11,500 research outputs found
Integrability Formulas. Part III
In this article, we give several differentiation and integrability formulas of composite trigonometric function.Li Bo - Qingdao University of Science and Technology, ChinaMa Na - Qingdao University of Science and Technology, ChinaCzesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697-702, 1990.Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703-709, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Yasunari Shidama. The Taylor expansions. Formalized Mathematics, 12(2):195-200, 2004.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Peng Wang and Bo Li. Several differentiation formulas of special functions. Part V. Formalized Mathematics, 15(3):73-79, 2007, doi:10.2478/v10037-007-0009-4.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998
Intertwining operator for Calogero-Moser-Sutherland system
We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian
associated with a configuration of vectors on the plane which is a union
of and root systems. The Hamiltonian depends on one parameter.
We find an intertwining operator between and the Calogero-Moser-Sutherland
Hamiltonian for the root system . This gives a quantum integral for of
order 6 in an explicit form thus establishing integrability of .Comment: 24 page
Classification of the Killing Vectors in Nonexpanding HH-Spaces with Lambda
Conformal Killing equations and their integrability conditions for
nonexpanding hyperheavenly spaces with Lambda are studied. Reduction of ten
Killing equations to one master equation is presented. Classification of
homothetic and isometric Killing vectors in nonexpanding hyperheavenly spaces
with Lambda and homothetic Killing vectors in heavenly spaces is given. Some
nonexpanding complex metrics of types [III,N]x[N] are found. A simple example
of Lorentzian real slice of the type [N]x[N] is explicitly given
Deforming Lie algebras to Frobenius integrable non-autonomous Hamiltonian systems
Motivated by the theory of Painlev\'e equations and associated hierarchies,
we study non-autonomous Hamiltonian systems that are Frobenius integrable. We
establish sufficient conditions under which a given finite-dimensional Lie
algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie
algebra of Frobenius integrable vector fields spanning the same distribution as
the original algebra. The results are applied to quasi-St\"ackel systems.Comment: 14 pages, no figures. We repaired Example 5 and also made some minor
amendments in the tex
Reproducing formulas for generalized translation invariant systems on locally compact abelian groups
In this paper we connect the well established discrete frame theory of
generalized shift invariant systems to a continuous frame theory. To do so, we
let , , be a countable family of closed, co-compact
subgroups of a second countable locally compact abelian group and study
systems of the form with generators in and with each
being a countable or an uncountable index set. We refer to systems of this form
as generalized translation invariant (GTI) systems. Many of the familiar
transforms, e.g., the wavelet, shearlet and Gabor transform, both their
discrete and continuous variants, are GTI systems. Under a technical
local integrability condition (-LIC) we characterize when GTI systems
constitute tight and dual frames that yield reproducing formulas for .
This generalizes results on generalized shift invariant systems, where each
is assumed to be countable and each is a uniform lattice in
, to the case of uncountably many generators and (not necessarily discrete)
closed, co-compact subgroups. Furthermore, even in the case of uniform lattices
, our characterizations improve known results since the class of GTI
systems satisfying the -LIC is strictly larger than the class of GTI
systems satisfying the previously used local integrability condition. As an
application of our characterization results, we obtain new characterizations of
translation invariant continuous frames and Gabor frames for . In
addition, we will see that the admissibility conditions for the continuous and
discrete wavelet and Gabor transform in are special cases
of the same general characterizing equations.Comment: Minor changes (v2). To appear in Trans. Amer. Math. So
Discrete asymptotic nets and W-congruences in Plucker line geometry
The asymptotic lattices and their transformations are studied within the line
geometry approach. It is shown that the discrete asymptotic nets are
represented by isotropic congruences in the Plucker quadric. On the basis of
the Lelieuvre-type representation of asymptotic lattices and of the discrete
analog of the Moutard transformation, it is constructed the discrete analog of
the W-congruences, which provide the Darboux-Backlund type transformation of
asymptotic lattices.The permutability theorems for the discrete Moutard
transformation and for the corresponding transformation of asymptotic lattices
are established as well. Moreover, it is proven that the discrete W-congruences
are represented by quadrilateral lattices in the quadric of Plucker. These
results generalize to a discrete level the classical line-geometric approach to
asymptotic nets and W-congruences, and incorporate the theory of asymptotic
lattices into more general theory of quadrilateral lattices and their
reductions.Comment: 28 pages, 4 figures; expanded Introduction, new Section, added
reference
Integrable Systems and Discrete Geometry
This is an expository article for Elsevier's Encyclopedia of Mathematical
Physics on the subject in the title. Comments/corrections welcome.Comment: 22 pages, 7 figure
Review of AdS/CFT Integrability, Chapter I.3: Long-range spin chains
In this contribution we briefly review recent developments in the theory of
long-range integrable spin chains. These spin chains constitute a natural
generalisation of the well-studied integrable nearest-neighbour chains and are
of particular relevance to the integrability in the AdS/CFT correspondence
since the dilatation operator in the asymptotic region is conjectured to be a
Hamiltonian of an integrable long-range psu spin chain.Comment: 17 pages, see also overview article arXiv:1012.3982, v2: references
to other chapters updated, v3: minor typos corrected, references adde
On the integrability of a system describing the stationary solutions in Bose--Fermi mixtures
We study the integrability of a Hamiltonian system describing the stationary
solutions in Bose--Fermi mixtures in one dimensional optical lattices. We prove
that the system is integrable only when it is separable. The proof is based on
the Differential Galois approach and Ziglin-Morales-Ramis method.Comment: 21 page
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