A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane
Jerry
L.
Bona;
S.
M.
Sun;
Bing-Yu
Zhang
427-490
Abstract: The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem \begin{displaymath}(0.1)\qquad\qquad\quad \left\{ \begin{array}{l} \eta _t+\eta ... ...uad \eta(0,t) =h(t),\end{array}\right. \qquad\qquad\qquad\quad \end{displaymath} studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data $\phi$ in the class $H^s(R^+)$ for $s>\frac34$ and boundary data $h$ in $H^{(1+s)/3}_{loc} (R^+)$, whereas global well-posedness is shown to hold for $\phi \in H^s (R^+) , h\in H^{\frac{7+3s}{12}}_{loc} (R^+)$ when $1\leq s\leq 3$, and for $\phi \in H^s(R^+) , h\in H^{(s+1)/3}_{loc} (R^+)$ when $s\geq 3$. In addition, it is shown that the correspondence that associates to initial data $\phi$and boundary data $h$ the unique solution $u$ of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.
Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries
B.
Canuto;
E.
Rosset;
S.
Vessella
491-535
Abstract: In this paper we obtain quantitative estimates of strong unique continuation for solutions to parabolic equations. We apply these results to prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain $\Omega$ in $\mathbb{R}^{n}$, from the knowledge of overdetermined boundary data for parabolic boundary value problems.
On the structure of spectra of periodic elliptic operators
Peter
Kuchment;
Sergei
Levendorskiî
537-569
Abstract: The paper discusses the problem of absolute continuity of spectra of periodic elliptic operators. A new result on absolute continuity for a matrix operator of Schrödinger type is obtained. It is shown that all types of operators for which the absolute continuity has previously been established can be reduced to this one. It is also discovered that some natural generalizations stumble upon an obstacle in the form of non-triviality of a certain analytic bundle on the two-dimensional torus.
A classification of hyperpolar and cohomogeneity one actions
Andreas
Kollross
571-612
Abstract: An isometric action of a compact Lie group on a Riemannian manifold is called hyperpolar if there exists a closed, connected submanifold that is flat in the induced metric and meets all orbits orthogonally. In this article, a classification of hyperpolar actions on the irreducible Riemannian symmetric spaces of compact type is given. Since on these symmetric spaces actions of cohomogeneity one are hyperpolar, i.e. normal geodesics are closed, we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions. This result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces.
Sufficient conditions for zero-one laws
Jason
P.
Bell
613-630
Abstract: We generalize a result of Bateman and Erdos concerning partitions, thereby answering a question of Compton. From this result it follows that if $\mathcal{K}$ is a class of finite relational structures that is closed under the formation of disjoint unions and the extraction of components, and if it has the property that the number of indecomposables of size $n$ is bounded above by a polynomial in $n$, then $\mathcal{K}$ has a monadic second order $0$-$1$ law. Moreover, we show that if a class of finite structures with the unique factorization property is closed under the formation of direct products and the extraction of indecomposable factors, and if it has the property that the number of indecomposables of size at most $n$ is bounded above by a polynomial in $\log n$, then this class has a first order $0$-$1$ law. These results cover all known natural examples of classes of structures that have been proved to have a logical $0$-$1$ law by Compton's method of analyzing generating functions.
Polynomials nonnegative on a grid and discrete optimization
Jean
B.
Lasserre
631-649
Abstract: We characterize the real-valued polynomials on $\mathbb R^n$that are nonnegative (not necessarily strictly positive) on a grid $\mathbb K$ of points of $\mathbb R^n$, in terms of a weighted sum of squares whose degree is bounded and known in advance. We also show that the mimimization of an arbitrary polynomial on $\mathbb K$ (a discrete optimization problem) reduces to a convex continuous optimization problem of fixed size. The case of concave polynomials is also investigated. The proof is based on a recent result of Curto and Fialkow on the $\mathbb K$-moment problem.
Skew Schubert functions and the Pieri formula for flag manifolds
Nantel
Bergeron;
Frank
Sottile
651-673
Abstract: We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric functions, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtainng a combinatorial chain construction of Schubert polynomials.
