In an earlier paper [3] we considered an inversion formula for the Gauss transformation G defined by

1.1

We noted there that formally G is inverted by,

1.2

and we showed that if e-D2 is interpreted via the power series for the exponential function, that is if

1.3

then under certain conditions on φ,

1.4

An orientation of a graph G is an assignment of a unique direction to each line of G. The result is called an oriented graph. Two orientations of a graph are regarded as equivalent if the resulting oriented graphs are isomorphic as directed graphs. For example, the graph C3 consisting of a cycle of length 3 (a triangle) shown in Figure 1(a), has exactly two orientations D1 and D2; see Figure 1(b) and (c).

J. Dixmier has observed [3, p. 1070] that a non-reflexive Banach space has non-rotund fourth conjugate space. It is the aim of this paper to improve Dixmier’s result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are non-smooth points of the third conjugate space.

We consider the class of polynomial differential systems of the form $\dot{x} =\,\lambda x\,-\,y\,+\,{{P}_{n}}\left( x,\,y \right)$ , $\dot{y} =\,x\,+\,\lambda y\,+\,{{Q}_{n}}\left( x,\,y \right)$ where ${{P}_{n}}$ and ${{Q}_{n}}$ are homogeneous polynomials of degree $n$ . For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence.

Let $X,Y$ be two Hilbert spaces, let E be a subset of $X,$ and let $G\colon E \to Y$ be a Lipschitz mapping. A famous theorem of Kirszbraun’s states that there exists $\tilde {G} : X \to Y$ with $\tilde {G}=G$ on E and $ \operatorname {\mathrm {Lip}}(\tilde {G})= \operatorname {\mathrm {Lip}}(G).$ In this note we show that in fact the function $\tilde {G}:=\nabla _Y( \operatorname {\mathrm {conv}} (g))( \cdot , 0)$, where

$$\begin{align*}g(x,y) = \inf_{z \in E} \Big\lbrace \langle G(z), y \rangle + \frac{\operatorname{\mathrm{Lip}}(G)}{2} \|(x-z,y)\|^2 \Big\rbrace + \frac{\operatorname{\mathrm{Lip}}(G)}{2}\|(x,y)\|^2, \end{align*}$$

$C^{1,1}$

Let μ∗ be an outer measure on (X, S) with σ- algebra S and let μ* be the inner measure induced by μ∗. A set M is a measurable cover of a set A ⊆ X if A ⊆ M, M is measurable, and μ∗ (M-A) = 0. We assume that every subset of X has a measurable cover; this holds, for example, if μ∗ is the outer measure induced by a measure which is σ- finite on X [2, theorem C, p. 50].

A graph with m(>1) vertices, each pair of distinct vertices connected by an edge, and also a graph obtained from such a graph by the process of subdividing edges through the insertion of new vertices of valency 2, will be denoted by ≪m, o≫. A graph obtained from a graph with m(>2) vertices in which each pair of distinct vertices are connected by an edge, by deleting n (≤ m-1) edges incident with one and the same vertex, and also a graph obtained from such a graph by the process of subdividing edges through the insertion of new vertices of valency 2, will be denoted by ≪m, n≫.

In [8] and subsequent papers, Jônsson (et al) developed a lattice identity which reflects precisely Desargues Law in projective geometry in that a projective geometry satisfies Desargues Law if and only if the geometry, qua lattice, satisfies this identity. This identity, appropriately called the Arguesian law, has become exceedingly important in recent investigations in the variety of modular lattices (see for example [2], [3], [9], and [12]). In this note, we supply two possible lattice identities for the Pappus' Law of projective geometry.

Katz and Sarnak predicted that the one level density of the zeros of a family of L-functions would fall into one of five categories. In this paper, we show that the one level density for L-functions attached to cubic Galois number fields falls into the category associated with unitary matrices.

We find conditions under which the numerator of the zeta-function of the curve y2+y = xd over Fp, where d — 2g +1 is a prime, d ≠ p, is irreducible over Q. This leads to the generalized Mersenne problem of "almost primality" of the number of points on the jacobian of such a curve over an extension of Fp, which has application to public key cryptography.

