Cubic form Theorem for Affine Immersions

1988 ◽  
Vol 13 (3-4) ◽  
pp. 338-362 ◽  
Author(s):  
Katsumi Nomizu ◽  
Ulrich Pinkall
Keyword(s):  
Cubic Form ◽  
Form Theorem ◽  
Affine Immersions

Structures of chemically stabilized ceramics

1993 ◽  
Vol 51 ◽  
pp. 924-925
Author(s):  
Pratibha L. Gai ◽  
M. A. Saltzberg ◽  
L.G. Hanna ◽  
S.C. Winchester
Keyword(s):  
High Temperature ◽  
Calcium Nitrate ◽  
Nitrate Hexahydrate ◽  
Chemical Route ◽  
Cubic Form ◽  
Tetragonal Form ◽  
Wet Chemical ◽  
Wet Chemical Route ◽  
Final Composition ◽  
Aluminium Nitrate Nonahydrate

Silica based ceramics are some of the most fundamental in crystal chemistry. The cristobalite form of silica has two modifications, α (low temperature, tetragonal form) and β (high temperature, cubic form). This paper describes our structural studies of unusual chemically stabilized cristobalite (CSC) material, a room temperature silica-based ceramic containing small amounts of dopants, prepared by a wet chemical route. It displays many of the structural charatcteristics of the high temperature β-cristobalite (∼270°C), but does not undergo phase inversion to α-cristobalite upon cooling. The Structure of α-cristobalite is well established, but that of β is not yet fully understood.Compositions with varying Ca/Al ratio and substitutions in cristobalite were prepared in the series, CaO:Al2O3:SiO2 : 3-x: x : 40, with x= 0-3. For CSC, a clear sol was prepared from Du Pont colloidal silica, Ludox AS-40®, aluminium nitrate nonahydrate, and calcium nitrate hexahydrate in proportions to form a final composition 1:2:40 composition.

scholarly journals The large-N limit of the 4d $$ \mathcal{N} $$ = 1 superconformal index

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Alejandro Cabo-Bizet ◽  
Davide Cassani ◽  
Dario Martelli ◽  
Sameer Murthy
Keyword(s):  
Effective Action ◽  
Black Hole Solution ◽  
Saddle Points ◽  
Unitary Matrix ◽  
Finite Abelian Groups ◽  
Superconformal Index ◽  
Cubic Form ◽  
Chemical Potentials ◽  
Large N ◽  
Superconformal Theories

Abstract We systematically analyze the large-N limit of the superconformal index of $$ \mathcal{N} $$ N = 1 superconformal theories having a quiver description. The index of these theories is known in terms of unitary matrix integrals, which we calculate using the recently-developed technique of elliptic extension. This technique allows us to easily evaluate the integral as a sum over saddle points of an effective action in the limit where the rank of the gauge group is infinite. For a generic quiver theory under consideration, we find a special family of saddles whose effective action takes a universal form controlled by the anomaly coefficients of the theory. This family includes the known supersymmetric black hole solution in the holographically dual AdS5 theories. We then analyze the index refined by turning on flavor chemical potentials. We show that, for a certain range of chemical potentials, the effective action again takes a universal cubic form that is controlled by the anomaly coefficients of the theory. Finally, we present a large class of solutions to the saddle-point equations which are labelled by group homomorphisms of finite abelian groups of order N into the torus.

Powder Diffraction Data of CdTexSe1-xSolid Solutions

1990 ◽  
Vol 5 (4) ◽  
pp. 206-209 ◽  
Author(s):  
N.A. Razik ◽  
G. Al-Barakati ◽  
S. Al-Heneti
Keyword(s):  
Powder Diffraction ◽  
Diffraction Data ◽  
Hexagonal Wurtzite Structure ◽  
Powder Diffraction Data ◽  
Lattice Constants ◽  
Cubic Form ◽  
Hexagonal Form ◽  
State Diffusion ◽  
Vegard’S Law ◽  
Image Position

AbstractCdTexSe1-xsolid solutions with (x) ranging between zero and one were prepared by solid state diffusion under vacuum and their precise lattice constants and X-ray powder diffraction data were determined. It was found that alloys with 0≤x≤0.4 possess the hexagonal wurtzite structure while those with 0.5≤x≤1.0 have the cubic zincblende structure. The lattice parameters obeyed Vegard's law according to the following formulaeExtrapolated lattice constants were a = 6.066(6) Å for the cubic form of CdSe and a = 4.563(6) Å and c = 7.502 (10) Å for the hexagonal form of CdTe.

