A characterization of p-bases of rings of constants

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.     

Comparison of volumes by means of the areas of central sections

The Busemann–Petty problem asks whether origin-symmetric convex bodies in Rⁿ with smaller areas of all central hyperplane sections necessarily have smaller n-dimensional volume. The solution was completed in the end of the 1990s, and the answer is affirmative if n4 and negative if n5. Since the answer is negative in most dimensions, it is natural to ask what information about the volumes of central sections of two bodies does allow to compare the n-dimensional volumes of these bodies in all dimensions. In this article we give an answer to this question in terms of certain powers of the Laplace operator applied to the section function of the body.     

Sums of sets of lattice points and unimodular coverings of polytopes

If P is a lattice polytope (that is, the convex hull of a finite set of lattice points in \({\mathbf{R}^n}\)), then every sum of h lattice points in P is a lattice point in the h-fold sumset hP. However, a lattice point in the h-fold sumset hP is not necessarily the sum of h lattice points in P. It is proved that if the polytope P is a union of unimodular simplices, then every lattice point in the h-fold sumset hP is the sum of h lattice points in P.     

Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ _β ⁻ whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 946–971, July, 2005.     

Maximal f-Vectors of Minkowski Sums of Large Numbers of Polytopes

It is known that in the Minkowski sum of r polytopes in dimension d, with r<d, the number of vertices of the sum can be as high as the product of the number of vertices in each summand. However, the number of vertices for sums of more polytopes was unknown so far.     

Behavior at infinity of solutions of an equation of Osserman-Keller type

We study the behavior at infinity of solutions of equations of the form Δu=u^p, where p>1, in dimensions n?3. In particular we extend results proved by Loewner and Nirenberg in Contribution to Analysis, 1974, pp. 245-272 for the case p=(n+2)/(n−2), n?3, to values of p in the range p>n/(n−2), n?3.     

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH ₂(M; ?) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

The evolution of the concept of homeomorphism

Topology, or analysis situs, has often been regarded as the study of those properties of point sets (in Euclidean space or in abstract spaces) that are invariant under “homeomorphisms.” Besides the modern concept of homeomorphism, at least three other concepts were used in this context during the late 19th and early 20th centuries, and regarded (by various mathematicians) as characterizing topology: deformations, diffeomorphisms, and continuous bijections. Poincaré, in particular, characterized analysis situs in terms of deformations in 1892 but in terms of diffeomorphisms in 1895. Eventually Kuratowski showed in 1921 that in the plane there can be a continuous bijection of P onto Q, and of Q onto P, without P and Q being homeomorphic.     

Combinatorics of normal sequences of braids

Many natural counting problems arise in connection with the normal form of braids—and seem to have not been much considered so far. Here we solve some of them. One of the noteworthy points is that a number of different induction schemes appear. The key technical ingredient is an analysis of the normality condition in terms of permutations and their descents, in the vein of the Solomon algebra. As was perfectly summarized by a referee, the main result asserts that the size of the automaton involved in the automatic structure of Bn associated with the normal form can be lowered from n! to p(n), the number of partitions of n.     

Approximation of functions of boundedp-variation by polynomials in terms of the faber-schauder system

In this paper the best polynomial approximation in terms of the system of Faber-Schauder functions in the spaceC _p [0, 1] is studied. The constant in the estimate of Jackson’s inequality for the best approximation in the metric ofC _p [0, 1] and the estimate of the modulus of continuity ω_1−1/p are refined. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 363–371, September, 1997. Translated by N. K. Kulman     

Order of the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

An exclusive-OR sum of pseudoproducts (ESPP) is a modufo-2 sum of products of affine (linear) Boolean functions. The length of an ESPP is defined as the number of summands in this sum; the length of a Boolean function in the class of ESPPs is the minimum length of an ESPP representing this function. The Shannon length function L ^ESPP(n) on the set of Boolean functions in the class of ESPPs is considered; it is defined as the maximum length of a Boolean function of n variables in the class of ESPPs. It is proved that L ^ESPP(n) = ? (2 ⁿ /n ²). The quantity L ^ESPP(n) also equals the least number l such that any Boolean function of n variables can be represented as a modulo-2 sum of at most l multiaffine functions.     

Contractions of infrainvariant systems of subgroups

We create a method which allows an arbitrary group G with an infrainvariant system ℒ(G) of subgroups to be embedded in a group G* with an infrainvariant system ℒ(G*) of subgroups, so that G _α^* ∩G ∈ ℒ(G) for every subgroup G _α^* ∩G ∈ ℒ(G*) and each factor B/A of a jump of subgroups in ℒ(G*) is isomorphic to a factor of a jump in ℒ(G), or to any specified group H. Using this method, we state new results on right-ordered groups. In particular, it is proved that every Conrad right-ordered group is embedded with preservation of order in a Conrad right-ordered group of Hahn type (i.e., a right-ordered group whose factors of jumps of convex subgroups are order isomorphic to the additive group ℝ); every right-ordered Smirnov group is embedded in a right-ordered Smirnov group of Hahn type; a new proof is given for the Holland–McCleary theorem on embedding every linearly ordered group in a linearly ordered group of Hahn type.     

