Let lambda is an element of R and let H = -1/2 partial derivative(2)(x) + q delta(0) be the one-dimensional Schr & ouml;dinger equation with a repulsive delta potential. We study the Cauchy problem for the nonlinear equation {i partial derivative(t)u(t,x)=Hu(t,x)+lambda divided by u(t,x)divided by(2)u(t,x),(t,x) is an element of R & times; R, u(0,x) = u(0)(x), in L-p-based spaces. Using the boundedness of wave operators, a characterization of Besov space adapted to H, and the cancellation property of the trilinear form T(v(1)(tau),v(2)(tau),v(3)(tau))=U(-tau)(U(-tau)v(1)(tau)U(tau)v2(tau)U(tau)v3(tau)) with U(tau)=e-itH, we demonstrate that under the linear transformation v(t)=U(-t)u(t), the problem is locally well-posed in L-p(R) for 1
Since the 1980s, it has been known that the smallest non-finitely based semigroups are of order six. Surprisingly, for involution semigroups, a nonfinitely based example of order five was recently discovered. In this article, it is confirmed that every involution semigroup of order four is finitely based. Since every involution semigroup of order three or less is already known to be finitely based, it follows that the smallest non-finitely based involution semigroups are of order five.
Given a complex manifold containing a relatively compact Z(q) domain, we give sufficient geometric conditions on the domain so that its L2-cohomology in degree (p, q) (known to be finite-dimensional) vanishes. The condition consists in the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms.
We analyze bounds for the sums of eigenvalues of the Stokes operator restricted to a bounded domain Omega subset of R(d )with d >= 2. We improve upon existing lower bound estimates due to Ilyin as well as those by the second author and S. Y & imath;ld & imath;r & imath;m Yolcu, while preserving sharpness in the sense of Weyl asymptotics.
We study a strong openness property for singular Hermitian vector bundles (E, h) that are Griffiths-semipositive.
Spun normal surfaces are a useful way of representing proper essential surfaces using ideal triangulations for 3-manifolds with tori boundaries. Here we consider spinning surfaces in the case of a 3-manifold with a nontrivial JSJ decomposition, where each of the JSJ components is hyperbolic. We prove that a proper essential surface E can be spun, so long as none of the JSJ components are bundles with fiber a subsurface of E and the ideal triangulation satisfies similar properties to a taut structure.
We investigate the differentiable structure on compact simply connected submanifolds in Riemannian manifolds under curvature pinching conditions. We prove a sharp differentiable sphere theorem that an n-dimensional compact simply connected submanifold Mn (n >= 5, n =/ 7, 8) in the sphere S-N(1/root c) (c > 0) with the second fundamental form A and the mean curvature vector H satisfying |A|(2) <= 4c + |H|(2 )/n-2 is diffeomorphic to the standard sphere. The similar differentiable sphere theorem also holds for compact simply connected submanifolds in the space form F-N(c) with c <= 0.
We explicitly compute the Rankin-Selberg type integral introduced by Piatetski-Shapiro over adeles for vector-valued Siegel cusp forms of square-free levels F0(N). On the way, for particular test functions in the Bessel models of irreducible admissible representations, exact evaluations of the local zeta integrals are given.
We use the Saloff-Coste Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of positive solutions to the nonlinear elliptic equation Delta (p)u + au(q) = 0 defined on a complete Riemannian manifold (M, g) with Ricci lower bound, where p > 1 is a constant and Delta (p)u = div(|del u|(p-2)del u) is the usual p-Laplace operator. Under certain assumptions on a, p and q, we derive some gradient estimates and Liouville type theorems for positive solutions to the above equation. In particular, under certain assumptions on a, p and q we show whether or not the exact Cheng-Yau log-gradient estimates for the positive solutions to Delta (p)u + au(q) = 0 on (M, g) with Ricci lower bound hold true is equivalent to whether or not the positive solutions to this equation fulfill Harnack inequality, and hence some new Cheng-Yau log-gradient estimates are established.
We prove Howe duality for an exceptional theta correspondence.To that end, we relate the K -types of corresponding representations by exploiting a pair of see-saw identities.
The Braverman-Kazhdan program, later refined by Ng & ocirc;, aims to understand Langlands L-functions attached to a reductive group G and a representation p: LGGLy, of its L-group, plus certain additional desiderata. Such pairs (G, p) are called Braverman-Kazhdan-Ng & ocirc; (BKN) pairs. We explain in this paper how it is enough to consider BKN pairs (G, p), in order to understand general Langlands L-functions. A key tool in the approach of Braverman and Kazhdan is a certain reductive monoid attached to p. There are two methods of constructing such a reductive monoid in the literature. We prove that the two methods yield the same monoid when (G, p) is a BKN pair.
Given a smooths-dimensional submanifoldSofRm+cand a smooth distribu-tionD superset of T Sof rankmalongS, we study the following geometric Cauchyproblem: to find anm-dimensional rank-ssubmanifoldMofRm+c(that is,anm-submanifold with constant index of relative nullitym-s) such thatM superset of SandT M|S=D. In particular, under some reasonable assumptionand using a constructive approach, we show that a solution exists and isunique in a neighborhood of S.
We classify unimodal isolated complete intersection singularities in arbitrarycharacteristic under contact equivalence. The classification overChas beendone by A. Dimca and C. G. Gibson. We continue and generalize theirwork. To complete the classification, we generalized the complete transversalmethod into positive characteristic field, which is also useful in many otherclassification problem.
We provide a collection of natural axioms centered around the symmetric forcing theorem, which yield the concept of symmetric extensions, avoiding the technicalities involved in standard presentations.
We prove that the homological and Balmer spectra in tensor-triangular geometry are functorial in certain definable functors, thereby providing an alternative perspective on functoriality in tensor-triangular geometry from the viewpoint of purity, and generalising current results in the literature.
We prove that equality in a sharp lower bound for the first p-eigenvalue of the Hodge Laplacian on closed submanifolds in space forms can occur only on topological spheres, assuming positivity.
Suppose that (gamma) over right arrow (t): = (gamma(1)(t), ...,gamma(n)(t))=(a(1)t(d1), ...,a(n)t(dn)), 1 <= d(1) center dot center dot center dot < d(n) is an element of Z a(i) not equal 0, is a homogeneous polynomial curve. We prove that whenever p(1), ..., p(n) > 1/p = Sigma(n)(j=1) 1/p(j) <= 1, there exists an absolute constant 0 < C = C-p1,C-...,C-pn < infinity such that parallel to sup(r>0) 1/r integral(r)(0) Pi(n)(i=1) vertical bar f(i)(x - gamma(1)(t))vertical bar dt parallel to(Lp(R)) <= C center dot Pi(n)(i=1) parallel to f(j)parallel to(Lpj)(R). Our main tool is a smoothing estimate, adapted from work of Kosz, Mirek, Peluse, Wan, and Wright.
We prove a number of structure and isomorphism results concerning the noncommutative Natsume-Olsen spheres S2n-1 theta deformed along a skew-symmetric matrix B 2 R. These include (a) the fact that two C *-algebras of the form S3 theta (R) Mn are isomorphic precisely in the obvious cases; (b) the fact that m and n are recoverable from the isomorphism class of C(S2m-1 theta)(R) Mn; (c) the PI character, PI degree and Azumaya loci of C(S2m-1 theta) for rational B, along with a realization of their centers as (function algebras of) branched cover of S2n-1; and (d) for rational B again, the topological finite generation of C(S2m-1 theta) over their centers, with algebraic finite generation equivalent to being classical (equivalently, Azumaya).
