An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

Lecture Notes of Tensor Network Contractions

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.Comment: 134 pages, 68 figures. In this version, the manuscript has been changed into the format of book; new sections about tensor network and quantum circuits have been adde

Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective

The theory of intrinsic volumes of convex cones has recently found striking applications in areas such as convex optimization and compressive sensing. This article provides a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Direct derivations of the General Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications are presented.Comment: Survey, 23 page

Exegesis of Sect. III.B from “Fundamentals of the Mechanics of Continua” by E. Hellinger

This is our third and last exegetic essay on the fundamental review article DIE ALLGEMEINEN ANSÄTZE DER MECHANIK DER KONTINUA in the Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Bd. IV-4, Hft. 5 (1913) by Ernst Hellinger which contains the translation and the commentary of the remaining text starting from p. 663. The six subsections, No. 9–15, deal with the applications of the previously developed conceptual tools to formulate: an effective theory of elasticity, the dynamics of ideal fluids, models for internal friction and elastic hysteresis, a theory of capillarity, optics, the fundamental equations of electrodynamics, an introduction of the thermodynamical foundations and the relationship between the theory of continua and the theory of relativity. Hellinger refers to relevant literature while consolidating in an effective way the contemporary knowledge in 1913. Considering notational differences as being irrelevant for the characterization of the presented scientific content, Hellinger's article shows that an effective compendium of a large part of the insights given in Truesdell and Toupin and Truesdell and Noll has already been available in 1913. We include in this paper an assessment of the different roles played by pioneers, who are innovating their scientific discipline, and by erudite scholars whose role consists in re-ordering existent knowledge and advertising to a wider audience the most important technical results already obtained in a given discipline

Fabricating ultrasensitive metal nano-structures with Langmuir-Blodgett technique to improve plasmonic response of SERS

