An explicit growth condition for middle-degree cuspidal cohomology of arithmetically defined quaternionic hyperbolic n-manifolds

Let ${G/\mathbb Q}$ be the simple algebraic group Sp(n, 1) and ${\Gamma=\Gamma(N)}$ a principal congruence subgroup of level N ≥ 3. Denote by K a maximal compact subgroup of the real Lie group ${G(\mathbb R)}$ . Then a double quotient ${\Gamma\backslash G(\mathbb R)/K}$ is called an arithmetically defined, quaternionic hyperbolic n-manifold. In this paper we give an explicit growth condition for the dimension of cuspidal cohomology ${H^{2n}_{cusp}(\Gamma\backslash G(\mathbb R)/K,E)}$ in terms of the underlying arithmetic structure of G and certain values of zeta-functions. These results rely on the work of Arakawa (Automorphic Forms of Several Variables: Taniguchi Symposium, Katata, 1983, eds. I. Satake and Y. Morita (Birkhäuser, Boston), pp. 1–48, 1984).     

Codimension reduction for real submanifolds of quaternionic hyperbolic spaces

We prove a reduction theorem of codimension for real submanifolds of quaternionic hyperbolic spaces as a quaternionic analogue corresponding to those in Cecil [2], Erbacher [4], Kawamoto [8], Kwon and the second author [10] and Okumura [13].     

A nonvanishing theorem for the cuspidal cohomology of SL2 over imaginary quadratic integers

Mathematische Annalen -     

n-Dimensional stable and unstable manifolds of hyperbolic singular point

Invariant manifold play an important role in the qualitative analysis of dynamical systems, such as in studying homoclinic orbit and heteroclinic orbit. This paper focuses on stable and unstable manifolds of hyperbolic singular points. For a type of n-dimensional quadratic system, such as Lorenz system, Chen system, Rossler system if n = 3, we provide the series expression of manifolds near the hyperbolic singular point, and proved its convergence using the proof of the formal power series. The expressions can be used to investigate the heteroclinic orbits and homoclinic orbits of hyperbolic singular points.     

An n log n Algorithm for Online BDD Refinement

Binary decision diagrams are in widespread use in verification systems for the canonical representation of finite functions. Here we consider multivalued BDDs, which represent functions of the form : ^ν →  , where is a finite set of leaves. We study a rather natural online BDD refinement problem: a partition of the leaves of several shared BDDs is gradually refined, and the equivalence of the BDDs under the current partition must be maintained in a discriminator table. We show that it can be solved in O(n log n) time if n bounds both the size of the BDDs and the total size of update operations. Our algorithm is based on an understanding of BDDs as the fixed points of an operator that in each step splits and gathers nodes. We apply our algorithm to show that automata BDD-represented transition functions can be minimized in time O(n · log n), where n is the total number of BDD nodes representing the automaton. This result is not an instance of Hopcroft's classical minimization algorithm, which breaks down for BDD-represented automata because of the BDD path compression property.     

Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods

Discretization of autonomous ordinary differential equationsby numerical methods might, for certain step sizes, generatesolution sequences not corresponding to the underlying flow—so-called‘spurious solutions’ or ‘ghost solutions’.In this paper we explain this phenomenon for the case of explicitRunge-Kutta methods by application of bifurcation theory fordiscrete dynamical systems. An important tool in our analysisis the domain of absolute stability, resulting from the applicationof the method to a linear test problem. We show that hyperbolicfixed points of the (nonlinear) differential equation are inheritedby the difference scheme induced by the numerical method whilethe stability type of these inherited genuine fixed points iscompletely determined by the method's domain of absolute stability.We prove that, for small step sizes, the inherited fixed pointsexhibit the correct stability type, and we compute the correspondinglimit step size. Moreover, we show in which way the bifurcationsoccurring at the limit step size are connected to the valuesof the stability function on the boundary of the domain of absolutestability, where we pay special attention to bifurcations leadingto spurious solutions. In order to explain a certain kind ofspurious fixed points which are not connected to the set ofgenuine fixed points, we interprete the domain of absolute stabilityas a Mandeibrot set and generalize this approach to nonlinearproblems.     

Computations of cuspidal cohomology of congruence subgroups of SL(3, Z)

Algorithms are presented which find a basis of the vector space of cuspidal cohomology of certain congruence subgroups of SL(3, $\text{Z}$) and which determine the action of the Hecke operators on this space. These algorithms were implemented on a computer. Four pairs of cuspidal classes were found with prime level less than 100. Tables are given of the eigenvalues of the first few Hecke operators on these classes.     

