scholarly journals Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 167
Author(s):  
Ke-Pao Lin ◽  
Yi-Fan Wang ◽  
Ruo-Yu Wang ◽  
Andrew Yang
Keyword(s):  
Trigonometric Polynomial ◽  
Algebraic Polynomial ◽  
Trigonometric Polynomials ◽  
Research Areas ◽  
Better Than

We face the problem to determine whether an algebraic polynomial is nonnegative in an interval the Yau Number Theoretic Conjecture and Yau Geometric Conjecture is proved. In this paper, we propose a new theorem to determine if an algebraic polynomial is nonnegative in an interval. It improves Wang-Yau Lemma for wider applications in light of Sturm’s Theorem. Many polynomials can use the new theorem but cannot use Sturm’s Theorem and Wang-Yau Lemma to judge whether they are nonnegative in an interval. New Theorem also performs better than Sturm’s Theorem when the number of terms and degree of polynomials increase. Main Theorem can be used for polynomials whose coefficients are parameters and to any interval we use. It helps us to find the roots of complicated polynomials. The problem of constructing nonnegative trigonometric polynomials in an interval is a classical, important problem and crucial to many research areas. We can convert a given trigonometric polynomial to an algebraic polynomial. Hence, our proposed new theorem affords a new way to solve this classical, important problem.

scholarly journals Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

2021 ◽  
Vol 19 (1) ◽  
pp. 1047-1055
Author(s):  
Zhihua Zhang
Keyword(s):  
Fourier Series ◽  
Trigonometric Polynomial ◽  
Algebraic Polynomial ◽  
Fourier Coefficients ◽  
Random Processes ◽  
Qualitative Theory ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Fourier Approximation ◽  
Random Differential Equations

Abstract Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

Possibilistic Clustering Methods for Interval-Valued Data

2014 ◽  
Vol 22 (02) ◽  
pp. 263-291 ◽  
Author(s):  
Bruno Almeida Pimentel ◽  
Renata M. C. R. De Souza
Keyword(s):  
Credit Card ◽  
Data Sets ◽  
Clustering Methods ◽  
Cluster Algorithms ◽  
Research Areas ◽  
Possibilistic Clustering ◽  
Membership Value ◽  
Type Data ◽  
Interval Type ◽  
Better Than

Outliers may have many anomalous causes, for example, credit card fraud, cyberintrusion or breakdown of a system. Several research areas and application domains have investigated this problem. The popular fuzzy c-means algorithm is sensitive to noise and outlying data. In contrast, the possibilistic partitioning methods are used to solve these problems and other ones. The goal of this paper is to introduce cluster algorithms for partitioning a set of symbolic interval-type data using the possibilistic approach. In addition, a new way of measuring the membership value, according to each feature, is proposed. Experiments with artificial and real symbolic interval-type data sets are used to evaluate the methods. The results of the proposed methods are better than the traditional soft clustering ones.

The average number of real zeros of a random trigonometric polynomial

1968 ◽  
Vol 64 (3) ◽  
pp. 721-730 ◽  
Author(s):  
Minaketan Das
Keyword(s):  
Trigonometric Polynomial ◽  
Mathematical Expectation ◽  
Random Variables ◽  
Trigonometric Polynomials ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Image Position ◽  
Mutually Independent

AbstractLet a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the formfor large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.

scholarly journals The Trigonometric Polynomial Like Bernstein Polynomial

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Xuli Han
Keyword(s):  
Trigonometric Polynomial ◽  
Bernstein Polynomial ◽  
Bernstein Polynomials ◽  
Trigonometric Polynomials ◽  
Polynomial Space ◽  
Polynomial Sequence ◽  
Symmetric Basis ◽  
Uniformly Convergent

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given.

scholarly journals Emerging Research Areas in Social Sciences Due to Covid-19

2020 ◽  
pp. 01-04 ◽  
Author(s):  
Muhammad Farooq
Keyword(s):  
Social Sciences ◽  
Mobile Learning ◽  
Technology Development ◽  
Public Awareness ◽  
Research Paper ◽  
Research Areas ◽  
The People ◽  
New Research ◽  
Better Than

Covid-19 has impacted all of the walks of life. It has changed the behavior of humans. Due to lockdown, people had the opportunity to rethink several perspectives of life. On ends of Lockdown, it is expected that, the customers will no more be the same, rushing towards product and reacting to every marketing advertisement. The Pandemic and Lockdown has taught the people to live with less. It also taught us, that technology development should be towards making humans life better. The term “Customer is king” has become more valuable in pandemic days. No matter, how many airlines one firm has, how much advance one economy is, if the health of humans is on danger, everything discontinues. This pandemic has taught us to focus more on creating the product, which improves humans’ life, do marketing and advertising that are customer centric. In education, more focus is required on mobile learning. The transportation dynamics have also change. The requirement of self-deriving cars has increased. The pandemic reiterated strongly that prevention is better than cure. Public awareness can sever more people than doctors' services in times of crisis. All these learning are will create new research areas in social sciences. This research paper highlights research areas for post- COVID developments.

