Let E be a compact set in the complex plane, $g_E$ be the Green function of the unbounded component of ${ℂ}_{\infty }\setminus E$ with pole at infinity and $Mₙ\left(E\right)=sup\left(\right||P^{\text{'}}{\left|\right|}_E{)/(\left|\right|P\left|\right|}_E)$ where the supremum is taken over all polynomials ${P|}_E\not\equiv 0$ of degree at most n, and ${\left|\right|f\left|\right|}_E=sup\left|f\right(z\left)\right|:z\in E$. The paper deals with recent results concerning a connection between the smoothness of $g_E$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence ${Mₙ\left(E\right)}_{n=1,2,...}$. Some additional conditions are given for special classes of sets.

A notion of a wide-sense Markov process $X_t$ of order k ≥ 1, $X_t\sim WM\left(k\right)$, is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of $X_t$ is the k-dimensional process $x_t=(X_{t-k+1},...,X_t)$. The covariance structure of $X_t\sim WM\left(k\right)$ is considered in the general case and in the periodic case. In the general case it is shown that $X_t\sim WM\left(k\right)$ iff $x_t$ is a k-dimensional WM(1) process and iff the covariance function of $x_t$ has the triangular...

We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $𝛷$ be a class of functions on the parameter space and consider estimating elements of $𝛷$ under quadratic loss. If the formal Bayes estimator of every function in $𝛷$ is admissible, then the prior is strongly admissible with respect to $𝛷$. Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with...

Suppose that $\{Xn:n\ge 0\}$ is a stationary Markov chain and $V$ is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if $Y_n:=N^{-1/2}{\sum }_{n=0}^{N}V\left(X_n\right)$ converge in law to a normal random variable, as $N\to +\infty$. For a stationary Markov chain with the $L^2$ spectral gap the theorem holds for all $V$ such that $V\left(X_0\right)$ is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables $V$ for which the CLT holds...

Asymptotic properties of the sequences (a) ${P_{\phi }^{j}g}_{j=1}^{\infty }$ and (b) ${j^{-1}{\sum }_{i=0}^{j-1}P{ⁱ}_{\phi }g}_{j=1}^{\infty }$, where $P_{\phi }:L¹\to L¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov...

Let Ω be an open subset of ${ℝ}^d$ with 0 ∈ Ω. Furthermore, let $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator with domain $C_c^{\infty }\left(\Omega \right)$ where the coefficients $c_{ij}\in W_{loc}^{1,\infty }\left(\Omega ̅\right)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=\left(c_{ij}\right)$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If ${\int }_0^{\infty }ds{s}^{d/2}{e}^{-\lambda \mu \left(s\right)²}<\infty$ for some λ > 0 where $\mu \left(s\right)={\int }_0^{s}dtc{\left(t\right)}^{-1/2}$ then we establish that $H_{\Omega }$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup....

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

In this paper we establish a decoupling feature of the random interlacement process ${ℐ}^u\subset {\mathbb{Z}}^d$ at level $u$, $d\ge 3$. Roughly speaking, we show that observations of ${ℐ}^u$ restricted to two disjoint subsets $A_1$ and $A_2$ of ${\mathbb{Z}}^d$ are approximately independent, once we add a sprinkling to the process ${ℐ}^u$ by slightly increasing the parameter $u$. Our results differ from previous ones in that we allow the mutual distance between the sets $A_1$ and $A_2$ to be much smaller than their diameters. We then provide an important application...

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space $𝙴$. For an open set $𝙶\subseteq 𝙴$, let ${\tau }_{𝙶}$ be the exit time of $X$ from $𝙶$ and $A^{𝙶}$ be the generator of the process killed when it leaves $𝙶$. Let $r:[0,\infty [\to [0,\infty [$ and $R\left(t\right)={\int }_0^{t}r\left(s\right)\mathrm{d}s$. We give necessary and sufficient conditions for ${𝔼}_{\mu }R\left({\tau }_{𝙶}\right)lt;\infty$ in terms of the behavior near the origin of the spectral measure of $-A^{𝙶}$. When $r\left(t\right)=t^l$, $l\ge 0$, by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order $l+1$ for ${\tau }_{𝙶}$...

Let $\beta >1$ be a non-integer. We consider expansions of the form ${\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, where the digits ${\left(d_i\right)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor \left(\beta -1\right)]$. We show existence and uniqueness of a $K_{\beta }$-invariant probability measure, absolutely continuous with respect to $m_p\otimes \lambda$, where $m_p$ is the Bernoulli measure on ${\{0,1\}}^{\mathbb{N}}$ with parameter $p$ ($0<p<1$) and $\lambda$ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta -1\left)\right]$. Furthermore, this measure is of the form $m_p\otimes {\mu }_{\beta ,p}$, where ${\mu }_{\beta ,p}$ is equivalent to $\lambda$. We prove that the measure of maximal entropy and $m_p\otimes \lambda$ are mutually...

