On R-duals and the duality principle in Gabor analysis

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et el., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames

Cyclicity in families of circle maps

In this paper we will study families of circle maps of the form x↦x+2πr+af(x)(mod2π) and investigate how many periodic trajectories maps from this family can have for a ‘typical’ function f provided the parameter a is small

Lattice vibrations and structural instability in Cesium near the cubic to tetragonal transition

Under pressure cesium undergoes a transition from a high-pressure fcc phase (Cs-II) to a collapsed fcc phase (Cs-III) near 4.2GPa. At 4.4GPa there follows a transition to the tetragonal Cs-IV phase. In order to investigate the lattice vibrations in the fcc phase and seek a possible dynamical instability of the lattice, the phonon spectra of fcc-Cs at volumes near the III-IV transition are calculated using Savrasov's density functional linear-response LMTO method. Compared with quasiharmonic model calculations including non-central interatomic forces up to second neighbours, at the volume $V/V_0= 0.44$ ($V_0$ is the experimental volume of bcc-Cs with $a_0$=6.048{\AA}), the linear-response calculations show soft intermediate wavelength $T_{[1\bar{1}0]}[{\xi}{\xi}0]$ phonons. Similar softening is also observed for short wavelength $L[\xi\xi\xi]$ and $L[00\xi]$ phonons and intermediate wavelength $L[\xi\xi\xi]$ phonons. The Born-von K\'{a}rm\'{a}n analysis of dispersion curves indicates that the interplanar force constants exhibit oscillating behaviours against plane spacing $n$ and the large softening of intermediate wavelength $T_{[1\bar{1}0]}[{\xi}{\xi}0]$ phonons results from a negative (110)-interplanar force-constant $\Phi_{n=2}$. The frequencies of the $T_{[1\bar{1}0]}[{\xi}{\xi}0]$ phonons with $\xi$ around 1/3 become imaginary and the fcc structure becomes dynamically unstable for volumes below $0.41V_0$. It is suggested that superstructures corresponding to the $\mathbf{q}{\neq}0$ soft mode should be present as a precursor of tetragonal Cs-IV structure.Comment: 12 pages, 5 figure

PROGRAMMED EFFECTS OF SURFACE WATER PRICE LEVELS ON U.S. AGRICULTURAL WATER USE AND PRODUCTION PATTERNS

Resource /Energy Economics and Policy,

Wrong on Red: The Constitutional Case Against Red-Light Cameras

This Note posits that red-light cameras dangerously reverse the presumption of innocence and deprive cited motorists of the fundamental right to confront their accusers. Cogent analysis of these weighty claims requires a narrowing of the field, both substantively and geographically. Though similar critiques may be mounted against other forms of automated enforcement technology, this Note focuses specifically on the constitutional implications of red-light cameras presently operating in Saint Louis, Missouri. Employing Saint Louis as an exemplar is useful in several ways. Unlike some communities throughout the region and nation, Saint Louis has not adopted other forms of automated traffic enforcement (e.g., photographic speed radar), rendering local law and commentary uniquely riveted on red-light cameras. Moreover, Saint Louis-area municipalities are the most active in Missouri—and among the most active in the nation—in adopting red-light camera ordinances. Though this Note views red-light cameras‘ legality through the lens of Missouri law, the analysis is capable of extrapolation to other states‘ red-light schemas to the extent that they parallel Missouri‘s system. Part I.A reviews the historical origins and jurisprudential development of two due process claims upon which courts might invalidate red-light camera ordinances: the presumption of innocence and the right to confrontation. With this constitutional context in mind, Part I.B–D tracks the history of automated enforcement technology in the United States, including legal challenges to red-light cameras and their predecessors. Part II analyzes the due process implications of red-light cameras and proposes that Missouri courts should extend their own relevant precedents and adopt those of other state courts addressing red-light camera ordinances and related schemas. On due process principles, Missouri courts should refuse to enforce local ordinances authorizing the installation and operation of red-light cameras

File Wrapper Estoppel

Patent law cases are rare in Washington. Thys v. Rivard, the most recent, concerned patent infringement and turned on two important rules of patent law: the doctrine of equivalents and the doctrine of file wrapper estoppel. Although the former of these rules had confronted the court on an earlier occasion, file wrapper estoppel was presented for the first time by the Rivard case. File wrapper estoppel, a rule of patent construction, derives its name from the Patent Office file wrapper containing the record of proceedings leading up to the grant of the letters patent. If, when the Patent Office examines the application for patent, it is found in the light of the prior art in the field that the applicant has drawn his claims too broadly for his actual invention they will be rejected. Should the rejection be made final, two alternatives are open to the applicant: appeal to the federal courts or amend the claims to meet the grounds of rejection. If the latter course is followed and the claims are narrowed to avoid conflict with the prior art cited against them, the applicant\u27s rights under the resulting patent are limited by file wrapper estoppel. That is, a court construing the patent will not extend its claims to the point where they become equal to or greater than those originally in the application. The theory is that the patentee is estopped from asserting an interpretation of his patent claims so broad as to recapture subject matter once cancelled from the application in order to secure allowance of the patent