A Primitive Non-symmetric 3-Class Association Scheme on 36 Elements with p11 = 0 Exists and is Unique

In this paper we construct a primitive, non-symmetric 3-class association scheme with parameters v = 36, v₁ = 7, p¹₁₁ = 0 and p²₁₁ = 4 and show that such a scheme is determined by its parameters.     

Onp-class groups of cyclic extensions of prime degreep of certain cyclotomic fields

Letp be an odd prime number, and letK be a cyclic extension of ℚ(ζ) of degreep, where ζ is a primitivep-th root of unity. LetC _K be thep-class group ofK, and letr _K be the minimal number of generators ofC _K ^1−σ as a module over Gal(K/ℚ(ζ)), were σ is a generator of Gal(K/ℚ(ζ)). This paper shows how likely it is forr _K = 0, 1, 2, … whenp=3, 5, or 7, and describes the obstacle to generalizing these results to regular primesp>7.     

Algebraic surfaces of general type withK 2=2p g−1,p g≥5

This paper mainly deals with minimal algebraic surfaces of general type withK ²=2p _g–1. We prove that forp _g7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces withp _g=5, 6. Lastly we show that whenp _g13 all these surfaces are simply-connected.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

PBIB designs with m associate classes induced by full rank matrices modulo pm

This paper presents series of PBIB designs with m associate classes in which the treatment set is a subset of the Z(p^m)-module of n × 1 vectors over the ring of integers modulo p^m, p any prime. The association scheme of this series of designs is determined by the Fuller canonical form under row equivalence of n × 2 matrices [a,b] for vectors a and b in the treatment set. The blocking procedure utilizes full rank s × n matrices over Z(p^m), 1 ? s ? n ? 2, n ? 3. For m = 2, n = 3, s =1 and for each prime p, each PBIB is regular divisible and yields a finite proper uniform projective Hjelmslev plane with parameters j = p and k = p(p + 1).     

Equally-weighted formulas for numerical differentiation

Equally-weighted formulas for numerical differentiation at a fixed pointx=a, which may be chosen to be 0 without loss in generality, are derived for (1) whereR _2n =0 whenf(x) is any (2n)^th degree polynomial. Equation (1) is equivalent to (2) ,r=1,2,..., 2n. By choosingf(x)=1/(z–x),x _i fori=1,..., n andx _i fori=n+1,..., 2n are shown to be roots ofg _n (z) andh _n (z) respectively, satisfying (3) . It is convenient to normalize withk=(m–1)!. LetP _s (z) denotez ^s · numerator of the (s+1)^th diagonal member of the Padé table fore ^x , frx=1/z, that numerator being a constant factor times the general Laguerre polynomialL _s ^–2s–1 (x), and letP _s (X _i )=0, i=1, ...,s. Then for anym, solutions to (1) are had, for2n=2ms, forx _i , i=1, ...,ms, andx _i , i=ms+1,..., 2ms, equal to all them ^th rootsX _i ^1/m and (–X _i )^1/m respectively, and they give {(2s+1)m–1}^th degree accuracy. For2sm2n(2s+1)m–1, these (2sm)-point solutions are proven to be the only ones giving (2n)^th degree accuracy. Thex _i 's in (1) always include complex values, except whenm=1, 2n=2. For2sm<2n(2s+1)m–1,g _n (z) andh _n (z) are (n–sm)-parameter families of polynomials whose roots include those ofg _ms (z) andh _ms (z) respectively, and whose remainingn–ms roots are the same forg _n (z) andh _n (z). Form>1, and either 2n<2m or(2s+1)m–1<2n<(2s+2)m, it is proven that there are no non-trivial solutions to (1), real or complex. Form=1(1)6, tables ofx _i are given to 15D, fori=1(1)2n, where 2n=2ms ands=1(1) [12/m], so that they are sufficient for attaining at least 24^th degree accuracy in (1).Presented at the Twelfth International Congress of Mathematicians, Stockholm, Sweden, August 15–22, 1962.General Dynamics/Astronautics. A Division of General Dynamics Corporation.     

Spline systems as bases in Hardy spaces

It is known that the spline system of orderm is an unconditional basis forH ^p[0,1] whenp>1/(m+2) and a Schauder basis whenp ≧1/(m+2). We show that these results are sharp.     

On construction ofT m -type PBIB designs

Summary Recently Saha and Das [10] constructed partially balanced incomplete block (PBIB) designs of two and more associate classes by using confounded designs for 2 ⁿ factorials. Several new designs of two associate classes were obtained through those methods. This paper generalizes one of the earlier methods of construction to obtain several series ofT _m -type (m≧2) PBIB designs, i.e., the designs havingm-dimensional triangular association schemes. Some more new designs of two associate classes (i.e.,T ₂-type) are obtained through the generalized methods of construction.     

Potentials and isometric embeddings inL 1

lp/3 does not embed isometrically inL ₁, forp>2. The question whenl _r ⁿ embeds isometrically inL _pis completely answered.     

