General solution to the problem of decorrelation of a linear combination of two random complex variables

2021 ◽  
Author(s):  
Liberovskiy Nikita Yurievich ◽  
Priputin Vladimir Sergeevich ◽  
Lobova Elizaveta Olegovna
Keyword(s):  
General Solution ◽  
Linear Combination ◽  
Complex Variables ◽  
Random Complex

Minimum uncertainty filters for pulses

Geophysics ◽  
1993 ◽  
Vol 58 (6) ◽  
pp. 853-862 ◽  
Author(s):  
E. Clark Trantham
Keyword(s):  
General Solution ◽  
Frequency Domain ◽  
Linear Combination ◽  
Linear Function ◽  
Gaussian Function ◽  
Hermite Functions ◽  
Filter Length ◽  
First Derivative ◽  
Minimum Uncertainty ◽  
Domain Constraints

The objective of this paper is to calculate filters with a minimum uncertainty, the product of filter length and bandwidth. The method is applicable to producing minimum uncertainty filters with time or frequency domain constraints on the filter. The calculus of variations is used to derive the conditions that minimize a filter’s uncertainty. The general solution is a linear combination of Hermite functions, where the Hermite functions are summed from low to high order until the filter’s constraints are met. Filters constrained to have zero amplitude at zero hertz have an uncertainty at least three times greater than expected from the uncertainty principle, and the minimum uncertainty filter is a first derivative Gaussian. For the previous filter, the minimum uncertainty high cut filter is a Gaussian function of frequency, but the minimum uncertainty low cut filter is a linear function of frequency.

scholarly journals Holographic M5 branes in AdS7 × S4

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Varun Gupta
Keyword(s):  
General Solution ◽  
Special Class ◽  
Gauge Theories ◽  
Holomorphic Functions ◽  
Boundary Region ◽  
Complex Variables ◽  
Minimal Amount ◽  
World Volume ◽  
The World ◽  
Supersymmetric Gauge

Abstract We study classical M5 brane solutions in the probe limit in the AdS7× S4 background geometry that preserve the minimal amount of supersymmetry. These solutions describe the holography of codimension-2 defects in the 6d boundary dual $$ \mathcal{N} $$ N = (0, 2) supersymmetric gauge theories. The general solution is described in terms of holomorphic functions that satisfy a scaling condition. We show the behavior of the world-volume of a special class of BPS solutions near the AdS boundary region can be characterized by general equations, which describe it as intersections of the zeros of holomorphic functions in three complex variables with a 5-sphere.

Determining the Most Significant Parametric Function for a Given Linear Hypothesis

1987 ◽  
Vol 12 (3) ◽  
pp. 225-233 ◽  
Author(s):  
J. Gary Lutz ◽  
Leigh A. Cundari
Keyword(s):  
General Solution ◽  
Statistical Model ◽  
Linear Combination ◽  
Model Parameters ◽  
Linear Hypothesis ◽  
Parametric Function ◽  
Linear Statistical Model ◽  
Pairwise Contrasts

After a hypothesis about some linear statistical model has been tested and rejected (e.g., in an ANOVA), many researchers employ the Scheffe procedure to locate the source(s) of the rejection. This procedure guarantees that there is at least one linear combination of the model parameters (consistent with the hypothesis) that is significantly different from its hypothesized value. This most significant parametric function is not always easy to find, however, because it may not manifest itself in simple functions (such as pairwise contrasts between groups) or in “obvious” functions (such as those suggested by the graph of an interaction). A general solution to this problem is presented along with a practical example of its application.

scholarly journals The Third Five-Parametric Hypergeometric Quantum-Mechanical Potential

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
T. A. Ishkhanyan ◽  
A. M. Ishkhanyan
Keyword(s):  
General Solution ◽  
Linear Combination ◽  
Schrodinger Equation ◽  
Short Range ◽  
Hypergeometric Functions ◽  
Variable Parameter ◽  
Fundamental Solutions ◽  
Quantum Mechanical ◽  
Square Root ◽  
The Third

We introduce the third five-parametric ordinary hypergeometric energy-independent quantum-mechanical potential, after the Eckart and Pöschl-Teller potentials, which is proportional to an arbitrary variable parameter and has a shape that is independent of that parameter. Depending on an involved parameter, the potential presents either a short-range singular well (which behaves as inverse square root at the origin and vanishes exponentially at infinity) or a smooth asymmetric step-barrier (with variable height and steepness). The general solution of the Schrödinger equation for this potential, which is a member of a general Heun family of potentials, is written through fundamental solutions each of which presents an irreducible linear combination of two Gauss ordinary hypergeometric functions.

