Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A₁,...,A_k\in V$ by $A₁*\cdots *A_k=A_{i₁}\cdots A_{iₘ}$. This includes the usual product $A₁*\cdots *A_k=A₁\cdots A_k$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying ${\mu }^m=1$ such that ϕ: V → V has...

Let $G_d$ denote the isometry group of ${ℝ}^d$. We prove that if G is a paradoxical subgroup of $G_d$ then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system ${ℱ}_d$ of Jordan domains with differentiable boundaries and of the same volume such that ${ℱ}_d$ has the cardinality of the continuum, and for every amenable subgroup G of $G_d$, the elements of ${ℱ}_d$ are not G-equidecomposable; moreover, their interiors are not G-equidecomposable...

Let $𝔄=\left(\begin{array}{cc}𝒜& ℳ\\ & ℬ\end{array}\right)$ be the triangular algebra consisting of unital algebras $𝒜$ and $ℬ$ over a commutative ring $R$ with identity $1$ and $ℳ$ be a unital $(𝒜,ℬ)$-bimodule. An additive subgroup $𝔏$ of $𝔄$ is said to be a Lie ideal of $𝔄$ if $[𝔏,𝔄]\subseteq 𝔏$. A non-central square closed Lie ideal $𝔏$ of $𝔄$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $𝔄$, every generalized Jordan triple higher derivation of $𝔏$ into $𝔄$ is a generalized higher derivation of $𝔏$ into $𝔄$. ...

In a recent paper we proved that there are at most finitely many complex numbers $t\ne 0,1$ such that the points $(2,\sqrt{2(2-t)})$ and $(3,\sqrt{6(3-t)})$ are both torsion on the Legendre elliptic curve defined by $y^2=x(x-1)(x-t)$. In a sequel we gave a generalization to any two points with coordinates algebraic over the field $\mathbf{Q}\left(t\right)$ and even over $\mathbf{C}\left(t\right)$. Here we reconsider the special case $(u,\sqrt{u(u-1)(u-t)})$ and $(v,\sqrt{v(v-1)(v-t)})$ with complex numbers $u$ and $v$.

Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁\ast \cdots \ast T_k$ on elements in ${}_i$, which covers the usual product $T₁\ast \cdots \ast T_k=T₁\cdots T_k$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $\sigma \left(\Phi \left(A₁\right)\ast \cdots \ast \Phi \left(A_k\right)\right)=\sigma \left(A₁\ast \cdots \ast A_k\right)$ whenever any one of $A_i$’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied...

If the space $𝒬$ of quadratic forms in ${ℝ}^n$ is splitted in a direct sum ${𝒬}_1\oplus ...\oplus {𝒬}_k$ and if $X$ and $Y$ are independent random variables of ${ℝ}^n$, assume that there exist a real number $a$ such that $E\left(X\right|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E\left(q\left(X\right)\right|X+Y)=b_{i}q(X+Y)$ for any $q$ in ${𝒬}_i.$ We prove that this happens only when $k=2$, when ${ℝ}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

Let $𝒜$ be a noncommutative prime ring equipped with an involution ‘$*$’, and let ${𝒬}_{ms}\left(𝒜\right)$ be the maximal symmetric ring of quotients of $𝒜$. Consider the additive maps $\mathscr{H}$ and $𝒯:𝒜\to {𝒬}_{ms}\left(𝒜\right)$. We prove the following under some inevitable torsion restrictions. (a) If $m$ and $n$ are fixed positive integers such that $(m+n)𝒯\left(a^2\right)=m𝒯\left(a\right)a^{*}+na𝒯\left(a\right)$ for all $a\in 𝒜$ and $(m+n)\mathscr{H}\left(a^2\right)=m\mathscr{H}\left(a\right)a^{*}+na𝒯\left(a\right)$ for all $a\in 𝒜$, then $\mathscr{H}=0$. (b) If $𝒯\left(aba\right)=a𝒯\left(b\right)a^{*}$ for all $a,b\in 𝒜$, then $𝒯=0$. Furthermore, we characterize Jordan left $\tau$-centralizers in semiprime rings admitting an anti-automorphism $\tau$. As applications, we find the...

Condizione necessaria e sufficiente affinché una funzione rapidamente decrescente di variabile reale sia uniformemente analitica è che per i suoi coefficienti ${\gamma }_0,{\gamma }_1,\mathrm{\dots }$ di Fourier-Hermite riesca ${\gamma }_m=0(e^{\sqrt{mt}})$ per $t>0$ abbastanza piccolo.

