Vector packing in finite dimensional vector spaces

Let m be an integer ? 2. The effect of crowding m unit vectors x₁,…,x_m into the real Euclidean space R_n of n dimensions is investigated. In particular, several upper bounds for the quantity min_i≠j∥x_i ? x_j∥ are obtained. These are simpler than any previously known and, at least in some cases, almost as sharp. The results have application to the so-called maximum-dispersal (or “misanthrope”) problem, an open problem recently popularized by Klee.     

An improved upper bound in the maximum dispersal problem

For integral? m?2, let x₁,…, x_m be any unit vectors in R_n, the real Euclidean space of n dimensions. We obtain an upper bound for the quantity min_i≠j|x_i-x_j| which, though not as simple, is uniformly sharper than one recently obtained by the author. The result has application to the so-called maximum-dispersal problem, an open problem recently popularized by Klee.     

Linear homotopy method for computing generalized tensor eigenpairs

Let m,m′, n be positive integers such that m ≠ m′. Let A be an mth order n-dimensional tensor, and let ? be an m′th order n-dimensional tensor. λ ∈ ? is called a ?-eigenvalue of A if A x^m?1 = λ?x^m′?1 and ?xm′= 1 for some x ∈ ?ⁿ\{0}. In this paper, we propose a linear homotopy method for solving this eigenproblem. We prove that the method finds all isolated ?-eigenpairs. Moreover, it is easy to implement. Numerical results are provided to show the efficiency of the proposed method.     

Nonuniform right definiteness

The multiparameter eigenvalue problem W_m(λ) x_m = x_m, $\text{W}{}_{\mathrm{m}}\text{(λ) = T}{}_{\mathrm{m}}\text{+}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}$, m = 1,…, k, where /gl /gE $\text{C}$^k, x_m is a nonzero element of the separable Hilbert space H_m, and T_m and V_mn are compact symmetric is studied. Various properties, including existence and uniqueness, of λ = λⁱ ? $\text{C}$^k for which the i_mth greatest eigenvalue of W_m(λⁱ) equals one are proved. “Right definiteness” is assumed, which means positivity of the determinant with (m, n)th entry (y_m, V_mny_m) for all nonzero y_m?H_m, m = 1 … k. This gives a “Klein oscillation theorem” for systems of an o.d.e. satisfying a definiteness condition that is usefully weaker than in previous such results. An expansion theorem in terms of the corresponding eigenvectors x_mⁱ is also given, thereby connecting the abstract oscillation theory with a result of Atkinson.     

On Certain C-Test Words for Free Groups

Let F_m be a free group of a finite rank m ≥ 2 and let X_i, Y_j be elements in F_m. A non-empty word w(x₁,…,x_n) is called a C-test word in n letters for F_m if, whenever (X₁,…,X_n) = w(Y₁,…,Y_n) ≠ 1, the two n-typles (X₁,…,X_n) and (Y₁,…,Y_n) are conjugate in F_m. In this paper we construct, for each n ≥ 2, a C-test word v_n(x₁,…,x_n) with the additional property that v_n(X₁,…,X_n) = 1 if and only if the subgroup of F_m generated by X₁,…,X_n is cyclic. Making use of such words v_m(x₁,…,x_m) and v_m + 1(x₁,…,x_m + 1), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u₁, u₂ ∈ F_m such that every endomorphism ψ of F_m with non-cyclic image is completely determined by ψ(u₁), ψ(u₂).     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.     

On Maillet determinant

For a positive integer m, let $\text{A = {1 ≤ a <}\text{m}\text{2}\text{| (a, m) = 1}}$ and let n = |A|. For an integer x, let R(x) be the least positive residue of x modulo m and if (x, m) = 1, let x′ be the inverse of x modulo m. If m is odd, then |R(ab′)|_a,b∈A = ?2^1?n(∏_χ(Σ_{a = 1}^{m ? 1}aχ(a))), where χ runs over all the odd Dirichlet characters modulo m.     

Optimal Ordering Policies for Inventory Systems with Emergency Ordering

This paper considers dynamic single- and multi-product inventory problems in which the demands in each period are independent and identically distributed random variables. The problems considered have the following common characteristics. At the beginning of each period two order quantities are determined for each product. A “normal order” quantity with a constant positive lead time of λ _n periods and an “emergency order” quantity with a lead time of λ _e periods, where λ _e = λ _n - 1. The ordering decisions are based on linear procurement costs for both methods of ordering and convex holding and penalty costs. The emergency ordering costs are assumed to be higher than the normal ordering costs. In addition, future costs are discounted.For the single-product problem the optimal ordering policy is shown to be the same for all periods with the exception of the last period in the N-period problem. For the multi-product problem the one- and N-period optimal ordering policy is characterized where it is assumed that there are resource constraints on the total amount that can be ordered or produced in each period.     

