On a functional equation generalizing the class of semistable distributions

With ?(p),p≥0 the Laplace-Stieltjes transform of some infinitely divisible probability distribution, we consider the solutions to the functional equation ?(p-e ^?pβΠ _i=1 ^m ?^γi (c _i p) for somem≥1,c _i>0, γ _i >0,i=1., …,m, β ε ®. We supply its complete solutions in terms of semistable distributions (the ones obtained whenm=1). We then show how to obtain these solutions as limit laws (r → ∞) of normalized Poisson sums of iid samples when the Poisson intensity λ(r) grows geometrically withr.     

On the multiplicative structure of odd perfect numbers

Starting with Euler's theorem that any odd perfect number n has the form n = p^ep_i^2e_i … p_k^2e_k, where p, p₁,…,p_k are distinct odd primes and p ≡ e ≡ 1 (mod 4), we show that extensive subsets of these numbers (so described) can be eliminated from consideration. A typical result says: if p^e, p_i^2e_i,…,p_r^2e_r are all of the prime-power divisors of such an n with p ≡ p_i ≡ 1 (mod 4), then the ordered set {e₁,…,e_r} contains an even number or odd number of odd numbers according as e ≡ pore ≡ p (mod 8).     

On some combinatorial questions in finite-dimensional spaces

Bounds are given for supinf∥Σ^p_i=1ν_i∥, where sup is taken over all set systems V₁,…, V_p of Rⁿ with 0∈∩^p_i=1convV_i and sup_v∈Vi∥ν∥?1 for i=1,…, p, and inf is taken over all possible choices ν_i∈V_i for i=1,…, p. Another similar problem is considered. The bounds are sharp.     

Representing the GCD as linear combination in non-PID rings

We prove the following fact: If finitely many elements p ₁,p ₂,…,p _n of a unique factorization domain are given such that the greatest common divisor of each pair (p _i ,p _j ) can be expressed as a linear combination of p _i and p _j , then the greatest common divisor of all the p _i ’s can also be expressed as a linear combination of p ₁,…,p _n . We prove an analogous statement in general commutative rings.     

Testing the homogeneity of small error rates: application to the sensitivity of different Elisa tests for aids antibodies

This note develops modifications of the chi-square and maximum likelihood tests for the homogeneity of the success probabilities p_i of k binomial variables when the number of trials, n_i, vary widely and all p_i are small. With N = Σn_i, and letting the n_i/N approach fixed fractions, γ_i may differ. The procedures are appied to the first licensing data on the sensitivility which includes the case where the γ_i may differ. The procedures are applied to the first licensing data on the sensitivity of the ELISA tests used to screen for the AIDS virus and it is shown that one of the test kits was significantly less sensitive than the others.     

On the representation of integers by p-adic diagonal forms

Let p be an odd prime, let d be a positive integer such that (d,p?1)=1, let r denote the p-adic valuation of d and let m=1+3+3²+…+3^r. It is shown that for every p-adic integer n the equation Σ_i=1^mX_i^d=n has a nontrivial p-adic solution. It is also shown that for all p-adic units a₁, a₂, a₃, a₄ and all p-adic integers n the equation Σ_i=1⁴a_iX_i^p=n has a nontrivial p-adic solution. A corollary to each of these results is that every p-adic integer is a sum of four pth powers of p-adic integers.     

Effectifs des diverses sequences de polyedres 3-valents convexes ayant moins de 11 faces et de polyedres sans face triangulaire ayant de 11 a 14 faces

A table is presented of various sequences of 3-valent convex polyhedra, a sequence being defined by p₃p₄???p_i??? where p_i is the number of faces with i edges. These sequences are given for polyhedra with 6, 7, 8, 9, 10 faces and the sequences where p₃=0 are given for polyhedra having 11, 12, 13, 14 faces. The number of polyhedra is also given for each sequence.     

A stability property of the unit vector basis ofl p

Let {e _n} be the unit vector basis ofl _p, l<p<∞, and letx _n=a_ne_n?b_nen+1. Necessary and sufficient conditions are given for the operatorT:l _p → span {x _n} defined byTe _i=x_i to be invertible.     

Analogues of eberhard's theorem for 4-valent 3-polytopes with involutory automorphisms

Let p_k(P) denote the number of k-gonal faces of the 3-polytope P. Necessary and sufficient conditions are found for a vector (p₃, p₄,…., p_m) to have the following property: There exists a 4-valent 3-polytope P with p_i(P)=p_i for all i≠4, which has a center, or a line, or a plane of symmetry.     

A conjecture of recski in combinatorial set theory

A p-cover of $\text{n = {1, 2,…,}\text{n}\text{}}$ is a family of subsets $\text{S}{}_{\mathrm{i}}\text{≠ ?}$ such that ∪ S_i = n and |S_i ∩ S_i| ? p for i ≠ j. We prove that for fixed p, the number of p-cover of n is O(n^p+1logn).     

