303,927 research outputs found
E. Blatter et C. Mc. Cann, The Bombay grasses, 1935
No full textE. Blatter et C. Mc. Cann, The Bombay grasses, 1935. In: Bulletin mensuel de la Société linnéenne de Lyon, 5ᵉ année, n°8, octobre 1936. p. 135
Extensions and Limits of the Specker-Blatter Theorem
No full textThe original Specker-Blatter Theorem (1983) was formulated for classes of structures of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set [n] is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker-Blatter Theorem does not hold for one quaternary relation (2003).
If the vocabulary allows a constant symbol c, there are n possible interpretations on [n] for c. We say that a constant c is hard-wired if c is always interpreted by the same element j ∈ [n]. In this paper we show:
(i) The Specker-Blatter Theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case.
(ii) The Specker-Blatter Theorem does not hold already for with one ternary relation definable in First Order Logic FOL. This was left open since 1983.
Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers B_{r,A}, restricted Stirling numbers of the second kind S_{r,A} or restricted Lah-numbers L_{r,A}. Here r is an non-negative integer and A is an ultimately periodic set of non-negative integers
The Specker–Blatter theorem does not hold for quaternary relations
No full textAbstractLet C be a class of relational structures. We denote by fC(n) the number of structures in C over the labeled set {0,…,n−1}. For any C definable in monadic second-order logic with unary and binary relation symbols only, E. Specker and C. Blatter showed that for every m∈N, the function fC satisfies a linear recurrence relation modulo m, and hence it is ultimately periodic modulo m. The case of ternary relation symbols, and more generally of arity k symbols for k⩾3, was left open.In this paper we show that for every m there is a class of structures Cm, which is definable even in first-order logic with one quaternary (arity four) relation symbol, such that fCm is not ultimately periodic modulo m. This shows that the Specker–Blatter Theorem does not hold for quaternary relations, leaving only the ternary case open
E. Blatter et C. Mc. Cann, The Bombay grasses, 1935
No full textE. Blatter et C. Mc. Cann, The Bombay grasses, 1935. In: Bulletin mensuel de la Société linnéenne de Lyon, 5ᵉ année, n°8, octobre 1936. p. 135
Paramagnetic nanographenes - derivatives of the Blatter radicals.
No full textCette thèse décrit la synthèse et la caractérisation de la structure électronique du 2-phényl-3H- [1,2,4] triazino [5,6,1-kl] phénoxazin-3-yle (radical Blatter planaire) et de ses dérivés par Ultra-Violet spectroscopie photoélectronique (UV-PES). La première partie décrit la synthèse et la structure électronique du radical Blatter et une série de dérivés substitués en C (10) de radicaux Blatter planaires contenant des substituants H, F, Cl, Br, CN, CF3 et OMe a été étudiée par spectroscopie UV photoélectronique en phase gazeuse. Les radicaux ont également été analysés par résonance paramagnétique électronique, UV vis et méthodes électrochimiques. L'interprétation des spectres de photoélectrons a été soutenue par des calculs quantiques utilisant la méthode DFT CAM-B3LYP / 6-311G (d, p). L'analyse comparative des énergies d'ionisation théorique et expérimentale est présentée. La deuxième partie du manuscrit traite de l'effet de l'expansion du système et de délocalisation de la délocalisation de spin sur les radicaux Blatter planaires contenant des anneaux de naphtalène, de phénanthrène et de pyrène. La structure électronique de ces radicaux a été étudiée par spectroscopie UV-photoélectronique et comparée aux résultats de calcul DFT obtenus au niveau de théorie CAM-B3LYP / 6-311G (d, p). La dernière partie présente quelques transformations de groupes fonctionnels pour les dérivés substitués en C (10) des radicaux Blatter planaires et leurs caractérisations spectroscopiques et électrochimiques, ainsi qu'une description de chimie physique étudiée par ordinateur.This thesis describes the synthesis and electronic structure characterization of 2-phenyl-3H-[1,2,4]triazino[5,6,1-kl]phenoxazin-3-yl (planar Blatter radical) and its derivatives by Ultra-Violet photoelectron spectroscopy (UV-PES). First part discribes synthesis and electronic structure of Blatter radical and a series of C(10)-substituted derivatives of planar Blatter radicals containing H, F, Cl, Br, CN, CF3 and OMe substituents was investigated by gas phase UV-photoelectron spectroscopy. The radicals were also analyzed by electron paramagnetic resonance, UV vis and electrochemical methods. The interpretation of photoelectron spectra was supported by quantum calculations using DFT CAM-B3LYP/6-311G(d,p) method. The comparative analysis of theoretical and the experimental ionization energies are presented. The second part of the manuscript deals with the effect of expansion of the -system and increased spin delocalization on the planar Blatter radicals containing naphthalene, phenanthrene and pyrene rings. The electronic structure of these radicals ware investigated by UV-photoelectron spectroscopy and compared to DFT computational results obtained at the CAM-B3LYP/6-311G(d,p) level of theory. The final part presentes some functional group transformations for C(10)-substituted derivatives of planar Blatter radicals and their spectroscopic and electrochemical characterizations, as well as computationally studied physical chemistry description
