Transformations of discrete closure systems

The workshop Discrete Differential Geometry , organized by Alexander I. Bobenko (Berlin), Richard W. Kenyon (Vancouver), John M. Sullivan (Berlin) and Günter M. Ziegler (Berlin), was held March 5th to March 11th, 2006. The meeting was very well attended, with almost 50 participants, from as far away as Australia and China. Discrete differential geometry is a new and active mathematical terrain where differential geometry (providing the classical theory for smooth manifolds) and discrete geometry (concerned with polytopes, simplicial complexes, etc.) meet and interact. Problems of discrete differential geometry also naturally appear in (and are relevant for) other areas of mathematics. Moreover, the process of discretizing notions, problems and methods from the smooth theory often brings out new connections and interrelations between different areas. The workshop at Oberwolfach brought together researchers with a wide variety of backgrounds, including of course discrete geometry and differential geometry, but also integrable systems, combinatorics, mathematical physics and geometry processing. The exchange of ideas among different subfields helped to build new bridges between these mathematical communities. Discrete differential geometry can be said to have arisen from the observation that when a notion from smooth geometry (such as the notion of a minimal surface) is discretized “properly”, the discrete objects are not merely approximations of the smooth ones, but have special properties of their own which make them form in some sense a coherent entity by themselves. The discrete theory would seem to be the more fundamental one: The smooth theory can always be recovered as a limit, while there seems to be no natural way to predict from the smooth theory which discretizations will have the nicest properties. One case where these ideas seem particularly well-developed is for geometries described by integrable systems. The notion of a discrete integrable system as given by consistency on a cubic lattice has already shed new light on classical, smooth integrable systems. Another theme which arose repeatedly during the workshop was that of circle patterns and sphere packings. These can be used to discretize conformal maps, isothermic surfaces, and elastic bending energy. Since a computer works with discrete representations of data, it is no surprise that many of the applications of discrete differential geometry are found within computer science, particularly in the areas of computational geometry, graphics and geometry processing. The workshop brought theoreticians together with people interested in these and other applications.

Symmetries and integrability of difference equations

A higher-order integrator approach is proposed to obtain an approximate discrete-time transfer function for uncertain continuous systems having interval uncertainties. Thanks to simple algebraic operations of this approach, the resulting discrete model is a rational function of the uncertain parameters. The problem of non-linearly coupled coefficients of exponential nature in the exact discrete-time transfer function is therefore circumvented. Furthermore, interval structure of the uncertain continuous-time system is preserved in the resulting discrete model by using this approach. Formulas to obtain the lower and upper bounds for the discrete interval system are derived, so that existing robust results in the discrete-time domain can be easily applied to the discretized system. Digital simulation and design for the continuous-time interval plant can then be performed based on the obtained discrete-time interval model.

Dynamic fracture of a one-dimensional chain of identical linear oscillators (masses connected by springs) is considered in the work. The system is supposed to consist of arbitrary but finite number of links and the first mass is supposed to be fixed. Two loading conditions are discussed: free oscillations of an initially statically prestressed chain and loading the chain with a short deformation pulse. Both problems are solved analytically for an arbitrary number of links. The obtained solutions are investigated and a dynamic fracture effect related to an explicitly discrete structure of the system is revealed: a deformation wave travelling through the chain is distorted and some links may be subjected to critical deformation. The obtained solutions for the chain are compared to the solutions of analogous problems stated for an elastic rod – a continuum counterpart of the considered discrete system. It is shown that the discussed fracture effect cannot be found in a continuous system.

This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco’s allow one to see more easily the connections between standard conditions on general cco’s, and furthermore, we show that these connections for cco’s can be even deduced from the corresponding ones for bco’s, when considering cco’s relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco’s via bco’s is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco’s, independent results on cco’s are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco’s also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory.

We present the first steps of a procedure which discretizes surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Bäcklund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalization of the integrable discrete Tzitzéica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretization of affine spheres in affine differential geometry.

it shows the possibility of modeling regimes of deterministic chaos in vibrations of discrete structures, which formed as special matrix forms of the Latin square. A mathematical model of such discrete systems, which based of an analysis is made of the conditions for combining the discrete structure and the requirements for transformations of the operator of its evolution, which ensure the achievement of such modes, is proposed. Results of imitation modeling of oscillatory processes on cycles of discrete system states, which forming similarity of attractor trajectories, are represented.

