Invariant Conditions Research Articles

Singularities of worldsheets along framed particle trajectories in Minkowski three-space

Modeling the worldsheets along the trajectory (a string) of a point particle (a 0-brane) as two timelike surfaces in 3-subspace of Minkowskian spacetime, respectively, the work aims at analyzing the topological structures of these two worldsheets along the trajectory of the point particle from the view point of singularity theory. Different from the regular curves, the traveling trajectory (modeled as framed timelike curve) of the particle is allowed to be singular, two worldsheets are generated by the traveling trajectories of the particle. As applications of singularity theory, we classify the singularities of these two worldsheets along the traveling trajectory of the particle. Using the approach of the unfolding theory in singularity theory, we find two new geometric invariants which are useful for characterizing the local topological structures of singularities of these two worldsheets along the particle. It is revealed that there exist cuspidal edge type and swallowtail type of singularities for these two worldsheets under the appropriate conditions of geometric invariants. Meanwhile, it is also pointed out that the types of singularities of these two worldsheets have a close relationships to the order of contacts between these two worldsheets and two timelike planes, respectively. Finally, some examples are presented to interpret our theoretical results.

Criterion for vanishing valley asymmetric transmission in dual-gated bilayer graphene

Abstract Realizing valley asymmetric transmission(VAT) for incoming valley unpolarized Dirac carriers across an electrostatically modulated confinement potential(ECP) step, which is located at the interface of the unipolar ($n-n'$) junction in dual-gated Bernal-bilayer graphene(BBG), has caught growing attention because it provides a platform to study the valley polarizer. Such a junction, however, seems to preclude from supporting vanishing VAT. Based on the effective two-band Hamiltonian of BBG with wave matching approach, we demonstrate theoretically that VAT through the $n$/$ n'$ interface vanishes in the following conditions: (i) the disappearing electrostatically modulated interlayer potential difference(EPD), or (ii) the invariance condition of the sum or difference of ECPs and EPDs, which physically corresponds to the zero band offset of the conduction or valence band in the band alignment of $n-n'$ junction. Guided by this, it is worth mentioning that the we can derive simple, analytical valley-independent expressions for the reflection and transmission probabilities. We further find the appearance of the nearly vanishing VAT for the four-band model in bilayer graphene with 'Mexican hat'-like band. As a byproduct, we apply these findings to suggest the best design for the valley-dependent selection of momenta based on dual-gated BBG connected to two pristine BBG electrodes in which ECPs of the two outer regions are distinguishable and show that the sign of valley polarization can be switched via varying EPD. We expect our results can provide guidance for future device applications relying on ballistic and valley-selective transport.

AdS3×S3$AdS_3 \times S^3$ Background From Poisson–Lie T‐Duality

AbstractThe author proceed to construct a dual pair for the background by applying non‐Abelian T‐duality (here as Poisson–Lie [PL] T‐duality on a semi‐Abelian double). By using a certain parametrization of the 4‐dimensional Lie group and by a suitable choice of spectator‐dependent matrices the original ‐model including the metric and a non‐trivial ‐field are constructed. The dual background constructed by means of the PL T‐duality with the spectators is an asymptotically flat one with a potential black hole interpretation supported by a non‐trivial ‐flux whose metric contains the true singularity with a single horizon. The question of classical integrability of the non‐Abelian T‐dual ‐models under consideration is addressed, and their corresponding Lax pairs are found, depending on some spectral parameters. Finally, the conformal invariance conditions of the models are checked up to two‐loop order, and it has been concluded that the resulting model is indeed a solution of supergravity.

On aggregation invariance of multinomial processing tree models

Multinomial processing tree (MPT) models are prominent and frequently used tools to model and measure cognitive processes underlying responses in many experimental paradigms. Although MPT models typically refer to cognitive processes within single individuals, they have often been applied to group data aggregated across individuals. We investigate the conditions under which MPT analyses of aggregate data make sense. After introducing the notions of structural and empirical aggregation invariance of MPT models, we show that any MPT model that holds at the level of single individuals must also hold at the aggregate level when it is both structurally and empirically aggregation invariant. Moreover, group-level parameters of aggregation-invariant MPT models are equivalent to the expected values (i.e., means) of the corresponding individual parameters. To investigate the robustness of MPT results for aggregate data when one or both invariance conditions are violated, we additionally performed a series of simulation studies, systematically manipulating (1) the sample sizes in different trees of the model, (2) model parameterization, (3) means and variances of crucial model parameters, and (4) their correlations with other parameters of the respective MPT model. Overall, our results show that MPT parameter estimates based on aggregate data are trustworthy under rather general conditions, provided that a few preconditions are met.

A New (3+1)-Dimensional Extension of the Kadomtsev–Petviashvili–Boussinesq-like Equation: Multiple-Soliton Solutions and Other Particular Solutions

In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). Exact analytical solutions were derived through the power series method, with a comprehensive proof of solution convergence. Employing the (G′/G2)-expansion method enabled the identification of trigonometric, exponential, and rational solutions of the equation. Furthermore, we established the auto-Bäcklund transformation of the equation. Multiple-soliton solutions were identified by utilizing Hirota’s bilinear method. The fundamental properties of these solutions were elucidated through graphical representations. Our results are of certain value to the interpretation of nonlinear problems.

