Pure three-valued Łukasiewiczian implication

1966 ◽  
Vol 31 (3) ◽  
pp. 399-405 ◽  
Author(s):  
Storrs McCall ◽  
R. K. Meyer
Keyword(s):  
Modus Ponens ◽  
The Matrix ◽  
Image Position

The matrix defining Łukasiewicz's three-valued logic, constructed in 1920 and described at length in [1], is the following: This matrix was axiomatized in 1931 by Wajsberg (see [6]), who showed that the following axioms together with the rules of substitution and modus ponens were sufficient:

Post completeness in modal logic

1972 ◽  
Vol 37 (4) ◽  
pp. 711-715 ◽  
Author(s):  
Krister Segerberg
Keyword(s):  
Modal Logic ◽  
Upper Bound ◽  
Modus Ponens ◽  
Proper Subset ◽  
Proper Extension ◽  
Image Position

Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the ruleA regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know thatThere is an obvious upper bound, too:Furthermore,.

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

Can the Number of D Electrons in Transition Metals be Measured from White Lines?

1997 ◽  
Vol 3 (S2) ◽  
pp. 957-958 ◽  
Author(s):  
P. Rez
Keyword(s):  
Transition Metals ◽  
Energy Loss ◽  
Electron Energy ◽  
Transition Elements ◽  
White Line ◽  
Line Intensities ◽  
Continuum Region ◽  
The Matrix ◽  
Image Position ◽  
Electron Energy Loss

Sharp peaks at threshold are a prominent feature of the L23 electron energy loss edges of both first and second row transition elements. Their intensity decreases monotonically as the atomic number increases across the period. It would therefore seem likely that the number of d electrons at a transition metal atom site and any variation with alloying could be measured from the L23 electron energy loss spectrum. Pearson measured the white line intensities for a series of both 3d and 4d transition metals. He normalised the white line intensity to the intensity in a continuum region 50eV wide starting 50eV above threshold. When this normalised intensity was plotted against the number of d electrons assumed for each elements he obtained a monotonie but non linear variation. The energy loss spectrum is given bywhich is a product of p<,the density of d states, and the matrix elements for transitions between 2p and d states.

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.

On a Normal Form of the Orthogonal Transformation III

1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Linear Space ◽  
Bilinear Form ◽  
Matrix Equation ◽  
Orthogonal Transformation ◽  
The Matrix ◽  
Image Position

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

Quantification Of Segregation Levels Using Xeds In The Stem

1999 ◽  
Vol 5 (S2) ◽  
pp. 146-147
Author(s):  
V. J. Keast ◽  
D. B. Williams
Keyword(s):  
Grain Boundary ◽  
Boundary Segregation ◽  
Scanning Transmission Electron Microscope ◽  
Factor V ◽  
X Ray ◽  
Interaction Volume ◽  
Probe Size ◽  
Scanning Transmission ◽  
The Matrix ◽  
Image Position

The quantification of grain boundary segregation levels, as measured with X-ray energy dispersive spectroscopy (XEDS) in a scanning transmission electron microscope (STEM), is dependent on the size and shape of the interaction volume. The segregation level T (in atoms/nm2) is related to the intensities of the characteristic peaks in the X-ray spectrum, Is and Im, bywhere ρ is the density of the matrix in atoms/nm3, Am and As are the atomic masses of the matrix and segregant respectively and ksm is the usual k-factor. The geometric factor, V/A, is the ratio of the volume of interaction to the area of the grain boundary inside in the interaction volume. Different models have been used to describe the interaction volume and these are illustrated in Fig. 1 and the appropriate expression for V/A is given in each case. In the simplest case, beam broadening is neglected and the interaction volume can be described as a cylinder with diameter equal to the probe size, d.

scholarly journals The normalizer of Γ0(N) in PSL(2, ℝ)

1990 ◽  
Vol 32 (3) ◽  
pp. 317-327 ◽  
Author(s):  
M. Akbas ◽  
D. Singerman
Keyword(s):  
Positive Integer ◽  
Modular Group ◽  
Prime Divisors ◽  
Möbius Transformations ◽  
The Matrix ◽  
Image Position ◽  
Mobius Transformations

Let Γ denote the modular group, consisting of the Möbius transformationsAs usual we denote the above transformation by the matrix remembering that V and – V represent the same transformation. If N is a positive integer we let Γ0(N) denote the transformations for which c ≡ 0 mod N. Then Γ0(N) is a subgroup of indexthe product being taken over all prime divisors of N.

Independence of Rose's axioms for m-valued implication

1969 ◽  
Vol 34 (2) ◽  
pp. 283-284 ◽  
Author(s):  
Ernest Edmonds
Keyword(s):  
Modus Ponens ◽  
Image Position

Rose has shown in [2] that the following axioms are sufficient, with modus ponens, for m-valued Łukasiewiczian implication.

scholarly journals Bifurcation for some quasilinear operators

2001 ◽  
Vol 131 (4) ◽  
pp. 733-765 ◽  
Author(s):  
David Arcoya ◽  
José Carmona ◽  
Benedetta Pellacci
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Complete Analysis ◽  
Multiplicity Results ◽  
Elliptic Boundary Value ◽  
The Matrix ◽  
Image Position ◽  
Quasilinear Elliptic

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem , where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

Inclusion of sets of regular summability matrices

1964 ◽  
Vol 60 (4) ◽  
pp. 705-712 ◽  
Author(s):  
J. W. Baker ◽  
G. M. Petersen
Keyword(s):  
Finite Subset ◽  
Null Sequence ◽  
Fixed Sequence ◽  
Summable Sequence ◽  
The Matrix ◽  
Summability Matrices ◽  
Image Position ◽  
Regular Summability ◽  
Bounded Sequences

1. In this paper we wish to discuss some problems which arise from a paper by Lorentz and Zeller; see (5). If {μn} is a fixed sequence monotonically increasing to infinity, and every sequence {sn} summed by both of the regular matrices A = (amn) and B = (bmn) and satisfying sn = O{μn) is summed to the same value by both matrices, the matrices are called (μn)-consistent. The two matrices are called consistent if they are (μn)-consistent for all {μn}, μn↗∞; they are b-consistent if the bounded sequences summed by both are summed to the same value by both. The matrix A is said to be (μn)-stronger than the matrix B, if every sequence {μn} that is B summable and satisfying sn = O(μn) is also A summable. The matrix A is stronger than B if every B summable sequence is A summable; A is b-stronger if every bounded B summable sequence is A summable. The symbol A -lim x denotes the value to which the sequence x = {xn} is summed by A; Am(x) is the transformationand A(x) is the sequence {Am(x)}. Let {A(i)}i ∈ I be any family, infinite or finite, of regular summability matrices. This family is called simultaneously consistent if, given any finite subset of I, say F, and any set of sequences {x(i)i ∈ F such that A(i) sums x(i) for each i in F, and such that is the null sequence, then .

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