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E\subset {ℂ}^N$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function ${\Phi }_E$. Moreover, we show that one of these extremal-like functions is equal to ${\Phi }_E$ if E is a nonpluripolar set with $li{m}_{n\to \infty }Mₙ{\left(E\right)}^{1/n}=1$ where $Mₙ\left(E\right):=sup{\left|\right|\left|gradP\right|\left|\right|}_E/{\left|\right|P\left|\right|}_E$, the supremum is taken over all polynomials P of N variables...

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

Let ${ℳ}^d$ be a d-dimensional normed space with norm ||·|| and let B be the unit ball in ${ℳ}^d$. Let us fix a Lebesgue measure $V_B$ in ${ℳ}^d$ with $V_B\left(B\right)=1$. This measure will play the role of the volume in ${ℳ}^d$. We consider an arbitrary simplex T in ${ℳ}^d$ with prescribed edge lengths. For the case d = 2, sharp upper and lower bounds of $V_B\left(T\right)$ are determined. For d ≥ 3 it is noticed that the tight lower bound of $V_B\left(T\right)$ is zero.

Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. ${ₙ}^c$) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ\in {ₙ}^c$ of the form $Pₙ\left(z\right):={\sum }_{j=0}^n{a}_j{z}_j$, $a_j\in C$, by $Sₙ\left(Pₙ\right)\left(z\right):={\sum }_{j=0}^{n}a{̃}_j{z}_j$, $a{̃}_j:=a_j\left|a_j\right|min|a_j|,1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{real}:=su{p}_{Pₙ\in ₙ}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|)$, $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{comp}:=su{p}_{Pₙ\in {ₙ}^c}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

Let $P\left(z\right)$ be a polynomial of degree $n$ having no zeros in $\left|z\right|<k$, $k\le 1$, and let $Q\left(z\right):=z^n\overline{P(1/\overline{z})}$. It was shown by Govil that if ${max}_{\left|z\right|=1}\left|P^{\text{'}}\left(z\right)\right|$ and ${max}_{\left|z\right|=1}\left|Q^{\text{'}}\left(z\right)\right|$ are attained at the same point of the unit circle $\left|z\right|=1$, then $$\underset{\left|z\right|=1}{max}|P^{\text{'}}\left(z\right)|\le \frac{n}{1+k^n}\underset{\left|z\right|=1}{max}\left|P\left(z\right)\right|.$$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${𝔽}_p\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

We consider nonsingular polynomial maps F = (P,Q): ℝ² → ℝ² under the following regularity condition at infinity $\left(J_{\infty }\right)$: There does not exist a sequence $(p_k,q_k)\subset ℂ²$ of complex singular points of F such that the imaginary parts $(\Im \left(p_k\right),\Im \left(q_k\right))$ tend to (0,0), the real parts $(\Re \left(p_k\right),\Re \left(q_k\right))$ tend to ∞ and $F(\Re \left(p_k\right),\Re \left(q_k\right)))\to a\in ℝ²$. It is shown that F is a global diffeomorphism of ℝ² if it satisfies Condition $\left(J_{\infty }\right)$ and if, in addition, the restriction of F to every real level set $P^{-1}\left(c\right)$ is proper for values of |c| large enough.

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{|}^p$1/p$,$aj ∈ ℂ$,$such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ and L > 0 let ${\kappa }_{\infty }(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0|\ge Lma{x}_{1\le j\le n}|a_j|$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1\surd (n/L)-1\le {\kappa }_{\infty }(n,L)\le c_2\surd (n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${\left|\nabla f\right|\ge C\left|f\right|}^{\varrho }$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1-R{(n,d)}^{-1}$ with $R(n,d)=d{(3d-3)}^{n-1}$.

Let $T_1,\cdots ,T_d$ be homogeneous trees with degrees $q_1+1,\cdots ,q_d+1\ge 3$, respectively. For each tree, let $𝔥:T_j\to \mathbb{Z}$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_1,\cdots ,T_d$ is the graph $\mathrm{𝖣𝖫}(q_1,\cdots ,q_d)$ consisting of all $d$-tuples $x_1\cdots x_d\in T_1\times \cdots \times T_d$ with $𝔥\left(x_1\right)+\cdots +𝔥\left(x_d\right)=0$, equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If $d=2$ and $q_1=q_2=q$ then we obtain a Cayley graph...

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{\text{'}}\left(x\right)={lim}_{t\to x+}{f}_{+}^{\text{'}}\left(t\right)$ and $f_{-}^{\text{'}}\left(x\right)={lim}_{t\to x-}{f}_{-}^{\text{'}}\left(t\right)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DS{C}_{\omega }$ of functions on $ℝ$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new...

We show that for a polynomial mapping $F=(f₁,...,fₘ):{ℂ}^n\to {ℂ}^m$ the Łojasiewicz exponent ${}_{\infty }\left(F\right)$ of F is attained on the set $z\in {ℂ}^n:f₁\left(z\right)·...·fₘ\left(z\right)=0$.

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.