A note on a conjecture on permanents

Let Kn denote the set of all n × n nonnegative matrices whose entries have sum n, and let ϕ be a real function on K_n defined by ϕ (X) = Πⁿ_i=1Σⁿ_j=1x_ij + Πⁿ_j=1Σⁿ_i=1x_ij − per X for X = [x_ij] ϵ K_n. A matrix A ϵ K_n is called a ϕ -maximizing matrix on K_n if ϕ (A) ⩾ ϕ (X) for all X ϵ K_n. It is conjectured that J_n = [1/n]_{n × n} is the unique ϕ-maximizing matrix on K_n. In this note, the following are proved: (i) If A is a positive ϕ-maximizing matrix, then A = J_n. (ii) If A is a row stochastic ϕ-maximizing matrix, then A = J_n. (iii) Every row sum and every column sum of a ϕ-maximizing matrix lies between 1 − √2·n!/nⁿ and 1 + (n − 1)√2·n!/nⁿ. (iv) For any p.s.d. symmetric A ϵ K_n, ϕ (A) ⩽ 2 − n!/nⁿ with equality iff A = J_n. (v) ϕ attains a strict local maximum on K_n at J_n.     

Hypoellipticity on the Heisenberg group

Let P be a left-invariant differential operator on the Heisenberg group Hⁿ, P homogeneous with respect to the dilations on Hⁿ. We show that a necessary and sufficient condition for the hypoellipticity of P is that π(P) be an injective operator for every irreducible unitary representation π of Hⁿ (except the trivial representation). Furthermore, hypoellipticity is preserved if the homogeneous operator P is perturbed by terms of lower order of homogeneity. (Homogeneity means homogeneity with respect to dilations of Hⁿ.) It is also shown that if P is homogeneous, left-invariant and hypoelliptic on Hⁿ, then its formal adjoint is hypoelliptic.     

A note on asymptotic values of quasiregular maps

In this note we give an example of a quasiregular map in ? ⁿ (n ≥ 3) of order of growth n ? 1 and whose set of asymptotic values is A ∪ {∞} for a given Suslin analytic set A ? ? ⁿ . Our example is a modification of Drasin’s construction in [4] of a quasiregular map with order of growth n ? 1 and set of asymptotic values ? ⁿ ∪ {∞}.     

Discrete analysis on a radial lattice

A discrete model for analytic functions is constructed using lattice points of the complex plane arranged in radial form. The discrete analytic functions are defined as solutions of a finite-difference approximation to the polar Cauchy-Riemann equations. The resulting discrete powerz⁽ⁿ⁾ (an analogue of zⁿ) has a simple algebraic form (a direct analogue of ?ⁿ exp{inθ}) and has some surprising properties. For example every discrete polynomial ∑₀^ma_nz⁽ⁿ⁾ has a factorization in terms of the zeros of its classical counterpart ∑₀^ma_nzⁿ every discrete entire function has a power series representation ∑ a_nz⁽ⁿ⁾.     

Reducibility of the families of 0-dimensional schemes on a variety

IfA is a regular local ring of dimensionr>2, over an algebraically closed fieldk, we show that the Hilbert scheme Hilb ⁿ A parametrizing ideals of colengthn inA(dim _k A/I=n) has dimension>cn ^2?2/r and is reducible, for alln>c′, wherec andc′ depend only onr. We conclude that ifV is a nonsingular projective variety of dimensionr>2, the Hilbert scheme Hilb ⁿ V parametrizing the 0-dimensional subschemes ofV having lengthn, is reducible for alln>c″(r). We may takec″(r) to be (1) $$102 ifr = 3,25 ifr = 4,35 ifr = 5,and\left( {1 + r} \right)\left( {{{1 + r} \mathord{\left/ {\vphantom {{1 + r} 4}} \right. \kern-\nulldelimiterspace} 4}} \right)ifr > 5.$$ The result answers in the negative a conjecture of Fogarty [1] but leaves open the question of the conjectured irreducibility of Hilb ⁿ A, whereA has dimension 2. Hilb ⁿ V is known to be irreducible ifV is a nonsingular surface (Hartshorne forP ₂, and Fogarty [1]). In all cases Hilb ⁿ V and Hilb ⁿ A are known to be connected (Hartshorne forP _r, and Fogarty [1]). The author is indebted to Hartshorne for suggesting that Hilb ⁿ A might be reducible ifr>2. The proof has 3 steps. We first show that ifV is a variety of dimensionr, then Hilb ⁿ V is irreducible only if it has dimensionr n. We then show that ifA is a regular local ring of dimensionr, Hilb ⁿ A can be irreducible only if it has dimension (r?1)(n?1). Finally in § 3 we construct a family of graded ideals of colengthn in the local ringA, and having dimensionc′ n^2?2/r. Since for largen this dimension is greater thanr n, and since Hilb ⁿ A?Hilb ⁿ V whenA is the local ring of a closed point onV, the proof is complete, except for (1), which follows from § 3, and the monotonicity of (dim Hilb ⁿ V?r n) (see (2)). In § 4, we comment on some related questions.     