The Minimum of a Binary Cubic Form (II)

1943 ◽  
Vol s1-18 (4) ◽  
pp. 210-217 ◽  
Author(s):  
L. J. Mordell
Keyword(s):  
Cubic Form

A normal form theorem for Lω1p, with applications

1982 ◽  
Vol 47 (3) ◽  
pp. 605-624 ◽  
Author(s):  
Douglas N. Hoover
Keyword(s):  
Normal Form ◽  
Probability Model ◽  
Random Variables ◽  
Interpolation Theorem ◽  
Complete Theory ◽  
De Finetti’S Theorem ◽  
Form Theorem ◽  
Normal Form Theorem ◽  
De Finetti's Theorem

AbstractWe show that every formula of Lω1P is equivalent to one which is a propositional combination of formulas with only one quantifier. It follows that the complete theory of a probability model is determined by the distribution of a family of random variables induced by the model. We characterize the class of distribution which can arise in such a way. We use these results together with a form of de Finetti’s theorem to prove an almost sure interpolation theorem for Lω1P.

An Experimental and Theoretical Investigation of Nonlinear Vibrations of Piezoelectrically-Driven Microcantilevers

2006 ◽  
Author(s):  
S. Nima Mahmoodi ◽  
Nader Jalili
Keyword(s):  
Natural Frequency ◽  
Piezoelectric Layer ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Closed Form Solution ◽  
Cubic Form ◽  
Dynamic Representation ◽  
Microcantilever Beam

The nonlinear vibrations of a piezoelectrically-driven microcantilever beam are experimentally and theoretically investigated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Practically, the first resonance of the beam is of interest, and hence, the microcantilever beam is modeled to obtain the natural frequency theoretically. The bending vibrations of the beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials. The nonlinear term appears in the form of quadratic due to presence of piezoelectric layer, and cubic form due to geometry of the beam (mainly due to the beam's inextensibility). Galerkin approximation is utilized to discretize the equations of motion. The obtained equation is simulated to find the natural frequency of the system. In addition, method of multiple scales is applied to the equations of motion to arrive at the closed-form solution for natural frequency of the system. The experimental results verify the theoretical findings very closely. It is, therefore, concluded that the nonlinear approach could provide better dynamic representation of the microcantilever than previous linear models.

Refinements of Vaught's normal from theorem

1979 ◽  
Vol 44 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Normal Form ◽  
Boolean Combination ◽  
Form Theorem ◽  
Central Notion ◽  
The Matrix ◽  
Normal Form Theorem ◽  
Skolem Theorem ◽  
Image Position ◽  
Countable Models

The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the formwhere the indices ki, ji range over given countable sets and the matrix conjuncts are, say, open -formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]—[15] and Vaught [20]. Other references are [1], [3]—[5], [10]—[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).Theorem 0.1. Let L = L0(R). For every -sentence ϕ there is an L0-game sentence Θ such that ⊨′ ∃Rϕ ↔ Θ.(A word about the notations: L0(R) denotes the language obtained from L0 by adding to it the sequence R of logical symbols which do not belong to L0; “⊨′α” means that α is true in all countable models.)0.1 can be restated as follows.Theorem 0.1′. For every-sentence ϕ there is an L0-game sentence Θ such that ⊨ϕ → Θ and for any-sentence ϕ if ⊨ϕ → ϕ and L′ ⋂ L ⊆ L0, then ⊨ Θ → ϕ.(We sketch the proof of the equivalence between 0.1 and 0.1′.0.1 implies 0.1′. This is obvious once we realize that game sentences and their negations satisfy the downward Löwenheim-Skolem theorem and hence, ⊨′α is equivalent to ⊨α whenever α is a boolean combination of and game sentences.

Normalization theorems for full first order classical natural deduction

1991 ◽  
Vol 56 (1) ◽  
pp. 129-149 ◽  
Author(s):  
Gunnar Stålmarck
Keyword(s):  
Normal Form ◽  
Natural Deduction ◽  
Full System ◽  
Strong Normalization ◽  
Logical Constants ◽  
Restricted Version ◽  
First Order ◽  
Form Theorem ◽  
Normalization Theorem ◽  
Normal Form Theorem

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.

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