Hamiltonian formalism for particles with a generalized rigidity

The Hamiltonian formalism is developed for mechanical systems described by reparametrization-invariant Lagrangians dependent on the external curvatures of a world line. The complete sets of constraints are constructed for the Lagrangians quadratic in the external curvatures, the Lagrangians proportional to an arbitrary external curvature, and the Lagrangians linear in the first two curvatures. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 1, pp. 130–139, October, 1998.     

Localization of the essential spectrum of the energy operators ofn-particle quantum systems in a magnetic field

Theorems on the localization of the essential spectrum of givenSO(2) symmetry forn-particle Hamiltonians in an external magnetic field are proved. The energy operators are studied for systems of arbitrary particles in the absence of an external potential field and for systems of identical particles in an external potential field. Some of the results were announced in earlier papers [1,2].Radiophysics Research Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No.1, pp. 94–112, October, 1993.     

Mathematical modeling of drug delivery from cylindrical implantable devices

A mechanistic mathematical model applicable to the controlled dispersed‐drug release from cylindrical device such as implantable drug delivery system was derived. Analytical solutions based on the pseudosteady state approximation are derived taken account an exact external medium volume. The model prediction is accurate when the initial drug load is higher than the drug solubility in the polymer. The results obtained are compared with the analytical solutions available in the literature. The equations are corroborated by comparison with experimental profiles reported in the literature for sink conditions and non sink conditions. The evolution of concentration distribution profiles is compared for different volume of external medium. A reduction in the volume of the external solution leads to an increase in the concentration on the surface of the device, which determines decreases in the release of drug. One criterion for determining whether the volume of external solution should be considered for the prediction of drug release from cylindrical devices is established. This criterion is based on establishing a maximum percentage error allowed in the values of amount of drug released. The usefulness of the model is focused in the design of implant for controlled release of drug into a small volume of external medium of release. Copyright © 2014 John Wiley & Sons, Ltd.     

On the use of external information in social network analysis

Network analysis focuses on links among interacting units (actors). Interactions are often derived from the presence of actors at events or activities (two-mode network) and this information is coded and arranged in a typical affiliation matrix. In addition to the relational data, interest may focus on external information gathered on both actors and events. Our aim is to explore the effect of external information on the formation of ties by setting a strategy able to decompose the original affiliation matrix by linear combinations of data vectors representing external information with a suitable matrix of coefficients. This allows to obtain peculiar relational data matrices that include the effect of external information. The derived adjacency matrices can then be analyzed from the network analysis perspective. In particular, we look for groups of structurally equivalent actors obtained through clustering methods. Illustrative examples and a real dataset in the framework of scientific collaboration will give a major insight into the proposed strategy.     

Strictly passive suppression of limit cycles in direct form digital filters with saturation nonlinearity: linear matrix inequality approach

In this paper, we propose a new linear matrix inequality criterion for suppression of limit cycles in state–space direct form digital filters with saturation arithmetic and external interference via a passivity approach. The passive approach is employed to reduce the effect of external interference on the digital filter. The criterion guarantees not only asymptotic stability but also passivity from the external interference to the output vector. This criterion is in the form of linear matrix inequality; hence, it is computationally tractable. An example shows the effectiveness of the proposed criterion. Copyright © 2013 John Wiley & Sons, Ltd.     

Experimental methods of investigating the state of stress of glass-reinforced plastic shells

A method of testing glass-reinforced plastic shells under external hydrostatic pressure is described together with a procedure for determining the elastic properties of the material of which the shell is composed. Test data on the strength and stability of shells subjected to external hydrostatic pressure are presented.Leningrad. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 157–163, January–February, 1969.     

