q-Deformed grassmann fields and the two-dimensional Ising model

We construct an exact representation of the Ising partition function in the form of the SL_q(2, R)-invariant functional integral for the lattice-free q-fermion field theory (q=–1). It is shown that the q-fermionization allows one to rewrite the partition function of the eight-vertex model in an external field through a functional integral with four-fermion interaction. To construct these representations, we define a lattice (l, q, s)-deformed Grassmann bispinor field and extend the Berezin integration rules to this field. At q=–1, l=s=1, we obtain the lattice q-fermion field which allows us to fermionize the two-dimensional Ising model. We show that the Gaussian integral over (q, s)-Grassmann variables is expressed through the (q, s)-deformed Pfaffian which is equal to square root of the determinant of some matrix at q=±1, s=±1.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 3, pp. 388–412. June, 1995.     

Spectral functions of zeros in the Besselq-functions

Zeta functions _v(z; q)= _n=1 [j_vn(q)]^–z and partition functions Z_v(t; q)=_n exp[–tj _vn ² (q)] related to the zeros j_vn(q) of the Bessel q-functions J_v(x; q) and J _v ⁽²⁾ (x; q) are studied and explicit formulas for _v(2n; q) at n=±1, ±2, ... are obtained. The poles of _v(z; q) in the complex plane and the corresponding residues are found. Asymptotics of the partition functions Z_v(t; q) at t 0 are investigated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 397–414, June, 1996.     

Functional representation of the partition function and phase transitions in the Hubbard model

The functional representation of the partition function for the Hubbard model with strong Coulomb repulsion is obtained in the form of an integral with respect to auxiliary Grassmann variables. The approximate study of the phase transitions shows:

1. for a small doping constant δ, the system is an antiferromagnetic dielectric;
2. for δ≥t/V, the system is a ferromagnetic metal if U/t>10 and superconducting metal if U/t? 10. Bibliography:13 titles.

     

Exceptional directions for a convex body

Let K be a convex body in Rⁿ andO be a point inside K. We examine the Grassmann manifold of k-planes passing throughO. We take as exceptional the planes intersecting K along a body having at least one (k – 1)-dimensional face such that it does not have points inside the hyperfaces of body K. We prove that in the Grassmann manifold G _k ⁿ the set of such exceptional planes is of measure zero.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 365–371, September, 1976.The author thanks V. A. Zalgaller for aid and advice on the work.     

First-Quantized Fermions in Compact Dimensions

We discuss a path integral representation for fermionic particles and strings in the spirit of V. Ya. Fainberg and the author (Nucl. Phys. B, 306, 659–676, (1998); Phys. Lett. B, 211, 81–85, (1988)). We concentrate on the problems arising when some target-space dimensions are compact. We consider the partition function for a fermionic particle at a finite temperature or in compact time in detail as an example. We demonstrate that a self-consistent definition of the path integral generally requires introducing nonvanishing background Wilson loops and that modulo some common problems for real fermions in the Grassmannian formulation, these loops can be interpreted as condensates of world-line fermions. Properties of the corresponding string-theory path integrals are also discussed.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.     

An integral-free representation of a complex with a fourfold inflection center at infinity

We obtain the differential equations of a complex ₄ each ray of which contains a fourfold inflection center at infinity. The study of its geometric properties culminates in an integral-free representation of it, i.e., a geometric method of constructing the complex ₄ is exhibited; the correctness of the representation so found is proved.The degree of arbitrariness in the complex ₄ — one function of one variable — is determined and a geometric interpretation of this function is given.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 76–81.     

A partition function representation through Grassmann variables

We propose a formula for a classical partition function Z _N that does not involve the Hamilton function of the system. In the general case, we avoid passing to canonical variables (p,x) at the price of extending the space of Lagrange variables (v,x) by introducing additional velocities ¯u, u, which are the generators of a Grassmann algebra. In this space, the partition function Z _N is the integral of a Gibbs-type distribution, whose explicit form is determined by the system Lagrange function. We calculate the partition function of a model system governed by the Darwin Lagrange function.     

Extensions of an inequality of Bonnesen toD-dimensional space and curvature conditions for convex bodies

For convex bodies inE ^d (d 3) with diameter 2 we consider inequalitiesW _i – W _d–1 +( - 1) W _d 0 (i = 0, , d – 2) whereW _j are the quermassintegrals. In addition, for a ball, equality is attained for a body of revolution for which the elementary symmetric functions _d–1–i of main curvature radii is constant. The inequality is actually proved fori = d – 2 by means of Weierstrass's fundamental theorem of the calculus of variations.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Some integral transforms with the hypergeometric function F4(α, β, γ, δ; x,y)

We investigate one of the most efficient methods for solving differential equations and boundary-value problems —the integral transform method. The properties of the Jacobi polynomial are used to construct a new integral transform with the hypergeometric function F ₄ in the kernel.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 26–30, 1986.     

