Profinite Structures are Retracts of Ultraproducts of Finite Structures

We establish the following model-theoretic characterization: profinite $L$-structures, the cofiltered limits of finite $L$-structures,are retracts of ultraproducts of finite $L$-structures. As a consequence, any elementary class of $L$-structures axiomatized by $L$-sentences of the form \forall \vec{x} (\psi_{0}(\vec{x}) \ra \psi_{1}(\vec{x})), where $\psi_{0}(\vec{x}),\psi_{1}(\vec{x})$ are existencial-positives $L$-formulas, is closed under the formation of profinite objects in the category {\bf L-mod}, the category of structures suitable for the language $L$ and $L$-homomorphisms

Some consequences of the Firoozbakht's conjecture

In this paper we present the statement of the Firoozbakht's conjecture, some of its consequences if it is proved and we show a consequence of Zhang's theorem concerning the Firoozbakht's conjecture

A Universal Algebraic Survey of C∞−\mathcal{C}^{\infty}-Rings

In this paper we present some basic results of the Universal Algebra of $\mathcal{C}^\infty$-rings which were nowhere to be found in the current literature. The outstanding book of I. Moerdijk and G. Reyes,[24], presents the basic (and advanced) facts about $\mathcal{C}^\infty$-rings, however such a presentation has no universal algebraic "flavour". We have been inspired to describe $\mathcal{C}^\infty$-rings through this viewpoint by D. Joyce in [15]. Our main goal here is to provide a comprehensive material with detailed proofs of many known "taken for granted" results and constructions used in the literature about $\mathcal{C}^\infty$-rings and their applications - such proofs either could not be found or were merely sketched. We present, in detail, the main constructions one can perform within this category, such as limits, products, homomorphic images, quotients, directed colimits, free objects and others, providing a "propaedeutic exposition" for the reader's benefit

Representation theory of logics: a categorial approach

The major concern in the study of categories of logics is to describe condition for preservation, under the a method of combination of logics, of meta-logical properties. Our complementary approach to this field is study the "global" aspects of categories of logics in the vein of the categories \Ss, \Ls, \cA_s studied in \cite{AFLM3}. All these categories have good properties however the category of logics \cL does not allow a good treatment of the "identity problem" for logics (\cite{Bez}): for instance, the presentations of "classical logics" (e.g., in the signature $\{\neg, \vee\}$ and $\{\neg',\rightarrow'\}$) are not \Ls-isomorphic. In this work, we sketch a possible way to overcome this "defect" (and anothers) by a mathematical device: a representation theory of logics obtained from category theoretic aspects on (Blok-Pigozzi) algebraizable logics. In this setting we propose the study of (left and right) "Morita equivalence" of logics and variants. We introduce the concepts of logics (left/right)-(stably) -Morita-equivalent and show that the presentations of classical logics are stably Morita equivalent but classical logics and intuitionist logics are not stably-Morita-equivalent: they are only stably-Morita-adjointly related.Comment: 10 page

Von Neumann Regular C∞−\mathcal{C}^{\infty}-Rings and Applications

In this paper we present the notion of a von Neumann regular $\mathcal{C}^{\infty}-$ring, we prove some results about them and we describe some of their properties. We prove, using two different methods, that the category of von Neumann regular $\mathcal{C}^{\infty}-$rings is a reflective subcategory of $\mathcal{C}^{\infty}{\rm \bf Rng}$. We prove that every homomorphism between Boolean algebras can be represented by a $\mathcal{C}^{\infty}-$ring homomorphism between von Neumann regular $\mathcal{C}^{\infty}-$rings.Comment: 72 page

Topics on Smooth Commutative Algebra

We present, in the same vein as in [20] and [21], some results of the so-called "Smooth (or $\mathcal{C}^\infty$) Commutative Algebra", a version of Commutative Algebra of $\mathcal{C}^{\infty}-$rings instead of ordinary commutative unital rings, looking for similar results to those one finds in the latter, and expanding some others presented in [20]. We give an explicit description of an adjunction between the categories $\mathcal{C}^\infty{\rm Rng}$ and ${\rm CRing}$, in order to study this "bridge". We present and prove many properties of the analog of the radical of an ideal of a ring (namely, the $\mathcal{C}^\infty$-radical of an ideal), saturation (which we define as "smooth saturation", inspired by [13]), rings of fractions ($\mathcal{C}^\infty$-rings of fractions, defined first by I. Moerdijk and G. Reyes in [20]), local rings (local $\mathcal{C}^\infty$-rings), reduced rings ($\mathcal{C}^\infty$-reduced $\mathcal{C}^\infty$-rings) and others. We also state and prove new results, such as ad hoc "Separation Theorems", similar to the ones we find in Commutative Algebra, and a stronger version (Theorem 6) of the Theorem 1.4 of [20], characterizing every $\mathcal{C}^\infty$-ring of fractions. We describe the fundamental concepts of Order Theory for $\mathcal{C}^\infty$-rings, proving that every $\mathcal{C}^\infty$-ring is semi-real, and we prove an important result on the strong interplay between the smooth Zariski spectrum and the real smooth spectrum of a $\mathcal{C}^\infty$-ring