By iterating the Bolyai-Rényi transformation $T\left(x\right)={(x+1)}^2(mod1)$, almost every real number $x\in [0,1)$ can be expanded as a continued radical expression $$x=-1+\sqrt{x_1+\sqrt{x_2+\cdots +\sqrt{x_n+\cdots }}}$$ with digits $x_n\in \{0,1,2\}$ for all $n\in \mathbb{N}$. For any real number $x\in [0,1)$ and digit $i\in \{0,1,2\}$, let $r_n(x,i)$ be the maximal length of consecutive $i$’s in the first $n$ digits of the Bolyai-Rényi expansion of $x$. We study the asymptotic behavior of the run-length function $r_n(x,i)$. We prove that for any digit $i\in \{0,1,2\}$, the Lebesgue measure of the set $$D\left(i\right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}\frac{r_n(x,i)}{logn}=\frac{1}{log{\theta }_i}\right\}$$ is $1$, where ${\theta }_i=1+\sqrt{4i+1}$. We also obtain that the level set $$E_{\alpha }\left(i\right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}\frac{r_n(x,i)}{logn}=\alpha \right\}$$ is of full Hausdorff...

For graphs $G$, $F_1$, $F_2$, we write $G\to (F_1,F_2)$ if for every red-blue colouring of the edge set of $G$ we have a red copy of $F_1$ or a blue copy of $F_2$ in $G$. The size Ramsey number $\widehat{r}(F_1,F_2)$ is the minimum number of edges of a graph $G$ such that $G\to (F_1,F_2)$. Erdős and Faudree proved that for the cycle $C_n$ of length $n$ and for $t\ge 2$ matchings $t{K}_2$, the size Ramsey number $\widehat{r}(t{K}_2,C_n)<n+(4t+3)\sqrt{n}$. We improve their upper bound for $t=2$ and $t=3$ by showing that $\widehat{r}(2K_2,C_n)\le n+2\sqrt{3n}+9$ for $n\ge 12$ and $\widehat{r}(3K_2,C_n)<n+6\sqrt{n}+9$ for $n\ge 25$.

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $\left|B\right|>\left|G\right|/k^{1/3}$ we have $B^3=G$. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a...

Let $d$ be a square-free positive integer and $h\left(d\right)$ be the class number of the real quadratic field $\mathbb{Q}\left(\sqrt{d}\right).$ We give an explicit lower bound for $h(n^2+r)$, where $r=1,4$. Ankeny and Chowla proved that if $g>1$ is a natural number and $d=n^{2g}+1$ is a square-free integer, then $g\mid h\left(d\right)$ whenever $n>4$. Applying our lower bounds, we show that there does not exist any natural number $n>1$ such that $h(n^{2g}+1)=g$. We also obtain a similar result for the family $\mathbb{Q}\left(\sqrt{n^{2g}+4}\right)$. As another application, we deduce some criteria for a class group of prime power order to be...

Let $ℛ$ be a semiprime ring with unity $e$ and $\phi$, $\varphi$ be automorphisms of $ℛ$. In this paper it is shown that if $ℛ$ satisfies $$2𝒟\left(x^n\right)=𝒟\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒟\left(x^{n-1}\right)$$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ is an ($\phi$, $\varphi$)-derivation. Moreover, this result makes it possible to prove that if $ℛ$ admits an additive mappings $𝒟,𝒢:ℛ\to ℛ$ satisfying the relations $$\begin{array}{c}2𝒟\left(x^n\right)=𝒟\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒢\left(x\right)+𝒢\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒢\left(x^{n-1}\right),\\ 2𝒢\left(x^n\right)=𝒢\left(x^{n-1}\right)\phi \left(x\right)+\varphi \left(x^{n-1}\right)𝒟\left(x\right)+𝒟\left(x\right)\phi \left(x^{n-1}\right)+\varphi \left(x\right)𝒟\left(x^{n-1}\right),\end{array}$$ for all $x\in ℛ$ and some fixed integer $n\ge 2$, then $𝒟$ and $𝒢$ are ($\phi$, $\varphi$)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras. ...

Let $G$ be a finite group. An element $g\in G$ is called a vanishing element if there exists an irreducible complex character $\chi$ of $G$ such that $\chi \left(g\right)=0$. Denote by $\mathrm{Vo}\left(G\right)$ the set of orders of vanishing elements of $G$. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that $\mathrm{Vo}\left(G\right)=\mathrm{Vo}\left(M\right)$ and $\left|G\right|=\left|M\right|$. Then $G\cong M$. We answer in affirmative this conjecture for $M=Sz\left(q\right)$, where $q=2^{2n+1}$ and either $q-1$, $q-\sqrt{2q}+1$ or $q+\sqrt{2q}+1$ is a prime number, and $M=F_4\left(q\right)$, where...

Let $S_n^{\left(2\right)}$ denote the iterated partial sums. That is, $S_n^{\left(2\right)}=S_1+S_2+\cdots +S_n$, where $S_i=X_1+X_2+\cdots +X_i$. Assuming $X_1,X_2,...,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{\left(2\right)}:=\mathbb{P}\left(\underset{1\le i\le n}{max}{S}_i^{\left(2\right)}lt;0\right)\le c\sqrt{\frac{𝔼|S_{n+1}|}{(n+1)𝔼|X_1|}},$$ with $c\le 6\sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{\left(2\right)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0lt;\gamma lt;1/4$ there exist integrable,...