GenASiS Mathematics : Object-oriented manifolds, operations, and solvers for large-scale physics simulations
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Cardall, Christian Y.↗
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Liouville-Type Theorems for Theories of Riemannian Almost Product Structures and Submersions
We prove Liouville type theorems, i.e., non-existence theorems for special classes of Riemannian almost product structures and submersions, that generalize well-known similar results of the theory of compact Riemannian manifolds.
97 MATHEMATICS AND COMPUTING↗
GenASiS Mathematics: Object-oriented manifolds, operations, and solvers for large-scale physics simulations (version 2)
We report GenASiS Mathematics provides modern Fortran classes furnishing extensible object-oriented functionality for the solution of fields governed by selected partial differential equations. The initial release included extensible object-oriented implementations of simple meshes and the evolution of generic conserved currents thereon. This revision - Version 2 of Mathematics - includes significant reorganization and streamlining of these classes, higher-order reconstruction by a different method, a Poisson solver, coarsening to avoid Courant time step limitations near coordinate singularities, and the offloading of computational kernels to GPUs.
97 MATHEMATICS AND COMPUTING↗
The design-to-cost manifold
A mathematical technique which both quantifies the design-to-cost process and the mass/complexity issue is presented. A simplified approximation of the PRICE H production-production cost is used to generate a dual set of differential equations which define the directions of maximum and minimum cost change over (mass, complexity) space. The equations are solved in closed form to obtain the one-dimensional design-to-cost and design-for-cost spaces. Preliminary results indicate that cost is relatively insensitive to changes in mass and that the reduction of complexity, both in the manufacturing process and in the spacecraft itself, is dominant in reducing cost. Two major objections to the design-to-cost method are discussed.
Dean, Edwin B.↗
Hidden symmetries in Sasaki–Einstein geometries
We describe a method for constructing Killing–Yano tensors on Sasaki spaces using their geometrical properties, without the need of solving intricate generalized Killing equations. We obtain the Killing–Yano tensors on toric Sasaki–Einstein spaces using the fact that the metric cones of these spaces are Calabi–Yau manifolds which in turn are described in terms of toric data. We extend the search of Killing–Yano tensors on mixed 3-Sasakian manifolds. We illustrate the method by explicit construction of Killing forms on some spaces of current interest.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Elliptic G-Operators on Manifolds with Isolated Singularities
In the present work we study elliptic operators on manifolds with singularities in the situation where the manifold is endowed with an action of a discrete group G. As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). The Fredholm property of elliptic elements is obtained.
97 MATHEMATICS AND COMPUTING↗
Integrable Dynamic Systems with Dissipation and Finitely Many Degrees of Freedom
We establish the integrability for some classes of dynamic systems on the tangent bundles of multidimensional manifolds. We consider the case where the force fields possess variable dissipation. An example of a four-dimensional manifold is discussed in detail.
97 MATHEMATICS AND COMPUTING↗
On the Stability of Toroidal Manifold for One Class of Dynamical Systems
We study the problem of stability of an invariant toroidal manifold for one class of linear extensions of a dynamical system on a torus. The obtained result is used to investigate the problem of existence of invariant manifold for a nonlinear system of differential equations.
97 MATHEMATICS AND COMPUTING↗
Mixed-mode oscillations in a three-timescale coupled Morris–Lecar system
Mixed-mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov–Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris–Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.
97 MATHEMATICS AND COMPUTING↗
The Effective Dynamics of the Volume Preserving Mean Curvature Flow
We consider the dynamics of small closed submanifolds (‘bubbles’) under the volume preserving mean curvature flow. We construct a map from (n+1)-dimensional Euclidean space into a given (n+1)-dimensional Riemannian manifold which characterizes the existence, stability and dynamics of constant mean curvature submanifolds. This is done in terms of a reduced area function on the Euclidean space, which is given constructively and can be computed perturbatively. This allows us to derive adiabatic and effective dynamics of the bubbles. The results can be mapped by rescaling to the dynamics of fixed size bubbles in almost Euclidean Riemannian manifolds.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Complexity of virtual 3-manifolds
Virtual 3-manifolds were introduced by Matveev in 2009 as natural generalizations of classical 3-manifolds. In this paper, we introduce a notion of complexity for a virtual 3-manifold. We investigate the values of the complexity for virtual 3-manifolds presented by special polyhedra with one or two 2-components. On the basis of these results, we establish the exact values of the complexity for a wide class of hyperbolic 3-manifolds with totally geodesic boundary. Bibliography: 24 titles. (paper)
97 MATHEMATICS AND COMPUTING↗
On the Optimal Stabilization of an Integral Manifold
By the method of Lyapunov functions, we study the problem of optimal stabilization of an analytically defined integral manifold in the class of stochastic differential equations in the case where random perturbations belong to the class of processes with independent increments.
