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.     

Characterizations of optimal scalings of matrices

A scaling of a non-negative, square matrixA ≠ 0 is a matrix of the formDAD ^?1, whereD is a non-negative, non-singular, diagonal, square matrix. For a non-negative, rectangular matrixB ≠ 0 we define a scaling to be a matrixCBE ^?1 whereC andE are non-negative, non-singular, diagonal, square matrices of the corresponding dimension. (For square matrices the latter definition allows more scalings.) A measure of the goodness of a scalingX is the maximal ratio of non-zero elements ofX. We characterize the minimal value of this measure over the set of all scalings of a given matrix. This is obtained in terms of cyclic products associated with a graph corresponding to the matrix. Our analysis is based on converting the scaling problem into a linear program. We then characterize the extreme points of the polytope which occurs in the linear program.     

Lattices of dominions of universal algebras

We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that any pair of homomorphisms f, g: A → M ∈ M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 26–45, January–February, 2007.     

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.     

Existence of normal Hall subgroups by means of orders of products

Let G be a finite group, let π be a set of primes and let p be a prime. We characterize the existence of a normal Hall π‐subgroup in G in terms of the order of products of certain elements of G. This theorem generalizes a characterization of A. Moretó and the second author by using the orders of products of elements for those groups having a normal Sylow p‐subgroup 6 . As a consequence, we also give a π‐decomposability criterion for a finite group also by means of the orders of products.     

Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces

On a complete metric space X, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subsetA. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset A: (1) A is the complete preimage of a closed subspace H under a mapping from X into the metric space Y; (2) A is the set of coincidence points of n (n > 1) mappings from X into Y; (3) A is the set of common fixed points of n mappings of X into itself (n = 1, 2, …). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space X, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set A. In particular, in case (2) for n = 2, we obtain a generalization of Arutyunov’s theorem on the coincidences of two mappings. In case (3) for n = 1, we obtain a generalization of the contraction mapping principle.     

Bounds for solid angles of lattices of rank three

We find sharp absolute constants C₁ and C₂ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C₁,C₂]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in RN with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R³. Such spherical configurations come up in connection with the kissing number problem.     

On Isotopies and Homologies of Subvarieties of Toric Varieties

In ℂⁿ we consider an algebraic surface Y and a finite collection of hypersurfaces S_i. Froissart’s theorem states that if Y and S_i are in general position in the projective compactification of ℂⁿ together with the hyperplane at infinity then for the homologies of Y \∪ S_i we have a special decomposition in terms of the homology of Y and all possible intersections of S_i in Y. We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of ℂn in which Y and S_i are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in Y leaving invariant all hypersurfaces Y ∩ S_k with the exception of one Y ∩ S_i, which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface Y the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.     

Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid

In this paper, we obtain a nonlinear Poisson structure and two first integrals in the problem of the plane motion of a circular cylinder and n point vortices in an ideal fluid. This problem is a priori not Hamiltonian; specifically, in the case n= 1 (i.e., in the problem of the interaction of a cylinder with a vortex) it is integrable.     

Fast computation of the modality of polygons

The modality of a vertex of a polygon is the number of minima or maxima in the distance function from the vertex under consideration to each of the other vertices taken in order around the polygon. Modality was defined by Avis, Toussaint, and Bhattacharya (Comput. Math Appl., 8 (1982), 153–156) and has been further studied by Toussaint and Bhattacharya (Technical Report, No. SOCS-81.3, School of Computer Science, McGill University, January 1981). These authors have defined modality for polygons, both convex and nonconvex, but have not given an algorithm to compute the modality of a polygon, other than the obvious algorithm derived from the definition, which is quadratic in the number of vertices of the polygon. Our paper gives a modality-determination algorithm for convex polygons whose running time is linear in the sum of the number of vertices and the total modality of the polygon. As a special case, we can test if a convex polygon is unimodal (a term introduced by Avis et al.) in time O(n) for a polygon with n vertices. This latter result is shown to be optimal to within a constant factor. We then extend our technique to nonconvex polygons and derive a modality-determination algorithm which is better than the obvious algorithm, but probably not optimal. We conclude by posing some open problems and indicating a connection between modality determination and a range query problem which has received some attention in the Computational Geometry literature.     

