On Banach spaces X for which Π2 (X,L 1) = N1(X,L 1)

We prove that every 2-summing operator from a Banach space X into an L ₁-space is nuclear if and only if X is isomorphic to a Hilbert space. Then we study the class of Banach spaces X for which Π₂(l ₂, X) = N ₁(l ₂, X).     

On the number of pairs of positive integers x, y ≤ H such that x 2 + y 2 + 1 is squarefree

It is not difficult to find an asymptotic formula for the number of pairs of positive integers x, y ≤ H such that x ² + y ² + 1 is squarefree. In the present paper we improve the estimate for the error term in this formula using the properties of certain exponential sums. A.Weils’s estimate for the Kloosterman sum plays the major role in our analysis.     

Characterization of {v μ+1 + 2v μ,v μ + 2v μ − 1;t,q}-min · hypers and its applications to error-correcting codes

Recently, Hamada [5] characterized all {v ₂ + 2v ₁,v ₁ + 2v ₀;t,q}-min · hypers for any integert 2 and any prime powerq 3 wherev _l = (q ^l – 1)/(q – 1) for any integerl 0. The purpose of this paper is to characterize all {v _{+ 1} + 2v ,v + 2v _{– 1};t,q}-min · hypers for any integerst, and any prime powerq such thatt 3, 2 t – 1 andq 5 and to characterize all (n, k, d; q)-codes meeting the Griesmer bound (1.1) for the casek 3, d = q ^k-1 – (2q ^-1 +q ) andq 5 using the results in Hamada [3, 4, 5].     

On the maximum size of Erdős-Ko-Rado sets in H(2d+1, q^2)

Erd?s-Ko-Rado sets in finite classical polar spaces are sets of generators that intersect pairwise non-trivially. We improve the known upper bound for Erd?s-Ko-Rado sets in \(H(2d+1, q^2)\) for \(d>2\) and \(d\) even from approximately \(q^{d^2+d}\) to \(q^{d^2+1}.\)      

On radially symmetric solutions of the equation ?Δu + u = |u| p-1 u. An ODE approach

Questions of the existence in a ball of radially symmetric solutions of the equation indicated in the title with the Dirichlet zero boundary conditions are studied in many publications and generally speaking, there was obtained more or less complete answer on these questions. It is known now that if the dimension of the space d????3 and 1 <?p?<?(d?+ 2)/(d ? 2) or if d?=?2 and p?>?1, then for any integer l??? 0 this problem in a ball or in the entire space ${x \in \mathbb {R}^d}$ has a radially symmetric solution with precisely l zeros as a function of r?=?|x|. If d??? 3 and p????(d?+?2)/(d ? 2), then the problem in the entire space has no nontrivial solution. For the first time, this problem was studied by a variant of the variational method. However, it is known to the specialists in the field that it is also interesting to obtain the same results by using methods of the qualitative theory of ODEs. In the present article, we shall give a simple proof of the result above in this way. An earlier proof of this result of the other authors is essentially more complicated than our one.     

On the Number of Solutions of sum from i=1 to k x_i~2≡0 (mod p)(1≤x_1

孙琦 《数学进展》1990,(4)

Critical determinant of the region ∣x∣p+∣y∣p<1 (supplement to the paper “on the application of the electronic computer to the proof of a conjecture of Minkowski from the geometry of numbers,” Zap. Nauchn. Sem. Lomi,71, 163–180, 1977)

Supplement to the paper cited in the title. One presents the results of the calculations on an electronic computer, by the method developed in that paper, proving Minkowski's conjecture regarding the critical determinant of the region x^p+y^p<1 for all real numbers p>2.035 and for 1.01p1.99.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 29–32, 1979.     

On the primitive divisors of the recurrent sequence u_(n+1)=(4cos~2(2π/7)-1)u_n-u_(n-1) with applications to group theory

Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos~2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the sequence has a primitive divisor in Z[2 cos(2π/7)].As a consequence we deduce that for any sufficiently large n there exists a prime power q such that the groupcan be generated by a pair x,y with χ~2=y~3=(xy)~7=1 and the order of the commutator[x,y]is exactly n.The latter result answers in affirmative a question of Holt and Plesken.     