A Note on Quantum Field Theory and P-brans innDimensions

Euclidean n−1 dimensional spheres are postulated as P-brans sweeping an n dimensional world sheet. It is conjectured that the resulting space, S ^(∞) , is a hierarchical one with an effective topological dimension equal to theexpectationvalue of n, where 0⩽n⩽∞. Numerical estimation shows that 〈n〉=〈dim_T S ^(∞) 〉 is very close to the topological dimensionof actualspacetime.     

Global hypoellipticity and global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ) where Tⁿ is the n-dimensional torus and s?1. We prove that if P is s-globally hypoelliptic in Tⁿ then its transposed operator tP is s-globally solvable in Tⁿ, thus extending to the Gevrey classes the well-known analogous result in the corresponding C^∞ class.     

An efficient pram algorithm for maximum-weight independent set on permutation graphs

An efficient parallel algorithm is presented to find a maximum weight independent set of a permutation graph which takesO (logn) time usingO (n ²/logn) processors on an EREW PRAM, provided the graph has at mostO (n) maximal independent sets. The best known parallel algorithm takesO (log² n) time andO (n ³/logn) processors on a CREW PRAM.     

A remark on full rank perfect codes

Any full rank perfect 1-error correcting binary code of length n=2^k-1 and with a kernel of dimension n-log(n+1)-m, where m is sufficiently large, may be used to construct a full rank perfect 1-error correcting binary code of length 2^m-1 and with a kernel of dimension n-log(n+1)-k. Especially we may construct full rank perfect 1-error correcting binary codes of length n=2^m-1 and with a kernel of dimension n-log(n+1)-4 for m=6,7,…,10.This result extends known results on the possibilities for the size of a kernel of a full rank perfect code.     

Adams theorem on Bernoulli numbers revisited

If we denote B_n to be nth Bernoulli number, then the classical result of Adams (J. Reine Angew. Math. 85 (1878) 269) says that p^?|n and (p−1)?n, then p^?|B_n where p is any odd prime p>3. We conjecture that if (p−1)?n, p^?|n and p?+1?n for any odd prime p>3, then the exact power of p dividing B_n is either ? or ?+1. The main purpose of this article is to prove that this conjecture is equivalent to two other unproven hypotheses involving Bernoulli numbers and to provide a positive answer to this conjecture for infinitely many n.     

Chebyshev Polynomials on Compact Sets

In connection with a problem of H. Widom it is shown that if a compact set K on the complex plane contains a smooth Jordan arc on its outer boundary, then the minimal norm of monic polynomials of degree n?=?1,2,... is at least (1?+?β)cap(K) ⁿ with some β?>?0, where cap(K) ⁿ would be the theoretical lower bound. It is also shown that the rate (1?+?o(1))cap(K) ⁿ is possible only for compact for which the unbounded component of the complement is simply connected. A related result for sets lying on the real line is also proven.     

