25 research outputs found
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 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 -points, th constrained quantization errors, the constrained
quantization dimensions, and the constrained quantization coefficients taking
different families of constraints for all . 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 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 is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
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 ,
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