Infinitesimal Properties Research Articles

Pairing Symmetries for Euclidean and Spherical Frameworks

We consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in {mathbb {R}}^d. In particular, for a graph G=(V,E) and a framework (G, p), we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in some subset Xsubset V realised on the equator and point-hyperplane frameworks with the vertices in X representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks.

Bol three-webs with torsion tensor of rank

The infinitesimal properties of multidimensional Bol three-webs with covariantly constant curvature tensor (webs ) are considered, and a foundation for classifying such webs in accordance with the rank of the torsion tensor is laid. For a three-web of rank Cartan's method is used to construct an adapted frame and find the corresponding system of (differential) structure equations. A three-web of rank is shown to have a normal subweb that is a group web; the corresponding factor web is a regular three-web. By integrating the structure equations new families of examples of multidimensional three-webs of special form and smooth Bol loops are discovered which are generalizations of a semidirect product of two Abelian Lie groups. Bibliography: 40 titles.

Stability Properties of the Value Function in an Infinite Horizon Optimal Control Problem

Properties of the value function are examined in an infinite horizon optimal control problem with an integrand index appearing in the quality functional with a discount factor. The properties are analyzed in the case when the payoff functional of the control system includes a quality index represented by an unbounded function. An upper estimate is given for the growth rate of the value function. Necessary and sufficient conditions are obtained to ensure that the value function satisfies the infinitesimal stability properties. The question of coincidence of the value function with the generalized minimax solution of the Hamilton–Jacobi–Bellman–Isaacs equation is discussed. The uniqueness of the corresponding minimax solution is shown. The growth asymptotic behavior of the value function is described for the logarithmic, power, and exponential quality functionals, which arise in economic and financial modeling. The obtained results can be used to construct grid approximation methods for the value function as the generalized minimax solution of the Hamilton–Jacobi–Bellman–Isaacs equation. These methods are effective tools in the modeling of economic growth processes.

Symmetries of semiclassical gauge systems

Semiclassical systems being symmetric under Lie group are studied. A state of a semiclassical system may be viewed as a set (X, f) of a classical state X and a quantum state f in the external classical background X. Therefore, the set of all semiclassical states may be considered as a bundle (semiclassical bundle). Its base {X} is the set of all classical states, while a fiber is a Hilbert space ℱX of quantum states in the external background X. Symmetry transformation of a semiclassical system may be viewed as an automorphism of the semiclassical bundle. Automorphism groups can be investigated with the help of sections of the bundle: to any automorphism of the bundle one assigns a transformation of section of the bundle. Infinitesimal properties of transformations of sections are investigated; correspondence between Lie groups and Lie algebras is discussed. For gauge theories, some points of the semiclassical bundle are identified: a gauge group acts on the bundle. For this case, only gauge-invariant sections of the semiclassical bundle are taken into account.

Continuous discretization of infinite-dimensional Hilbert spaces

In quantum theory, observables with a continuous spectrum are known to be fundamentally different from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for discrete variables have been translated to continuous ones, this is not the case in general. For instance, despite their importance, no experimental demonstration of nonlocality exists in the continuous variables regime. Attempts to bridge this gap and put continuous variables on a closer footing to discrete ones used dichotomization. However, this approach considers only discrete properties of the continuum, and its infinitesimal properties are not fully exploited. Here we show that it is possible to manipulate, detect and classify continuous variable states using observables with a continuous spectrum revealing properties and symmetries which are analogous to finite discrete systems. Our approach leads to an operational way to define and adapt, to arbitrary continuous quantum systems, quantum protocols and algorithms typical to discrete systems.

Tropical linear-fractional programming and parametric mean payoff games

Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to study the tropical analogue of the classical linear-fractional programming problem. We construct an associated parametric mean payoff game problem, and show that the optimality of a given point, or the unboundedness of the problem, can be certified by exhibiting a strategy for one of the players having certain infinitesimal properties (involving the value of the game and its derivative) that we characterize combinatorially. We use this idea to design a Newton-like algorithm to solve tropical linear-fractional programming problems, by reduction to a sequence of auxiliary mean payoff game problems.

