Papers by Ricardo Bianconi
Santalo's encyclopedy is the state of "the art" in integral geometry. The text surveys some exten... more Santalo's encyclopedy is the state of "the art" in integral geometry. The text surveys some extensions of basic results in euclidean integral geometry to Lorentz and de Sitter spaces. We propose a new geometrical proof using light rays and Lorentz isometries of a Cauchy-Crofton like formula for time-like curves.
We present some results and open problems related to expansions of the field of real numbers by h... more We present some results and open problems related to expansions of the field of real numbers by hypergeometric and related functions focussing on definability and model completeness questions. In particular, we prove the strong model completeness for expansions of the field of real numbers by the exponential, arctangent and hypergeometric functions. We pay special attention to the expansion of the real field by the real and imaginary parts of the hypergeometric function F(1/2,1/2;1;z) because of its close relation to modular functions.
In this work is we prove model completeness for the expansion of the real field by the Weierstras... more In this work is we prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. In order to obtain this result we need to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasimodular form E2. We prove some auxiliary model completeness results with the same functions composed with appropriate change of variables. In the conclusion we make some remarks about the noneffectiveness of our proof and the difficulties to be overcome to obtain an effective model completeness result.
We treat the convergence of the Integrals of a sequence of truncations of a Henstock-Kurzweil Int... more We treat the convergence of the Integrals of a sequence of truncations of a Henstock-Kurzweil Integrable Function and show some strange phenomena.
We prove the model completeness of expansions of the reals by restricted elliptic and abelian fun... more We prove the model completeness of expansions of the reals by restricted elliptic and abelian functions. We make use of an auxiliary structure admitting quantifier elimination, where the basic relations are strongly definable in the original structure.
We prove the existence of a bound to the number of components of an ∀-definable set in the reals,... more We prove the existence of a bound to the number of components of an ∀-definable set in the reals, using Pfaffian functions, and give some applications.
We prove that if {S_1,S'_1,...,S_n,S'_n} is a collection of distinct spheres in R^m with common e... more We prove that if {S_1,S'_1,...,S_n,S'_n} is a collection of distinct spheres in R^m with common exterior, and g_i are Möbius transformations such that for each i, S_i is the isometric sphere of g_i and S'_i is the isometric sphere of g_i^(-1) and such that maps points of contact of S_i , to points of contact of S'_i, then the group G generated by the g_i's is Kleinian.
We prove that a discrete ordered structure is not interpretable in a densely ordered o-minimal st... more We prove that a discrete ordered structure is not interpretable in a densely ordered o-minimal structure.
We present some recent results and problems concerning definable sets and functions in o-minimal ... more We present some recent results and problems concerning definable sets and functions in o-minimal expansions of the field of real numbers by analytic or smooth functions. We deal with geometry (cell decomposition), Schanuel's conjecture and complex analytic varieties.
We show that if β∈R is not in the field generated by α1,…,αn, then no restriction of the function... more We show that if β∈R is not in the field generated by α1,…,αn, then no restriction of the function x^β to an interval is definable in 〈R,+,−,⋅,0,1,<,x^α1,…,x^αn〉. We also prove that if the real and imaginary parts of a complex analytic function are definable in Rexp or in the expansion of by functions x^α, for irrational α, then they are already definable in View the MathML source. We conclude with some conjectures and open questions.
We present a two-dimensional nonabsolute gauge integral which satisfies several convergence theor... more We present a two-dimensional nonabsolute gauge integral which satisfies several convergence theorems and a general divergence theorem, and at the same time admits a change of variables formula valid up to affine transformations - thus applicable to piecewise linear surfaces. Our approach is based on a modification of the M1-integral presented in [6], using triangle-based partitions.
We present theories of bounded arithmetic and weak analysis whose provably total functions (with ... more We present theories of bounded arithmetic and weak analysis whose provably total functions (with appropriate graphs) are the polyspace computable functions. More precisely, inspired in Ferreira’s systems PTCA, Σb1-NIA and BTFA in the polytime framework, we propose analogue theories concerning polyspace computability. Since the techniques we employ in the characterization of PSPACE via formal systems (e.g. Herbrand’s theorem, cut-elimination theorem and the expansion of models) are similar to the ones involved in the polytime setting, we focus on what is specific of polyspace and explains the lift from PTIME to PSPACE.
We prove that if A is an infinite von Neumann algebra (i. e., the identity can be decomposed as a... more We prove that if A is an infinite von Neumann algebra (i. e., the identity can be decomposed as a sum of a sequence of pairwise disjoint projections, all equivalent to the identity) then the cyclic cohomology of A vanishes. We show that the method of the proof applies to certain algebras of infinite matrices.
We prove a strong form of model completenes for expansions of the field of real numbers by (the r... more We prove a strong form of model completenes for expansions of the field of real numbers by (the real and imaginary parts of) the modular function J, by the modular forms E4 and E6 and quasimodular form E2 defined in the usual fundamental domain, and the restricted sine function and the (unrestricted) exponential function. This is done using ideas of Peterzil and Starchenko's paper [12] on the uniform definability of ℘ function in Ran (and of the modular function J). In the conclusion we pose some open problems related to this work.
Czechoslovak Mathematical Journal, 1999
A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integr... more A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integral approach. This allows us to study the HL class of functions from [9]. The theory is recovered together with a few new results.
Boletim Da Sociedade Brasileira De Matematica, 1995
We construct geometrically finite free Kleinian groups acting onS 3 whose limit sets are wild Can... more We construct geometrically finite free Kleinian groups acting onS 3 whose limit sets are wild Cantor sets.
Annals of Pure and Applied Logic, 2001
Schanuel&#x27;s Conjecture is the statement: if x1,…,xn∈C are linearly independent over Q, th... more Schanuel&#x27;s Conjecture is the statement: if x1,…,xn∈C are linearly independent over Q, then the transcendence degree of Q(x1,…,xn,exp(x1),…,exp(xn)) over Q is at least n. Here we prove that this is true if instead we take infinitesimal elements from any ultrapower of C, and in fact from any nonarchimedean model of the theory of the expansion of the field of real numbers by restricted analytic functions.
Journal of Symbolic Logic, 1997
Acta Mathematicae Applicatae Sinica-english Series, 2002
The Kurzweil-Henstock integral formalism is applied to establish the existence of solutions to th... more The Kurzweil-Henstock integral formalism is applied to establish the existence of solutions to the linear integral equations of Volterra-type 1$$ x{\left( t \right)} + \;{}^{ * }{\int_{{\left[ {a,t} \right]}} {\alpha {\left( s \right)}x{\left( s \right)}ds = f{\left( t \right)}} },\;t \in {\left[ {a,b} \right]}, $$ where the functions are Banach-space valued. Special theorems on existence of solutions concerning the Lebesgue integral setting are obtained. These sharpen earlier results.
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Papers by Ricardo Bianconi
The aim of the South American Journal of Logic is to promote logic in all its aspects: philosophical, mathematical, computational, historical by publishing high quality peer-reviewed papers.
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