Set Of Equations For Functions Research Articles

D’Alembert function for exact non-smooth modal analysis of the bar in unilateral contact

Non-smooth modal analysis is an extension of modal analysis to non-smooth systems, prone to unilateral contact conditions for instance. The problem of a one-dimensional bar subject to unilateral contact on its boundary has been previously investigated numerically and the corresponding spectrum of vibration could be partially explored. In the present work, the non-smooth modal analysis of the above system is reformulated as a set of functional equations through the use of both d’Alembert solution to the wave equation and the method of steps for Neutral Delay Differential Equations. The system features a strong internal resonance condition and it is established that irrational and rational periods of vibration should be carefully distinguished. For irrational periods, it was previously proven that the displacement field of the non-smooth modes of vibration is characterized with piecewise-linear functions in space and time and such a motion is unique for a prescribed energy. However, for rational periods, which are the subject of this work, new periodic solutions are found analytically. Findings consist of families of iso-periodic solutions with piecewise-smooth displacement fields in space and time and continua of piecewise-smooth periodic solutions of the same energy and frequency.

A Method for Calculating the Characteristics of Leaky Wave Antennas Implemented by Means of the Open Comb Transmission Lines

An electrodynamic method is developed for the analysis and the parametric synthesis of the microwave and millimeter-wave leaky wave antennas implemented by means of the planar dielectric waveguide loaded by a conductive comb array. The specified approach is based on the classical scheme of separating the partial domains, but it differs by the procedures it uses to simulate the metal comb ridges by the grooves of zero depth. It allows reducing the set of functional equations to the set of linear algebraic equations with the relatively simple coefficients using a single Fourier expansion. The introduced method is adjusted for the analysis of apertures with combs containing several rectangular grooves within the spacing, while their mouths can belong to one or two differing planes. The validity of the method and the reliability of the results obtained on its basis are confirmed experimentally.

Quantum spectral curve for the η-deformed AdS5 × S5 superstring

The spectral problem for the AdS5×S5 superstring and its dual planar maximally supersymmetric Yang–Mills theory can be efficiently solved through a set of functional equations known as the quantum spectral curve. We discuss how the same concepts apply to the η-deformed AdS5×S5 superstring, an integrable deformation of the AdS5×S5 superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the AdS5×S5 superstring, like the relation between the XXZ and XXX spin chains, or the sausage and the S2 sigma models for instance. We derive the quantum spectral curve for the η-deformed string by reformulating the corresponding ground-state thermodynamic Bethe ansatz equations as an analytic Y system, and map this to an analytic T system which upon suitable gauge fixing leads to a Pμ system – the quantum spectral curve. We then discuss constraints on the asymptotics of this system to single out particular excited states. At the spectral level the η-deformed string and its quantum spectral curve interpolate between the AdS5×S5 superstring and a superstring on “mirror” AdS5×S5, reflecting a more general relationship between the spectral and thermodynamic data of the η-deformed string. In particular, the spectral problem of the mirror AdS5×S5 string, and the thermodynamics of the undeformed AdS5×S5 string, are described by a second rational limit of our trigonometric quantum spectral curve, distinct from the regular undeformed limit.

A variational method for designing adaptive bandlimited wavelets

This paper proposes a new method of estimating an orthogonal bounded spectrum wavelet from a given signal. The method is based on minimizing the distance of the wavelet and the given signal in the sense of the metric of the L2 space. In this method, the amplitude and phase of the mother wavelet are optimized simultaneously. The Lagrange multipliers technique is applied to consider the constraints due to the bounded spectrum wavelet and orthogonality conditions. The variational method reduced the optimal matching problem to the solution of a set of functional equations for the amplitude and phase of the optimal-matched bandlimited wavelet spectrum. Continuous functional equations are written with respect to Fourier coefficients of phase of transfer function of the quadrate low-pass filter at the sampled frequencies, and asset of discrete algebraic equations allows us to design the wavelet directly from the signal of interest. To demonstrate the performance of the presented method in this paper, it is employed to determine the matched wavelets of some specified signals.

A variational method for designing wavelets to match a specified signal

In this paper, we have developed an efficient method for obtaining an orthonormal wavelet that is matched to a given signal. The error between the wavelet and the given signal is minimized subject to the constraints of the amplitude and phase of the band-limited wavelet spectrum. To consider the constraints of the minimization problem, the Lagrange multipliers technique is applied. Variational method reduced the optimal matching problem to the solution of a set of functional equations for the amplitude and phase of the wavelet spectrum. Continuous functional equations were written in terms of Fourier coefficients of the phase of the transfer function of the quadrature low pass filter at the sampled frequencies. Consequently, a set of discrete algebraic equations allows us to design the wavelet directly from the signal of interest. Specific examples are given for demonstrating the performance of wavelet matching equations for both known orthonormal wavelets and arbitrary signals.

