211 research outputs found
Global convergence of a non-convex Douglas-Rachford iteration
We establish a region of convergence for the proto-typical non-convex
Douglas-Rachford iteration which finds a point on the intersection of a line
and a circle. Previous work on the non-convex iteration [2] was only able to
establish local convergence, and was ineffective in that no explicit region of
convergence could be given
On one master integral for three-loop on-shell HQET propagator diagrams with mass
An exact expression for the master integral I_2 arising in three-loop
on-shell HQET propagator diagrams with mass is derived and its analytical
expansion in the dimensional regularization parameter epsilon is given.Comment: 6 pages, 1 figure; v3: completely re-written, 2 new authors, many new
results, additional reference
Strong duality in conic linear programming: facial reduction and extended duals
The facial reduction algorithm of Borwein and Wolkowicz and the extended dual
of Ramana provide a strong dual for the conic linear program in the absence of any constraint qualification. The facial
reduction algorithm solves a sequence of auxiliary optimization problems to
obtain such a dual. Ramana's dual is applicable when (P) is a semidefinite
program (SDP) and is an explicit SDP itself. Ramana, Tuncel, and Wolkowicz
showed that these approaches are closely related; in particular, they proved
the correctness of Ramana's dual using certificates from a facial reduction
algorithm.
Here we give a clear and self-contained exposition of facial reduction, of
extended duals, and generalize Ramana's dual:
-- we state a simple facial reduction algorithm and prove its correctness;
and
-- building on this algorithm we construct a family of extended duals when
is a {\em nice} cone. This class of cones includes the semidefinite cone
and other important cones.Comment: A previous version of this paper appeared as "A simple derivation of
a facial reduction algorithm and extended dual systems", technical report,
Columbia University, 2000, available from
http://www.unc.edu/~pataki/papers/fr.pdf Jonfest, a conference in honor of
Jonathan Borwein's 60th birthday, 201
Low energy expansion of the four-particle genus-one amplitude in type II superstring theory
A diagrammatic expansion of coefficients in the low-momentum expansion of the
genus-one four-particle amplitude in type II superstring theory is developed.
This is applied to determine coefficients up to order s^6R^4 (where s is a
Mandelstam invariant and R^4 the linearized super-curvature), and partial
results are obtained beyond that order. This involves integrating powers of the
scalar propagator on a toroidal world-sheet, as well as integrating over the
modulus of the torus. At any given order in s the coefficients of these terms
are given by rational numbers multiplying multiple zeta values (or
Euler--Zagier sums) that, up to the order studied here, reduce to products of
Riemann zeta values. We are careful to disentangle the analytic pieces from
logarithmic threshold terms, which involves a discussion of the conditions
imposed by unitarity. We further consider the compactification of the amplitude
on a circle of radius r, which results in a plethora of terms that are
power-behaved in r. These coefficients provide boundary `data' that must be
matched by any non-perturbative expression for the low-energy expansion of the
four-graviton amplitude.
The paper includes an appendix by Don Zagier.Comment: JHEP style. 6 eps figures. 50 page
Lagrange Duality in Set Optimization
Based on the complete-lattice approach, a new Lagrangian duality theory for
set-valued optimization problems is presented. In contrast to previous
approaches, set-valued versions for the known scalar formulas involving infimum
and supremum are obtained. In particular, a strong duality theorem, which
includes the existence of the dual solution, is given under very weak
assumptions: The ordering cone may have an empty interior or may not be
pointed. "Saddle sets" replace the usual notion of saddle points for the
Lagrangian, and this concept is proven to be sufficient to show the equivalence
between the existence of primal/dual solutions and strong duality on the one
hand and the existence of a saddle set for the Lagrangian on the other hand
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
We continue the study of the construction of analytical coefficients of the
epsilon-expansion of hypergeometric functions and their connection with Feynman
diagrams. In this paper, we show the following results:
Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth
(see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions.
Theorem B: The epsilon expansion of a hypergeometric function with one
half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the
harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are
ratios of polynomials. Some extra materials are available via the www at this
http://theor.jinr.ru/~kalmykov/hypergeom/hyper.htmlComment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected
and a few references added; v3: few references added
Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators
In this note we provide regularity conditions of closedness type which
guarantee some surjectivity results concerning the sum of two maximal monotone
operators by using representative functions. The first regularity condition we
give guarantees the surjectivity of the monotone operator , where and and are maximal monotone operators on
the reflexive Banach space . Then, this is used to obtain sufficient
conditions for the surjectivity of and for the situation when belongs
to the range of . Several special cases are discussed, some of them
delivering interesting byproducts.Comment: 11 pages, no figure
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