Aspects of Aperiodic Order

The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called “classical separation of variables theorem” for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.

This chapter is about the tiling dynamical systems approach to the study of aperiodic order. We compare and contrast four related types of systems: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional \(\mathbb {Z}^d\)-systems, and multidimensional tiling systems. Aperiodically ordered structures are often hierarchical in nature, and there are a number of different yet related ways to define them. We will focus on what we are calling “supertile construction methods”: symbolic substitution in one and many dimensions, S-adic sequences, self-similar and pseudo-self-similar tilings, and fusion rules. The techniques of dynamical analysis of these systems are discussed and a number of results are surveyed. We conclude with a discussion of the spectral theory of supertile systems from both the dynamical and diffraction perspectives.

Harmonic analysis on [formula omitted]: Part II

Consider a continuous flow on a locally compact, separable, metric space.If the set of nonperiodic recurrent points is nowhere dense, then there is an open, dense, invariant subset of the phase space which has a Hausdorff orbit space.A separatrix is defined to be a trajectory which is in the closure of the set of trajectories at which the orbit space is not Hausdorff.If the flow is completely unstable, then the set of points which lie on sparatrices is nowhere dense in the phase space.Introduction.Orbit spaces and their topological properties arise in a natural way when the trajectorial structure of a continuous flow is studied, e.g., in [2],[4], and [6].One of the fundamental concepts of trajectorial structure is that of a parallelizable flow.Parallelizable flows have been studied and characterized in [1], [2], and [7, 2.4].If the flow is parallelizable, then the natural mapping of the phase space onto the orbit space is the projection of a fiber bundle.One immediate consequence of parallelizability is that the orbit space is Hausdorff.If a flow on a locally compact, separable, metric space has a Hausdorff orbit space, then there is a natural generalization of the fiber bundle structure for parallelizable flows [6].Hence, flows with Hausdorff orbit spaces are a natural generalization of flows which are parallelizable.Flows with Hausdorff orbit spaces also arise in the study of completely unstable flows [4].Clearly, it is very restive to assume that a flow has a Hausdorff orbit space.In this paper we will show that if a flow on a locally compact, separable, metric space has a nowhere dense set of nonperiodic recurrent points, then there is an open, dense, invariant subset U of the phase space such that the orbit space with respect to U is Hausdorff.A separatrix is a trajectory which is in the closure of the set of trajectories at which the orbit space is not Hausdorff.In the case that the flow is completely unstable, we will show that the set of points which lie on sparatrices is nowhere dense in the phase-space.1. Basic definitions and properties of continuous flow.Throughout this paper X will be a locally compact, separable, metric space and R, R+ will denote the reals and nonnegative reals, respectively.A continuous flow tt on A-is a continuous mapping of X X R onto X

A spherical topological manifold of dimension n- 1 forms a prototile on its cover, the (n- 1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. A basis for the harmonic analysis on the (n- 1)-sphere is given by the spherical harmonics that transform according to irreducible representations of the orthogonal group. Multiplicity and selection rules appear in the form of reduction of group representations. The deck transformations form a subgroup and so the representations of the orthogonal group can be reduced to those of this subgroup. Upon reducing to the identity representation of the subgroup, the reduced subset of spherical harmonics becomes periodic on the tiling and tunes the harmonic analysis on the (n-1)-sphere to the manifold. A particular class of spherical 3-manifolds arises from the Platonic polyhedra. The harmonic analysis on the Poincaré dodecahedral 3-manifold was analyzed along these lines. For comparison we construct here the harmonic analysis on simplicial spherical manifolds of dimension n=1, 2, 3. Harmonic analysis can be applied to the cosmic microwave background observed in astrophysics. Selection rules found in this analysis can detect the multiple connectivity of spherical 3-manifolds on the space part of cosmic space–time.

