-NONUNIFORM MULTIRESOLUTION ANALYSIS

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal.158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. This gap was filled by Gabardo and Nashed [11] by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in \(L^2(\mathbb R)\). In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.

Non-dense orbits on homogeneous spaces and applications to geometry and number theory

Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

Non-SUSY heterotic string vacua of Gepner models with vanishing cosmological constant

We study a natural generalization of the results given in K. Aoyama and Y. Sugawara, Prog. Theor. Exp. Phys. 2020, 103B01 (2020) to heterotic strings. Namely, starting from the generic Gepner models for Calabi –Yau three-folds, we construct non-SUSY heterotic string vacua with vanishing cosmological constant at the one-loop level. We especially focus on asymmetric orbifolding based on some discrete subgroup of the chiral $U(1)$ action which acts on both the Gepner model and the $SO(32)$ or $E_8\times E_8$ sector. We present a classification of the relevant orbifold models leading to the string vacua with the properties mentioned above. In some cases, the desired vacua can be constructed in a manner quite similar to those given in the previous paper for the type II string, in which the orbifold groups contain two generators with discrete torsions. On the other hand, we also have simpler models that are just realized as asymmetric orbifolds of cyclic groups with only one generator.

Construction of nonuniform periodic wavelet frames on non-Archimedean fields

In real life applications not all signals are obtained by uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. Gabardo and Nashed, and Gabardo and Yu filled this gap by the concept of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which the associated translation set \(\Lambda= \{0,r/N\}+2\mathbb{Z}\) is no longer a discrete subgroup of \(\mathbb{R}\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we introduce a notion of nonuniform periodic wavelet frame on non-Archimedean field. Using the Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.

Directions in orbits of geometrically finite hyperbolic subgroups

We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather they may equidistribute on a fractal subset. Understanding the behavior of these orbits near the boundary is central to Patterson–Sullivan theory and much further work. Our theorem characterises the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish a formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which also applies in the lattice case.

Taxometric evidence for a dimensional latent structure of hypnotic suggestibility

Hypnotic suggestibility denotes a capacity to respond positively to direct verbal suggestions in an involuntary manner in the context of hypnosis. Elucidating the characteristics of this ability has bearing on responsiveness to suggestions in a variety of clinical and non-clinical contexts. A considerable amount of research has focused on a small subgroup of individuals who display strong responsiveness to hypnotic suggestions. However, it remains poorly understood whether these highly suggestible individuals constitute a discrete subgroup (taxon) that is characterized by a qualitatively distinct mode of responding from the remainder of the population or whether hypnotic suggestibility is better modelled as a dimensional ability. In this study, we applied taxometric analysis, a statistical method for distinguishing between dimensional and categorical models of a psychological ability, to behavioural and involuntariness subscale scores of the Harvard Group Scale of Hypnotic Susceptibility Scale: Form A (HGSHS:A) in a sample of neurotypical individuals (N=584). Analyses of HGSHS:A behavioural and involuntariness subscale scores with different a priori taxon base rates yielded consistent evidence for a dimensional structure. These results suggest that hypnotic suggestibility, as measured by the HGSHS:A, is dimensional and have implications for current understanding of individual differences in responsiveness to direct verbal suggestions.

Imaging in Pancreatic Neuroendocrine Tumors

Pancreatic neuroendocrine tumors (NETs) form a discrete subgroup of pancreatic neoplasms. They are rarer than pancreatic adenocarcinomas but need to be differentiated from other pancreatic tumors and pathologies as they carry a better prognosis. Imaging plays a central role in detecting, characterizing, and staging of pancreatic NETs as they tend to have typical radiological features. A lot of advancements have taken place in the field of imaging and theranostics which have revolutionized their detection and management. In this article we shall review the various imaging techniques available to the radiologist, salient imaging features of different types of pancreatic NETs, staging and grading systems, as well as a brief overview of their management.

Measure Rigidity for Horospherical Subgroups of Groups Acting on Trees

We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma \leq G$ a discrete subgroup. For a large class of groups $G$, we give a classification of the probability measures on $G/\Gamma $ invariant under horospherical subgroups. When $\Gamma $ is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund’s theorem for geometrically finite quotients. Finally, we show equidistribution of large compact orbits.

Inequalities for nonuniform wavelet frames

Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set {\Lambda=\{0,r/N\}+2\mathbb{Z}} is no longer a discrete subgroup of {\mathbb{R}} but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system {\{\psi_{j,\lambda}(x)=(2N)^{j/2}\psi((2N)^{j}x-\lambda),\,j\in\mathbb{Z},\,% \lambda\in\Lambda\}} to be a frame for {L^{2}(\mathbb{R})} . The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system.