On the images of horodisks under holomorphic self-maps of the unit disk

Suppose that f is a holomorphic self map of the unit disk ${\mathbb{D}}$ . Recently several monotonicity results related to the image of smaller disks under f have been proved. These results extend the classical Schwarz lemma in various ways. We prove analogous monotonicity results in the context of Julia’s boundary Schwarz lemma. A horodisk is a disk internally tangent to the unit circle. For positive ${\lambda}$ , we denote by ${H_{\lambda}}$ the disk of radius ${\lambda/(1\,+\,\lambda)}$ centered at the point ${1/(1\,+\,\lambda)}$ . This is a horodisk that touches the unit circle at the point 1. Suppose that f(1) = 1 (in the sense of radial limit) and denote by ${f^{\prime}(1)}$ the angular derivative. By Julia’s lemma ${f(H_{\lambda})\,\subset H_{{\lambda}f^{\prime}(1)}}$ . Let ${\Psi_f(\lambda)\,=\,\inf\,\{\rho > 0 : f(H_{\lambda}) \subset H_\rho\}}$ . We show that the function ${\Psi_f(\lambda)/\lambda}$ is a decreasing function of ${\lambda}$ and that ${\lim_{\lambda\,\to\,0+} \Psi_f(\lambda)/\lambda = f^\prime(1)}$ . This result implies that the constant ${f^\prime(1)}$ in Julia’s lemma is the best possible.     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

Convergence of the eigenvalues of the p-Laplace operator as p goes to 1

Let $(\lambda ^k_p)_k$ be the usual sequence of min-max eigenvalues for the $p$ -Laplace operator with $p\in (1,\infty )$ and let $(\lambda ^k_1)_k$ be the corresponding sequence of eigenvalues of the 1-Laplace operator. For bounded $\Omega \subseteq \mathbb{R }^n$ with Lipschitz boundary the convergence $\lambda ^k_p\rightarrow \lambda ^k_1$ as $p\rightarrow 1$ is shown for all $k\in \mathbb{N }$ . The proof uses an approximation of $BV(\Omega )$ -functions by $C_0^\infty (\Omega )$ -functions in the sense of strict convergence on $\mathbb{R }^n$ .     

Ranks of backward shift invariant subspaces of the Hardy space over the bidisk

Let $\{\varphi _n(z)\}_{n\ge 0}$ be a sequence of inner functions satisfying that $\zeta _n(z):=\varphi _n(z)/\varphi _{n+1}(z)\in H^\infty (z)$ for every $n\ge 0$ and $\{\varphi _n(z)\}_{n\ge 0}$ has no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace $\mathcal{M }$ of $H^2(\mathbb{D }^2)$ . The ranks of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }_z$ and $\mathcal{F }^*_z$ respectively are determined, where $\mathcal{F }_z$ is the fringe operator on $\mathcal{M }\ominus w\mathcal{M }$ . Let $\mathcal{N }= H^2(\mathbb{D }^2)\ominus \mathcal{M }$ . It is also proved that the rank of $\mathcal{M }\ominus w\mathcal{M }$ for $\mathcal{F }^*_z$ equals to the rank of $\mathcal{N }$ for $T^*_z$ and $T^*_w$ .     

A contour line of the continuum Gaussian free field

Consider an instance $h$ of the Gaussian free field on a simply connected planar domain $D$ with boundary conditions $-\lambda $ on one boundary arc and $\lambda $ on the complementary arc, where $\lambda $ is the special constant $\sqrt{\pi /8}$ . We argue that even though $h$ is defined only as a random distribution, and not as a function, it has a well-defined zero level line $\gamma $ connecting the endpoints of these arcs, and the law of $\gamma $ is $\mathrm{SLE}(4)$ . We construct $\gamma $ in two ways: as the limit of the chordal zero contour lines of the projections of $h$ onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. We also show that, as a function of $h, \gamma $ is “local” (it does not change when $h$ is modified away from $\gamma $ ) and derive some general properties of local sets.     

