Algebraic Bethe Ansatz for a seven-vertex model

The paper is devoted to construction of algebraic Bethe Ansatz for a seven-vertex model. R-matrix of the system is obtained by means of twist from the six-vertex model considered by us earlier. The presence of the seventh nonzero element in the R-matrix complicates the situation. In particular, commutation relations of elements of the monodromy matrix become more complicated in comparison with the six-vertex model. We construct algebraic Bethe Ansatz by introducing a new operator that is the difference of two operators on the main diagonal of the monodromy matrix. The eigenstates and spectrum of the system are found. This is the first step on the way of comparison of systems with six-and seven-vertex R-matrices, respectively. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 178–186.     

Determinant Representations for Scalar Products in the Algebraic Bethe Ansatz

Theoretical and Mathematical Physics - We study integrable models with gl(2|1) symmetry that are solvable by the nested algebraic Bethe ansatz. We obtain a new determinant representation for scalar...     

Bethe Ansatz for the Universal Weight Function

We consider universal off-shell Bethe vectors given in terms of Drinfeld realization of the algebra [10, 12]. We investigate ordering properties of the product of the transfer matrix and these vectors. We derive that these vectors are eigenvectors of the transfer matrix if their Bethe parameters satisfy the universal Bethe equations [1]. Submitted: October 30, 2008. Accepted: February 16, 2009.     

Nested Bethe Ansatz for the RTT Algebra of sp(4) Type

Theoretical and Mathematical Physics - We show how to formulate the algebraic nested Bethe ansatz for RTT algebras with an R-matrix of the sp(4) type. We obtain the Bethe vectors and Bethe...     

Algebraic Bethe ansatz for $$\mathfrak o_{2n+1}$$ -invariant integrable models

Theoretical and Mathematical Physics - We study the class of $$ \mathfrak{o} _{2n+1}$$ -invariant quantum integrable models in the framework of the algebraic Bethe ansatz and propose a construction...     

SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces

We study SO(2)-invariant minimal and constant mean curvature surfaces in R³ endowed with a homogenous Riemannian metric whose group of isometries has dimension greater or equal to 4. Work partially supported by 40% and 60% Italian M.U.R.S.T. funds.     

Trigonometric Approximation of SO(N) Loops

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a Hölder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order $\mathcal{O}(n^{-\alpha+\epsilon})This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N), N≥3. We prove that the best approximation of an SO(N) loop Q(t) belonging to a H?lder class Lip _α , α>1, by a polynomial SO(N) loop of degree ≤n is of order O(n^-a+e)\mathcal{O}(n^{-\alpha+\epsilon}) for n≥k, where k=k(Q) is determined by topological properties of the loop and ε>0 is arbitrarily small. The convergence rate is therefore ε-close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N) loops which incorporates the loops’ topological aspects.     

Algebraic multilevel iteration method for Stieltjes matrices

The numerical solution of elliptic selfadjoint second-order boundary value problems leads to a class of linear systems of equations with symmetric, positive definite, large and sparse matrices which can be solved iteratively using a preconditioned version of some algorithm. Such differential equations originate from various applications such as heat conducting and electromagnetics. Systems of equations of similar type can also arise in the finite element analysis of structures. We discuss a recursive method constructing preconditioners to a symmetric, positive definite matrix. An algebraic multilevel technique based on partitioning of the matrix in two by two matrix block form, approximating some of these by other matrices with more simple sparsity structure and using the corresponding Schur complement as a matrix on the lower level, is considered. The quality of the preconditioners is improved by special matrix polynomials which recursively connect the preconditioners on every two adjoining levels. Upper and lower bounds for the degree of the polynomials are derived as conditions for a computational complexity of optimal order for each level and for an optimal rate of convergence, respectively. The method is an extended and more accurate algebraic formulation of a method for nine-point and mixed five- and nine-point difference matrices, presented in some previous papers.     

Algebraic multigrid for complex symmetric matrices and applications

This paper is concerned with the numerical study of an algebraic multigrid preconditioner for complex symmetric system matrices. We use several different Krylov subspace methods as an outer iteration, namely the QMR method proposed by Freund and Nachtigal, the BiCGCR method of Clemens, and the CSYM method of Bunse-Gerstner and Stoever. In addition, we compare the results with the standard Jacobi preconditioner for complex symmetric problems. We test our approach on the numerical simulation of high-voltage insulators.     

The 1/N expansion in the two-dimensional supersymmetric O(N)xO(K)-invariant O(N)/O(N−K) model and dynamical generation of massive vector particles

Republication Engineering and Technical Center for the Renewal and Strengthening of Parts of Machines and Mechanisms, Tomsk, Siberian Branch, USSR Academy of Sciences; Erevan Physics Institute; State University, Tomsk. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 76, No. 3, pp. 371–378, September, 1988.     

SO(N) invariant Wess-Zumino action and its quantization

A consistent quantization procedure of anomalous chiral models is discussed. It is based on a modification of the classical action by adding Wess-Zumino terms. The SO(3) invariant WZ action for the SU(3) model is constructed. Quantization of the corresponding modified theory is considered in detail for N=3.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 270–291, November, 1995.     

CHIRAL RING OF Sp（N） AND SO（N） SUPERSYMMETRIC GAUGE THEORY IN FOUR DIMENSIONS＊＊

The chiral ring of classical supersymmetric Yang-Mills theory with gauge group Sp(N) or SO(N) is computed, extending previous work (of Cachazo, Douglas, Seiberg, and the author) for SU(N). The result is that, as has been conjectured, the ring is generated by the usual glueball superfield S - Tr WαWα, with the relation Sh = 0, h being the dual Coxeter number. Though this proposition has important implications for the behavior of the quantum theory, the statement and (for the most part) the proofs amount to assertions about Lie groups with no direct reference to gauge theory.     

On the algebra of multidimensional integral operators with homogeneous SO(n)-invariant kernels and weakly radially oscillating coefficients

In this paper, we study the Banach algebra B generated by multidimensional integral operators whose kernels are homogeneous functions of degree (?n) invariant with respect to the rotation group SO(n) and by the operators of multiplication by radial weakly oscillating functions. A symbolic calculus is developed for the algebra 25. The Fredholm property and the formula for calculating the index are described in terms of this calculus.     

Algebraic Boundaries of \mathrm{SO}(2)-Orbitopes

Let $X\subset \mathbb{A }^{2r}$ X ? A 2 r be a real curve embedded into an even-dimensional affine space. We characterise when the $r$ r th secant variety to $X$ X is an irreducible component of the algebraic boundary of the convex hull of the real points $X(\mathbb{R })$ X ( R ) of $X$ X . This fact is then applied to $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes and to the so called Barvinok–Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $4$ 4 -dimensional $\mathrm{SO}(2)$ SO ( 2 ) -orbitopes, we find all irreducible components of their algebraic boundary.