Abstract: We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of Sukochev and Zanin (J Oper Theory, 2019) and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry. We provide a variant of Connes’ trace theorem which applies to certain noncommutative settings, with a minimum of technical preliminaries. Our approach allows us to consider operators with non-smooth symbols, and we demonstrate the power of our approach by extending Connes’ trace theorem to operators with non-smooth symbols in three examples: the Lie group \(\mathrm {SU}(2)\) , noncommutative tori and Moyal planes. PubDate: 2019-03-22
Abstract: Abstract We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are \({\mathrm{CAT}}(0)\) . This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper \({\mathrm{CAT}}(0)\) in terms of the growth of the Dehn function at all scales. We finally obtain a generalization of this result of Lytchak and the author. Namely, we show that if the Dehn function of a proper, geodesic metric space is sufficiently close to the Euclidean Dehn function up to some scale then the space is not far (in a suitable sense) from being \({\mathrm{CAT}}(0)\) up to that scale. PubDate: 2019-03-21
Abstract: Abstract This paper is devoted to a classical variational problem for planar elastic curves of clamped endpoints, so-called Euler’s elastica problem. We investigate a straightening limit that means enlarging the distance of the endpoints, and obtain several new results concerning properties of least energy solutions. In particular we reach a first uniqueness result that assumes no symmetry. As a key ingredient we develop a foundational singular perturbation theory for the modified total squared curvature energy. It turns out that our energy has almost the same variational structure as a phase transition energy of Modica–Mortola type at the level of a first order singular limit. PubDate: 2019-03-14
Abstract: Abstract Let \((\mathsf {G},\mathsf {X})\) be a Shimura datum of Hodge type. Let p be an odd prime such that \(\mathsf {G}_{\mathbb {Q}_p}\) splits after a tamely ramified extension and \(p\not \mid \pi _1(\mathsf {G}^\mathrm{der}) \) . Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and \(\mathsf {G}_{\mathbb {Q}_p}\) is either unramified or residually split, we prove the generalisation of Mantovan’s formula for the l-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction modulo p of the Kisin–Pappas integral model. PubDate: 2019-03-09
Abstract: Abstract In this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite. This extends the recent extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued estimates including \(\ell ^\infty \) spaces and, in particular, we are able to reprove all the vector-valued bounds for the bilinear Hilbert transform obtained through the helicoidal method of Benea and Muscalu. Moreover, our result is quantitative and, in particular, allows us to extend quantitative estimates obtained from sparse domination in the Banach space setting to the quasi-Banach space setting. Our proof does not rely on any off-diagonal extrapolation results and we develop a multilinear version of the Rubio de Francia algorithm adapted to the multisublinear Hardy–Littlewood maximal operator. As a corollary, we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak-type and \({{\,\mathrm{BMO}\,}}\) spaces. PubDate: 2019-03-02
Abstract: Abstract The central result of this paper is a refinement of Hida’s duality theorem between ordinary \({\varLambda }\) -adic modular forms and the universal ordinary Hecke algebra. In particular, we give a sufficient condition for this duality to be integral with respect to particular submodules of the space of ordinary \({\varLambda }\) -adic modular forms. This refinement allows us to give a simple proof that the universal ordinary cuspidal Hecke algebra modulo Eisenstein ideal is isomorphic to the Iwasawa algebra modulo an ideal related to the Kubota-Leopoldt p-adic L-function. The motivation behind these results stems from a proof of the Iwasawa main conjecture over \({\mathbb {Q}}\) by Ohta. While simple and elegant, Ohta’s proof requires some restrictive hypotheses which we are able to remove using our results. PubDate: 2019-02-27
Abstract: Abstract We continue to investigate the race between prime numbers in many residue classes modulo q, assuming the standard conjectures GRH and LI. We show that provided \(n/\log q \rightarrow \infty \) as \(q \rightarrow \infty \) , we can find n competitor classes modulo q so that the corresponding n-way prime number race is extremely biased. This improves on the previous range \(n \geqslant \varphi (q)^{\epsilon }\) , and (together with an existing result of Harper and Lamzouri) establishes that the transition from all n-way races being asymptotically unbiased, to biased races existing, occurs when \(n = (\log q)^{1+o(1)}\) . The proofs involve finding biases in certain auxiliary races that are easier to analyse than a full n-way race. An important ingredient is a quantitative, moderate deviation, multi-dimensional Gaussian approximation theorem, which we prove using a Lindeberg type method. PubDate: 2019-02-21
Abstract: Abstract We prove a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by Keum, Oguiso and Zhang for complex projective varieties. PubDate: 2019-02-19
Abstract: Abstract The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in \(H^\sigma \) with \(\sigma \in [1,\frac{3}{2}[\) . In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities. PubDate: 2019-02-15
