# All Questions

**0**

votes

**2**answers

16 views

### Summation mollifier to ensure a certain alternating series has the correct value

I would like the function $f(n,M)$, where $n$ and $M$ are integers and $n\le M$, so that $f$ satisfies the following two conditions:
(1) $\sum_{n=0}^M (-1)^n f(n,M) = \tfrac{1}{2}$
(2) $f(n,M) ...

**1**

vote

**0**answers

6 views

### IVP accuracy - scheme accuracy Vs. derivative accuracy?

General Question: If I have an IVP with periodic and continuous initial condition, which rules the accuracy of the scheme - the manner in which we approximate spatial derivative or the acuuracy of the ...

**1**

vote

**0**answers

16 views

### Reference needed for Hilbert-Schmidt result regarding basis of $V \subset H$

I am seeking a reference that says:
If $V \subset H \subset V^*$ is a Gelfand triple with all spaces Hilbert spaces and if $V \subset H$ is a compact embedding, then there is a basis of $V$ which ...

**2**

votes

**1**answer

25 views

### Bound the degree of the generator of polynomial ring

Suppose we are given two polynomial rings $R_1$ and $R_2$ by presenting their generators, $S_1$ and $S_2$, where $S_i$ are finite set of $m_i$ variables, $i.e.$, $S_i\subset ...

**2**

votes

**0**answers

25 views

### analogy between connections and $\ell$-adic sheaves: what happens with the residue?

There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is ...

**1**

vote

**1**answer

89 views

### Why $( \infty , n)$-categories are useful too?

I know that mathematicians are trying to construct adequate models for $( \infty, n)$-categories. Although, it seems to be an interesting task, I would like to know some explicity examples where this ...

**0**

votes

**0**answers

22 views

### Vanishing of the module of differentials of a extension of perfect fields

Let $L|F$ be a extension of perfect fields of characteristic $p$, $\phi_F:F \to F_{\phi}$, $\phi_L:L \to L_{\phi}$ the Frobenius isomorphisms ($F_{\phi}=F$ but considered as $F$-algebra via $\phi_F$). ...

**2**

votes

**0**answers

17 views

### Limiting probabilities for two-player game drawing random uniform numbers

Consider this simple 2-person game I just made up:
Player A goes gets to draw a uniform U[0,1] number up to X times. At any time, he may either keep his number, or draw a brand new uniform number. ...

**0**

votes

**2**answers

72 views

### Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q

Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be ...

**-3**

votes

**0**answers

44 views

### 2d crank-nicolson [on hold]

I require your assistance in creating the system of equations for a 2-dimensional heat diffution equation, using a finite differences crank nicolson scheme. Can you direct me to a solved example, or ...

**7**

votes

**1**answer

57 views

### Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...

**21**

votes

**1**answer

309 views

### Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?

**0**

votes

**0**answers

34 views

### What is the (mixed strategies) equilibrium of this game?

Given a weight vector $w\in [0,1]^d$ such that $\sum w_i=1$, the game goes as follows:
Two players, $X,Y$ choose strategies $x,y\in [0,1]^d$ such that $\sum x_i = \sum y_i = 1$.
The utility (profit) ...

**0**

votes

**0**answers

27 views

### Is $[u_1,u_2]$ an edge of the polytope $conv(F)$? [on hold]

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$.
I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$.
Given $u_1, u_2 \in F$, is the ...

**5**

votes

**1**answer

76 views

### Modular polynomials for elliptic curves point counting

The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...

**0**

votes

**0**answers

56 views

### On monotonicity of the roots of polynomials

I feel that the question that I am posting is not a general research level question but probably some special form of it where the behaviors of the roots of polynomials are studied.
Here is my ...

**-1**

votes

**0**answers

45 views

### Colon ideal and Artin-Rees lemma [on hold]

Let $a$ and $b$ be ideals of a Noetherian integral domain $R$. Prove that there exists a natural number $r$ such that $(a^n:b) = a^{n-r}(a^r:b)$ for $n > r$. (homework of nagata's local rings, page ...

**0**

votes

**0**answers

13 views

### Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following:
Suppose we have a uniformly parabolic equation with holder ...

**-1**

votes

**0**answers

113 views

### Two Questions on $\pi(x)$ [on hold]

I have recently came to know about this conjecture.
The questions that naturally came to my mind are,
$\forall$ $x,y \geq2$ and $x,y \in \mathbb{N}$ prove that $\pi(x) \pi(y) \leq \pi(xy)$ ...

**1**

vote

**0**answers

86 views

### Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question:
Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification
$$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$
between the space of symmetric order ...

**0**

votes

**0**answers

44 views

### When is the Ad (Adjoint Representation) Morphism a Closed Map

Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...

**2**

votes

**3**answers

200 views

### Is there an integral point in the group generated by an rational point?

Let $E$ be an elliptic curve over rational field. Let $P=(a/d^2,b/d^3)\in E(\mathbb{Q})$ and
$$G=\{P,2P,3P,4P,\cdots\}.$$
Is there an integral point $Q\in G?$

**2**

votes

**1**answer

71 views

### Can this simple integral be zero for a Jordan curve?

The following simple problem came up while doing some unrelated research.
Does there exist a Jordan curve $\gamma : [0,2\pi] \to \mathbb{C}$ of positive orientation, lets say $C^1$-smooth (just to ...

**0**

votes

**0**answers

89 views

### extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set ...

**0**

votes

**0**answers

38 views

### Standard form for partitions of $Z_n$

Let $A$ and $B$ be partitions of $Z_n$. Suppose we say that $A$ and $B$ are equivalent if $A=Bx+y$ for some $x\in\{1, -1\}$ and $y\in Z_n$. In other words, two partitions are equivalent if one can be ...