scholarly journals Some inequalities for trigonometric polynomials

1985 ◽  
Vol 39 (2) ◽  
pp. 216-226 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Trigonometric Polynomial ◽  
Trigonometric Polynomials ◽  
Classical Inequality

AbstractWe obtain various refinements and generalizations of a classical inequality of S. N. Bernstein on trigonometric polynomials. Some of the results take into account the size of one or more of the coefficients of the trigonometric polynomial in question. The results are obtained using interpolation formulas.

scholarly journals Factorization properties of subrings in trigonometric polynomial rings

2009 ◽  
Vol 86 (100) ◽  
pp. 123-131
Author(s):  
Tariq Shah ◽  
Ehsan Ullah
Keyword(s):  
Trigonometric Polynomial ◽  
Commutative Ring ◽  
Polynomial Rings ◽  
Trigonometric Polynomials ◽  
Ring Extension ◽  
Irreducible Elements ◽  
Euclidean Domain

We explore the subrings in trigonometric polynomial rings and their factorization properties. Consider the ring S' of complex trigonometric polynomials over the field Q(i) (see [11]). We construct the subrings S'1 , S'0 of S' such that S'1 ?S'0 ?S'. Then S'1 is a Euclidean domain, whereas S'0 is a Noetherian HFD. We also characterize the irreducible elements of S'1, S'0 and discuss among these structures the condition: Let A ?B be a unitary (commutative) ring extension. For each x ? B there exist x' ?U(B) and x'' ? A such that x = x'x''. .

scholarly journals Estimation of deviation of continuous $2\pi$-periodic functions from corresponding trigonometric polynomials of S.N. Bernstein's type

2021 ◽  
pp. 43
Author(s):  
N.Ya. Yatsenko
Keyword(s):  
Periodic Function ◽  
Trigonometric Polynomial ◽  
Modulus Of Continuity ◽  
Trigonometric Polynomials ◽  
Periodic Functions

We have established the estimation of deviation of continuous $2\pi$-periodic function $f(x)$ from the trigonometric polynomial of S.N. Bernstein's type that corresponds to it, by the modulus of continuity of the function $f(x)$.

scholarly journals Gene Editing in Rabbits: Unique Opportunities for Translational Biomedical Research

2021 ◽  
Vol 12 ◽  
Author(s):  
Jie Xu ◽  
Jifeng Zhang ◽  
Dongshan Yang ◽  
Jun Song ◽  
Brooke Pallas ◽  
...  
Keyword(s):  
Animal Model ◽  
Biomedical Research ◽  
Gene Editing ◽  
Gene Knockout ◽  
Ocular Diseases ◽  
The Past ◽  
Research Areas ◽  
Large Size ◽  
Transgenic Rabbits ◽  
Better Than

The rabbit is a classic animal model for biomedical research, but the production of gene targeted transgenic rabbits had been extremely challenging until the recent advent of gene editing tools. More than fifty gene knockout or knock-in rabbit models have been reported in the past decade. Gene edited (GE) rabbit models, compared to their counterpart mouse models, may offer unique opportunities in translational biomedical research attributed primarily to their relatively large size and long lifespan. More importantly, GE rabbit models have been found to mimic several disease pathologies better than their mouse counterparts particularly in fields focused on genetically inherited diseases, cardiovascular diseases, ocular diseases, and others. In this review we present selected examples of research areas where GE rabbit models are expected to make immediate contributions to the understanding of the pathophysiology of human disease, and support the development of novel therapeutics.

An Application of a Theorem of J. Czipszer and G. Freud to a Problem of Simultaneous Approximation

1969 ◽  
Vol 12 (2) ◽  
pp. 193-201
Author(s):  
A.K. Varma
Keyword(s):  
Trigonometric Polynomial ◽  
Simultaneous Approximation ◽  
Trigonometric Polynomials

In an earlier work [12] we considered the case of (0, 2, 3) interpolation by trigonometric polynomials at the points , i = 0, 1,…, n-1. By (0, 2, 3) interpolation we mean the problem of finding a trigonometric polynomial of suitable order whose values, second and third derivatives are prescribed at some given points. An interesting distinction between the (0, 2) interpolation studied by the Hungarian mathematician O.

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