Special algebraic surfaces with an infinite set of planes of oblique symmetry

We establish a constraint on the mutual location of three linear spans of the Gorbits of the directions of symmetry for an algebraic hypersurface in E^m with a symmetry group G generated by reflections relative to hyperplanes.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 52–55, 1990.     

On algebraic surfaces with an infinite set of planes of skew symmetry

We consider (m-1)-dimensional noncylindrical algebraic surfaces in a real spaceR ^m with infinite symmetry groupsG generated by skew reflections with respect to planes, along with the relative locations of linear hulls of fourG-orbits of the directions of symmetry. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 69–74. 1993.     

Algebraic surfaces with an infinite set of planes of oblique symmetry. I

Translated from Ukrainskii Geometricheskii Sbornik, No. 32, pp. 47–60, 1989.     

Algebraic surfaces with an infinite set of planes of oblique symmetry. II

Suppose that G is an infinite group generated by obliquereflections with respect to hyperplanes in the real spaceE ^m and that theµ _j-planes II^µj = II^dj II^j (j = G(u) -orbits of directions of symmetryu (hyperplanes of symmetry conjugate to vectors II^dj pass through _j-planesII ^j). A complete solution of the problem of the mutual position of three II^j is given. Systems of generatrices of the rings of invariants of a series of groups G are found.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 42–51, 1991.     

On some classes of algebraic surfaces with an infinite set of subspaces of skew symmetry

Special classes of (m – 1)-dimensional algebraic surfaces F ⁿ in a space E^m with inifinite set ofsubspaces of skew symmetry (in particular, orthogonal) are studied. It is assumed that directions of symmetry, as a rule, are asymptotic for F _n .Translated from Dinamicheskie Sistemy, No. 8, pp. 119–126, 1989.     

Some cubic surfaces with an infinite set of planes of skew-symmetry

The symmetry of special cubic surfaces in Euclidean spaces of dimensions 7, 8, and 9 is studied.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 37–41.     

A non-linear algebraic matroid with infinite characteristic set

The algebraic characteristic set X_alg(M) of a matroid M is the set of characteristics of all fields F over which M has an algebraic representation. This characteristicsset X_lin(M) for vector representations is defined similarly. P. Vamox has observed that if X_lin (M) is infinite, then 0 ϵ X_lin (M). We show by an example that the corresponding results for algebraic representations does not hold.     

On the general equation of an algebraic surface having the symmetry group of the polyhedron 421

Translated from Ukrainskiii Geometricheskii Sbornik, Issue 28, 1985, pp. 70–74.     

Minimal surfaces with the area growth of two planes: The case of infinite symmetry

We prove that a connected properly immersed minimal surface in with infinite symmetry group and area growth constant less than is a plane, a catenoid, or a Scherk singly-periodic minimal surface. As a consequence, the Scherk minimal surfaces are the only connected periodic minimal desingularizations of the intersections of two planes.

     

The algebraic structure of the set of solutions to the Thue equation

Let Fn be a binary form with integral coefficients of degree n?2, let d denote the greatest common divisor of all non-zero coefficients of Fn, and let h?2 be an integer. We prove that if d=1 then the Thue equation (T) Fn(x,y)=h has relatively few solutions: if A is a subset of the set T(Fn,h) of all solutions to (T), with r:=card(A)?n+1, then

(#)

h divides the numberΔ(A):=_∏1?k<l?rδ(ξk,ξl),

where ξk=〈xk,yk〉∈A, 1?k?r, and δ(ξk,ξl)=xkyl−xlyk. As a corollary we obtain that if h is a prime number then, under weak assumptions on Fn, there is a partition of T(Fn,h) into at most n subsets maximal with respect to condition (#).     

Evaluation of infinite series involving special products and their algebraic characterization

The aim of the paper is the investigation of special infinite series of the form

The complete set of symmetry operators of the schrödinger equation

The complete set of symmetry operators of an arbitrary order associated with the Schrödinger equation is found. It is shown that this equation is invariant with respect to a 28-dimensional Lie algebra, realized in the class of differential operators of the second order. Higher-order symmetries of the Levi-Leblond equation are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1521–1526, November, 1991.     

Solution of an infinite system of algebraic equations

A system of infinite algebraic equations is solved explicitly using a Carleman-type problem.     

Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

The conjugation problem on an infinite set of intervals

It is known that if path of integration consists of a finite number of intervals, then: (1) in the case of a Fredholm-type kernel, the index of the Fredholm operator is zero; (2) in the case of a Cauchy-type kernel, the index of the singular integration operator is a finite number (possible zero). Study of the conjugate boundary-value problem on an infinite set of intervals brings out new facts. The following may be noted: (1) A homogeneous boundary-value problem is always solvable in the classK, which is a natural generalization of that of piecewise analytic functions [1]. (2) Associated (conjugated) homogeneous boundary-value problems have any number of linearly independent solutions in the associated (conjugated) classes, so that the notion of class index is no longer relevant. (3) Associated (conjugated) homogeneous singular integral equations have any number of linearly independent solutions in the associated (conjugated) spacesL _p, L_q, p^?1+q^?1=1, so that the notion of operator index is no longer relevant The general theory of the problems under consideration is satisfactorily illustrated by the simplest case—a set of intervals on the real axis. For this reason the line of discontinuities (integration path) in the present paper is part of the real axis. The paper generalizes the results of [2–4]. Relevant work includes [5].     

Discretisation of an infinite delay equation

In this paper, a Banach phase space containing and contained in is defined with which existence of a solution and convergence of a discrete scheme are proved for an infinite delay differential equation.

     

A complete set of symmetry operators for the Dirac equation

A complete set of symmetry operators of arbitrary finite order admitted by the Dirac equation is found. The algebraic structure of this set is investigated and subsets of symmetry operators that form bases of Lie algebras and superalgebras are isolated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1388–1398, October, 1991.     

Generalized Kadomtsev-Petviashvili equation with an infinite-dimensional symmetry algebra

A generalized Kadomtsev-Petviashvili equation, describing water waves in oceans of varying depth, density and vorticity is discussed. A priori, it involves 9 arbitrary functions of one, or two variables. The conditions are determined under which the equation allows an infinite-dimensional symmetry algebra. This algebra can involve up to three arbitrary functions of time. It depends on precisely three such functions if and only if it is completely integrable.