This is the proceedings of theworkshop on recent developments in ergodic theory and dynamical systemson March 2011and March 2012 at the University of North Carolina at Chapel Hill. Thearticles in this volume cover several aspects of vibrant research in ergodic theory and dynamical systems. It contains contributions to Teichmuller dynamics, interval exchange transformations, continued fractions, return times averages, Furstenberg Fractals, fractal geometry of non-uniformly hyperbolic horseshoes, convergence along the sequence of squares, adic and horocycle flows, and topological flows. These contributions illustrate the connections between ergodic theory and dynamical systems, number theory, harmonic analysis, probability, andalgebra. Two surveys are included which give a nice introduction for interested young or senior researcher to some active research areas. Overall this volume provides a very useful blend of techniques and methods as well as directions of research on general convergence phenomena in ergodic theory and dynamical systems.

This paper explores the concept of Generalized Roughness in LA-Semigroups and its applications in various mathematical disciplines. We highlight the fundamental properties and structures of Generalized Roughness, examining its relationships with Fuzzy Lie Algebras, Order Theory, Lattice Structures, Algebraic Structures, and Categorical Perspectives. Moreover, we investigate the potential of mathematical modeling, optimization techniques, data analysis, and machine learning in the context of Generalized Roughness. Our findings reveal important results in Generalized Roughness, such as the preservation of roughness under the fuzzy equivalence relation and the composition of roughness sets. We demonstrate the significance of Generalized Roughness in the context of order theory and lattice structures, presenting key propositions and a theorem that elucidate its properties and relationships. Furthermore, we explore the applications of Generalized Roughness in mathematical modeling and optimization, highlighting the optimization of roughness measures, parameter estimation, and decision-making processes related to LA-Semigroup operations. We showcase how mathematical techniques can enhance understanding and utilization of LA-Semigroups in practical scenarios. Lastly, we delve into the role of data analysis and machine learning in uncovering patterns, relationships, and predictive models in Generalized Roughness. By leveraging these techniques, we provide examples and insights into how data analysis and machine learning can contribute to enhancing our understanding of LA-Semigroup behavior and supporting decision-making processes.

This article addresses the extension of classical spectral theory to the setting of non-separable Banach spaces (NSBS), a field that remains underdeveloped despite its increasing relevance in functional analysis and mathematical physics. While spectral theory in separable Banach spaces has been thoroughly investigated—particularly through the work of Fredholm, Riesz, and others—the transition to nonseparable spaces presents both technical and conceptual challenges. These arise mainly from the absence of countable dense subsets and orthonormal or Schauder bases, which are instrumental in formulating and proving classical theorems. The main objective of this work is to provide a rigorous and systematic framework for spectral theory in NSBS, focusing on the classification of spectral components, the generalisation of key theorems, and the topological and algebraic behaviour of operators acting on such spaces. We begin by recalling the foundations of spectral theory in the classical (separable) case and proceed to identify the structural modifications required in the non-separable context. A central part of the analysis is devoted to the residual and continuous spectra, whose properties exhibit significant deviations from their counterparts in separable settings. We propose new definitions, prove generalised versions of the spectral theorem, and characterise the structure of the spectrum under various topological assumptions. Particular attention is given to the stability of the spectrum under perturbations—both compact and non-compact—and to the conditions under which the classical spectral decomposition remains valid. Several illustrative examples are provided to highlight the influence of non-separability on spectral behaviour. The results presented here contribute to a deeper understanding of spectral phenomena in infinitedimensional analysis, particularly in cases where separability cannot be assumed. They offer theoretical tools for future work in operator theory, interpolation, partial differential equations, inverse problems, and the theory of operator algebras. This article thereby lays the groundwork for further mathematical developments in non-separable environments and suggests new directions for interdisciplinary applications.

Uncertain stochastic calculus is a relatively new sub discipline of mathematics. This branch of mathematical sciences seeks to develop models that capture aleatory and epistemic features of generic uncertainty in dynamical systems. The growth of uncertain stochastic theory has given birth to a novel class of differential equations called uncertain stochastic differential equations (USDEs). Exact and analytic solutions to this family of differential equations are not always available. In such cases, numerical analysis provides a gateway to approximate solutions. This paper examines a Runge-Kutta method for solving USDEs. Before examining and applying the Runge-Kutta method, the paper states and proves the existence and uniqueness theorem. The Runge-Kutta method is then applied to solve an American call option pricing problem. This numerical algorithm proves to be effective and efficient because it produces almost the same results as compared to Chen's analytic formula and the classical Black-Scholes model.

