Unanswered Questions
49,209 questions with no upvoted or accepted answers
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Open Jackson network with deterministic arrivals.
Dear Friends,
Is there any known Jackson-like theorem for an open Jackson network with deterministic arrivals?
Thanks,
Michael.
- 85
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319
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Estimating a multinomial sum
I have the following sum
\begin{equation}
\sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda}
\...
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1k
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Necessary condition for a graph to be Non-Hamiltonian
Let us denote the edges incident on vertices of valence 2 as "required" as these edges has to be covered by a Hamiltonian circuit, if one exists on that (undirected) graph. Given a graph on which a ...
- 53
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667
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Looking for product of symmetric polynomials evaluated at roots of unity
Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + (n-...
- 13.9k
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0
answers
314
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faithful representation of locally compact group
I have been thinking about existence of faithful representation of locally compact groups. This representation exists for example for compact lie groups. But I am curious to know if one can say some ...
- 71
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289
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Good and/or standard notation for the abelianization of a Lie algebra
I'd like to solicit good notations for the abelianization of a Lie algebra $\mathfrak g$. One could write $\mathfrak g/[\mathfrak g,\mathfrak g]$, or even $H_1(\mathfrak g)$ but I'd like something ...
- 4,898
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301
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L-series method, how far can it go?
Using some suitable L-series for some appropriate ray class group, one can find the Dirichlet density of some set of primes. One can conclude that this set of prime is infinite as long as the density ...
- 468
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325
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Introductory text book for Linear Recurrence Sequences
What is a good introductory text for linear recurrence sequences?
What all are the necessary prerequisite for it? (My background is in Euclidean Fourier Analysis.) After browsing through several ...
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445
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Value of coefficient in estimation of computational complexity of polynomial division algorithm
Do you know value of coefficient $C$ at $C*n*log(n)$ in $O(n*log(n))$ estimation of complexity of polynomial division algorithm?
It would be great if you give me links to paper with information about ...
- 424
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306
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A finite direct product of Azumaya algebras may not be Azumaya?
Looking for an example, if it exists, of a finite set of Azumaya algebras $A_i$ such that $\oplus A_i$ is not Azumaya.
- 59
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183
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Standard system of parameters and an example
Let $(R,m)$ be a local Noetherian ring. A system of parameters $\bf{x}$$:=x_{1}, \dots, x_{d}$ is a standard system of parameters if $(\bf{x})H^{i}_{m}(R/(x_{1}, \dots, x_{j}))=0$ holds for all non-...
- 113
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279
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Tropical varieties correspondence to varieties over a non-archimedean valuation field.
I am a mathematical physicist and I am studying certain discrete dynamical systems defined in terms of piecewise linear mappings, which may be expressed in terms of expressions over the max-plus semi-...
- 106
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499
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Integrating the product of two functions one of which has a positive non-integer power
I'm looking to integrate several functions having the form
$\int_0^T \frac{ sin(\omega \tau) }{\omega} \tau^{2H} d\tau$
where $2H \ge 0$ but may not be an integer. I'd like to know if the machinery ...
- 661
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343
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Can KL divergence go to 0, but $E[\log(p/q)^2]$ diverge in certain cases?
Let $p(x)$ be a fixed distribution over a discrete space.
Let $A, C > 0$ be constants.
Let $\epsilon > 0$. Can we find an example of a distribution
$q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\...
- 1
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votes
0
answers
273
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Ext groups and flat modules on a K3 surface
Let $S$ be a $K3$ surface and $X$ the moduli space of some stable sheaves on it. Let $G$ be the universal family on $X\times S$ and $F$ the ideal of section of $X\times S\to X$. Knowing that for every ...
- 773
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436
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External tensor product of quasi coherent sheaf
When is the following map possible?
$A\boxtimes A(\mathcal{G}\times\mathcal{G})\rightarrow A(\mathcal{G}) \otimes A(\mathcal{G})$;
where $\mathcal{G}$ is a group scheme, $A$ is a quasi coherent sheaf(...
