On the entire self-shrinking solutions to Lagrangian mean curvature flow

The authors prove that the logarithmic Monge?CAmpère flow with uniformly bound and convex initial data satisfies uniform decay estimates away from time t?=?0. Then applying the decay estimates, we conclude that every entire classical strictly convex solution of the equation $$ \det D^{2}u=\exp\left\{n\left(-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} \right)\right\}, $$ should be a quadratic polynomial if the inferior limit of the smallest eigenvalue of the function |x|² D ² u at infinity has an uniform positive lower bound larger than 2(1 ? 1/n). Using a similar method, we can prove that every classical convex or concave solution of the equation $$ \sum_{i=1}^{n} \arctan\lambda_{i}=-u+\frac{1}{2} \sum_{i=1}^{n}x_{i} \frac{\partial u}{\partial x_{i}} $$ must be a quadratic polynomial, where ?? _i are the eigenvalues of the Hessian D ² u.     

Exponential laws for ultrametric partially differentiable functions and applications

We establish exponential laws for certain spaces of differentiable functions over a valued field $\mathbb{K}$ . For example, we show that $$C^{(\alpha ,\beta )} \left( {U \times V,E} \right) \cong C^\alpha \left( {U,C^\beta \left( {V,E} \right)} \right)$$ if α ∈ (?₀ ∪ {∞}) ⁿ , β ∈ (?₀ ∪ {∞}) ^m , $U \subseteq \mathbb{K}^n$ and $V \subseteq \mathbb{K}^m$ are open (or suitable more general) subsets, and E is a topological vector space. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by Enno Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C ^r -functions on (? _p ) ⁿ in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.     

Reduced Load Equivalence under Subexponentiality

The stationary workload W _A+B ^φ of a queue with capacity φ loaded by two independent processes A and B is investigated. When the probability of load deviation in process A decays slower than both in B and $e^{ - \sqrt x } $ , we show that W _A+B ^φ is asymptotically equal to the reduced load queue W _A ^φ?b , where b is the mean rate of B. Given that this property does not hold when both processes have lighter than $e^{ - \sqrt x } $ deviation decay rates, our result establishes the criticality of $e^{ - \sqrt x } $ in the functional behavior of the workload distribution. Furthermore, using the same methodology, we show that under an equivalent set of conditions the results on sampling at subexponential times hold.     

Cycle Lemma, Parking Functions and Related Multigraphs

For positive integers a and b, an ${(a, \overline{b})}$ -parking function of length n is a sequence (p ₁, . . . , p _n ) of nonnegative integers whose weakly increasing order q ₁ ≤ . . . ≤ q _n satisfies the condition q _i  < a + (i ? 1)b. In this paper, we give a new proof of the enumeration formula for ${(a, \overline{b})}$ -parking functions by using of the cycle lemma for words, which leads to some enumerative results for the ${(a, \overline{b})}$ -parking functions with some restrictions such as symmetric property and periodic property. Based on a bijection between ${(a, \overline{b})}$ -parking functions and rooted forests, we enumerate combinatorially the ${(a, \overline{b})}$ -parking functions with identical initial terms and symmetric ${(a, \overline{b})}$ -parking functions with respect to the middle term. Moreover, we derive the critical group of a multigraph that is closely related to ${(a, \overline{b})}$ -parking functions.     

Global heat kernel estimates for fractional Laplacians in unbounded open sets

In this paper, we derive global sharp heat kernel estimates for symmetric ??-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C ^1,1 open sets in ${\mathbb R^d}$ : half-space-like open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for p _D (t, x, y) for all t?>?0 and ${x, y\,{\in}\,D}$ . Our approach is based on the idea that for x and y in D far from the boundary and t sufficiently large, we can compare p _D (t, x, y) to the heat kernel in a well understood open set: either a half-space or ${\mathbb R^d}$ ; while for the general case we can reduce them to the above case by pushing x and y inside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets.     