A functional model associated with a generalized Nevanlinna pair

Let $ \mathcal{L} $ be a Hilbert space, and let $ \mathcal{H} $ be a Pontryagin space. For every self-adjoint linear relation $ \tilde{A} $ in $ \mathcal{H} \oplus \mathcal{L} $ , the pair $ \{ I + \lambda \psi (\lambda ),\,\psi (\lambda )\} $ where $ \psi (\lambda ) $ is the compressed resolvent of $ \tilde{A} $ , is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some self-adjoint linear relation $ \tilde{A} $ in the above sense. A functional model for this selfadjoint linear relation $ \tilde{A} $ is constructed.     

Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

SIP: critical value functions have finite modulus of non-convexity

We consider semi-infinite programming problems ${{\rm SIP}(z)}$ depending on a finite dimensional parameter ${z \in \mathbb{R}^p}$ . Provided that ${\bar{x}}$ is a strongly stable stationary point of ${{\rm SIP}(\bar{z})}$ , there exists a locally unique and continuous stationary point mapping ${z \mapsto x(z)}$ . This defines the local critical value function ${\varphi(z) := f(x(z); z)}$ , where ${x \mapsto f(x; z)}$ denotes the objective function of ${{\rm SIP}(z)}$ for a given parameter vector ${z\in \mathbb{R}^p}$ . We show that ${\varphi}$ is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of ${\varphi}$ .     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

On generalized Dhombres equations with nonconstant polynomial solutions in the complex plane

We characterize functional equations of the form ${f(zf(z))=f(z)^{k+1},z\in\mathbb {C}}$ , with ${k\in\mathbb N}$ , like those generalized Dhombres equations ${f(zf(z))=\varphi (f(z))}$ , ${z\in\mathbb C}$ , with given entire function ${\varphi}$ , which have a nonconstant polynomial solution f.     

Geodesics,distance, and the CAT(0) property for the manifold of Riemannian metrics

Let $\mathcal F ^a_\lambda $ be the PBW degeneration of the flag varieties of type $A_{n-1}$ . These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$ . We prove that $\mathcal F ^a_\lambda $ have rational singularities, are normal and locally complete intersections, and construct a desingularization $R_\lambda $ of $\mathcal F ^a_\lambda $ . The varieties $R_\lambda $ can be viewed as towers of successive $\mathbb{P }^1$ -fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties $R_\lambda $ are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for $\mathcal F ^a_\lambda $ . Using the Atiyah–Bott–Lefschetz formula for $R_\lambda $ , we compute the $q$ -characters of the highest weight $\mathfrak sl _n$ -modules.     

Harmonic mappings and conformal minimal immersions of Riemann surfaces into {\mathbb {R}^{\rm N}}

We prove that for any open Riemann surface ${\mathcal{N}}$ , natural number N ≥ 3, non-constant harmonic map ${h:\mathcal{N} \to \mathbb{R}}$ ^N?2 and holomorphic 2-form ${\mathfrak{H}}$ on ${\mathcal{N}}$ , there exists a weakly complete harmonic map ${X=(X_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ with Hopf differential ${\mathfrak{H}}$ and ${(X_j)_{j=3,\ldots,{\sc N}}=h.}$ In particular, there exists a complete conformal minimal immersion ${Y=(Y_j)_{j=1,\ldots,{\sc N}}:\mathcal{N} \to \mathbb{R}^{\sc N}}$ such that ${(Y_j)_{j=3,\ldots,{\sc N}}=h}$ . As some consequences of these results (1) there exist complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss map is non-degenerate and fails to intersect N hyperplanes of ${\mathbb{CP}^{{\sc N}-1}}$ in general position. (2) There exist complete non-proper embedded minimal surfaces in ${\mathbb{R}^{\sc N},}$ ${\forall\,{\sc N} >3 .}$      

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .