An Exposition of Fractional Spurs Resulting From Nonlinear Distortion

Quantization Noise ◽

Linear Combinations ◽

Upper Limit ◽

Comprehensive Theory ◽

Free Behavior ◽

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.

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

Electronic Journal of Linear Algebra ◽

10.13001/ela.2021.5049 ◽

2021 ◽

Vol 37 ◽

pp. 359-369

Author(s):

Marko Kostadinov

Keyword(s):

Linear Combination ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Linear Combinations ◽

Special Cases ◽

Sufficient And Necessary Conditions ◽

Alpha Beta ◽

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.

Linear combinations of polynomials with three-term recurrence

Publications de l Institut Mathematique ◽

10.2298/pim2124029t ◽

2021 ◽

Vol 110 (124) ◽

pp. 29-40

Author(s):

Khang Tran ◽

Maverick Zhang

Keyword(s):

Linear Combination ◽

Chebyshev Polynomials ◽

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.

Necessary and sufficient conditions for arbitrary linear combinations of solutions of a class of nonlinear partial differential equations to satisfy the differential equation

Nonlinear Analysis and Differential Equations ◽

10.12988/nade.2021.91133 ◽

2021 ◽

Vol 9 (1) ◽

pp. 15-23

Author(s):

Namik Yener

Keyword(s):

Differential Equation ◽

Partial Differential Equations ◽

Differential Equations ◽

Nonlinear Partial Differential Equations ◽

Sufficient Conditions ◽

Linear Combinations ◽

Partial Differential ◽

On idempotency of linear combinations of a quadratic or a cubic matrix and an arbitrary matrix

Filomat ◽

10.2298/fil1910161s ◽

2019 ◽

Vol 33 (10) ◽

pp. 3161-3185

Author(s):

Murat Sarduvan ◽

Nurgül Kalaycı

Keyword(s):

Sufficient Conditions ◽

Complex Numbers ◽

Linear Combinations ◽

Arbitrary Matrix ◽

Let A be a quadratic or a cubic n x n nonzero matrix and B be an arbitrary n x n nonzero matrix. In this study, we have established necessary and sufficient conditions for the idempotency of the linear combinations of the form aA + bB, under the some certain conditions imposed on A and B, where a, b are nonzero complex numbers.

Multivariate Convex Orderings, Dependence, and Stochastic Equality

Journal of Applied Probability ◽

10.1239/jap/1032192554 ◽

1998 ◽

Vol 35 (1) ◽

pp. 93-103 ◽

Cited By ~ 23

Author(s):

Marco Scarsini

Keyword(s):

Sufficient Conditions ◽

Random Variables ◽

Multivariate Normal ◽

Random Vectors ◽

Linear Combinations ◽

Convex Ordering ◽

Normal Distributions ◽

Necessary And Sufficient ◽

Stochastic Equality

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

Multivariate Convex Orderings, Dependence, and Stochastic Equality

Journal of Applied Probability ◽

10.1017/s0021900200014704 ◽

1998 ◽

Vol 35 (01) ◽

pp. 93-103 ◽

Cited By ~ 12

Author(s):

Marco Scarsini

Keyword(s):

Sufficient Conditions ◽

Random Variables ◽

Multivariate Normal ◽

Random Vectors ◽

Linear Combinations ◽

Convex Ordering ◽

Normal Distributions ◽

Necessary And Sufficient ◽

Stochastic Equality

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).

The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.