Abstract:
The main idea of the present paper is to compute the spectrum and the fine spectrum of the generalized difference operator over the sequence spaces . The operator denotes a triangular sequential band matrix defined by with for , where or , ; the set nonnegative integers and is either a constant or strictly decreasing sequence of positive real numbers satisfying certain conditions. Finally, we obtain the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of the operator over the sequence spaces and . These results are more general and comprehensive than the spectrum of the difference operators , , , , and and include some other special cases such as the spectrum of the operators , , and over the sequence spaces or . 1. Introduction, Preliminaries, and Definitions In analysis, operator theory is one of the important branch of mathematics which has vast applications in the field applied science and engineering. Operator theory deals with the study related to different properties of operators such as their inverse, spectrum, and fine spectrum. Since the spectrum of a bounded linear operator generalizes the notion of eigen values of the corresponding matrix, therefore, the study of spectrum of an operator takes a prominent position in solving many scientific and engineering problems. Hence, mathematicians and researchers have devoted their works in achieving new ideas and concepts in the concerned field. For instance, the fine spectrum of the Cesàro operator on the sequence space for has been studied by Gonzalez [1]. Okutoyi [2] computed the spectrum of the Cesàro operator over the sequence space . The fine spectra of the Cesàro operator over the sequence space have been determined by Akhmedov and Ba？ar [3]. Akhmedov and Ba？ar [4, 5] have studied the fine spectrum of the difference operator over the sequence spaces and , where . Altay and Ba？ar [6] have determined the fine spectrum of the difference operator over the sequence spaces , for . The fine spectrum of the difference operator over the sequence spaces and was investigated by Kayaduman and Furkan [7]. Srivastava and Kumar [8] have examined the fine spectrum of the generalized difference operator over the sequence space . Recently, the spectrum of the generalized difference operator over the sequence spaces and has been studied by Dutta and Baliarsingh [9, 10], respectively. The main focus of this paper is to define the difference operator and establish its spectral characterization with respect to the Goldberg’s classifications. Let be either constant or strictly

Abstract:
In the present work the generalized weighted mean difference operator has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences and . For any two sequences and of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator is defined by with for all . Furthermore, we compute the spectrum and the fine spectrum of the operator over the sequence space . In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space . 1. Introduction, Preliminaries, and Definitions Let and be two bounded sequences of either constant or strictly decreasing positive real numbers such that and for all , and By and , we denote the spaces of all absolutely summable and p-bounded variation series, respectively. Also, by , , and , we denote the spaces of all bounded, convergent, and null sequences, respectively. The main perception of this paper is to introduce the weighted mean difference operator as follows. Let be any sequence in , and we define the weighted mean difference transform of by where denotes the set of nonnegative integers and we assume throughout that any term with negative subscript is zero. Instead of writing (3), the operator can be expressed as a lower triangular matrix , where Equivalently, in componentwise the triangle can be represented by The main objective of this paper is to determine the spectrum of the operator over the basic sequence space . The operator has been studied by Polat et al. [1] in detail by introducing the difference sequence spaces , , and . In the existing literature several researchers have been actively engaged in finding the spectrum and fine spectrum of different bounded linear operators over various sequence spaces. The spectrum of weighted mean operator has been studied by Rhoades [2], whereas that of the difference operator over the sequence spaces for and , has been studied by Altay and Ba？ar [3, 4]. Kayaduman and Furkan [5] have determined the fine spectrum of the difference operator over the sequence spaces and and on generalizing these results, Srivastava and Kumar [6, 7] have determined the fine spectrum of the operator over the sequence spaces and , where is a sequence of either constant or strictly deceasing sequence of reals satisfying certain conditions. Dutta and Baliarsingh [8–10] have computed the spectrum of the operator ( ) and over the sequence spaces , , and , respectively. The fine spectrum of the generalized

Abstract:
We evaluate Shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials. The Bialynicki-Birula-Mycielski (BBM) inequality has also been tested for a number of states.

Abstract:
We study the Hartree ground state of a dipolar condensate of atoms or molecules in an three-dimensional anisotropic geometry and at T=0. We determine the stability of the condensate as a function of the aspect ratios of the trap frequencies and of the dipolar strength. We find numerically a rich phase space structure characterized by various structures of the ground-state density profile.