Separation properties at p for the topological category of Cauchy spaces

An explicit characterization of each of the separation properties?T _i , i=0,1, $\mathop {\mathrm {Pre}}\nolimits T_{2}$ , and T ₂ at a point p is given in the topological category of Cauchy spaces. Moreover, specific relationships that arise among the various?T _i , i=0,1, $\mathop {\mathrm {Pre}}\nolimits T_{2}$ , and T ₂ structures at p are examined in this category. Finally, we investigate the relationships between generalized separation properties and separation properties at a point p in this category.     

Chromatic numbers of the strong product of odd cycles

The problem of determining the chromatic numbers of the strong product of cycles is considered. A construction is given proving χ (G) = 2^p +1 for a product of p odd cycles of lengths at least 2^p +1. Several consequences are discussed. In particular, it is proved that the strong product of p factors has chromatic number at most 2^p +1 provided that each factor admits a homomorphism to sufficiently long odd cycle C_mi, m_i ≥ 2^p +1.     

On the distributions of the form ∑i(δpi−δni)

We present some properties of the distributions of the form T=∑_i(δ_pi?δ_ni), with ∑_id(p_i,n_i)<∞, which arise in the 3-d Ginzburg–Landau problem studied by Bourgain, Brezis and Mironescu (C. R. Acad. Sci. Paris, Ser. I 331 (2000) 119–124). We show that there always exists an irreducible representation of T. We also extend a result of Smets (C. R. Acad. Sci. Paris, Ser. I 334 (2002) 371–374) which says that T is a measure iff T can be written as a finite sum of dipoles. To cite this article: A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A Katona-type proof of an Erd?s-Ko-Rado-type theorem

Let p?1/2 and let μ_p be the product measure on {0,1}ⁿ, where μ_p(x)=p^∑x_i(1-p)^n-∑x_i. Let A⊂{0,1}ⁿ be an intersecting family, i.e. for every x,y∈A there exists 1?i?n such that x_i=y_i=1. Then μ_p(A)?p. Our proof uses a probabilistic trick first applied by Katona to prove the Erd?s-Ko-Rado theorem.     

The failure of the Hardy inequality and interpolation of intersections

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=?1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L _p(w₀), N∩L_p(w₁)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L _p(w₀),N∩L _p (w₁)), which turns out to be essentially different from theK-functional for (L _p(w₀), L_p(w₁)), even for the case whenN∩L _p(w_i) is dense inL _p(w_i) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.     

Profit-based latency problems on the line

We consider a latency problem with a profit p_i for each client. When serving a client, a revenue of p_i−t is collected. The goal is to find routes for the servers such that total collected revenue is maximized. We study the complexity of different variants of this problem on the line.     

Number of Zeros of Diagonal Polynomials over Finite Fields

Let F be a finite field with q=pf elements, where p is a prime. Let N be the number of solutions (x1,…,xn) of the equation c1xd11+···+cnxdnn=c over the finite fields, where d1∣q−1, ciϵF*(i=1, 2,…,n), and cϵF. In this paper, we prove that if b1 is the least integer such that b1≥∑ni=1 (f/ri) (Di, p−1)/(p−1), then q[b1/f]−1∣N, where ri is the least integer such that di∣pri−1, Didi=pri−1, the (Di, p−1) denotes the greatest common divisor of Di and p−1, [b1/f] denotes the integer part of b1/f. If di=d, then this result is an improvement of the theorem that pb∣N, where b is an integer less than n/d, obtained by J. Ax (1969, Amer. J. Math.86, 255–261) and D. Wan (1988, Proc. AMS103, 1049–1052), under a certain natural restriction on d and n.     

Simultaneous estimation of gamma scale parameter under entropy loss: Bayesian approach

LetX ₁,...,X _p bep(≥ 2) independent random variables, where eachX _i has a gamma distribution withk _i andθ _i . The problem is to simultaneously estimatep gammar parametersθ _i under entropy loss where the parameters are believed priori. Hierarchical Bayes (HB) and empirical Bayes(EB) estimators are investigated. Next, computer simulation is studied to compute the risk percentage improvement of the HB, EB and the estimator of Dey et al.(1987) compared to MVUE ofθ.     

The complexity of the fixed point property

It is NP-complete to determine whether a given ordered set has a fixed point free order-preserving self-map. On the way to this result, we establish the NP-completeness of a related problem: Given ordered sets P and Q with t-tuples (p ₁, ... , p _t) and (q ₁, ... , q _t) from P and Q respectively, is there an order-preserving map f: P→Q satisfying f(p _i)≥q _i for each i=1, ... , t?