[eng] Nanoparticle self-assembly is a versatile and coherent strategy for the development of functional nanostructured materials, offering low-cost and scalable methods that can be fine-tuned for many different specific application. Functionalized nanoparticles could be spread at the interface of liquid/gas by means of self-assembly. In this work, we demonstrate a pathway for the fabrication of tailorable quasi two-dimensional lattices of gold nanoparticles with several core sizes and shapes (5, 10, 20, 30, 40 nm and nano-urchin) to be used in surface enhanced Raman scattering (SERS) detection of biomolecules. Upon spreading gold nanoparticles at the water/air interface in this research, we used the Langmuir-Blodgett technique as a way of making supra-molecular and nano-structure assembly in ultrathin films with a controlled layered and spatial structure, which have many envisioned technological applications for several branches of science as well as to develop SERS substrates. Monolayers of gold particles were transferred at a target lateral density using the Langmuir–Blodgett technique. Once gold nanoparticles were firmly adhered to the substrate, we used electroless plating to let the nanoparticle grow, thus tuning the plasmonic response and leading to SERS enhancement. Compared to direct deposition, chemical deposition or lithographic methods, our protocol enables to obtain consistent results and much higher coverages of Au nanoparticles thanks to the active control of the surface pressure of the spread monolayers. Prepared substrates were analyzed with different techniques such as UV/VIS spectroscopy, SEM and TEM microscopy. We have demonstrated that, for a given particle size, the enhancement for SERS detection of a referent analyte, 4-MBA, can be tuned by controlling the packing density of the nanoparticles at the water/air interface by adjusting the surface pressure using the Langmuir film balance setup. The other factor which affects the SERS signals is the thickness of a subsequent gold layer, deposited and tuned by using electroless plating. After finding the optimum conditions of surface pressure and electroplating time for 10 nm gold nanoparticles, we entered the second phase of the experiments to unveil the effect of gold nanoparticle size on our study. SERS data of different- sized nanoparticles did not prove that bigger particles result in better SERS signals. However, urchin-shaped gold nanoparticles have shown more intense signals in comparison with spherical nanoparticles. For the same conditions of preparation, we achieved better result by using urchin-shaped nanoparticles. In order to test our substrate efficiency to detect more substances, in the final step of this research, we investigated and performed tests on Thiram and Carbaryl as water-polluting molecules that are widely using as pesticide compounds. We performed several SERS measurements with different substrates and studied the effect of gold nanoparticles shape, contact time between substrate and pollutant solution, substrate functionalization with thiol groups, and effect of pollutant solution concentration on SERS signals. Finally, we report the limit of pollutant detection with our prepared substrates.[cat] El procés d’autoassemblatge de nanopartícules és una estratègia versàtil i coherent per al desenvolupament de materials nanoestructurats funcionals, que ofereix mètodes de baix cost i escalables que es poden ajustar per a diferents aplicacions específiques. Les nanopartícules funcionalitzades es podrien estendre a la interfície de líquid / gas mitjançant un fenomen d’autoassemblatge. En aquest treball, demostrem una via per a la fabricació de xarxes quasi bidimensionals adaptables de nanopartícules d’or amb diverses mides i formes de nucli (5, 10, 20, 30, 40 nm i nano-eriçó) que s’utilitzaran per a la detecció de biomolècules en dispersió Raman de superfície millorada (SERS). En estendre nanopartícules d’or a la interfície aigua / aire en aquesta investigació, hem utilitzat la tècnica Langmuir-Blodgett com una manera de fer un assemblatge de nanoestructures en pel·lícules ultrafines amb una estructura espacial i de capes controlades, que tenen moltes aplicacions tecnològiques potencials en diverses branques de la ciència, com ara substrats SERS. Les monocapes de partícules d'or es transfereixen, a una densitat lateral determinada, sobre substrats de vidre o sílice mitjançant la tècnica LB. Una vegada que les nanopartícules d'or s’han adherit fermament al substrat, vam utilitzar un mètode d’electrodeposició d’or sense electròlisi per fer créixer le nanopartícules, ajustant així la resposta plasmònica i la millora del SERS. En comparació amb la deposició directa, la deposició química o els mètodes litogràfics, el nostre protocol permet obtenir resultats consistents i una cobertura molt més gran de nanopartícules d’or gràcies al control actiu de la pressió superficial de la monocapa estesa sobre la interfície aigua/aire. Els substrats preparats es van analitzar amb diferents tècniques com l'espectroscòpia UV / VIS, microscòpia SEM i TEM. Hem demostrat que, per a una mida de partícula determinada, la millora per a la detecció SERS d’un analit referent, 4-MBA, es pot ajustar controlant la densitat d’empaquetament de les nanopartícules a la interfície aigua / aire ajustant la pressió superficial mitjançant la balança de pel·lícules de Langmuir. L’altre factor que afecta els senyals SERS és el gruix de la capa d’or dipositada posteriorment mitjançant una tècnica no-electrolítica. Després de trobar els paràmetres òptims de pressió superficial i temps d’electrodeposició per a nanopartícules d’or de 10 nm, vam entrar a la segona fase de l’estudi per descobrir l’efecte de la mida de les nanopartícules d’or en el nostre sistema. Tot i que les dades SERS de nanopartícules de diferent mida no van indicar que les partícules més grans donessin millors senyals SERS, les nanopartícules d'or en forma d’eriçó han mostrat senyals més intensos en comparació amb les nanopartícules esfèriques. En les mateixes condicions de preparació, vam obtenir un millor resultat mitjançant l’ús de nanopartícules de forma d’eriçó. Per tal de provar l’eficiència del nostre substrat per detectar més substàncies, en el darrer pas d’aquesta investigació vam investigar i realitzar proves emprant Thiram i Carbaryl com a molècules contaminants de l’aigua que s’utilitzen àmpliament com a compostos pesticides. Hem realitzat diverses mesures de SERS amb substrats diferents i hem estudiat l’efecte de a) la forma de les nanopartícules d’or, b) el temps de contacte entre el substrat i la solució contaminant, c) la funcionalització del substrat amb grups tiol i d) l’efecte de la concentració de solució contaminant en els senyals SERS. Finalment, informem del límit de detecció d’aquests contaminants amb els nostres substrats nanoestructurats

Fuzzy Mathematics

This book provides a timely overview of topics in fuzzy mathematics. It lays the foundation for further research and applications in a broad range of areas. It contains break-through analysis on how results from the many variations and extensions of fuzzy set theory can be obtained from known results of traditional fuzzy set theory. The book contains not only theoretical results, but a wide range of applications in areas such as decision analysis, optimal allocation in possibilistics and mixed models, pattern classification, credibility measures, algorithms for modeling uncertain data, and numerical methods for solving fuzzy linear systems. The book offers an excellent reference for advanced undergraduate and graduate students in applied and theoretical fuzzy mathematics. Researchers and referees in fuzzy set theory will find the book to be of extreme value