All Questions
5,076 questions with no upvoted or accepted answers
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Need any information about an affine lattice
Motivation - I was thinking about calculating the integrals from An interesting integral expression for $\pi^n$? using old plain Riemann sums. There, one needs integrating over that part of $[0,1]^n$ ...
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77
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Reference request: what is the definition of $H^1(\Omega)$ with $\Omega=(0,1)\times\mathbb{T}$?
Denote the 1-D torus as $\mathbb{T}:=\mathbb{R}/\mathbb{Z}$. Using Fourier series, one can define the Sobolev space $H^k(\mathbb{T})$ (see for instance this note from Wikipedia). On the other hand, ...
user14319
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266
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Completing a dyadic sum
Suppose I knew the behaviour of a given sum in every other interval, for example:
$$
\sum_{\substack{0\leq a \leq x\\ a\equiv 1 (k)}} \sum_{x/(a+k/2)< b \leq x/a} f(b) \sim g(x),
$$
for any $x>1$...
- 3,799
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416
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Solving the equation $\operatorname{Powerset}(X)=\varnothing$
There are (at least) two variants of this question.
Is it possible to modify the axioms of set theory, without arriving at obvious contradiction, in such a way that in a model of the theory there is ...
- 18.4k
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61
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Heat trace asymptotic coefficients for conformal metrics $\widetilde{g}=e^{f}g$ surfaces
As is well known $\sum e^{-\lambda_{k}t}\approx(4\pi t)^{dim(M)/2}\sum a_{j}t^{j}$, where $a_{j}$ are geometric properties of manifold M.
Moreover, the arbitrary order coefficients don't have closed ...
- 5,474
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58
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in search of convergent daughter sequences
Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$.
Question. Is there a subsequence $\{f_{...
- 43.2k
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Does this parameter of two sequence have a name?
Let $a=a_1,a_2,...,a_n$ and $b=b_1,b_2,...,b_n$ be two sequence of real numbers. Define
$$D(a,b) := \sum_{\{i,j\}}\sum_{\{k,l\}}{(|a_i-a_j||b_k-b_l|-|a_k-a_l||b_i-b_j|)^2} .$$
Does this parameter ...
- 519
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211
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On the number of $y$-smooth numbers less than a given magnitude
I'm trying to provide a full-fledged proof of the estimate
$$\Psi(x,x^{1/u}) = x \, \rho(u)+ O\left(\frac{x}{\log x}\right) \qquad (*)$$
via the sly inductive approach commonly attributed to A. ...
- 8,842
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191
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Fiberwise injective resolution of coherent sheaf
Let $k$ be an algebraically closed field (of characteristic zero) and $X, Y$ be projective $k$-varieties. Let $F$ be a coherent sheaf on $X \times_k Y$, flat over $Y$. Does there exists a coherent $\...
- 2,323
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353
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Inner products on abelian groups and general modules
Where can I find a discussion about inner products on something more general than vector spaces, and to which extent should one attempt such a generalization?
My particular interest is in abelian ...
- 229
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80
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Good cohomological setting for binary operations on arithmetical functions
Is there currently a good abstract theory (derived from algebraic geometry and cohomological theories) to study binary operations on arithmetical functions like the Dirichlet convolution $$f\star g = \...
- 1,017
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Shortest hyperpath algorithm in intuitionistic fuzzy hypergraphs
I was looking for an algorithm to calculate the shortest hyperpath in intuitionistic fuzzy hypergraphs and I found only this article (which propose two algorithms).
Are there any others algorithms ...
- 101
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328
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Infinite groups in which every element is a commutator
Finite groups in which every element is a commutator are considered in many works. How about infinite group case? Are there any recent results or constructions related to infinite groups in which ...
- 379
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105
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specific qi on free groups
Let $F_n$ be the free group on $n$ generators, $n>1$.
If $\phi$ is a quasi-isometry (or a bijective bilipschitz equivalence) on $F_n$, then what can we say about the explicit form of $\phi$?
In ...
- 1,528
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Isomorphism between Sobolev space and its dual for elliptic system
Given a domain $\Omega\subset\mathbb{R}^d$, suppose that the boundary is sufficiently regular. Then by Gröger's regularity theory we know that the following operator
\begin{equation}
\nabla\cdot\nabla ...
