4,599 research outputs found
A Pointwise a-priori Estimate for the d-bar Neumann Problem on Weakly Pseudoconvex Domains
We introduce a new integral representation formula in the d-bar Neumann
Theory on weakly pseudoconvex domains which satisfies certain estimates
analogous to the basic L^2 estimate. It is expected that more complete
estimates can be obtained in case the boundary is of finite type
Regular Representations of Time-Frequency Groups
In this paper, we study the Plancherel measure of a class of non-connected
nilpotent groups which is of special interest in Gabor theory. Let be a
time-frequency group. More precisely, that is ,
are translations and modulations operators acting in
and is a non-singular matrix. We compute the
Plancherel measure of the left regular representation of which is denoted
by The action of on induces a representation
which we call a Gabor representation. Motivated by the admissibility of this
representation, we compute the decomposition of into direct integral of
irreducible representations by providing a precise description of the unitary
dual and its Plancherel measure. As a result, we generalize Hartmut F\"uhr's
results which are only obtained for the restricted case where ,
and Even in the case where is not type I, we
are able to obtain a decomposition of the left regular representation of
into a direct integral decomposition of irreducible representations when .
Some interesting applications to Gabor theory are given as well. For example,
when is an integral matrix, we are able to obtain a direct integral
decomposition of the Gabor representation of $G.
Functional Equations and Fourier Analysis
By exploring the relations among functional equations, harmonic analysis and
representation theory, we give a unified and very accessible approach to solve
three important functional equations -- the d'Alembert equation, the Wilson
equation, and the d'Alembert long equation, on compact groups.Comment: 8 pages, to appear in CM
Bells of Mindfulness: An Online Mindfulness Meditation Course to Promote Mindfulness Meditation for PhD Students
Over the last 20 years, there has been growing evidence of mental health issues in doctoral candidates worldwide (Zhang et al., 2022; Barry et al., 2019; Gewin, 2012; Radison & DiGeronimo, 2005). Practicing mindfulness meditation, which is one way to cope with stress and anxiety (Kabat-Zinn, 1991), could be a useful practice for these PhD students. However, despite all the evidence that suggests the health benefits of having a regular meditation routine, motivating graduate students to practice meditation can be challenging (Franco, 2020). This study addresses this challenge by assessing a 5-week mindfulness meditation course designed to support graduate students in developing a habit of practicing mindfulness meditation. Graduate students in PhD degree programs, many of whom worked and/or had families, were recruited to participate in a 5-week online mindfulness meditation course. Principles from social cognitive learning theory, particularly self-efficacy, guided course structure and activities, helping to better understand and interpret participants\u27 experiences and growth throughout the course. Interviews were conducted mid- and post-course to find out how effective the online course was in helping participants to make a habit of practicing mindfulness meditation and to understand what factors of the course were most effective in changing their mindfulness meditation practice. Participants took the Self-Efficacy for Mindfulness Meditation Practice surveys, pre-, mid-, and post-course to inform qualitative data from interviews
Quantization Of Spin Direction For Solitary Waves In A Uniform Magnetic Field
It is known that there are nonlinear wave equations with localized solitary
wave solutions. Some of these solitary waves are stable (with respect to a
small perturbation of initial data) and have nonzero spin (nonzero intrinsic
angular momentum in the center of momentum frame). In this paper we consider
vector-valued solitary wave solutions to a nonlinear Klein-Gordon equation and
investigate the behavior of these spinning solitary waves under the influence
of an externally imposed uniform magnetic field. We find that the only
stationary spinning solitary wave solutions have spin parallel or anti-parallel
to the magnetic field direction.Comment: 4 page
Axially Symmetric Cosmological Mesonic Stiff Fluid Models in Lyra's Geometry
In this paper, we obtained a new class of axially symmetric cosmological
mesonic stiff fluid models in the context of Lyra's geometry. Expressions for
the energy, pressure and the massless scalar field are derived by considering
the time dependent displacement field. We found that the mesonic scalar field
depends on only coordinate. Some physical properties of the obtained models
are discussed.Comment: 13 pages, no figures, typos correcte
Generalized Satisfiability Problems via Operator Assignments
Schaefer introduced a framework for generalized satisfiability problems on
the Boolean domain and characterized the computational complexity of such
problems. We investigate an algebraization of Schaefer's framework in which the
Fourier transform is used to represent constraints by multilinear polynomials
in a unique way. The polynomial representation of constraints gives rise to a
relaxation of the notion of satisfiability in which the values to variables are
linear operators on some Hilbert space. For the case of constraints given by a
system of linear equations over the two-element field, this relaxation has
received considerable attention in the foundations of quantum mechanics, where
such constructions as the Mermin-Peres magic square show that there are systems
that have no solutions in the Boolean domain, but have solutions via operator
assignments on some finite-dimensional Hilbert space. We obtain a complete
characterization of the classes of Boolean relations for which there is a gap
between satisfiability in the Boolean domain and the relaxation of
satisfiability via operator assignments. To establish our main result, we adapt
the notion of primitive-positive definability (pp-definability) to our setting,
a notion that has been used extensively in the study of constraint satisfaction
problems. Here, we show that pp-definability gives rise to gadget reductions
that preserve satisfiability gaps. We also present several additional
applications of this method. In particular and perhaps surprisingly, we show
that the relaxed notion of pp-definability in which the quantified variables
are allowed to range over operator assignments gives no additional expressive
power in defining Boolean relations
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