Constrained quantization for the Cantor distribution

In this paper, we generalize the notion of unconstrained quantization of the classical Cantor distribution to constrained quantization and give a general definition of constrained quantization. Toward this, we calculate the optimal sets of $n$-points, $n$th constrained quantization errors, the constrained quantization dimensions, and the constrained quantization coefficients taking different families of constraints for all $n\in \mathbb N$. The results in this paper show that both the constrained quantization dimension and the constrained quantization coefficient for the Cantor distribution depend on the underlying constraints. It also shows that the constrained quantization coefficient for the Cantor distribution can exist and be equal to the constrained quantization dimension. These facts are not true in the unconstrained quantization for the Cantor distribution.Comment: arXiv admin note: text overlap with arXiv:2305.1111

Constrained quantization for the Cantor distribution with a family of constraints

In this paper, for a given family of constraints and the classical Cantor distribution we determine the optimal sets of n-points, nth constrained quantization errors for all positive integers n. We also calculate the constrained quantization dimension and the constrained quantization coefficient, and see that the constrained quantization dimension D(P) exists as a finite positive number, but the D(P)-dimensional constrained quantization coefficient does not exist

Constrained quantization for a uniform distribution with respect to a family of constraints

In this paper, with respect to a family of constraints for a uniform probability distribution we determine the optimal sets of n-points and the nth constrained quantization errors for all positive integers n. We also calculate the constrained quantization dimension and the constrained quantization coefficient. The work in this paper shows that the constrained quantization dimension of an absolutely continuous probability measure depends on the family of constraints and is not always equal to the Euclidean dimension of the underlying space where the support of the probability measure is defined

Constrained quantization for probability distributions

In this paper, for a Borel probability measure P on a Euclidean space Rk, we extend the definitions of nth unconstrained quantization error, unconstrained quantization dimension, and unconstrained quantization coefficient, which traditionally in the literature known as nth quantization error, quantization dimension, and quantization coefficient, to the definitions of nth constrained quantization error, constrained quantization dimension, and constrained quantization coefficient. The work in this paper extends the theory of quantization and opens a new area of research. In unconstrained quantization, the elements in an optimal set are the conditional expectations in their own Voronoi regions, and it is not true in constrained quantization. In unconstrained quantization, if the support of P contains infinitely many elements, then an optimal set of n-means always contains exactly n elements, and it is not true in constrained quantization. It is known that the unconstrained quantization dimension for an absolutely continuous probability measure equals the Euclidean dimension of the underlying space. In this paper, we show that this fact is not true as well for the constrained quantization dimension. It is known that the unconstrained quantization coefficient for an absolutely continuous probability measure exists as a unique finite positive number. From work in this paper, it can be seen that the constrained quantization coefficient for an absolutely continuous probability measure can be any nonnegative number depending on the constraint that occurs in the definition of nth constrained quantization error

Discovering the roots: Uniform closure results for algebraic classes under factoring

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\ldots,x_n)$ of size $s$ we prove that each factor has size at most a polynomial in: $s$ and the degree of the squarefree part of $f$. Consequently, if $f_1$ is a $2^{\Omega(n)}$-hard polynomial then any nonzero multiple $\prod_{i} f_i^{e_i}$ is equally hard for arbitrary positive $e_i$'s, assuming that $\sum_i \text{deg}(f_i)$ is at most $2^{O(n)}$. It is an old open question whether the class of poly($n$)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial $f$ of degree $n^{O(1)}$ and formula (resp. ABP) size $n^{O(\log n)}$ we can find a similar size formula (resp. ABP) factor in randomized poly($n^{\log n}$)-time. Consequently, if determinant requires $n^{\Omega(\log n)}$ size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation $\tau$, $f(\tau\overline{x})$ completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure

Mechanical compression regulates tumor spheroid invasion into a 3D collagen matrix

Uncontrolled growth of tumor cells in confined spaces leads to the accumulation of compressive stress within the tumor. Although the effects of tension within 3D extracellular matrices on tumor growth and invasion are well established, the role of compression in tumor mechanics and invasion is largely unexplored. In this study, we modified a Transwell assay such that it provides constant compressive loads to spheroids embedded within a collagen matrix. We used microscopic imaging to follow the single cell dynamics of the cells within the spheroids, as well as invasion into the 3D extracellular matrices (EMCs). Our experimental results showed that malignant breast tumor (MDA-MB-231) and non-tumorigenic epithelial (MCF10A) spheroids responded differently to a constant compression. Cells within the malignant spheroids became more motile within the spheroids and invaded more into the ECM under compression; whereas cells within non-tumorigenic MCF10A spheroids became less motile within the spheroids and did not display apparent detachment from the spheroids under compression. These findings suggest that compression may play differential roles in healthy and pathogenic epithelial tissues and highlights the importance of tumor mechanics and invasion.Comment: 10 pages, 5 figure and 3 supplementary figure

Comparative Study of RDBMS, NOSQL and Graph Databases

The paper aims at analysis and comparison of various forms of databases particularly computer database Management System (RDBMS), Not solely SQL (NOSQL), Graph Databases. The Structured source language is employed by applications to access computer database systems containing informative during a semi declarative language whereas NOSQL databases area unit supported the key-value pairs. Graph info uses graph structures for resolution queries and to represent and store knowledge

Predicting Academic Performance: A Systematic Literature Review

The ability to predict student performance in a course or program creates opportunities to improve educational outcomes. With effective performance prediction approaches, instructors can allocate resources and instruction more accurately. Research in this area seeks to identify features that can be used to make predictions, to identify algorithms that can improve predictions, and to quantify aspects of student performance. Moreover, research in predicting student performance seeks to determine interrelated features and to identify the underlying reasons why certain features work better than others. This working group report presents a systematic literature review of work in the area of predicting student performance. Our analysis shows a clearly increasing amount of research in this area, as well as an increasing variety of techniques used. At the same time, the review uncovered a number of issues with research quality that drives a need for the community to provide more detailed reporting of methods and results and to increase efforts to validate and replicate work.Peer reviewe

Towards the integration of multiple classifier pertaining to the Student's performance prediction

Accurate predictions of students’ academic performance at early stages of the degree programme helps in identification of the weak students and enable management to take the corrective actions to prevent them from failure. Existing single classifier based predictive modelling is not easily scalable from one context to another context, Moreover, a predictive model developed for a particular course at a particular institution may not be valid for a different course at the same institution or any other institution. With this necessity, the notion of the integrated multiple classifiers for the predictions of students’ academic performance is proposed in this article. The integrated classifier consists of three complementary algorithms, namely Decision Tree, K-Nearest Neighbour, and Aggregating One-Dependence Estimators (AODE). A product of probability combining rule is employed to integrate the multiple classifiers for the prediction of academic performance of the engineering students. This approach provides a generalized solution for student performance prediction. The proposed method has been applied and compared on three student performance datasets using t-test. The proposed method is also compared with KSTAR, OneR, ZeroR, Naive Bayes, and NB tree classifiers as well as with the individual classifiers