Completely J —positive linear systems of finite order

Completely J — positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J — positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J — symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J — symmetric realizations are discussed.     

Meson correlation functions for a model with nonlocal four-fermion interaction

We study meson correlation functions for several models with a four-fermion quark interaction. We show that although the system average energy and quark condensate, as previously noted, are singular, the meson observables are finite, completely recognizable, and comparable to the experimental data on the energy scale. This permits using a wide set of Hamiltonians to model nonequilibrium states of quark and hadronic systems, which is a relevant problem in studying the physics of ultrarelativistic heavy-ion collisions. We obtain analytic expressions for the meson correlation functions in the Keldysh model.     

Analytic properties of Rényi’s invariant density

Under certain conditions a many-to-one transformation of the unit interval onto itself possesses a finite invariant ergodic measure equivalent to Lebesgue measure. The purpose of this paper is to investigate these conditions and to show how differentiable and analytic properties of the invariant density are inherited from the original transformation.     

Analytic properties of the S-matrix for interactions with Yukawa-potential tails

We give an explicit analytic expression for the S-matrix in the case of an arbitrary central interaction inside a sphere of finite radius with a Yukawa-potential tail at large distances. The method uses the completeness of the wave functions outside the finite sphere and also the unitarity and the symmetry conditions for the S-matrix. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 113–128, April, 2006.     

Theorems of Helly's Type for Partially-closed Half-spaces

In this article,under quite weaker conditions,theorems of Helly's type for (not necessarily finite) family of partially closed half-spaces are presented in both analytic and geometric forms.Examples are also provided to show that these conditions cannot be omitted in general.     

Fixed points of analytic self-mappings of Riemann surfaces

The existence of fixed points for an analytic self-mapping of a Riemann surface often permits strong conclusions about the mapping. For hyperbolic Riemann surfaces fixed point conditions that imply an analytic self-mapping is actually a conformal automorphism are given. For instance, an analytic self-mapping of a hyperbolic Riemann surface with two fixed points must be a conformal automorphism of finite order. On the other hand, for surfaces of finite genus estimates of the order of a conformal automorphism are obtained from fixed point information. For example, on a Riemann surface of genus g a conformal automorphism with 2g+3 fixed points is the identity.     

Optimality conditions and duality in continuous programming II. The linear problem revisited

This is Part II of a two-part paper; the purpose of this two-part paper is (a) to develop new concepts and techniques in the theory of infinite-dimensional programming, and (b) to obtain fruitful applications in continuous time programming. In Part II the continuous time version of Farkas' theorem developed in Part I serves as the foundation for the duality theory for a broad class of linear continuous time programming problems distinct from those previously examined. In particular, we establish duality under analytic conditions, e.g., whether the given functions are measurable or continuous, that are weaker, and algebraic conditions that are more general, than those previously imposed. The new class of problems arising from these conditions allows for several important resource allocation problems previously excluded from consideration. In addition, an assumption needed to prove the Kuhn-Tucker theorem for the nonlinear problem of Part I is shown in the linear case to be completely analogous to the well-known Slater condition utilized in finite-dimensional programming theory. An example is given that exhibits the essential role of the constraint qualification in linear continuous time programming, a result at variance with the theory in finite dimensions but consistent with other results concerning linear programs in infinite-dimensional spaces.     

Sharp critical thresholds for a class of nonlocal traffic flow models

We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.     

Limit boundary conditions for finite volume approximations of some physical problems

In the industrial context, finite volume schemes are used to compute an approximation of the solution of a system of equations set on a certain domain. When this domain is bounded, some numerical boundary conditions have to be implemented in order to complete the computation of the finite volume scheme. This is a tricky step in the elaboration of the scheme, which is still not mastered. In fact, at a closer sight, it appears that there is a deep interaction between the understanding of the physical phenomena at the boundary of the domain and the implementation of the numerical boundary conditions. Unfortunately, this link is not always completely intelligible and a reason for this lack of clarity is the fact that, whereas the continuous equation satisfied by the limit of the numerical solution is known, the boundary conditions satisfied by this very limit are not well-understood. The purpose of this paper is to clarify this point in three industrial situations of one-dimensional two-phase flows.     

Rudin orthogonality problem on the Bergman space

In this paper, we study the Rudin orthogonality problem on the Bergman space, which is to characterize those functions bounded analytic on the unit disk whose powers form an orthogonal set in the Bergman space of the unit disk. We completely solve the problem if those functions are univalent in the unit disk or analytic in a neighborhood of the closed unit disk. As a consequence, it is shown that an analytic multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift of finite multiplicity n if and only if its symbol is a constant multiple of the n-th power of a Möbius transform, which was obtained via the Hardy space theory of the bidisk in Sun et al. (2008) [10].     

Handling errors as they arise in whole-class interactions

There has been a long history of research into errors and their role in the teaching and learning of mathematics. This research has led to a change to pedagogical recommendations from avoiding errors to explicitly using them in lessons. In this study, 22 mathematics lessons were video-recorded and transcribed. A conversation analytic (CA) approach was then taken to examine how mathematical errors are treated by teachers and students when they arise in interaction. Despite pedagogical recommendations, in these interactions, errors continue to be predominantly treated as something to avoid. There is a tension between the affective aspects of managing errors in interactions and the cognitive aspects. Close examination of classroom interactions enable us to see how these tensions are managed both by teachers and students.     

