A note on the complete rotational invariance of biradial solutions to semilinear elliptic equations

We investigate symmetry properties of solutions to equations of the form $$-\Delta u = \frac{a}{|x|^2} u + f(|x|, u)$$ in R^N for $N \geq 4$, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with low Morse index (namely 0 or 1) and which are biradial (i.e. are invariant under the action of a toric group of rotations), are in fact completely radial. A similar result holds for the semilinear Laplace-Beltrami equations on the sphere. Furthermore, we show that the condition on the Morse index is sharp. Finally we apply the result in order to estimate best constants of Sobolev type inequalities with different symmetry constraints

Singular boundary behaviour and large solutions for fractional elliptic equations

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace–Poisson equation with zero boundary data. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equations driven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function

Principal toroidal bundles over Cauchy-Riemann products

The main result we obtain is that given π:N→M a Ts-subbundle of the generalized Hopf fibration π¯:H2n+s→ℂPn over a Cauchy-Riemann product i:M⊆ℂPn, i.e. j:N⊆H2n+s is a diffeomorphism on fibres and π¯∘j=i∘π, if s is even and N is a closed submanifold tangent to the structure vectors of the canonical ℊ-structure on H2n+s then N is a Cauchy-Riemann submanifold whose Chen class is non-vanishing

Transforming growth factor-β/Smad - signalling pathway and conjunctival remodelling in vernal keratoconjunctivitis

Vernal keratoconjunctivitis (VKC) is a chronic ocular allergic inflammation characterized by corneal complications and the formation of giant papillae. Sma- and Mad-related proteins (Smad) modulate extracellular matrix gene expression during wound healing, inflammation and tissue remodelling.Objective To investigate the relationship between allergic inflammation and TGF-β/Smad signalling pathway, expression in VKC patients and in primary cultured conjunctival fibroblasts exposed to mediators found previously over-expressed in VKC.Methods Smad-2, -3, -7, phospho-(p)Smads, TGF-β1 and -β2 were evaluated in the conjunctiva of normal subjects (CT) and VKC patients by immunohistochemistry. The expression of Smads, pro-collagen I (PIP), TGF-β1, -β2, mitogen-activated protein kinase (p38/MAPK), c-Jun N-terminal kinase (JNK) and extracellular signal-regulated kinase (ERK1/2) were also determined in conjunctival fibroblast cultures exposed to histamine, IL-4, -13, TGF-β1, IFN-γ and TNF-α using immunostaining or RT-PCR.Results Immunostaining for Smad-2, -3, pSmad-2, -3, TGF-β1, -β2 and PIP was significantly increased in VKC stroma compared with CT. In conjunctival fibroblast cultures, Smad-3 and PIP were stimulated by histamine, IL-4, -13 and TGF-β1 exposure, while PIP was reduced by IFN-γ, and TNF-α mRNA expression of Smad-3 was increased by histamine, while Smad-7 was reduced by IL-4. In addition, histamine, IL-4 and TNF-α increased JNK and ERK1/2 expression.Conclusion and Clinical Relevance The TGF-β/Smad signalling pathway is over-expressed in VKC tissues and modulated in conjunctival fibroblasts by histamine, IL-4, TGF-β1 and TNF-α. These mechanisms may be involved in fibrillar collagen production, giant papillae formation and tissue remodelling typical of VKC and might provide new therapeutic targets for its treatment. © 2010 Blackwell Publishing Ltd

An alternative construction of B-M and B-T unitals in Desarguesian planes

We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $\Pi$ of $PG(2,q^2)$ with the following three properties: 1. points of $\Pi$ are those of $PG(2,q^2)$; 2. lines of $\Pi$ are certain lines and conics of $PG(2,q^2)$; 3. the points in $U$ form a non-classical B-M unital in $\Pi$. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.Comment: Keywords: unital, desarguesian plane 11 pages; ISSN: 0012-365

In vitro and in vivo biological performance of modified gellan gum-based hydrogels for nucleus pulposus tissue engineering

