On the determination of the coefficients in the viscoelasticity equations

For the integrodifferential viscoelasticity equations, we study the problem of determining the coefficients of the equations and the kernels occurring in the integral terms of the system of equations. The density of the medium is assumed to be given. We suppose that the inhomogeneity support of the sought functions is included in some compact domain B ₀. We consider a series of inverse problems in which an impulse source is concentrated at the points y of the boundary of B ₀. The point y is the parameter of the problem. The given information about the solution is the trace of the solution to the Cauchy problem with zero initial data. This trace is given on the boundary of B ₀ for all y ∈ ?B ₀ and for a finite time interval. The main result of the article consists in obtaining uniqueness theorems for a solution to the initial inverse problem.     

Inverse problem of simultaneous determination of two coefficients in a parabolic equation

We establish conditions for the unique existence of a solution of the inverse problem of simultaneous determination of two unknown coefficients in a parabolic equation. One of these coefficients is the leading coefficient that depends on time, and the other coefficient depends on a space variable.     

Mathematical modeling numerical determination of coefficients in some mathematical models of nonisothermal sorption dynamics

We consider inverse problems for mathematical models of sorption dynamics that allow for diffusion, heat balance, and two types of kinetics assuming temperature-dependent kinetic coefficients. Two methods for numerical solution of inverse problems are proposed. Their efficiency is investigated by computer experiments.     

Graphoanalytical method of calculating the mechanical characteristics of a material from the stress relaxation at constant strain

A graphoanalytical method of calculating the stress relaxation curves at constant strain is described. By its means the constants characterizing the mechanical properties of polymers can be easily and rapidly determined.Mekhanika Polimerov, Vol. 1, No. 6, pp. 15–19, 1965     

Inverse problem for determination of time-dependent coefficients of a parabolic equation in a free-boundary domain

We establish conditions for unique solvability of the inverse problem of determination of time-dependent coefficients of a one-dimensional parabolic equation in a free-boundary domain.     

Quadratic sums of Gaussian q-binomial coefficients and Fibonomial coefficients

The Ramanujan Journal - In this paper we evaluate quadratic sums of Gaussian q-binomial coefficients with two additional parameters. We obtain a general summation theorem using a combination of...     

Statistical interpretation of rheological equations

The author examines one of the forms of elementary micromechanisms of formation of irreversible or high-elastic strains due to bending of the polymer chains with restricted internal rotation. He describes a method for the statistical calculation of the elementary micromechanisms that develop under the influence of temperature fluctuations, stress, stress rate, and mean hydrostatic pressure and together constitute the measured macrostrain. The use of the concept of asymmetric media and the introduction of a coefficient of asymmetry, of the structural mechanical state make possible the refinement of the mathematical model of a polymer material and, in particular, permit not only a phenomenological but also a physical statistical treatment of the know Maxwell, Kelvin and Maxwell-Thomson rheological equations.Mekhanika Polimerov, Vol. 2, No. 2, pp. 197–213, 1966     

A representation of the Mayer expansion coefficients and virial coefficients

All-Union Correspondence Institute of the Foodstuffs Industry. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 2, pp. 279–289, August, 1990.     

Structure and rheological properties of polymers

The viscoelastic properties of linear flexible-chain polymers with a narrow distribution, for which M > 5M_c (M_c corresponds to the formation of a three-dimensional entanglement network), their mixtures, and concentrated solutions are examined. It is established that under the influence of deformation the polymer may undergo a transition to the rubbery state, which thus defines the limit of its flow state; this transition is also observed in mixtures and concentrated solutions of high-molecular-weight polymers with a narrow distribution. The relative simplicity of the rheological properties of linear high-molecular-weight polymers and their mixtures is determined by the sharpness of the transition to the rubbery state. It has been found that in mixtures of high-molecular-weight polymers the apparent viscosity mechanism associated with a decrease in dissipative losses on transition of the high-molecular-weight components to the rubbery state is dominant; on a broad range of molecular weights (M > M_c), and moreover for polymer solutions, the decrease in entanglement network density under the influence of deformation acquires considerable importance. It is established that the separate effect of the high-molecular-weight components on the viscoelastic properties of their mixtures contradicts the idea of a random network of macromolecular chains. Attention is drawn to the temperature dependence of the viscosity of polymers with a narrow distribution and the dynamic properties of their mixtures. Problems of theoretical and practical interest associated with the particular rheological properties of polymer systems at high deformation rates are defined.     

Dependence of Hilbert coefficients

Let M be a finitely generated module of dimension d and depth t over a Noetherian local ring (A, \({\mathfrak{m}}\)) and I an \({\mathfrak{m}}\)-primary ideal. In the main result it is shown that the last t Hilbert coefficients \({e_{d-t+1}(I,M),\ldots, e_{d}(I,M)}\) are bounded below and above in terms of the first d ? t + 1 Hilbert coefficients \({e_{0}(I,M),\ldots,e_{d-t}(I,M)}\) and d.     

Negation of binomial coefficients

We extend the concept of a binomial coefficient to all integer values of its parameters. Our approach is purely algebraic, but we show that it is equivalent to the evaluation of binomial coefficients by means of the Γ-function. In particular, we prove that the traditional rule of “negation” is wrong and should be substituted by a slightly more complex rule. We also show that the “cross product” rule remains valid for the extended definition.     

Factorisation of Littlewood-Richardson coefficients

The hive model is used to show that the saturation of any essential Horn inequality leads to the factorisation of Littlewood-Richardson coefficients. The proof is based on the use of combinatorial objects known as puzzles. These are shown not only to account for the origin of Horn inequalities, but also to determine the constraints on hives that lead to factorisation. Defining a primitive Littlewood-Richardson coefficient to be one for which all essential Horn inequalities are strict, it is shown that every Littlewood-Richardson coefficient can be expressed as a product of primitive coefficients. Precisely the same result is shown to apply to the polynomials defined by stretched Littlewood-Richardson coefficients.     

Clebsch-Gordan quantum coefficients

On the basis of the investigation of the representations of the quantum group SL(2), one finds the qanalogues of the Majumdar, Racah-Fock, and van der Waerden formulas for the Clebsch-Gordan quantum coefficients. Their symmetry and particular values are investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 168, pp. 67–84, 1988.