4,492 research outputs found
Explicit formulas for the exponentials of some special matrices
AbstractThe matrix exponential plays a very important role in many fields of mathematics and physics. It can be computed by many methods. This work is devoted to the study of some explicit formulas for computing eA, where A is a special square matrix. The main results are based on the convergent power series of eA. Examples and applications are given
Exact Solutions to the Sine-Gordon Equation
A systematic method is presented to provide various equivalent solution
formulas for exact solutions to the sine-Gordon equation. Such solutions are
analytic in the spatial variable and the temporal variable and they
are exponentially asymptotic to integer multiples of as
The solution formulas are expressed explicitly in terms of a real triplet of
constant matrices. The method presented is generalizable to other integrable
evolution equations where the inverse scattering transform is applied via the
use of a Marchenko integral equation. By expressing the kernel of that
Marchenko equation as a matrix exponential in terms of the matrix triplet and
by exploiting the separability of that kernel, an exact solution formula to the
Marchenko equation is derived, yielding various equivalent exact solution
formulas for the sine-Gordon equation.Comment: 43 page
Light-Cone Expansion of the Dirac Sea in the Presence of Chiral and Scalar Potentials
We study the Dirac sea in the presence of external chiral and
scalar/pseudoscalar potentials. In preparation, a method is developed for
calculating the advanced and retarded Green's functions in an expansion around
the light cone. For this, we first expand all Feynman diagrams and then
explicitly sum up the perturbation series. The light-cone expansion expresses
the Green's functions as an infinite sum of line integrals over the external
potential and its partial derivatives.
The Dirac sea is decomposed into a causal and a non-causal contribution. The
causal contribution has a light-cone expansion which is closely related to the
light-cone expansion of the Green's functions; it describes the singular
behavior of the Dirac sea in terms of nested line integrals along the light
cone. The non-causal contribution, on the other hand, is, to every order in
perturbation theory, a smooth function in position space.Comment: 59 pages, LaTeX (published version
Pseudo-Exponential-Type Solutions of Wave Equations Depending on Several Variables
Using matrix identities, we construct explicit pseudo-exponential-type
solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two
variables and of nonlinear wave equations depending on three variables
Point Symmetries of Generalized Toda Field Theories
A class of two-dimensional field theories with exponential interactions is
introduced. The interaction depends on two ``coupling'' matrices and is
sufficiently general to include all Toda field theories existing in the
literature. Lie point symmetries of these theories are found for an infinite,
semi-infinite and finite number of fields. Special attention is accorded to
conformal invariance and its breaking.Comment: 25 pages, no figures, Latex fil
Free-Field Representation of Group Element for Simple Quantum Group
A representation of the group element (also known as ``universal -matrix'') which satisfies , is given in the form where , and and
are the generators of quantum group associated respectively with
Cartan algebra and the {\it simple} roots. The ``free fields'' $\chi,\
\vec\phi,\ \psi\psi^{(s)}\psi^{(s')} =
q^{-\vec\alpha_{i(s)} \vec\alpha_{i(s')}} \psi^{(s')}\psi^{(s)}, &
\chi^{(s)}\chi^{(s')} = q^{-\vec\alpha_{i(s)}\vec\alpha_{i(s')}}
\chi^{(s')}\chi^{(s)}& {\rm for} \ s<s', \\ q^{\vec h\vec\phi}\psi^{(s)} =
q^{\vec h\vec\alpha_{i(s)}} \psi^{(s)}q^{\vec h\vec\phi}, & q^{\vec
h\vec\phi}\chi^{(s)} = q^{\vec h \vec\alpha_{i(s)}}\chi^{(s)}q^{\vec
h\vec\phi}, & \\ &\psi^{(s)} \chi^{(s')} = \chi^{(s')}\psi^{(s)} & {\rm for\
any}\ s,s'.d_Ggg \rightarrow g'\cdot g''{\cal
R}{\cal R} (g\otimes I)(I\otimes g) =
(I\otimes g)(g\otimes I){\cal R}$Comment: 68 page
Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras
Methods of construction of the composition function, left- and
right-invariant vector fields and differential 1-forms of a Lie group from the
structure constants of the associated Lie algebra are proposed. It is shown
that in the second canonical coordinates these problems are reduced to the
matrix inversions and matrix exponentiations, and the composition function can
be represented in quadratures. Moreover, it is proven that the transition
function from the first canonical coordinates to the second canonical
coordinates can be found by quadratures
Decimated generalized Prony systems
We continue studying robustness of solving algebraic systems of Prony type
(also known as the exponential fitting systems), which appear prominently in
many areas of mathematics, in particular modern "sub-Nyquist" sampling
theories. We show that by considering these systems at arithmetic progressions
(or "decimating" them), one can achieve better performance in the presence of
noise. We also show that the corresponding lower bounds are closely related to
well-known estimates, obtained for similar problems but in different contexts
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