Hilbert functions of socle ideals

In this paper, we explore a relationship between Hilbert functions and the irreducible decompositions of ideals in local rings. Applications are given to characterize the regularity, Gorensteinness, Cohen-Macaulayness and sequentially Cohen-Macaulayness of local rings.Comment: arXiv admin note: substantial text overlap with arXiv:1504.0604

Face Alignment Using Active Shape Model And Support Vector Machine

The Active Shape Model (ASM) is one of the most popular local texture models for face alignment. It applies in many fields such as locating facial features in the image, face synthesis, etc. However, the experimental results show that the accuracy of the classical ASM for some applications is not high. This paper suggests some improvements on the classical ASM to increase the performance of the model in the application: face alignment. Four of our major improvements include: i) building a model combining Sobel filter and the 2-D profile in searching face in image; ii) applying Canny algorithm for the enhancement edge on image; iii) Support Vector Machine (SVM) is used to classify landmarks on face, in order to determine exactly location of these landmarks support for ASM; iv)automatically adjust 2-D profile in the multi-level model based on the size of the input image. The experimental results on Caltech face database and Technical University of Denmark database (imm_face) show that our proposed improvement leads to far better performance.Comment: 11 pages and 11 figure

Asymptotic Behaviour of Parameter Ideals in Generalized Cohen-Macaulay Modules

The purpose of this paper is to give affirmative answers to two open questions as follows. Let (R, \m) be a generalized Cohen-Macaulay Noetherian local ring. Both questions, the first question was raised by M. Rogers \cite {R} and the second one is due to S. Goto and H. Sakurai \cite {GS1}, ask whether for every parameter ideal \q contained in a high enough power of the maximal ideal \m the following statements are true: (1) The index of reducibility N_R(\q;R) is independent of the choice of \q; and (2) I^2=\q I, where I=\q:_R\m.Comment: 12 page

Parametric Decomposition of Powers of Parameter Ideals and Sequentially Cohen-Macaulay Modules

Let $M$ be a finitely generated module of dimension $d$ over a Noetherian local ring (R,\m) and \q the parameter ideal generated by a system of parameters \x = (x_1,..., x_d) of $M$. For each positive integer $n$, set $$\Lambda_{d,n}=\{\alpha =(\alpha_1,...,\alpha_d)\in\Bbb{Z}^d|\alpha_i\geqslant 1, \forall 1\leqslant i\leqslant d \text{and} \sum\limits_{i=1}^d\alpha_i=d+n-1\}$$ and \qa = (x_1^{\alpha_1},...,x_d^{\alpha_d}). Then we prove in this note that $M$ is a sequentially Cohen-Macaulay module if and only if there exists a certain system of parameters \x such that the equality \q^nM=\pd holds true for all $n$. As an application of this result, we can compute the Hilbert-Samuel polynomial of a sequentially Cohen-Macaulay module with respect to certain parameter idealsComment: 10 page

On the index of reducibility in Noetherian modules

Let $M$ be a finitely generated module over a Noetherian ring $R$ and $N$ a submodule. The index of reducibility ir$_M(N)$ is the number of irreducible submodules that appear in an irredundant irreducible decomposition of $N$ (this number is well defined by a classical result of Emmy Noether). Then the main results of this paper are: (1) $\mathrm{ir}_M(N) = \sum_{{\frak p} \in \mathrm{Ass}_R(M/N)} \dim_{k(\frak p)} \mathrm{Soc}(M/N)_{\frak p}$; (2) For an irredundant primary decomposition of $N = Q_1 \cap \cdots \cap Q_n$, where $Q_i$ is $\frak p_i$-primary, then $\mathrm{ir}_M(N) = \mathrm{ir}_M(Q_1) + \cdots + \mathrm{ir}_M(Q_n)$ if and only if $Q_i$ is a $\frak p_i$-maximal embedded component of $N$ for all embedded associated prime ideals $\frak p_i$ of $N$; (3) For an ideal $I$ of $R$ there exists a polynomial $\mathrm{Ir}_{M,I}(n)$ such that $\mathrm{Ir}_{M,I}(n)=\mathrm{ir}_M(I^nM)$ for $n\gg 0$. Moreover, $\mathrm{bight}_M(I)-1\le \deg(\mathrm{Ir}_{M,I}(n))\le \ell_M(I)-1$; (4) If $(R, \frak m)$ is local, $M$ is Cohen-Macaulay if and only if there exist an integer $l$ and a parameter ideal $\frak q$ of $M$ contained in $\frak m^l$ such that $\mathrm{ir}_M({\frak q}M)=\dim_k\mathrm{Soc}(H^d_{\frak m}(M))$, where $d=\dim M$.Comment: 14 pages, To appear in J. Pure Appl. Algebr

