scholarly journals Hyers-Ulam Stability of Lagrange’s Mean Value Points in Two Variables

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 216 ◽  
Author(s):  
Soon-Mo Jung ◽  
Ji-Hye Kim
Keyword(s):  
Efficient Algorithm ◽  
Mean Value ◽  
Two Dimensional ◽  
Ulam Stability

Using a theorem of Ulam and Hyers, we will prove the Hyers-Ulam stability of two-dimensional Lagrange’s mean value points ( η , ξ ) which satisfy the equation, f ( u , v ) − f ( p , q ) = ( u − p ) f x ( η , ξ ) + ( v − q ) f y ( η , ξ ) , where ( p , q ) and ( u , v ) are distinct points in the plane. Moreover, we introduce an efficient algorithm for applying our main result in practical use.

scholarly journals Hyers-Ulam Stability of Lagrange's Mean Value Points in Two Variables

2018 ◽  
Author(s):  
Soon-Mo Jung ◽  
Ji-Hye Kim
Keyword(s):  
Efficient Algorithm ◽  
Mean Value ◽  
Two Dimensional ◽  
Ulam Stability

Using a theorem of Ulam and Hyers, we will prove the Hyers-Ulam stability of two-dimensional Lagrange's mean value points $(\eta, \xi)$ which satisfy the equation, $f(u, v) - f(p, q) = (u-p) f_x(\eta, \xi) + (v-q) f_y(\eta, \xi)$, where $(p, q)$ and $(u, v)$ are distinct points in the plane. Moreover, we introduce an efficient algorithm for applying our main result in practical use.

scholarly journals Hyers–Ulam Stability of Two-Dimensional Flett’s Mean Value Points

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 733
Author(s):  
Soon-Mo Jung ◽  
Ji-Hye Kim ◽  
Young Woo Nam
Keyword(s):  
Differentiable Function ◽  
Line Segment ◽  
Mean Value ◽  
Two Dimensional ◽  
Ulam Stability ◽  
Intermediate Point

If a differentiable function f : [ a , b ] → R and a point η ∈ [ a , b ] satisfy f ( η ) - f ( a ) = f ′ ( η ) ( η - a ) , then the point η is called a Flett’s mean value point of f in [ a , b ] . The concept of Flett’s mean value points can be generalized to the 2-dimensional Flett’s mean value points as follows: For the different points r ^ and s ^ of R × R , let L be the line segment joining r ^ and s ^ . If a partially differentiable function f : R × R → R and an intermediate point ω ^ ∈ L satisfy f ( ω ^ ) - f ( r ^ ) = ω ^ - r ^ , f ′ ( ω ^ ) , then the point ω ^ is called a 2-dimensional Flett’s mean value point of f in L. In this paper, we will prove the Hyers–Ulam stability of 2-dimensional Flett’s mean value points.

scholarly journals An Efficient Signal Reconstruction Algorithm for Stepped Frequency MIMO-SAR in the Spotlight and Sliding Spotlight Modes

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Jiajia Zhang ◽  
Guangcai Sun ◽  
Mengdao Xing ◽  
Zheng Bao ◽  
Fang Zhou
Keyword(s):  
Efficient Algorithm ◽  
Low Cost ◽  
Signal Reconstruction ◽  
Multiple Input Multiple Output ◽  
Reconstruction Algorithm ◽  
Synthetic Aperture ◽  
Two Dimensional ◽  
Multiple Input ◽  
Input Multiple Output ◽  
Stepped Frequency

Multiple-input multiple-output (MIMO) synthetic aperture radar (SAR) using stepped frequency (SF) waveforms enables a high two-dimensional (2D) resolution with wider imaging swath at relatively low cost. However, only the stripmap mode has been discussed for SF MIMO-SAR. This paper presents an efficient algorithm to reconstruct the signal of SF MIMO-SAR in the spotlight and sliding spotlight modes, which includes Doppler ambiguity resolving algorithm based on subaperture division and an improved frequency-domain bandwidth synthesis (FBS) method. Both simulated and constructed data are used to validate the effectiveness of the proposed algorithm.

scholarly journals NEGATIVE ENERGY DENSITIES IN QUANTUM FIELD THEORY

2010 ◽  
Vol 25 (11) ◽  
pp. 2355-2363 ◽  
Author(s):  
L. H. FORD
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Energy Density ◽  
Negative Energy ◽  
Mean Value ◽  
Quantum Field ◽  
Two Dimensional ◽  
Vacuum Fluctuations ◽  
Negative Energy Density ◽  
Dimensional Spacetime

Quantum field theory allows for the suppression of vacuum fluctuations, leading to sub-vacuum phenomena. One of these is the appearance of local negative energy density. Selected aspects of negative energy will be reviewed, including the quantum inequalities which limit its magnitude and duration. However, these inequalities allow the possibility that negative energy and related effects might be observable. Some recent proposals for experiments to search for sub-vacuum phenomena will be discussed. Fluctuations of the energy density around its mean value will also be considered, and some recent results on a probability distribution for the energy density in two dimensional spacetime are summarized.

On the envelope of a Gaussian random field

1978 ◽  
Vol 15 (3) ◽  
pp. 502-513 ◽  
Author(s):  
R. J. Adler
Keyword(s):  
Random Field ◽  
Spectral Theory ◽  
Gaussian Random Field ◽  
Sample Path ◽  
Mean Value ◽  
Two Dimensional ◽  
Qualitative Information ◽  
The Mean

For homogeneous, two-dimensional random field ξ(t), t ∈ R2 we develop the ‘half' spectral theory sufficient to rigorously define its envelope η (t). We then specialise to the case of ξ Gaussian, which implies η is Rayleigh, and consider the mean value of a certain characteristic of the sets {t:η(t) ≧ u} (u ≧ 0). From this we deduce some qualitative information about the sample path behaviour of the Rayleigh field η .

scholarly journals Asymptotics for multilinear averages of multiplicative functions

2016 ◽  
Vol 161 (1) ◽  
pp. 87-101 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
BERNARD HOST
Keyword(s):  
Unit Disc ◽  
Convergence Result ◽  
Mean Value ◽  
Arithmetic Mean ◽  
Mean Value Theorem ◽  
Two Dimensional ◽  
Multiplicative Functions ◽  
Structural Result ◽  
Convergence Results ◽  
The Mean

AbstractA celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

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