This paper proves two main results. First, it is shown that if Σ is a smooth manifold homeomorphic to the standard n-torus T n = R n Z n and H is a real-analytically completely integrable convex hamiltonian on T ∗Σ, then Σ is diffeomorphic to Tn. Second, it is proven that for some topological 7-manifolds, the cotangent bundle of each smooth structure admits a real-analytically completely integrable riemannian metric hamiltonian.
Taimanov @cite_27 has proven that if a compact manifold @math admits a real-analytically completely integrable geodesic flow, then @math is almost abelian of rank at most @math ; @math ; and there is an injection @math where @math . These constraints are ineffective for exotic tori.
@cite_27
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A set S of vector fields on a differentiable manifold M is said to be completely controllable if for every pair @math of points of M there exists a trajectory of S from m to @math . Here a trajectory of S is a curve which is an integral curve of some @math or a finite concatenation of such curves so that, in general, a trajectory of S run in reverse is no longer a trajectory. Our main theorem is: on every connected paracompact manifold of class @math , @math , or @math , there exists a completely controllable set S consisting of two vector fields of class @math . A standard result in metric geometry is that every compact geodesic metric space can be approximated arbitrarily well by finite metric graphs in the Gromov-Hausdorff sense. It is well known that the first Betti number of the approximating graphs may blow up as the approximation gets finer. In our work, given a compact geodesic metric space @math , we define a sequence @math of non-negative real numbers by @math By construction, and the above result, this is a non-increasing sequence with limit @math . We study this sequence and its rates of decay with @math . We also identify a precise relationship between the sequence and the first Vietoris-Rips persistence barcode of @math . Furthermore, we specifically analyze @math and find upper and lower bounds based on hyperbolicity and other metric invariants. As a consequence of the tools we develop, our work also provides a Gromov-Hausdorff stability result for the Reeb construction on geodesic metric spaces with respect to the function given by distance to a reference point. Let @math denote the cover time of a two dimensional compact manifold @math by a Wiener sausage of radius @math . We prove that @math is tight, where @math denotes the area of @math . Given a polygon @math , for two points @math and @math contained in the polygon, their is the length of the shortest @math -path within @math . A of radius @math centered at a point @math is the set of points in @math whose geodesic distance to @math is at most @math . We present a polynomial time @math -approximation algorithm for finding a densest geodesic unit disk packing in @math . Allowing arbitrary radii but constraining the number of disks to be @math , we present a @math -approximation algorithm for finding a packing in @math with @math geodesic disks whose minimum radius is maximized. We then turn our focus on of @math and present a @math -approximation algorithm for covering @math with @math geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are @math -hard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius. We study identifiability for bilinear inverse problems under sparsity and subspace constraints. We show that, up to a global scaling ambiguity, almost all such maps are injective on the set of pairs of sparse vectors if the number of measurements @math exceeds @math , where @math and @math denote the sparsity of the two input vectors, and injective on the set of pairs of vectors lying in known subspaces of dimensions @math and @math if @math . We also prove that both these bounds are tight in the sense that one cannot have injectivity for a smaller number of measurements. Our proof technique draws from algebraic geometry. As an application we derive optimal identifiability conditions for the deconvolution problem, thus improving on recent work of [1]. Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on @math with i.i.d. exponential weights on the vertices. Fix two points @math and @math for some @math , and consider the maximal paths @math and @math starting at @math and @math respectively to the point @math for @math . Our object of study is the point of coalescence, i.e., the point @math with smallest @math . We establish that the distance to coalescence @math scales as @math , by showing the upper tail bound @math for some @math . We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction @math starting from @math and @math , we establish the optimal tail estimate @math , for the point of coalescence @math . This answers a question left open by Pimentel (Ann. Probab., 2016) who proved the corresponding lower bound. In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose @math is a lattice in semisimple Lie group, all of whose factors have rank @math or higher. Let @math be a smooth @math -action on a compact nilmanifold @math that lifts to an action on the universal cover. If the linear data @math of @math contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of @math and @math , on a finite-index subgroup of @math . If @math is a @math action and contains an Anosov element, then the semiconjugacy is a @math conjugacy. As a corollary, we obtain @math global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank @math or higher. We also obtain global rigidity of Anosov actions of @math on @math for @math and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds. Let @math be a primitive Hilbert modular form of parallel weight @math and level @math for the totally real field @math , and let @math be a rational prime coprime to @math . If @math is ordinary at @math and @math is a CM extension of @math of relative discriminant @math prime to @math , we give an explicit construction of the @math -adic Rankin-Selberg @math -function @math . When the sign of its functional equation is @math , we show, under the assumption that all primes @math are principal ideals of @math which split in @math , that its central derivative is given by the @math -adic height of a Heegner point on the abelian variety @math associated with @math . This @math -adic Gross--Zagier formula generalises the result obtained by Perrin-Riou when @math and @math satisfies the so-called Heegner condition. We deduce applications to both the @math -adic and the classical Birch and Swinnerton-Dyer conjectures for @math . Let @math be a compact Riemannian submanifold of @math of dimension @math and let @math be a sample of i.i.d. points in @math with uniform distribution. We study the random operators @math where @math is the Gaussian kernel and @math as @math Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of @math @math (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations @math for smooth functions @math both pointwise and uniformly in @math and @math (here @math and @math is the Riemannian volume measure). In particular, we show that for any class @math of three times differentiable functions on @math