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We review the non-trivial issue of the relativistic description of a quantum mechanical system that, contrary to a common belief, kept theoreticians busy from the end of 1920s to (at least) mid 1940s. Starting by the well-known wo...
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We review the non-trivial issue of the relativistic description of a quantum mechanical system that, contrary to a common belief, kept theoreticians busy from the end of 1920s to (at least) mid 1940s. Starting by the well-known works by Klein-Gordon and Dirac, we then give an account of the main results achieved by a variety of different authors, ranging from de Broglie to Proca, Majorana, Fierz-Pauli, Kemmer, Rarita-Schwinger and many others.A particular interest comes out for the general problem of the description of particles with. arbitrary spin, introduced (and solved) by Majorana as early as 1932, and later reconsidered, within a different approach, by Dirac in 1936 and by Fierz-Pauli in 1939. The final settlement of the problem in 1945 by Bhabha, who came back to the general ideas introduced by Majorana in 1932, is discussed as well, and, by making recourse also to unpublished documents by Majorana, we are able to reconstruct the line of reasoning behind the Majorana and the Bhabha equations, as well as its evolution. Intriguingly enough, such an evolution was. identical in the two authors, the difference being just the period of time required for that: probably few weeks in one case (Majorana), while more than ten years in the other one (Bhabha), with the contribution of several intermediate authors.The important unpublished contributions by Majorana anticipated later results obtained, in a more involved way, by de Broglie (1934) and by Duffin and Kemmer (1938-9), and testify the intermediate steps in the line of reasoning that led to the paper published in 1932 by Majorana, while Bhabha took benefit of the corresponding (later) published literature. Majorana's paper of 1932, in fact, contrary to the more complicated Dirac-Fierz-Pauli formalism, resulted to be very difficult to fully understand (probably for its pregnant meaning and latent physical and mathematical content): as is clear from his letters, even Pauli (who suggested its reading to Bhabha) took about one year in 1940-1 to understand it. This just testifies for the difficulty of the problem, and for the depth of Majorana's reasoning and results.The relevance for present day research of the issue here reviewed is outlined as well.
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This paper presents the first purely algebraic characterization of classes ofpartial algebras definable by a set of strong equations. This result was posible due to newtools such as invariant congruences, i.e. a generalization of ...
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This paper presents the first purely algebraic characterization of classes ofpartial algebras definable by a set of strong equations. This result was posible due to newtools such as invariant congruences, i.e. a generalization of the notion of a fully invariantcongruence, and extension of algebras, specific for strong equations.
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In this paper we consider the family of systems (2c + 1)U~2 — 2cV~2 = μ and (c — 2)U~2 — cZ~2 = — 2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic ...
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In this paper we consider the family of systems (2c + 1)U~2 — 2cV~2 = μ and (c — 2)U~2 — cZ~2 = — 2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field K = Q((-d)~(1/2)). We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation X~4 - 4cX~3Y + (6c + 2)X~2Y~2 + 4cXY~3 + Y~4 = μ and solve it by the method of Tzanakis under the same assumptions.
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We propose a new type of solution to the ultradiscrete hungry Lotka-Volterra (uhLV) equation. For the solution, the periodic phase is introduced into the known soliton and the extended soliton becomes a traveling wave showing a pe...
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We propose a new type of solution to the ultradiscrete hungry Lotka-Volterra (uhLV) equation. For the solution, the periodic phase is introduced into the known soliton and the extended soliton becomes a traveling wave showing a periodic variation. We call this type of wave a 'periodic phase soliton' (PPS). The solution has two forms of expression: one is the 'perturbation form' and the other is the 'ultradiscrete permanent form'. We analyze the interaction among PPSs and solitons. Moreover, we give the outline of proof to show that the solution satisfies the bilinear equation of the uhLV equation.
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We analyze a method of approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so-called artificial compressibilit...
