• Volume 126, Issue 1

February 2016,   pages  1-141

• Some sufficient conditions for Hamiltonian property in terms of Wiener-type invariants

The Wiener-type invariants of a simple connected graph 𝐺 = (𝑉,𝐸) can be expressed in terms of the quantities $W_{f} = \sum_{\{u,v\}\subseteq V} f (d_{G}(u, v))$ for various choices of the function 𝑓(𝑥), where 𝑑𝐺(𝑢, 𝑣) is the distance between vertices 𝑢 and 𝑣 in 𝐺. In this paper, we give some sufficient conditions for a connected graph to be Hamiltonian, a connected graph to be traceable, and a connected bipartite graph to be Hamiltonian in terms of the Wiener-type invariants.

• Outer-2-independent domination in graphs

We initiate the study of outer-2-independent domination in graphs. An outer-2-independent dominating set of a graph 𝐺 is a set 𝐷 of vertices of 𝐺 such that every vertex of 𝑉 (𝐺)\𝐷 has a neighbor in 𝐷 and the maximum vertex degree of the subgraph induced by 𝑉 (𝐺)\𝐷 is at most one. The outer-2-independent domination number of a graph 𝐺 is the minimum cardinality of an outer-2-independent dominating set of 𝐺. We show that if a graph has minimum degree at least two, then its outer-2-independent domination number equals the number of vertices minus the 2-independence number. Then we investigate the outer-2-independent domination in graphs with minimum degree one. We also prove the Vizing-type conjecture for outer-2-independent domination and disprove the Vizing-type conjecture for outer-connected domination.

• Some identities involving convolutions of Dirichlet characters and the Möbius function

In this paper, we present some identities involving convolutions of Dirichlet characters and the Möbius function, which are related to a well known identity of Ramanujan, Hardy and Littlewood.

• On the normality of orbit closures which are hypersurfaces

Let 𝑁 be a quiver representation with non-zero admissible annihilator. In this paper, we prove the normality of the orbit closure $\bar{\mathcal{O}}_{N}$ when it is a hypersurface. The result thus gives new examples of normal orbit closures of quiver representations.

• Test elements of direct sums and free products of free Lie algebras

We give a characterization of test elements of a direct sum of free Lie algebras in terms of test elements of the factors. In addition, we construct certain types of test elements and we prove that in a free product of free Lie algebras, product of the homogeneous test elements of the factors is also a test element.

• A new characterization for some extensions of PSL(2, 𝑞) for some 𝑞 by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2, 𝑝2) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2, 𝑝2) of order 2|PSL(2, 𝑝2)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let 𝑝 be an odd prime number and 𝑞 = 𝑝 or 𝑞 = 𝑝3. Let 𝑀 be a finite group such that |𝑀| = ℎ|PSL(2, 𝑞), where ℎ is a divisor of |Out(PSL(2, 𝑞))|. Also suppose that 𝑀 has an irreducible character of degree 𝑞 and 2𝑝 does not divide the degree of any irreducible character of 𝑀. As the main result of this paper we prove that 𝑀 has a unique nonabelian composition factor which is isomorphic to PSL(2, 𝑞). As a consequence of our result we prove that 𝑀 is uniquely determined by its order and some information on its character degrees which implies that 𝑀 is uniquely determined by the structure of its complex group algebra.

• The probability that a pair of group elements is autoconjugate

Let 𝑔 and ℎ be arbitrary elements of a given finite group 𝐺. Then 𝑔 and ℎ are said to be autoconjugate if there exists some automorphism 𝛼 of 𝐺 such that ℎ = 𝑔𝛼. In this article, we construct some sharp bounds for the probability that two random elements of 𝐺 are autoconjugate, denoted by $\mathcal{P}_{a}(G)$. It is also shown that $\mathcal{P}_{a}(G)|G|$ depends only on the autoisoclinism class of 𝐺.

• Sharpened forms of the generalized Schwarz inequality on the boundary

In this paper, a boundary version of the Schwarz inequality is investigated. We obtain more general results at the boundary. If we know the second coefficient in the expansion of the function $f(z) = 1 + c_{p}z^{p} + c_{p+1}z^{p+1} \ldots$, then we obtain new inequalities of the Schwarz inequality at boundary by taking into account $c_{p+1}$ and zeros of the function 𝑓(𝑧) − 1. The sharpness of these inequalities is also proved.

• Exponential function method for solving nonlinear ordinary differential equations with constant coefficients on a semi-infinite domain

A new approach, named the exponential function method (EFM) is used to obtain solutions to nonlinear ordinary differential equations with constant coefficients in a semi-infinite domain. The form of the solutions of these problems is considered to be an expansion of exponential functions with unknown coefficients. The derivative and product operational matrices arising from substituting in the proposed functions convert the solutions of these problems into an iterative method for finding the unknown coefficients. The method is applied to two problems: viscous flow due to a stretching sheet with surface slip and suction; and mageto hydrodynamic (MHD) flow of an incompressible viscous fluid over a stretching sheet. The two resulting solutions are compared against some standard methods which demonstrates the validity and applicability of the new approach.

• Null controllability of the viscous Camassa–Holm equation with moving control

In this paper, we study the null controllability of the viscous Camassa–Holm equation on the one-dimensional torus. By using a moving distributed control, we obtain that the system is null controllable for a given data with certain regularity.

• Optimal control problem for the extended Fisher–Kolmogorov equation

In this paper, the optimal control problem for the extended Fisher–Kolmogorov equation is studied. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.

• Smooth Frechet subalgebras of 𝐶*-algebras defined by first order differential seminorms

The differential structure in a 𝐶*-algebra defined by a dense Frechet subalgebra whose topology is defined by a sequence of differential seminorms of order 1 is investigated. This includes differential Arens–Michael decomposition, spectral invariance, closure under functional calculi as well as intrinsic spectral description. A large number of examples of such Frechet algebras are exhibited; and the smooth structure defined by an unbounded self-adjoint Hilbert space operator is discussed.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 127 | Issue 4
September 2017

• # Proceedings – Mathematical Sciences | News

© 2017 Indian Academy of Sciences, Bengaluru.