46,543 research outputs found
Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
We give examples of infinite order rational transformations that leave linear
differential equations covariant. These examples are non-trivial yet simple
enough illustrations of exact representations of the renormalization group. We
first illustrate covariance properties on order-two linear differential
operators associated with identities relating the same hypergeometric
function with different rational pullbacks. We provide two new and more general
results of the previous covariance by rational functions: a new Heun function
example and a higher genus hypergeometric function example. We then
focus on identities relating the same hypergeometric function with two
different algebraic pullback transformations: such remarkable identities
correspond to modular forms, the algebraic transformations being solution of
another differentially algebraic Schwarzian equation that emerged in a paper by
Casale. Further, we show that the first differentially algebraic equation can
be seen as a subcase of the last Schwarzian differential condition, the
restriction corresponding to a factorization condition of some associated
order-two linear differential operator. Finally, we also explore
generalizations of these results, for instance, to , hypergeometric
functions, and show that one just reduces to the previous cases through
a Clausen identity.
In a hypergeometric framework the Schwarzian condition encapsulates
all the modular forms and modular equations of the theory of elliptic curves,
but these two conditions are actually richer than elliptic curves or
hypergeometric functions, as can be seen on the Heun and higher genus example.
This work is a strong incentive to develop more differentially algebraic
symmetry analysis in physics.Comment: 43 page
Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras
The present text surveys some relevant situations and results where basic
Module Theory interacts with computational aspects of operator algebras. We
tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be
published in Lecture Notes in Computer Scienc
Exactly Solvable Quantum Mechanics
A comprehensive review of exactly solvable quantum mechanics is presented
with the emphasis of the recently discovered multi-indexed orthogonal
polynomials.
The main subjects to be discussed are the factorised Hamiltonians, the
general structure of the solution spaces of the Schroedinger equation (Crum's
theorem and its modifications), the shape invariance, the exact solvability in
the Schroedinger picture as well as in the Heisenberg picture, the
creation/annihilation operators and the dynamical symmetry algebras, coherent
states, various deformation schemes (multiple Darboux transformations) and the
infinite families of multi-indexed orthogonal polynomials, the exceptional
orthogonal polynomials, and deformed exactly solvable scattering problems.Comment: LaTeX 48 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1104.047
New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum
quasi-Yang-Baxter algebras being simple but non-trivial deformations of
ordinary algebras of monodromy matrices realize a new type of quantum dynamical
symmetries and find an unexpected and remarkable applications in quantum
inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter
algebras the standard procedure of QISM one obtains new wide classes of quantum
models which, being integrable (i.e. having enough number of commuting
integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic
Bethe ansatz solution for arbitrarily large but limited parts of the spectrum).
These quasi-exactly solvable models naturally arise as deformations of known
exactly solvable ones. A general theory of such deformations is proposed. The
correspondence ``Yangian --- quasi-Yangian'' and `` spin models ---
quasi- spin models'' is discussed in detail. We also construct the
classical conterparts of quasi-Yang-Baxter algebras and show that they
naturally lead to new classes of classical integrable models. We conjecture
that these models are quasi-exactly solvable in the sense of classical inverse
scattering method, i.e. admit only partial construction of action-angle
variables.Comment: 49 pages, LaTe
Free nilpotent and -type Lie algebras. Combinatorial and orthogonal designs
The aim of our paper is to construct pseudo -type algebras from the
covering free nilpotent two-step Lie algebra as the quotient algebra by an
ideal. We propose an explicit algorithm of construction of such an ideal by
making use of a non-degenerate scalar product. Moreover, as a bypass result, we
recover the existence of a rational structure on pseudo -type algebras,
which implies the existence of lattices on the corresponding pseudo -type
Lie groups. Our approach substantially uses combinatorics and reveals the
interplay of pseudo -type algebras with combinatorial and orthogonal
designs. One of the key tools is the family of Hurwitz-Radon orthogonal
matrices
Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K
We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of
higher order q-recurrence equations with rational coefficients. We extend a
method for finding a bound on the maximal power of t in the denominator of
arbitrary rational solutions y(t) as well as a method for bounding the degree
of polynomial solutions from the scalar case to the systems case. The approach
is direct and does not rely on uncoupling or reduction to a first order system.
Unlike in the scalar case this usually requires an initial transformation of
the system.Comment: 8 page
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