TheQ-operator for root-of-unity symmetry in the six-vertex model

We construct the explicit $Q$-operator incorporated with the $sl_2$-loop-algebra symmetry of the six-vertex model at roots of unity. The functional relations involving the $Q$-operator, the six-vertex transfer matrix and fusion matrices are derived from the Bethe equation, parallel to the Onsager-algebra-symmetry discussion in the superintegrable $N$-state chiral Potts model. We show that the whole set of functional equations is valid for the $Q$-operator. Direct calculations in certain cases are also given here for clearer illustration about the nature of the $Q$-operator in the symmetry study of root-of-unity six-vertex model from the functional-relation aspect.

On Spherically Symmetric Minimally Coupled Brane Worlds

For a static, spherically symmetric brane in the framework of the RS2 concept, we study the conditions under which the 4D tensor E μv, arising from the 5D Weyl tensor, vanishes on the brane. Gravity on the brane is then decoupled from the bulk geometry, it is the so-called minimally coupled brane world (MCBW). Assuming E μv =0 in the whole bulk, we try to solve the 5D Einstein equations G AB + Λ5gAB =0 and obtain an overdetermined set of equations for functions of the radial coordinate. Some special solutions are found, among which are the well-known “black string” solution with the Schwarzschild metric on the brane and its generalizations with Schwarzschild–(A)dS on-brane metrics. It is concluded that a MCBW can be embedded, in general, in a bulk where Eμv is not identically zero but only vanishes on the brane. We also present some previous results on the general properties of scalar fields on the brane and give an example of a wormhole supported by a scalar field in a MCBW.

The complete set of Casimir constants of the motion in magnetohydrodynamics

A set of functional equations that all Casimir constants of the motion must satisfy is derived. A simple technique is developed that can be used to obtain partial information on the functional dependence of the Casimirs on the state variables. Together with additional known information, this is sufficient to determine all the Casimirs for a magnetohydrodynamic plasma with nested magnetic flux surfaces, as well as for some other magnetic configurations.

The Theory of Conserving Approximations and the Density Functional Theory in Approximations for Nuclear Matter

Self-consistent relativistic approximations for nuclear matter are discussed in terms of the theory of conserving approximations and a self-consistent Green’s function approach. Moreover, the nuclear density functional theory is explained in terms of the theory of conserving approximations. The analysis is applied to a relativistic field-theoretical model for nuclear matter, in order to construct self-consistent Dirac-Hartree-Fock and Dirac-Ring approximations that maintain the Hugenholtz-Van Hove theorem and Landau’s hypothesis for quasiparticles. A self-consistent set of functional equations for self-energies is obtained by the stationary condition required by the theory of conserving approximations. The exact self-energy solutions are derived, and retardation corrections that maintain the HugenholtzVan Hove theorem and Landau’s hypothesis for quasiparticles are generated explicitly in the self-energies. This approach is a direct method in order to construct self-consistent approximations for nuclear matter, and the results are applied to Fermi-liquid properties of nuclear matter and properties of neutron stars.

A package for symbolic solution of real functional equations of real variables

This paper discusses the problems associated with the symbolic treatment of functional equations and presents a Mathematica package for the solution of real functional equations of real variables. The package includes a minimal basic database which contains a reduced set of functional equations with its four components: equation, domain, class and the corresponding solution. The word minimal is used in the sense that any equation that is solvable by the system using non-searching methods is excluded from the database. The package incorporates a searching algorithm which can solve functional equations independently of their notation and their algebraic representation. Not only general solutions but particular and candidate solutions are dealt with. This implies a careful analysis of domains and classes. The package includes some methods for solving functional equations, which are used when the input functional equations are not found in the database. Some methods have been implemented internally and some are in an external package. Finally, some examples illustrate the use of the package.

On the Liouville three-point function

The recently proposed expression for the general three point function of exponential fields in quantum Liouville theory on the sphere is considered. By exploiting locality or crossing symmetry in the case of those four-point functions, which may be expressed in terms of hypergeometric functions, a set of functional equations is found for the general three point function. It is shown that the expression proposed by the Zamolodchikovs solves these functional equations and that under certain assumptions the solution is unique.