Chaos theory can be used for quantitative dynamics of uncertainty and finding order in its perturbances. Chaos theory is largely a colloquial notion and is referred to as an analysis of non-linear dynamical systems. The studies devoted to linear dynamical systems and theories of complexity involve studies of turbulence, or, more precisely, the transfer from stability to turbulence. The dynamical system by its nature does not follow long-term forecasts. There are two reasons for the unpredictability. Dynamical systems involve both feedback and critical levels. To paraphrase, one can say that a dynamical system forms a system of non-linear feedback. The most important properties of the system include: sensitivity to the change of the initial conditions, occurrence of critical points and a fractal dimension. Non-linear dynamical systems tend to have more than one solution. The case often is that the number of solutions is huge or even infinite. A visual representation of the data forms a finite space called the phase space of the system. The number of dimensions in the space is determined by the number of variable presents in the system. If there are two or three variables, it is possible to visually examine the data. In case of a larger number of dimensions, the data are examined with the use of mathematical data. Another term used in non-linear dynamical systems is the attractor. This is an area of a equilibrium of non-linear system within a time series. A system which aims to achieve an equilibrium in the form of a single value has a point attractor. Besides, there are phase attractors, which form periodic cycles or orbits in space forming a limit cycle. An attractor, in which none of the points in the space overlaps and whose orbits do not cross but both originate in the same area of phase space, is called strange attractor. Such attractors in contrast to point attractors are non-periodic and predominantly have a fractal dimension [10,11].

Various new nonembeddability results (mainly into L/sub 1/) are proved via Fourier analysis. In particular, it is shown that the edit distance on {0, 1}/sup d/ has L/sub 1/ distortion (log d)/sup 1/2 - o(1)/. We also give new lower bounds on the L/sub 1/ distortion of quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.

The study of geodesic flows on homogenous spaces is an area of research that has yielded some fascinating developments. This book, first published in 2000, focuses on many of these, and one of its highlights is an elementary and complete proof (due to Margulis and Dani) of Oppenheim's conjecture. Also included here: an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; Ledrappier's example of a mixing action which is not a mixing of all orders. The treatment is as self-contained and elementary as possible. It should appeal to graduate students and researchers interested in dynamical systems, harmonic analysis, differential geometry, Lie theory and number theory

In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation ( i ∂ t + Δ ) u = V ( x ) u (i\partial _t+\Delta )u=V(x)u . In general the problem is to find a class of potentials V V which allows the unique continuation. The key point of our work is to make a direct link between the problem and the weighted L 2 L^2 resolvent estimates ‖ ( − Δ − z ) − 1 f ‖ L 2 ( | V | ) ≤ C ‖ f ‖ L 2 ( | V | − 1 ) \|(-\Delta -z)^{-1}f\|_{L^2(|V|)}\leq C\|f\|_{L^2(|V|^{-1})} . We carry it out in an abstract way, and thereby we do not need to deal with each of the potential classes. To do so, we will make use of the limiting absorption principle and Kato H H -smoothing theorem in spectral theory, and employ some tools from harmonic analysis. Once the resolvent estimate is set up for a potential class, from our abstract theory the unique continuation would follow from the same potential class. Also, it turns out that there can be no dented surface on the boundary of the maximal open zero set of the solution u u . In this regard, another main issue for us is to know which class of potentials allows the resolvent estimate. We establish such a new class which contains previously known ones, and we will also apply it to the problem of well-posedness for the equation.

We give an overview of several works of Victor Katsnelson published in 1965–1970, and pertaining to the complex and harmonic analysis and the spectral theory.

Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory