Construction Of Class Research Articles

Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise

In this paper we aim at generalizing the results of A. K. Zvonkin (That removes the drift, 22, 129, 41) and A. Y. Veretennikov (Theory Probab. Appl., 24, 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

The spectrum of resolvable Bose triple systems

A classical construction of Bose produces a Steiner triple system of order 3n from a symmetric, idempotent latin square of order n, which exists whenever n is odd. In an application to access-balancing in storage systems, certain Bose triple systems play a central role. A natural question arises: For which orders v does there exist a resolvable Bose triple system? Elementary counting establishes the necessary condition that v≡9(mod18). For specific Bose triple systems that optimize an access metric, we show that v≡9(mod18) is also sufficient.

Sets of real numbers closed under Turing equivalence: applications to fields, orders and automorphisms

In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with 1-generic degrees.

On the symplectic structure deformations related to the Monge–Ampère equation on the Kähler manifoldP2(ℂ)

UDC 517.9 We analyze the cohomology structure of the fundamental two-form deformation related to a modified Monge–Ampère type on the complex Kähler manifoldP2(ℂ). Based on the Levi-Civita connection and the related vector-field deformation of the fundamental two-form, we construct a hierarchy of bilinear symmetric forms on the tangent bundle of the K\"{a}hler manifoldP2(ℂ),that generate Hermitian metrics on it and corresponding solutions to the Monge–Ampère-type equation. The classical fundamental two-form construction on the complex Kähler manifoldP2(ℂ)is generalized and the related metric deformations are discussed.

Ordering regular languages and automata: Complexity

Given an order of the underlying alphabet, we can lift it to the states of a finite deterministic automaton: to compare states we use the order of the strings reaching them. When the order on strings is the co-lexicographic one and this order turns out to be total, the DFA is called Wheeler. This recently introduced class of automata—the Wheeler automata—constitute an important data-structure for languages, since it allows the design and implementation of a very efficient tool-set of storage mechanisms for the transition function, supporting a large variety of substring queries.In this context it is natural to consider the class of regular languages accepted by Wheeler automata, i.e. the Wheeler languages. An inspiring result in this area is the following: it has been shown that, as opposed to the general case, the classic determinization by powerset construction is polynomial on Wheeler automata. As a consequence, most classical problems, when considered on this class of automata, turn out to be “easy”—that is, solvable in polynomial time.In this paper we consider computational problems related to Wheelerness, but starting from non-deterministic automata. We also consider the case of reduced non-deterministic ones—a class of NFAs where recognizing Wheelerness is still polynomial, as for DFAs. Our collection of results shows that moving towards non-determinism is, in most cases, a dangerous path leading quickly to intractability.Moreover, we start a study of “state complexity” related to Wheeler DFAs and languages, proving that the classic construction for the intersection of languages turns out to be computationally simpler on Wheeler DFAs than in the general case. We also provide a construction for the minimum Wheeler DFA recognizing a given Wheeler language.

Stratifications and foliations in phase portraits of gene network models.

Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section "Investigation of functions and plot of their graphs" of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations.

Character Development in Higher Education Using Classical Archetypes

Since classical times, character development has played a key role in the proper functioning of democracies. This article outlines a modern reimagining of a classical archetypal character construction as a novel method for character development. The author presents examples of its operation in higher educational settings and in professional education, together with examples that show students using the method to frame ethical and professional issues in a character context. The novel method is shown to have potential to play a role in overcoming factors, some often unspoken, hindering greater establishment of character development in modern education.

Xiyanglou – “Buildings of the Western Ocean”: Splendor, Poverty and Lessons of the Ruin

The paper gives a complex analysis of the Xiyanglou palace and park complex (“Buildings of the Western Ocean”), located on the current northwestern outskirts of Beijing, designed, and built in 1747–1751 by European masters, led by the Italian Jesuit artist Giuseppe Castiglione (Lang Shining, 1688–1766). Castiglione, best known as a painter, acted in this case as a visionary architect and designer who skillfully integrated the “European part” into the already existing Yuanmingyuan palace complex and received for this his main recognition at the court of the Manchu Emperor Qianlong, who personally oversaw the project. Using the example of the syncretic Sino-European style of the Xiyanglou architectural complex, incorporating European Baroque and Rococo styles and classical Chinese construction principles, the paper demonstrates the European masters’ practical application of the adaptation principles developed by Matteo Ricci, the leader of the Jesuit mission in China when preaching Christianity in China. Separately, the innovative buildings in the Chinese style located in Yuanming Yuan, which incorporated Xiyanglou, are briefly analyzed, the background of the design and creation of Xiyanglou is considered, detailed descriptions of the buildings of the Western Quarter are given, and the fate of the palace and park complex is traced. The author concludes that the Western Quarter, which became a testing ground where were practiced the principles of Occidental art, acted as a focus of mutual influence of European and Chinese artistic trends.

