Symmetric Domains Research Articles

Assessing Permutationally Invariant Polynomial and Symmetric Gradient Domain Machine Learning Potential Energy Surfaces for H3O2.

The singly hydrated hydroxide anion OH-(H2O) is of central importance to a detailed molecular understanding of water; therefore, there is strong motivation to develop a highly accurate potential to describe this anion. While this is a small molecule, it is necessary to have an extensive data set of energies and, if possible, forces to span several important stationary points. Here, we assess two machine-learned potentials, one using the symmetric gradient domain machine learning (sGDML) method and one based on permutationally invariant polynomials (PIPs). These are successors to a PIP potential energy surface (PES) reported in 2004. We describe the details of both fitting methods and then compare the two PESs with respect to precision, properties, and speed of evaluation. While the precision of the potentials is similar, the PIP PES is much faster to evaluate for energies and energies plus gradient than the sGDML one. Diffusion Monte Carlo calculations of the ground vibrational state, using both potentials, produce similar large anharmonic downshift of the zero-point energy compared to the harmonic approximation of the PIP and sGDML potentials. The computational time for these calculations using the sGDML PES is roughly 300 times greater than using the PIP one.

Sharp coefficient inequalities of starlike functions connected with secant hyperbolic function

This article comprises the study of class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document} that represents the class of normalized analytic functions f satisfying ςf′(z)/f(ς)≺sech(ς)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\varsigma \\mathsf{f}}^{\\prime }(z)/\\mathsf{f}( {\\varsigma })\\prec \\sec h ( \\varsigma ) $\\end{document}. The geometry of functions of class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document} is star-shaped, which is confined in the symmetric domain of a secant hyperbolic function. We find sharp coefficient results and sharp Hankel determinants of order two and three for functions in the class SE∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}_{E}^{\\ast }$\\end{document}. We also investigate the same sharp results for inverse coefficients.

Identifying the source term and the initial value simultaneously for Caputo–Hadamard fractional diffusion equation on spherically symmetric domain

Identifying the source term and the initial value simultaneously for Caputo–Hadamard fractional diffusion equation on spherically symmetric domain

An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution

Abstract A T T -symmetric univalent function is a complex valued function that is conformally mapping the unit disk onto itself and satisfies the symmetry condition ϕ [ T ] ( ζ ) = [ ϕ ( ζ T ) ] 1 ∕ T {\phi }^{\left[T]}\left(\zeta )={\left[\phi \left({\zeta }^{T})]}^{1/T} for all ζ \zeta in the unit disk. In other words, it is a complex function that preserves the unit disk’s shape and orientation and is symmetric about the unit circle. They are used in the study of geometric function theory and the theory of univalent functions. In recent effort, we extend the class of fractional anomalous diffusion equations in a symmetric complex domain. we aim to present the analytic univalent solution for such a class using special functions technique. Our analysis and comparative findings are further supported by the geometric simulations for the univalent solution such as the convexity and starlikeness of the diffusion. As a consequence of illustration of a list of conditions yielding the univalent solutions (normalize analytic function in the open unit disk), the normalization of diffusion shape is achieved.

A simplified approach to the holomorphic discrete series

Expository article on semisimple Lie groups of Hermitian type and their unitary representations known as the holomorphic discrete series. The realization of the symmetric spaces associated to the groups as bounded symmetric domains is described. The representations in question are defined by holomorphic induction and realized on spaces of vector-valued holomorphic functions on the domain. A key question is whether the induction process yields a non-zero space. It is answered by Harish-Chandra’s condition, for which a complete proof is given.

Curvature conservation and conduction modulation for symmetric charged ferroelectric domain walls

Curvature conservation is a common feature for the physical properties with topological constraints, from the macroscopic cosmic string loops to the microscopic quantized vortex rings in the superfluid. Interestingly, ferroelectric domain-wall curvature is important to determine its conduction, which is the key parameter for designing domain-wall nanoelectronic devices. Here, we demonstrate the curvature conservation of the charged domain walls confined in ferroelectric topological structures with specific domain symmetry by combining piezoelectric force microscopy, conducting atomic force microscopy, and phase-field simulations. Significantly, the intrinsic conductivity of the charged domain walls exhibits an inverse functional relationship with the local domain-wall curvature in the ferroelectric BiFeO3 nano-islands, which can also be precisely and continuously modulated by applying an electric field. This work provides us with an insightful understanding of the conduction mechanism of charged domain walls confined in ferroelectric topological structures with a high degree of spatial symmetry, which paves the way for the application of charged domain walls as flexible conducting channels with continuously changing resistance states to be used for ferroelectric domain-wall memristors.