Chebyshev polynomials on a system of curves

This paper is devoted to the problem of how close can one get with the n-th Chebyshev numbers of a compact set ?? to the theoretical lower bound cap(??) ⁿ . It is shown that for a system of m ?? 2 analytic curves, there is always a subsequence for which the Chebyshev numbers are at least (1 + ??)cap(??) ⁿ , while for another subsequence they are at most (1 + O(n ^?1/(m?1)))cap(??) ⁿ . It is also shown that the last estimate is optimal. We also discuss how well a system of curves can be approximated by lemniscates in Hausdorff metric. The proofs are based on potential theoretical arguments. Simultaneous Diophantine approximation of harmonic measures lies in the background. To achieve the correct rate, a perturbation of the multi-valued complex Green??s function is introduced which makes the n-th power of its exponential single-valued and which allows the construction of Faber-like polynomials on multiply connected domains.     

Yamabe Flow and Myers Type Theorem on Complete Manifolds

In this paper, we prove the following Myers type theorem: If (M ⁿ ,g), n≥3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc≥?Rg, where R>0 is the scalar curvature and ?>0 is a uniform constant, then M ⁿ must be compact.     

Sharp upper bounds on the number of the scattering poles

We study the scattering poles of a compactly supported “black box” perturbations of the Laplacian in Rⁿ, n odd. We prove a sharp upper bound of the counting function N(r) modulo o(rⁿ) in terms of the counting function of the reference operator in the smallest ball around the black box. In the most interesting cases, we prove a bound of the type N(r)?A_nrⁿ+o(rⁿ) with an explicit A_n. We prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula.     

A note on the differentiable structure of generalized idempotents

For a fixed n > 2, we study the set Λ of generalized idempotents, which are operators satisfying T ⁿ⁺¹ = T. Also the subsets Λ _? , of operators such that T ^n?1 is the Moore-Penrose pseudo-inverse of T, and Λ*, of operators such that T ^n?1 = T* (known as generalized projections) are studied. The local smooth structure of these sets is examined.     

A note on the similarity depth

In this note we demonstrate the existence of E0L forms F and G which are n-similar, i.e. $\text{L}$ⁿ(F) = $\text{L}$ⁿ(G) but $\text{L}$ⁿ⁺¹(F)≠$\text{L}$ⁿ⁺¹(G) for n ∈ {2, 3}. This partially solves an open problem from [3].     

Author index for volume 39

The distribution γ(c, n) of de Bruijn sequences of order n and linear complexity c is investigated. Some new results are proved on the distribution of de Bruijn sequences of low complexity, i.e., their complexity is between 2^n?1 + n and 2^n?1 + 2^n?2. It is proved that for n ? 5 and 2^n?1 + n?c<2^n?1 + 2^n?2, γ(c, n) ≡ 0 (mod 4). It is shown that for n ? 11, γ(2^n?1 + n, n) > 0. It is also proved that γ(2^n?1 + 2^n?2, n) ? 4γ(2^n?2 ? 1, n ? 2) and we give a recursive method to generate de Bruijn sequences of complexity 2^n?1 + 2^n?2.     

Almost everywhere convergence of sequences of multiplier operators on local fields

LetK ⁿ be then-dimensional vector space over a local fieldK. Two maximal multiplier theorems onL ^p (K ⁿ ) are proved for certain multiplier operator sequences associated with regularization and dilation respectively. Consequently the a. e. convergence of such multiplier operator sequences is obtained. This sharpens Taibleson’s main result and applies to several important singular integral operators onK ⁿ .     

Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle

We consider the class P _n ^* of algebraic polynomials of a complex variable with complex coefficients of degree at most n with real constant terms. In this class we estimate the uniform norm of a polynomial P _n ∈ P _n ^* on the circle Γ_r = z ∈ ?: ¦z¦ = r of radius r = 1 in terms of the norm of its real part on the unit circle Γ₁ More precisely, we study the best constant μ(r, n) in the inequality ||P_n||C(Γ_r) ≤ μ(r,n)||Re P_n||C(Γ₁). We prove that μ(r,n) = rⁿ for rⁿ⁺² ? r ⁿ ? 3r² ? 4r + 1 ≥ 0. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality μ(r, n) = r ⁿ is valid for r sufficiently close to 1.     

Commutativity-preserving operators on symmetric matrices

If φ is a nonsingular linear operator on n × n symmetric matrices over a formally real field, n ? 3, and if φ preserves commutativity, then φ(A) = cU^-1AU + f(A)I, where U^tU and c are scalar and ? is a linear functional. This is an extension of a result of Chan and Lim.