Geometric Theory Research Articles

Geometric spectral theory of quantum graphs

Geometric spectral theory of quantum graphs

Mutations of noncommutative crepant resolutions in geometric invariant theory

Let X be a generic quasi-symmetric representation of a connected reductive group G. The GIT quotient stack X=[Xss(ℓ)/G]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {X}=[X^\ ext {ss}(\\ell )/G]$$\\end{document} with respect to a generic ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} is a (stacky) crepant resolution of the affine quotient X/G, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of X/G. Halpern-Leistner and Sam showed that the derived category Db(cohX)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extrm{D}}^{\ extrm{b}}}({\ ext {coh}}\\mathfrak {X})$$\\end{document} is equivalent to certain subcategories of Db(coh[X/G])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extrm{D}}^{\ extrm{b}}}({\ ext {coh}}[X/G])$$\\end{document}, which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When G is a torus, it turns out that the exchanges are nothing but iterated Iyama–Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama–Wemyss mutations induce a group action of the fundamental group π1(P1\\{0,1,∞})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi _1(\\mathbb {P}^1\\,\\backslash \\{0,1,\\infty \\})$$\\end{document} on the derived category of a Calabi-Yau complete intersection in a weighted projective space.

Dynamics of the epidemiological Predator-Prey system in advective environments.

This paper aims to establish the existence of traveling wave solutions connecting different equilibria for a spatial eco-epidemiological predator-prey system in advective environments. After applying the traveling wave coordinates, these solutions correspond to heteroclinic orbits in phase space. We investigate the existence of the traveling wave solution connecting from a boundary equilibrium to a co-existence equilibrium by using a shooting method. Different from the techniques introduced by Huang, we directly prove the convergence of the solution to a co-existence equilibrium by constructing a special bounded set. Furthermore, the Lyapunov-type function we constructed does not need the condition of bounded below. Our approach provides a different way to study the existence of traveling wave solutions about the co-existence equilibrium. The existence of traveling wave solutions between co-existence equilibria are proved by utilizing the qualitative theory and the geometric singular perturbation theory. Some other open questions of interest are also discussed in the paper.

Geometric Properties Connected with a Certain Multiplier Integral q−Analogue Operator

The topic concerning the introduction and investigation of new classes of analytic functions using subordination techniques for obtaining certain geometric properties alongside coefficient estimates and inclusion relations is enriched by the results of the present investigation. The prolific tools of quantum calculus applied in geometric function theory are employed for the investigation of a new class of analytic functions introduced by applying a previously defined generalized q−integral operator and the concept of subordination. Investigations are conducted on the new class, including coefficient estimates, integral representation for the functions of the class, linear combinations, forms of the weighted and arithmetic means involving functions from the class, and the estimation of the integral means results.

Constraining Extended Teleparallel Gravity via Cosmography: A Model-independent Approach

As a classical approach, the dynamics of the Universe, influenced by its dark components, are unveiled through prior modifications of Einstein’s equations. Cosmography, on the other hand, is a highly efficient tool for reconstructing any modified theory in a model-independent manner. By employing kinematic variables, it offers a profound explanation for cosmic expansion. Although the cosmographical approach has been highly successful in several geometric theories in recent years, it has not been extensively explored in coupled gravities. With this in mind, we intend to constrain an extended teleparallel gravity model, f(T,T) , through cosmographic parameters. We utilize Taylor series expansion, assuming a minimally coupled form, to constrain the unknowns involved in the series. To achieve this, we conduct a Markov Chain Monte Carlo (MCMC) analysis using three different data sets (cosmic chronometers, baryon acoustic oscillations, and Pantheon+SH0ES). The constrained results obtained from MCMC are then compared and verified using various cosmological parameters. Finally, we compare the resulting models with three well-known f(T,T) models.

Laplacian2Mesh: Laplacian-Based Mesh Understanding.

Geometric deep learning has sparked a rising interest in computer graphics to perform shape understanding tasks, such as shape classification and semantic segmentation. When the input is a polygonal surface, one has to suffer from the irregular mesh structure. Motivated by the geometric spectral theory, we introduce Laplacian2Mesh, a novel and flexible convolutional neural network (CNN) framework for coping with irregular triangle meshes (vertices may have any valence). By mapping the input mesh surface to the multi-dimensional Laplacian-Beltrami space, Laplacian2Mesh enables one to perform shape analysis tasks directly using the mature CNNs, without the need to deal with the irregular connectivity of the mesh structure. We further define a mesh pooling operation such that the receptive field of the network can be expanded while retaining the original vertex set as well as the connections between them. Besides, we introduce a channel-wise self-attention block to learn the individual importance of feature ingredients. Laplacian2Mesh not only decouples the geometry from the irregular connectivity of the mesh structure but also better captures the global features that are central to shape classification and segmentation. Extensive tests on various datasets demonstrate the effectiveness and efficiency of Laplacian2Mesh, particularly in terms of the capability of being vulnerable to noise to fulfill various learning tasks.

