# All Questions

116 views

### Multiplicative domains and conditional expectations

Let $M$ be an injective von Neumann subalgebra of $B(H)$. For a completely positive map $\phi:B(H)\to B(H)$, let $Mult(\phi)$ denote be the multiplicative domain of $\phi$. For any conditional ...
80 views

### Maximize combinatorial sum for boolean function

I am trying to maximize the function $$S(f)=\sum_{j=0}^{n-\frac{n-1}{t}}(-1)^j{n-\frac{n-1}{t}\choose{j}}\sum_{i=0}^{\frac{n-1}{t}}(-1)^{f(i-j)}(t-1)^i{\frac{n-1}{t}\choose{i}}$$ for a function ...
45 views

### solve for three unknowns. [on hold]

Apple cost 97 dollars. Orange cost 56 and lemon cost 3. The total amount spent is 16047 dollars and total fruits bought is 240 and each one is bought atleast one. Calculate how many of each have been ...
76 views

### Rationally building bridges from Jacquet-Langlands to Langlands functoriality conjectures

For now I mainly worked on very classical proofs (viz. Bolte & Johansson, Bergeron) of the Jacquet-Langlands correspondence, but I hope to be able to understand in what this special case lead ...
96 views

### The (global) theory of Borel equivalence relations

What do we know about the complexity of the theory of Borel equivalence relations, with the Borel reducibility order $\leq_B$? That is, let $\mathcal{B}$ be the set of all Borel equivalence ...
196 views

### If $G$ is compact, $H \leq G$ open, $V$ an irreducible $H$-rep, is $\text{Ind}_H^G$ semisimple?

Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a finite-dimensional topological ...
114 views

### Graphs where each edge belongs to the same number of 1-factors

Let $G$ be a simple connected graph that has at least one 1-factor. We'll define: $G$ has property A iff it is edge-transitive. $G$ has property B iff each edge belongs to the same number of ...
80 views

### Estimate maximal coefficient of a polynomial from a circle containing all roots

Suppose I have a polynomial $$p(x)=\sum_{i=0}^n p_ix^i.$$ For simplicity furthermore assume $p_n=1$. As it is well known we may use Gershgorin circles to give an upper bound for the absolute ...
65 views

### On existence of right units with control of their norms

Does, there exist a Banach algebra with a family of right units with norms converging to 1, but without right unit of norm 1?
93 views

### Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [on hold]

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?
155 views

### Ergodic theory reference for converging sequences of matrices

I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone ...
186 views

### Is the Tate conjecture known for etale covers of products of curves

Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...
64 views

### Property of $\limsup$ in $\ell_1$ [on hold]

Let $w$ be any element in $\ell_1$, and $(w_n)$ be a bounded sequence in $\ell_1$ that converge to 0 pointwise. I want to prove ...
242 views

### Lower bound for a prime gap occurring infinitely often

In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound ...
65 views

### In the definition of the Heegard Floer surgery exact triangle, what exactly is the correspondence between Whitney triangles and periodic domains?

I'm reading Osváth-Szabó's notes on Heegard Floer homology, in particular about the surgery exact triangle. On page 14 (numbered 42 on the document), they describe an isomorphism between the space of ...
481 views

### Tight prime bounds

This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...
46 views

### Tensor product of commutators vs. commutator in a tensor product

Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and ...
182 views

### Integer points on $y^2=x^2-x^3+x^4$

Does the Diophantine equation $y^2=x^2-x^3+x^4$ have solutions other than $x=1,y=1$? Interestingly, the Diophantine equation $y^2=x^2-x^3+x^5$ has such solutions: $x=3,y=15$, $x=5,y=55$, ...
465 views

### History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
53 views

### Cusp points in Alexandrov spaces

Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...
34 views

### Looking for N-dimensional spheres in the configuration space of the colorful Tverberg problem

Here we use standard notation for Tverberg's theorem: Dimension $d$, number of partition blocks $r$, and $N=(r-1)(d+1)$. The configuration space of Tverberg's theorem is the simplicial complex ...
46 views

### Are lightface \Delta-1-1 classes of reals describable with hyperarthmetic formulae?

I'd appreciate if someone could check my reasoning. Suppose $S$ is a lightface $\Delta^1_1$ class of reals. I want to argue that there is a computable $\Delta^0_\alpha$ formula $\phi(Y)$, for ...
56 views

