Existence of Perfect 4-Deletion-Correcting Codes with Length Six

By a T ^*(2, k, v)-code we mean a perfect4-deletion-correcting code of length 6 over an alphabet of size v, which is capable of correcting anycombination of up to 4 deletions and/or insertions of letters that occur in transmission of codewords. Thethird author (DCC Vol. 23, No. 1) presented a combinatorial construction for such codes and prove thata T ^*(2, 6, v)-code exists for all positive integers v 3 (mod 5), with 12 possible exceptions of v. In this paper, the notion of a directedgroup divisible quasidesign is introduced and used to show that a T ^*(2, 6,v)-code exists for all positive integers v 3 (mod 5), except possiblyfor v {173, 178, 203, 208}. The 12 missing cases for T ^*(2,6, v)-codes with v 3 (mod 5) are also provided, thereby the existenceproblem for T ^*(2, 6, v)-codes is almost complete.     

Resolvable balanced incomplete block designs with subdesigns of block size 4

In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly (v,4,1) RBIBDs. The problem we investigate is the existence of (v,4,1) RBIBDs containing a (w,4,1) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB(v,w). The necessary conditions for the existence of an IRB(v,w) are that v?4w and . We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever and whenever w?1852.We also give some results on pairwise balanced designs on v points containing (at least one) block of size w, i.e., a (v,{K,w^*},1)-PBD.If the list of permitted block sizes, K₅, contains all integers of size 5 or more, and v,w∈K₅, then a necessary condition on this PBD is v?4w+1. We show this condition is not sufficient for any w?5 and give the complete spectrum (in v) for 5?w?8, as well as showing the condition v?5w is sufficient with some definite exceptions for w=5 and 6, and some possible exceptions when w=15, namely 77?v?79. The existence of this PBD implies the existence of an IRB(12v+4,12w+4).If the list of permitted block sizes, K₁₍₄₎, contains all integers , and v,w∈K₁₍₄₎, then a necessary condition on this PBD is v?4w+1. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w?1037. The existence of this PBD implies the existence of an IRB(3v+1,3w+1).     

A splitting theorem for certain cohomology theories associated to BP* (−)

Let P(n)^*(–) be Brown-Peterson cohomology modulo I_n and put B(n)^*(–)=P(n)^*(–)[1/v_n]. In this note we construct a canonical multiplicative and idempotent operation _n in a suitable completion (n)^*(–) of B(n)^*(–) which has the property that its image is canonically isomorphic to the n-th Morava K-theory K(n)^*(–). In particular, the ring theory K(n)^*(–) is contained as a direct summand in the theory (n)^*(–). A similar result is not true before completing. pleting. Because the completion map B (n)^*(–) (n)^*(–) is injective, the above splitting theorem contains also information about B(n)^*(–). The proof of the theorem depends on a result about the behaviour of formal groups of finite height over complete graded F_p.     

An infinite order operator on the lattice of varieties of completely regular semigroups

Let be a variety of completely regular semigroups. Define C ^* to be the class of all completely regular semigroupsS whose least full and self-conjugate subsemigroupC ^*(S) belongs to . ThenC ^* is an operator on the lattice of varieties of completely regular semigroups. In this note we show that the order ofC ^* is infinite. This fact yields that the Mal'cev project is not associative on . We describe (C ^*)¹, andi 0, in terms of -invariant normal subgroups of the free group over a countably infinite set. The lattice theoretic properties ofC ^* are also studied.Presented by W. Taylor.     

A Combinatorial Construction for Perfect Deletion-Correcting Codes

By a T ^*(2, k, v)-code we mean a perfect(k-2)-deletion-correcting code of length k over an alphabet ofsize v, which is capable of correcting any combination of up to(k-2) deletions and insertions of letters occured in transmission ofcodewords. In this paper, we provide a combinatorial construction forT ^*(2, k, v-codes. As an application, we show that aT ^*(2, 6, v-code exists for all positive integersv 3 (mod 5), with at most 12 possible exceptions of v. In theprocedure, a result on incomplete directed BIBDs is also established which is ofinterest in its own right.     

On the number of limit cycles in perturbations of a two dimensional nonlinear differential system

Summary For the nonlinear system , which has a family { _h } of closed orbits, we consider perturbations of the type , whereP andQ are arbitrary polynomials. The abelian integralsA(h) corresponding to this family { _h } are investigated. By deriving differential equations forA(h) and proving monotonicity for quotients of abelian integrals, we obtain results on the number of zeros of abelian integrals and, hence, on the number of closed orbits _h which persist as limit cycles of the perturbed system (^*). In particular, a uniqueness theorem for limit cycles of (^*) with quadratic polynomialsP, Q is proved. Moreover, whenP, Q are of arbitrary degree, a lower bound for the possible number of limit cycles of (^*) is derived.     

