On the carnot theorem in the theory of impulsive motion of mechanical systems : PMM vol. 43, no. 6, 1979, pp. 1114–1116

The kinetic energy variation of a system subjected to impact under conditions of the generalized Carnot theorem [1] is investigated. Proof of the theorem is given in the case of acatastatic links. It is shown that the second part of that theorem extends to systems whose interaction with the links is defined by properties which include the known conditions of ideality of the latter and, also, of zero virtual work of link reaction impulses during the regeneration time.     

Topological imitation of a colored link with the same Dehn surgery manifold

By the topological imitation theory, we construct, from a given colored link, a new colored link with the same Dehn surgery manifold. In particular, we construct a link with a distinguished coloring whose Dehn surgery manifold is a given closed connected oriented 3-manifold except the 3-sphere. As a result, we can naturally generalize the difference between the Gordon–Luecke theorem and the property P conjecture to a difference between a link version of the Gordon–Luecke theorem and the Poincaré conjecture. Similarly, we construct a link with a π₁-distinguished coloring whose Dehn surgery manifold is a given non-simply-connected closed connected oriented 3-manifold. We also construct a link with just two colorings whose Dehn surgery manifolds are the 3-sphere.     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem.     

The Balian–Low theorem and noncommutative tori

We point out a link between the theorem of Balian and Low on the non-existence of well-localized Gabor–Riesz bases and a constant curvature connection on projective modules over noncommutative tori.     

Generalized Budan–Fourier theorem and virtual roots

In this Note we give a proof of a generalized version of the classical Budan–Fourier theorem, interpreting sign variations in the derivatives in terms of virtual roots.     

Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain

We prove an existence theorem for the Boltzmann–Fermi–Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.     

A new proof of an Engelbert–Schmidt type zero–one law for time-homogeneous diffusions

In this paper we give a new proof to an Engelbert–Schmidt type zero–one law for time-homogeneous diffusions, which provides deterministic criteria for the convergence of integral functional of diffusions. Our proof is based on a slightly stronger assumption than that in Mijatović and Urusov (2012a) and utilizes stochastic time change and Feller’s test of explosions. It does not rely on advanced methods such as the first Ray–Knight theorem, William’s theorem, Shepp’s dichotomy result for Gaussian processes or Jeulin’s lemma as in the previous literature (see Mijatović and Urusov (2012a) for a pointer to the literature). The new proof has an intuitive interpretation as we link the integral functional to the explosion time of an associated diffusion process.     

The polynomial invariants of twisted links

A twisted link is a generalization of a virtual link, which is related to a link diagram on a closed, possibly non-orientable surface. In this paper we generalize the Miyazawa polynomial invariant of a virtual link to an invariant of a twisted link in two formulae one of which is introduced by A. Ishii and the other by the author.     

Symmetry via spherical reflection

In 1955 Alexandrov [2, 7, p. 147] introduced the method of planar reflection as a procedure for proving symmetry of closed surfaces of constant mean curvature. In subsequent years, a number of refinements were given, and the method was applied successfully to other problems in differential geometry and partial differential equations, cf. [10, 5, 12, 9, 3, 12]. In the present paper, we introduce a new procedure for obtaining symmetry which may be considered an extension of the planar reflection method to reflection about spheres. In the interest of clearly delineating the new features of the method, in this initial presentation we give a new proof of the original Alexandrov theorem: A closed embedded surface of constant mean curvature in Euclidean space is a round sphere. Our proof of this result is not simpler than that of Alexandrov; it is, however, different in some crucial respects. Our justification for introducing the new procedure lies in its applicability to configurations that are not accessible to planar reflection arguments. In particular, we use the new method in [8] to prove the non-existence in certain cases of embedded liquid bridges joining two planes that meet to form a wedge domain, a result whose interest derives in turn largely from the fact that under the same (physical) boundary conditions embedded bridges do exist and can be stable when the planes are parallel. The method of spherical reflection as presented below relies on a new maximum principle related to surface geometry, which we believe to have independent interest. Further, the method utilizes a peculiar characterization of spherical surfaces by aggregate symmetry, which should also have interest for other problems arising in the global differential geometry of surfaces.     

Splittings of finitely generated groups over two-ended subgroups

We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of the JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated group which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.

     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. Research partly supported by KBN grant 2 P03A 002 14 (1998). Received June 5, 2001; in revised form March 4, 2002     

Polynomial invariants and quandles of twisted links

Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link.     

The rack space

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James bundle.

We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James-Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links.

     

Joint cumulative distribution functions for Dempster–Shafer belief structures using copulas

We first introduce the Dempster–Shafer belief structure and highlight its role in the representation of information about a random variable for which our knowledge of the probabilities is interval-valued. We investigate the formation of the cumulative distribution function (CDF) for these types of variables. It is noted that this is also interval-valued and is expressible in terms of plausibility and belief measures. The class of aggregation operators known as copulas are introduced and a number of their properties are provided. We discuss Sklar’s theorem, which provides for the use of copulas in the formulation of joint CDFs from the marginal CDFs of classic random variables. We then look to extend these ideas to the case of joining the marginal CDFs associated with Dempster–Shafer belief structures. Finally we look at the formulation CDFs obtained from functions of multiple D–S belief structures.     

Reflection quotients in Riemannian geometry. A geometric converse to Chevalley's theorem

Chevalley's theorem and its converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that, in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely, that a freely-generated polynomial subring is closed with respect to the gradient product.

     

Quasifactors of ergodic systems with positive entropy

The relation between the two notions, quasifactors and joinings, is investigated and the notion of a joining quasifactor is introduced. We clarify the close connection between quasifactors and symmetric infinite selfjoinings which arises from de Finetti-Hewitt-Savage theorem. Unlike the zero-entropy case where quasifactors seems to preserve some properties of their parent system, it is shown that any ergodic system of positive entropy admits all ergodic systems of positive entropy as quasifactors. A restricted version of this result is obtained for joining quasifactors.     

Noethers theorem

Summary We shall here generalise a geometrical version of Noethers theorem due to F. Severi. This is essentially done by replacing the notion of virtual multiplicity by that of «imaginary multiplicity» to be introduced below. The proof of this generalised Noethers theorem is based on the use of the resultant of the two curves in question.     

THE MORSE CRITICAL GROUPS OF THE MINMAX THEOREM

The author gives a new min-max theorem and studies the local homology propertiesof critical points that is got by min-max theorem.In particular,this result includes manymin-max theorem of link type,which can be applicable to nonlinear analysis.     

Reflection symmetry and symmetrizability of Hilbert space operators

A general factorization theorem for symmetrizable operators relating their spectra to spectra of selfadjoint operators induced by minimal factorizations is established. Its modified version essentially improves and completes a theorem of Jorgensen, which concerns diagonalizing operators with reflection symmetry.

     

On the convergence of 2-dimensional markov chains in quadrants with boundary reflection

The aim of this paper is to prove a limit theorem on the weak convergence of a family of rescaled Markov chains in a quadrant with boundary reflection. The limiting process is specified in terms of solutions of a certain submartingale problem in the style used by Varadhan and Williams. The obtained result is then applied to the problem of approximating an arbitrary Brownian motion with oblique reflection in a wedge by a family of Markov chains.