The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

A norm |⋅| on a Banach space X is locally uniformly rotund (LUR) if lim |xn — x| = 0 for every xn, x ∈ X for which lim2|x|2 + 2 |xn|2-|xn+xn|2 = 0. It is shown that a Banach space X admits an equivalent LUR norm provided there is a bounded linear operator T of X into c0(Γ) such that T* c*0(Γ) is norm dense in X*. This is the case e.g. if X* is weakly compactly generated (WCG).

The main result of this paper is the following: any weighted Riemannian manifold $(M,g,\unicode[STIX]{x1D707})$, i.e., a Riemannian manifold $(M,g)$ endowed with a generic non-negative Radon measure $\unicode[STIX]{x1D707}$, is infinitesimally Hilbertian, which means that its associated Sobolev space $W^{1,2}(M,g,\unicode[STIX]{x1D707})$ is a Hilbert space.

We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold $(M,F,\unicode[STIX]{x1D707})$ can be isometrically embedded into the space of all measurable sections of the tangent bundle of $M$ that are $2$-integrable with respect to $\unicode[STIX]{x1D707}$.

By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.

Let $R$ be a semiprime ring with center $Z\left( R \right)$. For $x,\,y\,\in \,R$, we denote by $\left[ x,\,y \right]\,=\,xy\,-\,yx$ the commutator of $x$ and $y$. If $\sigma $ is a non-identity automorphism of $R$ such that

1

$$\left[ \left[ \cdot \cdot \cdot \,\left[ \left[ \sigma \left( {{x}^{n0}} \right),\,{{x}^{n1}} \right],\,{{x}^{n2}} \right],\cdot \cdot \cdot \right],\,{{x}^{nk}} \right]\,=\,0$$

for all $x\,\in \,R$, where ${{n}_{0}},\,{{n}_{1}},\,{{n}_{2}},\,...,\,{{n}_{k}}$ are fixed positive integers, then there exists a map $\mu \,:\,R\,\to \,Z\left( R \right)$ such that $\sigma \left( x \right)\,=\,x\,+\,\mu \left( x \right)$ for all $x\,\in \,R$. In particular, when $R$ is a prime ring, $R$ is commutative.

In 1960, Z. Opial [20] proved the following interesting integral inequality:

Theorem A. If u is a continuously differentiable function on [0, b], and if u(0) = u(b) = 0, and u(x)>0 for x∊(0, b), then

1

where the constant b/4 is the best possible.

It is known that among all plane convex domains of given constant width Reuleaux triangles have minimal and circular discs have maximal area. Some estimates are given concerning the following associated stability problem: If K is a convex domain of constant width w and if the area of K differs at most ∊ from the area of a Reuleaux triangle or a circular disc of width w, how close (in terms of the Hausdorff distance) is K to a Reuleaux triangle or a circular disc? Another result concerns the deviation of a convex domain M of diameter d from a convex domain of constant width if the perimeter of M is close to πd.

We study a class of subgroups of $\text{PS}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ which can be characterized in different ways, such as congruence groups, modular forms, modular curves, elliptic surfaces, lattices and graphs.

In this paper, we discuss monotonicity formulae of various entropy functionals under various rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue of a family of geometric operators $-4\Delta \,+\,kR$ is monotonic along the normalized Ricci flow for all $k\,\ge \,1$ provided the initial manifold has nonpositive total scalar curvature.

Let δ1,δ2 be two derivations of a C*-algebra. We characterize when δ1δ2 is a derivation, a compact, or a weakly compact operator

The orthogonal latin squares displayed in [1] and [2] have the property that their row permutations are transformed amongst themselves by a permutation of order 7. In this note I present three examples of orthogonal latin squares of order 10 whose row permutations are transformed amongst themselves by a permutation of order 9.

The well-known Falkner–Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\text{ }\!\!\lambda\!\!\text{ }\!\!\pi\!\!\text{ /2}$, where $\text{ }\!\!\lambda\!\!\text{ }\,\in \,\mathbb{R}$ is a parameter involved in the equation. It is known that there exists ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,<\,0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\text{ }\!\!\lambda\!\!\text{ }\,\ge \,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ and has no positive solutions for $\text{ }\!\!\lambda\!\!\text{ }\,<\,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ . The known numerical result shows ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,=\,-0.1988$ . In this paper, ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,\in \,[-0.4,\,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner–Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner–Skan equation.

We consider round-robin tournaments of n players in which, at each encounter, the winner is awarded 1 point and the loser 0 (ties are excluded).

Let

1

be the n scores, ordered in a non-decreasing sequence. Clearly

2

In the sequel we construct simple, unital, separable, stable, amenable ${{C}^{*}}$-algebras for which the ordered ${{K}_{0}}$-group is strongly perforated and group isomorphic to $Z$. The particular order structures to be constructed will be described in detail below, and all known results of this type will be generalised.

In [4], A. Sade defines the singular direct product for quasigroups. In this paper we use the singular direct product to construct quasigroups satisfying the identity x(xy)=yx. In particular, we show that the singular direct product preserves the identity x(xy)=yx.

Let (X1, d1) and (X2, d2) be metric spaces. A mapping/: X1→X2 is said to be a Lipschitz mapping (with respect to d1 and d2) if and only if (*)d2(f(x), f(y))≤λ⋅ d1(x, y) for all x, y∈X1, where λ is a fixed real number. The constant λ is called a Lipschitz constant for f. If (*) is satisfied for λ=l, then f is called nonexpansive (see, for example, [21]) and if (*), again with λ=1, is replaced by a strict inequality for all x≠y, then f is called contractive [1]. If x∊X1 and X1=X2, then the sequence where f1(x)=f(x) and fn(x)=f(fn-1(x)) for each n>1, is called the sequence of iterates of f at x.

A presentation of a group G is an exact sequence of groups

where F is a free group. Let l→S ⊆ F→G→1 be another presentation of G involving the same free group F.

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

The Pólya-Vinogradov inequality states that for any non-principal character x modulo q and any N ≧ 1,

where c is an absolute constant. We show that (*) holds with c = 2/(3π2) + o(1) in the case x is primitive and x (— 1) =1 with c = 1/(3π) + o(l) in the case x is primitive and x(— 1) = — 1- This improves by a factor 2/3 the previously best-known values for these constants.

Let y be the solution of the equation

where A, B, C, λ and η aie complex numbers and It is shown that y has exponential order equal to one if A ≠ 0 and if y is not a polynomial; otherwise, y has exponential order equal to zero. In the latter case, y and all of its derivatives are unbounded on any ray.

We give a simple characterization of coefficient multipliers from the mixed norm space Hp,q,α, 2 ≤ p ≤ ∞, into Hu,v,β, 0 ≤ u ≤ 2, which includes the main results of Wojtaszczyk in [5]. We also calculate multipliers from the Hardy space Hp, 2 ≤ p ≤ ∞, into Hq, 0 < q ≤ 2.