- 223
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197
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'Adelic torus' not arising from a rational torus
Let $G$ be a reductive group over a global field $F$, and $\gamma$ a strongly regular semi-simple element of $G(F)$. Then the centralizer $G_\gamma$ is defines an $F$-torus $T$, and hence by base ...
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131
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terminology: monotone maps of posets such that the image of a lower set is a lower set
How are called in combinatorics
monotone maps of partially ordered sets such that the image of a lower set is a lower set, i.e. closed (or open) maps of finite topologies? Is there a classification ...
- 113
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69
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optimal frame for a Riemann metric
In his excellent book, Harmonic maps, conservation laws and moving frames, Helein proves the existence of conformal coordinates on a surface, looking for an optimal frame. Let $\mathbb{D}$ equipped ...
- 914
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79
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Geodesic flows on affine two-dimensional tori
I am looking for a reference here. Consider a two-dimensional torus $\mathrm{T}^2 =S^1 \times S^1$ together with an affine structure, that is a $(Aff(\mathbb{R}^2), \mathbb{R}^2)$-structure. Such a ...
- 2,696
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272
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Fixed-point iteration depending on a parameter
Let $f\colon X\times \mathbb{R}\to X, (x,\varepsilon)\mapsto y$, with $X$ open, be a continuous function in both arguments. Consider the following fixed-point iteration
\begin{align}
x_{k+1} = f(x_k,\...
- 2,712
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68
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Simplified the series expansion $\sum_{n=0}^{\infty} \, (n+a)^{^\alpha}\, L^{(1)}_{n}(a)$?
Any reference that we can find the following series expansion
$$f_{\alpha}(a) = \sum_{n=0}^{\infty} \, (n+a)^{^\alpha}\, L^{(1)}_{n}(a),$$
where $a>0, \alpha>0$ and $L^{(1)}_{n}(.)$ are the ...
- 650
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268
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Dimension of the Severi Variety
Edit: In short, I know how to get a lower bound given that nodal curves are dense, and I know how to show nodal curves are dense given the lower bound, I can't find a reference that proves either fact ...
- 1,537
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182
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Is the $B$-tensor power of flat $A$-modules, $A$-flat?
Let $k$ be an algebraically closed field and $A, B$ be two finitely generated $k$-algebras. Suppose $B$ is flat over $A$. Let $M$ be a finitely generated $B$-module which is flat over $A$. Is it true ...
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68
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$H$ self-adjoint with mass gap, $P≥0,Ω∈D(P),H+λP$ self-adjoint $⟹$ for $λ$ small, $H+λP$ has gap?
Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
- 221
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376
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Regularity for a div-curl system
Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...
- 161
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236
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Laplace transform (or characteristic functional) of atomic random measure
A random (nonnegative Radon) measure $M$ (on $\mathbb R^n$, say) has its law characterized by the Laplace transform $\mathbb E\exp(-\int \varphi(x)\ M(dx))$, $\varphi\in C_c^+(\mathbb R^n)$ (...
- 3,085
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77
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Intersection Of Valentine Convex Sets
A set X is said to be m-convex , m integer >=2, if for each set of m points at least one of the associated line segments lies in X.
A 3-convex set is sometimes also known as Valentine convex after ...
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307
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Graph Coloring: Two adjacent vertices share same color
Consider, subgraphs $G_1, G_2,...... G_x$ of graph $G$. Each subgraph has $k$ vertices.
Now, Fix subgraph $G_1$ and consider another subgraph $G_k$ where $1 <k \le x$.
The edge set ...
- 267
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57
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Where can I find this article of Doléans-Dade?
I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade.
I could not find a pdf version online, and my university library does not have a printed version.
Thank ...
- 630
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121
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Generalizing Integration by parts for general bounded continous measure
Consider a probability measure $d\mu = w(t) dt$ with $w(t)\in L^1(I)$, $I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions $f,g:I\ \to \mathbb{R}$ so that an ...
- 3,574
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Fastest algorithm to compute isogeny
Let $E/GF(p)$ and $E'/GF(p')$ are two isogenous elliptic curves($\#E=\#E'$). We know that there exist the map
$$\psi : E \to E'$$
Suppose that we haven't any information about degree of $\psi$.