On the long-time behaviour of solutions to unforced evolution Navier–Stokes equations under Navier boundary conditions

We study the asymptotic behaviour of the solutions to Navier–Stokes unforced equations under Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest. The paper draws its main motivation from celebrated results by Foias and Saut (1984) under Dirichlet conditions; here the choice of the boundary conditions requires carefully considering the geometry of the domain Ω, due to the possible lack of the Poincaré inequality in presence of symmetries. In non-axially symmetric domains we show the validity of the Foias–Saut result about the limit at infinity of the Dirichlet quotient, in axially symmetric domains we provide two invariants of the flow which completely characterize the motion and we prove that the Foias–Saut result holds for initial data belonging to one of the invariants.

Holomorphicity of totally geodesic Kobayashi isometry between bounded symmetric domains

AbstractIn this paper, we study the holomorphicity of totally geodesic Kobayashi isometric embeddings between bounded symmetric domains. First, we show that for a ‐smooth totally geodesic Kobayashi isometric embedding where , are bounded symmetric domains, if is irreducible and or more generally, for any tangent vector of , then is either holomorphic or anti‐holomorphic. Second, we characterize Kobayashi isometries from a reducible bounded symmetric domain to itself.

Nuclear Quantum Effects in Hydroxide Hydrate Along the H-Bond Bifurcation Pathway.

Path integral (PI) simulations are used to explore nuclear quantum effects (NQEs) in hydroxide hydrate and its perdeuterated isotopomer along the H-bond bifurcation pathway. Toward this, a new potential energy surface using the symmetric gradient domain machine learning method with ab initio data at the CCSD(T)/aug-cc-pVTZ level is built. From PI umbrella sampling (US) simulations, free energy profiles along the bifurcation coordinate are explored as a function of temperature. At ambient temperature, the bifurcation barrier is increased upon inclusion of NQEs. At low temperatures in the deep tunneling regime, the barrier is strongly decreased and flattened. These trends are examined, and the role of the O-O distance is also investigated through two-dimensional US simulations.

Impurity-doped stable domain walls in spherically symmetric spacetimes

In this work, radially symmetric domain wall solutions in the presence of impurities are investigated for both flat and curved D+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D+1$$\\end{document} spacetimes, with geometry generated by a rotationally invariant background metric. We have examined the constraints placed upon the model by Derrick’s theorem, and found out the Bogomol’nyi bound and equations of the symmetric restriction to this theory. Impurity-doped versions of a ϕ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^4$$\\end{document} model in two dimensions and a model with logarithmic potential in a Schwarzschild background have been explicitly worked out. The resulting configurations have been compared with those found in the homogeneous version of the theory, so that the effect of impurities in the form of solutions may be better appreciated. We have also generalized to higher dimensions some of the results that had been presented in the recent literature. These results relate to the possibility of BPS-preserving impurities, which we have found to still exist in the spacetimes considered in this work. We also investigate ways in which these results may be extended in a curved background.

On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function FK to Symmetric Domains

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function FK with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks.

On a class of infinite semipositone problems for ( p , q ) Laplace operator

We analyze a non-linear elliptic boundary value problem that involves ( p , q ) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in ( 0 , ∞ ) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving ( p , q ) Laplace operator when the parameter λ is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.

Edge states in non-Hermitian composite acoustic Su Schrieffer Heeger chains

Non-Hermiticity alone can trigger topological phase transition in physical systems. Here, we construct different unit cells in an acoustic Su Schrieffer Heeger chain with different distributions of onsite losses. We theoretically and numerically investigate the different edge modes that can occur at the domain walls of different finite chains. Three types of edge modes are identified. The first type comes from the topology of the unit cells. The second type comes from the local Parity symmetry at the interface, which are cavity modes. The third one comes from the Parity-Time symmetric domain wall. The robustness against coupling disorder is then examined, confirming the robustness of the topologically protected modes. The evolution with increasing disorder of the interface modes due to the Parity-Time symmetric domain wall is singular as they appear first as more robust than the cavity modes before diverging. These results show the ability of the onsite losses ingredient to control wavefields.

Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces

We consider three different questions related to the Steklov and mixed Steklov problems on surfaces. These questions are connected by the techniques that we use to study them, which exploit symmetry in various ways even though the surfaces we study do not necessarily have inherent symmetry. In the spirit of the celebrated Hersch-Payne-Schiffer and Weinstock inequalities for Steklov eigenvalues, we obtain a sharp isoperimetric inequality for the mixed Steklov eigenvalues considering the interplay between the eigenvalues of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues. In 1980, Bandle showed that the unit disk maximizes the kth nonzero normalized Steklov eigenvalue on simply connected domains with rotational symmetry of order p when k≤p−1. We discuss whether the disk remains the maximizer in the class of simply connected rotationally symmetric domains when k≥p. In particular, we show that as k→∞, the upper bound converges to the Hersch-Payne-Schiffer upper bound. We give full asymptotics for mixed Steklov problems on arbitrary surfaces, assuming some conditions at the meeting points of the Steklov boundary with the Dirichlet or Neumann boundary.

Mechanical manipulation for ordered topological defects.

Randomly distributed topological defects created during the spontaneous symmetry breaking are the fingerprints to trace the evolution of symmetry, range of interaction, and order parameters in condensed matter systems. However, the effective mean to manipulate topological defects into ordered form is elusive due to the topological protection. Here, we establish a strategy to effectively align the topological domain networks in hexagonal manganites through a mechanical approach. It is found that the nanoindentation strain gives rise to a threefold Magnus-type force distribution, leading to a sixfold symmetric domain pattern by driving the vortex and antivortex in opposite directions. On the basis of this rationale, sizeable mono-chirality topological stripe is readily achieved by expanding the nanoindentation to scratch, directly transferring the randomly distributed topological defects into an ordered form. This discovery provides a mechanical strategy to manipulate topological protected domains not only on ferroelectrics but also on ferromagnets/antiferromagnets and ferroelastics.

On the Non-degeneracy of the Robin Function for the Fractional Laplacian on Symmetric Domains

On the Non-degeneracy of the Robin Function for the Fractional Laplacian on Symmetric Domains

Weyl algebras for quantum homogeneous spaces

We present a new family of quantum Weyl algebras where the polynomial part is the quantum analog of functions on homogeneous spaces corresponding to symmetric matrices, skew symmetric matrices, and the entire space of matrices of a given size. The construction uses twisted tensor products and their deformations combined with invariance properties derived from quantum symmetric pairs. These quantum Weyl algebras admit Uq(glN)-module algebra structures compatible with standard ones on the polynomial part, have relations that are expressed nicely via matrices, and are closely related to an algebra arising in the theory of quantum bounded symmetric domains.

Regularity of CR maps into uniformly pseudoconvex hypersurfaces and applications to proper holomorphic maps

We study regularity properties of CR maps in positive codimension valued in pseudoconvex manifolds which carry a nontrivial Levi foliation. We introduce an invariant which can be used to deduce that any sufficiently regular CR map from a minimal manifold into such a foliated target is either generically smooth or geometrically highly constrained, and to show generic smoothness of sufficiently regular CR transversal CR maps between pseudoconvex hypersurfaces. As an application, we discuss boundary regularity of proper holomorphic maps into bounded symmetric domains.

Fractional operators on the bounded symmetric domains of the Bergman spaces

<abstract><p>Mathematics has several uses for operators on bounded symmetric domains of Bergman spaces including complex geometry, functional analysis, harmonic analysis and operator theory. They offer instruments for examining the interaction between complex function theory and the underlying domain geometry. Here, we extend the Atangana-Baleanu fractional differential operator acting on a special type of class of analytic functions with the $ m $-fold symmetry characteristic in a bounded symmetric domain (we suggest the open unit disk). We explore the most significant geometric properties, including convexity and star-likeness. The boundedness in the weighted Bergman and the convex Bergman spaces associated with a bounded symmetric domain is investigated. A dual relations exist in these spaces. The subordination and superordination inequalities are presented. Our method is based on Young's convolution inequality.</p></abstract>

P-VALENTLY BAZILEVIˇ C FUNCTIONS DEFINED WITH HIGHER ORDER DERIVATIVES

The object of this paper is to derive various properties and characters for a class of p-valently Bazileviˇ c functions defined with higher order derivatives. Using the technique of Briot-Bouquet differential subordination we obtained several results, some of them presented here generalize and improve a few earlier results, and several others are new ones involving differential inequalities. These last implications deal with the symmetric open disc and symmetric conic domains with respect to the real axe. Our results extends and improves many other previous ones, and some of them are the best possible under the given assumptions.