Predicting transient dynamics in a model of reed musical instrument with slowly time-varying control parameter.

When playing a self-sustained reed instrument (such as the clarinet), initial acoustical transients (at the beginning of a note) are known to be of crucial importance. Nevertheless, they have been mostly overlooked in the literature on musical instruments. We investigate here the dynamic behavior of a simple model of reed instrument with a time-varying blowing pressure accounting for attack transients performed by the musician. In practice, this means studying a one-dimensional non-autonomous dynamical system obtained by slowly varying in time the bifurcation parameter (the blowing pressure) of the corresponding autonomous systems, i.e., whose bifurcation parameter is constant. In this context, the study focuses on the case for which the time-varying blowing pressure crosses the bistability domain (with the coexistence of a periodic solution and an equilibrium) of the corresponding autonomous model. Considering the time-varying blowing pressure as a new (slow) state variable, the considered non-autonomous one-dimensional system becomes an autonomous two-dimensional fast-slow system. In the bistability domain, the latter has attracting manifolds associated with two stable branches of the bifurcation diagram of the system with constant parameter. In the framework of the geometric singular perturbation theory, we show that a single solution of the two-dimensional fast-slow system can be used to describe the global system behavior. Indeed, this allows us to determine, depending on the initial conditions and rate of change of the blowing pressure, which manifold is approached when the bistability domain is crossed and to predict whether a sound is produced during transient as a function of the musician's control.

KCC theory for a type of nonlinear damped SD oscillators

Smooth and discontinuous (SD) oscillators are the nonlinear models that will exhibit SD dynamics due to continuous variation of the smooth parameter. Kosambi–Cartan–Chern (KCC) theory is a differential geometric theory that describes the deviations from the entire trajectory of the variational equation to the nearby trajectory. This paper presents a completely new dynamical analysis of a type of SD oscillators with nonlinear damping based on the KCC theory. First, the paper gives five KCC geometrical invariants of SD oscillators in the smooth and non-smooth cases, respectively. The results show that the geometric quantities in the non-smooth case cannot be derived directly from the geometric quantities in the smooth case. Second, from these geometric quantities, KCC stability of SD oscillators trajectory at any point except [Formula: see text] is analyzed. Lastly, numerical results show that in some regions that deviate from the equilibrium point of system, the system will exhibit complex and variable dynamical behavior due to small changes in parameters. This paper shows that the KCC theory is also a useful tool in the dynamical analysis of non-smooth systems.

Nonmetric geometric flows and quasicrystalline topological phases for dark energy and dark matter in f(Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(Q)$$\\end{document} cosmology

We elaborate on nonmetric geometric flow theory and metric-affine gravity with applications in modern cosmology. Two main motivations for our research follow from the facts that (1) cosmological models for f(Q) modified gravity theories, MGTs, are efficient for describing recent observational data provided by the James Webb Space Telescope; and (2) the statistical thermodynamic properties of such nonmetric locally anisotropic cosmological models can be studied using generalizations of the concept of G. Perelman entropy. We derive nonmetric distorted R. Hamilton and Ricci soliton equations in such canonical nonholonomic variables when corresponding systems of nonlinear PDEs can be decoupled and integrated in general off-diagonal forms. This is possible if we develop and apply the anholonomic frame and connection deformation method involving corresponding types of generating functions and generating sources encoding nonmetric distortions. Using such generic off-diagonal solutions (when the coefficients of metrics and connections may depend generically on all spacetime coordinates), we model accelerating cosmological scenarios with quasi-periodic gravitational and (effective) matter fields; and study topological and nonlinear geometric properties of respective dark energy and dark matter, DE and DM, models. As explicit examples, we analyze some classes of nonlinear symmetries defining topological quasicrystal, QC, phases which can modified to generate other types of quasi-periodic and locally anisotropic structures. The conditions when such nonlinear systems possess a behaviour which is similar to that of the Lambda cold dark matter (Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM) scenario are stated. We conclude that nonmetric geometric and cosmological flows can be considered as an alternative to the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM concordance models and speculate on how such theories can be elaborated. This is a partner work with generalizations and applications of the results published by Bubuianu, Vacaru et all. EPJC 84 (2024) 211; 80 (2020) 639; 78 (2018) 393; 78 (2018) 969; and CQG 35 (2018) 245009.

