Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
244,024
result(s) for
"Mathematics and Statistics"
Sort by:
Unirational threefolds with no universal codimension 2 cycle
We prove that the general quartic double solid with
k
≤
7
nodes does not admit a Chow theoretic decomposition of the diagonal, (or equivalently has a nontrivial universal
CH
0
group,) and the same holds if we replace in this statement “Chow theoretic” by “cohomological”. In particular, it is not stably rational. We also deduce that the general quartic double solid with seven nodes does not admit a universal codimension
2
cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibers of its Jacobian by a family of
1
-cycles. This finally implies that its third unramified cohomology group is not universally trivial.
Journal Article
Machine translation
\"The proposed project on machine translation will be based on the above pedagogy, through the study of phenomena, formalization, and then elucidation of the techniques. Case studies, examples, and historical perspectives will be used extensively to cover the material. The primary aim of this book is to provide an accessible text book on machine translation covering lucidly the foundations, insights, and case studies for practical concerns. The book would also point towards where the field is currently and heading towards in the future\"-- Provided by publisher.
Book
The Local Langlands Correspondence for GLn over p-adic fields
We extend our methods from Scholze (Invent. Math.
2012
, doi:
10.1007/s00222-012-0419-y
) to reprove the Local Langlands Correspondence for GL
n
over
p
-adic fields as well as the existence of
ℓ
-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties,
2001
).
In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455,
2000
), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties,
2001
), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Éc. Norm. Super. 21(4), 497–544,
1988
). Instead, we make use of a previous result from Scholze (Invent. Math.
2012
, doi:
10.1007/s00222-012-0419-y
) describing the inertia-invariant nearby cycles in certain regular situations.
Journal Article
Nuclear dimension of simple C∗-algebras
We compute the nuclear dimension of separable, simple, unital, nuclear, Z-stable C∗-algebras. This makes classification accessible from Z-stability and in particular brings large classes of C∗-algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme.
Journal Article
On Falconer’s distance set problem in the plane
If E⊂R2 is a compact set of Hausdorff dimension greater than 5 / 4, we prove that there is a point x∈E so that the set of distances {|x-y|}y∈E has positive Lebesgue measure.
Journal Article
The globalization theorem for the Curvature-Dimension condition
The Lott–Sturm–Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space (X,d,m) (so that (supp(m),d) is a length-space and m(X)<∞) verifying the local Curvature-Dimension condition CDloc(K,N) with parameters K∈R and N∈(1,∞), also verifies the global Curvature-Dimension condition CD(K,N). In other words, the Curvature-Dimension condition enjoys the globalization (or local-to-global) property, answering a question which had remained open since the beginning of the theory. For the proof, we establish an equivalence between L1- and L2-optimal-transport–based interpolation. The challenge is not merely a technical one, and several new conceptual ingredients which are of independent interest are developed: an explicit change-of-variables formula for densities of Wasserstein geodesics depending on a second-order temporal derivative of associated Kantorovich potentials; a surprising third-order theory for the latter Kantorovich potentials, which holds in complete generality on any proper geodesic space; and a certain rigidity property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. As a consequence, numerous variants of the Curvature-Dimension condition proposed by various authors throughout the years are shown to, in fact, all be equivalent in the above setting, thereby unifying the theory.
Journal Article
Nonuniformly elliptic Schauder theory
Local Schauder theory holds in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic problems are locally Hölder continuous if so are their coefficients.
Journal Article
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next
Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.
Journal Article