On a new invariant of finitely generated modules over local rings

Let $M$ be a finitely generated module on a local ring $R$ and \F: M_0\subset M_1\subset...\subset M_t=M a filtration of submodules of $M$ such that $d_o<d_1< ... <d_t=d$, where $d_i=\dim M_i$. This paper is concerned with a non-negative integer $p_\mathcal F(M)$ which is defined as the least degree of all polynomials in $n_1, ..., n_d$ bounding above the function $$\ell(M/(x_1^{n_1}, ..., x_d^{n_d})M)-\sum_{i=0}^tn_1...n_{d_i}e(x_1,..., x_{d_i};M_i).$$ We prove that $p_\mathcal F(M)$ is independent of the choices of good systems of parameters $\underline x=x_1, ..., x_d$. When \F is the dimension filtration of $M$ we also present some relations between p_\F(M) and the polynomial type of each $M_i/M_{i-1}$ and the dimension of the non-sequentially Cohen-Macaulay locus of $M$.Comment: To appear in the Journal of Algebra and its Application

When are the Rees algebras of parameter ideals almost Gorenstein graded rings?

Let $A$ be a Cohen-Macaulay local ring with $\operatorname{dim} A = d\ge 3$, possessing the canonical module ${\mathrm K}_A$. Let $a_1, a_2, \ldots, a_r$ $(3 \le r \le d)$ be a subsystem of parameters of $A$ and set $Q= (a_1, a_2, \ldots, a_r)$. It is shown that if the Rees algebra ${\mathcal R}(Q)$ of $Q$ is an almost Gorenstein graded ring, then $A$ is a regular local ring and $a_1, a_2, \ldots, a_r$ is a part of a regular system of parameters of $A$.Comment: 9 page

On Hilbert coefficients and sequentially Cohen-Macaulay rings

In this paper, we explore the relation between the index of reducibility and the Hilbert coefficients in local rings. Consequently, the main result of this study provides a characterization of a sequentially Cohen-Macaulay ring in terms of its Hilbert coefficients for non-parameter ideals. As corollaries to the main theorem, we obtain characterizations of a Gorenstein/Cohen-Macaulay ring in terms of its Chern coefficients for parameter ideals.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1508.0280

Pseudo-Frobenius numbers versus defining ideals in numerical semigroup rings

The structure of the defining ideal of the semigroup ring $k[H]$ of a numerical semigroup $H$ over a field $k$ is described, when the pseudo-Frobenius numbers of $H$ are multiples of a fixed integer

Techniques to Leverage Potential-based Mechanisms in Human-centered Sensing Systems

Human-centered sensing systems have grown up fast, with more than 1 billion devices and $250 billion in sales revenue. These systems face common challenges in fine-tuning the sensing methods to be robust, scalable, and integrable. Thus, we have explored the potential-based sensing fundamentals where the human body and its connected devices can be modelled as a circuit of either resistor, capacitor, or inductor. Any small changes in this circuit can be detected with the corresponding potential-based sensors. The potential-based mechanisms are simple in integration and have a lot of room to improve in perspective of performance and system efficiency. I propose a body of work that explores the combination of the analysis of devices, human anatomy and physiology; and three approaches to leverage the fundamentals of the potential-based sensors in different applications: (1) muscle analysis and capacitive sensing in human-computer interactions, (2) autonomous nervous system analysis and impedance sensing in mobile healthcare, and (3) afferent nerve analysis and resistive sensing in sensorized prosthetic hand. The expected impact of this work is to provide researchers and engineers with the capabilities to utilize the research methodology and the fundamentals of potential-based sensing methods to enrich our understanding of human and their connected devices.</p