Discrete Clifford Analysis

Design of Continuous and Discrete Optimal Control Systems with Prescribed Transient Response

We develop a discrete version of Clifford analysis, i.e., a higher-dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤm grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases are considered, involving so-called forward and backward basis vectors, controlling the support of the involved operators. Following a proper definition of a discrete Dirac operator and of some topological concepts, function theoretic results amongst which Stokes' theorem, Cauchy's theorem and a Cauchy integral formula are established.

The idea of time scales calculus’ theory was initiated and introduced by Hilger (1988) in his PhD thesis order to unify discret and continuous analysis and to expend the discrete and continous theories to cases ``in between''. Since then, mathematical research in this field has exceeded more than 1000 publications and a lot of applications in the fields of science, i.e., operations research, economics, physics, engineering, statistics, finance and biology. Ostrowski proved an inequality to estimate the absolute deviation of a differentiable function from its integral mean. This result was obtained by applying the Montgomery identity. In the present paper we derive a generalization of the Montgomery identity to the various time scale versions such as discrete case, continuous case and the case of quantum calculus, by obtaining this generalization of Montgomery identity we would prove our results about the generalization of the Ostrowski inequalities (without weighted case) to the several time scales such as discrete case, continuous case and the case of quantum calculus and recapture the several published results of different authors of various papers and thus unify corresponding discrete version and continuous version. Similarly we would also derive our results about the generalization of the Ostrowski inequalities (weighted case) to the different time scales such as discrete case and continuous case and recapture the different published results of several authors of various papers and thus unify corresponding discrete version and continuous version. Moreover, we would use our obtained results (without weighted case) to the case of quantum calculus.

Modelling non-negative discrete valued for time series analysis was introduced by previous researchers, that is Integer-Valued Autoregressive (INAR) model. The model involves both the parameter index and state space are discrete. Various phenomena within the scope of discrete cases are not only affected by previous discrete observations, but also affected by the discrete observations at surrounding locations in previous time. As a result, this model involves two parameter indices namely time and space, and also state space which are discrete. In continuous cases, a model has been introduced to overcome this problem, namely Generalized Space Time Autoregressive (GSTAR) model (discrete parameter index and continuous state space). Using this model to solve the discrete case produces a continuous forecast value, so it needs a discretization process. In the time series model for discrete cases, thinning operators have been introduced as a substitute for discretization in a continuous time series. The thinning operator paradigm in the discrete time series will be developed in the space-time model proposed in this paper. This paper will also introduce a new model, namely a space-time model for cases related to discrete cases. This means that the index of time and location parameters, as well as the state space used is a discrete state space. The properties that apply to the model are adopted from the INAR model and the GSTAR model for continuous cases. The main focus of this paper is the development of thinning operators from time series to space-time models. This paper also provides a modeling procedure using the GSTAR model for discrete cases. The result of this paper is the formation of a GSTAR model with a discrete state space.

This paper investigates the problem of Hankel norm model reduction for linear systems with time-varying delay in the state. For a given stable system, our attention is focused on the construction of reduced-order model, which guarantees the corresponding error system to be asymptotically stable and has a specified Hankel norm error performance. Two different approaches are proposed to solve this problem. One casts the model reduction into a convex optimization problem by using a linearization procedure, and the other is based on the cone complementarity linearization idea, which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints. Both continuous and discrete time cases are considered. A numerical example is provided to show the effectiveness of the proposed theory.

In a recent paper [Trans. Amer. Math. Soc. 378 (2025), pp. 851–883], the concept of generalized partial-slice monogenic (or regular) function was introduced over Clifford algebras. The present paper shall extend the study of generalized partial-slice monogenic functions from the associative case of Clifford algebras to non-associative alternative algebras, such as octonions. The new class of functions encompasses the regular functions [Rend. Sem. Mat. Univ. Padova 50 (1973), pp. 251–267] and slice regular functions [Rocky Mountain J. Math. 40 (2010), no. 1, 225–241] over octonions, indeed both appear in the theory as special cases. In the non-associative setting of octonions, we shall develop some fundamental properties such as identity theorem, Representation Formula, Cauchy (and Cauchy-Pompeiu) integral formula, maximum modulus principle, Fueter polynomials, Taylor series expansion. As a complement, the paper also introduces and discusses the notion of generalized partial-slice (and regular) functions. Although the study is limited to the case of octonions, it is clear from the statements and the arguments in the proofs that the results hold more in general in real alternative algebras equipped with a notion of conjugation.

The paper studies the asymptotic behavior of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given by these functionals. It is shown that in the discrete sampling case additive functionals have the same asymptotic distribution as the corresponding integral functionals for the continuous functional data case. These results are applied to obtain non central limit theorems for weighted additive functionals of random fields. As the majority of known results concern the discrete sampling case the developed methodology helps in translating these results to functional data without deriving them again. Numerical studies suggest that the theoretical findings are valid for wider classes of long-range dependent data.

An interesting issue in moving objects databases is to find similar trajectories of moving objects. Previous work on this topic focuses on movement patterns (trajectories with time dimension) of moving objects, rather than spatial shapes (trajectories without time dimension) of their trajectories. In this paper we propose a simple and effective way to compare spatial shapes of moving object trajectories. We introduce a new distance function based on ``one way distance'' (OWD). Algorithms for evaluating OWD in both continuous (piece wise linear) and discrete (grid representation) cases are developed. An index structure for OWD in grid representation, which guarantees no false dismissals, is also given to improve the efficiency of similarity search. Empirical studies show that OWD out-performs existent methods not only in precision, but also in efficiency. And the results of OWD in continuous case can be approximated by discrete case efficiently.

