The Geometry of the Lie Algebra of the Orthogonal Group O(ℝ4)

In the six-dimensional space of bivectors, a Lie product similar to the standard vector product in is introduced. The Lie algebra constructed is proved to be isomorphic to the Lie algebra of the orthogonal group , and the isomorphism is a canonical isometry between and the space of antisymmetric operators in . Bibliography: 2 titles.     

The Convex Hull of Several Compacta

Let K₀,K₁,...,K_m be nonempty compact sets in . Then the convex hulls with r₀=0 form a convex family of sets parametrized by . For m=1, the volume Volconv(K₀(K₁+r)) is a convex function of . Bibliography: 5 titles.     

A Higher-Order Analog of the Linking Number for a Pair of Divergence-Free Vector Fields

Pairs B, of divergence-free vector fields with compact support in are considered higher-order analog M(B, c (of order 3) of the Gauss helicity number H(B, )= , curl(A)=B; (of order 1) is constructed, which is invariant under volume-preserving diffeomorphisms. An integral expression for M is given. A degree-four polynomial m(B(x₁), B(x₂), ( ₁), ( ₂)), x₁, x₂, _{1 2} , is defined, which is symmetric in the first and second pairs of variables separately. M is the average value of m over arbitrary configurations of points. Several conjectures clarifying the geometric meaning of the invariant and relating it to invariants of knots and links are stated. Bibliography: 11 titles.     

Transversals of Families of Translations of a Planar Convex Compact

Let be a family of translations of a convex compact set such that every two elements of have a common point. Then there exist three points such that each element of contains one of these points. This answers in the affirmative an old question by Grünbaum. Bibliography: 13 titles.     

Problems on Extremal Decomposition of the Riemann Sphere

We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let be a system of distinct points on and let be the family of all systems of nonoverlapping simply connected domains on such that . Let

Boundary Estimates for Solutions of the Parabolic Free Boundary Problem

Let u and solve the problem

Intersections of Convex Bodies

Let be nonempty convex bodies in . Let be vectors in , let , and let . Then is a convex set, and the family of sets is concave. Let . Then for the mean cross-sectional measures W_v (\Phi (\rho )), , the functions are concave on D. (Note that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca% WGxbWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiabfA6agjaacIcacqaH% bpGCcaGGPaGaaiykaiabg2da9iaabAfacaqGVbGaaeiBamaaBaaale% aatCvAUfKttLearyqr1ngBPrgaiuGacqWFRbWAaeqaaOGaeuOPdyKa% aiikaiabeg8aYjaacMcaaaa!4EE7!\[W_0 (\Phi (\rho )) = {\text{Vol}}_k\Phi (\rho )\] is the k-volume.) Bibliography: 2 titles.     

Cobordisms of Embeddings with Codimension Two

We single out the obstruction for a closed -null-homologous submanifold of codimension 2 to be the boundary of a submanifold of codimension 1. As an application, we calculate the groups of cobordisms of embeddings of nonoriented n-manifolds in the Euclidean (n+2)-space for n=3 and 4. Namely, we show that and . A specific generator of the former group is explicitly given. Bibliography: 5 titles.     

An Extremal Property of the Rellot Triangle

Let be a planar set having unit constant width and piecewise -smooth boundary. Then the area of the set of the points belonging to at least three diameters of K is at most , and the area of the set of the points belonging to a unique diameter of K is at least . In both cases, an equality is attained only if K is the Rellot triangle. Bibliography: 2 titles.     

Sectional Curvatures and the Separation Set of the Complex Projective Space in Its Plücker Model

The Plücker embedding of the complex projective space in the Grassmannian of bivectors is used for proving several theorems on the relationship between the complex structure of and its Riemannian geometry. It is shown that the separation set of in the Plücker model is the face of for a certain calibration. Bibliography: 11 titles.     

Geometry of Two- and Three-Dimensional Minkowski Spaces

A class of centrally-symmetric convex 12-topes (12-hedrons) in is described such that for an arbitrary prescribed norm on each polyhedron in the class can be inscribed in (circumscribed about) the -ball via an affine transformation, and this can be done with large degree of freedom. Bibliography: 5 titles.     

Geodesics on Faces of Calibrations of Degree Two

It is proved that faces of the unit spheres of the mass and comass norms are totally geodesic submanifolds in the manifolds of the extreme points of the spheres. A canonical embedding of the complex projective space in the Plücker model of the Grassmannian is described, and some of the properties of the embedding are proved. As an application of these results, the 2-dimensional sections in having minimal curvature are characterized geometrically. Bibliography: 16 titles.     

Test Rank for Some Free Polynilpotent Groups

We prove a theorem on possible test rank values for groups of the form . It is shown that test rank of a free polynilpotent group is equal to or , for any and every collection of classes. Moreover, for and .     

Two Multidimensional Analogs of the F. and M.~Riesz Theorem

A new collection of Riesz sets for and is exhibited. The results are new even in the one-dimensional case. The asymptotic estimate of Fourier multipliers on finite measures in the space used in the proof is interesting in itself. Bibliography: 3 titles.     

A Uniqueness Theorem for the Dual Problem Associated to a Variational Problem with Linear Growth

Uniqueness is proved for solutions of the dual problem that is associated with the minimum problem among the mappings with prescribed Dirichlet boundary data and for smooth strictly convex integrands f of linear growth. No further assumptions on f or its conjugate function are imposed, in particular, is not assumed to be strictly convex. A special solution of the dual problem is seen to be a mapping into the image of , which immediately implies uniqueness. Bibliography: 13 titles.     

Geometry of the Sphere of Calibrations of Degree Two

A canonical bijection between the set of extreme points of the comass-unit sphere and the manifold of orthogonal complex structures in is described, under which unitary bases correspond to decompositions of the forms realizing the conjugate norm of the mass. This correspondence is used for obtaining a classification of faces of the sphere and the known classification of faces of the set polar to . Bibliography: 10 titles.     

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. III

In the canonical smooth fiber bundles endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups (1, n) and construct (1, n)-invariant generalized differential-geometric connections. In both regular and special cases of the representations of the considered groups (1, n), we find all affine nonholonomic , and _1,2-connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations (1, n) and also describe the corresponding (1, n)invariant planar connections.     

Invariants of Knots and Links on T-Polyhedra

Any link in is isotopic to a knot lying on the polyhedron . A nontrivial theory of links and knots on T is developed. The main result is a construction of an isotopy invariant, which can distinguish pairs of knots on T that are isotopic as knots in . Bibliography: 3 titles.     

Power Invariants of a Union of Coaxial Prisms

The paper is an addition to the paper of Yu. I. Babenko and V. A. Zalgaller published in the same volume. It gives a condition under which the set of all vertices of several coaxial prisms inscribed in a sphere in has power invariants I₁,...,I_n. A finite set in with 11 invariants is constructed. It is also proved that unions of prisms yield finite sets in with any preassigned number n of invariants with alternating signs. Bibliography: 5 titles.     

Temperature-Dependence of Equilibrium States of a Two-Phase Elastic Medium with the Zero Surface Tension Coefficient

We consider the energy functional of a two-phase elastic medium with quadratic energy densities defined for such that ,where is a measurable characteristic function. Under some natural conditions on the data of the problem, we prove the existence of an interval (t ^-,t ⁺) of the change of temperature such that the energy functional has only a minimizer such that for or such that t^ + $$ " align="middle" border="0"> . The energy functional has no minimizers such that or if . We derive two-sided estimates for the numbers in terms of the characteristics of the two-phase elastic medium and the boundary condition. Bibliography: 3 titles.