Критерии технического состояния бетонных бун

The classical constructions of concrete boones on rocky and eroded bases used as part of engineering protection structures of the roadbed are considered. Concrete boones are designed to preserve the artificial beach and to create conditions for its formation. In order to determine the defects of both the structures as a whole and their elements, a survey of marine coastal protection structures (concrete boones) was carried out on the section of the Tuapse - Adler railway line of the North Caucasus Railway. Experimental studies of the interaction of waves with concrete boones and the beach on hydrowave models in a wave pool have been conducted. Criteria for the technical condition of concrete boone elements by types of defects have been developed. The division of concrete boones into individual elements according to the degree of weight is carried out. For each significant element of the coastal protection structure, criteria of technical condition by types of defects were obtained. In order to assess the technical condition of concrete boones and determine the necessary measures to ensure their operability, the algorithm is proposed in the form of a scheme. The development of quantitative indicators enables to assess the technical condition of concrete boones and, thus, improve the operation safety of the railway track on the pressing offshore sections. The proposed criteria are taken into account in the development of GOST R 59241-2020 «Coastal protection structures. Rules of inspection and monitoring of technical condition».

Ранги, спектры и их динамика для семейств константных обогащений теорий

Constant or nonessential extensions of elementary theories provide a productive tool for the study and structural description of models of these theories, which is widely used in Model Theory and its applications, both for various stable and ordered theories, countable and uncountable theories, algebraic, geometric and relational structures and theories. Families of constants are used in Henkin’s classical construction of model building for consistent families of formulas, for the classification of uncountable and countable models of complete theories, and for some dynamic possibilities of countable spectra of ordered Ehrenfeucht theories. The paper describes the possibilities of ranks and degrees for families of constant extensions of theories. Rank links are established for families of theories with CantorBendixson ranks for given theories. It is shown that the 𝑒-minimality of a family of constant expansions of the theory is equivalent to the existence and uniqueness of a nonprincipal type with a given number of variables. In particular, for strongly minimal theories this means that the non-principal 1-type is unique over an appropriate tuple. Relations between 𝑒-spectra of families of constant expansions of theories and ranks and degrees are established. A model-theoretic characterization of the existence of the least generating set is obtained. It is also proved that any inessential finite expansion of an o-minimal Ehrenfeucht theory preserves the Ehrenfeucht property, and this is true for constant expansions of dense spherically ordered theories. For the expansions under consideration, the dynamics of the values of countable spectra is described.

Radially symmetric non-isentropic Euler flows: Continuous blowup with positive pressure

We establish the existence of radial self-similar Euler flows in which a continuous incoming wave generates a blowup of primary (undifferentiated) flow variables. A key point is that the solutions have a strictly positive pressure field, in contrast to Guderley's classic construction of converging shock waves. In Guderley's solutions, a converging shock invades a quiescent region at zero pressure (due to vanishing temperature), and the velocity and pressure in its immediate wake become unbounded at the time of collapse. It is reasonable that the lack of upstream counter-pressure is conducive to large speeds, with concomitant large amplitudes. Based on Guderley's original solutions, it is therefore unclear if it is the zero-pressure region that is responsible for blowup. The same applies to self-similar Euler flows describing radial cavity flow. Our results demonstrate that the geometric mechanism of wave focusing is sufficiently strong on its own to drive unbounded growth. We propagate the solution beyond blowup and observe numerically that there are two distinct possibilities depending on the incoming flow: either an expanding spherical shock wave is generated, or the flow propagates in a continuous manner. Focusing on the former case, we show that the resulting flows define global admissible weak solutions to the full, multi-d compressible Euler system. These solutions have the unusual property that the flow is isentropic in each of the two regions separated by the shock.

Statistical structures arising in null submanifolds

We show a link between affine differential geometry and null submanifolds in a semi-Riemannian manifold via statistical structures. Once a rigging for a null submanifold is fixed, we can construct a semi-Riemannian metric on it. This metric and the induced connection constitute a statistical structure on the null submanifold in some cases. We study the statistical structures arising in this way. We also construct statistical structures on a null hypersurface in the Lorentz–Minkowski space using the null second fundamental form. This extends the classical construction to the null case.

Exponentially larger affine and projective caps.

In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach and by Ellenberg and Gijswijt), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulusp. Moreover, we show that for all primes with , the new construction leads to an exponentially larger growth of the affine and projective caps in and . For example, when , the existence of caps with growth follows from a three-dimensional example of Bose, and the only improvement had been to by Edel, based on a six-dimensional example. We improve this lower bound to .

Reflection length with two parameters in the asymptotic representation theory of type B/C and applications

We introduce a two-parameter function ϕq+,q− on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length. We show that this signed reflection function ϕq+,q− is positive definite if and only if it is an extreme character of the infinite hyperoctahedral group and we classify the corresponding set of parameters q+,q−. We construct the corresponding representations through a natural action of the hyperoctahedral group B(n) on the tensor product of n copies of a vector space, which gives a two-parameter analog of the classical construction of Schur–Weyl.We apply our classification to construct a cyclic Fock space of type B generalizing the one-parameter construction in type A found previously by Bożejko and Guta. We also construct a new Gaussian operator acting on the cyclic Fock space of type B and we relate its moments with the Askey–Wimp–Kerov distribution by using the notion of cycles on pair-partitions, which we introduce here. Finally, we explain how to solve the analogous problem for the Coxeter groups of type D by using our main result.

