The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

Traces of Permuting n-Additive Maps and Permuting n-Derivations of Rings

Let R be a prime ring with center Z(R). For a fixed positive integer n, a permuting n-additive map ${\Delta : R^n \to R}$ is known to be permuting n-derivation if ${\Delta(x_1, x_2, \ldots, x_i x'_{i},\ldots, x_n) = \Delta(x_1, x_2, \ldots, x_i, \ldots, x_n)x'_i + x_i \Delta(x_1, x_2, \ldots, x'_i, \ldots, x_n)}$ holds for all ${x_i, x'_i \in R}$ . A mapping ${\delta : R \to R}$ defined by δ(x) = Δ(x, x, . . . ,x) for all ${x \in R}$ is said to be the trace of Δ. In the present paper, we have proved that a ring R is commutative if there exists a permuting n-additive map ${\Delta : R^n \to R}$ such that ${xy + \delta(xy) = yx + \delta(yx), xy- \delta(xy) = yx - \delta(yx), xy - yx = \delta(x) \pm \delta(y)}$ and ${xy + yx = \delta(x) \pm \delta(y)}$ holds for all ${x, y \in R}$ . Further, we have proved that if R is a prime ring with suitable torsion restriction then R is commutative if there exist non-zero permuting n-derivations Δ₁ and Δ₂ from ${R^n \to R}$ such that Δ₁(δ ₂(x), x, . . . ,x) =  0 for all ${x \in R,}$ where δ ₂ is the trace of Δ₂. Finally, it is shown that in a prime ring R of suitable torsion restriction, if ${\Delta_1, \Delta_2 : R^n \longrightarrow R}$ are non-zero permuting n-derivations with traces δ ₁, δ ₂, respectively, and ${B : R^n \longrightarrow R}$ is a permuting n-additive map with trace f such that δ ₁ δ ₂(x) =  f(x) holds for all ${x \in R}$ , then R is commutative.     

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.     

On the Linear Dependence of a Finite Set of Additive Functions

In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ .     

Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.     

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n?>?7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σ _i has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${{\hat \Sigma}_i \subset \mathbb{R}^n}$ , then $$ \int\limits_{\Sigma_i} H \ d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} \ d {\hat \sigma} $$ where H is the mean curvature of Σ _i in (Ω, g), ${\hat{H}}$ is the Euclidean mean curvature of ${{\hat \Sigma}_i}$ in ${\mathbb{R}^n}$ , and where d σ and ${d {\hat \sigma}}$ denote the respective volume forms. Moreover, equality holds for some boundary component Σ _i if, and only if, (Ω, g) is isometric to a domain in ${\mathbb{R}^n}$ . In the proof, we make use of a foliation of the exterior of the ${\hat \Sigma_i}$ ’s in ${\mathbb{R}^n}$ by the ${\frac{H}{R}}$ -flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).     

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 ${\alpha\in(0, 2]}$ .     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Noncommutative Multivariable Operator Theory

In this paper, we study noncommutative domains ${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$ generated by positive regular free holomorphic functions f and certain classes of n-tuples ${\varphi = (\varphi_1, \ldots, \varphi_n)}$ of formal power series in noncommutative indeterminates Z ₁, . . . , Z _n . Noncommutative Poisson transforms are employed to show that each abstract domain ${\mathbb{D}_f^\varphi}$ has a universal model consisting of multiplication operators (M _Z1, . . . , M _{Z n} ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M _Z1, . . . , M _{Z n} and show that all pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ are compressions of ${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$ to their coinvariant subspaces. We show that the eigenvectors of ${M_{Z_1}^*, \ldots, M_{Z_n}^*}$ are precisely the noncommutative Poisson kernels ${\Gamma_\lambda}$ associated with the elements ${\lambda}$ of the scalar domain ${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra ${H^\infty(\mathbb{D}_f^\varphi)}$ . We introduce the characteristic function of an n-tuple ${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$ , present a model for pure n-tuples of operators in the noncommutative domain ${\mathbb{D}_f^\varphi(\mathcal{H})}$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ .     

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 .     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G =  (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n =  2 ^p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C ₄]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C ₄], where G is a graph of order n and n is an arbitrary natural number.     

Conservative fragments of {{S}^{1}_{2}} and {{R}^{1}_{2}}

Conservative subtheories of ${{R}^{1}_{2}}$ and ${{S}^{1}_{2}}$ are presented. For ${{S}^{1}_{2}}$ , a slight tightening of Je?ábek??s result (Math Logic Q 52(6):613?C624, 2006) that ${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$ is presented: It is shown that ${T^{0}_{2}}$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this ${\forall\Sigma^{b}_{1}}$ -theory, we define a ${\forall\Sigma^{b}_{0}}$ -theory, ${T^{-1}_{2}}$ , for the ${\forall\Sigma^{b}_{0}}$ -consequences of ${S^{1}_{2}}$ . We show ${T^{-1}_{2}}$ is weak by showing it cannot ${\Sigma^{b}_{0}}$ -define division by 3. We then consider what would be the analogous ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ based on Pollett (Ann Pure Appl Logic 100:189?C245, 1999. It is shown that this theory, ${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$ , also cannot ${\Sigma^{b}_{0}}$ -define division by 3. On the other hand, we show that ${{S}^{0}_{2}+open_{\{||id||\}}}$ -COMP is a ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ . Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium?? 98, 262?C279, 2000) and show that ${\hat{C}^{0}_{2}}$ is ${\forall\hat\Sigma^{b}_{1}}$ -conservative over a theory based on open _cl-comprehension.     

Some new classes of Kadison-Singer lattices in Hilbert spaces

Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspace{(,):∈H}.Let L be the closed lattice in the strong operator topology generated by the projections(E 00 0),{(E 00 0):E∈N}and Q.We show that L is a Kadison-Singer lattice with trivial commutant,i.e.,L′=CI.Furthermore,we similarly construct some Kadison-Singer lattices in the matrix algebras M2n(C)and M2n.1(C).     

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

Randomly weighted sums of dependent subexponential random variables*

We consider the randomly weighted sums $ \sum\nolimits_{k = 1}^n {{\theta_k}{X_k},n \geqslant 1} $ , where $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ are n real-valued random variables with subexponential distributions, and $ \left\{ {{\theta_k},1 \leqslant k \leqslant n} \right\} $ are other n random variables independent of $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ and satisfying $ a \leqslant \theta \leqslant b $ for some $ 0 < a \leqslant b < \infty $ and all $ 1 \leqslant k \leqslant n $ . For $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ satisfying some dependent structures, we prove that $$ {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant m \leqslant n} \sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant k \leqslant n} {\theta_k}{X_k} > x} \right)\sim \sum\limits_{k = 1}^m {{\text{P}}\left( {{\theta_k}{X_k} > x} \right)} $$ as x??????.     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$