First-Order Twistor Lifts

The use of twistor methods in the study of Jacobi fields has proved quite fruitful, leading to a series of results. L. Lemaire and J. C. Wood proved several properties of Jacobi fields along harmonic maps from the two-sphere to the complex projective plane and to the three- and four-dimensional spheres, by carefully relating the infinitesimal deformations of the harmonic maps to those of the holomorphic data describing them. In order to advance this programme, we prove a series of relations between infinitesimal properties of the map and those of its twistor lift. Namely, we prove that isotropy and harmonicity to first order of the map correspond to holomorphicity to first order of its lift into the twistor space, relatively to the standard almost complex structures and . This is done by obtaining first-order analogues of classical twistorial constructions.

Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes

This a first step to develop a theory of smooth, étale, and unramified morphisms between Noetherian formal schemes. Our main tool is the complete module of differentials, which is, a coherent sheaf whenever the map of formal schemes is of pseudofinite type. Among our results, we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitable closed subsets. We characterize unramifiedness by the vanishing of the module of differentials. Also we see that a smooth morphism of Noetherian formal schemes is flat and its module of differentials is locally free. The article closes with a version of Zariski's Jacobian criterion.

Nonsmooth analysis on smooth manifolds

We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function. A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

Material derivatives of higher dimension in geophysical fluid dynamics

The familiar operator D/Dt in fluid dynamics defines the material derivative for a fluid particle with dimension zero. In this paper we define and use macroscopic or multidimensional material derivatives D 1 /Dt, D 2 /Dt and D 3 /Dt. They are the material derivatives of infinitesimal properties of the fluid having dimensions, i. e. when particles build a line, a surface area, or a volume. Simple rules between the three operators are presented that avoid complicated calculations in fluid dynamics. For example, these operators are invariant with respect to solid rotations of coordinate systems. We rewrite a number of equations of fluid dynamics in terms of these operators and show that simple identities involving these operators already contain the structure of known vorticity theorems, especially those given by Hans Ertel. One application deals with the circulation of eddy velocities in atmospheric turbulence, showing that this circutation maybe an almost material invariant with time. Further possible applications (e.g., in electrodynamics of fluids and in radiation hydrodynamics) are also suggested.

Differential Inclusions and Monotonicity Conditions for Nonsmooth Lyapunov Functions

In this paper we address the problem of characterizing the infinitesimal properties of functions which are nonincreasing along all the trajectories of a differential inclusion. In particular, we extend the condition based on the proximal gradient to the case of semicontinuous functions and Lipschitz continuous differential inclusions. Moreover, we show that the same criterion applies also in the case of Lipschitz continuous functions and continuous differential inclusions.

Calendar of forthcoming conference and eventsnumerable Markov processes from infinitesimal properties

Calendar of forthcoming conference and eventsnumerable Markov processes from infinitesimal properties

Density based classes of priors: Infinitesimal properties and approximations

We define a notion of local sensitivity for classes of prior densities. The local sensitivity measures how much a small change in the prior affects the posterior. We find an explicit expression for this local sensitivity and we use it as a diagnostic for robustness. We also show that the local sensitivity can be used to find simple, accurate approximations to upper and lower bounds of posterior expectations.

Differential properties of integral funnels and stable bridges

Left derivatives of a multivalued mapping, similar to those previously introduced in /1, 2/, are applied to investigate the necessary and sufficient conditions for the integral funnel of a differential inclusion. In terms of the construction developed for analysing the value function of a differential game (see, e.g., /3, 4/) and generalized solutions of the Hamilton-Jacobi equation (viscosity solutions in the sense of /5, 6/), we describe the integral funnel using viscosity solutions of Bellman's equation. The properties of stable bridges are investigated using left derivatives. Necessary and sufficient conditions for stable bridges are stated ∗∗ ∗∗ Detailed proofs of the propositions formulated in this paper can be found in GUSEINOV KH.G. and USHAKOV V.N., Infinitesimal Properties of Integral Funnels and Stable Bridges, Baku, 1988. Unpublished manuscript available from VINITI, 6.05.88, 3571-V88. .