Jacobi Stability Analysis of Liu System: Detecting Chaos

By utilizing the Kosambi–Cartan–Chern (KCC) geometric theory, this paper is dedicated to providing novel insights into the Liu dynamical system, which stands out as one of the most distinctive and noteworthy nonlinear dynamical systems. Firstly, five important geometrical invariants of the system are obtained by associating the nonlinear connection with the Berwald connection. Secondly, in terms of the eigenvalues of the deviation curvature tensor, the Jacobi stability of the Liu dynamical system at fixed points is investigated, which indicates that three fixed points are Jacobi unstable. The Jacobi stability of the system is analyzed and compared with that of Lyapunov stability. Lastly, the dynamical behavior of components of the deviation vector is studied, which serves to geometrically delineate the chaotic behavior of the system near the origin. The onset of chaos for the Liu dynamical system is obtained. This work provides an analysis of the Jacobi stability of the Liu dynamical system, serving as a useful reference for future chaotic system research.

Моделирование конструкций сборного абразивного инструмента

Introduction. Grinding is one of the most common types of finishing. It allows the production of surfaces with the required quality parameters and is one of the most available and productive methods for machining high-strength and difficult-to-machine materials. Grinding wheels represent the most prevalent application of grinding technology in mechanical engineering. The use of this abrasive tool helps to increase processing productivity by ensuring the removal of a significant layer of material. In addition, grinding wheels have a longer service life and are widely used in the implementation of hybrid technologies based on the combination of mechanical (abrasive), electrical, chemical, and thermal effects in various combinations. A variety of tool body shapes and types of abrasives allow the use of wheels in a wide variety of production areas. One of the ways to analyze and design a new tool is numerical simulation. In this research, graphic modeling was selected as the most appropriate method for representing the future design of the tool. This approach allows for a more straightforward conceptualization process compared to other modeling techniques. The purpose of the work is to simulate a modular abrasive tool in order to analyze and synthesize structures to increase the efficiency of tool support for the manufacture of products made of high-strength and difficult-to-process materials using traditional or hybrid processing technologies. Research methodology. Theoretical studies are carried out using the basic principles of system analysis, geometric theory of surface formation, cutting tool design, graph theory, mathematical and computer simulation. To solve the problem, we have studied the available designs of modular grinding wheels. There has also been the analysis of the types of abrasive parts, methods of fastening of the abrasive cutting part on the wheel’s body, the materials used for the manufacture of the body, the characteristics of the body of the wheel, and fastening schemes. Results and discussions. A simulation technique based on graphic modelling theory has been developed. A comprehensive investigation of the existing design of the grinding wheel has enabled the identification of the key structural elements that define its design. The data obtained has been used to create a generalized graphic simulation of a modular abrasive tool. This simulation integrates all the components and displays a conditional constructive relationship between them. The developed design methodology was tested on an example of two designs of modular grinding wheels. The theoretical studies established that the design efficiency of modular abrasive tools can be increased by 2–4 times by using the developed simulation technique.

X-ray fluorescence detection system for trace elements in oil based on graphite curved crystal technology

Curved crystal technology is a key technology in the field of X-ray fluorescence trace detection. To detect trace sulfur, chlorine, and silicon in oil, the key parameters of Johann-type graphite curved crystal design were calculated based on geometrical optics theory and mathematical modeling, and a curved crystal optical system was designed. Subsequently, the hardware, firmware, software, and mechanical structure were designed according to specific requirements. Following the development of the relevant equipment, laboratory tests were performed on standard petroleum samples. Experimental results show that technology based on bent crystals can significantly reduce the detection limit of the instrument, and the detection limit of sulfur, chlorine, and silicon can reach below 1 ppm. In accordance with the experimental results, possible interference factors were analyzed, and suitable solutions were proposed. The performance of the proposed system was thus directly verified, showcasing its applicability in detecting trace elements.

Precise analysis of geometrical nonlinear mechanical behavior of shear deformable laminated curved composite beams in local coordinates via mixed type finite elements

In the present article, geometrically nonlinear analyses of curved laminated composite beams are carried out using Mixed Finite Element Method (MFEM). Timoshenko beam theory is used to consider shear influence which significantly effects the structural behavior of composite layers. Energy-based functional is generally needed to perform geometric nonlinear analyzes of layered composites. However, obtaining energy-based functional in geometric nonlinear theory is quite challenging due to the difficulty of solving mathematically nonlinear differential equations. In this study, this situation is overcome and a functional suitable for performing geometric nonlinear analyzes of layered composite materials is obtained. Incremental formulation procedure is utilized as a solution method for nonlinear finite element analyzes. This is an innovative application of the non-linear mixed finite element method and has the ability to perform geometric nonlinear analysis of curved laminated composite beams with perfect precision. By obtaining the functional in local coordinates, it is possible to solve special problems in various fields such as construction, machinery, aerospace and biomechanics.

