If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group ${}_f\subseteq G*H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism $p_f{=p|}_f{:}_f\to G$. A right inverse (section) $G{\to }_f$ of $p_f$ is called a coaction on G. In this paper we study ${}_f$ and the sections of $p_f$. We consider the following topics: the structure of ${}_f$ as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and...

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 ∈...

It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

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 $ℐ$ be a coherent subsheaf of a locally free sheaf $𝒪\left(E_0\right)$ and suppose that $ℱ=𝒪\left(E_0\right)/ℐ$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $ℱ$ we construct a vector-valued Coleff-Herrera current $\mu$ with support on the variety associated to $ℱ$ such that $\phi$ is in $ℐ$ if and only if $\mu \phi =0$. Such a current $\mu$ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....

For $\left(P_k\right)$ being Rademacher, Fermion or q-Gaussian (-1 ≤ q ≤ 0) operators, we find the optimal constants $C_{2n}$, n∈ ℕ, in the inequality $\parallel {\sum }_{k=1}^N{A}_k\otimes P_k{\parallel }_{2n}\le {\left[C_{2n}\right]}^{1/2n}max{\parallel ({\sum }_{k=1}^{N}A{*}_k{A}_k}^{1/2}{\parallel }_{L_{2n}},\parallel ({\sum }_{k=1}^N{A}_{k}A{*}_k$1/2∥L2n$,$valid for all finite sequences of operators $\left(A_k\right)$ in the non-commutative $L_{2n}$ space related to a semifinite von Neumann algebra with trace. In particular, $C_{2n}=(2nr-1)!!$ for the Rademacher and Fermion sequences.

The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_{n,r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r{(1-x)}^r$, r ≥ 0, and let $P_{n,r}^{\mathbb{Z}}\subset P_{n,r}$ be the set of polynomials with integer coefficients. Let $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ be the maximal distance of elements of $P_{n,r}$ from $P_{n,r}^{\mathbb{Z}}$ in $L_p(0,1)$. We give rather precise quantitative estimates of $\mu (P_{n,r}^{\mathbb{Z}};L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ for p ≠ 2. It follows that $\mu (P_{n,r}^{\mathbb{Z}};L_p)\asymp n^{-2r-2/p}$ as n → ∞. The results...

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$, then $f(x)$ has at least one root a with the following property: if $\mu \le k\le n$, where $\mu$ is the multiplicity of $\alpha$, then $f^{(k)}(\alpha )\ne 0$ (such a root is said to be a "free" root of $f(x)$). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$) with coefficients in a field of positive characteristic $p>n$ (Sudbery's Conjecture). In this paper it...

We provide a deep investigation of the notions of - and ${}^{\infty }$-hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for - and ${}^{\infty }$-hypoellipticity are given

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

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...

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f\left(u\right)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f\left(u\right)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f\left(u\right)$ which is $C^1$ and satisfies $f\left(0\right)=0$, we show that the omega limit set $\omega \left(u\right)$ of every bounded positive solution is determined by a stationary...

Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by $T^{†}$ the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies $\Phi \left(A\right)\Phi \left(B\right)=\Phi \left(B\right)\Phi {\left(A\right)}^{†}$ for all A, B ∈ ℬ(H) with $AB=B{A}^{†}$ if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that $\Phi \left(A\right)=cUA{U}^{†}$ for all A ∈ ℬ(H).

A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character...

Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${\left({ϵ}_i\right)}_{i=1}^{\infty }\in {0,1}^{\mathbb{N}}$ a β-expansion for x if $x={\sum }_{i=1}^{\infty }{ϵ}_i{\beta }^{-i}$. We call a finite sequence ${\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $\Psi :\mathbb{N}\to {ℝ}_{\ge 0}$, we introduce the following subset of ℝ: $W_{\beta }\left(\Psi \right):={\bigcap }_{m=1}^{\infty }{\bigcup }_{n=m}^{\infty }{\bigcup }_{{\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n}[{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right),{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)+\Psi \left(n\right)]$In other words, $W_{\beta }\left(\Psi \right)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0\le x-{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)\le \Psi \left(n\right)$. When ${\sum }_{n=1}^{\infty }{2}^n\Psi \left(n\right)<\infty$, the Borel-Cantelli lemma tells us that the Lebesgue measure...

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...

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 $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

If $f(x)$ is a polynomial with coefficients in the field of complex numbers, of positive degree $n$, then $f(x)$ has at least one root a with the following property: if $\mu \le k\le n$, where $\mu$ is the multiplicity of $\alpha$, then $f^{(k)}(\alpha )\ne 0$ (such a root is said to be a "free" root of $f(x)$). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree $n$) with coefficients in a field of positive characteristic $p>n$ (Sudbery's Conjecture). In this paper it...

CONTENTSIntroduction................................................................................................................................... 5 Notations.......................................................................................................................... 5§ 1. Preliminaries........................................................................................................................ 6 1. Right invertible operators.....................................................................................................