Tensor product varieties and crystals: $GL$ case
Anton
Malkin
675-704
Abstract: A geometric theory of tensor product for $\mathfrak{gl}_{N}$-crystals is described. In particular, the role of Spaltenstein varieties in the tensor product is explained, and thus a direct (non-combinatorial) proof of the fact that the number of irreducible components of a Spaltenstein variety is equal to a Littlewood-Richardson coefficient (i.e. certain tensor product multiplicity) is obtained.
Tenth order mock theta functions in Ramanujan's lost notebook (IV)
Youn-Seo
Choi
705-733
Abstract: Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.
The A-polynomial from the noncommutative viewpoint
Charles
Frohman;
Razvan
Gelca;
Walter
LoFaro
735-747
Abstract: The paper introduces a noncommutative generalization of the A-polynomial of a knot. This is done using the Kauffman bracket skein module of the knot complement, and is based on the relationship between skein modules and character varieties. The construction is possible because the Kauffman bracket skein algebra of the cylinder over the torus is a subalgebra of the noncommutative torus. The generalized version of the A-polynomial, called the noncommutative A-ideal, consists of a finitely generated ideal of polynomials in the quantum plane. Some properties of the noncommutative A-ideal and its relationships with the A-polynomial and the Jones polynomial are discussed. The paper concludes with the description of the examples of the unknot, and the right- and left-handed trefoil knots.
New bases for Triebel-Lizorkin and Besov spaces
G.
Kyriazis;
P.
Petrushev
749-776
Abstract: We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the $L_p$, $H_p$, potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a very general way. In particular, if $\Phi$ is any sufficiently smooth and rapidly decaying function, then our method constructs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function $\Phi$. Typical examples of such $\Phi$'s are the rational function $\Phi (\cdot) = (1 + \vert\cdot\vert^2)^{-N}$ and the Gaussian function $\Phi (\cdot) = e^{-\vert\cdot\vert^2}.$ This paper also shows how the new bases can be utilized in nonlinear approximation.
Symmetric approximation of frames and bases in Hilbert spaces
Michael
Frank;
Vern
I.
Paulsen;
Terry
R.
Tiballi
777-793
Abstract: We introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. We prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames. Even in the case of the symmetric orthogonalization of bases, our techniques and results are new. A crucial role is played by whether or not a certain operator related to the initial frame or basis is Hilbert-Schmidt.
Transverse surfaces and attractors for 3-flows
W.
J.
Colmenarez;
C.
A.
Morales
795-806
Abstract: We prove that a hyperbolic strange attractor of a three-dimensional vector field is a suspension if it exhibits a transverse surface over which the unstable manifold induces a lamination without closed leaves. We also study the topological equivalence of singular attractors exhibiting transverse surfaces for three-dimensional vector fields.
Induced operators on symmetry classes of tensors
Chi-Kwong
Li;
Alexandru
Zaharia
807-836
Abstract: Let $V$ be an $n$-dimensional Hilbert space. Suppose $H$ is a subgroup of the symmetric group of degree $m$, and $\chi: H \rightarrow \mathbb C$ is a character of degree 1 on $H$. Consider the symmetrizer on the tensor space $\bigotimes^m V$ \begin{displaymath}S(v_1\otimes \cdots \otimes v_m) = {1\over \vert H\vert}\sum... ... v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(m)} \end{displaymath} defined by $H$ and $\chi$. The vector space \begin{displaymath}V_\chi^m(H) = S(\bigotimes^m V) \end{displaymath} is a subspace of $\bigotimes^m V$, called the symmetry class of tensors over $V$ associated with $H$ and $\chi$. The elements in $V_\chi^m(H)$ of the form $S(v_1\otimes \cdots \otimes v_m)$ are called decomposable tensors and are denoted by $v_1*\cdots * v_m$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $K(T)$ acting on $V_\chi^m(H)$ satisfying \begin{displaymath}K(T) v_1* \dots *v_m = Tv_1* \cdots * Tv_m. \end{displaymath} In this paper, several basic problems on induced operators are studied.
On inversion of the Bessel and Gelfand transforms
Masaaki
Furusawa;
Joseph
A.
Shalika
837-852
Abstract: We construct the Plancherel measure corresponding to the Bessel model on the split special orthogonal group of $\text{odd degree}\ge 5$ and the Whittaker model on a connected split reductive group in general. As an application we prove the inversion formula which expresses the related integral transform in terms of the Satake transform.