Analysis of Tensor Approximation Schemes for Continuous Functions

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weights.

Convergence of Newton Raphson Method and its Variants

In Numerical Analysis and various uses, including operation testing and processing, Newton’s method may be a fundamental technique. We research the history of the methodology, its core theories, the outcomes of integration, changes, they’re worldwide actions. We consider process implementations for various groups of optimization issues, like unrestrained optimization, problems limited by equality, convex programming, and methods for interior points. Some extensions are quickly addressed (non-smooth concerns, continuous analogue, Smale’s effect, etc.), whereas some others are presented in additional depth (e.g., variations of the worldwide convergence method). The numerical analysis highlights the quicker convergence of Newton’s approach obtained with this update. This updated sort of Newton-Raphson is comparatively straightforward and reliable; it’d be more probable to converge into an answer than either the upper order strategies (4th and 6th degree) or the tactic of Newton itself. Our dissertation could be about the Convergence of the Newton-Raphson Method which is a way to quickly find an honest approximation for the basis of a real-valued function g(m) = 0. The derivation of the Newton Raphson formula, examples, uses, advantages, and downwards of the Newton Raphson Method has also been discussed during this dissertation.

A Continuous Analogue of Lattice Path Enumeration

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.

Study of nonlinear Pachpatte’s inequalities on time scales

In this paper, we develop a fundamental dynamic inequality, a generalization of comparison theorem and reproduce the proofs of some nonlinear integral Pachpatte’s inequalities by using their continuous analogue. We also unify and extend these improved integral Pachpatte’s inequalities and their corresponding discrete analogues on arbitrary time scales. The results are used to make qualitative analysis of higher order dynamic equations.

Simultaneous parameter estimation and variable selection via the logit-normal continuous analogue of the spike-and-slab prior

We introduce a Bayesian prior distribution, the logit-normal continuous analogue of the spike-and-slab, which enables flexible parameter estimation and variable/model selection in a variety of settings. We demonstrate its use and efficacy in three case studies—a simulation study and two studies on real biological data from the fields of metabolomics and genomics. The prior allows the use of classical statistical models, which are easily interpretable and well known to applied scientists, but performs comparably to common machine learning methods in terms of generalizability to previously unseen data.

Continous analogues for the binomial coefficients and the Catalan numbers

Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing these analogues can be applied to a wide variety of combinatorial sequences. As an application we develop a continuous analogue for the binomial distribution.

CUTTING A PART FROM MANY MEASURES

Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^{d}$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint.We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$ . Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geqslant 2$ , $d\geqslant 2$ , $c\geqslant d$ with $m\geqslant n(c-d)+d$ are integers, and $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^{d}$ , we prove that there exists a partition of $\mathbb{R}^{d}$ into $n$ convex pieces which equiparts the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$ , and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$ .

Continuous Description of Discrete Biological Data

The applicability of differential equations to description of integer values dynamics in bio-informatics is investigated. It is shown that a differential model may be interpreted as a continuous analogue of a stochastic flow. The method of construction of a quasi-Poisson flow on the base of multi-dimension differential equations is proposed. Mathematical correctness of the algorithm is proven. The system has been studied by a computer simulation and a discrete nature of processes has been taken into account. The proposed schema has been applied to the classical Volterra's models, which are widely used for description of biological systems. It has been demonstrated that although behaviour of discrete and continuous models is similar, some essential qualitative and quantitative differences in their dynamics take place.