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Search: All articles in the CMB digital archive with keyword classical

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1. CMB Online first

Roche, Alan; Vinroot, C. Ryan
A factorization result for classical and similitude groups
For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well known result of Mœglin, Vignéras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representation to its dual.

Keywords:classical group, similitude group, involution, $p$-adic group, dual of representation
Categories:20G15, 22E50

2. CMB Online first

Alhasanat, Ahmad; Ou, Chunhua
Periodic steady-state solutions of a liquid film model via a classical method
In this paper, periodic steady-state of a liquid film flowing over a periodic uneven wall is investigated via a classical method. Specifically, we analyze a long-wave model that is valid at the near-critical Reynolds number. For the periodic wall surface, we construct an iteration scheme in terms of an integral form of the original steady-state problem. The uniform convergence of the scheme is proved so that we can derive the existence and the uniqueness, as well as the asymptotic formula, of the periodic solutions.

Keywords:film flow, classical methods, asymptotic analysis
Categories:34E05, 34E10, 34E15

3. CMB 2016 (vol 59 pp. 734)

Dimassi, Mouez
Semi-classical Asymptotics for Schrödinger Operator with Oscillating Decaying Potential
We study the distribution of the discrete spectrum of the Schrödinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$.

Keywords:periodic Schrödinger operator, semi-classical asymptotics, effective Hamiltonian, asymptotic expansion, spectral shift function
Categories:81Q10, 35P20, 47A55, 47N50, 81Q15

4. CMB 2015 (vol 58 pp. 877)

Zaatra, Mohamed
Generating Some Symmetric Semi-classical Orthogonal Polynomials
We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation $x^{2}\sigma u = -\lambda v$, where $(\sigma f)(x)=f(x^{2})$ and the odd moments of $u$ are $0$, is also regular and semi-classical form for every complex $\lambda $ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example.

Keywords:orthogonal polynomials, quadratic decomposition, semi-classical forms, structure relation
Categories:33C45, 42C05

5. CMB 2011 (vol 56 pp. 3)

Aïssiou, Tayeb
Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori
We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in the proof.

Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits
Categories:58G25, 81Q50, 35P20, 42B05

6. CMB 2011 (vol 55 pp. 736)

Hernández, Eduardo; O'Regan, Donal
Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations
In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

Keywords:neutral equations, mild solutions, classical solutions
Categories:35R10, 34K40, 34K30

7. CMB 2006 (vol 49 pp. 82)

Gogatishvili, Amiran; Pick, Luboš
Embeddings and Duality Theorem for Weak Classical Lorentz Spaces
We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$ \left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t), $$ where $$ f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds. $$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$ \|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t), $$ where $$ f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds. $$

Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality
Categories:26D10, 46E20

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