and Applied Analysis 3 where f ∈ C J × E × E × E, E , Ik, Ik ∈ C E, E , and the operators T , S are given by Tu t ∫ t 0 k t, s u s ds, Su t ∫2π 0 k1 t, s u s ds, 1.5 with k ∈ C D,R , D { t, s ∈ R2 : 0 ≤ s ≤ t ≤ 2π}, k1 ∈ C J × J,R . By applying the monotone iterative technique and cone theory based on a comparison result, the author obtained an existence theorem of minimal and maximal solutions...

The q-deformed supersymmetric t-J model on a semi-infinite lattice is diagonalized by using the level-one vertex operators of the quantum affine superalgebra Uq[ ̂ sl(2|1)]. We give the bosonization of the boundary states. We give an integral expression of the correlation functions of the boundary model, and derive the difference equations which they satisfy. E-mail: [email protected] E-mail:...

If we do gradient descent with η * (t), then almost surely, the algorithm converges (for the quadratic model). To prove that, we follow classical techniques based on Lyapunov stability theory (Bucy, 1965). Notice that the expected loss follows E J θ (t+1) | θ (t) = 1 2 h · E (1 − η * h)(θ (t) − θ *) + η * hσξ 2 + σ 2 = 1 2 h (1 − η * h) 2 (θ (t) − θ *) 2 + (η *) 2 h 2 σ 2 + σ 2 = 1 2 h σ 2 (θ (...

// perform f i l t e r i n g vector s e l e c t edFea tu r e s ; s e l e c t o r−> f i l t e r ( posFeatureVectors , negFeatureVectors , s e l e c t edFea tu r e s ) ; // trim the f e a t u r e v e c t o r s o f p o s i t i v e examples for (unsigned i = 0 ; i < posFeatureVectors . s i z e ( ) ; i++) { for ( int j = posFeatureVectors [ i ] . s i z e ( ) − 1 ; j >= 0 ; j−−) { i f ( ! s e l...

and Applied Analysis 3 H2 for all t > 0 and i, j ∈ {1, 2, . . . , n}, there exist constants η > 0, λ > 0 and ξi > 0 such that ∫∞ 0 ∣ ∣Kij u ∣ ∣edu < ∞, −η > − [ ci t − λe−ληi t − ηi t ci t ( λ c i e λη i )] ei t ξi n ∑ j 1 L̃j (∣ ∣aij t ∣ ∣eij t a ijηi t ci t e ληi t e ij ) ξj n ∑ j 1 Lj ∫∞ 0 ∣ ∣Kij u ∣ ∣edu (∣ ∣bij t ∣ ∣ b ijηi t ci t e ληi t ) ξj ; 1.5 H3 Ii t O e−λt t → ±∞ , i 1, 2, . . . , n...

and Applied Analysis 3 To prove our main results, for any h ∈ C J, E , we consider the Neumann boundary value problem NBVP of linear impulsive differential equation in E: −u′′ t Mu t h t , t ∈ J ′, −Δu′|t tk yk, k 1, 2, . . . , m, u′ 0 u′ 1 θ, 2.3 where M > 0, yk ∈ E, k 1, 2, . . . , m. Lemma 2.4. For any h ∈ C J, E , M > 0, and yk ∈ E, k 1, 2, . . . , m, the linear NBVP 2.3 has a unique soluti...