MP-algebras with relative types

In this paper we define anMP-algebra with relative types (B, s, t ₁ t ₂,t ₃) where (B,s) is a monadic algebra and wheret ₁,t ₂,t ₃ arerelative types fromB to itself satisfying:t ₁ (ps(q))=t _i (p)s(q),s(t _i (p))=t _i (p),s(p)t ₁ (p)t ₂ (p)t ₃ (p), ifi j thent _i (p)t _j (p)=0, ifp q) thent ₃ (p) =t ₃ (q) andt ₂(p) t₃ (p) t ₂ (q) t ₃ (q), ifp q =0 thent _i ,(p) t _j (q) t _k (pq) withk=min(i + j, 3). Every relation algebra has anMP-algebra with relative types associated with it. We prove by Givant's results that everyMP-algebra with relative types arises in this way from some relation algebra generated by its rectangles.Presented by B. Jónsson.     

On the Minimum Size of Some Minihypers and Related Linear Codes

We denote by m_r,q(s) the minimum value of f for which an {f, _r-2+s ; r,q }-minihyper exists for r 3, 1 s q–1, where _j=(q^j+1–1)/(q–1). It is proved that m_3,q(s)=₁(₁+s) for many cases (e.g., for all q 4 when ) and that m_r,q(s) _r-1+s₁+q for 1 s q – 1,~q 3,~r 4. The nonexistence of some [n,k,n+s–q^k-2]_q codes attaining the Griesmer bound is given as an application.AMS classification: 94B27, 94B05, 51E22, 51E21     

Monotone subsequences in (0, 1)-matrices

A (0, 1)-matrix contains anS ₀(k) if it has 0-cells (i, j ₁), (i + 1,j ₂),..., (i + k – 1,j _k) for somei andj ₁ < ... < j_k, and it contains anS ₁(k) if it has 1-cells (i ₁,j), (i ₂,j + 1),...,(i _k ,j + k – 1) for somej andi ₁ < ... <i _k . We prove that ifM is anm × n rectangular (0, 1)-matrix with 1 m n whose largestk for anS ₀(k) isk ₀ m, thenM must have anS ₁(k) withk m/(k ₀ + 1). Similarly, ifM is anm × m lower-triangular matrix whose largestk for anS ₀(k) (in the cells on or below the main diagonal) isk ₀ m, thenM has anS ₁(k) withk m/(k ₀ + 1). Moreover, these results are best-possible.     

Splitting with Continuous Control in Algebraic K-Theory

In this work, the continuously controlled assembly map in algebraic K-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups that satisfy certain geometric conditions. The group is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K₀(k) is proved to be isomorphic to the colimit of K₀(kH) over the finite subgroups H of , when is a virtually polycyclic group and k is a field of characteristic zero.     

Classes of graphs which approximate the complete euclidean graph

LetS be a set ofN points in the Euclidean plane, and letd(p, q) be the Euclidean distance between pointsp andq inS. LetG(S) be a Euclidean graph based onS and letG(p, q) be the length of the shortest path inG(S) betweenp andq. We say a Euclidean graphG(S)t-approximates the complete Euclidean graph if, for everyp, q S, G(p, q)/d(p, q) t. In this paper we present two classes of graphs which closely approximate the complete Euclidean graph. We first consider the graph of the Delaunay triangulation ofS, DT(S). We show that DT(S) (2/(3 cos(/6)) 2.42)-approximates the complete Euclidean graph. Secondly, we define(S), the fixed-angle-graph (a type of geometric neighbor graph) and show that(S) ((1/cos)(1/(1–tan)))-approximates the complete Euclidean graph.     

The Domination Parameters of Cubic Graphs

Let ir(G), (G), i(G), ₀(G), (G) and IR(G) be the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number of a graph G, respectively. In this paper we show that for any nonnegative integers k₁, k₂, k₃, k₄, k₅ there exists a cubic graph G satisfying the following conditions: (G) – ir(G) k₁, i(G) – (G) k₂, ₀(G) – i(G) > k₃, (G) – ₀(G) – k₄, and IR(G) – (G) – k₅. This result settles a problem posed in [9].Supported by the INTAS and the Belarus Government (Project INTAS-BELARUS 97-0093).Supported by RUTCOR.     

Presentations for Certain Chevalley Groups

Let G = P SL_n(K), n 3, K a division ring or D_n(K), n 4 or E_n(K), 6 n 8, K a field. Then two types of presentations for G are given. In the first, G is generated by SL₂(K)'s which satisfies relations according to the Dynkin diagram of G. In the second, G is generated by a set {A_i | i I } of Abelian groups, which satisfy relations similar to the root subgroups of G.     

Un théorème sur les séries de Dirichlet

We considerk Dirichlet series a _j (n)n ^–s (1jk),k2. We suppose that for eachj the series a _j (n)n ^–s converges fors=s _j =j+it _j , and that Max j<1/(k–1). We prove that the (Dirichlet) product of these series converges uniformly on every bounded segment of the line es = (₁+...+ _k )/k+1–1/k and we estimate the rate of convergence. The number 1–1/k cannot be replaced by a smaller one.     