Hochschild cohomology for periodic algebras of polynomial growth

We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra.     

An asymptotic minimax theorem of order n

The asymptotic minimax theorem of LeCam and Hájek is refined by inclusion of terms of order n^−1/2. This renders more precise informations about the local properties of superefficient estimator-sequences.     

An example of a finitely defined solvable group without the maximality condition for normal subgroups

There is a finitely defined solvable group which does not satisfy the maximality condition for normal subgroups. This theorem gives a negative answer to one of the questions raised by P. Hall.Translated from Matematicheskie Zametki, Vol.l2,No. 3, pp. 287–293, September, 1972.     

A bound for the condition of a hyperbolic eigenvector matrix

The hyperbolic eigenvector matrix is a matrix X which simultaneously diagonalizes the pair (H,J), where H is Hermitian positive definite and J = diag(±1) such that X*HX = Δ and X*JX = J. We prove that the spectral condition of X, κ(X), is bounded byK(X)√minK(D^*HD), where the minimum is taken over all non-singular matrices D which commute with J. This bound is attainable and it can be simply computed. Similar results hold for other signature matrices J, like in the discretized Klein—Gordon equation.     

An n-dimensional Borg–Levinson theorem for singular potentials

We prove in this paper that the boundary spectral data, i.e. the Dirichlet eigenvalues and normal derivatives of the eigenfunctions at the boundary uniquely determines a potential in L^p on bounded domains. This result generalizes the result of Nachman, Sylvester and Uhlmann to unbounded potentials. This result can be viewed as a generalization of the classical one-dimensional Borg–Levinson theorem.     

An extension of Luque’s growth condition

This work gives an extension of Luque’s growth condition ensuring a locally linear rate of the proximal point algorithm for a maximal monotone operator.     

On the stability of an explicit difference scheme for hyperbolic equations with nonlocal boundary conditions

We consider the stability of an explicit finite-difference scheme for a linear hyperbolic equation with nonlocal integral boundary conditions. By studying the spectrum of the transition matrix of the explicit three-layer difference scheme, we obtain a sufficient condition for stability in a special norm.     

An explicit/implicit Galerkin domain decomposition procedure for parabolic integro-differential equations

In this paper, a two-dimensional fractional advection-dispersion equation (2D-FADE) with variable coefficients on a finite domain is considered. We use a new technique of combination of the Alternating Directions Implicit-Euler method (ADI-Euler), the unshifted Grünwald formula for the advection term, the right-shifted Grünwald formula for the diffusion term, and a Richardson extrapolation to establish an unconditionally stable second order accurate difference method. Stability, consistency and convergence of the ADI-Euler method for 2D-FADE are examined. A numerical example with known exact solution is also presented, and the behavior of the error is analyzed to verify the order of convergence of the ADI-Euler method and the extrapolated ADI-Euler method.     

An effective algorithm for the cohomology ring of symplectic reductions

Let G be a compact torus acting on a compact symplectic manifold M in a Hamiltonian fashion, and T a subtorus of G. We prove that the kernel of is generated by a small number of classes satisfying very explicit restriction properties. Our main tool is the equivariant Kirwan map, a natural map from the G-equivariant cohomology of M to the G/T-equivariant cohomology of the symplectic reduction of M by T . We show this map is surjective. This is an equivariant version of the well-known result that the (nonequivariant) Kirwan map is surjective. We also compute the kernel of the equivariant Kirwan map, generalizing the result due to Tolman and Weitsman [TW] in the case T = G and allowing us to apply their methods inductively. This result is new even in the case that dim T = 1. We close with a worked example: the cohomology ring of the product of two , quotiented by the diagonal 2-torus action. Submitted: September 2001, Revised: December 2001, Revised: February 2002.     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

An efficient explicit/implicit domain decomposition method for convection‐diffusion equations

A nonoverlapping domain decomposition method for some time‐dependent convection‐diffusion equations is presented. It combines predictor‐corrector technique, modified upwind differences with explicit/implicit coupling to provide intrinsic parallelism, and unconditional stability while improving the accuracy. Both rigorous mathematical analysis and numerical experiments are carried out to illustrate the stability, accuracy, and parallelism. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010