Comparing Comparisons: Infinite Sums vs. Partial Sums

1993 ◽  
Vol 66 (5) ◽  
pp. 326
Author(s):  
Kendall Richards
Keyword(s):  
Partial Sums ◽  
Infinite Sums

Comparing Comparisons: Infinite Sums vs. Partial Sums

1993 ◽  
Vol 66 (5) ◽  
pp. 326-327
Author(s):  
Kendall Richards
Keyword(s):  
Partial Sums ◽  
Infinite Sums

Convergence of lower records and infinite divisibility

2003 ◽  
Vol 40 (04) ◽  
pp. 865-880 ◽  
Author(s):  
Arup Bose ◽  
Sreela Gangopadhyay ◽  
Anish Sarkar ◽  
Arindam Sengupta
Keyword(s):  
Random Variable ◽  
Infinite Divisibility ◽  
Partial Sums ◽  
Necessary And Sufficient Condition ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Infinite Sums ◽  
Necessary And Sufficient ◽  
Lower Records

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

scholarly journals The infinite sums of reciprocals and the partial sums of Chebyshev polynomials

2021 ◽  
Vol 7 (1) ◽  
pp. 334-348
Author(s):  
Fan Yang ◽  
◽  
Yang Li
Keyword(s):  
Chebyshev Polynomials ◽  
Partial Sums ◽  
Floor Function ◽  
Infinite Sums

<abstract><p>In this paper, the infinite sums of reciprocals and the partial sums derived from Chebyshev polynomials are studied. For the infinite sums of reciprocals, we apply the floor function to the reciprocals of these sums to obtain some new and interesting identities involving the Chebyshev polynomials. Simultaneously, we get several identities about the partial sums of Chebyshev polynomials by the relation of two types of Chebyshev polynomials.</p></abstract>

Convergence of lower records and infinite divisibility

2003 ◽  
Vol 40 (4) ◽  
pp. 865-880 ◽  
Author(s):  
Arup Bose ◽  
Sreela Gangopadhyay ◽  
Anish Sarkar ◽  
Arindam Sengupta
Keyword(s):  
Random Variable ◽  
Infinite Divisibility ◽  
Partial Sums ◽  
Necessary And Sufficient Condition ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Infinite Sums ◽  
Necessary And Sufficient ◽  
Lower Records

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

An Application of Infinite Sums and Products Relating to Spectral Synthesis

2019 ◽  
Vol 31 (1) ◽  
pp. 1-13
Author(s):  
Melanie Henthorn-Baker
Keyword(s):  
Spectral Synthesis ◽  
Infinite Sums

On Rational Approximation of Markov Functions by Partial Sums of Fourier Series on a Chebyshev–Markov System

2020 ◽  
Vol 108 (3-4) ◽  
pp. 566-578
Author(s):  
E. A. Rovba ◽  
P. G. Potseiko
Keyword(s):  
Fourier Series ◽  
Rational Approximation ◽  
Partial Sums ◽  
Markov System

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

Using Butterfly-patterned Partial Sums to Draw from Discrete Distributions

2019 ◽  
Vol 6 (4) ◽  
pp. 1-30
Author(s):  
Guy L. Steele Jr. ◽  
Jean-Baptiste Tristan
Keyword(s):  
Discrete Distributions ◽  
Partial Sums

Fast Bilinear Algorithms for Symmetric Tensor Contractions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Edgar Solomonik ◽  
James Demmel
Keyword(s):  
Symmetric Matrix ◽  
Matrix Multiplication ◽  
Computational Cost ◽  
Symmetric Tensor ◽  
Partial Sums ◽  
Symmetric Tensors ◽  
Outer Product ◽  
Bilinear Complexity ◽  
Special Cases ◽  
Tensor Contractions

AbstractIn matrix-vector multiplication, matrix symmetry does not permit a straightforward reduction in computational cost. More generally, in contractions of symmetric tensors, the symmetries are not preserved in the usual algebraic form of contraction algorithms. We introduce an algorithm that reduces the bilinear complexity (number of computed elementwise products) for most types of symmetric tensor contractions. In particular, it lowers the bilinear complexity of symmetrized contractions of symmetric tensors of order {s+v} and {v+t} by a factor of {\frac{(s+t+v)!}{s!t!v!}} to leading order. The algorithm computes a symmetric tensor of bilinear products, then subtracts unwanted parts of its partial sums. Special cases of this algorithm provide improvements to the bilinear complexity of the multiplication of a symmetric matrix and a vector, the symmetrized vector outer product, and the symmetrized product of symmetric matrices. While the algorithm requires more additions for each elementwise product, the total number of operations is in some cases less than classical algorithms, for tensors of any size. We provide a round-off error analysis of the algorithm and demonstrate that the error is not too large in practice. Finally, we provide an optimized implementation for one variant of the symmetry-preserving algorithm, which achieves speedups of up to 4.58\times for a particular tensor contraction, relative to a classical approach that casts the problem as a matrix-matrix multiplication.

Sign in / Sign up

Export Citation Format

Share Document