Normally flat deformations

We study flat families Z / T Z/T , together with a section σ : T → Z \sigma :T \to Z such that the normal cone to the image of σ \sigma in Z is flat over T. Such a family is called a “normally flat deformation (along σ \sigma )"; it corresponds intuitively to a deformation of a singularity which preserves the Hilbert-Samuel function. We construct the versal normally flat deformation of an isolated singularity (X,x) in terms of the flat strata of the relative jets of the “usual” versal deformation of X. We give explicit criteria, in terms of equations, for a flat family to be normally flat along a given section. These criteria are applied to demonstrate the smoothness of normally flat deformation theoryand of the canonical map from it to the cone deformation theory of the tangent cone-in the case of strict complete intersections. Finally we study the tangent space to the normally flat deformation theory, expressing it as the sum of two spaces: The first is a piece of a certain filtration of the tangent space to the usual deformation theory of X; the second is the tangent space to the special fibre of the canonical map N → S N \to S , where N (resp. S) is the parameter space for the versal normally flat deformation of (X, x) (resp. for the versal deformation of X). We discuss the relation of this second space to infinitesimal properties of sections.

The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.

The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.

On Infinitesimal Properties of Closed Sets of Arbitrary Dimension

1. In the general combinatory theory' the topology of closed sets of arbitrary dimension has reached an entirely new stage, and at the present time there remain only a few parts of general topology as yet not brought within the scope of combinatory methods.2 Despite the close connection there exists a fundamental difference between the combinatorial and general parts of topology, which manifests itself in the complete absence of elements of a set of a simplicial or cellular character, with the exception of the 0-dimensional (points). The possibility of applying combinatory methods in general topology is based entirely on approximation of sets by complexes and cycles, and this also explains the character of the results, which express chiefly properties of a set in the large. The combinatory invariants of complexes are properties in the large, and as such they have been extended to sets by means of approximation. At the same time scarcely any progress has been made in our knowledge of local or infinitesimal properties of sets, or in other words of their properties in a point. The existence of such properties depending on the main invariants of a set in the large, which may also be said to be properties in a point of integral origin,3 was established in some recent papers of mine. In continuation of my work on this problem I intend to give in this paper some new results of a general nature.

Societies and Academies

PARIS Academy of Sciences, August 12 (C.R., 201, 413–436). The President announced the death of Antoine Guntz, Correspondant for the Section of Chemistry. HANS SCHWERDTFEGER: Functions of matrices. BORIS KATJFMANN: The infinitesimal properties of closed ensembles of arbitrary dimension. RICHARD BRAUER: The integral invariants of varieties representative of simple closed Lie groups. GEORGES BOURION: The limit functions of the partial sums of an integral series at the frontier of the circle of convergence. MARCUS BRUTZCUS: The appreciation, a priori, of the value of a commercial combustible for motors. RENE DUBRISAY: The action of sulphur on silver. The blackening of silver can be produced by sulphur without the intervention of a sulphur compound. The increase in the rapidity of the action caused by a high vacuum may be partly due to the increased rate of diffusion of the sulphur vapour, and partly to the removal of a layer of protective gases on the silver. JEAN CALVET, JEAN J. TRILLAT and MILOSLAV PAIC: The recrystalHsation of pure aluminium. Application of the X-ray method to the study of the velocity of crystallisation of aluminium containing 99-9986 per cent of the metal. At 0° C, slight traces of crystallisation appear after 12 hours, and this is still incomplete after 336 hours. At 100 ° C, recrystallisation is complete after one minute, and at higher temperatures is practically instantaneous. CHARLES DTJFRAISSE and MARCEL GERARD: Dissociable organic oxides and the anthracene structure. The existence of a photo-oxide of anthracene: its thermal decomposition. According to the theory developed from the study of the rubenes, anthracene should absorb oxygen rapidly under the action of light, forming a compound decomposing on heating but without emitting oxygen. The results of experiments with anthracene are given, fully confirming these views. RENE SALGTJES: Erythrocytes, haemoglobin and the globular value in the course of cancerous affections in birds. LEON VELLTJZ: The comparative action of the bile acids on the tetanus and diphtheria toxins: the special properties of lithocholic acid. For the same poly cyclic structure, the neutralisation of the diphtheria toxin depends on the number of alcohol groups, whilst this substitution is without influence on the neutralisation of the tetanus toxin. Lithocholic acid is the most energetic agent known as regards neutralisation of the toxin of diphtheria. RAYMOND-HAMET: The non-modification of the sympathicolytic activity of yohimbine by the introduction of a double bond in the molecule of this alkaloid.

Infinitesimal Properties of Lines in S₄ with Applications to Circles in S₃

Infinitesimal Properties of Lines in S₄ with Applications to Circles in S₃