z-Scan characterization of zwitterionic chromophores for optoelectronic switching

Single beam z-scan measurements have been made on films containing amorphous polycarbonate and the zwitterionic chromophore, PYR-3, that has a very high second order nonlinear optical (NLO) figure of merit. The third order NLO figure of merit is ≈1.6 at 1030 nm and is comparable to that found in organic compounds optimized for high n ₂ values. The two-photon absorption coefficient is 2.1×10⁻¹² m/W, which is very low and advantageous for NLO device applications. The third order NLO refractive index is −1.4×10⁻¹⁸ m²/W.     

Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider

$?\mathrm{div}A(x,\mathrm{?}u)=f\in L^1(\mathrm{\Omega }),$

on a Lipschitz bounded domain in ${\mathbb{R}}^N$. The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate $M^{?}$ (neither ${\mathrm{\Delta }}_2$, nor ${\mathrm{?}}_2$). The condition we impose is log-Hölder continuity of M, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.     

Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving A-Laplacian ?Δ _A u = ?divA(?u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset \({\Omega \subseteq \mathbb{R}^n}\) . Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the form $$\int_\Omega F_{\bar{A}}(|\xi|) \mu_1(dx) \leq \int_\Omega \bar{A}(|\nabla \xi|)\mu_2(dx),$$ where \({\bar{A}(t)}\) is a Young function related to A and satisfying Δ′-condition, while \({F_{\bar{A}}(t) = 1/(\bar{A}(1/t))}\) . Examples involving \({\bar{A}(t) = t^p{\rm log}^\alpha(2+t), p \geq 1, \alpha \geq 0}\) are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30–50, 2013), where we dealt with inequality ?Δ _p u ≥ Φ, leading to Hardy and Hardy–Poincaré inequalities with the best constants.     

Vector-valued Fourier multipliers on symmetric spaces of the noncompact type

LetX be a Riemannian symmetric space of the noncompact type. We prove the multiplier theorem for the Helgason-Fourier transform and the vector valued function spacesL _p (X, l _q ). As a consequence we get the inequalities of the Littlewood-Paley type forL _p (X) spaces.Research supported by K.B.N. Grant 210519101 (Poland).     

Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains \(\Omega \) obtained by different approaches. In particular, we consider three settings \(\mathcal {M}_{u,p}(\Omega )\), \(\mathbb {M}_{u,p}(\Omega )\) and \(\mathfrak {M}_{u,p}(\Omega )\), where \(0<p\le u<\infty \), commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes \(\mathfrak {E}_{\mathsf {G}}(\mathcal {M}_{u,p}(\Omega ))\) as well as \(\mathfrak {E}_{\mathsf {G}}(\mathfrak {M}_{u,p}(\Omega ))\), and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether \(\frac{n}{u}\ge \frac{1}{p}\) or \(\frac{n}{u} < \frac{1}{p}\) plays a decisive role when it comes to the behaviour of these spaces.     

A geometric criterion for compactness of invariant subspaces

Let M be a non-compact homogeneous Riemannian manifold, and let Ω be a compact subgroup of isometries of M. We show, under general conditions, that the Ω-invariant subspace A _Ω of a normed vector space ${A\hookrightarrow L^q(M)}$ A ? L q ( M ) is compactly embedded into L ^q (M) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity.     

Strangeness enhancement and strangeness suppression in nuclear collisions at 200 GeV/N

Data on yields of strange and negative hadrons in full phase space produced in proton-nucleus and central S+S collisions at 200 GeV/nucleon are reviewed. The total number of produced strange and nonstrange valence quark-antiquark pairs is determined on the basis of the hadron multiplicity at freezout. It is shown that the strangeness production factor λ _s is increased by a factor of 2 in central S+S collisions, whereas no increase is observed in proton-nucleus collisions.     

On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

We deal with decay and boundedness properties of elements of radial subspaces of homogeneous Besov and Triebel-Lizorkin spaces. For the region of parameters which are of interest for us these homogeneous spaces are larger than the inhomogeneous counterparts. By switching from the inhomogeneous spaces to the homogeneous classes the properties of the radial elements change. Our investigations are based on the atomic decompositions for radial subspaces in the sense of Epperson and Frazier (J.?Fourier Anal Appl. 1:311?C353, 1995). Finally, we apply these results for deriving some assertions on compact embeddings on unbounded domains.