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We analyze a method of approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so-called artificial compressibility method adapted to the MHD system. By exploiting the wave equation structure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the soleinoidal component of the approximating velocity and magnetic fields is relatively compact and converges strongly to a weak solution of the MHD equation.
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The generalized Gaussian distribution with location parameter , scale parameter sigma, and shape parameter p contains the Laplace, normal, and uniform distributions as particular cases for p = 1, 2, +, respectively. Derivations of...
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The generalized Gaussian distribution with location parameter , scale parameter sigma, and shape parameter p contains the Laplace, normal, and uniform distributions as particular cases for p = 1, 2, +, respectively. Derivations of the true maximum-likelihood estimators of and sigma for these special cases are popular exercises in many university courses. Here, we show how the true maximum-likelihood estimators of and sigma can be derived for p = 3, 4, 5. The derivations involve solving of quadratic, cubic, and quartic equations.
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In this paper we investigate local and global existence as well as asymptotic behavior of the solution for a class of abstract; (hyperbolic) quasilinear equations perturbed by bounded delay operators. We assume that the leading op...
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In this paper we investigate local and global existence as well as asymptotic behavior of the solution for a class of abstract; (hyperbolic) quasilinear equations perturbed by bounded delay operators. We assume that the leading operator is of bounded variation in time. In the last section, the abstract results are applied on a heat conduction model.
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Linear Volterra-type integral equations with kernels having a series expansion in the first variable have series solutions with coefficients given iteratively. Their resolvents may be expanded likewise. The associated homogeneous ...
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Linear Volterra-type integral equations with kernels having a series expansion in the first variable have series solutions with coefficients given iteratively. Their resolvents may be expanded likewise. The associated homogeneous equation Kf = f generally has Frobenius series solutions when the kernel is singular, whereas Kf =0 generally has such solutions regardless of singularity: the proviso in each case is that associated ‘‘indicial equation’’ has solutions.
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In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n ? Z with |xy| > 1, n ≥ 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger...
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In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n ? Z with |xy| > 1, n ≥ 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger ranges of A, B, C. We give among other things all the solutions for A = C = 1, B < 235 (cf. Theorem 1), and for C= 1, A, B ≤ 50, with six explicitly given exceptions (A, B, n) (cf. Theorem 3). The equations under consideration are solved by combining powerful techniques, including Prey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, classical cyclotomy and computational approaches to Thue equations of low degree. Along the way, we derive a new result on the solvability of binomial Thue equations (cf. Theorem 6) which is crucial in the proof of our Theorems 1 and 2. Some important applications of our theorems will be given in a forthcoming paper.
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The description of the change in characteristic temperatures of thermomechanical and viscoelastic properties of polymers and elastomers with deformation frequency or of temperature dependence of polymer p
The description of the change in characteristic temperatures of thermomechanical and viscoelastic properties of polymers and elastomers with deformation frequency or of temperature dependence of polymer properties is widely achieved by two equations: (1) the Williams‐Landel‐Ferry (WLF) equation and (2) the Arrhenius equation. Mostly the WLF equation is used. Often the distinction between the two descriptions is based on the argument: if volume processes play the key role, then WLF equation is the right one, if thermally activated processes play the key role, then Arrhenius equation is the right one. Both equations are based on the activation of processes, and always the temperature is the variable, which activate the processes. Both descriptions are methods to parameterize the temperature dependence of properties or the change of characteristic temperatures, as glass‐rubber transition temperature, with deformation rate. Also, the so‐called ‘volume processes’ are controlled by temperature, but the thermal activation can be small in energy to initiate the change in spatial position from one site to another for a molecule. This means both descriptions should be congruent. In this article, the congruence is shown and the relation between WLF parameters and Arrhenius parameters will be established. For this, a slight modification of the usual Arrhenius equation is necessary. Also, other descriptions are discussed in short: Doolittle equation and Vogel‐Fulcher equation, they were or are used to describe the change of viscosity with temperature in melts or solutions.
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