On the family of translations of a tight probability measure on a topological group

Summary Let P be a tight probability measure on an Abelian normal topological group and the family of all its translations ({P(t ^–1)}). We shall investigate the closed convex hull of the set , where the closure is taken in the weak topology. Theorem 3 shows that a probability measure Q is an element of the closed convex hull of if and only if there exists a probability measure R such that Q=P*R, where P*     

Inequalities of Putnam and Berger-Shaw forp-quasihyponormal operators

For an operatorT satisfying thatT ^*(T ^* T–TT ^*)T0, we shall show that and, moreover, tr itT isn-multicyclic.For an operatorT satisfying thatT ^* {(T ^* T) ^p –(TT ^*) ^p }T0 for somep (0, 1], we shall show that and, moreover, ifT isn-multicyclic.     

The Existence of Kirkman Squares—Doubly Resolvable (v,3,1)-BIBDs

A Kirkman square with index , latinicity , block size k, and v points, KS _k (v;,), is a t×t array (t=(v–1)/(k–1)) defined on a v-set V such that (1) every point of V is contained in precisely cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k,)-BIBD. For =1, the existence of a KS _k (v; , ) is equivalent to the existence of a doubly resolvable (v, k, )-BIBD. The spectrum of KS ₂ (v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with =1. We show that there exist KS ₃ (v; 1, 1) for , v=3 and v27 with at most 23 possible exceptions for v.     

A general descent framework for the monotone variational inequality problem

We present a framework for descent algorithms that solve the monotone variational inequality problem VIP _v which consists in finding a solutionv ^* _v satisfyings(v ^*)^T(v–v ^*)0, for allv _v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.This research was supported in part by individual operating grants from NSERC.     

Semi-Invertible Extensions and Asymptotic Homomorphisms

We consider the semi-group Ext(A, B) of extensions of a separable C ^*-algebra A by a stable C ^*-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semi-group contains the group Ext^–1/2(A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext^–1/2(A, B) is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from C A to M(B) that map S A C( ) A into B.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equationTx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution belongs to the range of the operator (T ^* T) ^v , o<v1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate ifv=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2v1.The final version of this work was written while M. T. Nair was a Visiting Fellow at the Centre for Mathematics and Its Applications, Australian National University, Canberra, Australia. The work of S. George was supported by a Senior Research Fellowship from CSIR, India.     

Décomposition spectrale dans des algèbres munies d'un star-produit

Summary The purpose of this paper is to study, in intrinsic way, the Moyal's product, defined in the flat space R ²ⁿ. This product is defined here with the twisted convolution and the Fourier transform. The S(R ²ⁿ) and L²(R ²ⁿ) spaces are_*5-algebras. Because of this definition, the_*V-product of some tempered distributions is defined. Let O _M ^v be the set of multiplication operators in S(R ²ⁿ). By transposition, the S(R ²ⁿ) space is a right-module on O _M ^v . The support of f_*v g is different from the support of f·g; under large enough hypotheses, there is a Taylor's formula for the star-product function of the v variable. The ^v space of the multiplication operators in L²(R ²ⁿ) is defined here as the space of tempered distributions, the image of which is the set of bounded operators in L²(R ²ⁿ) by the Weyl map. After the study of ^v space, it is possible to show the spectral resolution of the real elements of ^v or of O _M ^v , which satisfies a, probably superfluous, hypothesis.     

Surface- and interface oscillations of a rotating viscoelastic liquid column of immiscible liquids

Summary The natural damped coupled frequencies of a solidly rotating visco-elastic infinite liquid column with no axial dependency (/z=0, two-dimensional problem) have been determined. The frequency equation is numerically evaluated for a single visco-elastic liquid column, where the influence of the tension parameterTa/v ², the relaxation parameter ^*/a ² and the rotational Reynoldsnumber =Taylornumber) has been determined. It was found that the liquid column becomes unstable for a rotational speed , which is much earlier than in the case of frictionless liquid, where . In addition the stability boundary does neither depend on the magnitude of the viscosity nor the Maxwell relaxation time ^* of the liquid. The complex frequencies are presented for the modem=2, where the cross-section of the liquid column assumes during its oscillation an elliptic shape.