The external constraint 4 nonempty part sandwich problem

List partitions generalize list colourings. Sandwich problems generalize recognition problems. The polynomial dichotomy (NP-complete versus polynomial) of list partition problems is solved for 4-dimensional partitions with the exception of one problem (the list stubborn problem) for which the complexity is known to be quasipolynomial. Every partition problem for 4 nonempty parts and only external constraints is known to be polynomial with the exception of one problem (the 2K₂-partition problem) for which the complexity of the corresponding list problem is known to be NP-complete. The present paper considers external constraint 4 nonempty part sandwich problems. We extend the tools developed for polynomial solutions of recognition problems obtaining polynomial solutions for most corresponding sandwich versions. We extend the tools developed for NP-complete reductions of sandwich partition problems obtaining the classification into NP-complete for some external constraint 4 nonempty part sandwich problems. On the other hand and additionally, we propose a general strategy for defining polynomial reductions from the 2K₂-partition problem to several external constraint 4 nonempty part sandwich problems, defining a class of 2K₂-hard problems. Finally, we discuss the complexity of the Skew Partition Sandwich Problem.     

Description of periodic extreme gibbs measures of some lattice models on the Cayley tree

The uniqueness of the translation-invariant extreme Gibbs measure for the antiferromagnetic Potts model with an external field and the existence of an uncountable number of extreme Gibbs measures for the Ising model with an external field on the Cayley tree are proved. The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. The periodic extreme Gibbs measures, which are invariant with respect to subgroups of index 2, are constructed for the Ising model with zero external field. From these measures, the existence of an uncountable number of nonperiodic extreme Gibbs measures for the antiferromagnatic Ising model follows. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 1, pp. 109–117, April, 1997.     

Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions

Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.     

Polarization of vacuum by an external magnetic field in the (2+1)-dimensional quantum electrodynamics with a nonzero fermion density

The Green's function of the Dirac equation with an external stationary homogeneous magnetic field in the (2+1)-dimensional quantum electrodynamics (QED ₂₊₁) with a nonzero fermion density is constructed. An expression for the polarization operator in an external stationary homogenous magnetic field with a nonzero chemical potential is derived in the one-loopQED ₂₊₁ approximation. The contribution of the induced Chern—Simons term to the polarization operator and the effective Lagrangian for the fermion density corresponding to the occupation of n relativistic Landau levels in an external magnetic field are calculated. An expression of the induced Chern—Simons term in a magnetic field for the case of a finite temperature and a nonzero chemical potential is obtained. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 132–151, October, 2000.     

Statistical mechanics of relaxation processes in alternating fields

Using the nonequilibrium statistical operator method and the projection technique, we derive the system of exact relaxation equations for a quantum system interacting with an alternating external field. These equations hold in the case where some of the basic dynamical variables describing a nonequilibrium state depend explicitly on time. We obtain the exact expression for the entropy production in an alternating external field. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 1, pp. 102–112, January, 2008.     

Self‐Directed Teacher Learning in Collaborative Contexts

Two related case studies of secondary mathematics teachers examine the roles and conditions helpful in initiating, directing, and/or supporting teachers' own professional development. Using multiple data sources from school‐based and professional settings, we applied analytic induction to identify patterns of similarities and differences in teachers' experiences and developmental trajectories. Findings revealed how the teachers served as brokers in their own development process and how attitude, attention, and awareness played important roles in teacher learning. As the teachers initiated learning processes, they demonstrated awareness of broader educational contexts that support learning by identifying external policies and priorities that aligned with teachers' attitudes. They then used these external priorities as boundary objects to broker for teachers' learning, including other teachers' learning. They attended to learning needs in directing a plan for learning activities, rather than using externally mandated goals for changing practice (e.g., an external focus on improving students' tests scores). These cases suggest that in structuring professional learning, opportunities to self‐identify learning needs should be part of the process, and external supports can become more powerful when teachers play a brokering role in relating these supports to their immediate contexts. Further implications for professional development processes are explored.     

Nonlinear vibrations of elastic beams with external general visco-elastic devices

Vibration damping for slender beams is achieved by applying devices at external points. The latter consist of general single visco-elastic springpot elements. An approximate nonlinear boundary value problem is found in frequency domain that holds for moderately large vibrations or for linear beams with external nonlinear devices, both in the vicinity of primary resonances. The interaction force of a so-called springpot element is expressed as a sum of two separate forces: the first develops due to the external loading function at the device location, and the second contribution arises due to an imposed time-harmonic support excitation with no other external forces acting on the structure. Finally the nonlinear frequency response function follows from a (nonlinear) algebraic equation where the influence of the springpot element appears as isolated parameter. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global eigenvalue distribution regime of random matrices with an anharmonic potential and an external source