Effects of boundary conditions on pattern formation in a nonlocal prey–predator model

Effects of periodic and Neumann boundary conditions on a nonlocal prey–predator model are investigated. Two types of kernel functions with finite supports are used to characterize the nonlocal interactions. These kernel functions are modified to handle the Neumann boundary condition. Numerical techniques to find the Turing and spatial-Hopf thresholds for Neumann boundary condition are also described. For a fixed range of nonlocal interaction with a given kernel function, Turing bifurcation curves corresponding to both the boundary conditions are close to each other. The same is true for the spatial-Hopf bifurcation curves too. However, the nonlinear solutions inside the Turing domain as well as spatial-Hopf domain depend on the boundary condition. Thus, boundary conditions play important roles in a nonlocal model of prey-predator interaction.     

Linear Waves and Nonlinear Wave Interactions in a Bounded Three‐Layer Fluid System

We investigate possible linear waves and nonlinear wave interactions in a bounded three‐layer fluid system using both analysis and numerical simulations. For sharp interfaces, we obtain analytic solutions for the admissible linear mode‐one parent/signature waves that exist in the system. For diffuse interfaces, we compute the overtaking interaction of nonlinear mode‐two solitary waves. Mathematically, owing to a small loss of energy to dispersive tails during the interaction, the waves are not solitons. However, this energy loss is extremely minute, and because the dispersively coupled waves in the system exhibit the three types of Lax KdV interactions, we conclude that for all intents and purposes the solitary waves exhibit soliton behavior.     

Error estimates for the finite element solution of quasilinear obstacle problems *

This paper focusses on quasilinear second order differential operators. Concerning their strong monotonicity and Lipschitz continuity sufficient conditions are stated which are both explicit and easy to show. Applying these functional analytic properties to nonlinear obstacle problems we prove error estimates in energy norm for the finite element method with

linear trial functions.     

SEPARABILITY CONDITIONS IN ACTS OVER MONOIDS

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong subact separability and complete separability. For each of these four separability conditions, we investigate which monoids have the property that all their (finitely generated) acts satisfy the condition. In particular, we prove that: all acts over a finite monoid are completely separable (and hence satisfy the other three separability conditions); all finitely generated acts over a finitely generated commutative monoid are residually finite and strongly subact separable (and hence weakly subact separable); all acts over a commutative idempotent monoid are residually finite and strongly subact separable; and all acts over a Clifford monoid are strongly subact separable.

     

Monodromy of certain Painlevé–VI transcendents and reflection groups

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ= and 2α=(2μ-1)² with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space. Oblatum 19-III-1999 & 25-XI-1999?Published online: 21 February 2000     

Modeling and simulation of the motion of nanoparticles in cylindrical capillaries allowing particle‐to‐wall interactions

The process of transporting nanoparticles at the blood vessels level stumbles upon various physical and physiological obstacles; therefore, a Mathematical modeling will provide a valuable means through which to understand better this complexity. In this paper, we consider the motion of nanoparticles in capillaries having cylindrical shapes (i.e., tubes of finite size). Under the assumption that these particles have spherical shapes, the motion of these particles reduces to the motion of their centers. Under these conditions, we derive the mathematical model, to describe the motion of these centers, from the equilibrium of the gravitational force, the hemodynamic force and the van der Waals interaction forces. We distinguish between the interaction between the particles and the interaction between each particle and the walls of the tube. Assuming that the minimum distance between the particles is large compared with the maximum radius R of the particles and hence neglecting the interactions between the particles, we derive simpler models for each particle taking into account the particles‐to‐wall interactions. At an error of order O(R) or O(R³)(depending if the particles are 'near' or 'very near' to the walls), we show that the horizontal component of each particle's displacement is solution of a nonlinear integral equation that we can solve via the fixed point theory. The vertical components of the displacement are computable in a straightforward manner as soon as the horizontal components are estimated. Finally, we support this theory with several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Underfitting and Overfitting Sets of Models Chosen by Order Selection Criteria

For a general class of order selection criteria, we establish analytic and non-asymptotic evaluations of both the underfitting and overfitting sets of selected models. These evaluations are further specified in various situations including regressions and autoregressions with finite or infinite variances. We also show how upper bounds for the misfitting probabilities and hence conditions ensuring the weak consistency can be derived from the given evaluations. Moreover, it is demonstrated how these evaluations, combined with a law of the iterated logarithm for some relevant statistic, can provide conditions ensuring the strong consistency of the model selection criterion used.     

A functional equation

The translational functional equation (1) with right-hand side analytic in a finite convex region D is investigated. It is proved that a solution exists which is analytic in a finite convex region G₁ determined by G. A corollary of this result is the existence of analytic solutions of differential-difference equations of finite order and of difference equations of infinite order with constant coefficients.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 733–742, June, 1969.     

Stability and bifurcations on a finite time interval in variational inequalities

Sufficient conditions for stability on a finite time interval are derived for a class of evolution variational inequalities with the use of Lyapunov functions and frequency-domain conditions. These inequalities are considered for Hilbert spaces and Sobolev spaces of infinite order in the sense of Yu.A. Dubinskii. We show how to use the stability result on a finite interval to characterize bifurcations. An algorithm for finding observation functionals is presented that uses an isomorphism between the algebra of pseudodifferential operators with constant coefficients whose symbols are real-analytic functions in some domain and an algebra of analytic matrix-valued functions.