A note on the CBC-DBD construction of lattice rules with general positive weights

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule for an s-dimensional integral is specified by its generating vectorz∈Zs and its number of points N. While there are many results on the existence of “good” rank-1 lattice rules, there are no explicit constructions of good generating vectors for dimensions s≥3. Therefore one resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors for integration of functions in weighted Korobov classes equipped with product weights. Here, we generalize this result to arbitrary positive weights, answering an open question from the paper of Ebert et al. We include a section on how the algorithm can be implemented in the case of POD weights, implying that the CBC-DBD construction is competitive with the classical CBC construction.

Deeply learning deep inelastic scattering kinematics

We study the use of deep learning techniques to reconstruct the kinematics of the neutral current deep inelastic scattering (DIS) process in electron–proton collisions. In particular, we use simulated data from the ZEUS experiment at the HERA accelerator facility, and train deep neural networks to reconstruct the kinematic variables Q^2 and x. Our approach is based on the information used in the classical construction methods, the measurements of the scattered lepton, and the hadronic final state in the detector, but is enhanced through correlations and patterns revealed with the simulated data sets. We show that, with the appropriate selection of a training set, the neural networks sufficiently surpass all classical reconstruction methods on most of the kinematic range considered. Rapid access to large samples of simulated data and the ability of neural networks to effectively extract information from large data sets, both suggest that deep learning techniques to reconstruct DIS kinematics can serve as a rigorous method to combine and outperform the classical reconstruction methods.

Optimal Bound on the Combinatorial Complexity of Approximating Polytopes

This article considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body K of unit diameter in Euclidean d -dimensional space (where d is a constant) and an error parameter ε > 0, the objective is to determine a convex polytope of low combinatorial complexity whose Hausdorff distance from K is at most ε. By combinatorial complexity , we mean the total number of faces of all dimensions. Classical constructions by Dudley and Bronshteyn/Ivanov show that O (1/ε ( d -1)/2 ) facets or vertices are possible, respectively, but neither achieves both bounds simultaneously. In this article, we show that it is possible to construct a polytope with O (1/ε ( d -1)/2 ) combinatorial complexity, which is optimal in the worst case. Our result is based on a new relationship between ε-width caps of a convex body and its polar body. Using this relationship, we are able to obtain a volume-sensitive bound on the number of approximating caps that are “essentially different.” We achieve our main result by combining this with a variant of the witness-collector method and a novel variable-thickness layered construction of the economical cap covering.

Doubling the Cube and Constructability in Higher Dimensions

Summary It is known that the three classical geometric construction problems introduced by the ancient Greeks: trisecting an angle, squaring a circle, and doubling a cube, cannot be solved using the Euclidean tools. However, ancient Greek mathematicians solved these three problems using other means. We present solutions to the doubling-the-cube problem using ideas that go beyond the Euclidean tools, and we consider generalizations to higher dimensions.

Toric Quasifolds

Toric quasifolds are highly singular spaces that were first introduced in order to address, from the symplectic viewpoint, the longstanding open problem of extending the classical constructions of toric geometry to those simple convex polytopes that are not rational. We illustrate toric quasifolds, and their atlases, by describing some notable examples. We conclude with a number of considerations.

Consolidation of Additive Manufactured Continuous Carbon Fiber Reinforced Polyamide 12 Composites and the Development of Process-Related Numerical Simulation Methods.

Additive manufacturing of high-performance polymers—such as PA12, PPS, PEEK, and PEKK—combined with industrial-grade carbon fibers with a high fiber volume ratio of up to 60% allows a weight reduction of over 40% compared to classic metal construction. Typically, these 3D-printed composites have a porosity of 10–30% depending on the material and the printing process parameters, which significantly reduces the quality of the part. Therefore, the additive manufacturing of load-bearing structural applications requires a proper consolidation after the printing process—the so-called ‘additive fusion technology’—allowing close to zero void content in the consolidated part. By means of the upfront digital modeling of the consolidation process, a highly optimized composite component can be produced while decreasing the number of expensive prototyping iterations. In this study, advanced numerical methods are presented to describe the consolidation process of additive manufactured continuous carbon fiber reinforced composite parts based on the polyamide 12 (PA12) matrix. The simulation of the additive fusion step/consolidation provides immediate accuracy in determining the final degree of crystallization, process-induced deformation and residual stresses, final engineering constants, as well as porosity. The developed simulation workflow is demonstrated and validated with experimental data from consolidation tests on the final porosity, thickness, and fiber–volume ratio.