The notion of a Boolean algebra with operators (BAO) was first defined by Jonsson and Tarski in 1951. Since that time, many varieties of BAOs, including modal algebras, closure algebras, monadic algebras, and of course relation algebras, have been studied. With the exception of relation algebras, these varieties each have one unary operator. This paper investigates a variety of BAOs with one binary operator. Begin with a semigroup S = $(S,\cdot).$ The complex algebra of S, denoted S$\sp+,$ is a Boolean algebra whose underlying set is the power set of S with set union, intersection and complementation as the Boolean algebra operations. The multiplication operation defined on the semigroup induces a normal, associative binary operation, '*', on the complex algebra as follows: for all subsets A and B contained in S, $A\* B=\{a\cdot b:a\in A, b\in B\}.$ Hence, the complex algebra of a semigroup is a BAO with one normal binary associative operator. Let S$\sp+$ be the class of all complex algebras of semigroups and consider the variety generated by the class S$\sp+,$ denoted ${\cal V}$(S$\sp+).$ The tools required to study this variety are developed, including the duality between BAOs and relational structures as it applies to ${\cal V}$(S$\sp+).$ A closure operator is defined which is used to determine homomorphic images of members of ${\cal V}$(S$\sp+).$ Theorems on the subdirectly irreducible and simples algebras in ${\cal V}$(S$\sp+)$ are proved. Next, the structure of ${\cal V}$(S$\sp+)$ is analyzed. The general problem of representing a BAO with one binary, normal, associative operator as a member of ${\cal V}$(S$\sp+)$ is discussed. Several examples and theorems concerning representation are presented. It is shown that the quasivariety generated by S$\sp+$ is strictly contained in ${\cal V}$(S$\sp+).$ Lastly, the structure of the lattice of subvarieties of ${\cal V}$(S$\sp+)$ is investigated. There are precisely two atoms in this lattice; each atom is generated by a two element algebra. Two infinite chains of varieties exist, with the smallest element in each chain a cover of exactly one atom. This leads to a discussion of certain splitting algebras and conjugate varieties. Finally, equations characterizing some important subvarieties are developed.

The ultimate goal of our study is to obtain suggestions for the development of devices capable of the automatic modeling of natural phenomena. It is therefore of fundamental importance to develop a theoretical basis for the description of their optimal performance. In the previous chapters it was stated that an empirical modeling of natural phenomena includes three main tasks: Estimation, storage, and application of a probability distribution. Each of these tasks can be optimized by using the methods described in the previous chapter. The aim of this section is to present the problems and the solutions related with an optimal storage of empirical information about a continuous probability distribution in a system comprised of a finite number of discrete memory units. Such a system can be considered as a basic building block of an automatic modeler of natural phenomena. For example, we can imagine the brain of a biological organism or the digital memory of a computer that continually obtains signals from its surroundings and it optimally stores the corresponding empirical information. The first problem is the estimation of the probability density function of a continuous variable from the empirical data. We have already seen that this can be solved using Parzen’s window. [2] The second problem is the storage of the continuous probability density in a discrete system. This task is generally related with a loss of information and thus the question arises, how one can minimize this loss.KeywordsWindow FunctionExcited NeuronReference VectorDiscrete Random VariableMemory UnitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

One technique to measure system performance is using discrete event simulation (DES), which models organizational structures and behavior. DES also allows testing of resource configurations to assess their impact on performance. To evaluate their efficiency, data envelopment analysis (DEA) can be used. However, current DES software does not automate DEA for efficiency evaluation, requiring separate analysis of performance measures and resource efficiency. This complicates finding the most efficient resource configuration, especially in healthcare systems. To address this, this paper proposes a framework combining DES and DEA for simpler analysis of their inputs and outputs. The framework automates data transfer mechanisms between DES outputs and DEA inputs and implements an integrated computational approach to DEA. To validate the framework, a case study in an emergency department was conducted, where complex interconnected processes are common, and optimizing resource allocation is critical for patient care and system performance. The case study analyzed 35 resource configurations to identify the most efficient one. The results demonstrated the framework’s potential to simplify resource analysis, identify optimal configurations, and enhance decision-making, thereby improving the system’s operational efficiency. The framework provides a robust, portable, and scalable solution applicable across diverse industries for effectively optimizing system performance and resource allocation.