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions. Starting with a Finsler spray $S$ and a holonomy invariant function ${\mathcal{P}}$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\unicode[STIX]{x1D706}{\mathcal{P}}{\mathcal{C}}$. We prove that for any holonomy invariant nontrivial function ${\mathcal{P}}$ and for almost every value $\unicode[STIX]{x1D706}\in \mathbb{R}$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray. In these cases, the holonomy invariant function ${\mathcal{P}}$ is necessarily one of the principal curvatures of the geodesic structure.

A nonstandard proof of the fact that a Banach space in which a ball is contained in the range of a countably additive measure is superreflexive is given. The proof is an application of a general method in which we first transfer certain standard objects to the nonstandard hull of a Banach space and then, using the quite well developed theory of nonstandard hulls, derive results about the objects in the original Banach space. It also provides us with an example of the applications of the theory of nonstandard hull valued measures.

In this paper we determine necessary and sufficient conditions for the square of a vertex-unicyclic graph to be Hamiltonian. The conditions are simple and easily checked. Further, we show that the square of a vertex-unicyclic graph is Hamiltonian if and only if it is vertex-pancyclic.

An F-measure on a Cartesian product of algebras of sets is a scalar-valued function which is a scalar measure independently in each coordinate. It is demonstrated that an F-measure on a product of algebras determines an F-measure on the product of the corresponding σ-algebras if and only if its Fréchet variation is finite. An analogous statement is obtained in a framework of fractional Cartesian products of algebras, and a measurement of p-variation of F-measures, based on Littlewood-type inequalities, is discussed.

This paper studies a pointwise definition of sets of uniqueness on compact, 0-dimensional, metric groups. It is shown that this definition is equivalent for closed sets to one based on supports of pseudo functions. An analog of Rajchman's theorem is given leading to examples of sets of uniqueness.

For a completely regular Hausdorff topological space X, let Z(X) denote the lattice of zero-sets of X. If T is a continuous map from X to Y, then there is a lattice homomorphism T” from Z(Y) to Z(X) induced by T which is defined by τ‘(A) = τ←(A). A characterization is given of those lattice homomorphisms from Z(Y) to Z(X) which are induced in the above way by a continuous function from X to Y.

New characterizations of weakly-continuous and θ-continuous functions are presented, and θ-continuity is applied to characterize H(i) spaces; a recent characterization of closed graph functions is utilized to characterize H-closed spaces. Noiri has shown that a function λ which is almost-continuous in the sense of Husain is weakly-continuous if cl(λ−1(W)) ⊂ λ−1(cl(W)) for all open W. It is established here that almost-continuity is superfluous in this statement.

In 2012, Gubeladze (Adv. Math. 2012) introduced the notion of $k$-convex-normal polytopes to show that integral polytopes all of whose edges are longer than $4d(d+1)$ have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no diòerence between $k$- and $(k+1)$-convex-normality (for $k\ge 3$) and improve the bound to $2d(d+1)$. In the second part we extend the definition to pairs of polytopes. Given two rational polytopes $P$ and $\text{Q}$, where the normal fan of $P$ is a refinement of the normal fan of $\text{Q}$, if every edge ${{e}_{P}}$ of $P$ is at least $d$ times as long as the corresponding face (edge or vertex) ${{e}_{\text{Q}}}$ of $\text{Q}$, then $(P+\text{Q})\cap {{\mathbb{Z}}^{d}}=(P\cap {{\mathbb{Z}}^{d}})+(\text{Q}\cap {{\mathbb{Z}}^{d}})$.

The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log \,\left| P \right|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log \,\left| P \right|$ for possibly different ${{P}^{'}}\text{s}$), multiple Mahler measure (involving products of $\log \,\left| P \right|$ for possibly different ${{P}^{'}}\text{s}$), and higher Mahler measure (involving ${{\log }^{k}}\,\left| P \right|$).

D. H. Lehmer initiated the study of the distribution of totatives, which are numbers coprime with a given integer. This led to various problems considered by P. Erdős, who made a conjecture on such distributions. We prove his conjecture by establishing a theorem on the ordering of residues.