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We study the classical Cucker-Smale model in continuous time with a positive time delay $τ$. As in the non-delayed case, unconditional flocking occurs when $β\leq 1/2$ for every $τ>0$. Furthermore, we prove the exponential decay for the diameter of the velocities.
In this paper, we consider a special V-line transform. It integrates a given function f over the V-lines whose centers are on a circle centered at the origin and the symmetric axes pass through the origin. We derive two sampling scheme of the transform: the standard and interlaced ones. We prove the an error estimate for the schemes, which is explicitly expressed in term of f.
We study residual polynomials, $R_{x_0,n}^{(\mathfrak{e})}$, $\mathfrak{e}\subset\mathbb{R}$, $x_0\in\mathbb{R}\setminus\mathfrak{e}$, which are the degree at most $n$ polynomials with $R(x_0)=1$ that minimize the $\sup$ norm on $\mathfrak{e}$. New are upper bounds on their norms (that are optimal in some cases) and Szegő--Widom asymptotics under fairly general circumstances. We also discuss several illuminating examples and some results in the complex case.
We study holographic operators associated with Rankin-Cohen brackets which are symmetry breaking operators for the restriction of tensor products of holomorphic discrete series of SL2(R). Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.
We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.
We study a connection problem between the fundamental systems of solutions at singular points $0$ and $1$ for the generalized hypergeometric equation which is satisfied by the generalized hypergeometric series ${}_nF_{n-1}$. In general, the local solution space around $x=1$ consists of one dimensional singular solution space and $n-1$ dimensional holomorphic solution space. Therefore in the case of $n\ge3$, the expression of connection matrix depends on the choice of the fundamental system of solutions at $x=1$. On the connection problem for ordinary differential equations, Schäfke and Schmidt (LNM 810, Springer, 1980) gave an impressive idea which focuses on the series expansion of fundamental system of solutions. We apply their idea to solve the connection problem for the generalized hypergeometric equation and derive the connection matrix.
This paper targets to study the effect of the Riemann-Liouville fractional integral operator on unbounded variation points of a continuous function. In particular, we show that the fractional integral preserves the bounded variation points of a function. We also prove that the fractional integral operator is a bounded linear operator on the class of bounded variation and continuous functions.
We give a new proof of $l^2$ decoupling for the parabola inspired from efficient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and differences between this new proof and efficient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling such as $l^2 L^2$ decoupling, Bernstein, and ball inflation come into play.
An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group.
The intermediate dimensions are a family of dimensions introduced in 2019 by Falconer, Fraser, and Kempton [arXiv:1811.06493] to interpolate between the Hausdorff dimension and the box dimension. To date, there are limited examples of explicit calculations of the intermediate dimensions of interesting sets. We calculate the intermediate dimensions of sets of concentric spheres converging to the origin in Euclidean spaces. We also consider related sets including isolated points on concentric spheres and attenuated topologist's sine curves.
Some analogues of the Schrödinger refined Strichartz inequalities (Du, Guth, Li and Zhang) are obtained for the wave equation. These are used to improve the best known $L^2$ fractal Strichartz inequalities for the wave equation in dimensions $d \geq 4$.
In 1984 Bedford and McMullen independently introduced a family of self-affine sets now known as \emph{Bedford-McMullen carpets}. Their work stimulated a lot of research in the areas of fractal geometry and non-conformal dynamics. In this survey article we discuss some aspects of Bedford-McMullen carpets, focusing mostly on dimension theory.
We show that integrating a polynomial of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous polynomials of degree j = 1, 2,. .. , t, each at a unique point $ξ$ j of the simplex. This new and very simple formula can be exploited in finite (and extended finite) element methods, as well as in other applications where such integrals are needed.
We construct $δ$-regular sets with $δ\ge \frac12$ for which the analog of the Bourgain--Dyatlov Fractal Uncertainty Principle fails for the Walsh--Fourier transform.
Given a bounded measurable function $σ$ on $\mathbb{R}^n$, we let $T_σ$ be the operator obtained by multiplication on the Fourier transform by $σ$. Let $0<s_1\le s_2\le \cdots \le s_n<1$ and $ψ$ be a Schwartz function on the real line whose Fourier transform $\widehatψ$ is supported in $[-2,-1/2]\cup[1/2,2]$ and which satisfies $\sum_{j \in \mathbb{Z}} \widehatψ\left(2^{-j} ξ\right)=1$ for all $ξ\neq 0$. In this work we sharpen the known forms of the Marcinkiewicz multiplier theorem by finding an almost optimal function space with the property that, if the function \begin{equation*} (ξ_1,\dots, ξ_n)\mapsto \prod_{i=1}^n (I-\partial_i^2)^{\frac {s_i}2} \Big[ \prod_{i=1}^n \widehatψ(ξ_i) σ(2^{j_1}ξ_1,\dots , 2^{j_n}ξ_n)\Big] \end{equation*} belongs to it uniformly in $j_1,\dots , j_n \in \mathbb Z$, then $T_σ$ is bounded on $ {L}^p(\mathbb R^n)$ when $ |\frac{1}{p}-\frac{1}{2} | < s_1$ and $1<p<\infty$. In the case where $s_i\neq s_{i+1}$ for all $i$, it was proved in [Grafakos, Israel J. Math., to appear] that the Lorentz space $L ^{\frac{1}{s_1},1} (\mathbb{R}^n) $ is the function space sought. In this work we address the significantly more difficult general case when for certain i
We give two Roth theorems, related to the nonlinear configuration $x$, $x+P_1(t)$, $x+P_2(t)$ involving two polynomials, for sets in $\mathbb{R}$ of positive density and of fractional dimensions. The proof uses Fourier analysis.