Tauberian Theorems for Weighted Means of Double Sequences

Let p := {p _j } _j=0 and q := {q _k } _k–0 be complex (or real) sequences with the property that P _m := _j–0 ^m p _j 0 for all m 0, Q _n := _k–0 ⁿ q _k 0 for all n 0, and both of {P _m } _m=0 and {Q _n } _n=0 are varying away from 1. Assume that {s _mn } is a double sequence in C(or one of R, a Banach space, and an ordered linear space), which is (N¯,p,q; ,) summable to a finite limit, where (,) =(1,1), (1,0), or (0,1). We give necessary and sufficient conditions under which {s _mn } converges in Pringsheim's sense. These conditions are weaker than the two-dimensional analogues of Landau's condition and Schmidt's slow decrease condition. Our results generalize and extend [1 4, 12 15]. We also solve the problems posed in [3, 13, 14].     

On blocking sets in affine planes

Denote by _q an affine plane of order q. In the desarguesian case _q=AG(2,q), q 5(q= p^h, p prime), we prove that the smallest cardinality of a blocking set is 2q–1. In any arbitrary affine plane _q (desarguesian or not) with q5, for any integer k with 2q–1 k(q–1)², we construct a blocking set S with ¦S¦=k. For an irreducible blocking set S of _q we determine the upper bound S [qq]+1. We prove that if _q contains a blocking set S which is irreducible with its complementary blocking set, then necessarily _q=AG(2, 4) and S is uniquely determined. Finally we introduce techniques to obtain blocking sets in AG(2, q) and in PG(2, q).Research partially supported by G.N.S.A.G.A. (CNR)     

Normal Rational Curves Over Prime Fields

It is shown that in the projective spaces PG(n,p),p prime, 2 n p-2, the normal rational curves are the only (p+1)-arcs fixed by a projective group G isomorphic to PSL(2,p).     

Random Lazy Random Walks on Arbitrary Finite Groups

This paper considers lazy random walks supported on a random subset of k elements of a finite group G with order n. If k=a log₂ n where a>1 is constant, then most such walks take no more than a multiple of log₂ n steps to get close to uniformly distributed on G. If k=log₂ n+f(n) where f(n) and f(n)/log₂ n0 as n, then most such walks take no more than a multiple of (log₂ n) ln(log₂ n) steps to get close to uniformly distributed. To get these results, this paper extends techniques of Erdös and Rényi and of Pak.     

Remarks on the perturbation of analytic matrix functions III

As in [N], [LN] the Newton diagram is used in order to get information about the first terms of the Puiseux expansions of the eigenvalues () of the perturbed matrix pencilT(, )=A()+B(, ) in the neighbourhood of an unperturbed eigenvalue () ofA(). In fact sufficient conditions are given which assure that the orders of these first terms correspond to the partial multiplicities of the eigenvalue ₀ ofA().     

Differentiable Connections with Poincar{e} Groups of Local Transformations. {III}

In the canonical smooth fiber bundles :ⁿ⁺¹ⁿ, we study generalized differentiable connections constructed by the author in papers [4] and [5]. Special emphasis is laid on the investigation of the behavior of these connections under local transformations of the classical Poincar{e} groups (1,n) and extended Poincar{e} groups canonically acting in the given connections. We found all first-order nonholonomic affine, ₁, ₂, and _1,2-connections with the groups (1,n) and of local transformations and also constructed classes of the corresponding invariant second-order connections.     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Blocking Structures of Hermitian Varieties

Given a hermitian variety H(d,q²) and an integer k (d–1)/2, a blocking set with respect to k-subspaces is a set of points of H(d,q²) that meets all k-subspaces of H(d,q²). If H(d,q²) is naturally embedded in PG(d,q²), then linear examples for such a blocking set are the ones that lie in a subspace of codimension k of PG(d,q²). Up to isomorphism there are k+1 non-isomorphic minimal linear blocking sets, and these have different cardinalities. In this paper it is shown for 1 k< (d–1)/2 that all sufficiently small minimal blocking sets of H(d,q²) with respect to k-subspaces are linear. For 1 k< d/2–3, it is even proved that the k+1 minimal linear blocking sets are smaller than all minimal non-linear ones.AMS Classification: 1991 MSC: 51E20, 51E21     

Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)

We investigate a connection between distance-regular graphs and U _q(sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let be a distance-regular graph with diameter d 3 and valency k 3, and assume is not isomorphic to the d-cube. Fix a vertex x of , and let (x) denote the Terwilliger algebra of with respect to x. Fix any complex number q {0, 1, –1}. Then is generated by certain matrices satisfying the defining relations of U _q(sl(2)) if and only if is bipartite and 2-homogeneous.     

Building Symmetric Designs With Building Sets

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q ^m+1-1)/(q-1), kq ^m ,q ^m), where m is any positive integer, (v, k, ) are parameters of an abelian difference set, and q = k ²/(k - ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v, k, ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9 · 4 ^d , thus obtaining seven infinite families of symmetric designs.     

On a Problem of D. H. Lehmer and Kloosterman Sums

Let q3 be an odd number, a be any fixed positive integer with (a, q)=1. For each integer b with 1b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bca (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bca (mod q) for 1b, c<q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.Received January 24, 2002; in revised form August 12, 2002 Published online February 28, 2003