Nature of mathematics

3D animation software is extensively used to create graphics for special effects sequences in live-action films, in animated feature film production and for the generation of computer games and virtual reality content. It is a commonly held view within the computer graphics community that 3D animation software is ―just another tool‖ for self-expression. This research questions that view by examining the inherent nature of the tool itself. In doing so, some of the philosophical assumptions embedded in the design of the new digital tool are revealed.

99p runthenight05 01-03-2023 6 2   Download

Lecture “Natural language processing - Chapter 2: Fundamental algorithms and mathematical models” has contents: Probability theory and Bayes theorems (Concepts in probability, Bayes theorems, application of the probability theory in NLP).

14p dien_vi01 21-11-2018 34 1   Download

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure µ ∈ M(G) is said to be idempotent if µ ∗ µ = µ, or alternatively if µ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure µ is idempotent if and only if the set {γ ∈ G : µ(γ) = 1} belongs to the coset ring of G, 1. Introduction Let

31p dontetvui 17-01-2013 73 8   Download

We construct an exhaustive submeasure that is not equivalent to a measure. This solves problems of J. von Neumann (1937) and D. Maharam (1947). Contents 1. Introduction 2. Roberts 3. Farah 4. The construction 5. The main estimate 6. Exhaustivity 7. Proof of Theorems 1.2 to 1.4 References 1. Introduction Consider a Boolean algebra B of sets.

30p dontetvui 17-01-2013 75 11   Download

A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall. 1. Introduction The classical isoperimetric inequality states that if E is a Borel set in Rn , n ≥ 2, with finite Lebesgue measure |E|, then the ball with the same volume has a lower perimeter, or, equivalently, that (1.1) 1/n nωn |E|(n−1)/n ≤ P (E) . Here P (E) denotes the distributional perimeter of E (which coincides with the classical (n − 1)-dimensional measure of ∂E when E has a smooth boundary) and ωn is the measure of the...

41p dontetvui 17-01-2013 65 8   Download

The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M have a nontrivial amount of symmetry. By this we mean that Isom(M ) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom(M ) : π1 (M )] = ∞. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal cover M . Our description of such M is given in Theorem 1.2 below. The proof of this theorem uses methods from Lie theory, harmonic maps,...

27p dontetvui 17-01-2013 54 7   Download

In this paper we give a new proof for the classification result in [3]. We show that isoparametric hypersurfaces with four distinct principal curvatures in spheres are of Clifford type provided that the multiplicities m1 , m2 of the principal curvatures satisfy m2 ≥ 2m1 − 1. This inequality is satisfied for all but five possible pairs (m1 , m2 ) with m1 ≤ m2 .

15p dontetvui 17-01-2013 64 7   Download

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: −∆p u = uq + µ, Fk [−u] = uq + µ, u ≥ 0, on Rn , or on a bounded domain Ω ⊂ Rn . Here ∆p is the p-Laplacian defined by ∆p u = div ( u| u|p−2 ), and Fk [u] is the k-Hessian defined as the sum of k × k principal minors of the Hessian matrix D2 u (k = 1, 2, . . . ,...

58p dontetvui 17-01-2013 58 10   Download

In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Rie2 1 mannian metric g. That is, for X = M × Rt , P = Dt − ∆M , and u ∈ Hloc (X) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WFb (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C ∞ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with...

65p dontetvui 17-01-2013 55 8   Download

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35]. 1.

48p dontetvui 17-01-2013 59 7   Download

We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance dissipation in this sense. ...

33p dontetvui 17-01-2013 49 9   Download

A long-standing conjecture due to Michael Freedman asserts that the 4-dimensional topological surgery conjecture fails for non-abelian free groups, or equivalently that a family of canonical examples of links (the generalized Borromean rings) are not A − B slice. A stronger version of the conjecture, that the Borromean rings are not even weakly A − B slice, where one drops the equivariant aspect of the problem, has been the main focus in the search for an obstruction to surgery.

21p dontetvui 17-01-2013 73 7   Download

Bowen’s formula relates the Hausdorff dimension of a conformal repeller to the zero of a ‘pressure’ function. We present an elementary, self-contained proof to show that Bowen’s formula holds for C 1 conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class.

55p dontetvui 17-01-2013 55 7   Download

We study unitary random matrix ensembles of the form −1 Zn,N | det M |2α e−N Tr V (M ) dM, where α −1/2 and V is such that the limiting mean eigenvalue density for n, N → ∞ and n/N → 1 vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight |x|2α e−N V (x) . ...

42p dontetvui 17-01-2013 73 7   Download

We determine the order of magnitude of H(x, y, z), the number of integers n ≤ x having a divisor in (y, z], for all x, y and z. We also study Hr (x, y, z), the number of integers n ≤ x having exactly r divisors in (y, z]. When r = 1 we establish the order of magnitude of H1 (x, y, z) for all x, y, z satisfying z ≤ x1/2−ε . For every r ≥ 2, C 1 and ε 0, we determine the order of magnitude of Hr (x, y, z) uniformly...

68p dontetvui 17-01-2013 46 8   Download

Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A−1 does not exceed Cn3/2 with probability close to 1. 1. Introduction Let A be an n × n matrix, whose entries are independent, identically distributed random variables. The spectral properties of such matrices, in particular invertibility, have been extensively studied (see, e.g. [M] and the survey [DS]).

28p dontetvui 17-01-2013 54 8   Download

We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a compact manifold of negative curvature – in fact, we only assume that the geodesic flow has the Anosov property. In the semi-classical limit, we prove that the Wigner measures associated to eigenfunctions have positive metric entropy. In particular, they cannot concentrate entirely on closed geodesics. 1. Introduction, statement of results We consider a compact Riemannian manifold M of dimension d ≥ 2, and assume that the geodesic flow (g t )t∈R , acting on the unit tangent bundle of M , has a “chaotic”...

43p dontetvui 17-01-2013 54 7   Download

If G is a locally compact group, then for each derivation D from L1 (G) into L1 (G) there is a bounded measure μ ∈ M (G) with D(a) = a ∗ μ − μ ∗ a for a ∈ L1 (G) (“derivation problem” of B. E. Johnson). Introduction Let A be a Banach algebra, E an A-bimodule. A linear mapping D : A → E is called a derivation, if D(a b) = a D(b) + D(a) b for all a, b ∈ A ([D, Def. 1.8.1]). For x ∈ E, we define the inner derivation adx :...

27p dontetvui 17-01-2013 46 8   Download

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe’s theorem, settling a long standing open problem in the nonlinear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and coarse embeddings, settling in particular the problem of classifying when Lp coarsely or uniformly embeds into Lq . We also prove a nonlinear analog of the Maurey-Pisier theorem, and use it to...

53p dontetvui 17-01-2013 65 8   Download

In this paper we extend the results obtained in [9], [10] to manifolds with SpinC -structures defined, near the boundary, by an almost complex structure. We show that on such a manifold with a strictly pseudoconvex boundary, there ¯ are modified ∂-Neumann boundary conditions defined by projection operators, Reo , which give subelliptic Fredholm problems for the SpinC -Dirac operator, + .eo . We introduce a generalization of Fredholm pairs to the “tame” category.

68p dontetvui 17-01-2013 64 8   Download