Uncertainty Principles and Balian-Low type Theorems in Principal Shift-Invariant Spaces

In this paper, we consider the time-frequency localization of the generator of a principal shift-invariant space on the real line which has additional shift-invariance. We prove that if a principal shift-invariant space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any $n\ge2$, there exist principal shift-invariant spaces on the real line that are also \nZ-invariant with an integrable orthonormal (or a Riesz) generator $\phi$, but $\phi$ satisfies $\int_{\mathbb R} |\phi(x)|^2 |x|^{1+\epsilon} dx=\infty$ for any $\epsilon>0$ and its Fourier transform $\hat\phi$ cannot decay as fast as $(1+|\xi|)^{-r}$ for any $r>1/2$. Examples are constructed to demonstrate that the above decay properties for the orthormal generator in the time domain and in the frequency domain are optimal

Uncertainty Principles in Finitely generated Shift-Invariant Spaces with additional invariance

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in $L^2(\R^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in $L^2(\R)$ whose generator and its translates form a Riesz basis.Comment: 12 page

On the foundations of thermodynamics

On the basis of new, concise foundations, this paper establishes the four laws of thermodynamics, the Maxwell relations, and the stability requirements for response functions, in a form applicable to global (homogeneous), local (hydrodynamic) and microlocal (kinetic) equilibrium. The present, self-contained treatment needs very little formal machinery and stays very close to the formulas as they are applied by the practicing physicist, chemist, or engineer. From a few basic assumptions, the full structure of phenomenological thermodynamics and of classical and quantum statistical mechanics is recovered. Care has been taken to keep the foundations free of subjective aspects (which traditionally creep in through information or probability). One might describe the paper as a uniform treatment of the nondynamical part of classical and quantum statistical mechanics ``without statistics'' (i.e., suitable for the definite descriptions of single objects) and ``without mechanics'' (i.e., independent of microscopic assumptions). When enriched by the traditional examples and applications, this paper may serve as the basis for a course on thermal physics.Comment: 78 page

Adventures in Theoretical Physics: Selected Papers of Stephen L. Adler -- Commentaries

These are the commentaries for a volume of reprints of my selected papers with commentaries that I am preparing for publication by World Scientific. Contents: Preface; (1)Early Years, and Condensed Matter Physics; (2) High Energy Neutrino Reactions, PCAC Relations, and Sum Rules; (3) Anomalies: Chiral Anomalies and Their Nonrenormalization, Perturbative Corrections to Scaling, and Trace Anomalies to All Orders; (4) Quantum Electrodynamics; (5) Particle Phenomenology and Neutral Currents; (6) Gravitation; (7) Non-Abelian Monopoles, Confinement Models, and Chiral Symmetry Breaking; (8) Overrelaxation for Monte-Carlo and Other Algorithms; (9) Quaternionic Quantum Mechanics, Trace Dynamics, and Emergent Quantum Theory; (10) Where Next?Comment: Latex 115p; Final version. Book version will differ in reference format and indexing; version 3 differs from version 2 by minor copy-editing correction