a way to know that for instance $\ldots ...

**10**

votes

**2**answers

424 views

### On a proposition in Hartshorne's paper “Ample vector bundles on curves”

In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field ...

**-7**

votes

**0**answers

91 views

### off-topic government question [closed]

We are estimating the spares requirement for a radar power supply. The power supply was designed with a mean (μ) life of 6500 hours. The standard deviation (σ) determined from testing is 750 hours. ...

**10**

votes

**0**answers

111 views

### Polytopes with few vertices and few facets

I recently realized that, for fixed $\alpha$ and $\beta$, the number of (combinatorial types of) $d$-polytopes with $\leq d+1+\alpha$ vertices and $\leq d+1+\beta$ facets is bounded by a constant that ...

**4**

votes

**1**answer

232 views

### When is a `1-form' with continuous coefficients exact?

Let $\Omega$ be a convex, bounded open subset of $\mathbb{R}^d$, and let $C^1(\bar \Omega)$ be usual space of continuous functions on $\bar \Omega$ which are $C^1$ in $\Omega$ and whose partials in ...

**0**

votes

**0**answers

26 views

### How to find out if a given sequence of orthogonal polynomials belongs to the Askey scheme?

I am studying some classes of orthogonal polynomials and want to find out which of them belong to the Askey scheme. To give a simple example consider the polynomials
$${p_n}(x,r) = \sum\limits_{k = ...

**1**

vote

**0**answers

65 views

### About the reduceness of the commuting scheme associated with a symmetric pair

my question is the following one:
Let $G$ be a connected reductive algebraic group over the field of complex numbers, and let $V$ be a linear representation of $G$ obtained as the isotropy ...

**0**

votes

**1**answer

144 views

### Orthogonal projection

Let $G$ be an operator with compact resolvent on a Hilbert space $H$ such that
$\ker G \neq \{0\}$.
Further let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} := G+P$.
My question is: ...

**1**

vote

**1**answer

26 views

### Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability:
One factor would be the condition number of the approximation operator. The other factor ...

**4**

votes

**2**answers

270 views

### What is the best reference for Spectral theory?

I'm studying Bernard Aupetit: A Primer on Spectral Theory
but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things?
Thank you.

**1**

vote

**0**answers

52 views

### Does this condition imply a polynomial is a product of linear factors

Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that $\forall\mu \in \Lambda^+$, ...

**3**

votes

**0**answers

73 views

### Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding
$$
\int_0^T L\left(\tfrac{1}{2} + it, f ...

**1**

vote

**0**answers

55 views

### Lower periodic subsets of groups and semigroups

Suppose that $A$ and $B$ are subsets of a group or semigroup. We call $A$ left
upper [resp. lower] $B$-periodic if $BA\subseteq A$
[resp. $A\subseteq BA$]. If $A$ is both left upper and
lower ...

**-3**

votes

**0**answers

57 views

### Game theory question Folk Theorem [closed]

I wonder what is the strategy is here.. I have calculated the potential result start from p1 choose C and p2 choose D at first time, and then they both confess. However someone points this is not the ...

**-3**

votes

**0**answers

56 views

### Verification of Gauss Bonnet Theorem on Beltrami pseudosphere and bent sphere patches [closed]

Given that boundary geodesic curvature k_g and Gauss curvature K are constant,
patch area = A and perimeter length = p.
$ K\, A + k_g\, p = 2 \pi $
For a flat circle patch $ k_g= 1/R, $
$ ...

**2**

votes

**1**answer

127 views

### Graph automorphism that swaps two pairs of nodes

Suppose we have two automorphisms on a graph $G$ such that each one swaps a separate pairs of vertices. Is it possible to construct (or prove the existence of) a third automorphism that swaps both ...

**9**

votes

**1**answer

195 views

### Subquotients in the Verma filtration on Verma modules

Let $\lambda$ be a dominant integral weight of $\mathfrak g$, a finite-dimensional reductive Lie algebra over $\mathbb C$. Let $M(w\cdot \lambda)$ denote the Verma module with high weight $w\cdot ...

**2**

votes

**1**answer

68 views

### Standard names and methods for this type of fitting minimization

In material science research, we have come across the following type of problem.
Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization
$$\eqalign{
...

**2**

votes

**1**answer

306 views

### Hopf-algebras in associative ring spectra

I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A_\infty$) ring spectra. One thought I had was to think of cocommutative ...