Extension of the ν-metric

We extend the ν-metric introduced by Vinnicombe in robust control theory for rational plants to the case of infinite-dimensional systems/classes of nonrational transfer functions.     

Hamiltonians associated with the third and fifth Painlevé equations

We obtain a Painlevé-type differential equation for the simplest rational Hamiltonian associated with the fifth Painlevé equation in the case γ ≠ 0, δ = 0. We prove the existence of Hamiltonians of a nonrational type associated with the fifth Painlevé equation in the case γ ≠ 0, δ = 0. We obtain a generalization of the Garnier and Okamoto formulas for rational Hamiltonians associated with the third Painlevé tequation.     

Del Pezzo surfaces with nonrational singularities

Normal algebraic surfacesX with the property rk(Div(X)⊗ℚ/≡)=1, numerically ample canonical classes, and nonrational singularities are classified. It is proved, in particular, that any such surfaceX is a contraction of an exceptional section of a (possibly singular) relatively minimal ruled surface with a nonrational base. Moreover, f is uniquely determined by the surfaceX. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 451–467, September, 1997. Translated by O. V. Sipacheva     

Double cubics and double quartics

We study a double cover branched over a smooth divisor such that R is cut on V by a hypersurface of degree 2(n−deg(V)), where n ≥ 8 and V is a smooth hypersurface of degree 3 or 4. We prove that X is nonrational and birationally superrigid.     

Certain examples of birationally rigid varieties pencil of double quadrics

We prove that double covers ofP ¹×Q, where is a nondegenerate quadric, and the branch divisor cut out by a hypersurface of the type (2λ, 2k−2) are birationally superrigid. In particular, they have only one structure of a Fano fibration. Therefore, they are nonrational. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya, Tematicheskie Obzory. Vol. 56. Algebraic Geometry-9, 1998.     

The genealogy of continuous-state branching processes with immigration

Recent works by J.F. Le Gall and Y. Le Jan [15] have extended the genealogical structure of Galton-Watson processes to continuous-state branching processes (CB). We are here interested in processes with immigration (CBI). The height process H which contains all the information about this genealogical structure is defined as a simple local time functional of a strong Markov process X ^*, called the genealogy-coding process (GCP). We first show its existence using It?’s synthesis theorem. We then give a pathwise construction of X ^* based on a Lévy process X with no negative jumps that does not drift to +∞ and whose Laplace exponent coincides with the branching mechanism, and an independent subordinator Y whose Laplace exponent coincides with the mechanism. We conclude the construction with proving that the local time process of H is a CBI-process. As an application, we derive the analogue of the classical Ray–Knight–Williams theorem for a general Lévy process with no negative jumps. Received: 28 January 2000 / Revised version: 5 February 2001 / Published online: 11 December 2001     

A Family of Complete Caps in PG (<Emphasis Type="Italic">n</Emphasis>, 2)

We give a combinatorial construction of a one-parameter and a two-parameter family of complete caps in finite projective spaces over GF(2). As an application of our construction we find, for each α ε[1.89,2], a sequence of complete caps in PG(n,2) whose sizes grow roughly as α ⁿ . We also discuss the relevance of our caps to the problem of finding the least dependent caps of a given cardinality in a given dimension. *Research partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). An erratum to this article is available at .     

On nonrational divisors over non-Gorenstein terminal singularities

Let (X, o) be a germ of a 3-dimensional terminal singularity of index m ≥ 2. If (X, o) has type cAx/4, cD/3-3, cD/2-2, or cE/2, then we assume that the standard equation of X in ℂ⁴/ℤ _m is nondegenerate with respect to its Newton diagram. Let π: Y → X be a resolution. We show that there are at most 2 nonrational divisors E _i , i = 1, 2, on Y such that π(E _i ) = o and the discrepancy a(E _i , X) is at most 1. When such divisors exist, we describe them as exceptional divisors of certain blowups of (X, o) and study their birational type. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 169–184, 2005.     

The orbit method for profinite groups and a p-adic analogue of Brown’s theorem

We develop an approach to the character theory of certain classes of finite and profinite groups based on the construction of a Lie algebra associated to such a group, but without making use of the notion of a polarization which is central to the classical orbit method. Instead, Kirillov’s character formula becomes the fundamental object of study. Our results are then used to produce an alternate proof of the orbit method classification of complex irreducible representations of p-groups of nilpotence class < p, where p is a prime, and of continuous complex irreducible representations of uniformly powerful pro-p-groups (with a certain modification for p = 2). As a main application, we give a quick and transparent proof of the p-adic analogue of Brown’s theorem, stating that for a nilpotent Lie group over ℚ_p the Fell topology on the set of isomorphism classes of its irreducible representations coincides with the quotient topology on the set of its coadjoint orbits. The research of M. B. was partially supported by NSF grant DMS-0401164.     

Explicit constructions of graphs without short cycles and low density codes

We give an explicit construction of regular graphs of degree 2r withn vertices and girth ≧c logn/logr. We use Cayley graphs of factor groups of free subgroups of the modular group. An application to low density codes is given.     