We consider ensembles of random Hermitian matrices with a distribution measure determined by a polynomial potential perturbed by an external source. We find the genus-zero algebraic function describing the limit mean density of eigenvalues in the case of an anharmonic potential and a diagonal external source with two symmetric eigenvalues. We discuss critical regimes where the density support changes the connectivity or increases the genus of the algebraic function and consequently obtain local universal asymptotic representations for the density at interior and boundary points of its support (in the generic cases). The investigation technique is based on an analysis of the asymptotic properties of multiple orthogonal polynomials, equilibrium problems for vector potentials with interaction matrices and external fields, and the matrix Riemann-Hilbert boundary value problem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 1, pp. 34–57, April, 2009.     

Degenerate plasma in a half-space under an external electric field

An analytic solution is found for the problem on the behavior of a collisional plasma in a variable external electric field. We elucidate the structure of the screened electric field and investigate the case where the frequency of the external field is close to the plasma oscillation frequency (resonance). We show that there are two near-surface layers where the behavior of the screened field differs essentially. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 487–502, June, 2006.     

Tracing Robert's use of representations to solve counting problems: A 16-year case study

This study describes how Robert used his external representations (formulae, notations, sketches, models, and figures) to solve progressively more challenging counting tasks over a 16-year period. In his explorations of counting tasks, Robert discovered intricate connections between solutions to problems that looked different on the surface. Using video data from Robert's problem solving, analyses of his solutions are presented that shed light on how he built new ideas from existing ideas and how he modified external representations to make new mathematical discoveries and provide justifications for his solutions.     

Thermoelastic deformation of a transversely isotropic compressed spheroid

An exact solution is presented for a static thermoelastic problem for a transversely isotropic compressed spheroid when an arbitrary temperature distribution is assigned on its surface. It is assumed that the surface of the spheroid is free of external forces. The general solution is expressed through four potential functions, each of which is harmonic in a certain coordinate system. The external and internal problems for the spheroid are investigated. The solution is constructed in the form of double series in products of trigonometric functions and associated Legendre functions.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 39–46, 1988.     

Propagation of unsteady waves in a hollow magnetoelastic cylinder in contact with liquid media

A numerical-analytic solution is constructed for the problem of magnetoelasticity for a hollow cylinder immersed in a liquid and loaded from inside by an impulse-type axisymmetric mechanical pressure. Nonconducting and compressible internal and external media have different densities and elastic moduli, with their motion described by wave equations. The hollow cylinder is assumed to be an ideal conductor, and its motion is described by a linearized system of equations of magnetoelasticity; on internal and external boundaries, the conditions of conjugation hold. The problem is solved by the method of integral Laplace transforms in the time domain, and the inverse transforms are found by numerical inversion. The solutions obtained for the bounded problem are compared with solutions for a simplified unbounded problem.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 83–87, 1987.     

Active control with delay of catastrophic motion and horseshoes chaos in a single well Duffing oscillator

In this paper, the control of escape and Melnikov chaos of an harmonically excited particle from a catastrophic (unbounded) single well φ⁴ potential is considered. In the linear limit, the range of the control gain parameter leading to good control is obtained and the effect of time delays on the control force is taken into account. The approximate critical external forcing amplitudes for catastrophe and chaos are obtained by using the energy and Melnikov methods. The control efficiency is found by analysing the behaviour of the external critical forcing amplitude of the controlled system as compared to that of the uncontrolled system.     

Hölder estimates for solutions of the Cauchy problem for the porous medium equation with external forces

We study the interior Hölder regularity problem for weak solutions of the porous medium equation with external forces. Since the porous medium equation is the typical example of degenerate parabolic equations, Hölder regularity is a delicate matter and does not follow by classical methods. Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed Hölder regularity for the model equation without external forces. DiBenedetto and Friedman showed the Hölder continuity of weak solutions with some integrability conditions of the external forces but they did not obtain the quantitative estimates. The quantitative estimates are important for studying the perturbation problem of the porous medium equation. We obtain the scale invariant Hölder estimates for weak solutions of the porous medium equations with the external forces. As a particular case, we recover the well known Hölder estimates for the linear heat equation.