pp a-b
pp 1-10
S Arumugam Karam Ebadi Martín Manrique
Let $G = (V,E)$ be a graph and let $f:V→ \{0, 1, 2\}$ be a function. A vertex 𝑢 is said to be protected with respect to 𝑓 if $f(u)> 0$ or $f(u)=0$ and 𝑢 is adjacent to a vertex with positive weight. The function 𝑓 is a co-Roman dominating function (CRDF) if: (i) every vertex in 𝑉 is protected, and (ii) each $v \in V$ with $f(v) > 0$ has a neighbor $u\in V$ with $f(u)=0$ such that the function $f_{vu}: V→ \{0,1,2\}$, defined by $f_{vu}(u)=1$, $f_{vu}(v)=f(v)-1$ and $f_{vu}(x)=f(x)$ for $x\in V\backslash \{u,v\}$ has no unprotected vertex. The weight of 𝑓 is $w(f)=𝛴_{v\in V} f(v)$. The co-Roman domination number of a graph 𝐺, denoted by $𝛾_{cr}(G)$, is the minimum weight of a co-Roman dominating function on 𝐺. In this paper we initiate a study of this parameter, present several basic results, as well as some applications and directions for further research. We also show that the decision problem for the co-Roman domination number is NP-complete, even when restricted to bipartite, chordal and planar graphs.
pp 11-20
Pullback and pushout crossed polymodules
In this paper, we introduce the concept of pullback and pushout crossed polymodules and we describe the construction of pullback and pushout crossed polymodules. In particular, by using the notion of fundamental relation, we obtain a crossed module from a pullback crossed polymodule.
pp 21-28
Reflexive modules with finite Gorenstein dimension with respect to a semidualizing module
Elham Tavasoli Maryam Salimi Siamak Yassemi
Let 𝑅 be a commutative Noetherian ring and let 𝐶 be a semidualizing 𝑅-module. It is shown that a finitely generated 𝑅-module 𝑀 with finite $G_{C}$-dimension is 𝐶-reflexive if and only if $M_{\mathfrak{p}}$ is $C_{\mathfrak{p}}$-reflexive for $\mathfrak{p}\in \text{Spec} (R)$ with depth $(R_{\mathfrak{p}})≤ 1$, and $G_{C_{\mathfrak{p}}} - \dim R_{\mathfrak{p}}(M_{\mathfrak{p}})≤ $ depth $(R_{\mathfrak{p}}) - 2$ for $\mathfrak{p}\in \text{Spec} (R)$ with depth $(R_{\mathfrak{p}})≥ 2$. As the ring 𝑅 itself is a semidualizing module, this result gives a generalization of a natural setting for extension of results due to Serre and Samuel (see Czech. Math. J. 62(3) (2012) 663-672 and Beiträge Algebra Geom. 50(2) (2009) 353-362). In addition, it is shown that over ring 𝑅 with $\dim R≤ n$, where 𝑛 ≥ 2 is an integer, $G_{D}-\dim_{R}(Hom_{R}(M,D))≤ n-2$ for every finitely generated 𝑅-module 𝑀 and a dualizing 𝑅-module 𝐷.
pp 29-36
Classification of non-solvable groups with a given property
Zeinab Foruzanfar Zohreh Mostaghim
In this paper, we classify the finite non-solvable groups satisfying the following property $P_{5}$: their orders of representatives are set-wise relatively prime for any 5 distinct non-central conjugacy classes.
pp 37-43
Regularity criteria for the 3D magneto-micropolar fluid equations via the direction of the velocity
We consider sufficient conditions to ensure the smoothness of solutions to 3D magneto-micropolar fluid equations. It involves only the direction of the velocity and the magnetic field. Our result extends to the cases of Navier–Stokes and MHD equations.