New Coins From Old: Computing With Unknown Bias

Suppose that we are given a function f : (0, 1)→(0,1) and, for some unknown p∈(0, 1), a sequence of independent tosses of a p-coin (i.e., a coin with probability p of “heads”). For which functions f is it possible to simulate an f(p)-coin? This question was raised by S. Asmussen and J. Propp. A simple simulation scheme for the constant function f(p)≡1/2 was described by von Neumann (1951); this scheme can be easily implemented using a finite automaton. We prove that in general, an f(p)-coin can be simulated by a finite automaton for all p ∈ (0, 1), if and only if f is a rational function over ℚ. We also show that if an f(p)-coin can be simulated by a pushdown automaton, then f is an algebraic function over ℚ; however, pushdown automata can simulate f(p)-coins for certain nonrational functions such as . These results complement the work of Keane and O’Brien (1994), who determined the functions f for which an f(p)-coin can be simulated when there are no computational restrictions on the simulation scheme. * Supported by a Miller Fellowship. † Supported in part by NSF Grant DMS-0104073 and by a Miller Professorship. ‡ This work is supported under a National Science Foundation Graduate Research Fellowship.     

Completely periodic directions and orbit closures of many pseudo-Anosov Teichmueller discs in Q(1,1,1,1)

In this paper, we investigate the closure of a large class of Teichmüller discs in the stratum Q(1, 1, 1, 1){\mathcal{Q}(1, 1, 1, 1)} or equivalently, in a GL⁺₂(\mathbbR){{\rm GL}^+_2(\mathbb{R})} -invariant locus L{\mathcal{L}} of translation surfaces of genus three. We describe a systematic way to prove that the GL⁺₂(\mathbbR){{\rm GL}^+_2(\mathbb{R})} -orbit closure of a translation surface in L{\mathcal{L}} is the whole locus L{\mathcal{L}} . The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratner’s theorem to unipotent subgroups acting on an “adequate” splitting. This analysis applies for example to all Teichmüller discs obtained by the Thurston–Veech’s construction with a trace field of degree three which are moreover “obviously not Veech”. We produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on L{\mathcal{L}} , contrary to the situation in genus two. We also study completely periodic directions on translation surfaces in L{\mathcal{L}} . For instance, we prove that completely periodic directions are dense on surfaces obtained by the Thurston–Veech’s construction.     

On improved asymptotic bounds for codes from global function fields

We analyse a binary cyclotomic sequence constructed via generalized cyclotomic classes by Bai et al. (IEEE Trans Inforem Theory 51: 1849–1853, 2005). First we determine the linear complexity of a natural generalization of this binary sequence to arbitrary prime fields. Secondly we consider k-error linear complexity and autocorrelation of these sequences and point out certain drawbacks of this construction. The results show that the parameters for the sequence construction must be carefully chosen in view of the respective application.      

Bulk Correlation Functions in 2d Quantum Gravity

We compute the bulk three- and four-point tachyon correlators in the 2d Liouville gravity with a nonrational matter central charge c < 1, using and comparing two approaches. The continuous CFT approach exploits the action on the tachyons of the ground-ring generators deformed with Liouville and matter screening charges. We derive a general formula for the matter three-point OPE structure constants as a by-product. The discrete formulation of the theory is a generalization of the ADE string theories, in which the target space is the semi-infinite chain of points. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 132–145, January, 2006.     

Factorization of a class of matrix‐functions with stable partial indices

A new effective method for factorization of a class of nonrational n × n matrix‐functions with stable partial indices is proposed. The method is a generalization of one recently proposed by the authors, which was valid for the canonical factorization only. The class of matrices being considered is motivated by their applicability to various problems. The properties and steps of the asymptotic procedure are discussed in detail. The efficiency of the procedure is highlighted by numerical results. Copyright © 2016 John Wiley & Sons, Ltd.     

On the resolution of 3-dimensional terminal singularities

We show that there is at most one nonrational exceptional divisor with discrepancy 1 over a three-dimensional terminal point of type cD. If such a divisor exists, then it is birationally isomorphic to the surface ¹ × C, where C is a hyperelliptic (for g(C) > 1) curve.     

Rigid modules over preprojective algebras

Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis. Mathematics Subject Classification (2000) 14M99, 16D70, 16E20, 16G20, 16G70, 17B37, 20G42     

Constructing pairs of dual bandlimited framelets with desired time localization

For sufficiently small translation parameters, we prove that any bandlimited function ψ, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame with a dual frame also having the wavelet structure. This dual frame is generated by a finite linear combination of dilations of ψ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction is based on characterizing equations for dual wavelet frames and relies on a technical condition. We exhibit a general class of function satisfying this condition; in particular, we construct piecewise polynomial functions satisfying the condition.      

Multipoint Taylor formulæ

Summary. The main result of this paper is an abstract version of the Kowalewski–Ciarlet–Wagschal multipoint Taylor formula for representing the pointwise error in multivariate Lagrange interpolation. Several applications of this result are given in the paper. The most important of these is the construction of a multipoint Taylor error formula for a general finite element, together with the corresponding –error bounds. Another application is the construction of a family of error formul? for linear interpolation (indexed by real measures of unit mass) which includes some recently obtained formul?. It is also shown how the problem of constructing an error formula for Lagrange interpolation from a D–invariant space of polynomials with the property that it involves only derivatives which annihilate the interpolating space can be reduced to the problem of finding such a formula for a ‘simpler’ one–point interpolation map. Received March 29, 1996 / Revised version received November 22, 1996     

Counting permutations with no long monotone subsequence via generating trees and the kernel method

We recover Gessel’s determinantal formula for the generating function of permutations with no ascending subsequence of length m+1. The starting point of our proof is the recursive construction of these permutations by insertion of the largest entry. This construction is of course extremely simple. The cost of this simplicity is that we need to take into account in the enumeration m−1 additional parameters—namely, the positions of the leftmost increasing subsequences of length i, for i=2,…,m. This yields for the generating function a functional equation with m−1 “catalytic” variables, and the heart of the paper is the solution of this equation.