A problem in prediction theory

Summary A new kind of ? prediction problem ? is defined and studied, first for certain special distributions and then generally. The general results are connected with conjugate trigonometric series. To Giovanni Sansone on his 70^th birth day. The authors acknowledge the support of theAlfred P. Sloan Foundation and the National Science Foundation, respectively.     

Locality problem in quantum theory

We discuss the locality problem in relativistic and nonrelativistic quantum theory. We show that there exists a formulation of quantum theory that, on one hand, preserves the mathematical apparatus of the standard quantum mechanics and, on the other hand, ensures the satisfaction of the locality condition for each individual event including the measurement procedure. As an example, we consider the scattering from two slits. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 327–343, May, 2008.     

A problem in elasticity theory

The problem of determining the dimensions of the transverse cross-sections of a beam from the given frequencies of its natural vibrations is examined. Frequency spectra are indicated that determine the dimenions of the transverse cross-sections of the beam uniquely, an effective procedure is presented for solving the inverse problem, and a uniqueness theorem is proved. The method of standard models /1/ is used to solve the inverse problem.     

Saturation theory converse problem

The following problem is solved: Given a class of functions K, it is required to show a saturated approximation process for which K is the saturation class.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 583–590, November, 1969.     

On a problem in probability theory

For continuous random variables, we study a problem similar to that considered earlier by one of the authors for discrete random variables. Let numbers $$N > 0, E > 0, 0 \leqslant \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _s $$ be given. Consider a random vector x = (x ₁, …, x _s), uniformly distributed on the set $$x_j \geqslant 0, j = 1, \ldots ,s; \sum\limits_{j = 1}^s {x_j = N} , \sum\limits_{j = 1}^s {\lambda _j x_j \leqslant E} .$$ We study the weak limit of x as s → ∞.     

Zonoid theory and Hilbert's fourth problem

A finite vector sum of line segments is termed a zonotope. A zonoid is a Blaschke-Hausdorff limit of zonotopes. A projective metric d on a convex subset of projective space is shown to be of negative type if and only if the spheres in any tangent space are polar duals of zonoids. It follows that metric arclength can be represented by a Crofton formula with respect to a positive measure on the hyperplanes if and only if d is of negative type. These ideas allow a nice characterization of this cone of metrics.     

An extremal problem in graph theory

It is proved that the maximum number of cut-vertices in a connected graph withn vertices andm edges is $$max\left\{ {q:m \leqq (_2^{n - q} ) + q} \right\}$$ All the extremal graphs are determined and the corresponding problem for cut-edges is also solved.     

Picard-vessiot theory and the Jacobian problem

A differential version of the classical theorem of Campbell is presented     

An optimal problem in graph theory

The following problem is solved: determine a point on a tree having the property that the sum of the products of the intensities of its vertices by the corresponding distances to that point is a minimum. The proposed algorithm is reduced to the stepwise application to the tree of truncation of its vertices. A feasible interpretation of the problem is given.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 355–359, September, 1971.     

An inverse problem in potential theory

Summary We consider the classical problem of finding the density of a material body embedded into a region S, when the potential generated by (possibly coinciding with S) is known outside (or on the surface of) S. In the set of such solutions we look for the density which has the smallestL ²-norm and we prove that belongs toL _H ²⁼ (), the space of square summable functions harmonic in . However is unstable, i.e. itsL ²-norm does not depend continuously upon the L₂-norm of the potential. We show how a continuous dependence may be restored by introducing mild restrictions on the set of admissible solutions.     

On a combinatorial problem in group theory

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X ²|<|X|², whereX ²={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism. Dedicated to John G. Thompson, the Wolf Prize Laureate in Mathematics for 1992 The first author wishes to thank the Department of Mathematics in the University of Napoli for their hospitality during the preparation of this paper.     

On a problem in additive number theory

For a set A, let P(A) be the set of all finite subset sums of A. We prove that if a sequence B={b ₁<b ₂<⋯} of integers satisfies b ₁≧11 and b _n+1≧3b _n +5 (n=1,2,…), then there exists a sequence of positive integers A={a ₁<a ₂<⋯} for which P(A)=ℕ∖B. On the other hand, if a sequence B={b ₁<b ₂<⋯} of positive integers satisfies either b ₁=10 or b ₂=3b ₁+4, then there is no sequence A of positive integers for which P(A)=ℕ∖B.