Geodesics on the regular tetrahedron and the cube

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.     

The Diameters Graph of the Root System E 8 is Uniquely Geometrisable

The root system E ₈ has 120 diameters (i.e., lines joining pairs of opposite roots) any two of which make an angle of sixty or ninety degrees. The graph of the title has these diameters as its vertices, where two vertices are adjacent if and only if the corresponding diameters are at right angles. Known results in the literature imply that this graph is geometrisable. In this paper we prove that, modulo automorphisms of the graph, there is a unique way to construct this geometry out of the given graph. Along the way, we observe that this graph is locally the orthogonal graph O(7, 2). We also prove that, modulo automorphisms of the polar space, there is a unique spread of O(7,2).     

Note on packing and weak-packing measures with Hausdorff functions

For every Hausdorff function we construct a compact metric space of finite positive weak-packing measure. Also we prove that for every non-doubling Hausdorff function there exists a compact metric space on which the packing and weak-packing measures are not equivalent.     

Geometric characterizations of centroids of simplices

We study the centroid of a simplex in space. Primary attention is paid to the relationships among the centroids of the different k-skeletons of a simplex in n-dimensional space. We prove that the 0-dimensional skeleton and the n-dimensional skeleton always have the same centroid. The centroids of the other skeleta are generically different (as we prove), but there are remarkable instances where they coincide in pairs. They never coincide in triples for regular pyramids.     

Derived Rees Matrix Semigroups as Semigroups of Transformations

An ordered pair (e,f) of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. Previously [1] we have established that there are four distinct types of skew pairs of idempotents. We have also described (as quotient semigroups of certain regular Rees matrix semigroups [2]) the structure of the smallest regular semigroups that contain precisely one skew pair of each of the four types, there being to within isomorphism ten such semigroups. These we call the derived Rees matrix semigroups. In the particular case of full transformation semigroups we proved in [3] that T_X contains all four skew pairs of idempotents if and only if |X| ≥ 6. Here we prove that T_X contains all ten derived Rees matrix semigroups if and only if |X| ≥ 7.     

Discretization of harmonic measures for foliated bundles

We prove in this Note that there is, for some foliated bundles, a bijective correspondence between Garnett?s harmonic measures and measures on the fiber that are stationary for some probability measure on the holonomy group. As a consequence, we show the uniqueness of the harmonic measure in the case of some foliations transverse to projective fiber bundles.     

Joint Similarities and Parameterizations for Dilations of Dual g -frame Pairs in Hilbert Spaces

In this paper, we give an operator parameterization for the set of dilations of a given pair of dual g-frames and the set of dilations of pairs of dual g-frames of a given g-frame. In particular, for the dilations of a given pair of dual g-frames, we introduce the concept of joint complementary g-frames and prove that the joint complementary g-frames of a pair of dual g-frames are unique in the sense of joint similarity, which then helps to obtain a sufficient condition such that the complementary g-frames of a g-frame are unique in the sense of similarity and show that the set of dilations of a given dual g-frame pair are parameterized by a set of invertible diagonal operators. For the dilations of pairs of dual g-frames, we prove that the set of dilations of pairs of dual g-frames are parameterized by a set of invertible upper triangular operators.     

关于吸引子的某些性质

Conley在[1]中讨论拓扑空间X上动力系统的吸引子——排斥子对时,给出了吸引子存在的两个充分条件．本文证明这两个条件实际上也是必要的,进而证明Conley所定义的吸引子是渐近稳定的．     

关于吸引子的某些性质

Conley在[1]中讨论拓扑空间X上动力系统的吸引子——排斥子对时，给出了吸引子存在的两个充分条件．本文证明这两个条件实际上也是必要的，进而证明Conley所定义的吸引子是渐近稳定的．     

Measures on minimally generated Boolean algebras

We investigate properties of minimally generated Boolean algebras. It is shown that all measures defined on such algebras are separable but not necessarily weakly uniformly regular. On the other hand, there exist Boolean algebras small in terms of measures which are not minimally generated. We prove that under CH a measure on a retractive Boolean algebra can be nonseparable. Some relevant examples are indicated. Also, we give two examples of spaces satisfying some kind of Efimov property.     

Relaxation in ferromagnetism: The rigid case

Summary We undertake the analysis of measure-valued magnetizations in the context of micromagnetics, i.e., parametrized measures coming from sequences of magnetizations, and show that there are no constraints, other than the natural restriction on the support, for this family of probability measures. As a consequence, we prove a general existence theorem for this relaxed formulation and explore relaxation in terms of the first moment of these generalized magnetizations.     

Insecurity for compact surfaces of positive genus

A pair of points in a riemannian manifold M is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in M are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.     

Non-harmonic Cones are Heisenberg Uniqueness Pairs for the Fourier Transform on $${\mathbb {R}}^n$$

In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on \({\mathbb {R}}^n\). We derive that \(\left( S^2, \text { paraboloid}\right) \) and \(\left( S^2, \text { geodesic of } S_r(o)\right) \) are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in \({\mathbb {R}}^3\). Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.     

No Borel Connections for the Unsplitting Relations

We prove that there is no Borel connection for non‐trivial pairs of unsplitting relations. This was conjectured in [3].     

On Numbers which are Mahler Measures

We investigate which algebraic numbers can be Mahler measures. Adler and Marcus showed that these must be Perron numbers. We prove that certain integer multiples of every Perron number are Mahler measures. The results of Boyd give some necessary conditions on Perron number to be a measure. These do not include reciprocal algebraic integers, so it would be of interest to find one which is not a Mahler measure. We prove a result in this direction. Finally, we show that for every non-negative integer k there is a cubic algebraic integer having norm 2 such that precisely the kth iteration of its Mahler measure is an integer.     

Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations

We solve the problem of describing compatible nonlocal Poisson brackets of hydrodynamic type. We prove that for nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type, there exist special local coordinates such that the metrics and the Weingarten operators of both brackets are diagonal. The nonlinear evolution equations describing all nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived in these special coordinates, and the integrability of these equations is proved using the inverse scattering transform. The Lax pairs with a spectral parameter for these equations are found. We construct various classes of integrable reductions of the derived equations. These classes of reductions are of an independent differential-geometric and applied interest. In particular, if one of the compatible Poisson brackets is local, we obtain integrable reductions of the classical Lamé equations describing all orthogonal curvilinear coordinate systems in a flat space; if one of the compatible brackets is generated by a constant-curvature metric, the corresponding equations describe integrable reductions of the equations for orthogonal curvilinear coordinate systems in a space of constant curvature.     

Choquet-type integral representation of polysupermedian measures

Some aspects of multi-parameter potential theory are developed: we give a Choquet-type integral representation for measures which are supermedian for a countable family of submarkovian resolvents of commuting kernels on a Radon measurable space. For the subclass of polysupermedian measures we prove a Riesz-type decomposition, and we show that there is a unique integral representation by minimal polysupermedian measures. The setting covers a variety of very different examples like random fields, measures on product spaces which are supermedian for resolvents on the factor spaces, and completely supermedian measures.     

多峰映射族中非双曲奇异吸引子的丰富性

本文考虑多峰映射族中非双曲奇异吸引子的丰富性,证明多维参数空间中存在正测度的参数集合,对应系统具有绝对连续的不变测度.     

Shift equivalence and the Conley index

In this paper we introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.

     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.