Abstract:
We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular we derive that in any local ring R of mixed characteristic p > 0, where p is a non-zero-divisor, if I is an ideal of finite projective dimension over R and p is in I or p is a non-zero-divisor on R/I, then every minimal generator of I is a non-zero-divisor. Hence if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a non-zero-divisor in R.

Abstract:
Given a minimal set of generators $\bold{x}$ of an ideal $I$ of height d in a regular local ring ($R, m, k$) we prove several cases for which the map $K_d(\bold{x}; R) \otimes k \to \Tor_d^R (R/I, k)$ is the 0-map. As a consequence of the order ideal conjecture we derive several cases for which $K_{d+i}(\bold{x}; R) \otimes k \to \Tor_{d+i}^R (R/I, k)$ are 0-maps for $i \ge 0$.

Abstract:
In this article first we prove that a special case of the order ideal conjecture, originating from the work of Evans and Griffith in equicharacteristic, implies the monomial conjecture due to M. Hochster. We derive a necessary and sufficient condition for the validity of this special case in terms certain syzygis of canonical modules of normal domains possessing free summands. We also prove some special cases of this observation.

Abstract:
Let $I$ be an ideal of height $d$ in a regular local ring $(R,m,k=R/m)$ of dimension $n$ and let $\Omega$ denote the cannonical module of $R/I$. In this paper we first prove the equivalence of the following: the non-vanishing of the edge homomorhpism $\eta_d: \ext{R}{n-d}{k,\Omega} \rightarrow \ext{R}{n}{k,R}$, the validity of the order ideal conjecture for regular local rings, and the validity of the monomial conjecture for all local rings. Next we prove several special cases of the order ideal concjecture/monomial conjecture.

Abstract:
Mathematical models in seismo-geochemical monitoring offer powerful tools for the study and exploration of complex dynamics associated with discharge of radon as the indicator of change of intense-deformed conditions of seismogenic layers or blocks within the lithosphere. Seismic precursory model of radon gas emanation in the process of earthquake prediction research aims to find out the distinct anomaly variation necessary to correlate radon gas with processes of preparation and realization of tectonic earthquakes in long-term and short-term forecasts tectonic earthquakes. The study involves a radon gas volume analytic model to find the correlation of radon fluctuations to stress drop under compression and dilatation strain condition. Here, we present a mathematical inference by observing radon gas emanation prior to the occurrence of earthquake that may reduce the uncertainties in models and updating their probability distributions in a Bayesian deterministic model. Using Bayesian melding theorem, we implement an inferential framework to understand the process of preparation of tectonic earthquake and concurrent occurrence of radon discharge during a tectonic earthquake phenomena. Bayesian melding for deterministic simulation models was augmented to make use of prior knowledge on correlations between model inputs. The background porosity is used as a priori information for analyzing the block subjected to inelastic strain. It can be inferred that use of probabilistic framework involving exhalation of radon may provide a scenario of earthquake occurrences on recession of the curve that represents a qualitative pattern of radon activity concentration drop, indicating associated stress change within the causative seismogenic fault. Using evidence analysis, we propose a joint conditional probability framework model simulation to understand how a single fracture may be affected in response to an external load and radon anomaly change that can be used to detect the slip, a predictable nature of the causative fault in the subsurface rock.

Abstract:
In this paper, we define and study \emph{quantum cyclic codes}, a generalisation of cyclic codes to the quantum setting. Previously studied examples of quantum cyclic codes were all quantum codes obtained from classical cyclic codes via the CSS construction. However, the codes that we study are much more general. In particular, we construct cyclic stabiliser codes with parameters $[[5,1,3]]$, $[[17,1,7]]$ and $[[17,9,3]]$, all of which are \emph{not} CSS. The $[[5,1,3]]$ code is the well known Laflamme code and to the best of our knowledge the other two are new examples. Our definition of cyclicity applies to non-stabiliser codes as well; in fact we show that the $((5,6,2))$ nonstabiliser first constructed by Rains\etal~ cite{rains97nonadditive} and latter by Arvind \etal~\cite{arvind:2004:nonstabilizer} is cyclic. We also study stabiliser codes of length $4^m +1$ over $\mathbb{F}_2$ for which we define a notation of BCH distance. Much like the Berlekamp decoding algorithm for classical BCH codes, we give efficient quantum algorithms to correct up to $\floor{\frac{d-1}{2}}$ errors when the BCH distance is $d$.