Inequalities and sphere-packing inl p

If 1<p<∞, there is a constantr _p <1/2 so that ifr>r _p only a bounded number of balls inl _p of radiusr can be packed into the unit ball ofl _p . We obtain the exact value of this bound for eachp andr as a consequence of several new inequalities relating the expressions Σλ _i λ _j ‖x _i –x _j ‖ ^p , Σλ _i ‖x _i ‖ ^p and Σλ _i ^/2 for sequences (x _i ) ₁ ⁿ ⊂l _p and (λ _i ) ₁ ⁿ ⊂R.     

The range of a projection of small norm inl 1 n

It is proved that there exists a positive function Φ(∈) defined for sufficiently small ∈ 〉 0 and satisfying lim_t→0 Φ(∈) =0 such that for any integersn 〉>0, ifQ is a projection ofl ₁ ⁿ onto ak-dimensional subspaceE with ‖|Q‖|≦1+∈ then there is an integerh〉=k(1−Φ(∈)) and anh-dimensional subspaceF ofE withd(F,l ₁ ^h ) 〈= 1+Φ (∈) whered(X, Y) denotes the Banach-Mazur distance between the Banach spacesX andY. Moreover, there is a projectionP ofl ₁ ⁿ ontoF with ‖|P‖| ≦1+Φ(∈). Author was partially supported by the N.S.F. Grant MCS 79-03042.     

On Besselian Schauder bases inl p (E)

By complex interpolation and tensor products, Schauder bases are constructed of the Banach sequence spacesl _p (E). In a general result, we study the Besselian property of the basis, and ifE is assumed to be theL _p (Lebesgue) andS _p (v. Neumann-Schatten) space, we obtain inequalities for the coefficient functionals associated to the basis which generalise other results given by Hausdorff—Young and Gohberg—Marcus. Finally, we construct non-Besselian and conditional bases ofl _p (E).     

Minimax testing of the hypothesis of independence for ellipsoids inl p

The paper continues the author's previous papers. The aim of it is to study the ellipsoids , which arises in problems of simple hypothesis testing. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 77–97, 1993. Translated by the author.     

Carathéodory balls and norm balls inH n ={z ε ℂ n : ‖z‖1<1}

It is shown that the only balls in the Carathéodory distance onH _n ={z ε ℂ ⁿ :‖z‖₁<1},n≥2, which are balls with respect to the ℓ₁ norm in ℂ ⁿ are those centered at the origin. In memory of Albert Pfluger The research was partially supported by the Fund for the Promotion of Research at the Technion.     

Carathéodory balls and norm balls of the domainsH n ={z ∈ ℂ n : |z 1|+…+|z n |<1}

In this short note we prove that the only Carathéodory balls in domains given in the title (forn≥2) which are the norm balls are the ones with the center at 0. It is a generalization of the result of B. Schwarz, who proved this theorem in casen=2.     

A class of ((p n−p m)(p n−1), p n−p m)-arcs in PG(2, p n)

A (k, d)-arc in PG(2, q) is a set of k points such that some d, but no d+1, of them are collinear. An outstanding problem is to find the maximum value of k for which a (k, d)-arc exists. A construction is given for a class of (k, p ⁿ–p ^m)-arcs in PG(2, p ⁿ). These arcs constitute a lower bound on the maximum possible value of k, and a subset of them is shown to be optimal.     

The log-behavior of $$\root n \of {p(n)}$$ and $$\root n \of {p(n)/n}$$p(n)/nn

Let p(n) denote the partition function and let \(\Delta \) be the difference operator with respect to n. In this paper, we obtain a lower bound for \(\Delta ^2\log \root n-1 \of {p(n-1)/(n-1)}\), leading to a proof of a conjecture of Sun on the log-convexity of \(\{\root n \of {p(n)/n}\}_{n\ge 60}\). Using the same argument, it can be shown that for any real number \(\alpha \), there exists an integer \(n(\alpha )\) such that the sequence \(\{\root n \of {p(n)/n^{\alpha }}\}_{n\ge n(\alpha )}\) is log-convex. Moreover, we show that \(\lim \limits _{n \rightarrow +\infty }n^{\frac{5}{2}}\Delta ^2\log \root n \of {p(n)}=3\pi /\sqrt{24}\). Finally, by finding an upper bound for \(\Delta ^2 \log \root n-1 \of {p(n-1)}\), we establish an inequality on the ratio \(\frac{\root n-1 \of {p(n-1)}}{\root n \of {p(n)}}\).     

An inner product and orthonormalization inl p -spaces and solution of infinite systems of infinite linear equations

Let the rows of an infinite square matrixM be elements ofl ^p -space (p>1) andX be an infinite column vector of unknowns andC an infinite column vector of real numbers. To our knowledge the solvability ofMX=C has nowhere been satisfactorily studied in the literature. This is also true of Riesz’classical work [2]. A reason for this is that not until recently [1] an appropriate inner product and the corresponding orthonormalization forp≠2 has been introduced. In this paper, based on [1], it is shown thatMX=C has a solution which is an element ofl ^q if and only if upon our process of orthonormalization of the rows ofM the system yields an equivalent systemAX=K where the rows ofA form an orthonormal sequence (in our sense) of elements ofl ^p andK becomes an element ofl ^q withp ⁻¹+q ⁻¹=1. A solution is then given byX=(A ^(q) (AA ^(q) )⁻¹)K whereA ^(q) is ourq-transpose ofA. This solution is the solution of the minimall ^q -norm. Otherwise, the obvious dual concept of the best approximating solution inl ^q -norm is introduced and obtained. 1980 Mathematics Subject Classification: Primary 46C10, Secondary 15A06