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

Engel type identities with derivations for right ideals in prime rings

Let R be a prime ring of char R≠2, d a non-zero derivation of R and ρ a non-zero right ideal of R such that [[d(x),d(y)]_n [y,x]_m] = 0 for all x,y ∈ ρ or [[d(x),d(y)]_n d[y,x]_m] = 0 for all x,y ∈ ρ, n, m ≥ 0 are fixed integers. If [ρ,ρ]ρ ≠ 0, then d(ρ)ρ = 0.     

Tight bounds for rational sums of squares over totally real fields

Let K be a totally real Galois number field. Hillar proved that if f ∈ ?[x ₁, ..., x _n ] is a sum of m squares in K[x ₁, ..., x _n ], then f is a sum of N(m) squares in ?[x ₁, ..., x _n ], where N(m) ≤ 2^[K:?]+1 · $ \left( {_2^{[K:\mathbb{Q}] + 1} } \right) $ · 4m. We show in fact that N(m) ≤ m + 4 $ \left\lceil {\tfrac{m} {4}} \right\rceil $ ([K: ?] ? 1), our proof being constructive too. Moreover, we give some examples where this bound is sharp, for instance in the case of quadratic extensions. We also extend our results to the setting of non-commutative polynomials over ?.     

A practical error formula for multivariate rational interpolation and approximation

We consider exact and approximate multivariate interpolation of a function f(x ₁?,?.?.?.?,?x _d ) by a rational function p _n,m /q _n,m (x ₁?,?.?.?.?,?x _d ) and develop an error formula for the difference f???p _n,m /q _n,m . The similarity with a well-known univariate formula for the error in rational interpolation is striking. Exact interpolation is through point values for f and approximate interpolation is through intervals bounding f. The latter allows for some measurement error on the function values, which is controlled and limited by the nature of the interval data. To achieve this result we make use of an error formula obtained for multivariate polynomial interpolation, which we first present in a more general form. The practical usefulness of the error formula in multivariate rational interpolation is illustrated by means of a 4-dimensional example, which is only one of the several problems we tested it on.     

n-Tuple colorings and associated graphs

The n-tuple graph coloring, which assigns to each vertex n colors, is defined together with its respective chromatic number x_n. It is proved that these numbers satisfy the inequality x_n ≥ 2 + x_n?1, and that equality holds only for bipartite graphs. Graphs $\text{G}{}_{\mathrm{n}}{}^{\mathrm{m}}$ are defined which play the same role for the n-tuple coloring that K_m plays for the conventional coloring. The chromatic numbers of various classes of graphs are also calculated.     

Abstract oscillation theorems for multiparameter eigenvalue problems

We prove abstract analogous of Klein's oscillation theorem by demonstrating the existence (and in some cases uniqueness) of eigenpairs with a given index for the multiparameter problem $\text{T}{}_{\mathrm{m}}\text{x}{}_{\mathrm{m}}\text{=}\text{∑}\text{n = 1}\text{k}\text{λ}{}_{\mathrm{n}}\text{V}{}_{\mathrm{mn}}\text{x}{}_{\mathrm{m}}$, 0 ≠ x_m?H_m, m = 1 … k. (1) Here T_m and V_mn are self-adjoint operators on Hilbert spaces H_m. The index is based on the number of negative eigenvalues of T_m ? ∑_{n = 1}^kλ_nV_mn and on the sign of the determinant δ₀ with (m, n)th entry (x_m, V_mnx_m). We assume that certain cofactors of δ₀ are positive, and we complement previous work of Sleeman on Sturm-Liouville systems, and of Binding and Browne on (1) in the case where δ₀ is positive.     

Synthesis of easy tested diagrams in the basis {&;, ∨, <Superscript>−</Superscript>} for systems of functions from some classes

A new method for synthesis of easy tested diagrams of functional elements in the basis {&;, ∨, ^?} is proposed for systems of m Boolean functions differing from constants and representable as disjunctive normal forms containing l variables x ₁,...,x _l , l > 0, without negations and the other variables x _l+1,...,x _n with negations only. All faults are assumed to be stack-at-1 faults at the outputs of elements. It is proved that the length of a complete verification test is not greater than min {m, l} for such diagrams.     

Reconstruction from vertex-switching

Let X be a graph with vertices x₁ ,…, x_n. Let X_i be the graph obtained by removing all edges {x_i, x_j} of X and inserting all nonedges {x_i, x_k}. If n ? 0 (mod 4), then X can be uniquely reconstructed from the unlabeled graphs X₁.…, X_n. If n = 4 the result is false, while for n = 4m≥8 the result remains open. The proof uses linear algebra and does not explicitly describe the reconstructed graph X.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.