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Zusammenfassung Es werden die gekoppelten gedämpften Frequenzen einer rotierenden visko-elastischen Flüssigkeitssäule ohne axiale Abhängigkeit (zweidimensionales Problem) aus nichtmischbaren Flüssigkeiten bestimmt. Die Frequenzgleichung einer einfachen viskoelastischen Säule wird numerisch ausgewertet, wobei der Einfluß des OberflächenspannungsparametersTa/v ², des Relaxationparameters ^* v/a ² und der rotierenden Reynoldszahl untersucht wird. Die Flüssigkeitssäule wird instabil bei , unabhängig von der Größe der Viskosität und der Relaxationszeit. Bei reibungsfreier Flüssigkeit tritt diese Instabilität erst für größeres auf. Es werden die komplexen Frequenzen für die elliptische Querschnitts-Schwingungsform m=2 berechnet.

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Die Veröffentlichung wurde aus Haushaltsmitteln der Universität der Bundeswehr München gefördert.     

Bi-invariant Integrals on GL(n) with Applications

In this paper measures and functions on GL(n) are called bi-invariant if they are invariant under left and right multiplication of their arguments. If v is any bi-invariant Borel measure on GL(n), then there exists a unique Borel measure v^* on D _{+ ≥}(n), the set of all diagonal matrices of rank n with positive non-increasing diagonal entries, such that holds for each v-integrable bi-invariant function f:GL(n) → IR. An explicit formula for v^* will be derived in case v equals the Lebesgue measure on GL(n) and the above integral formula will be applied to concrete integration problems. In particular, if v is a probability measure, then v^* can be interpreted as the distribution of the singular value vector. This fact will be used to derive a stochastic version of a theorem from perturbation theory concerning the numerical computation of the polar decomposition.     

Existence of T*(3, 4,v)‐codes

A word of length k over an alphabet Q of size v is a vector of length k with coordinates taken from Q. Let Q^*₄ be the set of all words of length 4 over Q. A T^*(3, 4, v)‐code over Q is a subset C^*? Q^*₄ such that every word of length 3 over Q occurs as a subword in exactly one word of C^*. Levenshtein has proved that a T^*(3, 4, vv)‐code exists for all even v. In this paper, the notion of a generalized candelabra t‐system is introduced and used to show that a T^*(3, 4, v)‐code exists for all odd v. Combining this with Levenshtein's result, the existence problem for a T^*(3,4, v)‐code is solved completely. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 42–53, 2005.     

Block-Transitive Point-Imprimitive t-Designs

We study block-transitive point-imprimitive t–(v, k, ) designs. It was showed by Cameron and Praeger that in such designs t = 2 or 3. In 1989, Delandtsheer and Doyen proved that a block-transitive point-imprimitive 2-design satisfies v (( ^k ₂)–1)². In this paper, we give a proof of the Cameron–Praeger conjecture which states that for t = 3 the stronger inequality v ( ^k ₂)+1 holds. We find two infinite families of 3-designs for which this bound is met. We also show that the above designs cannot have = 1, and that = 2 is possible only if v attains its maximal value, and various other restrictions are met.     

Designs on vector spaces constructed using quadratic forms

Let N be the set of nonnegative integers, let , t, v be in N and let K be a subset of N, let V be a v-dimensional vector space over the finite field GF(q), and let W _Kbe the set of subspaces of V whose dimensions belong to K. A t-[v, K, ; q]-design on V is a mapping : W _K N such that for every t-dimensional subspace, T, of V, we have (B)=. We construct t-[v, {t, t+1}, ; q-designs on the vector space GF(q ^v) over GF(q) for t2, v odd, and q ^t(q–1)² equal to the number of nondegenerate quadratic forms in t+1 variables over GF(q). Moreover, the vast majority of blocks of these designs have dimension t+1. We also construct nontrivial 2-[v, k, ; q]-designs for v odd and 3kv–3 and 3-[v, 4, q ⁶+q ⁵+q ⁴; q]-designs for v even. The distribution of subspaces in the designs is determined by the distribution of the pairs (Q, a) where Q is a nondegenerate quadratic form in k variables with coefficients in GF(q) and a is a vector with elements in GF(q ^v) such that Q(a)=0.This research was partly supported by NSA grant #MDA 904-88-H-2034.     

Uniformly resolvable designs with index one, block sizes three and five and up to five parallel classes with blocks of size five

Each parallel class of a uniformly resolvable design (URD) contains blocks of only one block size k (denoted k-pc). The number of k-pcs is denoted r_k. The necessary conditions for URDs with v points, index one, blocks of size 3 and 5, and r₃,r₅>0, are . If r_k>1, then v≥k², and r₃=(v−1−4⋅r₅)/2. For r₅=1 these URDs are known as group divisible designs. We prove that these necessary conditions are sufficient for r₅=3 except possibly v=105, and for r₅=2,4,5 with possible exceptions (v=105,165,285,345) New labeled frames and labeled URDs, which give new URDs as ingredient designs for recursive constructions, are the key in the proofs.