Invariant cones for jump-diffusions in infinite dimensions

In this paper we provide sufficient conditions for stochastic invariance of closed convex cones for stochastic partial differential equations (SPDEs) of jump-diffusion type, and clarify when these conditions are necessary. We emphasize that the diffusion coefficient of the SPDE does not need to be smooth, an assumption which is frequently imposed when dealing with stochastic invariance problems. Our results apply to the positive cone of abstract L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-spaces. Furthermore, we present a series of applications, where we investigate SPDEs arising in natural sciences and economics.

Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.

A note on invariant manifolds for stochastic partial differential equations in the framework of the variational approach

In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a connection to SPDEs in continuously embedded spaces.

Convergence rates under a range invariance condition with application to electrical impedance tomography

Convergence rates under a range invariance condition with application to electrical impedance tomography

Homogenization of quasi-periodic conformal architectured materials and applications to chiral lattices

In this study, we propose to extend asymptotic periodic homogenization for non-periodic continuous microstructured media, assuming that the non-periodic geometry (called quasi-periodic) can be designed by a conformal planar transformation of a periodic parent domain architectured media with periodically disposed unit cells. Conformal transformations are shown to play a privileged role in the design of circular macroscopic heterogeneous domains tessellated with non-periodic unit cells, obtained from a periodic parent domain architectured with these unit cells. The conditions for conformal invariance are established, leading to the general form of conformal transformation in their dependencies upon the periodic coordinates. It is shown that any conformal map can be decomposed into the product of an isotropic dilatation function of the first periodic spatial position of decreasing exponential type and a rotation characterized by an angular function linear in the second periodic position. A general theory of quasi-periodic homogenization in the framework of conformal transformations is established for the first time, leading to an expression of the tensor of quasi-periodic moduli which is fully evaluated from the solution of the elasticity boundary value problem posed over the periodic unit cell. The influence of microcurvature distortion of individual unit cells on their effective properties is evaluated. Closed-form solutions are confronted to numerical examples issued from the implementation of circular periodicity in a finite element solver, showing overall a good agreement with the identified homogenized moduli.

Domain generalization via causal fine-grained feature decomposition and learning

Domain generalization aims to accurately predict unknown data using models trained by known domain data. Learning domain-invariant representations based on causal inference is one of the popular directions in domain generalization. However, existing domain generalization models based on causal inference cannot correctly determine the invariance conditions of the data. It is because of the interdependent relationship between domain-invariant features and confounding factors that ultimately leads to difficulty in obtaining truly invariant representations by the model. In this paper, we propose a novel domain generalization method called causal fine-grained feature decomposition and learning (CFFDL), which aims to eliminate latent confounding factors and learn causal domain-invariant representations in image classification tasks. Specifically, we design a feature decomposition module based on mutual information, which can decompose deep features into fine-grained feature factors and achieve factor independence between different dimensions, thereby helping to eliminate confounding factors. Furthermore, we introduce a causal representations learning module that can effectively filter and extract relevant causal features of the prediction task while eliminating the influence of confounding factors, improving the model’s performance on domain generalization. Extensive experiments on three domain generalization datasets VLCS, PACS and Office–Home show that our method outperforms the current state-of-the-art models, proving its effectiveness and superiority on domain generalization tasks of image classification.

Robust Invariance Conditions of Uncertain Linear Discrete Time Systems Based on Semidefinite Programming Duality

This article proposes a novel robust invariance condition for uncertain linear discrete-time systems with state and control constraints, utilizing a method of semidefinite programming duality. The approach involves approximating the robust invariant set for these systems by tackling the dual problem associated with semidefinite programming. Central to this method is the formulation of a dual programming through the application of adjoint mapping. From the standpoint of semidefinite programming dual optimization, the paper presents a novel linear matrix inequality (LMI) conditions pertinent to robust positive invariance. Illustrative examples are incorporated to elucidate the findings.

Quantization of resistivity as consequence of symmetry invariance

The Schrödinger equation for a spinless electron under the influence of a constant electromagnetic field is analyzed based on the conserved operators of the system when the magnetic field is described by Landau’s gauge. For this specific situation, the Lorentz force can be recovered if two conserved generalized momentum operators are considered: one along the x-axis and the second along the y-axis. From the general solution obtained, a time-dependent ground state is constructed characterized by quantized resistivity proportional to integer multiples of von Klitzing’s constant when an invariance condition under a unitary transform is satisfied.