A somewhat general class of situations, that include Kolmogorov-Smirnov type results as special cases, is considered. These situations, which are described in the following sections, are required to have uniquely determined probability properties when the sample values used are from continuous populations of any nature. If the populations sampled are discrete, however, these probability values are not uniquely determined. This paper shows that the values for the continuous case represent bounds for the values that occur in any discrete case. The method used to show that these bound relations hold consists in noting that any discrete data situation can be interpreted as a situation involving the grouping of continuous data. Then bound relationships are established between the values of probabilities for the grouped data situations and the corresponding ungrouped data situations, which are the situations considered for the case of the continuous data. These bounds on probabilities for discrete data cases should be useful for practical applications. In practice, all data are discrete (due to limitations in measurement accuracy).

Relevance. The practical importance of the tasks of utility networks design, namely, the problems of optimizing the structure of the main pipeline according to given criteria, such as efficiency, reliability etc., under conditions of limitations, for example, the compatibility of various types of utilities. Since the main pipeline is laid on the ground with various physical and geological factors, natural and situational conditions, it is advisable to take the reliability of its operation as a global criterion. The task of network optimization is proposed in the form of displaying the main pipeline along the selected routes in three-dimensional space, which considers various existing communications and objects, as well as elevation marks of the area. The paper presents the problems of optimizing networks, both in the continuous case and in the discrete case, and also studies various indicators of the reliability of the operation of the main pipeline. Aim. To develop a model for laying the main pipeline in three-dimensional space, considering the reliability of the pipeline transport; to conduct a comparative analysis for various reliability indicators and topologies of the main pipeline. Objects. Utility communications and networks laid in three-dimensional space. Methods. Calculus of variations, discrete optimization methods, graph theory and hypernet theory methods, network reliability analysis methods. Results. The task of optimizing the main pipeline transport is given taking into account its nesting along the route in three-dimensional space with the choice of an optimization criterion (economic efficiency, reliability, etc.). The problem is presented in the form of continuous and discrete formulations, which is important for its development both within the theory of the calculus of variations and discrete optimization. In this work, the problem was studied within the framework of the theory of graphs and hypernets, which allow, firstly, taking into account the nesting of one structure (main pipeline) into another (a discrete analogue of three-dimensional space) and, secondly, clearly illustrating the results of numerical experiments. It is shown that under the conditions of a given variants for laying a secondary network along the primary channels, various optimal structures are obtained when considering various reliability indicators as a criterion, which can be used to implement a design solution for the construction and operation of pipeline transport for various purposes.

In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the well-known Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether's theorem is presented. An example was discussed to illustrate these results.

The maximum and minimum problems of functions which take only discrete values are often omitted from the usual treatises. Now, we are interested not only in the continuous case, and we shall pause for a moment at the discrete case, in order that we should obtain some indications of the situation.

As an analogue of the massless field equations in Euclidean space, we consider the so-called generalized Cauchy–Riemann equations introduced by E. Stein and G. Weiss. In the spin 1/2 case these equations reduce to the Dirac equation for spin 1/2 fields, which was thoroughly and intensively studied in Clifford analysis. For general spin it was recently shown that, in dimension 4, homogenous solutions form irreducible Spin modules. The next step then is to describe the corresponding Fischer decomposition, i.e. an irreducible decomposition of the space of spinor fields, which is well-known for spin 1/2 and for spin 1. The main aim of the present paper is to describe, still in dimension 4, the Fischer decomposition for spin 3/2.

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.

We propose a discrete surface theory in ℝ3 that unites the most prevalent versions of discrete special parametrizations. Our theory encapsulates a large class of discrete surfaces given by a Lax representation and, in particular, the one-parameter associated families of constant curvature surfaces. Our theory is not restricted to integrable geometries, but extends to a general surface theory. A quad net is a map from a strongly regular polytopal cell decomposition of a surface with all faces being quadrilaterals into ℝ3 with nonvanishing straight edges. A polytopal cell decomposition is strongly regular if each edge connects distinct vertices and meets at most two faces. Notice, in particular, that nonplanar faces are admissible. In discrete differential geometry, quad nets are understood as discretizations of parametrized surfaces [9, 10, 13]. In this agenda many classes of special surfaces have been discretized using algebro-geometric approaches for integrable geometry—originally using discrete analogues of soliton theory techniques (e.g., discrete Lax pairs and finite-gap integration [6]) to construct nets, but more recently using the notion of 3D consistency (reviewed in [10]). As in the smooth setting, these approaches have been successfully applied to space forms (see, e.g., [15–17, 22]). As an example consider the case of K-surfaces, i.e., surfaces of constant negative Gauß curvature. The integrability equations of classical surface theory are equivalent to the famous sine-Gordon equation [1, 20]. In an integrable discretization, the sine-Gordon equation becomes a finite difference equation for which integrability is encoded by a certain closing condition around a 3D cube. Both in the smooth and discrete settings, integrability is bound to specific choices of parameterizations, such as asymptotic line parametrizations for K-surfaces. In this way different classes of surfaces, such as minimal surfaces or surfaces of constant mean curvature (CMC), lead to different partial differential equations and give rise to different parametrizations. In the discrete case, this is reflected by developments that treat different special surfaces by disparate approaches. These integrable discretizations maintain characteristic properties of their smooth counterparts (e.g., the transformation theory of Darboux, Bäcklund, Bianchi, etc.). What has been lacking, however, is a unified discrete theory that lifts the restriction of special surface parametrizations. Indeed, different from the case of classical smooth surface theory, existing literature does not provide a general discrete theory for quad nets.