Equivariant closure operators and trisp closure maps

A trisp closure map ? is a special map on the vertices of a trisp (triangulated space) T with the property that T collapses onto the subtrisp induced by the image of ?. We study the interaction between trisp closure maps and group operations on the trisp, and give conditions such that the quotient map is again a trisp closure map. Special attention is on the case that the trisp is the nerve of an acyclic category, and the relationship between trisp closure maps and closure operators on posets is studied.     

Collapsing unicolored graphs

A graph is said to be unicolored if it is colored by nonnegative integers so that adjacent points have colors that differ in absolute value by one. A unicolored graph is collapsible if it has a 1-factor that does not contain a 1-factor of any bicolored cycle. We show that a regular CW complex K cell collapses to a subcomplex O if and only if its relative unicolored incidence graph collapses. We consider the 1-factors and the bicolored cycles of unicolored incidence graphs and their relationship to the relative homology and homotopy properties of the pair of cell complexes.     

Restrictions on symplectic fibrations

Moduli spaces of stable pseudoholomorphic curves can be defined parametrically, i.e., over total spaces of symplectic fibrations. This imposes several restrictions on the spectral sequence of a symplectic fibration. We prove, among others, that under certain assumptions the spectral sequence collapses at E₂. In the appendix, we prove nontriviality of certain Gromov-Witten invariant for blow-ups. As an application we obtain that any Hamiltonian fibration with the blow-up of  along four dimensional submanifold as a fibre c-splits. That is its spectral sequence collapses.     

ON THE STRUCTURE OF COHOMOLOGY RINGS OF p-NILPOTENT LIE ALGEBRAS

In this paper the authors investigate the structure of the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose E ₂-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen–Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as a ring with the E ₂-term. We present criteria for the collapsing of this spectral sequence and provide some examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.     

Ricci Flow on a Class of Noncompact Warped Product Manifolds

We consider the Ricci flow on noncompact \(n+1\)-dimensional manifolds M with symmetries, corresponding to warped product manifolds \(\mathbb {R}\times T^n\) with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate \(|{{\mathrm{Rm}}}|(p,t) \le C/t\) for some \(C > 0\) and all \(p \in M, t \in (1,\infty )\) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as \(t \rightarrow \infty \)) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the (n-dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.     

Guarantees for the success frequency of an algorithm for finding Dodgson-election winners

In the year 1876 the mathematician Charles Dodgson, who wrote fiction under the now more famous name of Lewis Carroll, devised a beautiful voting system that has long fascinated political scientists. However, determining the winner of a Dodgson election is known to be complete for the Θ ₂ ^p level of the polynomial hierarchy. This implies that unless P=NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates—although the number of voters may still be polynomial in the number of candidates—a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner.     

Edge scaling limits for a family of non-Hermitian random matrix ensembles

A family of random matrix ensembles interpolating between the Ginibre ensemble of n × n matrices with iid centered complex Gaussian entries and the Gaussian unitary ensemble (GUE) is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n ^?1/3. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy–Widom distributions.     

A kernel for automorphic L-functions

The full multiple Dirichlet series of an automorphic cusp form is defined, in classical language, as a Dirichlet series of several complex variables over all the Fourier coefficients of the cusp form. It is different from the L-function of Godement and Jacquet, which is defined as a Dirichlet series in one complex variable over a one-dimensional array of the Fourier coefficients. In GL(2) and GL(3), the two notions are simply related. In this paper, we construct a kernel function that gives the full multiple Dirichlet series of automorphic cusp forms on GL(n,R). The kernel function is a new Poincaré series. Specifically, the inner product of a cusp form with this Poincaré series is the product of the full multiple Dirichlet series of the form times a function that is essentially the Mellin transform of Jacquet's Whittaker function. In the proof, the full multiple Dirichlet series is produced by applying the Lipschitz summation formula several times and by an integral which collapses the sum over SL(n−1,Z) in the Fourier expansion of the cusp form.     

Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes

In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes.To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lovász complex of a graph. Then, for an arbitrary lattice L we describe a formal deformation of the barycentric subdivision of the atom crosscut complex Γ(L) to its order complex . We proceed by proving that the complex of sets bounded from below J(L) can also be collapsed to .Finally, as a pinnacle of our project, we apply all these results to certain graph complexes. Namely, by describing an explicit formal deformation, we prove that, for any graph G, the neighborhood complex N(G) and the polyhedral complex Hom(K₂,G) have the same simple homotopy type in the sense of Whitehead.     