Alternative Gibbs measure for fertile three-state Hard-Core models on a Cayley tree

ABSTRACT We consider fertile three-state Hard-Core (HC) models on a Cayley tree. It is known that there exist four types of such models: ‘wrench’, ‘wand’, ‘hinge’ and ‘pipe’. In cases ‘wand’ and ‘hinge’ on a Cayley tree of arbitrary order a complete description of translation-invariant Gibbs measures is obtained. The concept of alternative Gibbs measure is introduced and in the case ‘wand’ translational invariance conditions for alternative Gibbs measures are found. Also, we show the existence of alternative Gibbs measures which are not translation-invariant.

The Role of the Structure of Matter in its Dynamics and Evolution

The article is devoted to the study of the role of the structure of the body in its dynamics and evolution. A system of potentially interacting material points was used as a body model. To study the role of body structures, the equation of motion of the system was used. It is shown how it is derived from the condition of invariance of the total energy of the system. This equation takes into account that the work of external forces goes not only to change the energy of motion of the system, but also to change its internal energy. As a result, it was found that taking into account the structural nature of bodies opens up the possibility of describing the processes of their evolution within the framework of the fundamental laws of physics. Based on the condition of infinite divisibility of matter, which follows from the equation of motion of the system, the general principles of organizing the structures of matter at all its hierarchical levels are considered. Examples are given that demonstrate the importance of taking into account the role of the structure of bodies for the development of physics.

Cyclic Codes over a Non-Commutative Non-Unital Ring

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as E=⟨a,b∣2a=2b=0,a2=a,b2=b,ab=a,ba=b⟩. This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, left self-dual, right self-dual, and linear complementary dual (LCD) cyclic codes over E. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over E and quasi-cyclic codes of length 2n over F2 is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived.

Group Classification of Second-Order Linear Neutral Differential Equations

In this paper, we shall extend the method of obtaining symmetries of ordinary differential equations to second-order non-homogeneous functional differential equations with variable coefficients. The existing research for delay differential equations defines a Lie-Bäcklund operator and uses the invariant manifold theorem to obtain the infinitesimal generators of the Lie group. However, we shall use a different approach that requires Taylor’s theorem for a function of several variables to obtain a Lie invariance condition and the determining equations for second-order functional differential equations. Certain standard results from the theory of ordinary differential equations have been employed to simplify the equation under study. The symmetry analysis of this equation was found to be non-trivial for arbitrary variable coefficients. In such cases, by selecting certain specific functions, arising in most practical models, we find the symmetries that are seen to be in terms of Bessel’s functions, Mathieu functions, etc. We then make a complete group classification of the second-order linear neutral differential equation, for which there is no existing literature.

Scaling limit of the staggered six-vertex model with $U_q\big(\mathfrak{sl}(2)\big)$ invariant boundary conditions

We study the scaling limit of a statistical system, which is a special case of the integrable inhomogeneous six-vertex model. It possesses U_q\big(\mathfrak{sl}(2)\big)Uq(𝔰𝔩(2)) invariance due to the choice of open boundary conditions imposed. An interesting feature of the lattice theory is that the spectrum of scaling dimensions contains a continuous component. By applying the ODE/IQFT correspondence and the method of the Baxter QQ operator the corresponding density of states is obtained. In addition, the partition function appearing in the scaling limit of the lattice model is computed, which may be of interest for the study of nonrational CFTs in the presence of boundaries. As a side result of the research, a simple formula for the matrix elements of the QQ operator for the general, integrable, inhomogeneous six-vertex model was discovered, that has not yet appeared in the literature. It is valid for a certain one parameter family of diagonal open boundary conditions in the sector with the zz-projection of the total spin operator being equal to zero.

A study of self-adjointness, Lie analysis, wave structures, and conservation laws of the completely generalized shallow water equation

This article explores the analysis of the completely generalized Hirota–Satsuma–Ito equation through Lie symmetry analysis. The equation under consideration represents a more comprehensive form of the (2+1)-dimensional HSI equation, encompassing four additional second-order derivative terms: Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, and Δ6Hιι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{6}H_{\\iota \\iota }$$\\end{document}, emerging from the inclusion of second-order dissipative-type elements. We calculate the infinitesimal generators and determine the symmetry group for each generator using the Lie group invariance condition. Employing the conjugacy classes of the Abelian algebra, we transform the considered equation into an ordinary differential equation through similarity reduction. Subsequently, we solve these ordinary differential equations to derive closed-form solutions for the completely generalized Hirota–Satsuma–Ito equation under certain conditions. For other scenarios, we utilize the extended direct algebraic method to obtain soliton solutions. Furthermore, we rigorously calculated the conserved quantities corresponding to each symmetry generator, the conservation laws of the model are established using the multiplier approach. Additionally, we present the graphical representation of selected solutions for specific values of the physical parameters of the equation under scrutiny.

The de Rham Cohomology Classes of Hemi-Slant Submanifolds in Locally Product Riemannian Manifolds

This paper aims to discuss the de Rham cohomology of hemi-slant submanifolds in locally product Riemannian manifolds. The integrability and geodesical invariance conditions of the distributions derived from the definition of a hemi-slant submanifold are given. The existence and non-triviality of de Rham cohomology classes of hemi-slant submanifolds are investigated. Finally, an example is presented.