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.     

On Morrey’s estimate of the Sobolev norms of solutions of elliptic equations

We give a complete proof of Morrey’s estimate for the W ^1,p -norm of a solution of a second-order elliptic equation on a domain in terms of the L ₁-norm of this solution. The dependence of the constant in this estimate on the coefficients of the equation is investigated.     

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.     

Stability of Blaschke's Characterization of Ellipsoids and Radon Norms

Roughly speaking, in this article we prove quantitative versions of the following statements: (i) If each shadow boundary of a convex body in E ^d (d≥ 3) under parallel illumination contains a curve which is almost planar and has the same shadow as the shadow boundary, then the body is approximately ellipsoidal. (ii) If for each diameter of a convex disk in E ² there is a diameter which is not far from being conjugate to the former, then the disk is close to a Radon disk. A consequence of these theorems is a stability result for the symmetry of orthogonality in finite-dimensional normed spaces. In addition, a stability result for the relation between different definitions of length in a normed plane is given. Received May 28, 1996, and in revised form September 5, 1996.     

Some properties of c-covers of a pair of Lie algebras

In [H. Safa and H. Arabyani, On c-nilpotent multiplier and c-covers of a pair of Lie algebras, Commun. Algebra 45(10) (2017), 4429–4434], we characterized the structure of the c-nilpotent multiplier of a pair of Lie algebras in terms of its c-covering pairs and discussed some results on the existence of c-covers of a pair of Lie algebras. In the present paper, it is shown under some conditions that a relative c-central extension of a pair of Lie algebras is a homomorphic image of a c-covering pair. Moreover, we prove that a c-cover of a pair of finite dimensional Lie algebras, under some assumptions, has a unique domain up to isomorphism and also that every perfect pair of Lie algebras admits at least one c-cover. Finally, we discuss a result concerning the isoclinism of c-covering pairs.     

Representation of MV-algebras by regular ultrapowers of [0, 1]

We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real interval [0,1]. This result also implies the existence, for any cardinal α, of a single MV-algebra in which all infinite MV-algebras of cardinality at most α embed. Recasting the above construction with iterated ultrapowers, we show how to construct such an algebra of values in a definable way, thus providing a sort of “canonical” set of values for the functional representation.     

Semigroups of matrices of intermediate growth

Finitely generated linear semigroups over a field K that have intermediate growth are considered. New classes of such semigroups are found and a conjecture on the equivalence of the subexponential growth of a finitely generated linear semigroup S and the nonexistence of free noncommutative subsemigroups in S, or equivalently the existence of a nontrivial identity satisfied in S, is stated. This ‘growth alternative’ conjecture is proved for linear semigroups of degree 2, 3 or 4. Certain results supporting the general conjecture are obtained. As the main tool, a new combinatorial property of groups is introduced and studied.     

Further study of growth of fractional-power semigroups

If the resolvent of a closed linear operator A in Banach space is defined and decays suitably in a region asymptotic, in a special sense, to a half-plane, then fractional powers ?(?A)^a generate semigroups, holomorphic in sectors of the complex plane. We show that the growth of such semigroups near the origin, or with angular approach to the edges of their sectors of definition, corresponds to the rate of decay of the resolvent of A and the extent of the resolvent set of A.     

Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions

We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height M > 0 [4] in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non–radial) minimizer in accordance with the results of [9]. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with n sides centered in the disc, and numerical experiments indicate that the natural number n > 2 is a non–decreasing function of M. The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height M.     

On End-Extensions of Models of ¬exp

Every model of IΔ₀ is the tally part of a model of the stringlanguage theory Th-FO (a main feature of which consists in having induction on notation restricted to certain AC⁰. sets). We show how to “smoothly” introduce in Th-FO the binary length function, whereby it is possible to make exponential assumptions in models of Th-FO. These considerations entail that every model of IΔ₀ + ¬exp is a proper initial segment of a model of Th-FO and that a modicum of bounded collection is true in these models. Mathematics Subject Classification: 03F30, 03C62, 68Q15.