pp 45-55
Multiplicative perturbations of local 𝐶-semigroups
In this paper, we establish some left and right multiplicative perturbation theorems concerning local 𝐶-semigroups when the generator 𝐴 of a perturbed local 𝐶-semigroup $S(\cdot)$ may not be densely defined and the perturbation operator 𝐵 is a bounded linear operator from $\overline{D(A)}$ into 𝑅(𝐶) such that 𝐶𝐵=𝐵𝐶 on $\overline{D(A)}$, which can be applied to obtain some additive perturbation theorems for local 𝐶-semigroups in which 𝐵 is a bounded linear operator from $[D(A)]$ into 𝑅(𝐶) such that 𝐶𝐵=𝐵𝐶 on $\overline{D(A)}$. We also show that the perturbations of a (local) 𝐶-semigroup $S(\cdot)$ are exponentially bounded (resp., norm continuous, locally Lipschitz continuous, or exponentially Lipschitz continuous) if $S(\cdot)$ is.
pp 57-70
Volume inequalities for Orlicz mean bodies
Changmin Du Lujun Guo Gangsong Leng
In this paper, the Orlicz mean body $H_{𝜙}K$ of a convex body 𝐾 is introduced. Using the notion of shadow system, we establish a sharp lower estimate for the volume ratio of $H_{𝜙}K$ and 𝐾.
pp 71-77
$L_{p}$-dual affine surface area forms of Busemann-Petty type problems
Associated with the notion of $L_{p}$-intersection body which was defined by Haberl, we research $L_{p}$-dual affine surface area forms of Busemann–Petty type problems.
pp 79-102
Strongly minimal triangulations of $(S^{3}× S^{1})^{\# 3}$ and $(S^{3}× S^{1})^{\# 3}$
A triangulated 𝑑-manifold 𝐾, satisfies the inequality $\binom{f_{0}(K)-d-1}{2}≥ \binom{d+2}{2}𝛽_{1}(K;\mathbb{Z}_{2})$ for $d≥ 3$. The triangulated 𝑑-manifolds that meet the bound with equality are called tight neighbourly. In this paper, we present tight neighbourly triangulations of 4-manifolds on 15 vertices with $\mathbb{Z}_{3}$ as an automorphism group. One such example was constructed by Bagchi and Datta (Discrete Math. 311 (2011) 986-995). We show that there are exactly 12 such triangulations up to isomorphism, 10 of which are orientable.
pp 103-111
Generalization of Samuelson’s inequality and location of eigenvalues
We prove a generalization of Samuelson’s inequality for higher order central moments. Bounds for the eigenvalues are obtained when a given complex 𝑛 × 𝑛 matrix has real eigenvalues. Likewise, we discuss bounds for the roots of polynomial equations.
pp 113-125
Differential operators on Hermite Sobolev spaces
In this paper, we compute the Hilbert space adjoint 𝜕* of the derivative operator 𝜕 on the Hermite Sobolev spaces $\mathcal{S}_{q}$. We use this calculation to give a different proof of the ‘monotonicity inequality’ for a class of differential operators (𝐿, 𝐴) for which the inequality was proved in Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2(4) (2009) 515–591. We also prove the monotonicity inequality for (𝐿, 𝐴), when these correspond to the Ornstein–Uhlenbeck diffusion.
pp 127-138
Let 𝐺 be one of the classical compact, simple, centre-less, connected Lie groups of rank 𝑛 with a maximal torus 𝑇, the Lie algebra $\mathcal{G}$ and let $\{E_{i},F_{i},H_{i},i=1,\ldots,n\}$ be tha standard set of generators corresponding to a basis of the root system. Consider the adjoint-orbit space $M=\{\text{Ad}_{g}(H_{1}), g\in G\}$, identified with the homogeneous space 𝐺/𝐿 where $L=\{g\in G : \text{Ad}_{g}(H_{1})=H_{1}\}$. We prove that the coordinate functions $f_{i}(g):=𝛾_{i}(\text{Ad}_{g}(H_{1}))$, $i=1,\ldots,n$, where $\{𝛾_{1},\ldots,𝛾_{n}\}$ is basis of $\mathcal{G}'$ are `quadratically independent' in the sense that they do not satisfy any nontrivial homogeneous quadratic relations among them. Using this, it is proved that there is no genuine compact quantum group which can act faithfully on $C(M)$ such that the action leaves invariant the linear span of the above coordinate functions. As a corollary, it is also shown that any compact quantum group having a faithful action on the noncommutative manifold obtained by Rieffel deformation of 𝑀 satisfying a similar `linearity' condition must be a Rieffel-Wang type deformation of some compact group.
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.