On the effective Putinar’s Positivstellensatz and moment approximation

Mathematical Programming - We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar’s Positivstellensatz over a compact...     

Interrelation between the convergence rates in von Neumann’s and Birkhoff’s ergodic theorems

In the L _p spaces, 1 < _p < ∞, we prove some inequalities for discrete and continuous times that make it possible to obtain the convergence rate in Birkhoff’s theorem in the presence of bounds on the convergence rate in von Neumann’s ergodic theorem belonging to a sufficiently large rate range. The exact operator analogs of these inequalities for contraction semigroups in L _p are given. These results also have the obvious exact analogs in the class of wide-sense stationary stochastic processes.     

A link between Wilson’s and d’Alembert’s functional equations

If \({f, g : G \to \mathbb{C}}\), f ≠ 0, is a solution of Wilson’s functional equation on a group G, then g is a d’Alembert function.     

On the relationship between Semi-Lagrangian and Lagrange–Galerkin schemes

Following a previous result stating their equivalence under constant advection speed, Semi-Lagrangian and Lagrange–Galerkin schemes are compared in this paper in the situation of variable coefficient advection equations. Once known that Semi-Lagrangian schemes can be proved to be equivalent to area-weighted Lagrange–Galerkin schemes via a suitable definition of the basis functions, we will further prove that area-weighted Lagrange–Galerkin schemes represent a “small” (more precisely, an $O(\Delta t$ )) perturbation of exact Lagrange–Galerkin schemes. This equivalence implies a general result of stability for Semi-Lagrangian schemes.     

A note on Burer’s copositive representation of mixed-binary QPs

In an important paper, Burer (Math. Program Ser. A 120:479–495, 2009) recently showed how to reformulate general mixed-binary quadratic optimization problems (QPs) into copositive programs where a linear functional is minimized over a linearly constrained subset of the cone of completely positive matrices. In this note we interpret the implication from a topological point of view, showing that the Minkowski sum of the lifted feasible set and the lifted recession cone gives exactly the closure of the former. We also discuss why feasibility of the copositive program implies feasibility of the original mixed-binary QP, which can be derived from the arguments in Burer (Math. Program Ser. A 120:479–495, 2009) without any further condition.     

A revised theorization of the relationship between teachers’ conceptions of mathematics and its teaching

We assembled the ideas about mathematics and about its teaching which were expressed by mathematicians and mathematics educators into two pairs of ‘official’ (collective) conceptions: mathematics is either static or dynamic, and mathematics teaching is either closed or open. These polar conceptions produce a 4-pair relationship between the conceptions of mathematics and its teaching. The adherence to official conceptions was tapped by a questionnaire encompassing 176 Israeli high school mathematics teachers, aimed at examining the relationship between their conceptions of mathematics and its teaching. The majority of these teachers either hold a single conception in one of the domains or do not adhere to any conception, and a quarter of them hold either the static-closed or dynamic-open pairs of conceptions that prevail among teachers in other countries. Consequently, we define a conception of an entity as a comprehensive and homogenous set of ideas about a particular characteristic or feature of that entity. Reality is that teachers practice their profession without adhering to any official conception, and perhaps are (/to be?/) praised for their reluctance to blindly adopt the clear-cut rigid official conceptions of mathematics and its teaching while maintaining their individual and independent blends of ideas.     

From spline approximation to Roth’s equation and Schur functors

Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree d and smoothness r on a generic triangulation of a planar simplicial complex Δ, for d ≥ 3r + 1. Schenck and Stiller conjectured that this formula actually holds for all d ≥ 2r + 1. Up to this moment there was not known a single example where one could show that the bound d ≥ 2r + 1 is sharp. However, in 2005, a possible such example was constructed to show that this bound is the best possible (i.e., the Alfeld–Schumaker formula does not hold if d = 2r), except that the proof that this formula actually works if d ≥ 2r + 1 has been a challenge until now when we finally show it to be true. The interesting subtle connections with representation theory, matrix theory and commutative and homological algebra seem to explain why this example presented such a challenge. Thus in this paper we present the first example when it is known that the bound d ≥ 2r + 1 is sharp for asserting the validity of the Alfeld–Schumaker formula.     

