1,665,730 research outputs found
Local cohomology modules of invariant rings
Let be a field and let be a regular domain containing . Let be
a finite subgroup of the group of automorphisms of . We assume that is
invertible in . Let be the ring of invariants of . Let be an
ideal in . Fix .
If is Gorenstein then,
\begin{enumerate} \item $injdim_{R^G} H^i_I(R^G) \leq \dim \ Supp \
H^i_I(R^G). \item
H^j_{\mathfrak{m}}(H^i_I(R^G))\mathfrak{m}R^G.
\item
\mu_j(P, H^i_I(R^G)) = \mu_j(P^\prime, H^i_{IR}(R))P^\primeRP. \end{enumerate}
We also prove that if PR^GR^G_P\mu_j(P, H^i_I(R^G)) jc\mu_j(P, H^i_I(R^G)) = 0 j < c\mu_j(P, H^i_I(R^G)) > 0j \geq c$.Comment: Some of the results in the previous version was already known by work
of N\'{u}\~{n}ez-Betancourt. Those results have been removed in this version.
Also some new results are adde
Rings of invariants of finite groups when the bad primes exist
Let R be a ring (not necessarily with 1) and G be a finite group of
automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not
p-torsion free, is called the set of bad primes. When the ring is |G|-torsion
free, i.e., B(R, G) is empty set, the properties of the rings R and R^G are
closely connected. The aim of the paper is to show that this is also true when
B(R, G) is not empty set under natural conditions on bad primes. In particular,
it is shown that the Jacobson radical (resp., the prime radical) of the ring
R^G is equal to the intersection of the Jacobson radical (resp., the prime
radical) of R with R^G; if the ring R is semiprime then so is R^G; if the trace
of the ring R is nilpotent then the ring itself is nilpotent; if R is a
semiprime ring then R is left Goldie iff the ring R^G is so, and in this case,
the ring of G-invariants of the left quotient ring of R is isomorphic to the
left quotient ring of R^G and
im (R^G)\leq
im (R)\leq |G|
im (R^G)
Gravastars in Gravity
In this paper, we discuss some feasible features of gravastar that were
firstly demonstrated by Mazur and Mottola. It is already established that
gravastar associates the de-Sitter spacetime in its inner sector with the
Schwarzschild geometry at its exterior through the thin shell possessing the
ultra-relativistic matter. We have explored the singularity free spherical
model with a particular equation of state under the influence of
gravity, where is the Ricci scalar and is the Gauss-Bonnet term. The
interior geometry is matched with a suitable exterior using Israel formalism.
Also, we discussed a feasible solution of gravastar which describes the other
physically sustainable factors under the influence of gravity.
Different realistic characteristics of the gravastar model are discussed, in
particular, shell's length, entropy, and energy. A significant role of this
particular gravity is examined for the sustainability of gravastar model.Comment: 21 pages, 4 figure
On a Conjecture of Randi\'{c} Index and Graph Radius
The Randi\'{c} index of a graph is defined as the sum of over all edges of , where is the degree of the
vertex in . The radius of a graph is the minimum graph
eccentricity of any graph vertex in . Fajtlowicz(1988) conjectures for all connected graph . A stronger version, , is
conjectured by Caporossi and Hansen(2000) for all connected graphs except even
paths. In this paper, we make use of Harmonic index , which is defined as
the sum of over all edges of , to show that
for any graph with cyclomatic number , and
for any tree except even paths. These results improve and
strengthen the known results on these conjectures
Energy conditions in gravity
Modified gravity is one of the most favorable candidates for explaining the
current accelerating expansion of the Universe. In this regard, we study the
viability of an alternative gravitational theory, namely , by imposing
energy conditions. We consider two forms of , commonly discussed in the
literature, which account for the stability of cosmological solutions. We
construct the inequalities obtained by energy conditions and specifically apply
the weak energy condition using the recent estimated values of the Hubble,
deceleration, jerk and snap parameters to probe the viability of the
above-mentioned forms of .Comment: 10 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1011.4159, arXiv:1012.0953 by other author
Restricted size Ramsey number for versus cycles
Let , and be simple graphs. We say if for
every -coloring of the edges of there exists a monochromatic or
in . The Ramsey number is defined as , while the restricted size Ramsey number is
defined as . In this paper we determine previously unknown restricted size Ramsey
numbers for . We also give new upper bound
for even
Randi\'c index, diameter and the average distance
The Randi\'c index of a graph , denoted by , is defined as the sum
of over all edges of , where denotes the
degree of a vertex in . In this paper, we partially solve two
conjectures on the Randi\'c index with relations to the diameter
and the average distance of a graph . We prove that for any
connected graph of order with minimum degree , if
, then ; if and , and
. Furthermore, for any arbitrary real number $\varepsilon \
(0<\varepsilon<1)\delta(G)\geq \varepsilon n\frac{R(G)}{D(G)}
\geq \frac{n-3+2\sqrt 2}{2n-2}R(G)\geq \mu(G)n$.Comment: 7 page
Reliability Polynomials and their Asymptotic Limits for Families of Graphs
We present exact calculations of reliability polynomials for lattice
strips of fixed widths and arbitrarily great length with
various boundary conditions. We introduce the notion of a reliability per
vertex, where denotes
the number of vertices in and denotes the formal limit . We calculate this exactly for various families of graphs. We
also study the zeros of in the complex plane and determine exactly
the asymptotic accumulation set of these zeros , across which
is nonanalytic.Comment: 56 pages, latex, 16 figures, version to appear in J. Stat. Phy
Constructions of Gravity from Some Expansions of the Universe
Here we propose the extended modified gravity theory named as
gravity where is the Ricci scalar, is the
Gauss-Bonnet invariant and is the trace of the stress-energy
tensor. We derive the gravitational field equations in
gravity by taking least action principle. Next we construct the
in terms of , and in de Sitter as
well as power law expansion. We also construct if the
expansion follows the finite time future singulary (big rip singularity). We
investigate the energy conditions in this modified theory of gravity and
examine the validities of all energy conditions. Finally, we analyze the
stability of the constructed modified gravity.Comment: 19 pages, 18 figure
On the Roman bondage number of a graph
A Roman dominating function on a graph is a function
such that every vertex with has at
least one neighbor with . The weight of a Roman dominating
function is the value . The minimum weight of a
Roman dominating function on a graph is called the Roman domination number,
denoted by . The Roman bondage number of a graph
with maximum degree at least two is the minimum cardinality of all sets
for which . In this paper, we
first show that the decision problem for determining is NP-hard
even for bipartite graphs and then we establish some sharp bounds for and characterizes all graphs attaining some of these bounds.Comment: 15 pages, 35 reference
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