Riesz Energy Problems with External Fields and Related Theory

Minimal Riesz Energy on the Sphere for Axis-supported External Fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere S in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y| with d−2 ≤ s < d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on S i...

Riesz spherical potentials with external elds and minimal energy points separation

In this paper we consider the minimal energy problem on the sphere S for Riesz potentials with external …elds. Fundamental existence, uniqueness, and characterization results are derived about the associated equilibrium measure. The discrete problem and the corresponding weighted Fekete points are investigated. As an application we obtain the separation of the minimal senergy points for d 2 < s...

Open Problems in Dynamics and Related Fields

the evaluation of language related engagment and task related engagment with the purpose of investigating the effect of metatalk and task typology

abstract while task-based instruction is considered as the most effective way to learn a language in the related literature, it is oversimplified on various grounds. different variables may affect how students are engaged with not only the language but also with the task itself. the present study was conducted to investigate language and task related engagement on the basis of the task typolog...

Almost Everywhere Convergence of Riesz Means Related to Schrödinger Operator with Constant Magnetic Fields

and Applied Analysis 3 Lemma 4. For λ > 0, one has 󵄩󵄩󵄩󵄩󵄩 K δ,l,j λ f (x) 󵄩󵄩󵄩󵄩󵄩 2 2 ≤ C2 −2M(j+l) δ 2M󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩 2 2 , (19) where the constant C is independent of λ and δ. Proof. With the method similar to the proof of Lemma 4 in [9], we write h(t) = φ(t) − φ(2t) and expandm into a Taylor series around λt. Then, ?̂? δ,l,j λ (t) = ∫m δ (λ(t − 2 −(j+l) δ 2 r λ )) ĥ (r) dr = ∫m δ (λt − 2 −(j+l) δ 2 ...