Transfinite approximation of Hindman’s theorem

Hindman’s Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the finite analogue stating that for any n, r, there is a k such that for any r-coloring of [1, k], there is a set of n integers all of whose finite sums belong to the same color.     

Dugundji’s theorem for cone metric spaces

In this work, we will prove the Dugundji extension theorem for the cone metric space. It is heavily reliant on the paracompactness of the cone topology that is proved by Ayse Sönmez in the paper Sönmez (2010) [11].     

Ramanujan’s inverse elliptic arc approximation

We suggest a continued fraction origin to Ramanujan’s approximation to $(\frac{a-b}{a+b})^{2}$ in terms of the arc length of an ellipse with semiaxes a and b.     

The monopolist’s problem: existence, relaxation, and approximation

We study a variational problem arising from a generalization of an economic model introduced by Rochet and Choné in [5]. In this model a monopolist proposes a set Y of products with price list . Each rational consumer chooses which product to buy by solving a personal minimum problem, taking into account his/her tastes and economic possibilities. The monopolist looks for the optimal price list which minimizes costs, hence maximizes the profit. This leads to a minimum problem for functionals (the “pessimistic cost expectation”) and (the “optimistic cost expectation”), which are in turn defined through two nested variational problems. We prove that the minimum of exists and coincides with the infimum of . We also provide a variational approximation of by smooth functionals defined in finite dimensional Euclidean spaces.Received: 2 March 2004, Accepted: 19 October 2004, Published online: 22 December 2004Mathematics Subject Classification (2000): 49J45, 91B     

A refinement of Ramanujan’s factorial approximation

We present a new proof of the monotonicity of the correction term θ _n in Ramanujan’s refinement of Stirling’s formula. Moreover we prove that θ _n is concave.     

Consistency of the intensional level of the Minimalist Foundation with Church’s thesis and axiom of choice

Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by Maietti and Sambin (in Crosilla, Schuster (eds) From sets and types to topology and analysis: practicable foundations for constructive mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for short MF, completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory à la Martin-Löf, called mTT. The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church’s thesis of a fragment of intensional Martin-Löf’s type theory, called \(\mathbf{MLtt}_1\), where mTT can be easily interpreted. Then to show the consistency of \(\mathbf{MLtt}_1\) with CT we interpret it within Feferman’s predicative theory of non-iterative fixpoints \(\widehat{ID_1}\) by extending the well known Kleene’s realizability semantics of intuitionistic arithmetics so that CT is trivially validated. More in detail the fragment \(\mathbf{MLtt}_1\) we interpret consists of first order intensional Martin-Löf’s type theory with one universe and with explicit substitution rules in place of usual equality rules preserving type constructors (hence without the so called \(\xi \)-rule which is not valid in our realizability semantics). A key difficulty encountered in our interpretation was to use the right interpretation of lambda abstraction in the applicative structure of natural numbers in order to model all the equality rules of \(\mathbf{MLtt}_1\) correctly. In particular the universe of \(\mathbf{MLtt}_1\) is modelled by means of \(\widehat{ID_1}\)-fixpoints following a technique due first to Aczel and used by Feferman and Beeson.     

Laguerre spectral approximation of Stokes’ first problem for third-grade fluid

A Laguerre–Galerkin method is proposed and analyzed for Quasilinear parabolic differential equation which arises from Stokes’ first problem for a third-grade fluid on a semi-infinite interval. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations to the transformed equations is developed and implemented. Effects of non-Newtonian parameters on the flow phenomena are analyzed and documented.     

A finite element approximation of the Griffith’s model in fracture mechanics

Summary. The Griffith model for the mechanics of fractures in brittle materials is consider in the weak formulation of SBD spaces. We suggest an approximation, in the sense of –convergence, by a sequence of discrete functionals defined on finite elements spaces over structured and adaptive triangulations. The quasi-static evolution for boundary value problems is also taken into account and some numerical results are shown. Mathematics Subject Classification (2000):65N30