On the reducibility of some composite polynomials over finite fields

Let g(x)?=?x ⁿ ?+?a _n-1 x ^n-1?+?. . .?+?a ₀ be an irreducible polynomial over ${\mathbb{F}_q}$ . Varshamov proved that for a?=?1 the composite polynomial g(x ^p ?ax?b) is irreducible over ${\mathbb{F}_q}$ if and only if ${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})\neq 0}$ . In this paper, we explicitly determine the factorization of the composite polynomial for the case a?=?1 and ${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})= 0}$ and for the case a?≠ 0, 1. A recursive construction of irreducible polynomials basing on this composition and a construction with the form ${g(x^{r^kp}-x^{r^k})}$ are also presented. Moreover, Cohen’s method of composing irreducible polynomials and linear fractions are considered, and we show a large number of irreducible polynomials can be obtained from a given irreducible polynomial of degree n provided that gcd(n, q ³ ? q)?=?1.     

Differential polynomials with dilations in the argument and normal families

We show that a family ${\mathcal{F}}$ of analytic functions in the unit disk ${\mathbb{D}}$ which satisfy a condition of the form $$ f^n(z)+P[f](xz)+b\ne 0 $$ for all ${f\in\mathcal{F}}$ and all ${z\in\mathbb{D}}$ (where n ?? 3, 0?<?|x| ?? 1, b ?? 0 and P is an arbitrary differential polynomial of degree at most n ? 2 with constant coefficients and without terms of degree 0) is normal at the origin. Under certain additional assumptions on P the same holds also for b?=?0. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman?CPang rescaling method. Furthermore we prove some corresponding results of Picard type for functions meromorphic in the plane.     

Ein Axiomatischer Aufbau der Ähnlichkeitsgeometrie

Let Φ?F fields. With respect to the complex number plane we call the elements of Fpoints, the subsets Φm^+b, m≠O,b?F, lines and the bijektions z?F→zm⁺b?F (direct)similitudes. Two noncollinear point-triplets (a₁,a₂,a₃) and (b₁,b₂,b₃) are said to besimilar triangles if there exists a similitude, mapping a₁ onto b₁ for i=1,2,3. Therefore, similarity is an equivalence. relation on the set of all triangles. In this paper, we characterize these geometries axiomatically, starting from incidence structures with an abstract equivalence relation — called similarity — on the set of all triangles by imposing successively similarity-axioms for triangles.     

On Estimating the Asymptotic Variance of Stationary Point Processes

We investigate a class of kernel estimators $\widehat{\sigma}^2_n$ of the asymptotic variance σ ² of a d-dimensional stationary point process $\Psi = \sum_{i\ge 1}\delta_{X_i}$ which can be observed in a cubic sampling window $W_n = [-n,n]^d\,$ . σ ² is defined by the asymptotic relation $Var(\Psi(W_n)) \sim \sigma^2 \,(2n)^d$ (as n →? ∞) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{red}(\cdot)$ has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of $\gamma^{(2)}_{red}(\cdot)$ outside of an expanding ball centered at the origin, we determine optimal bandwidths b _n (up to a constant) minimizing the mean squared error of $\widehat{\sigma}^2_n$ . The case when $\gamma^{(2)}_{red}(\cdot)$ has bounded support is of particular interest. Further we suggest an isotropised estimator $\widetilde{\sigma}^2_n$ suitable for motion-invariant point processes and compare its properties with $\widehat{\sigma}^2_n$ . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of $\widehat{\sigma}^2_n$ for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b _n .     

The □<Subscript><Emphasis Type="Italic">b</Emphasis></Subscript>-Heat Equation on Quadric Manifolds

In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □ _b -heat equation on quadric submanifolds M?? ⁿ ×? ^m . As a consequence, we can also compute the heat kernel associated with the weighted \(\overline{\partial}\)-equation in ? ⁿ when the weight is given by exp?(?φ(z,z)?λ) where φ:? ⁿ ×? ⁿ →? ^m is a quadratic, sesquilinear form and λ∈? ^m . Our method involves the representation theory of the Lie group M and the group Fourier transform.     

L 2 existence theorems for the\bar \partial _b - Neumann problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL ² existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN _b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL ²(ω) coefficients. The proof depends on theL ^p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN _b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN _b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN _b is not a compact operator onL ²(ω).     

