scholarly journals Mean value theorems for a class of density-like arithmetic functions

2020 ◽  
pp. 1-15
Author(s):  
Lucas Reis
Keyword(s):  
Finite Field ◽  
Arithmetic Function ◽  
Field Extension ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Mean Value Theorem ◽  
Primitive Elements ◽  
Mean Values ◽  
Mean Value Theorems ◽  
Nonzero Mean

This paper provides a mean value theorem for arithmetic functions [Formula: see text] defined by [Formula: see text] where [Formula: see text] is an arithmetic function taking values in [Formula: see text] and satisfying some generic conditions. As an application of our main result, we prove that the density [Formula: see text] (respectively, [Formula: see text]) of normal (respectively, primitive) elements in the finite field extension [Formula: see text] of [Formula: see text] are arithmetic functions of (nonzero) mean values.

scholarly journals Generalizations of Lagrange and Cauchy Mean-Value Theorems

2010 ◽  
Vol 43 (4) ◽  
Author(s):  
Janusz Matkowski
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Value Theorems

AbstractSome generalizations of the Lagrange Mean-Value Theorem and Cauchy Mean-Value Theorem are proved and the extensions of the corresponding classes of means are presented.

scholarly journals On some mean value theorems of the differential calculus

1971 ◽  
Vol 5 (2) ◽  
pp. 227-238 ◽  
Author(s):  
J.B. Diaz ◽  
R. Výborný
Keyword(s):  
Differential Calculus ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Value Theorems ◽  
Partial Converse ◽  
Generalized Mean ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Special Case

A general mean value theorem, for real valued functions, is proved. This mean value theorem contains, as a special case, the result that for any, suitably restricted, function f defined on [a, b], there always exists a number c in (a, b) such that f(c) − f(a) = f′(c)(c−a). A partial converse of the general mean value theorem is given. A similar generalized mean value theorem, for vector valued functions, is also established.

Numerical Analysis of a Strip With a Hole Subjected to Cyclic Loading

1990 ◽  
Vol 112 (4) ◽  
pp. 471-476
Author(s):  
H. Ishikawa ◽  
K. Sasaki
Keyword(s):  
Numerical Analysis ◽  
Cyclic Loading ◽  
Cyclic Load ◽  
Mean Value ◽  
304 Stainless Steel ◽  
Mean Values ◽  
Nonzero Mean ◽  
Number Of Cycles ◽  
Type 304 ◽  
Type 304 Stainless Steel

This paper deals with the problem of a strip with a hole subjected to cyclic loading at the ends. For the numerical analysis, a constitutive model incorporating the motion of the center of the yield surface is implemented in a finite element method. The distributions of strain and stress during cyclic loading are discussed in detail both for the zero and nonzero mean values of cyclic load. To verify the results of the computer simulation, an experiment on a strip with a hole of type 304 stainless steel is carried out. Results of the experiment show that during cyclic loading with nonzero mean value of cyclic load, strain at the bottom of the circular hole is ratchetted with increase in number of cycles, and the stress approaches that due to the zero mean value of cyclic load.

scholarly journals On the fractional probabilistic Taylor's and mean value theorems

2016 ◽  
Vol 19 (4) ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Alessandra Meoli
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Value Theorems

AbstractIn order to develop certain fractional probabilistic analogues of Taylor's theorem and mean value theorem, we introduce the

scholarly journals Restricted mean value theorems and the metric theory of restricted Weyl sums

2020 ◽  
Author(s):  
Changhao Chen ◽  
Igor E Shparlinski
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Square Root ◽  
Mean Values ◽  
Line Segments ◽  
Exceptional Sets ◽  
Natural Measure ◽  
Measure Spaces ◽  
The Mean ◽  
Almost All

Abstract We study the behaviour of Weyl sums on a subset ${\mathcal X}\subseteq [0,1)^d$ with a natural measure µ on ${\mathcal X}$. For certain measure spaces $({\mathcal X}, \mu),$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for µ-almost all points of ${\mathcal X}$ the Weyl sums satisfy the square root cancellation law. Moreover, we characterize the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $({\mathcal X}, \mu)$. We obtain general results, which we refine for some special spaces ${\mathcal X}$ such as spheres, moment curves and line segments.

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