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 certain classes of hyperbolic 3-manifolds

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links. Received: 27 November 1998 / Accepted: 12 May 1999     

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 a certain nonlocal problem for a hyperbolic equation

The author proves the existence and uniqueness of a generalized solution of a nonlocal problem with an integral condition for a hyperbolic equation with n spatial variables. This work is a continuation of the studies started in [3–5], where the solvability problem of nonlocal problems with an integral condition was studied for hyperbolic equations on the plane. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

On the zero sets of certain entire functions

We consider the class of entire functions of the form

where are polynomials and are entire functions. We prove that the zero-set of such an , if infinite, cannot be contained in a ray. But for every region containing the positive ray there is an example of with infinite zero-set which is contained in this region.

     

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 the addition of convex sets in the hyperbolic plane

Analogue to the definition $K + L := \bigcup_{x\in K}(x + L)$ of the Minkowski addition in the euclidean geometry it is proposed to define the (noncommutative) addition $K \vdash L := \bigcup_{0\, \leqsl\, \rho\,\leqsl\, a(\varphi),0\,\leqsl\,\varphi\,<\, 2\pi}T_{\rho}^{(\varphi)}(L)$ for compact, convex and smoothly bounded sets K and L in the hyperbolic plane $\Omega$ (Kleins model). Here $\rho = a(\varphi)$ is the representation of the boundary $\partial$ K in geodesic polar coordinates and $T_{\rho}^{(\varphi)}$ is the hyperbolic translation of $\Omega$ of length $\rho$ along the line through the origin o of direction $\varphi$. In general this addition does not preserve convexity but nevertheless we may prove as main results: (1) $o \in$ int $K, o \in$ int L and K,L horocyclic convex imply the strict convexity of $K \vdash L$, and (2) in this case there exists a hyperbolic mixed volume $V_h(K,L)$ of K and L which has a representation by a suitable integral over the unit circle.     

Local maximality of hyperbolic sets

Two properties of a hyperbolic set F are discussed: its local maximality and the property that, in any neighborhood U ⊃ F, there exists a locally maximal set F′ that contains F (we suggest calling the latter property local premaximality). Although both these properties of the set F are related to the behavior of trajectories outside F, it turns out that, in the class of hyperbolic sets, the presence or absence of these properties is determined by the interior dynamics on F.     

On certain homomorphisms of restriction algebras of symmetric sets

LetG be a locally compact abelian group and Γ its dual group. For any closedH ⊆G denote the algebra of restrictions toH of Fourier transforms of functions inL ¹(Γ) byA(H). This paper considers certain Cantor like sets inR and ΠZ _m(j) and gives some necessary algebraic criterion fornatural isomorphisms of their restriction algebras. This work was supported mainly by the U.S. National Science Foundation Graduate Fellowship Program. The author wishes to thank Paul Cohen, Karel de Leeuw, and Yitzhak Katznelson for their counsel.     

On central difference sets in certain non-abelian 2-groups

In this note, we define the class of finite groups of Suzuki type, which are non-abelian groups of exponent 4 and class 2 with special properties. A group G of Suzuki type with |G|=2^2s always possesses a non-trivial difference set. We show that if s is odd, G possesses a central difference set, whereas if s is even, G has no non-trivial central difference set.     

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 theL p-L q-estimates for the solutions of certain hyperbolic equations

We establish conditions, which should be imposed on the exponentsp andq and the dimensionalityn of a space in order that theL _p-L _q-estimates should hold for the solutions of the Cauchy problem for a second-order hyperbolic equation with constant coefficients.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 992–1008, July, 1993.