Abstract: Abstract Given a locally integrable structure \({\mathcal {V}}\) over a smooth manifold \(\varOmega \) and given \(p\in \varOmega \) we define the Borel map of \({\mathcal {V}}\) atp as the map which assigns to the germ of a smooth solution of \({\mathcal {V}}\) at p its formal Taylor power series at p. In this work we continue the study initiated in Barostichi et al. (Math. Nachr. 286(14–15):1439–1451, 2013), Della Sala and Lamel (Int J Math 24(11):1350091, 2013) and present new results regarding the Borel map. We prove a general necessary condition for the surjectivity of the Borel map to hold and also, after developing some new devices, we study some classes of CR structures for which its surjectivity is valid. In the final sections we show how the Borel map can be applied to the study of the algebra of germs of solutions of \({\mathcal {V}}\) at p. PubDate: 2019-02-15
Abstract: Abstract We show that Aomoto’s q-deformation of de Rham cohomology arises as a natural cohomology theory for \(\Lambda \) -rings. Moreover, Scholze’s \((q-1)\) -adic completion of q-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach p-power roots of q, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze. PubDate: 2019-02-05
Abstract: Abstract We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric. PubDate: 2019-02-01
Abstract: Abstract We study the complexity of approximating complex zero sets of certain n-variate exponential sums. We show that the real part, R, of such a zero set can be approximated by the \((n-1)\) -dimensional skeleton, T, of a polyhedral subdivision of \(\mathbb {R}^n\) . In particular, we give an explicit upper bound on the Hausdorff distance: \(\Delta (R,T) =O\left( t^{3.5}/\delta \right) \) , where t and \(\delta \) are respectively the number of terms and the minimal spacing of the frequencies of g. On the side of computational complexity, we show that even the \(n=2\) case of the membership problem for R is undecidable in the Blum-Shub-Smale model over \(\mathbb {R}\) , whereas membership and distance queries for our polyhedral approximation T can be decided in polynomial-time for any fixed n. PubDate: 2019-01-31
Abstract: Abstract In this paper, we show that any n-dimensional \(\kappa \) -noncolla-psed steady (gradient) Kähler–Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in Deng and Zhu (Trans Am Math Soc 370(4):2855–2877, 2018). PubDate: 2019-01-30
Abstract: Abstract We formulate and prove a generalization of the Atiyah-Singer family index theorem in the context of the theory of spaces of manifolds à la Madsen, Tillmann, Weiss, Galatius and Randal-Williams. Our results are for Dirac-type operators linear over arbitrary \(C^*\) -algebras. PubDate: 2019-01-29
Abstract: Abstract We show that a K-approximate subgroup A of a residually nilpotent group G is contained in boundedly many cosets of a finite-by-nilpotent subgroup, the nilpotent factor of which is of bounded step. Combined with an earlier result of the author, this implies that A is contained in boundedly many translates of a coset nilprogression of bounded rank and step. The bounds are effective and depend only on K; in particular, if G is nilpotent they do not depend on the step of G. As an application we show that there is some absolute constant c such that if G is a residually nilpotent group, and if there is an integer \(n>1\) such that the ball of radius n in some Cayley graph of G has cardinality bounded by \(n^{c\log \log n}\) , then G is virtually \((\log n)\) -step nilpotent. PubDate: 2019-01-28
Abstract: Abstract The p-Laplacian problem with the exponent of nonlinearity p depending on the solution u itself is considered in this work. Both situations when p(u) is a local quantity or when p(u) is nonlocal are studied here. For the associated boundary-value local problem, we prove the existence of weak solutions by using a singular perturbation technique. We also prove the existence of weak solutions to the nonlocal version of the associated boundary-value problem. The issue of uniqueness for these problems is addressed in this work as well, in particular by working out the uniqueness for a one dimensional local problem and by showing that the uniqueness is easily lost in the nonlocal problem. PubDate: 2019-01-28
Abstract: Abstract We study the problem of obtaining asymptotic formulas for the sums \(\sum _{X < n \le 2X} d_k(n) d_l(n+h)\) and \(\sum _{X < n \le 2X} \Lambda (n) d_k(n+h)\) , where \(\Lambda \) is the von Mangoldt function, \(d_k\) is the \(k^{{\text {th}}}\) divisor function, X is large and \(k \ge l \ge 2\) are integers. We show that for almost all \(h \in [-H, H]\) with \(H = (\log X)^{10000 k \log k}\) , the expected asymptotic estimate holds. In our previous paper we were able to deal also with the case of \(\Lambda (n) \Lambda (n + h)\) and we obtained better estimates for the error terms at the price of having to take \(H = X^{8/33 + \varepsilon }\) . PubDate: 2019-01-21
Abstract: Abstract We consider nonlinear elastic wave equations generalizing Gol’dberg’s five constants model. We analyze the nonlinear interaction of two distorted plane waves and characterize the possible nonlinear responses. Using the boundary measurements of the nonlinear responses, we solve the inverse problem of determining elastic parameters from the displacement-to-traction map. PubDate: 2019-01-18
Abstract: Abstract We develop Nevanlinna’s theory for a class of holomorphic maps when the source is a disc. Such maps appear in the theory of foliations by Riemann Surfaces. PubDate: 2019-01-18
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