Where do Fractions Encounter their Equivalents? – Can this Encounter Take Place in Elementary-School?

The concept of equivalence class plays a significant role in the structure of Rational Numbers. Piaget taught that in order to help elementary school children develop mathematical concepts, concrete objects and concrete reflection-enhancing-activities are needed. The “Shemesh” software was specially designed for learning equivalence-classes of fractions. The software offers concrete representations of such classes, as well as activities which cannot be constructed without a computer. In a discrete Cartesian system students construct points on the grid and learn to identify each such point as a fraction-numeral (a denominator-numerator pair). The children then learn to construct sets of such points, all of which are located on a line through the origin point. They learn to identify the line with the set of its constituent equivalent fractions. Subsequently, they investigate other phenomena and constructions in such systems, developing these constructions into additional fraction concepts. These concrete constructions can be used in solving traditional fraction problems as well as in broadening the scope of fraction meaning. Fifth-graders who used “Shemesh” in their learning activities were clinically interviewed several months after the learning sessions ended. These interviews revealed evidence indicating initial actual development of the desired mathematical concepts. This revised version was published online in July 2006 with corrections to the Cover Date.     

Approximate equivalence in von Neumann algebras

One formulation of D. Voiculescu's theorem on approximate unitary equivalence is that two unital representations π and ρ of a separable C*-algebra are approximately unitarily equivalent if and only if rank οπ = rank ορ. We study the analog when the ranges of π and ρ are contained in a von Neumann algebra R, the unitaries inducing the approximate equivalence must come from R, and "rank" is replaced with "R -rank" (defined as the Murray-von Neumann equivalence of the range projection).     

Langlands duality for representations of quantum groups

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.     

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.     

When Two Circles Determine a Triangle. Discovering and Proving a Geometrical Condition in a Computer Environment

Visualization of mathematical relationships enables students to formulate conjectures as well as to search for mathematical arguments to support these conjectures. In this project students are asked to discover the sufficient and necessary condition so that two circles form the circumscribed and inscribed circle of a triangle and investigate how this condition effects the type of triangle in general and its perimeter in particular. Its open-ended form of the task is a departure from the usual phrasing of textbook’s exercises “show that…”.     

Ahlfors type estimates for perimeter measures in Carnot-Carathéodory spaces

We study the relationship between the geometry of hypersurfaces in a Carnot-Carathéodory (CC) space and the Ahlfors regularity of the corresponding perimeter measure. To this end we establish comparison theorems for perimeter estimates between an hypersurface and its tangent space, and between a CC geometry and its “tangent” Carnot group structure.     

The “free” quantum brownian particle as a non-fock linear bosonic system

We quantize the Langevin equation for a “free” Brownian particle. The corresponding linear bosonic system possesses infrared singularities and is therefore non-Fock. We construct the physical representations of fields using the generalized stationary states. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 116. No. 2, pp. 201–214. August, 1998.     

Time-dependent deformations of powers of the wave operator

Conditions under which the linear differential operators of the second order are equivalent to operators not containing “friction” (first partial derivatives) are investigated. One can construct iso-Huygens deformations for powers of the wave operator with time-dependent coefficients. The fundamental solutions of these deformations and conditions under which the Huygens principle holds are found. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 213–228.     

A problem with directional derivative in the theory of galvanomagnetic effects

We construct an asymptotics of the solution the Laplace equation in a “long” rectangle with the directional derivative given on its “long sides” and Dirichlet data on its “short sides.” By using the asymptotics, we calculate one of the integral characteristics, namely, the magnetoresistance. We obtain new formulas for the low-magnetic field magnetoresistance. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 520–532, April, 1999.     

On the nonresonance property of linear differential-algebraic systems

We consider a system of linear ordinary differential equations in which the coefficient matrix multiplying the derivative of the unknown vector function is identically singular. For systems with constant and variable coefficients, we obtain nonresonance criteria (criteria for bounded-input bounded-output stability). For single-input control systems, we consider the problem of synthesizing a nonresonant system in the stationary and nonstationary cases. An arbitrarily high unsolvability index is admitted. The analysis is carried out under assumptions providing the existence of a so-called “equivalent form” with separated “algebraic” and “differential” components.     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

