Monotonicity preserving transformations of MOT and SEP

Recently, Beiglböck and Juillet (2016) and Beiglböck et al. (2015) established that optimizers to the martingale optimal transport problem (MOT) are concentrated on $c$-monotone sets. In this article we characterize monotonicity preserving transformations revealing certain symmetries between optimizers of MOT for different cost functions. Due to the intimate connection of MOT and the Skorokhod embedding problem (SEP) these transformations are also monotonicity preserving and disclose symmetries for certain solutions to the optimal SEP. Furthermore, the SEP picture allows to easily understand the geometry of these transformations once we have established the SEP counterparts to the known solutions of MOT based on the monotonicity principle for SEP which in turn allows to directly read off the structure of the MOT optimizers.     

Partial Characterizations of 1‐Perfectly Orientable Graphs

We study the class of 1‐perfectly orientable graphs, that is, graphs having an orientation in which every out‐neighborhood induces a tournament. 1‐perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this article, we develop several results on 1‐perfectly orientable graphs. In particular, we (i) give a characterization of 1‐perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1‐perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1‐perfectly orientable graphs, and (iv) characterize the class of 1‐perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1‐perfectly orientable cobipartite graphs coincides with the class of cobipartite circular arc graphs.     

Grammar functors and covers: From non-left-recursive to greibach normal form grammars

Attention is paid to structure preserving properties of transformations from a non-left-recursive context-free grammar to a Greibach normal form grammar. It is demonstrated that such a transformation cannot only be ambiguity preserving, but also both cover and functor relations between grammars or their associated syntax-categories can be obtained from such a transformation.     

Generating quadrangulations of surfaces with minimum degree at least 3

In this article, we show that all quadrangulations of the sphere with minimum degree at least 3 can be constructed from the pseudo‐double wheels, preserving the minimum degree at least 3, by a sequence of two kinds of transformations called “vertex‐splitting” and “4‐cycle addition.” We also consider such generating theorems for other closed surfaces. These theorems can be translated into those of 4‐regular graphs on surfaces by taking duals. © 1999 John Wiley & Sons, In. J Graph Theory 30: 223–234, 1999     

Efficient local structure‐preserving schemes for the RLW‐type equation

The multisymplectic schemes have been used in numerical simulations for the RLW‐type equation successfully. They well preserve the local geometric property, but not other local conservation laws. In this article, we propose three novel efficient local structure‐preserving schemes for the RLW‐type equation, which preserve the local energy exactly on any time‐space region and can produce richer information of the original problem. The schemes will be mass‐ and energy‐preserving as the equation is imposed on appropriate boundary conditions. Numerical experiments are presented to verify the efficiency and invariant‐preserving property of the schemes. Comparisons with the existing nonconservative schemes are made to show the behavior of the energy affects the behavior of the solution.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1678–1691, 2017     

Stability approach to selecting the number of principal components

Principal component analysis (PCA) is a canonical tool that reduces data dimensionality by finding linear transformations that project the data into a lower dimensional subspace while preserving the variability of the data. Selecting the number of principal components (PC) is essential but challenging for PCA since it represents an unsupervised learning problem without a clear target label at the sample level. In this article, we propose a new method to determine the optimal number of PCs based on the stability of the space spanned by PCs. A series of analyses with both synthetic data and real data demonstrates the superior performance of the proposed method.     

A Copula-Based Method to Build Diffusion Models with Prescribed Marginal and Serial Dependence

This paper investigates the probabilistic properties that determine the existence of space-time transformations between diffusion processes. We prove that two diffusions are related by a monotone space-time transformation if and only if they share the same serial dependence. The serial dependence of a diffusion process is studied by means of its copula density and the effect of monotone and non-monotone space-time transformations on the copula density is discussed. This approach provides a methodology to build diffusion models by freely combining prescribed marginal behaviors and temporal dependence structures. Explicit expressions of copula densities are provided for tractable models.     

