381 research outputs found
Logic may be simple. Logic, congruence and algebra
This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The notion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacement theorem and algebraization of logic are studied (including MacLane-Curry’s theorem and a discussion about Curry’s algebras). We also examine how these concepts are related to such notions as semantics, truth-functionality and bivalence. We argue that a logic, which is simple, can deserve the name logic and that the opposite view is connected with a reductionist perspective (reduction of logic to algebra)
On polynomially integrable Birkhoff billiards on surfaces of constant curvature
We present a solution of the algebraic version of Birkhoff Conjecture on
integrable billiards. Namely we show that every polynomially integrable real
bounded convex planar billiard with smooth boundary is an ellipse. We extend
this result to billiards with piecewise-smooth and not necessarily convex
boundary on arbitrary two-dimensional surface of constant curvature: plane,
sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane
or a (pseudo-) sphere in equipped with appropriate quadratic
form. Namely, we show that a billiard is polynomially integrable, if and only
if its boundary is a union of confocal conical arcs and appropriate geodesic
segments. We also present a complexification of these results. These are joint
results of Mikhail Bialy, Andrey Mironov and the author. The proof is split
into two parts. The first part is given by Bialy and Mironov in their two joint
papers. They considered the tautological projection of the boundary to
and studied its orthogonal-polar dual curve, which is piecewise
algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's
theorem, it suffices to show that each non-linear complex irreducible component
of the dual curve is a conic. They have proved that all its singularities and
inflection points (if any) lie in the projectivized zero locus of the
corresponding quadratic form on . The present paper provides the
second part of the proof: we show that each above irreducible component is a
conic and finish the solution of the Algebraic Birkhoff Conjecture in constant
curvature.Comment: To appear in the Journal of the European Mathematical Society (JEMS),
69 pages, 2 figures. A shorter proof of Theorem 4.24. Minor precisions and
misprint correction
An analysis of the logic of Riesz Spaces with strong unit
We study \L ukasiewicz logic enriched with a scalar multiplication with
scalars taken in . Its algebraic models, called {\em Riesz MV-algebras},
are, up to isomorphism, unit intervals of Riesz spaces with a strong unit
endowed with an appropriate structure. When only rational scalars are
considered, one gets the class of {\em DMV-algebras} and a corresponding
logical system. Our research follows two objectives. The first one is to deepen
the connections between functional analysis and the logic of Riesz MV-algebras.
The second one is to study the finitely presented MV-algebras, DMV-algebras and
Riesz MV-algebras, connecting them from logical, algebraic and geometric
perspective
Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations
AbstractThis paper generalizes many-sorted algebra (MSA) to order-sorted algebra (OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including quotient, homomorphism, and initiality theorems. The paper's major mathematical results include a notion of OSA deduction, a completeness theorem for it, and an OSA Birkhoff variety theorem. We also develop conditional OSA, including initiality, completeness, and McKinsey-Malcev quasivariety theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like stack and list, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions
Lower Bounds by Birkhoff Interpolation
International audienceIn this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such a representation must be at least of order d. This is clearly optimal up to a constant factor. Previous lower bounds for this problem were only of order Ω(√ d), and were obtained from arguments based on Wronskian determinants and "shifted derivatives." We obtain this improvement thanks to a new lower bound method based on Birkhoff interpolation (also known as "lacunary polynomial interpolation")
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