Mappings preserving the area equality of hyperbolic triangles are motions

It is shown that a mapping ${\varphi: \mathfrak{A}\rightarrow \mathfrak{B}}It is shown that a mapping j: \mathfrakA? \mathfrakB{\varphi: \mathfrak{A}\rightarrow \mathfrak{B}} between models \mathfrakA{\mathfrak{A}} and \mathfrakB{\mathfrak{B}} of elementary plane hyperbolic geometry, coordinatized by Euclidean ordered fields, that maps triangles having the same area and sharing a side into triangles that have the same property, must be a hyperbolic motion onto j(\mathfrakA){\varphi(\mathfrak{A})}. The relations that Tarski and Szmielew used as primitives for geometry, the equidistance relation ≡ and the betweenness relation B are shown to be positively existentially definable in terms of the quaternary relation Δ, with Δ(abcd) standing for “the triangles abc and abd have the same area.”     

On the interaction of δ-shocks and contact vacuum states

We study the pressureless gas equations, with piecewise constant initial data. In the immediate solution, δ-shocks and contact vacuum states arise and even meet (interact) eventually. A solution beyond the “interaction” is constructed. It shows that the δ-shock will continue with the velocity it attained instantaneously before the time of interaction, and similarly, the contact vacuum state will move past the δ-shock with a velocity value prior to the interaction. We call this the “no-effect-from-interaction” solution. We prove that this solution satisfies a family of convex entropies (in the Lax’s sense). Next, we construct an infinitely large family of weak solutions to the “interaction”. Suppose further that any of these solutions satisfy a convex entropy, it is necessary and suffcient that these solutions reduce to only the “no-effect-from-interaction” solution. In [1], Bouchut constructed another entropy satisfying solution. As with other previous papers, it is obvious that it will not be sufficient that a “correct” solution satisfies a convex entropy, in a non-strictly hyperbolic conservation laws system.     

Direct problem of calculating the magnetic dipole field in a quasilayered medium

A method is developed for calculating the electromagnetic field of a magnetic dipole in a quasilayered two-dimensional medium. The quasi-three-dimensional problem is reduced to a two-dimensional problem for the Fourier-transformed electromagnetic field. An equivalent system of integral equations on the layer boundaries is obtained. This research was partially supported by the State Scientific-Technical Program “Future Information Technologies” (grant No. 0201.06.010) and the Interuniversity Scientific Program “Russian Universities: Basic Research.” Translated from Chislennye Metody v Matematicheskoi Fizike, Moscow State University, pp. 94–110, 1998.     

Relation of orbital integrals on SO(5) and PGL(2)

We relate the “Fourier” orbital integrals of corresponding spherical functions on thep-adic groups SO(5) and PGL(2). The correspondence is defined by a “lifting” of representations of these groups. This is a local “fundamental lemma” needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture.     

Free-field resolutions of unitary representations of theN=2 virasoro algebra: I1

We construct a free-field resolution of unitary representations of theN=2 superconformal algebra. The resolution is degenerated by the action of one fermionic screening operator (which generates the “gravitational descendants” in the field theory picture). The irreducible representation is singled out from the free-field space as the cohomology of this screening operator. this paper is written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 1, pp. 89–109, October, 1999.     

Linear systems on singular curves

We start this work by studying free linear systems on singular curves and related base point free linear systems on the non-singular model. We apply these results to the study of pencils of small degree on non-singular curves. We also prove a “base point free pencil trick” which holds for any (possibly) singular curve. Received: 15 June 1998     

Heteroclinic cycles in bifurcation problems withO(3) symmetry and the spherical Bénard problem

Summary It has been known since a paper of Armbruster and Chossat ([AC91]) that robust heteroclinic cycles between equilibria can bifurcate in differential systems which are invariant under the action of the groupO(3) defined as the sum of its “natural” irreducible representations of degrees 1 and 2 (i.e., of dimensions 3 and 5). Moreover, these cycles can be seen numerically in the simulation of the amplitude equations resulting from a center manifold reduction of the Bénard problem in a nonrotating spherical shell with suitable aspect ratio ([FH86]). In the present work we first generalize the results of [AC91] to the interactions of irreducible representations of degrees ℓ and ℓ+1 for any ℓ>0. Heteroclinic cycles of various types are shown to exist under certain “generic” conditions and are classified. We show in particular that these conditions are satisfied in most cases when the differential system proceeds from a ℓ, ℓ+1 mode interaction bifurcation in the spherical Bénard problem.