On the number of pseudo-triangulations of certain point sets

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically n¹²nΘ(1) pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.     

On the number of divisors of binomial coefficients

This paper deals with the problems of the upper and lower orders of growth of the ratios of the divisor functions of “adjacent” binomial coefficients, i.e., of the numbers of combinations of the form C _n ^k and C _n ^k+1 or C _n ^k and C _n+1 ^k . The suprema and infima of the corresponding ratios are obtained.     

On certain hyperbolic sets

Two invariant sets F of certain diffeomorphisms S that were described by A. Fathi, S. Crovisier, and T. Fisher as examples of hyperbolic sets with the property (unexpected at that time) that, in some neighborhood of such an F, there is no locally maximal set containing F are considered. It is proved that this property, although referring to the behavior of the orbits of S near F, is ultimately determined in the examples mentioned above by a combination of a certain explicitly stated intrinsic property of the action of S on F with the hyperbolicity of F. (This means that if a hyperbolic set F′ for a diffeomorphism S′ is equivariantly homeomorphic to a Fathi-Crovisier or a Fisher set, then F′ has a neighborhood in which S′ has no locally maximal set containing F′.)     

Extremal theorems on divisors of a number

For every positive integer N, we determine the maximum sizes of collections C and C' of divisors of N subject to the following restrictions. I any two numbers in C are coprime, then their least common multiple must be N. If any k numbers (k arbitrary) in C′ have the greatest common divisor 1, then their least common multiple must be N. For the special case that N is square free, the maximum sizes of C and C' have previously been determined. Since every number can be thought of as a multiset of primes, this work can be regarded as an extension of theorems on families of finite sets to families of multisets.     

On certain distinguished spectral sets

Sunto Ogni insieme parzialmente ordinato (X, ) ammette varie T₀-topologie «compatibili» con , nel senso che coincide con l'ordine indotto da su X (cf. Hochster [30]). Tra tali topologie la meno fine è la COP(= closures of points) -topologia, cioè la topologia meno fine per la quale l'insieme {y X: xy} è chiuso per ogni x X. La più fine è L(= left)-topologia discreta di Alexandroff, X^L, avente come base per gli aperti gli insiemi {y X: yx} al variare di x in X. In questo lavoro sono date condizioni su X affinchè questo abbia una struttura topologica spettrale noetheriana. Inoltre, vengono caratterizzati gli insiemi parzialmente ordinati X per i quali X^L è uno spazio spettrale; vengono anche caratterizzati gli spazi spettrali Y la cui topologia coincide con la L-topologia associata all'ordine indotto. La topologia dell' «ordine opposto » (opposite-order topology di Hochster, op. cit. Prop. 8) determina una «dualità» tra le L-topologie spettrali e le topologie spettrali noetheriane. Se Y=Spec (A) con la topologia di Zariski, allora Y^H coincide con la O(=ordine) topologia di G. Picavet.

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This work was supported in part by grants from the University of Tennessee at Knoxville and the Istituto Matematico, Università di Roma.

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This work was performed under the auspices of the GNSAGA of the CNR.

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This research was funded by a grant from the Research Council of the Graduate School, University of Missouri-Columbia and, also, by the Istituto Matematico, Università di Roma.     

On compact sets with a certain affine invariant

We give a complete characterization of finite-dimensional compact sets with the following property: all of their images under affine operators are symmetric (that is, have symmetry planes of certain dimensions). We also study the noncompact case; namely, we reveal a class of unbounded closed sets with this property and conjecture that this class is complete.     

On certain other sets of integers

We show that if A ⊂ {1,...,N} contains no non-trivial three-term arithmetic progressions then |A| = O(N/log^3/4−o(1) N).     

On the number of different prime divisors of element orders

We prove that the number of different prime divisors of the order of a finite group is bounded by a polynomial function of the maximum of the number of different prime divisors of the element orders. This improves a result of J. Zhang.

     

An external theorem on divisors of a number

Main result: Let ? be a collection of divisors of N = pe11?penn (e1 = minei for all i? {1,…, n}) such that if any two numbers in ? are coprime then their common multiple is N and maximal with respect to this condition, i.e. there is no other collection satisfying this condition which properly contains ?. We prove that ? consists of at least $\text{e}{}_1\text{П}{}^{\mathrm{n}}{}_{\mathrm{i\; =\; 2}}\text{(e}{}_{\mathrm{i}}\text{+ 1)?(e}{}_1\text{? 2)}$ elements.     

On sets not containing arithmetic progressions of a certain kind

The density of sets not containing a diagonal, a special type of arithmetic progression, is investigated. A lower bound on this density is established which sharpens a result of Alspach, Brown, and Hell (J. London Math. Soc.13 (2) (1976), 226–334.     

On finite groups with a certain number of centralizers

LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)?A₅, then #Cent(G)=22 or 32. Moroever, we prove that A₅ is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A₅ in terms of the number of centralizers     

Revisiting the average number of divisors of a quadratic polynomial

Monatshefte für Mathematik - In this work, we employ a well known result due to Zagier to derive an asymptotic formula for the average number of divisors of a quadratic polynomial of the form...     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

On the number of maximal sum-free sets

It is shown that the set contains at most maximal sum-free subsets, provided is large enough.