The identities of two circular functions are used, through a geometric construction onto a unit circle representing the z-domain of discrete systems, to generate a set of planar triangles from which both edge length and area can be used to build and model a 3D data structure of the system.

The problem of robust performance analysis for the delta operator formulated uncertain systems is considered. Sufficient conditions for D-stability robustness of the unified delta operator systems with structured uncertainty are presented in terms of matrix measure approach, which the desired closed-loop poles are constrained to locate within a prespecified D-region. The proposed results can unify related conclusions of continuous time and discrete time systems into the delta operator framework.

Reactive systems exhibit ongoing, possibly nonterminating, interaction with the environment. Real-time systems are reactive systems that must satisfy quantitative timing constraints. This paper presents a structured compositional design method for discrete real-time systems that can be used to combat the combinatorial explosion of states in the verification of large systems. A composition rule describes how the correctness of the system can be determined from the correctness of its modules, without knowledge of their internal structure. The advantage of compositional verification is clear. Each module is both simpler and smaller than the system itself. Composition requires the use of both model-checking and deductive techniques. A refinement rule guarantees that specifications of high-level modules are preserved by their implementations. The StateTime toolset is used to automate parts of compositional designs using a combination of model-checking and simulation. The design method is illustrated using a reactor shutdown system that cannot be verified using the StateTime toolset (due to the combinatorial explosion of states) without compositional reasoning. The reactor example also illustrates the use of the refinement rule.

Equational properties of Kleene algebras of relations with conversion

From September 27th to 29th, 2024, Harbin, China welcomed an academic feast - the 2024 2nd International Conference on Applied Statistics, Modeling and Advanced Algorithms (ASMA 2024). The conference gathers together about 100 famous scholars, experts, engineers and technicians as well as technology developers at home and abroad to discuss the latest research results and development trends in the fields of applied statistics, modeling and advanced algorithms. The ASMA conference series holds a significant place in the academic calendar, serving as a platform for sharing cutting-edge research, fostering collaborations, and driving innovation. The inaugural ASMA conference sets a high benchmark, and it is our pleasure to continue this tradition with ASMA 2024, which aims to build upon the success of its predecessor by exploring new frontiers and deepening our understanding of these intricate and ever-evolving disciplines. As an important event for domestic and international academics, ASMA 2024 attracts the attention of researchers and scholars from around the world. Experts and scholars from international institutions such as King’s College London (UK), and University of Jaen (Spain) gathered with scholars from China’s Xiamen University, Guangdong University of Science and Technology, Harbin University of Science and Technology, etc. to share their scientific research results and practical experiences in the domains of applied statistics, modeling and advanced algorithms and their relevant disciplines. In addition, many elites from the business community also participated in the conference enthusiastically, bringing a wealth of practical cases and cutting-edge thinking to the conference. The main agenda of ASMA 2024 is meticulously designed to encompass a broad spectrum of topics, including but not limited to reinforcement learning and fuzzy model-based control, output feedback adaptive boundary control for distributed parameter systems, quadratic estimation for discrete non-Gaussian systems, and secure state estimation using encryption and decryption strategies. These themes are meticulously chosen to reflect the current trends and future directions in the fields of applied statistics, modeling, and advanced algorithms. The conference also features a diverse array of presentations, including keynote speeches, oral presentations, poster sessions, and panel discussions. During these presentations, the participants actively interacted with the speakers and each other, colliding with a lot of new sparks of ideas. In addition to the wonderful academic reports and discussions, this conference also includes a number of high-quality academic contributions, forming a rich collection of papers. These papers cover a wide range of aspects, including statistical mechanics, data mining, partial differential equations, modeling simulation input, numerical analysis, natural computing, etc., with both theoretical discussions and practical analyses, fully demonstrating the latest research dynamics and academic level in these fields. Finally, we would like to express our special thanks to all the participants for their hard work and contributions. It is their active participation and in-depth exchanges that made this conference a complete success. Meanwhile, we would also like to acknowledge the conference organizers for their careful preparations and thoughtful arrangements, which provides a good platform for academic exchanges and cooperation among the participants. The Committee of ASMA 2024 List of Committee Member is available in this Pdf.