Branching point processes with independent transformations

We investigate branching point processes within an arbitrary number of point types, based on the generating functional technique. Every point can be characterized by a set of parameters such as time, space, energy, etc. The analyses have been done for a random number of identical stages with independent transformations. The set of functional equations has been obtained for the determination of either the statistical characteristics of the resulting point process or the parameters of the separate stage process. The proposed approach lets us describe a wide range of random point processes and find mechanisms of their formation. We considered some specific models of electron multiplication in a microchannel plate, a photomultiplier tube and an avalanche photodiode.

Block-scaling of value-iteration for discounted Markov renewal programming

The functional equations of infinite horizon discounted Markov renewal programming arev*=Tv* whereT is a monotone contraction operator. This paper shows how to accelerate convergence of the value-iteration schemev(n+1)=Tv(n) by a block-scaling step whereby all states in a given group have theirv i (n) scaled by a common scale factor. A similar method exists when the relative valuesv* i −v*1 are computed iteratively. In both cases, the block scaling factors are solution to a set of functional equations which has similar structure to a discounted Markov renewal program, and can itself be solved by successive approximation, policy iteration, or linear programming.

Dissecting orthoschemes into orthoschemes

We exhibit a dissection, with one degree of freedom, of an arbitrary orthoscheme in Euclidean, spherical or hyperbolic d-space into d+1 orthoschemes (Section 2); this can be interpreted as a set of relations in the scissors congruence group or, weaker, as a set of functional equations for the volume. Besides special cases where the dissection is into mutually congruent parts, we obtain, in the spherical case and for a special value of the parameter, scissors congruence formulae similar to Schlafli's period formulae for the spherical orthoscheme volume (see Section 5). In Section 6 we use the dissection to explain the structure of the volume formula for asymptotic hyperbolic 3-orthoschemes due to Lobachevsky. Finally, in Section 7, by exploiting symmetries, we show that two systems of special volume relations of Schlafli (in spherical d-space) and Coxeter (for all three geometries in dimension 3) hold even on the level of dissection. In particular, it seems that all the presently known exact values for the volume of special spherical 3-simplexes hold, independently of Schlafli's differential formula, as consequences of scissors congruence relations.

Lippmann–Schwinger equations for a scalar macroscopic field in an anisotropic stratified medium

We develop a general approach to solve the transmission problem for a scalar macroscopic field in an anisotropic stratified medium. The method is based on a chainlike set of functional equations of the Lippmann−Schwinger type. A typical example of the fields under consideration is the wave field of a particle in the effective mass tensor approximation.

Exact Analysis of Shielded Microstrip Lines and Bilateral Fin Lines, Jul. 1981

An exact analysis is presented for shielded microstrip and bilateral fin lines. The method of analysis is based on function-theoretic approach to solve a set of functional equations in the Fourier transform domain, representing the independent excitation of LSE and LSM modes in the systems without strip or fin conductors. The solution is obtained in the form of highly convergent systems of algebraic equations, which allow the accurate calculation of fields and the electrical parameters of these fines at arbitrary frequencies.

Duality for the Lorentz force in loop space

A set of functional equations is derived for the dual of the Lorentz force defined in the loop space. The duality is manifest for Polvakov's and newly obtained equations. Physical implications are briefly discussed.

Green's functions of scalar macroscopic fields in anisotropic stratified media

The general approach, based on a chainlike set of functional equations of Dyson's type in the mixed coordinate-wave representation, is developed for solving dynamic or static problems for scalar macroscopic fields in anisotropic stratified media. The theory is applicable for a variety of fields such as the electric, temperature, concentration, or the wave fields of particles in the effective-mass tensor approximation.

Functional equations in emitting atmospheres

Starting with the equation of transfer in a plane-parallel inhomogeneous atmosphere which emits and scatters radiation anisotropically, we obtain a set of functional equations which govern the relations between the internal and external (emergent) intensities. Since the functional equations are generalization of the principles of invariance of Chandrasekhar, the latter can be reduced from the former as a special case. Some applications of the functional equations are outlined.

Optimal Strategies for a Class of Constrained Sequential Problems

This paper considers and unifies two sequential problems which have been extensively discussed. A class of sequential problems is proposed that includes both. An arbitrary partial ordering constraint is permitted to restrict possible strategies. An algorithm is proposed for finding the optimal strategy, and we prove that a strategy is optimal for the class of problems if and only if it can be found by the algorithm. The main tool is a set of functional equations in strategy space.