Horizons that gyre and gimble: a differential characterization of null hypersurfaces

Motivated by the thermodynamics of black hole solutions conformal to stationary solutions, we study the geometric invariant theory of null hypersurfaces. It is well-known that a null hypersurface in a Lorentzian manifold can be treated as a Carrollian geometry. Additional structure can be added to this geometry by choosing a connection which yields a Carrollian manifold. In the literature various authors have introduced Koszul connections to study the study the physics on these hypersurfaces. In this paper we examine the various Carrollian geometries and their relationship to null hypersurface embeddings. We specify the geometric data required to construct a rigid Carrollian geometry, and we argue that a connection with torsion is the most natural object to study Carrollian manifolds. We then use this connection to develop a hypersurface calculus suitable for a study of intrinsic and extrinsic differential invariants on embedded null hypersurfaces; motivating examples are given, including geometric invariants preserved under conformal transformations.

Asymptotic Conformality and Polygonal Approximation

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal maps. The paper provides an approximative characterization of local conformality and its connection with univalent polynomials. Also, some other quantitative applications of this connection are given.

Hp-Multigrid Preconditioner for a Divergence-Conforming HDG Scheme for the Incompressible Flow Problems

In this study, we present an hp-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier–Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The hp-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we develop a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix–Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput Methods Appl Mech Engrg 311:304–326, 2022), which allows for the direct application of geometric multigrid theory on the Crouzeix–Raviart discretization. The numerical experiments demonstrate the robustness of the proposed hp-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi and Olshanskii (SIAM J Sci Comput 28:2095–2113, 2006) and Farrell et al. (SIAM J Sci Comput 41:A3073–A3096, 2019), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier–Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to 103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^3$$\\end{document}.

Antiferroelectric Ceramics for Energy-Efficient Capacitors by Theory-Guided Discovery.

Antiferroelectric ceramics, via the electric-field-induced antiferroelectric (AFE)-ferroelectric (FE) phase transitions, show great promise for high-energy-density capacitors. Yet, currently, only 70-80% energy release is found during a charge-discharge cycle. Here, for PbZrO3-based oxides, geometric nonlinear theory of martensitic phase transitions is applied (first used to guide supercompatible shape-memory alloys) to predict the reversibility of the AFE-FE transition by using density-functional theory to assess AFE/FE interfacial lattice-mismatch strain that assures ultralow electric hysteresis and extended fatigue lifetime. A good correlation of mismatch strain with electric hysteresis, hence, with energy efficiency of AFE capacitors is observed. Guided by theory, high-throughput material search is conducted and AFE compositions with a near-perfect charge-discharge energy efficiency (98.2%), i.e., near-zero hysteresis are discovered. And the fatigue life of the capacitor reaches 79.5 million charge-discharge cycles, a factor of 80 enhancement over AFE ceramics with large electric hysteresis.

Certain New Applications of Symmetric q-Calculus for New Subclasses of Multivalent Functions Associated with the Cardioid Domain

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies.

Sufficient Conditions for Linear Operators Related to Confluent Hypergeometric Function and Generalized Bessel Function of the First Kind to Belong to a Certain Class of Analytic Functions

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases.

A geometric framework for distributed frequency models

Geometric control theory, developed by Basile and Marro, and independently, by Wonham and Morse in the 1970s revolves around characterizing the properties of finite dimensional, linear and time-invariant systems using geometry. Some examples of these properties are invariance, controllability and observability. The task addressed in this paper is to develop the geometric tools for fractional systems using the diffusive representation (also known as the distributed frequency) model. The mathematics involved in this approach is different from the classical case as fractional systems are inherently infinite dimensional. Unlike the integer order derivative, the fractional derivative is a non-local operator, and so the evolution of the so-called pseudo-state depends on not just its current value but also its past history. Thus, the notion of an initial condition for a fractional system can come in different forms. This leads to different kinds of fractional derivative operators. With some of these operators, the semigroup property is lost. With the distributed frequency model, the initial condition comes in the form of an initialization function. This takes care of the infinite dimensional nature of fractional systems. Furthermore, the distributed frequency model retains the semigroup property. This is useful in developing invariance and controlled invariance for fractional systems. Moreover, these properties of fractional systems are verified numerically using a higher order finite-dimensional approximation, which retains all the geometric properties of the distributed frequency model.