97 MATHEMATICS AND COMPUTING↗
Integrable Systems with Dissipation and Two and Three Degrees of Freedom
We establish the integrability for some classes of dynamic systems on the tangent bundles of two– and three-dimensional manifolds (systems with two and three degrees of freedom). The force fields possess the so-called variable dissipation and generalize those considered in the previous publications of the author.
97 MATHEMATICS AND COMPUTING↗
Multiparameter Quantum Group and Quantum Minkowski Space-Time
—We construct representations of the quantum algebras U{sub q,q}(gl(n)) and U{sub q,q}(sl(n)) which depend on n(n−1)/ 2+1 deformation parameters q,q{sub ij} (1⩽i<j⩽n ) which is the maximal possible number in the case of GL(n). The representations act on the space of formal power series of n(n−1)/ 2 non-commuting variables which generate quantum flag manifolds of GL{sub qq}(n), SL{sub qq}(n). For n=4 we consider in detail the multiparameter quantum Minkowski space-time.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Quasitoric manifolds and small covers over properly coloured polytopes: immersions and embeddings
We construct small covers and quasitoric manifolds over n-dimensional simple polytopes which allow proper colourings of facets with n colours. We calculate the Stiefel-Whitney classes of these manifolds as obstructions to immersions and embeddings into Euclidean spaces. The largest dimension required for embedding is achieved in the case n is a power of two.Bibliography: 11 titles.
97 MATHEMATICS AND COMPUTING↗
Extended Hamiltonians and shift, ladder functions and operators
In recent years, many natural Hamiltonian systems, classical and quantum, with constants of motion of high degree, or symmetry operators of high order, have been found and studied. Most of these Hamiltonians, in the classical case, can be included in the family of extended Hamiltonians, geometrically characterized by the structure of warped manifold of their configuration manifold. For the extended Hamiltonians, the characteristic constants of motion of high degree are polynomial in the momenta of determined form. We consider here a different form of the constants of motion, based on the factorization procedure developed for systems of two degrees of freedom by S. Kuru, J. Negro and others. We show that an important subclass of the extended Hamiltonians, with arbitrary dimension, admits factorized constants of motion and we determine their expression. The classical constants can be polynomial or non-polynomial in the momenta, but the factorization procedure allows, in a type of extended Hamiltonians, their quantization via shift and ladder operators, for systems of any finite dimension. - Highlights: • A factorization procedure is proposed for first integrals of high degree of extended Hamiltonian systems. • The extended Hamiltonians are defined on warped manifolds of any finite dimension. • Polynomial first integrals are quantized into factorized polynomial symmetry operators of Laplace–Beltrami-type quantum Hamiltonians.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
On volumes of classical supermanifolds
We consider the volumes of classical supermanifolds (such as the supersphere, complex projective superspace, Stiefel and Grassmann supermanifolds) with respect to natural metrics or symplectic structures. We show that the formulae for the volumes of these supermanifolds can be obtained from the formulae for the volumes of the corresponding ordinary manifolds (under some universal normalization of the volume) by analytic continuation with respect to parameters. The volumes of nontrivial supermanifolds may be identically equal to zero. In the 1970s Berezin showed that the total Haar measure of the unitary supergroup U(n|m) vanishes except in the cases m=0 and n=0, when the supergroup is the ordinary unitary group U(n) or U(m). Some time ago Witten conjectured that the Liouville volume of a compact even symplectic supermanifold is always equal to zero (except for ordinary manifolds). We give counterexamples to this conjecture, present a simple explanation of Berezin's theorem, and generalize this theorem to the Stiefel supermanifold V{sub r|s}(C{sup n|m}). We mention a connection with recent work of Mkrtchyan and Veselov on universal formulae in Lie algebra theory. Bibliography: 32 titles. (paper)
97 MATHEMATICS AND COMPUTING↗
Dynamical aspects in the quantizer–dequantizer formalism
The use of the quantizer–dequantizer formalism to describe the evolution of a quantum system is reconsidered. We show that it is possible to embed a manifold in the space of quantum states of a given auxiliary system by means of an appropriate quantizer–dequantizer system. If this manifold of states is invariant with respect to some unitary evolution, the quantizer–dequantizer system provides a classical-like realization of such dynamics, which in general is non linear. Integrability properties are also discussed. Weyl systems and generalized coherent states are used as a simple illustration of these ideas.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