- 153
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555
views
étale cohomology with values in the $\ell$-torsion of an Abelian scheme
Let $S/\mathbf{F}_q$ be a $d$-dimensional smooth projective variety and $A/S$ be an Abelian scheme. Is there an easy description of $H^0(S, A(\ell)(d-1))$?` ($A(\ell)$ = union of $A_{\ell^n}$)
- 227
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answers
608
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Orthogonal Projections in Lie Theory
I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
- 157
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154
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Finding the bottleneck in a chain of functions
I have a problem that involves finding a bottleneck. It appears to me to be a linear bottleneck assignment problem, but recognizing (and solving) such problems is far outside my area of expertise. If ...
- 1
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answers
2k
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Similarity between two tree structures whose edges are weights
Given two trees with weighted edges, what kind of similarity or distance measures are there between these two trees? Note that the number of nodes in each tree need not be the same. Are there any ...
- 111
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189
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Packing Icons Onto A screen
You are trying to pack icons onto a screen that is divided into n horizontal rows of uniformly varying size. The rows narrow by a fixed ratio as one goes up the screen from the bottom. Since the icons ...
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297
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higher direct image: proof of the proper case.
Hi.
Let $f:X→S$ be a proper, open, surjective morphism of complex reduced spaces with constant fiber dimension n (or universally open morphism with n-fibers between locally noetherian excellent ...
- 435
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138
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Why do I not use post hoc tests with a 2 x 2 factorial?
I know this is an obvious answer. I am probably over thinking what I'm doing, but I cannot recall. Does it have to do with not having enough variables to compare the various means?
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187
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Are sieves in locally small categories still sets?
In "Sheaves in Geometry and Logic", M&M define a sieve of an object $C$ as a downward-closed set of arrows $S$ with codomain $C$. They go on to say that for a locally small category, a sieve of $C$...
- 1,872
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362
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Gradient of the energy functional in $H^{1,2}$-norm
I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
- 2,935
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320
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A result about Fredholm operator
When I read the article "Index Theory" in Handbook of global analysis, I meet a result as below(Corollary 2.13):
If every $F_0\in \mathcal {F}(H_1,H_2)$, there is an open neighborhood $U_0\subseteq \...
- 381
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368
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the implicit function theorem in subsets of R$^2$
As the implicit function theorem shows, if
(i)Function F is continuous in the region D$\subseteq R^2$;
(ii)F($x_0,y_0)=0,P_0(x_0,y_0)\in$D;
(iii)There is a continuous partial derivative $F_y$(x,y)=...
- 57
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165
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Support sets along a ring homomorphism.
Let $(R,m)$ and $(S,n)$ be commutative noetherian local rings, and $f: R\rightarrow S$ be a local homomorphism (i.e., $f(m) \subseteq n$) with $S$ flat as $R$-module. If $M$ is a finite generated $R$-...
- 1,207
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562
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Reducing two variable linear Diophantine equation to modular inversion
I'm in the field of secure multiparty computation using Homomrphic encryption or secret sharing. I want to implement a secure protocol to compute the GCD of two encrypted numbers.
To calculate the ...
- 193
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answers
176
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Representation of the group of automorphisms on the holomorphic forms
Let $X$ be a compact Riemann surface and $G = Aut(X)$ be its group of automorphisms (biholomorphisms between $X$ and $X$). It is known that $G$ acts on the space $Harm(X)$ of all harmonic forms and ...
- 1,271
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520
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Motivation of proof of Riemann-Roch for elliptic curve and generalizations
Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all ...
- 15.4k
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333
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Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is hausdorff
Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is Hausdorff? Note: here $\bar{M_{g,n}}$ is not the Deligne-Mumford space in the usual algebraic ...
- 1,499
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830
views
Finding components of a preimage
Let $f:X\to Y$ be a degree $d$ morphism of complex projective varieties, and let $V\subset Y$ an irreducible subvariety, $W$ its preimage under $f$. I want to find all of the components of $W$.
...
- 16k
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votes
0
answers
145
views
unstability of a scroll 2
For the purposes of this question I will work over $\mathbb{C}$. Consider on $T=\mathbb{P}^1$ the bundle $E=O_{T}^{\oplus 3}\oplus O_T(-1)^{\oplus 2}$ and $\mathbb{P}E_T$ the associated projective ...
