scholarly journals Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1550
Author(s):  
Jan L. Cieśliński ◽  
Dzianis Zhalukevich
Keyword(s):  
Vector Space ◽  
Special Relativity ◽  
Holomorphic Functions ◽  
Lorentz Symmetry ◽  
Dimensional Case ◽  
Hypercomplex Numbers ◽  
Explicit Formulas ◽  
Elliptic Case ◽  
Hyperbolic Case ◽  
All Solutions

Scators form a vector space endowed with a non-distributive product, in the hyperbolic case, have physical applications related to some deformations of special relativity (breaking the Lorentz symmetry) while the elliptic case leads to new examples of hypercomplex numbers and related notions of holomorphicity. Until now, only a few particular cases of scator holomorphic functions have been found. In this paper we obtain all solutions of the generalized Cauchy–Riemann system which describes analogues of holomorphic functions in the (1+2)-dimensional scator space.

scholarly journals Erlangen Programme at Large 3.2 Ladder Operators in Hypercomplex Mechanics

10.14311/1402 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
V. V. Kisil
Keyword(s):  
Quantum Mechanics ◽  
Representation Theory ◽  
Harmonic Oscillator ◽  
Correspondence Principle ◽  
Free Particle ◽  
Hypercomplex Numbers ◽  
Ladder Operators ◽  
Elliptic Case ◽  
Hyperbolic Case ◽  
Parabolic Case

We revise the construction of creation/annihilation operators in quantum mechanics based on the representation theory of the Heisenberg and symplectic groups. Besides the standard harmonic oscillator (the elliptic case) we similarly treat the repulsive oscillator (hyperbolic case) and the free particle (the parabolic case). The respective hypercomplex numbers turn out to be handy on this occasion. This provides a further illustration to the Similarity and Correspondence Principle.

scholarly journals Period polynomials and explicit formulas for Hecke operators on Γ0(2)

2009 ◽  
Vol 146 (2) ◽  
pp. 321-350 ◽  
Author(s):  
SHINJI FUKUHARA ◽  
YIFAN YANG
Keyword(s):  
Vector Space ◽  
Explicit Form ◽  
Congruence Subgroup ◽  
Bernoulli Polynomials ◽  
Affirmative Answer ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Isomorphism Theorem ◽  
Explicit Formulas ◽  
Space Of Cusp Forms

AbstractLet Sw+2(Γ0(N)) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).

scholarly journals KINEMATICAL SOLUTION OF THE UHE-COSMIC-RAY PUZZLE WITHOUT A PREFERRED CLASS OF INERTIAL OBSERVERS

2003 ◽  
Vol 12 (07) ◽  
pp. 1211-1226 ◽  
Author(s):  
GIOVANNI AMELINO-CAMELIA
Keyword(s):  
Special Relativity ◽  
Particle Physics ◽  
Planck Scale ◽  
Cosmic Ray ◽  
Lorentz Symmetry ◽  
Relativity Theory ◽  
Lorentz Transformations ◽  
Doubly Special Relativity ◽  
Energy Scenario ◽  
Special Relativity Theory

Among the possible explanations for the puzzling observations of cosmic rays above the GZK cutoff there is growing interest in the ones that represent kinematical solutions, based either on general formulations of particle physics with small violations of Lorentz symmetry or on a quantum-gravity-motivated scheme for the breakdown of Lorentz symmetry. An unappealing aspect of these cosmic-ray-puzzle solutions is that they require the existence of a preferred class of inertial observers. Here I propose a new kinematical solution of the cosmic-ray puzzle, which does not require the existence of a preferred class of inertial observers. My proposal is a new example of a type of relativistic theories, the so-called "doubly-special-relativity" theories, which have already been studied extensively over the last two years. The core ingredient of the proposal is a deformation of Lorentz transformations in which also the Planck scale Ep (in addition to the speed-of-light scale c) is described as an invariant. Just like the introduction of the invariant c requires a deformation of the Galileian transformations into the Lorentz transformations, the introduction of the invariant Ep requires a deformation of the Lorentz transformations, but there is no special class of inertial observers. The Pierre Auger Observatory and the GLAST space telescope should play a key role in future developments of these investigations. I also emphasize that the doubly-special-relativity theory here proposed, besides providing a solution for the cosmic-ray puzzle, is also the first doubly-special-relativity theory with a natural description of macroscopic bodies, and may find applications in the context of a recently-proposed dark-energy scenario.

