We use Kaufman’s spinor method to calculate the bulk, surface and corner free energies ¢ f f f f , , , b s s c of the anisotropic square lattice zero-field Ising model for the ordered ferromagnetic case. For ¢ f f f , , b s s our results of course agree with the early work of Onsager, McCoy and Wu. We also find agreement with the conjectures made by Vernier and Jacobsen (VJ) for the isotropic c...

Let F be an arbitrary context-free grammar form and (F) the family of grammars defined by F. For each grammar G in d(F), the derivation complexity function 6, on the language of G, is defined for each word x as the number of steps in a minimal G-derivation of x. It is shown that derivations may always be speeded up by any constant factor n, in the sense that for each positive integer n, an equi...

This paper discusses the Origin of Basic Formula for Fourier Series f(t)=a0/2+∑∞k=1 [ak cos⁡(kπt/L)+bk sin⁡(kπt/L) ] To obtain this model, author derives basic formula from Elementary Linear Algebra. In derivation, we use best approximation: least squares method whose details will be elaborated in until Series.

the present paper, through the linguistic analysis of mulla sadra's philosophical views, has shown that a single linguistic model can be applied to the analysis of all his philosophical views. to this end, based on two linguistic theories regarding the semantic structure of derivation, it is illustrated that mulla sadra’s philosophical views conform to the linguistic model of semantic simp...

sarhaddi balochi dialect, a language variety of western (rakhshani) balochi, employs derivation through affixation as one of its word formation processes. the purpose of this article is to present a synchronic description of the way(s) different derivational affixes function in making complex words in sarhaddi balochi as spoken in granchin[1] district located about 35kms to the southeast of kha...

we show that  higher derivations on a hilbert$c^{*}-$module associated with the cauchy functional equation satisfying generalized hyers--ulam stability.

Let C be an algebraically closed field with trivial derivation and let F denote the differential rational field C〈Yij〉, with Yij , 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n, i ≤ j, differentially independent over C. We show that there is a Picard-Vessiot extension E ⊃ F for a matrix equation X ′ = XA(Yij), with differential Galois group SOn, with the property that if F is any differential field with field of co...

there are several kinds of hyperrings, for example, krasnerhyperrings, multiplicative hyperring, general hyperrings and$h_v$-rings. in a multiplicative hyperring, the multiplication isa hyperoperation, while the addition is a binary operation. in this paper, the notion of derivation on multiplicative hyperrings is introduced and some related properties are investigated. {bf keywords:} multiplic...

Let i (G) be the number of induced i-cubes of a graph G. Then the cubes polynomial c(G; x) of G is introduced as P i0 i (G)x i. It is shown that any function f with two related, natural properties, is up to the factor f(K 1 ; x) the cubes polynomial. The derivation @ G of a median graph G is also introduced and it is proved that the cubes polynomial is the only function f with the property f 0 ...

a0 exp ✓ c0 + b 2 0 4a0 ◆ (1) Z R exp a1⌧ 1 a2⌧ 2 + b12⌧1⌧2 + b1⌧1 + b2⌧2 + c1 d⌧ = 2⇡ p 4a1a2 b12 exp ✓ c1 + a1b 2 1 + a2b 2 2 + b1b12b2 4a1a2 b12 ◆ (2) where a0, a1, a2 > 0 and b0, b1, b2, b12, c0, c1 2 R are constants. 1 Derivation of the kernel K f |h — given the filter function h The covariance function K f |h(t1, t2) of the random variable f |h, t 2 R is given by K f |h(t1, t2) = E [f(t1)...