A new class of bent and hyper-bent Boolean functions in polynomial forms

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over ${\mathbb{F}_{2^{n}}}$ (n = 2m) having the form ${f(x) = tr_{o(s_1)} (a x^ {s_1}) + tr_{o(s_2)} (b x^{s_2})}$ where o(s _i ) denotes the cardinality of the cyclotomic class of 2 modulo 2 ⁿ ? 1 which contains s _i and whose coefficients a and b are, respectively in ${F_{2^{o(s_1)}}}$ and ${F_{2^{o(s_2)}}}$ . Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case. We prove that the exponents s ₁ = 2 ^m ? 1 and ${s_2={\frac {2^n-1}3}}$ , where ${a\in\mathbb{F}_{2^{n}}}$ (a ?? 0) and ${b\in\mathbb{F}_{4}}$ provide a construction of bent functions over ${\mathbb{F}_{2^{n}}}$ with optimum algebraic degree. For m odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums. We generalize the result for functions whose exponent s ₁ is of the form r(2 ^m ? 1) where r is co-prime with 2 ^m  + 1. The corresponding bent functions are also hyper-bent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

Analytic Extension of Smooth Functions

Let F be a closed proper subset of ?ⁿ and let ?* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ? ∈ ?* (?ⁿ) there is ${\widetilde f} \in {\cal E}_{*}(\rm R ^{n})$ which is real analytic on ?ⁿF and such that ?^a ? ¦ _F = ?^a ? ¦ _F for any a ∈ ?₀ ⁿ. For bounded ultradifferentiable functions ? we can obtain ${\widetilde f}$ by means of a continuous linear operator.     

A new “dinv” arising from the two part case of the shuffle conjecture

For a symmetric function F, the eigen-operator Δ _F acts on the modified Macdonald basis of the ring of symmetric functions by $\Delta_{F} \tilde{H}_{\mu}= F[B_{\mu}] \tilde{H}_{\mu}$ . In a recent paper (Int. Math. Res. Not. 11:525–560, 2004), J. Haglund showed that the expression $\langle\Delta_{h_{J}} E_{n,k}, e_{n}\rangle$ q,t-enumerates the parking functions whose diagonal word is in the shuffle 12?J∪∪J+1?J+n with k of the cars J+1,…,J+n in the main diagonal including car J+n in the cell (1,1) by t ^area q ^dinv. In view of some recent conjectures of Haglund–Morse–Zabrocki (Can. J. Math., doi:10.4153/CJM-2011-078-4, 2011), it is natural to conjecture that replacing E _n,k by the modified Hall–Littlewood functions $\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots\mathbf{C}_{p_{k}} 1$ would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting the parking function touches the diagonal according to the composition p=(p ₁,p ₂,…,p _k ). We prove this conjecture by deriving a recursion for the polynomial $\langle\Delta_{h_{J}} \mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}} 1 , e_{n}\rangle $ , using this recursion to construct a new $\operatorname{dinv}$ statistic (which we denote $\operatorname{ndinv}$ ), then showing that this polynomial enumerates the latter parking functions by $t^{\operatorname{area}} q^{\operatorname{ndinv}}$ .     

An analog of the weyl decomposition of the spaceL q (ω, ℝ n ) for a first-order differential operatorfor a first-order differential operator

We prove the following theorem: Suppose the function f(x) belongs toL _q (ω, ? ⁿ ), ω ? ? ^m , q∈(1, ∞), and satisfies the inequality $$|\int\limits_\omega {(f(x),{\mathbf{ }}v(x)){\mathbf{ }}dx| \leqslant \mu ||} v||'_q ,{\mathbf{ }}\tfrac{1}{q} + \tfrac{1}{{q'}} = 1,$$ for all n-dimensional vector-valued functions in the kernel of a scalar-valued first-order differential operator £ for which the second-order operatorLL ^* is elliptic. Then there exists a function p(x)∈W _q ¹ (ω) such that $$||f(x) - \mathfrak{L}^* p(x)||q \leqslant C_q \mu .$$ Bibliography: 6 titles.     

On the generation of certain bundles over P n

In this paper we prove the following Theorem. Over an algebraically closed of characteristic ≠2 for anyn≥2 there exists an integera _n such that fora≥a _n andb≥≥max{(n ²+6n+11)a/3(n+3)+2/(n+3), a+3} the kernel of the generic morphism fromb O _P ⁿ (2) intoa O _P ⁿ (3) is a locally free sheaf generated by its global sections. The extends to bundles on P ⁿ an analogous result on P ³ (see [D,2.3]). The arguments are applications of basic ideas of the so-called «voie ouest» (see [EH]).