with uniformly bounded derivatives @math as soon as @math and also prove asymptotic normality of @math (functional CLT) for a fixed @math and uniformly in @math Given a large edge-weighted network @math with @math terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of @math is to construct a : a small network @math with the same @math terminals, in which the minimum cut value between every bipartition of terminals is the same as in @math . This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS '98], who proved that such @math of size at most @math always exists. Obviously, by having access to the smaller network @math , certain computations involving cuts can be carried out much more efficiently. We provide several new bounds, which together narrow the previously known gap from doubly-exponential to only singly-exponential, both for planar and for general graphs. Our first and main result is that every @math -terminal planar network admits a mimicking network @math of size @math , which is moreover a minor of @math . On the other hand, some planar networks @math require @math . For general networks, we show that certain bipartite graphs only admit mimicking networks of size @math , and moreover, every data structure that stores the minimum cut value between all bipartitions of the terminals must use @math machine words. An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least @math -fraction of the constraints given a @math -satisfiable instance, where @math as @math . Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction problem admits an efficient robust algorithm. This paper confirms their conjecture. We prove that any smooth action of @math , @math , on an @math -dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e., isomorphic up to a finite permutation to an affine action on the torus or on its factor by @math . Furthermore this isomorphism has nice geometric properties; in particular, it is smooth in the sense of Whitney on a set whose complement has arbitrarily small measure. We further derive restrictions on topology of manifolds that may admit such actions, for example, excluding spheres and obtaining lower estimate on the first Betti number in the odd-dimensional case. Consider a convex relaxation @math of a pseudo-boolean function @math . We say that the relaxation is totally half-integral if @math is a polyhedral function with half-integral extreme points @math , and this property is preserved after adding an arbitrary combination of constraints of the form @math , @math , and @math where @math is a constant. A well-known example is the roof duality relaxation for quadratic pseudo-boolean functions @math . We argue that total half-integrality is a natural requirement for generalizations of roof duality to arbitrary pseudo-boolean functions. Our contributions are as follows. First, we provide a complete characterization of totally half-integral relaxations @math by establishing a one-to-one correspondence with bisubmodular functions . Second, we give a new characterization of bisubmodular functions. Finally, we show some relationships between general totally half-integral relaxations and relaxations based on the roof duality. One can describe an @math -dimensional noncommutative torus by means of an antisymmetric @math -matrix @math . We construct an action of the group @math on the space of antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched. A theoretical and computational framework for computing intrinsic distance functions and geodesics on submanifolds of @math given by point clouds is introduced and developed in this paper. The basic idea is that, as shown here, intrinsic distance functions and geodesics on general co-dimension submanifolds of @math can be accurately approximated by extrinsic Euclidean ones computed inside a thin offset band surrounding the manifold. This permits the use of computationally optimal algorithms for computing distance functions in Cartesian grids. We use these algorithms, modified to deal with spaces with boundaries, and obtain a computationally optimal approach also for the case of intrinsic distance functions on submanifolds of @math . For point clouds, the offset band is constructed without the need to explicitly find the underlying manifold, thereby computing intrinsic distance functions and geodesics on point clouds while skipping the manifold reconstruction step. The case of poin... In this work we establish sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings: compact manifolds, operators on lattices, domains in @math of finite measure, and conditions for operators on @math given in terms of anharmonic oscillators. We also give examples in the settings of compact sub-Riemannian manifolds, contact manifolds, strictly pseudo-convex CR manifolds, and (sub-)Laplacians on compact Lie groups. The sign-rank of a matrix @math with @math entries is the least rank of a real matrix @math with @math for all @math . We obtain the first exponential lower bound on the sign-rank of a function in @math . Namely, let @math . We show that the matrix @math has sign-rank @math . This in particular implies that @math , which solves a longstanding open problem in communication complexity posed by Babai, Frankl, and Simon [Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), 1986, pp. 337-347]. Our result additionally implies a lower bound in learning theory. Specifically, let @math be functions such that every DNF formula @math of polynomial size has the representation @math for some reals @math . We prove that then @math , which essentially matches an upper bound of @math , due to Klivans and Servedio [J. Comput. System Sci., 68 (2004), pp. 303-318]. Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in @math . This substantially generalizes and strengthens the results of Krause and Pudlak [Theoret. Comput. Sci., 174 (1997), pp. 137-156]. We consider the problem of learning a one-hidden-layer neural network: we assume the input @math is from Gaussian distribution and the label @math , where @math is a nonnegative vector in @math with @math , @math is a full-rank weight matrix, and @math is a noise vector. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function @math whose landscape is guaranteed to have the following properties: 1. All local minima of @math are also global minima. 2. All global minima of @math correspond to the ground truth parameters. 3. The value and gradient of @math can be estimated using samples. With these properties, stochastic gradient descent on @math provably converges to the global minimum and learn the ground-truth parameters. We also prove finite sample complexity result and validate the results by simulations. This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . We give sufficient conditions on a symmetric tensor @math to satisfy the following equality: the symmetric rank of @math , denoted as @math , is equal to the rank of @math , denoted as @math . This is done by considering the rank of the unfolded @math viewed as a matrix @math . The condition is @math . In particular, @math for @math for the cases @math . We discuss the analogues of the above results for border rank and best approximations of symmetric tensors.