1
vote
0answers
12 views

Is there a simple proof that Milnor $K_2$ of a number field is torsion?

This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays ...
0
votes
0answers
9 views

Minimum degree of rational form

Given a positive integer $g$, what is the minimum degree of the rational function that represents the real function $P(x)$ where $P(x)=\frac{x- x\mod g}{g}+1$ at $x \in ...
1
vote
0answers
15 views

Certain algebraic variete defining by multiplicative functional equation

In this question $B(M_{n}(\mathbb{C}))$ and $B^{2}(M_{n}(\mathbb{C}))$ are the linear space of all linear and 2-linear $M_{n}(\mathbb{C})$-valued maps on $M_{n}(\mathbb{C})$, respectively. ...
0
votes
0answers
11 views

Minimum degree rational function interpolation

Find a rational function $R(x)$ such that: $1)$ For $i\in\{1,\dots,g\}$, $x_{i}=x_{i-1}+g$ with $x_0=0$. $2)$ For $i\in\{0,\dots,g-1\}$ $R(x_i)=R(x_i+1)=\dots=R(x_i+g-1)=i+1$. $3)$ $R(x_g)=g+1$. ...
0
votes
2answers
13 views

Generalised “projection” of a metric space

Assume we have $n$ points $p_0\ldots p_{n-1}$ which form a discrete metric space $V$ with metric $d$. Can we define a function $f:V\rightarrow \mathbb{R}$ with $f(p_0) = 0$, $f(p_1) = d(p_0,p_1)$ and ...
-1
votes
1answer
48 views

Solutions to zeta derivatives

In this paper, Guilherme França and André LeClair show that $$\gamma\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma$ are imaginary parts ...
0
votes
0answers
11 views

Visibility kernels of embedded graphs

Let $G$ be a connected graph embedded in the plane with all edges straight segments. For $\alpha \in (0,\pi)$, define an $\alpha$-path as a path in $G$ with all turns at vertices within ...
2
votes
0answers
31 views

Is such a map null-homotopic?

Suppose I have (semi-infinite) chain complexes $$ \cdots \rightarrow A_i \rightarrow A_{i+1}\rightarrow \cdots$$ $$ \cdots \rightarrow B_i \rightarrow B_{i+1}\rightarrow \cdots$$ over an additive ...
1
vote
0answers
28 views

Better Sobolev inequality holds in this case when assuming doubling and Poincare inequality?

Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$. Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in ...
0
votes
0answers
27 views

Can the Gaussian be covered by restricted recurrences?

Relaxation of the second question here. Let $a(n)$ be recurrence of the form $a(n)=f(n,a(n-1)\ldots(a(n-k))$ with fixed initial terms. (Observe that it might depend on $n$). $f$ may contain ...
3
votes
1answer
66 views

Fano varieties of cubic threefolds

Let $X$ be a smooth cubic threefold over $\mathbb{C}$. Let $F(X)$ denote the Fano variety of lines in $X$, which is a smooth surface of general type. Is this class of surfaces distingushed ...
-6
votes
0answers
48 views

The Riemann Zeta Function Works [on hold]

1 - Any counterexamples known for the Riemann Zeta Function? 2 - How to generalize the following? Here we have the visualization of the Riemann Zeta Function 3D Plot and the plane. We can observe ...
-2
votes
0answers
20 views

Can this specific Mixed-Integer Linear Program constraint be expressed? [on hold]

Thanks for your time. I have a linear program and no idea how I could express a form of constraint and even if it's possible. Maybe someone here know a solution. A company assembly and sells a ...
1
vote
0answers
112 views

What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
3
votes
0answers
149 views

Does Nelson try to prove PA inconsistent directly?

Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a ...
3
votes
1answer
141 views

Question on effective Mordell conjecture

Suppose $F(x,y,z)$ is a homogeneous polynomial over $\mathbb{Q}$, where $C:F(x,y,z)=0$ is a curve of genus $g\geq 2$. Question: Faltings proved that $C$ has finite many rational points. Suppose that ...
4
votes
1answer
59 views

almost huge embeddings and stationary correctness

Suppose $\kappa$ is an almost huge cardinal. Using the characterization of Theorem 24.11 in Kanamori's book, one can show that if $j : V \to M$ is an embedding derived from an almost-huge tower of ...
-2
votes
0answers
61 views

Centralizer of element in group PSL(2,F_p) [migrated]

Is it true, that $\forall g\in PSL(2,F_p)\setminus\{e\}$, $Z(g)$ is Abelian? I think that this is true, but i can't find a simple proof.
-1
votes
1answer
17 views

Convert constraint to do convex optimization or use Lagrange multiplier method

$w_1, w_2, w_3 ... w_n$ are the weights I need to find I have the following constraint: $|w_1| + |w_2| + .. |w_n| <= 5$ That is the sum of the absolute values of the weights has to be less than ...
4
votes
0answers
35 views

Estimate of the sum Taylor's coefficients

Let $f(x) = \begin{cases}\ln\frac{x}{e^x-1}, \quad x > 0; \\ 0, \quad\qquad x=0; \\ \ln\frac{-x}{e^x-1}, \quad x < 0. \end{cases}$ Power series in 0: $f(x) = \sum_{n=1}^{\infty} a_n x^n = ...
-1
votes
0answers
35 views

Lower bound for sum of binomial coefficients without summation

I am new here. So, I will be happy if can somebody help me to find an answer for this proof. I want to prove this lower bound: $$ \log {{n \choose n_1}} \leq nlog {n} - n_1log(n_1) - ...
2
votes
1answer
49 views

Curvatures preserved under the Kahler-Ricci flow

Maybe it is a trivial question. Is there any obvious reason that non-negative holomorphic bisectional curvature is preserved by (normalized) Kahler-Ricci flow, but non-negative Ricci curvature is not ...
3
votes
0answers
46 views

Some questions on analytic vectors and the integrability of Lie-algebra representations

I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...
8
votes
1answer
124 views

Is there an $L$ like inner model for $\sf Z$?

Godel proved the consistency of the axiom of choice with the axioms of $\sf ZF$ by showing that given any model of $\sf ZF$, there is a definable class which satisfies $\sf ZFC$. The proof uses a lot ...
10
votes
1answer
137 views

Who first noticed that Stirling numbers of the second kind count partitions?

When the Stirling numbers of the second kind were introduced by James Stirling in 1730, it was not combinatorially; rather, the numbers ${n \brace k}$ were defined via the polynomial identity $$ x^n = ...
1
vote
0answers
40 views

The number of blocks in Szemerédi Regularity Lemma

In mathematics, the Szemerédi regularity lemma states that every large enough graph can be divided into subsets of about the same size so that the edges between different subsets behave almost ...
2
votes
1answer
74 views

Resolving nodes of a quintic CY 3-fold

Let's consider the following quintic 3-fold $X$: \begin{equation} \{(x_i) \in \mathbb{P}^4 \ | \ x_1f(x)-x_2g(x)=0\} \end{equation} for generic homogeneous polynomials $f(x),g(x)$ of degree four. It ...
0
votes
0answers
66 views

Understanding Prof. Keevash's proof on the “Existence of Designs” [on hold]

I have tried to read Prof. Kevvash's paper on the "Existence of designs". I am finding it very tough to read it linearly. I am comfortable with the nibble ideas and probabilistic methods in general. ...
3
votes
1answer
37 views

the eigenvalues of a generalized circulant matrix

A $2k\times 2k$ circulant matrix $\ C$ takes the form \begin{align} C= \begin{bmatrix} c_0 & c_{2k-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{2k-1} & & c_{2} ...
2
votes
0answers
32 views

Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with ...
-4
votes
0answers
31 views

Compare time it takes to travel a curve and a line [on hold]

First posted in Math Stack Exchange: http://math.stackexchange.com/questions/989690/compare-time-it-takes-to-travel-a-curve-and-a-line?noredirect=1#comment2025799_989690 Suppose you have a right ...
5
votes
1answer
196 views

Polynomial recurrence relation covering the integers (and then Gaussian integers)

Say that a polynomial recurrence relation (my terminology) for $f_i$ is: $k$ initial conditions setting $f_1,\ldots,f_k$ to integers ($\in \mathbb{Z})$. A recurrence equation of the form $f_i =$ a ...
3
votes
0answers
55 views

Number of orbits of $\mathrm{SL}_2(\mathcal O_A)$ on $\mathbf P^1(A)$ when $A$ is a quaternion algebra