Asymptotic behavior for nonlocal dispersal equations

This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λ≤λ₁(Ω), the solution decays to zero as t→∞; while for λ>λ₁(Ω), the solution converges to the unique positive stationary solution as t→∞. In addition, we show that the solution blows up under some conditions.     

Forcing notions in inner models

There is a partial order \({\mathbb{P}}\) preserving stationary subsets of ω ₁ and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω ₁ over V also collapses ω ₁ over \({V^{\mathbb{P}}}\) . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V[c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B.     

On the structure of solutions to a class of quasilinear elliptic Neumann problems

We study the structure of positive solutions to the equation ?^mΔ_mu-u^m-1+f(u)=0 with homogeneous Neumann boundary condition. First, we show the existence of a mountain-pass solution and find that as ?→0⁺ the mountain-pass solution develops into a spike-layer solution. Second, we prove that there is an uniform upper bound independent of ? for any positive solution to our problem. We also present a Harnack-type inequality for the positive solutions. Finally, we show that if 1<m?2 holds and ? is sufficiently large, any positive solution must be a constant.     

Fully developed laminar slip and no-slip flow in rough microtubes

The effect of surface roughness on developed laminar flow in microtubes is investigated. The tube boundary is defined by \({r=R\left[{1+\varepsilon\, {\rm sin}\left( {\lambda \theta }\right)}\right]}\), with R representing the reference radius and \({\varepsilon}\) and λ the roughness parameters. The momentum equation is solved using Fourier–Galerkin–Tau method with slip at the boundary. A novel semi-analytical method is developed to predict friction factor and pressure drop in corrugated rough microtubes for continuum flow and slip flow that are not restricted to small values of \({\varepsilon \lambda }\) . The analytical solution collapses onto the perturbation solution ofDuan and Muzychka (J. Fluids Eng., 130:031102, 2008) for small enough values of \({\varepsilon \lambda }\) .     

Existence results for a class of second order evolution inclusions and its corresponding first order evolution inclusions

We deal with an abstract second order nonlinear evolution inclusion with its principal part having a small parameter ?. We prove the existence of a weak solution when the nonlinearity F is convex as well as nonconvex valued. Then we study the asymptotic behavior of a sequence of solutions {u _? } when ? → 0. We prove that there exists a limit function u, and u is a solution of the corresponding first order evolution inclusion.     

Gain of regularity for the KP-I equation

In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ? possesses certain regularity and sufficient decay as x→∞, then the solution u(t) will be smoother than ? for 0<t?T where T is the existence time of the solution.     

A generalized Newton algorithm using higher-order derivatives

This paper presents a general optimization algorithm using first-order torth order derivatives to find the optimum of anr-continuously differentiable function of many variables. This algorithm collapses to the Newton-Raphson algorithm when only first- and second-order derivatives are used. The computation of the required higher-order derivatives are readily available through thetable algorithm. The generalized CES production function is used as an example.     

On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

Semiclassical measure for the solution of the dissipative Helmholtz equation

We study the semiclassical measure for the solution of the high-frequency Helmholtz equation in Rn with non-constant absorption index and a source term concentrated on a bounded submanifold of Rn. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption index is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagrangian distributions.     

The generation,propagation, and extinction of multiphases in the KdV zero‐dispersion limit

We study the multiphases in the KdV zero‐dispersion limit. These phases are governed by the Whitham equations, which are 2g + 1 quasi‐linear hyperbolic equations where g is the number of phases. We are interested in both the interaction of two single phases and the breaking of a single phase for general initial data. We analyze in detail how a double phase is generated from the interaction or breaking, how it propagates in space‐time, and how it collapses to a single phase in a finite time. The Whitham equations are known to be integrable via a hodograph transform. The crucial step in our approach is to formulate the hodograph transform in terms of the Euler‐Poisson‐Darboux solutions. Under our scheme, the zeros of the Jacobian of the transform are given by the zeros of the Euler‐Poisson‐Darboux solution. Hence, the problem of inverting the hodograph transform to give the Whitham solution reduces to that of counting the zeros of the Euler‐Poisson‐Darboux solution. © 2002 Wiley Periodicals, Inc.     

Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

In this work, we study the problem of mean-variance hedging with a random horizon T∧τ, where T is a deterministic constant and τ is a jump time of the underlying asset price process. We first formulate this problem as a stochastic control problem and relate it to a system of BSDEs with a jump. We then provide a verification theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from filtration enlargement theory.