**0**

votes

**1**answer

24 views

### Number of double cosets of a Young subgroup

Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$.
Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...

**0**

votes

**0**answers

21 views

### Strongly regular graphs with strongly regular edge colorings

Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $(v,c k, c \lambda, c \mu)$ (simple) strongly regular graph that can be given an edge coloring with $c$ ...

**0**

votes

**0**answers

41 views

### Probability of that the sum is even [on hold]

There are $2n$ letters which are randomly placed into $2n$ envelopes (each envelope can have only one letter). The letters and the envelopes are numbered from $1$ to $2n$. What is the probability that ...

**3**

votes

**0**answers

114 views

### Not too classical group characters

When teaching group theory, one has too speak of characters, which are morphisms $\chi:G\rightarrow k^\times$ into the multiplicative group of a field $k$ (mind that I don't mean representation ...

**2**

votes

**1**answer

74 views

### Characterization of a subset of [0,1] $III$

I have a question related to the previous one.
Characterization of a subset of [0,1] $II$
Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e.
$t_n$ is said to converge to ...

**0**

votes

**0**answers

35 views

### Number of possible wall positions in the game Quoridor [on hold]

I'm trying to figure out how many possible board positions there are for the game Quoridor. I think sorting out the legal positions from the illegal positions will be difficult, so to start I'm trying ...

**2**

votes

**0**answers

66 views

### Dubins car shortest paths: Decidable?

A Dubins car follows a
Dubins path
in $\mathbb{R}^2$, with constant wheel speed and
limited turning radius.
It is known that the shortest Dubins path in the absence
of obstacles follows circular
arcs ...

**1**

vote

**0**answers

40 views

### Lifting a quadratic system to a non vanishing vector field on $S^{3}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...

**1**

vote

**0**answers

46 views

### Ordered statistics CDF

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...

**2**

votes

**0**answers

29 views

### Removable sets for simply connectedness of a differentiable manifold

I am sorry that my question might be stupid for experts, but I really do not know the answer.
Let $M$ be a smooth $n$-manifold. We assume that there exists a distance $d$ on $M$ such that $(M,d)$ is ...

**-1**

votes

**1**answer

47 views

### Maximum size of set of points with distance bounded from below

I am interesting in finding a reference for a result of the following type:
Suppose $D \subset \Bbb{R}^n$ is a bounded open set and $\delta>0$. Then the size $M$ of a family of points $F = ...

**3**

votes

**0**answers

63 views

### Explicit non observability example for the wave equation?

Is there a simple (one dimensional, radial, by separation of variable...) example of non observability for the linear wave for a non constant in space velocity, in a simple domain?
By this I mean:
...

**3**

votes

**1**answer

172 views

### Existence of partitions

Good morning everybody.
I would like to know if anybody is aware of nontrivial results of the following form : if a family $\mathcal I$ of subsets of $\mathbb N$ satisfies such and such assumption, ...

**6**

votes

**0**answers

53 views

### Can one detect smoothness of $k$-forms with $k$-dimensional manifolds

Fix an integer $k\ge 1$. Let $X$ be a smooth manifold. It is well-known, that a real valued function on $X$ is smooth if its restriction along all smooth maps $M\to X$ for manifolds of dimension $\le ...

**0**

votes

**0**answers

31 views

### Showing that a particular function from a locally compact Hausdorff group $ G $ to a $ C^{*} $-algebra $ A $ is Bochner-measurable

Suppose that we have the following data:
A $ C^{*} $-algebra $ A $.
A locally compact Hausdorff group $ G $.
A strongly Borel mapping $ \alpha: G \to \text{Aut}(A) $, the automorphism group of $ A ...

**1**

vote

**0**answers

14 views

### Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as
...

**4**

votes

**0**answers

266 views

### What is the *smallest* number used in a mathematical paper? [on hold]

Numbers such as Skewe's Number and Graham's Number and a few others are relatively well-known in mathematical folklore as being among the largest, if not the largest, numbers encountered in proper ...

**0**

votes

**0**answers

87 views

### A simple question in Hitchin's paper “The Geometry of Three-forms in Six Dimensions”

I am reading Hitchin's beautiful paper "The Geometry of Three-forms in Six Dimensions".
Everything goes smooth up to now except for a tiny problem in Section 6.2, which can be formulated as follows. ...

**3**

votes

**2**answers

191 views

### Classification of countable posets?

Is there a classification of countable posets where between each two comparable elements there is a third element between them?

**3**

votes

**1**answer

112 views

### A question about some notation involving the exclamation mark

What does the symbol ‘!’ signify? Is it $ \text{argmin} $? For example, $ \| A x - y \| = \min! $.

**-1**

votes

**0**answers

38 views

### Do the Hypersurfaces satisfy mean curvature $H$ is nonegative and $\langle F(p)-q_0,\nu(p)\rangle \geq 0$ are graphs? [on hold]

Let $F:M^n\to\mathbb{R}^{n+1}$ be the noncompact immersed hypersurface. Is the following ture? If the mean curvature $H$ is nonegative and there exists an fixed vector $q_0$ such that $\langle ...

**0**

votes

**0**answers

26 views

### Help in finding the probability density function

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...

**10**

votes

**3**answers

980 views

### (Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...

**6**

votes

**2**answers

85 views

### Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$.
If $L=M$ ($2$-dimensional square lattice), it is known (e.g., by Peierls' argument or Onsager's ...

**0**

votes

**1**answer

58 views

### Ways to order an algebraic extension

In following post, I describe the "classical" example of $\mathbb{Q}(\sqrt{2})$ that can be ordered in two distinct ways.
More generally, if $(k,P)$ is an ordered field, $R$ a real closure of $(k,P)$ ...