Most active questions
782 questions from the last 30 days
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Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation
Consider the following semilinear problem:
$$
\begin{cases}
- \Delta u + u = u |u|^{p - 2}
&\text{in} ~ \mathbb{R}^N;
\\
u (x) \to 0 &\text{as} ~ |x| \to \infty,
\end{cases}
$$
where $N \geq 2$...
- 144
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27
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Number variance of random points (and deviations for empirical processes)
Let $X_1, X_2, \dots$ be i.i.d. random variables having uniform distribution on $[0,1]$. Write $I_{t,x}$ for the indicator function of an interval of length $x$ with center $t$. Consider
$$
V(N,x) = \...
- 2,217
3
votes
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answers
40
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Cubic version of Kan loop group
Is there a version of the Kan loop group that is based on cubic rather than simplicial objects?
- 401
0
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96
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Length of generic intersection in local ring
Let $(R, \mathfrak{m})$ be a regular local ring and let $I\subset R$ be an ideal of coheight 1. Let $a \in \mathfrak{m}\setminus \mathfrak{m}^2$.
If $a$ that is not a zero divisor of $R/I$ we have ...
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28
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How to calculate the vertices of a convex polytope (k-DOP)
I am currently reading Christer Ericson's Real-Time Collision Detection Book. The topic I'm particularly interested in, is the chapter about Discrete-orientation Polytopes (k-DOPs). In his words "...
- 1
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22
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Alignment of unit vectors under graph-neighbor constraints with a global vector
Statement
Let $G = (V, E)$ be a connected, unweighted, and undirected graph with $n$ nodes, represented by its adjacency matrix $A$. Suppose each node $i$ is associated with a unit vector $ \mathbf{v}...
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24
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Characterisation of a family of continuous martingales
I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that
$$X_0=0\quad \mbox{ and } \quad\...
- 1,399
-1
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0
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35
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Different definition of Feller semi-group
(This is a crosspost of a question on MathStackExchange which did not receive any answer.)
Let $E$ be a locally compact metric space, let $C_0(E)$ be the set of real-valued continuous functions of $E$ ...
0
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37
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separator and vertex-connectivity
A definition of "separator" is the following: Let $G$ is an $n$-vertex graph, then $S\subseteq V(G)$ is a separator if there is a partition $V=A\cup B\cup S$ such that $|A|,|B|\le 2n/3$ and ...
- 281
1
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answers
45
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Proper Pisot n-tuples
Recall that x is a Pisot number if it is real and x>1, while all of its conjugates have magnitude less than 1. Then $\{(x)^k\}$ (where $\{\cdot\}$ is the fractional part of x) approaches 0 ...
- 680
1
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0
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6
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Finding Commonalities and Differences in Directed Hierarchical Graph
I have a directed graph with hierarchical structure, multiple nodes and edge types. One node can have different in- and outgoing connections. There would also be more details to some of the nodes -- e....
- 11
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35
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Describing the boundary of the feasible direction cone to a convex open subset of $\mathbb{R}^n$ at a boundary point: connection via subdifferential?
Let $U\subset \mathbb{R}^n$ be a convex, open set with nonempty boundary. Let $x_0\in \partial{U}.$ We can describe $U$ locally near $x_0$ as a super level set of a suitable continuous concave ...
- 1,467
1
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answers
23
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Uniform bound on the first moment for a perturbed advection-diffusion equation
I am studying the solution $u = u(t,x)$ to the following problem on the positive half-line:
$$
\begin{cases}
u_t = u_{xx} + u_x - \frac{1}{1+t}(xu)_x, & \quad t > 0, \, x > 0, \\[2mm]
-u_x = ...
- 43
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42
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How can I solve this non-local optimization problem?
I would like to find a continuous function $u:[0,1]\to \mathbb{R}$, which is a minimizer of the following functional
$$ F(u) = \frac{1}{2}\int_0^1 \int_0^1 \left( \frac{u(x)-u(y)}{x-y}\right)^2\mathrm{...
1
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0
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138
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What is a quantum condensed space?
Due to a categorical equivalence involving compact topological spaces and unital commutative $\mathrm{C}^*$-algebras, there is a practise involved in so-called quantum mathematics where a ...
