### Elementary calculus in chevalley groups over rings

JPRM-Vol. 1 (2013), Issue 1, pp. 79 – 95 Open Access Full-Text PDF
Alexei Stepanov
Abstract: The article studies structure theory of Chevalley groups over commutative rings. Main results of the article are relative dilation and local-global principles. and an economic set of generators of relative elementary subgroup. These statements proved by computations with elementary unipotents (hence the title) are very important in further development of the subject. No restrictions on the ground ring or the root system $$Φ$$ are imposed except that the rank of $$Φ$$ is not less than 2. The results improve previous results in the area. The article contains a brief survey of the subject, some gaps in proofs or incorrect references are discussed. Proofs of some known related results are substantially simplified.

### Construction of middle nuclear square loops

JPRM-Vol. 1 (2013), Issue 1, pp. 72 – 78 Open Access Full-Text PDF
Amir Khan, Muhammad Shah, Asif Ali
Abstract: Middle nuclear square loops are loops satisfying $$x(y(zz)) =(xy)(zz)$$ for all $$x, y$$ and $$z$$. We construct an infinite family of nonassociative noncommutative middle nuclear square loops whose smallest member is of order 12.

### Comaximal factorization graphs in integral

JPRM-Vol. 1 (2013), Issue 1, pp. 65 – 71 Open Access Full-Text PDF
Shafiq Ur Rehman
Abstract: In [1], I. Beck introduced the idea of a zero divisor graph of a commutative ring and later in [2], J. Coykendall and J. Maney generalized this idea to study factorization in integral domains. They defined irreducible divisor graphs and used these irreducible divisor graphs to characterize UFDs. We define comaximal factorization graphs and use these graphs to characterize UCFDs defined in [3]. We also study that, in certain cases, comaximal factorization graph is formed by joining r copies of thecomplete graph $$K_m$$ with one copy of complete graph $$K_n$$ in common.

### Withdrawal and drainage of generalized second grade fluid on vertical cylinder with slip conditions

JPRM-Vol. 1 (2013), Issue 1, pp. 51 – 64 Open Access Full-Text PDF
M. Farooq, M. T. Rahim, S. Islam, A. M. Siddiqui
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.

### Weight characterization of the boundedness for the riemann-liouville discrete transform

JPRM-Vol. 1 (2013), Issue 1, pp. 34 – 50 Open Access Full-Text PDF
Alexander Meskhi, Ghulam Murtaza
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.

### Exact wiener indices of the strong product of graphs

JPRM-Vol. 1 (2013), Issue 1, pp. 18 – 33 Open Access Full-Text PDF
K. Pattabiraman
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.

### On grothendieck-lidskii trace formulas and applications to approximation properties

JPRM-Vol. 1 (2013), Issue 1, pp. 11 – 17 Open Access Full-Text PDF
Qaisar Latif
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 $$trace(u) =\sum µ_k$$.

### Weakened condition for the stability to solutions of parabolic equations with “maxima”

JPRM-Vol. 1 (2013), Issue 1, pp. 01 – 10 Open Access Full-Text PDF
D. Kolev, T. Donchev, K. Nakagawa
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.