The Beckman-Quarles theorem for mappings from ℝ2 to $${\mathbb{F}}^2 $$ , where F is a subfield of a commutative field extending ℝ, where F is a subfield of a commutative field extending ℝ

Let F be a subfield of a commutative field extending ℝ. Let We say thatf : preserves distanced ≥ 0 if for eachx,y ∈ ℝ ∣x- y∣= d implies ϕ₂(f(x),f(y)) = d² . We prove that each unit-distance preserving mappingf : has a formI o (ρ,ρ), where is a field homomorphism and is an affine mapping with orthogonal linear part.     

On automorphisms of distance-regular graphs with intersection arrays {2<Emphasis Type="Italic">r</Emphasis> + 1, 2<Emphasis Type="Italic">r</Emphasis> − 2, 1; 1, 2, 2<Emphasis Type="Italic">r</Emphasis> + 1}

Let Γ be an antipodal graph with intersection array {2r+1, 2r?2, 1; 1, 2, 2r+1}, where 2r(r + 1) ≤ 4096. If 2r + 1 is a prime power, then Mathon’s scheme provides the existence of an arc-transitive graph with this intersection array. Note that 2r + 1 is not a prime power only for r ∈ {7, 17, 19, 22, 25, 27, 31, 32, 37, 38, 42, 43}. We study automorphisms of hypothetical distance-regular graphs with the specified values of r. The cases r ∈ {7, 17, 19} were considered earlier. We prove that, if Γ is a vertex-symmetric graph with intersection array {2r + 1, 2r ? 2, 1; 1, 2, 2r +1}, 2r + 1 is not a prime power, and r ≤ 43, then r = 25, 27, or 31.     

Corrigendum to “Computing of the number of right coideal subalgebras of Uq(so2n+1)” [J. Algebra 341 (1) (2011) 279–296]

Adjoint functors and projectivization in representation theory of partially ordered sets are used to generalize the algorithms of differentiation by a maximal and by a minimal point. Conceptual explanations are given for the combinatorial construction of the derived set and for the differentiation functor.     

Under which conditions is the Jacobi space L_{w^{(a,b)} }^p [ - 1,1] subset of L_{w^{(\alpha ,\beta )} }^1 [ - 1,1] ?

Exact conditions for α, β, a, b > ?1 and 1 ≤ p ≤ ∞ are determined under which the inclusion property $L_{w^{(a,b)} }^p [ - 1,1]$ ? $L_{w^{(\alpha ,\beta )} }^1 [ - 1,1]$ is valid. It is shown that the conditions characterize the inclusion property. The paper concludes with some results, in which the inclusion property can be detected in relation with estimates of Jacobi differential operators and with Muckenhoupt’s transplantation theorems and multiplier theorems for Jacobi series.     

The Rν-generalized solution of a boundary value problem with a singularity belongs to the space W2,ν+β2+κ+1/κ+2 (Ω, δ)

In the present paper, for a boundary value problem with noncoordinated degeneration of the data and a singularity in the solution, we show that the R _ν -generalized solution belongs to the weighted space W _2,ν+gb ^2+κ+1/κ+2 (Ω, δ)(κ > 0). Original Russian Text ? V.A. Rukavishnikov, E.V. Kuznetsova, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 6, pp. 894–898.     

Hochschild cohomology of algebras of quaternionic type: The family Q(2ℬ)<Subscript>1</Subscript>(<Emphasis Type="Italic">k,s, a,c</Emphasis>) over a field of characteristic different from 2

In a previous joint paper of the author with A.I. Generalov and S.O. Ivanov, the Hochschild cohomology algebra of quaternionic-type algebras from the family Q(2ℬ)₁ over an algebraically closed field of characteristic 2 was calculated. In this paper, the Hochschild cohomology groups of algebras from this family over an algebraically closed field of characteristic different from 2 are calculated. As a corollary, the additive structure of the Hochschild cohomology of algebras of type Q(2 $ A $ A ) over a field of characteristic not 2 is described.     

On Polarized 3-Folds (X,L) Such That <lowercase > h < /lowercase >0(L) = 2 and the Sectional Genus of (X,L) is Equal to the Irregularity of X

Let X be a smooth complex projective variety of dimension 3 and let L be an ample line bundle on X. In this article, we give a characterization of (X, L) with g(X, L) = q(X) and h⁰(L) = 2, where g(X, L) (resp. q(X)) denotes the sectional genus of (X, L) (resp. the irregularity of X).