In a recent paper [5] a smooth function f : [0; 1] --> R with all derivatives vanishing at 0 has been considered and a global condition, showing that f is indeed identically 0, has been presented. The purpose of this note is to replace the classical Fundamental Theorem of Calculus for the Riemann integral, as it has been used in [5], with a weaker form going back to Volterra [7], which is little known. Therefore the proof we propose in this paper turns to be important also from the teaching point of view, as long as in literature there are very few examples in which explicitly the lower integral and the upper integral of a function appear (usually the assumption that the function is Riemann-integrable is required).
For $h>0$ and positive integers $m$, $d$, such that $m>d/2$, we study non-stationary interpolation at the points of the scaled grid $h\mathbb{Z}^d$ via the Matérn kernel $Φ_{m,d}$---the fundamental solution of $(1-Δ)^m$ in $\mathbb{R}^d$. We prove that the Lebesgue constants of the corresponding interpolation operators are uniformly bounded as $h\to0$ and deduce the convergence rate $O(h^{2m})$ for the scaled interpolation scheme. We also provide convergence results for approximation with Matérn and related compactly supported polyharmonic kernels.
We consider weighted operators acting on $L^p(\mathbb{R}^d)$ and show that they depend continuously on the weight $w\in A_p(\mathbb{R}^d)$ in the operator topology. Then, we use this result to estimate $L^p_w(\mathbb{T})$ norm of polynomials orthogonal on the unit circle when the weight $w$ belongs to Muckenhoupt class $A_2(\mathbb{T})$ and $p>2$. The asymptotics of the polynomial entropy is obtained as an application.
We define in this work a notion of Young differential inclusion $$ dz_t \in F(z_t)dx_t, $$ for an $α$-Holder control $x$, with $α>1/2$, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, $γ$-Hölder continuous set-valued map on the interval $[0,1]$ has a selection with finite $p$-variation, for $p>1/γ$. We also give a notion of solution to the rough differential inclusion $$ dz_t \in F(z_t)dt + G(z_t)d{\bf X}_t, $$ for an $α$-Holder rough path $\bf X$ with $α\in \left(\frac{1}{3},\frac{1}{2}\right]$, a set-valued map $F$ and a single-valued one form $G$. Then, we prove the existence of a solution to the inclusion when $F$ is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.
We prove that any non-complete orthonormal system in a Hilbert space can be transformed into a basis by small perturbations.
We address a fundamental issue in the nonparametric inference for systems of interacting particles: the identifiability of the interaction functions. We prove that the interaction functions are identifiable for a class of first-order stochastic systems, including linear systems with general initial laws and nonlinear systems with stationary distributions. We show that a coercivity condition is sufficient for identifiability and becomes necessary when the number of particles approaches infinity. The coercivity is equivalent to the strict positivity of related integral operators, which we prove by showing that their integral kernels are strictly positive definite by using Müntz type theorems.
We give new identities for some symmetric polynomials. As applications of these identities, we obtain some formulas for a higher order analogue of Fibonacci and Lucas numbers.
Discrete analogs of the Lebedev-Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels ${\rm Re} K_{α+in}(x), {\rm Im} K_{α+in}(x), x >0, n \in \mathbb{N}, |α| < 1,\ i $ is the imaginary unit and $K_ν(z)$ is the modified Bessel function. The corresponding inversion formulas for suitable functions and sequences in terms of these series and integrals are established when $α= \pm 1/2$. The case $α=0$ reduces to the Kontorovich-Lebedev transform.
People in this topic
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J. Yu contributes to research discovery and scholarly infrastructure.
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Dong Wang contributes to research discovery and scholarly infrastructure.
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Rahul Kumar contributes to research discovery and scholarly infrastructure.
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R. Garg contributes to research discovery and scholarly infrastructure.
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Andronikos Paliathanasis contributes to research discovery and scholarly infrastructure.
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Stefan Steinerberger contributes to research discovery and scholarly infrastructure.
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Zheng Zhu contributes to research discovery and scholarly infrastructure.
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Xiaofeng Wang contributes to research discovery and scholarly infrastructure.
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Zakhar Kabluchko contributes to research discovery and scholarly infrastructure.
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Ting Chen contributes to research discovery and scholarly infrastructure.
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Xing Wang contributes to research discovery and scholarly infrastructure.
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Xiaodong Wang contributes to research discovery and scholarly infrastructure.
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