Towards a good notion of categories of logics

We consider (finitary, propositional) logics through the original use of Category Theory: the study of the "sociology of mathematical objects", aligning us with a recent, and growing, trend of study logics through its relations with other logics (e.g. process of combinations of logics as bring [Gab] and possible translation semantics [Car]). So will be objects of study the classes of logics, i.e. categories whose objects are logical systems (i.e., a signature with a Tarskian consequence relation) and the morphisms are related to (some concept of) translations between these systems. The present work provides the first steps of a project of considering categories of logical systems satisfying simultaneously certain natural requirements: it seems that in the literature ([AFLM1], [AFLM2], [AFLM3], [BC], [BCC1], [BCC2], [CG], [FC]) this is achieved only partially.Comment: 16 page

A global approach to AECs

In this work we present some general categorial ideas on Abstract Elementary Classes (AECs) %\cite{She}, inspired by the totality of AECs of the form $(Mod(T), \preceq)$, for a first-order theory T: (i) we define a natural notion of (funtorial) morphism between AECs; (ii) explore the following constructions of AECs: "generalized" theories, pullbacks of AECs, (Galois) types as AECs; (iii) apply categorial and topological ideas to encode model-theoretic notions on spaces of types %(see Michael Lieberman Phd thesis) ; (iv) present the "local" axiom for AECs here called "local Robinson's property" and an application (Robinson's diagram method); (v) introduce the category $AEC$ of Grothendieck's gluings of all AECs (with change of basis); (vi) introduce the "global" axioms of "tranversal Robinson's property" (TRP) and "global Robinson's property" (GRP) and prove that TRP is equivalent to GRP and GRP entails a natural version of Craig interpolation property.Comment: 10 page

Bayesian Restoration of Audio Degraded by Low-Frequency Pulses Modeled via Gaussian Process

A common defect found when reproducing old vinyl and gramophone recordings with mechanical devices are the long pulses with significant low-frequency content caused by the interaction of the arm-needle system with deep scratches or even breakages on the media surface. Previous approaches to their suppression on digital counterparts of the recordings depend on a prior estimation of the pulse location, usually performed via heuristic methods. This paper proposes a novel Bayesian approach capable of jointly estimating the pulse location; interpolating the almost annihilated signal underlying the strong discontinuity that initiates the pulse; and also estimating the long pulse tail by a simple Gaussian Process, allowing its suppression from the corrupted signal. The posterior distribution for the model parameters as well for the pulse is explored via Markov-Chain Monte Carlo (MCMC) algorithms. Controlled experiments indicate that the proposed method, while requiring significantly less user intervention, achieves perceptual results similar to those of previous approaches and performs well when dealing with naturally degraded signals.Comment: 14 pages, 7 figures, 4 tables. Submitted to IEEE Journal of Selected Topics in Signal Processing - Special Issue "Reconstruction of audio from incomplete or highly degraded observations

Variable Neighborhood Search for the Bin Packing Problem with Compatible Categories

Bin Packing with Conflicts (BPC) are problems in which items with compatibility constraints must be packed in the least number of bins, not exceeding the capacity of the bins and ensuring that non-conflicting items are packed in each bin. In this work, we introduce the Bin Packing Problem with Compatible Categories (BPCC), a variant of the BPC in which items belong to conflicting or compatible categories, in opposition to the item-by-item incompatibility found in previous literature. It is a common problem in the context of last mile distribution to nanostores located in densely populated areas. To efficiently solve real-life sized instances of the problem, we propose a Variable Neighborhood Search (VNS) metaheuristic algorithm. Computational experiments suggest that the algorithm yields good solutions in very short times while compared to linear integer programming running on a high-performance computing environment.Comment: 2018 SCALE Latin American Conference, Boston, USA, April 15th-16th, 201