Models for measure preserving transformations

Many recent results about the classification problem for ergodic measure preserving transformations involve global considerations about spaces of measure preserving transformations. This paper surveys recent joint work with Dan Rudolph and Benjamin Weiss in determining when various spaces of measure preserving transformations are equivalent in the sense of conjugacy preserving Borel isomorphism and in having the same generic dynamical properties.     

CHARACTERISTIC FORMS OF TRANSFORMATIONS

In this paper,the author defines boundary preserving transformations and provesthat they are homeomorphisms;defines interior preserving transformations and provesthat they usually are open imbedding;and defines the co-continuous transformations,which have not been discussed in continuous,dosed and open transformations.Thecharacteristic forms of transformations are most important in the discussion,and thereare 17 cases for homeomorphism.All spaces considered are connected T_1.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.     

Transformations of Grassmannians and automorphisms of classical groups

We study transformations preserving certain linear structure in Grassmannians and give a generalization of the Fundamental Theorem of Projective Geometry. This result is closely related to the geometrical interpretation of automorphisms of classical groups. Received: 27 September 2001.     

General Algebra and Linear Transformations Preserving Matrix Invariants

The interrelations between the theory of linear transformations preserving matrix invariants and different branches of mathematics are surveyed here. The preferences are given for those methods and motivations to study these transformations that arise from general algebra.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 83–101, 2003.     

Bi-orthogonal systems on the unit circle, regular semi-classical weights and integrable systems — II

We derive the Christoffel–Geronimus–Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities {z₁,…,z_M} the bi-orthogonal system is known to be monodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel–Geronimus–Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system — the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota–Miwa equations are derived for the τ-functions or equivalently Toeplitz determinants of the system.     

Multiple recurrence of Markov shifts and other infinite measure preserving transformations

We discuss the concept of multiple recurrence, considering an ergodic version of a conjecture of Erdős. This conjecture applies to infinite measure preserving transformations. We prove a result stronger than the ergodic conjecture for the class of Markov shifts and show by example that our stronger result is not true for all measure preserving transformations.     

Invariants of conformal transformations of almost contact metric structures

The so-called structure tensors of almost contact metric structures, which play a key role in the geometry of almost contact metric structures, are explicitly calculated. The transformations of these tensors under conformal transformations of almost contact metric structures are described. The results obtained are used to study the behavior of the most interesting classes of almost contact structures under conformal transformations.     

SUSY structures on deformed supermanifolds

We construct a geometric structure on deformed supermanifolds as a certain subalgebra of the vector fields. In the classical limit we obtain a decoupling of the infinitesimal odd and even transformations, whereas in the semiclassical limit the result is a representation of the supersymmetry algebra. In the case that the structure is mass preserving we describe all high energy corrections to this algebra.     

Phenomenologically symmetric local lie groups of transformations of the space <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript>

In this paper we define a phenomenologically symmetric local Lie group of transformations of an arbitrary-dimensional space. We take as a basis the axiom scheme of the theory of physical structures. Phenomenologically symmetric groups of transformations are nondegenerate both with respect to coordinates and to parameters. We obtain a multipoint invariant of this group of transformations and relate it with Ward quasigroups. We define a substructure of a physical structure as a certain phenomenologically symmetric subgroup of transformations. We establish a criterion for the phenomenological symmetry of the Lie group of transformations and prove the uniqueness of a structure with the minimal rank. We also introduce the notion of a phenomenologically symmetric product of physical structures.     

Normal ergodic actions

The von Neumann-Halmos theory of ergodic transformations with discrete spectrum makes use of the duality theory of locally compact abelian groups to characterize those transformations preserving a probability measure, which are defined by a rotation on a compact abelian group. We use the recently developed duality between general locally compact groups and Hopf-von Neumann algebras to characterize those actions of a locally compact group, preserving a σ-finite measure, which are defined by a dense embedding in another group. They are characterized by the property of normality, previously introduced by the author, and motivated by Mackey's theory of virtual groups. The discrete spectrum theory is readily seen to come out as the special case in which the invariant measure is finite.