scholarly journals Forms, functional calculus, cosine functions and perturbation

2007 ◽  
Author(s):  
Wolfgang Arendt ◽  
Charles J. K. Batty
Keyword(s):  
Functional Calculus ◽  
Cosine Functions

scholarly journals Regularized functional calculi, semigroups, and cosine functions for pseudodifferential operators

1997 ◽  
Vol 2 (1-2) ◽  
pp. 121-136 ◽  
Author(s):  
Ralph Delaubenfels ◽  
Yansong Lei
Keyword(s):  
Pseudodifferential Operator ◽  
Functional Calculus ◽  
Pseudodifferential Operators ◽  
Explicit Representation ◽  
Stieltjes Transform ◽  
Strongly Continuous Groups ◽  
Regularized Semigroups ◽  
Cosine Functions

LetiAj(1≤j≤n)be generators of commuting bounded strongly continuous groups,A≡(A1,A2,…,An). We show that, whenfhas sufficiently many polynomially bounded derivatives, then there existk,r>0such thatf(A)has a(1+|A|2)−r-regularizedBCk(f(Rn))functional calculus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, whenf(Rn)⫅R, then, for appropriatek,r,t↦(1−it)−ke−itf(A)(1+|A|2)−ris a Fourier-Stieltjes transform, and whenf(Rn)⫅[0,∞), thent↦(1+t)−ke−tf(A)(1+|A|2)−ris a Laplace-Stieltjes transform. WithA≡i(D1,…,Dn),f(A)is a pseudodifferential operator onLp(Rn)(1≤p<∞)orBUC(Rn).

Properties of the Local Functional Calculus

2002 ◽  
Vol 102 (2) ◽  
pp. 215-225
Author(s):  
Teresa Bermύdez ◽  
Manuel González ◽  
Antonio Martinόn
Keyword(s):  
Functional Calculus ◽  
Local Functional

Simulation of continuous random process according to the specified sequence of extremes

2020 ◽  
Vol 86 (7) ◽  
pp. 65-71
Author(s):  
I. V. Gadolina ◽  
R. I. Zainetdinov ◽  
T. P. Gryzlova ◽  
I. M. Petrova
Keyword(s):  
Spectral Density ◽  
Random Process ◽  
Continuous Process ◽  
Continuous Processes ◽  
Regression Formula ◽  
Bending Stresses ◽  
Development And Validation ◽  
Irregular Loading ◽  
Cosine Functions ◽  
Maximum Minimum

A method has been developed for converting a discrete sequence of extrema into a continuous process. The relevancy of the problem is attributed to the necessity of an approximate estimation of spectral density in in testing materials and structures under random (irregular) loading. A great bulk of available experimental data thus can be used in development and validation of calculation methods for assessing durability in the multi-cycle region. Postulating the continuity of random stress processes and their first derivative we propose to connect piecewise the available starting points (namely, the extrema of the random process) with half-cosine functions under the condition of compatibility at the points of extrema. A distinctive feature of the method is the provision of 100% coincidence of the values and sequences of extrema in the initial discrete and simulated continuous processes. The issue of choosing the magnitude of half-periods for these half-cosine functions is addressed on the basis of information obtained from the analysis of real stress records in the form of a regression equation linking half-periods and half-ranges for some realizations of the random process for transport vehicles. The regression dependences of the half-periods and semi-ranges of bending stresses (part of a railway train) and torsion (torsion shaft of a tracked vehicle) are shown as an example. An analysis of the correlation of two random variables (half-periods and half-ranges) according to empirical data has shown that the correlation exists and is significant for the observed number of points thus providing the basis for using the regression formula for an approximate choice of the frequency composition of the process. Moreover, the lower restrictions are imposed on the number of points (at least 5) in the half-period. Since the extrema of the initial and simulated processes coincide in accordance with the principle of the proposed simulation, the distribution of the amplitudes of complete cycles, as well as the results of schematization by other known methods are identical, therefore, the estimate of the durability by hypotheses based on a linear one is also identical. The validation of the method consists in consideration of the chain: 1) the initial continuous process; 2) the discrete process of extrema; 3) simulated continuous process according to the proposed method. Auxiliary distributions, such as distributions of maximum, minimum and average cycle values also coincide in accordance with the principle of modeling. The method is proposed to be used in analysis of the comparability of two competing approaches in assessing the loading in the problems of assessing durability, namely: those that use cycle-counting methods and methods based on the spectral density of processes. Since the spectral densities of the processes can differ due to an approximate choice of the frequencies on the basis of a regression formula, methods on their base can give estimates of the durability that differ from those obtained by schematization methods. To study this phenomenon, further computational experiments are required. The developed method can be very useful for the experiment design.

Some Mellin-type Definite Integrals Involving Sine and Cosine Functions

2019 ◽  
Vol 10 (1) ◽  
pp. 222-237
Author(s):  
M. I. Qureshi ◽  
Kaleem A. Quraishi ◽  
Dilshad Ahamad
Keyword(s):  
Definite Integrals ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Improved PSO algorithm based on cosine functions and its simulation

2013 ◽  
Vol 33 (2) ◽  
pp. 319-322
Author(s):  
Min ZHANG ◽  
Qiang HUANG ◽  
Zhouzhao XU ◽  
Baizhuang JIANG
Keyword(s):  
Pso Algorithm ◽  
Cosine Functions ◽  
Improved Pso

scholarly journals The spectral theorem for well-bounded operators

1993 ◽  
Vol 54 (3) ◽  
pp. 334-351 ◽  
Author(s):  
Ian Doust ◽  
Qiu Bozhou
Keyword(s):  
Functional Calculus ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Compact Interval ◽  
Spectral Theorem ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

A local functional calculus

1986 ◽  
Vol 9 (2) ◽  
pp. 218-236 ◽  
Author(s):  
Paul McGuire
Keyword(s):  
Functional Calculus ◽  
Local Functional
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