scholarly journals Some Remarks on Symmetric and Frobenius Algebras

1960 ◽  
Vol 16 ◽  
pp. 65-71 ◽  
Author(s):  
J. P. Jans
Keyword(s):  
Vector Space ◽  
Space Dimension ◽  
Symmetric Algebra ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Finite Dimensional Case

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

Bifurcations of Limit Cycles From Infinity in Quadratic Systems

2002 ◽  
Vol 54 (5) ◽  
pp. 1038-1064 ◽  
Author(s):  
Lubomir Gavrilov ◽  
Iliya D. Iliev
Keyword(s):  
Vector Space ◽  
Critical Point ◽  
Limit Cycles ◽  
Displacement Function ◽  
Abelian Integrals ◽  
Puiseux Series ◽  
Bifurcation Of Limit Cycles ◽  
Elliptic Case ◽  
Quadratic Systems ◽  
Critical Point At Infinity

AbstractWe investigate the bifurcation of limit cycles in one-parameter unfoldings of quadractic differential systems in the plane having a degenerate critical point at infinity. It is shown that there are three types of quadratic systems possessing an elliptic critical point which bifurcates from infinity together with eventual limit cycles around it. We establish that these limit cycles can be studied by performing a degenerate transformation which brings the system to a small perturbation of certain well-known reversible systems having a center. The corresponding displacement function is then expanded in a Puiseux series with respect to the small parameter and its coefficients are expressed in terms of Abelian integrals. Finally, we investigate in more detail four of the cases, among them the elliptic case (Bogdanov-Takens system) and the isochronous center S3. We show that in each of these cases the corresponding vector space of bifurcation functions has the Chebishev property: the number of the zeros of each function is less than the dimension of the vector space. To prove this we construct the bifurcation diagram of zeros of certain Abelian integrals in a complex domain.

On the number of L2-solutions of second order linear differential equations

1978 ◽  
Vol 80 (1-2) ◽  
pp. 1-13 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Differential Equations ◽  
Vector Space ◽  
Function Space ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
All Solutions ◽  
Linear Differential

SynopsisLet d denote the dimension of the vector space consisting of all solutions of the equation − (p(t)y′)′ + q(t)y = 0, a ≤ t < ∞; that lie in the function space L2[a, ∞). By means of certain bounds on the solutions of this equation, sufficiency criteria are obtained for the cases d = 0 and d = 2.

scholarly journals HAMILTONIAN FORMALISM AND SPACE–TIME SYMMETRIES IN GENERIC DSR MODELS

2006 ◽  
Vol 15 (06) ◽  
pp. 925-935 ◽  
Author(s):  
S. MIGNEMI
Keyword(s):  
Phase Space ◽  
Special Relativity ◽  
Hamiltonian Formalism ◽  
Lorentz Symmetry ◽  
Space Time ◽  
Generic Models ◽  
Definition Of ◽  
Classical Phase Space

We study the structure of the classical phase space of generic models of deformed special relativity, which gives rise to a definition of velocity consistent with the deformed Lorentz symmetry. In this way we can also determine the laws of transformation of space–time coordinates.

Asymptotic expansions for the coefficient functions that arise in turning-point problems

1987 ◽  
Vol 410 (1838) ◽  
pp. 35-60 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Turning Point ◽  
Bessel Functions ◽  
Holomorphic Functions ◽  
Standard Theory ◽  
Large Parameter ◽  
Uniform Asymptotic Expansion ◽  
All Solutions ◽  
Linear Differential

We study the uniform asymptotic expansion for a large parameter u of solutions of second-order linear differential equations in domains of the complex plane which include a turning point. The standard theory of Olver demonstrates the existence of three such expansions, corresponding to solutions of the form Ai ( u 2/3 ζ e 2/3απi ) Ʃ n s = 0 A s (ζ)/ u 2 s + u -2 d / d ζ Ai ( u 2/3 ζ e 2/3απi ) Ʃ n -1 s =0 B s (ζ)/u 2 s + ϵ ( α ) n ( u , ζ) for α = 0, 1, 2, with bounds on ϵ ( α ) n . We proceed differently, by showing that the set of all solutions of the differential equation is of the form Ai ( u 2/3 ζ) A ( u , ζ) + u -2 (d/dζ) Ai ( u 2/3 ζ) B ( u , ζ), where Ai denotes any situation of Airy's equation. The coefficent functions A ( u , ζ) and B ( u , ζ) are the focus of our attention : we show that for sufficiently large u they are holomorphic functions of ζ in a domain including the turning point, and as functions of u are described asymptotically by descending power series in u 2 , with explicit error bounds. We apply our theory to Bessel functions.

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