On the Choquet Charge of δ-superharmonic Functions

Let u v be positive superharmonic functions in a general potential-theoretic setting, where these functions have a Choquet-type integral representation by minimal such functions with Choquet charges (i.e. representing measures) and , respectively. We show that on the contact set {u – v = 0} of the -superharmonic function u – v, if this set is properly interpreted as the set of those minimal superharmonic functions s which satisfy lim sup _{T s} v/u = 1 for the co-fine neighborhood filter T _s associated with s. In the setting of classical potential theory for Laplace's equation this result improves on results obtained by Fuglede in 1992.     

Dissecting the Stanley partition function

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p_i(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π^′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p₀(n)-p₂(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that

(i)

and that for sufficiently large n

(ii)

In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:

Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.     

On Some Geometric Representations of GL N (𝔬)

We study a family of complex representations of the group GL _n (𝔬), where 𝔬 is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL _n (F) to its maximal compact subgroup GL _n (𝔬). We compute explicitly the transition matrix between a geometric basis of the Hecke algebra associated with the representation and an algebraic basis that consists of its minimal idempotents. The transition matrix involves combinatorial invariants of lattices of submodules of finite 𝔬-modules. The idempotents are p-adic analogs of the multivariable Jacobi polynomials.     

A Note on an Error Bound for the AOR Method

Suppose Ax = b is a system of linear equations where the matrix A is symmetric positive definite and consistently ordered. A bound for the norm of the errors _k = x– x ^k of the AOR method in terms of the norms of _k = x ^k–x ^k–1 and _k+1 = x ^k+1–x ^k and their inner product is derived.     

A q-Analogue of the Euler Gamma Integral

We discuss q-analogues of the Euler reflection formula and the Euler gamma integral. The central role here is played by the Ramanujan q-extension of the Euler integral representation for the gamma function, which allows deriving the Mellin integral transformations for the q-polynomials of Stieltjes–Wigert, Rogers–Szegö, Laguerre, and Wall, for the alternative q-polynomials of Charlier, and for the little q-polynomials of Jacobi.     

The nuclear method in potential scattering theory

The perturbation of a Schrödinger operator H₀ with an arbitrary bounded potential function q, decreasing sufficiently fast at infinity, is considered. With the aid of results of the nuclear theory, for the corresponding pair of Hamiltonians H₀, H=H₀+q, one establishes the existence and the completeness of the wave operators. Generalizations are given to a wider class of unperturbed operators H₀, and also to perturbations by firstorder differential operators. In addition, perturbations by integral operators of Fourier type are investigated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 171, pp. 12–35, 1989.     

Gauge-Invariant Regularization of Quantum Field Theory on the Light Front

We briefly describe problems of the Hamiltonian approach for quantizing gauge fields on the light front for space–time bounded by the inequality |x ^–|L with periodic boundary conditions in the variable x ^– imposed on all fields (the DLCQ method). With these restrictions, we consider the gauge-invariant ultraviolet regularization by passing to a lattice in transverse coordinates. We remove the remaining ultraviolet divergences in the longitudinal momentum p _– by imposing a gauge-invariant finite-mode regularization. It turns out that the canonical formalism on the light front with such a regularization imposed does not contain the usual most complicated second-class constraints between zero and nonzero modes of fields. The described scheme can be used both to regularize the standard gauge theory and to provide a gauge-invariant formulation of effective low-energy models on the light front. Because the manifest Lorentz invariance is broken in our formalism, the vacuum state is poorly defined. We discuss this problem, in particular, in relation to the problem of passing to the continuous space limit.     

On the periodic solutions of the second-order wave equations. V

It is established that the linear problemu _u –a ² u _xx =g(x,t),u(0,t) =u(x, t + T) =u(x,t) is always solvable in the function spaceA = {g:g(x,t) =g(x,t+T) =g( –x,t) = –g(–x,t)} provided thataTq = (2p – 1) and (2p – 1,q) = 1, wherepandq are integer numbers. To prove this statement, an exact solution is constructed in the form of an integral operator, which is used to prove the existence of a solution of a periodic boundary-value problem for a nonlinear second-order wave equation. The results obtained can be used when studying the solutions to nonlinear boundary-value problems by asymptotic methods.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1115–1121, August, 1993.     

On parabolic Whittaker functions II

We propose a Givental-type stationary phase integral representation for the restricted Gr _m,N -Whittaker function, which is expected to describe the (S ¹×U _N )-equivariant Gromov-Witten invariants of the Grassmann variety Gr _m,N . Our key tool is a generalization of the Whittaker model for principal series U(gl _N )-modules, and its realization in the space of functions of totally positive unipotent matrices. In particular, our construction involves a representation theoretic derivation of the Batyrev-Ciocan-Fontanine-Kim-van Straten toric degeneration of Gr _m,N .