A Unified Formalism of Two-Dimensional Anisotropic Elasticity, Piezoelectricity and Unsymmetric Laminated Plates

2005 ◽  
Vol 72 (3) ◽  
pp. 422-431
Author(s):  
Wan-Lee Yin
Keyword(s):  
General Solution ◽  
Constitutive Relations ◽  
Anisotropic Elasticity ◽  
Complex Variables ◽  
Laminated Plates ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Eigenvalues And Eigenvectors ◽  
Infinite Domain ◽  
Anisotropic Plates

A unified formalism is presented for theoretical analysis of plane anisotropic elasticity and piezoelectricity, unsymmetric anisotropic plates, and other two-dimensional problems of continua with linear constitutive relations. Complex variables are used to reduce the governing differential equations to algebraic equations. The constitutive relation then yields an eigenrelation, which is easily solved explicitly for the material eigenvalues and eigenvectors. The latter have polynomial expressions in terms of the eigenvalues. When the eigenvectors are combined after multiplication by arbitrary analytic functions containing the corresponding eigenvalues, one obtains the two-dimensional general solution. Important results, including the orthogonality of the eigenvectors, the expressions of the pseudometrics and the intrinsic tensors, are established here for nondegenerate materials, including the case of all distinct eigenvalues. Green’s functions of the infinite domain, and of the semi-infinite domain with interior or edge singularities, are determined explicitly for the most general types of point loads and discontinuities (dislocations).

General Solutions of Anisotropic Laminated Plates

2003 ◽  
Vol 70 (4) ◽  
pp. 496-504 ◽  
Author(s):  
W.-L. Yin
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Analytic Functions ◽  
Infinite Plate ◽  
Elliptical Hole ◽  
Complex Variables ◽  
Laminated Plates ◽  
Eigenvalues And Eigenvectors ◽  
Closed Form Solutions ◽  
Repeated Eigenvalues

Anisotropic laminates with bending-stretching coupling possess eigensolutions that are analytic functions of the complex variables x+μky, where the eigenvalues μk and the corresponding eigenvectors are determined in the present analysis, along with the higher-order eigenvectors associated with repeated eigenvalues of degenerate laminates. The analysis and the resulting expressions are greatly simplified by using a mixed formulation involving a new set of elasticity matrices A*, B*, and D*. There are 11 distinct types of laminates, each with a different expression of the general solution. For an infinite plate with an elliptical hole subjected to uniform in-plane forces and moments at infinity, closed-form solutions are obtained for all types of anisotropic laminates in terms of the eigenvalues and eigenvectors.

Determining the Most Significant Parametric Function for a Given Linear Multivariate Hypothesis

1989 ◽  
Vol 14 (1) ◽  
pp. 21-28 ◽  
Author(s):  
J. Gary Lutz ◽  
Leigh A. Cundari
Keyword(s):  
General Solution ◽  
Linear Combination ◽  
Model Parameters ◽  
Test Procedures ◽  
Parametric Function ◽  
Multivariate Statistical ◽  
Global Test ◽  
Simultaneous Test ◽  
Dependent Variables ◽  
Multivariate Statistical Model

After a hypothesis about some linear multivariate statistical model has been tested and rejected (e.g., in a MANOVA), many researchers employ simultaneous test procedures to locate the source(s) of the rejection. If the global test was conducted using Roy’s largest root criterion, then this procedure guarantees at least one linear combination of the model parameters relative to some linear combination of the dependent variables that is significantly different from its hypothesized value. This most significant parametric function is not always easy to find, however, because it may not manifest itself in simple or “obvious” functions. A general solution to this problem is presented along with a practical example of its application.

The Role of VLBI in the Establishment of Coordinate Systems

1975 ◽  
Vol 26 ◽  
pp. 293-295 ◽  
Author(s):  
I. Zhongolovitch
Keyword(s):  
General Solution ◽  
Future Development ◽  
Rational Approach ◽  
Radio Telescopes ◽  
Coordinate Systems ◽  
Tectonic Plates ◽  
Astronomical Observations ◽  
Optical Instruments ◽  
Fundamental Polyhedron

Considering the future development and general solution of the problem under consideration and also the high precision attainable by astronomical observations, the following procedure may be the most rational approach:1. On the main tectonic plates of the Earth’s crust, powerful movable radio telescopes should be mounted at the same points where standard optical instruments are installed. There should be two stations separated by a distance of about 6 to 8000 kilometers on each plate. Thus, we obtain a fundamental polyhedron embracing the whole Earth with about 10 to 12 apexes, and with its sides represented by VLBI.

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

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