### Basis for the rational functions [on hold]

The rational functions $f(x)$ are given by $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials. What bases for the rational functions are generally used in numerical analysis?
51 views

### Topology : Study on Separation Properties [on hold]

I want an example of a completely regular space which is not a normal space. I have tried a lot but am unable to construct any example
225 views

### Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
94 views

### Modular form, number of divisors [duplicate]

The Fourier expansion of Eisenstein series $E_k$ $(k \ge 4)$, which are modular forms, as well as the quasimodular $E_2$, involves powers-of-divisors $\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$. Is there ...
235 views

### Cardinality of $C^*([0,1])$ [on hold]

What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?
78 views

### k to 12 philippines [closed]

i just want to ask this question, hope you will answer..what will happen to students who didn't undergo the senior high school in k to 12, students who graduated in the old curriculum in ched the not ...
41 views

### NonLinear Maps and homogeneity [closed]

An example of a function $\phi : R^2 \to R$ such that $\phi(av) = a \phi(v)$ but $\phi$ is not linear. So I know that I need to find a function that has linear homogeneity but doesn't have the ...
73 views

### $G$-invariant part of products of determinants of minors

Let $G = SL_n$; then for any tuple $\lambda$ such that $\sum \lambda_i = n$, define $f_\lambda(g)$ as the product of the determinants of successive minors of lengths $\lambda_i$ of $g$ (e.g. for ...
173 views

### Factors of the polynomial $X^n-a$

I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a ...
71 views

### Is there a group-theoretic proof of the Riemann rearrangement theorem? [closed]

The analytic proofs are easy to understand but they don't explain why commutativity breaks down when you go from finite sums to infinite sums of real numbers. I suspect that this involves the action ...
34 views

### Generators for lattice polytopes

Let $V$ be a finite set of integral vectors in $\mathbb{R}_n$. Consider the real (positive) convex cone $\mathbb{R}^+V$ spanned by $V$. We can assume that all vectors in $V$ are extremal in ...
844 views

### Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
36 views

### Compact imbedding - reference request

I am looking for reference to the following imbedding theorem Theorem For any $s>1/2$ fractional Sobolev space $W^{s}_2(0,1)$ imbeds compactly into $C([0,1])$. I know how to prove it but I need ...
74 views

### How subset is a set is proved in ZF system? [closed]

I guess that all subset is a set is guaranteed by the axiom of separation in ZF system. Otherwise the notion of power-set will not make sense. But I wander how it's proved. I guess that the prove ...
113 views

### Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
654 views

### Removing an article from arxiv [on hold]

I put up my paper on arxiv before sending it for submission. But now the journal I wish to submit it to is not accepting it since its already been published (or because its publicly available). Is ...
200 views

### The embeddings $S^{6} \to S^{7}$ and $S^{7}\to S^{8}$

Is the standard smooth structure of $S^{7}$ the only structure for which each of the following canonical embedding is an smooth embedding?: 1)$S^{6}\to S^{7}$ 2)$S^{7}\to S^{8}$
125 views

### Universal cover of elliptic curve without a point

This is something that I perhaps would be expected to know, but don't. Let $E_\tau$ be the elliptic curve ${\mathbb C}/({\mathbb Z +\mathbb Z} \tau)$. Consider the complement of a point in $E_\tau$, ...
66 views

### Degree of join of two varieties‏

Let $X\subset\mathbb{P}^n$ be an irreducible variety of degree $d$ and dimension $n-4$. Let $L\subset X$ be a line of mulitplicity $m$ in $X$. Assume that $m+1\leq d$ so that the general line spanned ...
246 views

### Serious introduction to the Langlands program for nonspecialist

I recently became interested in the Langlands program and hope to learn more. For context, I am an analytic number theorist but have some light background in algebraic number theory and modular ...
30 views

### Any software that can symmetrize input sets?

Is there any software that contains symmetrization techniques ex. polarization, Steiner Symmetrization etc. I suppose not. Which software would you suggest for rigid transformations? Thank you
### Computing the Chern class of $S^6$ [on hold]
I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...