ICrA Over Ordered Pairs Applied to ABC Optimization Results

Author(s):  
Olympia Roeva ◽  
Dafina Zoteva
Keyword(s):  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).


Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 39
Author(s):  
Miroslav Hristov ◽  
Atanas Ilchev ◽  
Diana Nedelcheva ◽  
Boyan Zlatanov

We generalize the notion of coupled fixed (or best proximity) points for cyclic ordered pairs of maps to p-cyclic ordered pairs of maps. We find sufficient conditions for the existence and uniqueness of the coupled fixed (or best proximity) points. We illustrate the results with an example that covers a wide class of maps.


Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 44
Author(s):  
Gana Gecheva ◽  
Miroslav Hristov ◽  
Diana Nedelcheva ◽  
Margarita Ruseva ◽  
Boyan Zlatanov

We have obtained a new class of ordered pairs of multivalued maps that have pairs of coupled fixed points. We illustrate the main result with two examples that cover a wide range of models. We apply the main result in models in duopoly markets to get a market equilibrium and in aquatic ecosystems, also to get an equilibrium.


1999 ◽  
Vol 9 (5) ◽  
pp. 545-567 ◽  
Author(s):  
LAWRENCE C. PAULSON

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).


1989 ◽  
Vol 67 (12) ◽  
pp. 2178-2187 ◽  
Author(s):  
Peter S. Martin ◽  
Keith Yates ◽  
Imre G. Csizmadia

RHF SCF 3-21G calculations are reported for the 1Σg+, 1A1, 3A′, and 1,3A″ states of simple substituted acetylenes (Y—C≡C—H, where Y = H, F, Cl,CH3, andCF3), the 1A1, 1A′, and 1.3A″ states of their Markovnikov (M) vinyl cations (Y—C+ = CH2), the 1A′ and 1.3A″ states of their anti-Markovnikov (aM) vinyl cations (YCH=C+H), and the corresponding hydrated vinyl cations. Equilibrium electronic structures and the mechanism of adiabatic protonation are described qualitatively via Lewis/resonance schematic representations of the species involved. Calculated proton affinities (PA) suggest that relative to ground state Y—C≡C—H (1Σ+/1A1), Y—C≡C—H* (1.3A″) is of greatly enhanced basicity with respect to protonation of both regiocenters. A graphical representation of the ordered pairs PA(M) versus PA(aM) as a function of substituent Y and electronic state, leads to the conclusions: (1) irrespective of both regiocenter (M/aM) and state (1Σ1+/1A1, 3A, 1.3A″) the PA's for Y—C≡C—H decrease in the order CH3 > H > Cl> F > CF3; (2) in proceeding from CH3C≡CH to CF3C≡CH, a change in protonation regiospecificity (M → aM) is experienced to approximately the same extent for both S0 and S1; (3) T2 exhibits no significant protonation regioselectivity. Critical comparison of the calculated results is made with available experimental data. An approximate picture of the energy profiles for the adiabatic hydrations of Y—C≡C—H via its ground, triplet and singlet states has been developed, based on the fixed points acetylene, vinyl cation and hydrated vinyl cation. Predicted relative reactivities of these three states are in excellent accord with available experimental data on rates of hydration. Keywords: excited states, proton transfer, photohydration.


2018 ◽  
Vol 72 (4) ◽  
pp. 428-434
Author(s):  
Francesca Busetto ◽  
Giulio Codognato ◽  
Simone Tonin

Author(s):  
Jun-Chul Bae ◽  
Jonathan Wickert

Abstract The free vibration of disk-hat structures, such as automotive brake rotors, is investigated analytically and through laboratory experimentation. Of particular interest are the role played by the hat element’s depth in influencing the three-dimensional vibration of the disk, and the manner in which the bending and in-plane modes of the disk alone evolve as a hat of increasing depth is incorporated in the model. The lower vibration modes of disk-hat structures are shown to be characterized by the numbers of nodal circles NC and diameters ND present on the disk, as well as the phase relationship between the disk’s transverse and radial displacements due to coupling with the hat element. Such modes map continuously back to the pure bending and in-plane modes of the disk alone, appear in ordered pairs, and can exist at close frequencies. Those characteristics are explored particularly with respect to sensitivities in the disk’s thickness and the hat’s depth with a view towards shifting particular natural frequencies, or minimizing transverse disk motion in certain vibration modes. Results obtained through analysis and measurement of a prototypical disk-hat structure are applied in a case study with a ventilated automotive brake rotor.


1961 ◽  
Vol 13 ◽  
pp. 217-220 ◽  
Author(s):  
C. Y. Lee

We will consider the following enumeration problem. Let A and B be finite sets with α and β elements in each set respectively. Let n be some positive integer such that n ≦ αβ. A subset S of the product set A × B of exactly n distinct ordered pairs (ai, bj) is said to be admissible if given any a ∈ A and b ∈ B, there exist elements (ai, bj) and (ak, bl) (they may be the same) in S such that ai = a and bl = b. We shall find here a generating function for the number N(α, β n) of distinct admissible subsets of A × B and from this generating function, an explicit expression for N(α, β n). In obtaining this result, the idea of a cut probability is used. This approach in a problem of enumeration may be of interest.


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