- 1
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0
answers
206
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Vector-valued valuations on lattices
There's a fair amount of work on valuations on (modular) lattices, by which I mean functions $v : \mathcal{L} \rightarrow R$ that satisfy the modular expression
$$v(x) + v(y) = v(x \wedge y) + v(x \...
- 4,462
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votes
0
answers
700
views
Questions on orbit properties of group action on varieties
Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...
- 2,709
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373
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Amenability of an "almost Hamiltonian" group
Here is another interesting question that I can't answer on my own.
Let $G$ be a countable, discrete group such that for any subgroup $H$ of $G$ and any element $s$ of $G$ we have $[H : sHt]$ is ...
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0
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574
views
What's the expected number of iterations for this process?
Each step of the process consists of choosing a random integer between 1 and the last number chosen this way. On average, how long does it take to obtain "1" as a result of this process for any given ...
- 25
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158
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Are the Dolbeault Operators for a Quotient Space Equivariant?
Let $G$ be a Lie Group and $H$ a closed subgroup such that $G/H$ (the set of right cosets) is a complex manifold manifold. Now $\Omega^1(G/H)$, the space of complex one forms, is a $H$-equivariant ...
- 3,409
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322
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When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?
It is easy to show that the following problems are equivalent.
a. When is $Pn^2-2an+\frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square?
and
b. When is $X^2-PY^2=k$ ...
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votes
1
answer
127
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Proving that there are no integral points on a union of hyperbolas
I have a curve C: (x^2±x-y^2+1)(x^2∓x-y^2) where x,y ∈ Z+ that I want to prove has no non-trivial integral points other than (0,0),(1,0),(0,±1). I am having a hard time coming up with a solution.
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1
answer
126
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Integer quadratic representation subject to discriminant minimization algorithm
Let $f(x)=ax^2+bx+c$ and $f(x)=n$. Is there an algorithm to choose $a,b,c$ such that the discriminant is minimized? Where $a,b,c,n,x$ are all integers.
More concretely, is there an algorithm to find $...
0
votes
1
answer
214
views
number of representations by sums of three squares (with coefficients)
There are formulas for counting the number of representations of a positive integer $N$ as a sum of three integer squares. What is a reference for
$$
\#\{(x,y,z)\in \mathbf{N}^3: 5^4 x^2+y^2+z^2=N\}
?$...
- 3,062
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1
answer
324
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Finding examples of functions which are infinite or undefined with current extensions of the expected value?
Preliminaries
Consider the expectations desribed in this paper, which is an extension of the Lebesgue density theorem; this paper which is an extension of the Hausdorff measure, using Hyperbolic ...
- 63
0
votes
1
answer
349
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Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
- 10.5k
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votes
1
answer
122
views
Existence of an eigenpair for d-bar operator in the unit disck
Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem:
$$ \overline{\...
- 4,115
0
votes
1
answer
181
views
A simple procedure to simulate multifractional Brownian motion paths
In a paper by Peltier and Vehel the multifractional Brownian motion (mBm) was defined for the first time, and they also give a procedure to simulate mBm sample paths. Briefly, mBm generalizes the ...
- 207
0
votes
1
answer
161
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Prove zero slope point is global maximum for constrained function with binomials. Without restriction, objective function is non-concave
How to prove the zero slope point is a global maximum in this non-concave program for a function with binomials?
I need to find the (global) maximum of the following constrained problem:
$$\max_{CAP} \...
- 49
0
votes
1
answer
153
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Difference of two optimization problem's optimal value
Let we have two following optimization problems:
\begin{align}
\text{(P1)}\quad \alpha_1 = \max_{x_1,\ldots,x_M} &\quad \sum_{m=1}^{M}\log(1+f_m(x_1,\ldots,x_M))\\
\textrm{s.t.} &\quad \...
- 287
0
votes
1
answer
123
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Recognizing perfect Cayley graphs as tensor products
It is known (and can easily be seen) that a unitary Cayley graph on $n=\prod_ip_i$, ($p_i$ distinct primes) vertices with $n$ square-free can be recognized as the tensor product of the graphs $K_{p_i}$...
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