Strictly antipodal sets

A subsetA ofE ³ is called strictly antipodal provided that for every pairX ₁,X ₂ of points ofA there is a pairH ₁,H ₂ of parallel supporting planes ofA such thatH _i ∩A={X _i}. The main result asserts that a strictly antipodal set has at most five points. This strengthens a recent result of Croft [2]. This research was supported in part by the United States Air Force under Grant No. AF-EOAR 63-63 and monitored by the European Office, Office of Aerospace Research.     

Borsuk’s antipodal theorem for approachable correspondences

We present an extension of Borsuk’s antipodal theorem (on the existence of a zero) to antipodally approachable correspondences defined on bounded, symmetric and balanced domains.     

A uniqueness theorem for Delaunay graphs

We establish a necessary and sufficient condition for the congruence of two isomorphic Delaunay graphs.     

Uniqueness theorem for convex surfaces

A proof is given of the following assertion: two closed convex analytic surfaces in three-dimensional Euclidean space are equal if their areas and lengths of boundaries of orthogonal projections onto any plane coincide.Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 115–117, July, 1969.     

Uniqueness theorem for harmonic functions

If the boundary values of a function, harmonic in a sphere, and its normal derivative decrease sufficiently fast to zero as a fixed point of the sphere is approached, then the corresponding function is identically zero. This note gives an unimprovable condition on the rate of decrease for which the stated uniqueness theorem holds.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 247–252, March, 1968.     

Uniqueness sets for unbounded spectra

For every set $S?\mathbb{R}$ of finite measure, we construct a system of exponentials ${\{e^{i\lambda t}\}}_{\lambda \in \Lambda }$ which is complete in $L^2(S)$ and such that the set of frequencies Λ has the critical density $D(\Lambda )=\mathrm{mes}(S)/2\pi$.     

Uniqueness sets for multiple Haar series

Criteria are found for the membership of multidimensional sets in the class of M-sets for multiple Haar series with various conditions imposed on the coefficients. Several generalizations of the uniqueness theorem are established.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 789–798, December, 1973.     

Uniqueness theorem for CR-functions on generating CR-manifolds

Translated from Matematicheskie Zametki, Vol. 48, No. 6, pp. 47–50, December, 1990.     

A combinatorial proof of the Borsuk-Ulam antipodal point theorem

We give a proof of Tucker’s Combinatorial Lemma that proves the fundamental nonexistence theorem: There exists no continuous map fromB ⁿ toS ^{n − 1} that maps antipodal points of∂B ⁿ to antipodal points ofS ^{n − 1}.     

Equivariant uniformization theorem for subanalytic sets

In this paper we prove an equivariant version of the uniformization theorem for closed subanalytic sets: Let G be a Lie group and let M be a proper real analytic G-manifold. Let X be a closed subanalytic G-invariant subset of M. We show that there exist a proper real analytic G-manifold N of the same dimension as X and a proper real analytic G-equivariant map such that .      

The projection theorem for spectral sets

LetG be a locally compact group with polynomial growth and symmetricL ¹-algebra andN a closed normal subgroup ofG. LetF be a closedG-invariant subset of Prim_* L ¹(N) andE={ker; with |N(k(F))=0}. We prove thatE is a spectral subset of Prim_* L ¹(G) ifF is spectral. Moreover we give the following application to the ideal theory ofL ¹(G). Suppose that, in addition,N is CCR andG/N is compact. Then all primary ideals inL ¹(G) are maximal, provided allG-orbits in Prim_* L ¹(N) are spectral.Dedicated to Professor Elmar Thoma on the occasion of his 60th birthday     

An intersection theorem for four sets

Fix integers n,r?4 and let F denote a family of r-sets of an n-element set. Suppose that for every four distinct A,B,C,D∈F with |A∪B∪C∪D|?2r, we have A∩B∩C∩D≠∅. We prove that for n sufficiently large, , with equality only if _?F∈FF≠∅. This is closely related to a problem of Katona and a result of Frankl and Füredi [P. Frankl, Z. Füredi, A new generalization of the Erd?s-Ko-Rado theorem, Combinatorica 3 (3-4) (1983) 341-349], who proved a similar statement for three sets. It has been conjectured by the author [D. Mubayi, Erd?s-Ko-Rado for three sets, J. Combin. Theory Ser. A, 113 (3) (2006) 547-550] that the same result holds for d sets (instead of just four), where d?r, and for all n?dr/(d−1). This exact result is obtained by first proving a stability result, namely that if |F| is close to then F is close to satisfying _?F∈FF≠∅. The stability theorem is analogous to, and motivated by the fundamental result of Erd?s and Simonovits for graphs.     

Uniqueness theorem for integral equations and its application

This paper is devoted to answering a question asked recently by Y. Li regarding geometrically interesting integral equations. The main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions. In the case such condition is satisfied, we will write down the exact solution. As its application of our method, we should show that the non-existence theory of the solutions of prescribed scalar curvature equation on Sn can be generalized to that of prescribed Branson-Paneitz Q-curvature equations on Sn.     

A realization theorem for sets of lengths

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By a result of G. Freiman and A. Geroldinger [G. Freiman, A. Geroldinger, An addition theorem and its arithmetical application, J. Number Theory 85 (1) (2000) 59-73] it is known that the set of lengths of factorizations of an algebraic integer (in the ring of integers of an algebraic number field), or more generally of an element of a Krull monoid with finite class group, has a certain structure: it is an almost arithmetical multiprogression for whose difference and bound only finitely many values are possible, and these depend just on the class group. We establish a sort of converse to this result, showing that for each choice of finitely many differences and of a bound there exists some number field such that each almost arithmetical multiprogression with one of these difference and that bound is up to shift the set of lengths of an algebraic integer of that number field. Moreover, we give an explicit sufficient condition on the class group of the number field for this to happen.

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