We investigate some properties of spherical means on the universal covering space of a compact Riemannian manifold. If the fundamental group is amenable then the greatest lower bounds of the spectrum of spherical Laplacians are equal to zero. If the fundamental group is nontransient so are geodesic random walks. We also give an isoperimetric inequality for spherical means.

We prove that the least perimeter $P(n)$ of a partition of a smooth, compact Riemannian surface into $n$ regions of equal area $A$ is asymptotic to $n/2$ times the perimeter of a planar regular hexagon of area $A$. Along the way, we derive tighter estimates for flat tori, Klein bottles, truncated cylinders, and Möbius bands.

The main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

Williamson’s theorem states that for any $2n \times 2n$ real positive definite matrix A, there exists a $2n \times 2n$ real symplectic matrix S such that $S^TAS=D \oplus D$, where D is an $n\times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of A. Let H be any $2n \times 2n$ real symmetric matrix such that the perturbed matrix $A+H$ is also positive definite. In this paper, we show that any symplectic matrix $\tilde {S}$ diagonalizing $A+H$ in Williamson’s theorem is of the form $\tilde {S}=S Q+\mathcal {O}(\|H\|)$, where Q is a $2n \times 2n$ real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that $\tilde {S}$ and S can be chosen so that $\|\tilde {S}-S\|=\mathcal {O}(\|H\|)$. Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45–58, 2017].

There have been a number of papers which give necessary conditions for a ring to be finite, and a few, most notably H. E. Bell [1], which do the same for near-rings. We wish to make a contribution to this latter theme. Most of Bell's results concern distributive near-rings. Our main contribution is to extend a number of these results to weakly distributive near-rings.

We give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.

In his paper "Necessary and sufficient statistics for a family of probability distributions", Dynkin (1951) establishes the important concept of rank for such a family with this conclusion: "If the rank is infinite, then the family has no non-trivial sufficient statistic in any size of sample." His concept of rank is based on a theorem, Theorem 2 described below, which has been pointed out by Brown (1964) to be invalid under its hypotheses. This note shows that Dynkin's Theorem 2 remains valid under its original hypotheses provided that the set (in Dynkin's notation) Δ - S is countable.

In this paper we will construct an example showing that the problem posed in [1] has a negative answer. Two more theorems on the subject treated in [1] will be included.

Let Io = [0, 1], R the reals, and let, for A ⊂ R, Ao be the interior of A. Let {xn} be a sequence in [0, 1> such that 0 = x1 < x2 < … and lim xn = 1. For each n, let In be closed interval having x as its midpoint (except for n = 1 in which case x1 is the left endpoint of I1) such that In ∩ Im= ϕ, and the metric density relative to Io of at 1 is zero. Let Jn be a closed interval in In concentric with In (except for n = 1, where J1 has x1 as its left endpoint) whose length is half that of In

In this paper, we find a lower bound on the number of cyclic function fields of prime degree $l$ whose class numbers are divisible by a given integer $n$. This generalizes a previous result of D. Cardon and R. Murty which gives a lower bound on the number of quadratic function fields with class numbers divisible by $n$.

On the set F of complex-valued arithmetic functions we construct an infinite family of convolutions, that is, binary operations ψ of the form

so that (F, +, ψ) is a commutative ring, for which the unity is unbounded. Here + denotes pointwise addition.

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell +1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\text{Br}_{p^{m}}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^{m}$. We also prove that the symbol length of the Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space ${\mathcal{J}}$ nor into its dual ${\mathcal{J}}^{\ast }$. It is a particular case of a more general result on the non-equi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic structure. This allows us to exhibit a coarse invariant for Banach spaces, namely the non-equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property ${\mathcal{Q}}$ of Kalton. We conclude with a remark on the coarse geometry of the James tree space ${\mathcal{J}}{\mathcal{T}}$ and of its predual.

We determine the asymptotic behavior of the higher dimensional Reidemeister torsion for the graph manifolds obtained by exceptional surgeries along twist knots. We show that all irreducible $\text{S}{{\text{L}}_{2}}(\mathbb{C})$-representations of the graph manifold are induced by irreducible metabelian representations of the twist knot group. We also give the set of the limits of the leading coeõcients in the higher dimensional Reidemeister torsion explicitly.

A Lidstone series provides a (formal) two-point expansion of a given function f(x) in terms of its derivatives of even order at the nodes 0 and 1 and takes the form.

For a given list assignment $L\,=\,\{L\left( v \right)\,:\,v\,\in \,V\left( G \right)\}$, a graph $G\,=\,\left( V,\,E \right)$ is $L$-colorable if there exists a proper coloring $c$ of $G$ such that $c\left( v \right)\,\in \,L\left( v \right)$ for all $v\,\in \,V$. If $G$ is $L$-colorable for every list assignment $L$ having $\left| L\left( v \right) \right|\,\ge \,k$ for all $v\,\in \,V$, then $G$ is said to be $k$-choosable. Montassier (Inform. Process. Lett. 99 (2006) 68-71) conjectured that every planar graph without cycles of length 4, 5, 6, is 3-choosable. In this paper, we prove that every planar graph without 5-, 6- and 10-cycles, and without two triangles at distance less than 3 is 3-choosable.

Let E be one of the spaces C(K) and L1, F be an arbitrary Banach space, p > 1, and (X, σ) be a space with a finite measure. We prove that E is isometric to a subspace of the Lebesgue-Bochner space LP(X; F) only if E is isometric to a subspace of F. Moreover, every isometry T from E into Lp(X; F) has the form Te(x) = h(x)U(x)e, e ∊ E, where h: X —> R is a measurable function and, for every x ∊ X, U(x) is an isometry from E to F

This paper deals with the relative trace formula in the case of base change. Two truncations of the kernel are introduced, both based on the ideas of Arthur, and their integrals are shown to be asymptotic to each other. We also consider products of the kernel with automorphic forms, as these appear when comparing trace formulae (see [5]).

A vector lattice F is said to be Dedekind α-complete, where α is a cardinal number, provided that each non-empty order bounded subset D of F satisfying card(D) ≤ α has a supremum. Several characterizations of this property are presented here.

A commutative semigroup G of continuous, selfmappings on (X, d) is called proximately nonexpansive on X if for every x in X and every (β > 0, there is a member g in G such that d(fg(x),fg(y)) ≤ (1 + β) d (x, y) for every f in G and y in X. For a uniformly convex Banach space it is shown that if G is a commutative semigroup of continuous selfmappings on X which is proximately nonexpansive, then a common fixed point exists if there is an x0 in X such that its orbit G(x0) is bounded. Furthermore, the asymptotic center of G(x0) is such a common fixed point.

In 1967, Powers [7] proved that there exists a one-parameter family of pairwise non-isomorphic hyperfinite factors of type III. Powers′ result on hyperfinite factors has been extended by Araki and Woods [1]. Connes [4], and Williams [11], with different proofs, showed that there exists a continuous family of mutually non-isomorphic non-hyperfinite factors of type III.