This is a reference request. Let $A$ be an anisotropic quaternion algebra over $\mathbf Q$. Let $\mathcal O_A$ be a maximal order in $A$. Then $\mathrm{SL}_2(A)$ acts transitively on the right on ...
0
votes
0answers
32 views

What is the state of art MIQP solver

I used Gurobi with a MIQP with 26 binary variables and 26*4 interaction term without any other constraint. The speed is very slow already.... I want to ask what is the state of art of MIQP solvers. ...
0
votes
0answers
29 views

Intersection points of closed curves inscribed in a convex polygon

Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...
-2
votes
0answers
82 views

Arranging books into bags [on hold]

I'm trying to find an algorithm to answer the following question (informal): given a (finite) set of distinct books of different (positive integer) sizes and a (finite) set of bags of different ...
-4
votes
0answers
48 views

In Dedekind' construction of real numbers, what's wrong with this understanding [on hold]

in this prove, every cut corresponds to a real number. and a cut is a subset of Q.and cut have these three properties.1.is not empty 2.if p belong to this cut,any qp so in my understanding, every cut ...
-5
votes
0answers
43 views

transitive actions of automorphism group [on hold]

Can a finite group be extended to a group whose group of automorphisms acts transitively on the first group. Or more generally, given any finite module, can we extend this module so that the group ...
-2
votes
0answers
44 views

The space $W = \{u \in L^2(0,T;V) : u_t \in L^2(0,T:V^*)\}$ without having identified $H$ and $H^*$

Let $V \subset H \subset V^*$ be a Gelfand triple with the Hilbert space $H$ identified with its dual space and $V$ a reflexive separable Banach space. Define $W = \{u \in L^2(0,T;V) : u_t \in ...
4
votes
1answer
87 views

A lower bound on the $L^2$ norm of a Dirichlet polynomial

The Question. Suppose $0 < \alpha < \beta$ are fixed, and $a_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below \begin{equation*} \int_0^{T} \Big| \sum_{\alpha T ...
3
votes
1answer
118 views

Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$

Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = ...
0
votes
1answer
133 views

How to solve $e^{f(x)} + a f(x) + bx = 0$ [on hold]

How should determine solutions to equations of this form? $$e^{-f(x)} + b f(x) = ax$$ Here $f(x)>0$ is real valued. Also $a>0$, $b>0$.
2
votes
0answers
42 views

Finite extension of K(x) with extra structure: definable over field of invariants?

Let $K$ be an algebraically closed field, and let $\sigma$ be an automorphism on $K$. Set $k=K^\sigma$. Consider the rational function field $K(x)$ and extend $\sigma$ to $K(x)$ by $\sigma(x)=x$, ...
-4
votes
0answers
22 views

How to solve the following integral [on hold]

Do you have any idea how to solve the following integral: $\int\limits_0^a {{e^{ - \frac{{by}}{{c - dy}} - ey}}dy}$, where $a$, $b$, $c$, $d$ and $e$ are constants? Thank you very much.
2
votes
2answers
71 views

Finite series with reciprocal factorials

I asked the question at MSE http://math.stackexchange.com/questions/982388/simple-finite-series-with-reciprocal-factorials but got no answer or comment (it is not a homework). I'm trying to find the ...
4
votes
0answers
164 views

Any counterexamples known for the Generalized Tate conjecture?

One can state the generalized Tate conjecture over arbitrary finitely generated fields; to this end one should just define Galois representation to be effective if the eignevalues of the actions of ...
3
votes
1answer
31 views

Polynomial (non-)embedding of a simplex in euclidean space

Let $\Delta$ be a standard $k$-simplex, and let $f:\Delta\to\mathbb R^N$ be a polynomial map with known numerical coefficients. What sort of practical computational algorithms can be used to ...
5
votes
2answers
241 views

$j$-invariants of elliptic curves over finite fields

Let $K$ be a finite field, and $\overline{K}$ its algebraic closure. It is well known that two curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant. If two such ...
12
votes
2answers
1k views

Is every number the sum of two cubes modulo p where p is a prime not equal to 7?

If p is a prime other than 7, can every integer be written as sum of two cubes modulo p? Has Waring's problem mod p for cubes been proved simply and directly? Thanks for your proof. Lemi
5
votes
2answers
197 views

Which groups are LERF?

A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...

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