If a many-body quantum system approaches thermal equilibrium from a generic initial state, then the expectation value 〈ψ(t)|Ai|ψ(t)〉, where |ψ(t)〉 is the system’s state vector and Ai is an experimentally accessible observable, should approach a constant value which is independent of the initial state, and equal to a thermal average of Ai at an appropriate temperature. We show that this is the c...

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b1 < b2 < · · · } of all products aiaj with ai, aj ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps bn+1 − bn ¿ α−3 and that for t ≥ 2 there are infinitely many t-gaps bn+t − bn ¿ t2α−4. Furthermore we prove that these estimates are best possible. We also discuss a r...

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {bn = aiaj , ai, aj ∈ A} when A has upper Banach density α > 0. We prove that there are infinitely many gaps bn+1 − bn ¿ α−3 and that for t ≥ 2 there are infinitely many t-gaps bn+t − bn ¿ t2α−4. Furthermore we prove that these estimates are best possible. We also discuss a related question about the cardi...

and Applied Analysis 3 Lemma 5. The operators S α and A α have the following properties: (i) for any fixed t ≥ 0 and any x ∈ X β , one has the operators S α (t) and A α (t) which are linear and bounded operators; that is, for any x ∈ X β , 󵄩󵄩󵄩󵄩Sα (t) x 󵄩󵄩󵄩󵄩β ≤ M‖x‖β, 󵄩󵄩󵄩󵄩Aα (t) x 󵄩󵄩󵄩󵄩β ≤ M Γ (α) ‖x‖ β ; (12) (ii) the operatorsS α (t) andA α (t) are strongly continuous for all t ≥ 0; (iii) S α (...

Let f : [0,1]×R2 → R be a function satisfying Carathéodory’s conditions and e(t) ∈ L1[0,1]. Let ξi ∈ (0,1), ai ∈ R, i = 1,2, . . . ,m− 2, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1 be given. This paper is concerned with the problem of existence of a solution for the m-point boundary value problem x′′(t) = f (t,x(t),x′(t))+e(t), 0 < t < 1; x(0) = 0, x′(1) =∑m−2 i=1 aix′(ξi). This paper gives conditions for ...

We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant n-dimensional subsystems ẋ = Aix+ Bi(x)u (x ∈ R , t ∈ R+ and u ∈ R mi ), where i ∈ {1, . . . , N} and a switching signal σ(·) : R+ → {1, . . . , N} which orchestrates switching between these subsystems above, where Ai ∈ R, n ≥ 1, N ≥ 2, mi ≥ 1, and where Bi(·) : R → Ri sat...

We studied argyrophilic proteins associated with nucleolar organizer regions (AgNOR) in non-small-cell cancer. We determined the area index (AI) and coefficient of variation (CV) of AgNOR. AI is associated with the key clinicomorphological parameters within the TNM system: T and N values, greatest tumor dimension up to 3 cm and more, disease stage, histogenesis, and tumor differentiation. CV is...

ÖZ Amaç: Araştırmada bireylerin COVID-19 korkularının, bağışıklaması için üretilen aşılarla ilgili tutumlarına etkisini belirlemek, bu etkide bilgi arama davranışlarının ve aşılar ile inançlarının aracı rolünü belirlemek amaçlanmıştır. Gereç Yöntem: beş bölümden oluşan online anket formu aracılığıyla veriler toplanmıştır. Araştırmanın evrenini Konya ilinde ikamet eden 18 yaş üstü bireyler oluşt...

The influence of birth history and mother's parenting to motor development have been widely studied . However the importance knowledge maternal self-efficacy is not yet clear. This study aims analyze effect knowledge, self-efficacy, practices children aged 1-2 years old. research explanatory. location was chosen purposively in 5 selected villages Panggang city, Karangturi Wonorejo Glagah Konang...