The Specker-Blatter Theorem Revisited: Generating Functions for Definable Classes of Structures
No full textIn this paper we study the generating function of classes of graphs and hypergraphs. For a class of labeled graphs C we denote by fC(n) the number ofstructures of size n. For C de nable in Monadic Second Order Logic with unary and binary relation symbols only, E.Specker and C. Blatter showed in 1981 that for every m 2 N, fC(n) satisfies a linear recurrence relation fC(n) = Pdm j=1 a (m) j fC(n; j)� over Zm, and hence is ultimately periodic for each m. To show this they introduced what we call the Specker-index of C and rst showed the theorem to hold for any C of nite Specker-index, and then showed that every C definable in Monadic Second Order Logic is indeed of finite Specker-index. E. Fischer showed in 2002 that the Specker-Blatter Theorem does not hold for quaternary relations. In this paper we show how the Specker-Blatter Theorem is related to Schützenberger's Theorem and the Myhill-Nerode criterion for the characterization of regular languages, and discuss in detail how the behavior of this generating function depends on the choice of constant and relation symbols allowed in the definition of C. Among the main results we havethe following: -- We consider n-ary relations of degree at most d, where each element a is related to at most d other elements by any of the relations. We show that the Specker-Blatter Theorem holds for those, irrespective ofthe arity of the relations involved. -- Every C de nable in Monadic Second Order Logic with (modular) Counting (CMSOL) is of finite Specker-index. This covers many new cases, for which such a recurrence relation was not known before. -- There are continuum many C of finite Specker-index. Hence, contrary to the Myhill-Nerode characterization of regular languages, the recognizable classes of graphs cannot be characterized by the finiteness of the Specker-index
Blatter Radicals as Bipolar Materials for Symmetric Redox-Flow Batteries
No full textRedox-active organic molecules are promising charge-storage materials for redox-flow batteries (RFBs), but material crossover between posolyte/negolyte and chemical degradation are limiting factors in the performance of all-organic RFBs. We demonstrate that the bipolar electrochemistry of 1,2,4-benzotriazin-4-yl (Blatter) radicals allows construction of batteries with symmetric electrolyte composition. Cyclic voltammetry shows that these radicals retain reversible bipolar electrochemistry also in the presence of water. The redox potentials of derivatives with a C(3)-CF3 substituent are least affected by water and, moreover, these compounds show >90% capacity retention after charge/discharge cycling in a static H-cell for seven days (ca. 100 cycles). Testing these materials in a flow regime at 0.1 M concentration of active material confirmed the high cycling stability under conditions relevant for RFB operation, and demonstrated that polarity inversion in a symmetric flow battery may be used to rebalance the cell. Chemical synthesis provides insight in the nature of the charged species by spectroscopy and (for the oxidized state) X-ray crystallography. The stability of these compounds in all three states of charge highlights the potential for application in symmetric organic redox-flow batteries
Unique continuous selections for metric projections of C(X) onto finite-dimensional vector subspaces, II
No full textAbstractBest approximation in C(X) by elements of a Chebyshev subspace is governed by Haar's theorem, the de la Vallée Poussin estimates, the alternation theorem, the Remez algorithm, and Mairhuber's theorem. J. Blatter (1990, J. Approx. Theory 61, 194–221) considered best approximation in C(X) by elements of a subspace whose metric projection has a unique continuous selection and extended Haar's theorem and Mairhuber's theorem to this situation. In the present paper we so extend the de la Vallée Poussin estimates, the alternation theorem, and the Remez algorithm
New Blatter-type radicals from a bench-stable carbene
No full textStable benzotriazinyl radicals (Blatter's radicals) recently attracted considerable interest as building blocks for functional materials. The existing strategies to derivatize Blatter's radicals are limited, however, and synthetic routes are complex. Here, we report that an inexpensive, commercially available, analytical reagent Nitron undergoes a previously unrecognized transformation in wet acetonitrile in the presence of air to yield a new Blatter-type radical with an amide group replacing a phenyl at the C(3)-position. This one-pot reaction of Nitron provides access to a range of previously inaccessible triazinyl radicals with excellent benchtop stabilities. Mechanistic investigation suggests that the reaction starts with a hydrolytic cleavage of the triazole ring followed by oxidative cyclization. Several derivatives of Nitron were prepared and converted into Blatter-type radicals to test the synthetic value of the new reaction. These results significantly expand the scope of using functionalized benzotriazinyls as stable radical building blocks
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