- 1,037
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43
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questions on stochastic kernels and pushforward operator
Let $f:X \rightarrow \Delta (Y)$ and $g:X \rightarrow \Delta (X)$ be two kernels. For any bounded measurable function $h_Y:Y \rightarrow \mathbb{R},$ define $F(h_Y):X \rightarrow \mathbb{R}$ such that ...
- 11
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40
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Optimizing over convex polynomials
I have a minimization problem which reads $\min\limits_P J(P)$, where the minimum is over convex polynomials in $n$ variables, with degree at most $d$, and $J$ is a function taking polynomials as ...
- 101
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votes
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16
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Longest TSP in the unitary disc
I have the unitary disc $D=\{(x,y) \in R^2: x^2 + y^2 \leq 1\} $, and an integer $n \geq 2$. I want to select $n$ points in $D$ to maximise the length shortest path that connects them all. In other ...
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14
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Chains with full range on a Boolean algebra with convex measure
Preliminaries. Let $X$ be a set and let $\mathcal A$ be a Boolean algebra of subsets of $X$ (i.e., $\mathcal A\subset 2^X$ such that $\mathcal A$ contains the empty set and is closed under finite ...
- 111
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19
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Third order estimate for linear elliptic equations
Let $\lambda < A < \Lambda$ be a constant symmetric matrix and $u$ be a $C^{\infty}$(elliptic regularity gives smooth solutions) solution of $\text{div} A \nabla u = 0$. Let $S_1$ be a sphere ...
- 455
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53
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Spectral theory of compact operator for quasi-Banach spaces
Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
- 938
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56
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Compactness and Leray-Schauder degree
What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
- 471
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34
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Bôcher's theorem for singularities on the boundary
Let $\Omega\subset\mathbb{R}^2$ be connected, open, bounded, and smooth. Suppose that $u\in C^0(\bar \Omega\setminus \{0\})\cap C^2(\Omega\setminus\{0\})$ is harmonic and positive in $\Omega$.
If $0\...
- 308
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14
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Sum of Simplex Volumes with Corners from Points in Convex Configuration
Question:
given $k,\,k>n$ points in convex configuration and general position in $n$ dimensional Euclidean space, i.e. no $n+1$ points of which are co-hyperplanar,
what can be said about how the ...
- 13.2k
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22
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Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?
I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
- 708
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54
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Complete list of functions known to satisfy the Siegel-Walfisz assumption
When I was reading Yitang Zhang's paper "Bounded Gaps between Primes" Text, on Page 1145, it is stated that "It should be remarked, by the Siegel-Walfisz theorem, that for all the
...
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11
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$k$-th power of spanning tree
I note that there are many papers which study $k$-th power of Hamilton cycle or path.
But there is no paper about $k$-th power of spanning tree with bounded degree (spanning tree with bounded degree ...
- 91
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0
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18
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Conditions on SDE coefficients for well-posedness of Fokker-Planck equation
Consider the following $n$-dimensional Ito-SDE:
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
What are the necessary regularity conditions on $\mu$ and $\sigma$ to ensure that the ...
- 111
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0
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62
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Order-convergence and interval topology on ${\cal P}(\omega)/(\text{fin})$
On any poset $(P, \leq)$ we can consider two different topologies that arise directly from the ordering relation.
1) Order convergence topolog $\tau_o(P)$ : By a set filter $\mathcal{F}$ on $P$ we ...
- 48.9k
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31
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Regularity of the eigenfunctions associated to perturbed laplacian on a compact manifold
Let $M$ be a closed manifold, I consider first order laplacian perturbation associated to a density $\rho \in \mathcal{C}^\infty(M)$ with $\rho > 0$ of the form :
$$
\Delta_{\rho} f = \Delta f + \...
0
votes
0
answers
21
views
Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
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24
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Imposing perfect slip boundary conditions in Stokes equations with Nitsche's method
I am working with a 2d mesh, given by a square with a circular hole placed in the middle. Let me call $\partial \Omega$ the boundary at the circle. I have a PDE for a vector field $\vec{v}$, and I ...
- 343