Journal of Prime Research in Mathematics

Abstract: This paper investigates the steady thin film flows of an incompressible Generalized second grade fluid under the influence of nonisothermal effects. These thin films are considered for two different problems, namely, withdrawal and drainage problems. The governing continuity and momentum equations are converted into ordinary differential equations. These equations are solved analytically. Expressions for the velocity profile, temperature distribution, volume flux, average velocity and shear stress are obtained in both the cases. Effects of different parameters on velocity and temperature are presented graphically.

Abstract: We establish necessary and sufficient conditions on a weight sequence \({v_j}^{∞}_{j}=1\) governing the boundedness for the Riemann-Liouville discrete transform \(I_α\) from \(l^p (\mathbb{N})\) to \(l^{q}_{vj}(N)\) (trace inequality), where \(0 < α < 1\). The derived conditions are of \(D\). Adams or Maz’ya–Verbitsky (pointwise) type.

Abstract: The Wiener index, denoted by \(W(G)\), of a connected graph \(G\) is the sum of all pairwise distances of vertices of the graph, that is, \(W(G) = \frac{1}{2} \sum_{u,v∈V (G)}d(u, v)\). In this paper, we obtain the Wiener index of the strong product of a path and a cycle and strong product of two cycles.

Abstract: The purpose of this short note is to consider the questions in connection with famous the Grothendieck-Lidskii trace formulas, to give an alternate proof of the main theorem from [10] and to show some of its applications to approximation properties:
Theorem: Let \(r ∈ (0, 1]\), \(1 ≤ p ≤ 2\), \(u ∈ X^{∗}|⊗_{r,p}X\) and \(u\) admits a representation \(u=\sum \lambda_{i}x_{i}{‘} ⊗x_{i}\) with \((λi) ∈ l_r,(x_{i}^{‘})\) bounded and \((x_i) ∈ l_{p’}^{w} (X)\). If \(1/r + 1/2 − 1/p = 1\), then the system \((µ_k)\) of all eigenvalues of the corresponding operator \(\widetilde{u}\) (written according to their algebraic multiplicities), is absolutely summable and \(trace(u) =\sum µ_k\).

Abstract: A class of reaction-diffusion equations with nonlinear reaction terms perturbed with a term containing ”maxima” under initial and boundary conditions is studied. The similar problems that have no ”maxima” have been studied during the last decade by many authors. It would be of interest the standard conditions for the reaction function to be weakened in the sense that the partial derivative of the reaction function, w.r.t. the unknown, to be bounded from above by a rational function containing \((1 + t) ^{−1}\) where \(t\) is the time. When we slightly weaken the standard condition imposed on the reaction function then the solution still decays to zero not necessarily in exponential order. Then we have no exponential stability for the solution of the considered problem. We establish a criterion for the nonexponential stability. The asymptotic behavior of the solutions when \(t → +∞\) is discussed as well. The parabolic problems with ”maxima” arise in many areas as the theory of automation control, mechanics, nuclear physics, biology and ecology.

Abstract: We recall first some basic facts on weighted homogeneous functions and filtrations in the ring A of formal power series. We introduce next their analogues for weighted homogeneous diffeomorphisms and vector fields. We show that the Milnor algebra is a complete invariant for the classification of weighted homogeneous polynomials with respect to right-equivalence, i.e. change of coordinates in the source and target by diffeomorphism.