Association schemes arising from bent functions

We give a construction of 3-class and 4-class association schemes from s-nonlinear and differentially 2 ^s -uniform functions, and a construction of p-class association schemes from weakly regular p-ary bent functions, where p is an odd prime.     

Sur la quasi-convexité des arcs trinomiaux

From the equationp ^n?k sinnθ?ρ ⁿ sin(n?k)θ=sinkθ we will show that the function σ=σ(θ) is increasing for the arcsA _m , obtained when one putsn=m, k=m?1 andm=3,4,5,… Next, we will study the arcsB _m obtained whenn=m, k=m?2 andm an odd integer larger than 3. In this case, σ(θ) will be shown to be a decreasing function. Finally, the Farey arcsF(p,q;r,s) are obtained whenn=s, k=q, s andq relatively prime. It will be proved that the function σ(θ) is strictly quasi-convex.     

Edge sets contained in circuits

A graphG withn vertices has propertyp(r, s) ifG contains a path of lengthr and if every such path is contained in a circuit of lengths. G. A. Dirac and C. Thomassen [Math. Ann.203 (1973), 65–75] determined graphs with propertyp(r,r+1). We determine the least number of edges in a graphG in order to insure thatG has propertyp(r,s), we determine the least number of edges possible in a connected graph with propertyp(r,s) forr=1 and alls, forr=k ands=k+2 whenk=2, 3, 4, and we give bounds in other cases. Some resulting extremal graphs are determined. We also consider a generalization of propertyp(2,s) in which it is required that each pair of edges is contained in a circuit of lengths. Some cases of this last property have been treated previously by U. S. R. Murty [inProof Techniques in Graph Theory, ed. F. Harary, Academic Press, New York, 1969, pp. 111–118].     

On a conjecture of R. E. Miles about the convex hull of random points

Denoty byp _d+i (B _d ,d+m) the probability that the convex hull ofd+m points chosen independently and uniformly from ad-dimensional ballB _d possessesd+i(i=1,...,m) vertices. We prove Mile's conjecture that, given any integerm, p _d+m (B _d ,d+m)»1 asd». This is obvious form=1 and was shown by Kingman form=2 and by Miles form=3. Further, a related result by Miles is generalized, and several consequences are deduced.Dedicated to Professor E. Halwaka on the occasion of his seventieth     

On the 2-class field tower of some imaginary biquadratic number fields

Let be an imaginary biquadratic number field with Cl_k,2, the 2-class group of k, isomorphic to Z/2Z × Z/2^mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1. 2000 Mathematics Subject Classification Primary—11R29     

利用有限局部环Z/PmZ上的2阶交错矩阵构作多个结合类的结合方案

本文取有限局部环Ｚ／Ｐ^mＺ上的全体２×２交错矩阵集作为处理的集合，构作了有ｍ个结合类的结合方案，并且计算出参数     

Rates of convex approximation in non-hilbert spaces

This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the lowest achievable error can be reduced as larger subsets ofS are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including, in particular, a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, forL _p spaces with 1<p<∞, theO (n ^−1/2) bounds of Barron and Jones are recovered whenp=2. One motivation for the questions studied here arises from the area of “artificial neural networks,” where the problem can be stated in terms of the growth in the number of “neurons” (the elements ofS) needed in order to achieve a desired error rate. The focus on non-Hilbert spaces is due to the desire to understand approximation in the more “robust” (resistant to exemplar noise)L _p, 1 ≤p<2, norms. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.     

Uniformly complementedl p n ’s in quasi-reflexive Banach spaces

It is shown that for every non-reflexive Banach spaceX withX ^**/X reflexive there exists a uniformly bounded sequence of projections {P _n } _n=1 ^∞ whose ranges are uniformly isomorphic to {l _p ⁿ } _{n = 1} ^∞ either forp=1, orp=2 or forp=∞. The proof uses knowledge of the transfinite dualX ^ω, ESA Schauder decompositions and proof of a similar statement for spaces with an unconditional basis due to Tzafriri.     

Difference Sets Corresponding to a Class of Symmetric Designs

We study difference sets with parameters(v, k, ) = (p ^s(r ^2m - 1)/(r - 1), p ^s-1 r ^2m-2 r - 1)r ^{2m -2}, where r = r ^s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z ₃ × Z ₉ × Z ₇.     

Estimates of the maximal value of angular code distance for 24 and 25 points on the unit sphere in ℝ4

The present paper is devoted to the well-known problem of determining the maximum number of elementsτ _m (s) of a sphericals-code (−1<-s<1) in Euclidean space ℝ ^m of dimensionm>-2; to be exact, here we consider the Delsarte functionw _m (s) related toτ _m (s) via the inequalityτ _m (s) ≤w _m (s). In this paper, the solution of the equationw _m (s)=N is obtained form=4 andN=24,25. As a consequence, we obtain the assertion that among any 25 (24) points on the unit sphere in the space ℝ⁴ there always exist two points with angular distance between them strictly less than 60.5° (61.41°). Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 483–503, October, 2000.