scholarly journals Linear combinations of polynomials with three-term recurrence

2021 ◽  
Vol 110 (124) ◽  
pp. 29-40
Author(s):  
Khang Tran ◽  
Maverick Zhang
Keyword(s):  
Linear Combination ◽  
Chebyshev Polynomials ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Zero Distribution ◽  
Necessary And Sufficient ◽  
Linear Polynomials

We study the zero distribution of the sum of the first n polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of az + b for a, b ? R, of Chebyshev polynomials. In particular, we find necessary and sufficient conditions on a, b such that this linear combination is hyperbolic.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

scholarly journals A Note on Parametric Surfaces in Minkowski 3-Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Esra Betul Koc Ozturk
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Ruled Surface ◽  
Necessary And Sufficient Conditions ◽  
Frenet Frame ◽  
Parametric Surface ◽  
Parametric Surfaces ◽  
Null Curve ◽  
Necessary And Sufficient

With the help of the Frenet frame of a given pseudo null curve, a family of parametric surfaces is expressed as a linear combination of this frame. The necessary and sufficient conditions are examined for that curve to be an isoparametric and asymptotic on the parametric surface. It is shown that there is not any cylindrical and developable ruled surface as a parametric surface. Also, some interesting examples are illustrated about these surfaces.

scholarly journals An Exposition of Fractional Spurs Resulting From Nonlinear Distortion

2021 ◽  
Author(s):  
Yann Donnelly ◽  
Michael Peter Kennedy
Keyword(s):  
Nonlinear Distortion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantization Noise ◽  
Linear Combinations ◽  
Upper Limit ◽  
Comprehensive Theory ◽  
Free Behavior ◽  
Necessary And Sufficient

The interaction between quantization noise intro-duced by the divider controller and memoryless nonlinearities in a fractional-N PLL causes fractional spurs to occur. This paper presents a comprehensive theory to explain why combinations of quantizers and memoryless nonlinearities produce fractional spurs. Necessary and sufficient conditions for spur-free behavior in the presence of an arbitrary memoryless nonlinearity or linear combinations of sets of arbitrary memoryless nonlinearities are derived. Finally, an upper limit on the number of nonlinearities for which a quantizer can exhibit spur-free performance is derived.

scholarly journals Hypersurface Family with a Common Isoasymptotic Curve

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Ergin Bayram ◽  
Emin Kasap
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Frenet Frame ◽  
Necessary And Sufficient ◽  
The Given

We handle the problem of finding a hypersurface family from a given asymptotic curve in R4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.

scholarly journals An Exposition of Fractional Spurs Resulting From Nonlinear Distortion

2021 ◽  
Author(s):  
Yann Donnelly ◽  
Michael Peter Kennedy
Keyword(s):  
Nonlinear Distortion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantization Noise ◽  
Linear Combinations ◽  
Upper Limit ◽  
Comprehensive Theory ◽  
Free Behavior ◽  
Necessary And Sufficient

The interaction between quantization noise intro-duced by the divider controller and memoryless nonlinearities in a fractional-N PLL causes fractional spurs to occur. This paper presents a comprehensive theory to explain why combinations of quantizers and memoryless nonlinearities produce fractional spurs. Necessary and sufficient conditions for spur-free behavior in the presence of an arbitrary memoryless nonlinearity or linear combinations of sets of arbitrary memoryless nonlinearities are derived. Finally, an upper limit on the number of nonlinearities for which a quantizer can exhibit spur-free performance is derived.

scholarly journals On a Certain Subclass of Analytic Functions Involving Integral Operator Defined by Polylogarithm Function

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 66
Author(s):  
Danyal Soybaş ◽  
Santosh B. Joshi ◽  
Haridas Pawar
Keyword(s):  
Integral Operator ◽  
Linear Combination ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Distortion Theorem ◽  
Necessary And Sufficient Conditions ◽  
Partial Sums ◽  
Polylogarithm Function ◽  
Necessary And Sufficient

In the present paper, we have introduced a new subclass of analytic functions involving integral operator defined by polylogarithm function. Necessary and sufficient conditions are obtained for this class. Further distortion theorem, linear combination and results on partial sums are investigated.

scholarly journals Surfaces family with common smarandache asymptotic curve according to bishop frame in euclidean 3-space

2016 ◽  
Vol 34 (1) ◽  
pp. 187-202 ◽  
Author(s):  
Gulnur Saffak Atalay ◽  
Emin Kasap
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bishop Frame ◽  
The Family ◽  
Necessary And Sufficient

In this paper, we analyzed the problem of constructing a family of surfaces from a given some special Smarandache curves in  Euclidean 3-space. Using the Bishop frame of the curve in Euclidean 3-space, we express the family of surfaces as a linear combination of the components of this frame, and derive the necessary and sufficient conditions for coefficients to satisfy both the asymptotic and isoparametric requirements. Finally, examples are given to show the family of surfaces with common Smarandache asymptotic curve.

scholarly journals Surfaces family with common Smarandache asymptotic curve

2016 ◽  
Vol 34 (1) ◽  
pp. 9-20
Author(s):  
Gulnur Saffak Atalay ◽  
Emin Kasap
Keyword(s):  
Linear Combination ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Frenet Frame ◽  
The Family ◽  
Necessary And Sufficient

In this paper, we analyzed the problem of constructing a family of surfaces from a given some special Smarandache curves in Euclidean 3-space. Using the Frenet frame of the curve in Euclidean 3-space, we express the family of surfaces as a linear combination of the components of this frame, and derive the necessary and sufficient conditions for coefficients to satisfy both the asymptotic and isoparametric requirements. Finally, examples are given to show the family of surfaces with common Smarandache curve.

scholarly journals Darboux Iso-Geodesic Special Curve in Euclidean Space

2019 ◽  
Vol 13 (9) ◽  
pp. 98
Author(s):  
M. M. Wageeda ◽  
E. M. Solouma ◽  
M. Bary
Keyword(s):  
Euclidean Space ◽  
Linear Combination ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Family ◽  
Darboux Frame ◽  
Necessary And Sufficient ◽  
Special Curve

In this paper, by using Darboux frame we scrutinize the issues of reconstructing surfaces with given some unusual Smarandache curves in Euclidean 3-space, we make manifest the family of surfaces as a linear combination of the components of this frame and derive the necessary and sufficient conditions for coefficients to satisfy both the iso-geodesic and iso-parametric requirements.

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