...
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116
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rationality of Fano 3fold $X_{18}$
I need a reference for an explicit proof of the rationality of the Fano 3-fold $X_{18}$. By explicit I mean by a sequence of explicit birational transformations.
Thank you!
- 3,779
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61
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Conceptual question about partitions in a given rectangular grid
Suppose we have a Young diagram $\lambda$ inside an $r \times n$ rectangular grid, i.e. $\lambda \subset [r] \times [n]$. If I were to add just one more box to $\lambda$, obtaining a new partition (...
- 349
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112
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Obtaining Hessian of the embedding from an induced metric
Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric ...
- 267
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170
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A priori estimate for diffraction problem for linear elliptic PDEs
I am looking for a reference to show how to obtain a priori estimate of the solution $u\in H^1$ and $u\in C^{2,\alpha}$ to the diffraction problem of linear elliptic equation.
I looked at ...
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134
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when is "fibering" preserved under homotopy equivalence
Suppose I have an oriented $F$ bundle over $B$ with total space $E$ (all of the three are closed manifolds) and i have a closed manifold $E'$ which is homotopy equivalent to $E$.Is there any condition ...
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373
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Harmonic function with Dirichlet boundary condition
Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
- 1
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Abelian centralizer groups (CA-groups)
I am searching for all information about CA-groups [abelian centralizer groups] and i just found a German book [Huppert] and Nilpotent Centralizer group of Suzuki in 44 pages and Group theory book of ...
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82
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What results exist for functions with regionally fluctuant fractal dimension?
I'm interested in functions that have a varying fractal dimension at different scales and/or regions. Has this been investigated in detail? I'd be interested in results and references in this area of ...
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691
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Suitable reference on smooth manifolds for qualifying exam study?
Is there a single suitable reference to study for the smooth manifold (geometry) half of a typical Topology/Geometry PhD preliminary exam at an average AMS group I school? (Note: I know details about ...
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231
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Is the complex structure on a del-Pezzo surface a regular complex structure?
Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost
complex structure $J$ is said to be $\textit{...
- 3,245
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173
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Minimum regular open set containing a given set in a T0 Alexandrov topological space
What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be first-...
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54
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Differentiation of $(u(t),v(t))_{L^2(\Omega)}$ when $u, v \in H^1(I\times \Omega)$
Let $I=(0,\infty)$. Consider $u, v \in L^2(I;H^1(\Omega))$ with $u_t, v_t \in L^2(I;L^2(\Omega))$ where $\Omega$ is a bounded doamin.
Is it true that
$$\frac{d}{dt}(u(t),v(t))_{L^2(\Omega)} = (u'(t), ...
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122
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How to solve $\sqrt{-1}\partial\bar{\partial}u=\omega$
I'm looking for references on the study of the equation $\sqrt{-1}\partial\bar{\partial}u=\omega$,especially when $\omega$ is a k\"ahler metric on $\Omega\setminus S$,where $\Omega\subset \mathbb{C}^n$...
- 11
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121
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Bound of Chebyshev function and zeros of zeta function
It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...
- 101
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410
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Set as a (strict) infinite-category?
First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold:
1) trying to ...
- 438
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391
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Conditions for splitting of short exact sequence?
Assume $K$ is a number field and $E$ is an elliptic curve defined over $K$.
Are there conditions under which the short exact sequence
$$0\rightarrow E (K)/mE (K)\rightarrow H^1_{Sel}(K,E_m)\...
- 1,805
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151
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book for help on problems with noetherian rings
Can you please introduce to me a book which would help me to prove the two following problems?
In a noetherian ring, every integrally closed ideal is unmixed.
Let $R$ be a noetherian ring, $P$ a ...
- 565
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490
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The following ODE global existence theorem reference?
There is an ODE existence theorem of the form:
Let $f:[a,b]\times \mathbb{R}^n \to \mathbb{R}^n$ be a Caratheodory function.
Suppose that there is a constant $c$ such that if $y$ is a solution, then $...
- 251
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234
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What is the symplectic manifold whose Delzant polytope is a trapezoid?
What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...
- 1