implying C = 1 and δ (p) = δ (p̃). (b) Note that assuming p = ωp� dp� 1 dp � 2 2� � 0� (2π) 3 2ωp� f (p) = (2π) 3 δ p + m θ p f (p) d4p� � 2 2� � 0� = 3 δ p� + m θ p� f (p�) (2) (2π) dp � 2 2� � 0� dp� 1 = 3 δ p + m θ p f (p�) = 3 f (p�) (2π) (2π) 2ωp� In the second line we renamed p → p�, in the third we used | det Λ| = 1, p�2 = p and θ p�0 = θ p for Lorentz transformations connected to the ide...

Jacobi’s θ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of θ(z, τ), for z, τ verifying certain conditions, with precision P in O(M(P ) √ P ) bit operations, where M(P ) denotes the number of operations needed to multiply two complex P -bit numbers. We generalize an algorithm which computes specific values of the θ function (the...

Recent reports about the Θ + (formerly known as Z +), a pentaquark state, have awakened public opinion. We report our findings in search of a Θ ++ state in the reaction γp→ Θ ++ K − from our analysis of CLAS data. The existence or nonexistence of this state might resolve the question of the isotensor states. ++ The constituent quark model does not exclude penta-quark systems. So far only bound ...

Previously, algorithms for training conditional random fields were derived from the second-order Taylor expansion of the log-likelihood of the training data and the key issue is to approximate the Hessian matrix, which usually requires O(n) computations per iteration. In this paper, we show that efficient off-line and online algorithms for training conditional random fields (CRF) can also be de...

where ǫ, the noise, is a random variable over some probability space (Ω,F ,P). The gradient of L, i.e. Lθ(θ) is not available, not even via physical measurements. To estimate it let θ be fixed, and take a random simultaneous perturbation (SP) of θ by a multiple of a vector ∆ = (∆1, ...,∆p) . Here ∆ is defined over an independent probability space (Ω,F ,P ). We assume that the components ∆i are ...

Let As,k(m) be the maximum number of distinct letters in any sequence which can be partitioned into m contiguous blocks of pairwise distinct letters, has at least k occurrences of every letter, and has no subsequence forming an alternation of length s. Nivasch (2010) proved that A5,2d+1(m) = θ(mαd(m)) for all fixed d ≥ 2. We show that As+1,s(m) = ( m−⌈ s 2 ⌉ ⌊ s 2 ⌋ ) for all s ≥ 2, A5,6(m) = θ...

We provide proofs for the theorems presented in the main paper and additional numerical examples. 1 Proofs of Theorems and Lemmas 1.1 Proof of Lemma 2.1 Proof First, we need to verify that the conditions in (3)–(7) are satisfied. It follows directly from (14) that F̄ (θ̂; θ̂) = F (θ̂) and ∇θF̄ |θ=θ̂ = ∇θF |θ=θ̂ so (4)–(5) are satisfied. Also, by construction the entries of D in (15) are non-negative s...

which are just module axioms (MM2) and (MM3). [In the case of the second axiom, we have to show that (r1r2)θ = (r1θ)r2. We are thinking of r′ as an element of the free R-module R, and r2 as an element of the ring R acting on this module.] The kernel of θ is {r ∈ R : mr = 0}. Clearly any element r ∈ Ann(M) satisfies this condition. Conversely, since m generates M, every element of M has the form...

Let Γ denote a distance-regular graph with diameter D ≥ 3, intersection numbers ai, bi, ci and Bose-Mesner algebra M. For θ ∈ C ∪∞ we define a 1 dimensional subspace of M which we call M(θ). If θ ∈ C then M(θ) consists of those Y in M such that (A−θI)Y ∈ CAD, where A (resp. AD) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ = ∞ then M(θ) = CAD. By a pseudo primitive idempotent f...

Let (M, θ) be a strictly pseudoconvex pseudo-Hermitian compact hypersurface in C in the sense of Webster [34] with a pseudo-Hermitian real oneform θ on M . Let Rθ be the Webster pseudo scalar curvature for M with respect to θ. By the solution of the CR Yamabe problem given by Jerison and Lee [18], Gamara and Yacoub [10] and Gamara [9] (for n = 1), there is a pseudo-Hermitian real one-form θ so ...