Codimensions of products and of intersections of verbally primeT-ideals

By Kemer’s theory [9],T idealsJ ₁ ∪…∪J _r andJ ₁ …J _r, where eachJ _i is verbally prime, are of fundamental importance in the theory of P.I. algebras. We calculate, approximately and asymptotically, the codimensions of suchT-ideals, thereby extending the corresponding results about matrix algebras. In all such cases, the exponential growth of the codimensions is calculated; in particular, it is always an integer. Partially supported by NSF grant DMS 9303230. Partially supported by NSF grant DMS 9101488.     

Reconstruction of the set of branches of a graph

It is proved that the set of branches of a graphG is reconstructible except in a very special case. More precisely the set of branches of a graphG is reconstructible unless all the following hold: (1) the pruned center ofG is a vertex or an edge, (2)G has exactly two branches, (3) one branch contains all the vertices of degree one ofG and the other branch contains exactly one end-block. This is the best possible result in the sense that in the special excluded case, the reconstruction of the set of branches is equivalent to the reconstruction of the graph itself.1991Mathematics Subject Classification. Primary 05C60.     

Fixed points of projectivities of prime order

This work begins with a review of the classical results for fixed points of projectivities in a projective plane over a general commutative field. The second section of this work features all the material necessary to prove the main result, which is presented in Theorem 2.8. It is shown that, in a finite projective plane of order q, there exists a projectivity g? of prime order p?>?3 if and only if p divides exactly one of the integers q ? 1, q, q?+?1, q ² + q + 1. Theorem 2.8 establishes a correspondence between the possible structures of points fixed by g?, as presented in Theorem 1.3, and the integer that is divisible by p. The special case of p = 2 is handled in Sect. 2.1, where it is shown that every involution is a harmonic homology for q odd and an elation for q even. The special case of p?=?3 is handled in Sect. 2.2, and Theorem 2.8 is adapted for p?=?3 and presented as Theorem 2.15. An application of Theorems 2.8 and 2.15 is determining the sizes of (n, r)-arcs that are stabilized by projectivities of prime order p in the finite projective plane of order q; in Sect. 3, this application is presented in Propositions 3.2 and 3.3.     

On the Number of Arrangements of Pseudolines

Given a simple arrangement of n pseudolines in the Euclidean plane, associate with line i the list σ _i of the lines crossing i in the order of the crossings on line i. is a permutation of . The vector (σ ₁ ,σ ₂ , ...,σ_n) is an encoding for the arrangement. Define if and , otherwise. Let , we show that the vector (τ ₁ , τ ₂ , ... , τ_n) is already an encoding. We use this encoding to improve the upper bound on the number of arrangements of n pseudolines to . Moreover, we have enumerated arrangements with 10 pseudolines. As a byproduct we determine their exact number and we can show that the maximal number of halving lines of 10 point in the plane is 13. Received December 20, 1995, and in revised form March 8, 1996.     

The special theory of relativity as a case study of the importance of the philosophy of science for the history of science

Summary 1. The philosophical justification of Einstein's conception of distant simultaneity as conventional depends on two cardinal physical assumptions which are stated. Awareness of these two assumptions poses the following historical problem: On what grounds didEinstein feel entitled to make them in 1905? In an endeavor to answer this question, the contribution of experimental results to Einstein's postulational achievement in the Special Theory of Relativity (?RT?) is examined. As a consequence, the author rejects (i)M. Polanyi's recent citation of the history of RT as evidence against an empiricist account of scientific knowledge, and (ii)G. Holton's assessment of the relevance of knowledge of the history of RT to the philosophical mastery of its logical foundations. 2. An analysis of theKennedy-Thorndike experiment is used to provide a refutation of the widespread belief that the aether-theoreticLorentz-Fitzgerald contraction hypothesis was ad hoc in the logical sense. A distinction is drawn between a logical and a psychological sense in which an auxiliary hypothesis can be ad hoc. 3. E. T. Whittaker's disparaging estimate ofEinstein's contributions to RT vis -à -vis those ofLorentz andPoincare is shown to rest on fundamental philosophical misunderstandings ofEinstein's conception of theLorentz transformations.Holton's maxim for the study of the history of RT is then tested in the light of his evaluation ofWhittaker's belittlement ofEinstein's role. To Enrico Bompiani on his scientific Jubilee     

On perimeters of sections of convex polytopes

In his book “Geometric Tomography” Richard Gardner asks the following question. Let P and Q be origin-symmetric convex bodies in R³ whose sections by any plane through the origin have equal perimeters. Is it true that P=Q? We show that the answer is “Yes” in the class of origin-symmetric convex polytopes. The problem is treated in the general case of Rn.     

On the number of regular vertices of the union of jordan regions

Let \C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in \C cross exactly twice, then their intersection points are called regular vertices of the arrangement \A(\C) . Let R(\C) denote the set of regular vertices on the boundary of the union of \C . We present several bounds on |R(\C)| , depending on the type of the sets of \C . (i) If each set of \C is convex, then |R(\C)|=O(n ^1.5+\eps ) for any \eps>0 . (ii) If no further assumptions are made on the sets of \C , then we show that there is a positive integer r that depends only on s such that |R(\C)|=O(n ^2-1/r ) . (iii) If \C consists of two collections \C ₁ and \C ₂ where \C ₁ is a collection of m convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and \C ₂ is a collection of polygons with a total of n sides, then |R(\C)|=O(m ^2/3 n ^2/3 +m +n) , and this bound is tight in the worst case. Received December 4, 1998, and in revised form June 3, 2000. Online publication Feburary 1, 2001.