Nature of mathematics

This work develops and defends a structural view of the nature of mathematics, which is used to explain a number of striking features of mathematics that have puzzled philosophers for centuries. It rejects the most widely held philosophical view of mathematics (Platonism), according to which mathematics is a science dealing with mathematical objects such as sets and numbers—objects which are believed not to exist in the physical world.
394p bimap_5 28122012 24 7 Download

The ﬁrst is the widely held view that mathematics is, somehow, innate11. Preservice teachers will often indicate that they do not see the need to learn the material being covered because, when the time comes that they actually need it, they will be able to dredge it up.
640p dacotaikhoan 25042013 36 8 Download

Đề tài " The space of embedded minimal surfaces of fixed genus in a 3manifold III; Planar domains "
Annals of Mathematics This paper is the third in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3manifold. In [CM3]–[CM5] we describe the case where the surfaces are topologically disks on any ﬁxed small scale. Although the focus of this paper, general planar domains, is more in line with [CM6], we will prove a result here (namely, Corollary III.
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Annals of Mathematics In our ﬁrst article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number ﬁelds, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups.
23p tuanloccuoi 04012013 33 5 Download

Annals of Mathematics By S. Artstein, V. Milman, and S. J. Szarek For two convex bodies K and T in Rn , the covering number of K by T , denoted N (K, T ), is deﬁned as the minimal number of translates of T needed to cover K. Let us denote by K ◦ the polar body of K and by D the euclidean unit ball in Rn . We prove that the two functions of t, N (K, tD) and N (D, tK ◦ ), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K ⊂...
17p tuanloccuoi 04012013 24 5 Download

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in ﬁnite time for a generic family of initial data. By adding a diﬀusion term the ﬁnite time singularity is prevented and the solutions exist globally in time. 1. Introduction In this paper we study the nature of the solutions to the following class of equations (1.1)
14p noel_noel 17012013 29 5 Download

Annals of Mathematics By Curtis T. McMullen* .Annals of Mathematics, 165 (2007), 397–456 Dynamics of SL2(R) over moduli space in genus two By Curtis T. McMullen* Abstract This paper classiﬁes orbit closures and invariant measures for the natural action of SL2 (R) on ΩM2 , the bundle of holomorphic 1forms over the moduli space of Riemann surfaces of genus two. Contents 1. Introduction 2. Dynamics and Lie groups 3. Riemann surfaces and holomorphic 1forms 4. Abelian varieties with real multiplication 5. Recognizing eigenforms 6. Algebraic sums of 1forms 7.
61p noel_noel 17012013 29 5 Download

We prove that if f (x) = n−1 ak xk is a polynomial with no cyclotomic k=0 factors whose coeﬃcients satisfy ak ≡ 1 mod 2 for 0 ≤ k 1 + log 3 , 2n resolving a conjecture of Schinzel and Zassenhaus [21] for this class of polynomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials
21p noel_noel 17012013 24 5 Download

Annals of Mathematics We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler’s equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a “physical condition”, related to the fact that the pressure of a ﬂuid has to be positive. ...
87p noel_noel 17012013 25 4 Download

Annals of Mathematics In this paper we will solve one of the central problems in dynamical systems: Theorem 1 (Density of hyperbolicity for real polynomials). Any real polynomial can be approximated by hyperbolic real polynomials of the same degree. Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of inﬁnity.
39p noel_noel 17012013 20 4 Download

That is all: just a computer procedure to approximate a real root. From the narrow perspective of treating mathematics as a tool to solve real life problems, this is of course suﬃcient. However, from the point of view of mathematics, shouldn’t a student be interested in roots of polynomials in general? Fourth degree? Odd degree? Other roots, once one is found? Rational roots? Total number of roots? Not every detail need be explained, but even the average student will have his life improved by the mere knowledge that there are such questions, often with answers, e.g.
334p dacotaikhoan 25042013 16 2 Download

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A note on naturally embedded ternary trees...
20p thulanh7 05102011 17 1 Download

Like all other sciences, physics is based on experimental observations and quantitative measurements. The main objective of physics is to ﬁnd the limited number of fundamental laws that govern natural phenomena and to use them to develop theories that can predict the results of future experiments. The fundamental laws used in developing theories are expressed in the language of mathematics, the tool that provides a bridge between theory and experiment
1333p gian_anh 18102012 109 23 Download

Stochastic Calculus of Variations (or Malliavin Calculus) consists, in brief, in constructing and exploiting natural differentiable structures on abstract probability spaces; in other words, Stochastic Calculus of Variations proceeds from a merging of differential calculus and probability theory. As optimization under a random environment is at the heart of mathematical finance, and as differential calculus is of paramount importance for the search of extrema, it is not surprising that Stochastic Calculus of Variations appears in mathematical finance.
147p thuymonguyen88 07052013 38 8 Download

We verify an old conjecture of G. P´lya and G. Szeg˝ saying that the o o regular ngon minimizes the logarithmic capacity among all ngons with a ﬁxed area. 1. Introduction The logarithmic capacity cap E of a compact set E in R2 , which we identify with the complex plane C, is deﬁned by (1.1) − log cap E = lim (g(z, ∞) − log z), z→∞ where g(z, ∞) denotes the Green function of a connected component Ω(E) ∞ of C \ E having singularity at z = ∞; see [4, Ch. 7], [7, §11.1]. By an ngon with...
28p tuanloccuoi 04012013 33 7 Download

We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c
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We study “ﬂat knot types” of geodesics on compact surfaces M 2 . For every ﬂat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening ﬂow on the space of immersed curves on M 2 . We conclude existence of closed geodesics with prescribed ﬂat knot types, provided the associated Conley index is nontrivial. 1. Introduction If M is a surface with a Riemannian metric g then closed geodesics on (M, g) are critical points of the length functional L(γ) = γ (x)dx deﬁned on the space of unparametrized C...
56p noel_noel 17012013 31 7 Download

The stochastic 2D NavierStokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L2 (T2 ). Unlike previous 0 works, this class is independent of the viscosity and the strength of the noise.
41p noel_noel 17012013 21 7 Download

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of LaneEmden 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 pLaplacian deﬁned by ∆p u = div ( u up−2 ), and Fk [u] is the kHessian deﬁned as the sum of k × k principal minors of the Hessian matrix D2 u (k = 1, 2, . . . ,...
58p dontetvui 17012013 32 7 Download

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complexvalued 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 CohenHelsonRudin 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
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