Ionic- (iGG-MA) and photo-crosslinked (phGG-MA) methacrylated gellan gum hydrogels have been proposed as biomaterials for supporting nucleus pulposus (NP) regeneration and/or repair. In this study, the mechanical stability and biocompatibility of these hydrogels have been evaluated in vitro. Human intervertebral disc cells obtained from herniated patients were cultured within both hydrogels, for 1–21 days. Dynamic mechanical analysis and biological characterization (Live/ Dead assay, ATP and DNA quantification, PCR and immunocytochemistry) were performed after specific times of culturing. The in vitro study showed that both cell loading and culturing time do not affect the mechanical properties of hydrogels. In addition, the iGG-MA and phGG-MA hydrogels showed to be effective on supporting cells encapsulation and viability up to 21 days of culturing. In vivo biocompatibility screening was also performed, by subcutaneous implantation of both hydrogels in Lewis rats for the period of 10 and 18 days. Haematoxylin & eosin staining revealed that the hydrogels do not elicit necrosis, calcification or acute inflammatory reaction. The present study demonstrates that the iGG-MA and phGG-MA hydrogels support cells encapsulation and viability, and are well-tolerated, stable and non-cytotoxic in vitro and in vivo, thus possessing promising features for finding application as viable NP substitutes

Mechanical performance and biocompatibility study of methacrylated Gellan gum hydrogels with potential for nucleus pulposus regeneration

Methacrylated gellan gum hydrogels, obtained either by ionic- (iGGMA) and photo-crosslinking (phGG-MA), have been investigated as potential biomaterials for supporting nucleus pulposus (NP) regeneration and/or repair [1,2]. In previous work, some advantages were attributed to GG-MA hydrogels, such as: (i) the possibility to control endothelial cells infiltration and blood vessel ingrowth’s, (ii) tunable and improved mechanical properties, and (iii) in situ gelation, within seconds to few minutes. In this study, the mechanical and biological performance of these hydrogels was firstly evaluated in vitro. Human intervertebral disc (hIVD) cells obtained from herniated patients were cultured within both hydrogels, for 1 up to 21 days. Dynamic mechanical analysis and biological characterization (calcein-AM staining, ATP and DNA quantification and PCR) were performed after specific times of culturing. A biocompatibility study was also performed in vivo, by subcutaneous implantation of acellular iGG-MA and phGG-MA hydrogels in Lewis rats for the period of 10 and 18 days. Tissue response to the hydrogels implantation was determined by histological analysis (haematoxylin-eosin staining). The in vitro study showed that both cell loading and culturing time do not have an effect on the mechanical properties of the hydrogels. Regarding their biological performance, the iGG-MA and phGG-MA hydrogels showed to be effective on supporting hIVD cells encapsulation and viability up to 21 days of culturing. Human IVD cells were homogeneously distributed within the hydrogels and maintained its round-shape morphology during culturing time. The in vivo biocompatibility study showed that iGG-MA and phGG-MA hydrogels do not elicit any deleterious effect, as denoted by the absence of necrosis and calcification, or acute inflammatory reaction. A thin fibrous capsule was observed around the implanted hydrogels. The results presented in this study indicate that the iGG-MA and phGG-MA hydrogels are stable in vitro and in vivo, support hIVD cells encapsulation and viability, and were found to be well-tolerated and non-cytotoxic in vivo, thus being potential candidates for NP regeneration

In vitro concurrent endothelial and osteogenic commitment of adipose-derived stem cells and their genomical analyses through CGH array: novel strategies to increase the succesfull engraftement of a tissue engineered bone grafts

In the field of tissue engineering, adult stem cells are increasingly recognized as an important tool for in vitro reconstructed tissue-engineered grafts. In the world of cell therapies, mesenchymal stem cells from bone marrow or adipose tissue are undoubtedly the most promising progenitors for tissue engineering applications. In this setting, adipose-derived stem cells (ASC) are generally similar to those derived from bone marrow and are most conveniently extracted from tissue removed in elective cosmetic liposuction procedures; they also show a great potential for endothelization. The aim of the present work was to investigate how the co-commitment into a vascular and bone phenotype of ASC could be a usefull tools for improving the in vitro and in vivo reconstruction of a vascularized bone graft. Human ASC obtained from abdominoplasty procedures were loaded in a hydroxyapatite clinical-grade scaffold, co-differentiated and tested for proliferation, cell distribution, and osteogenic and vasculogenic gene expression. The chromosomal stability of the cultures was investigated using the CGH array for 3D cultures. ASC adhesion, distribution, proliferation and gene expression not only demonstrated a full osteogenic and vasculogenic commitment in vitro and in vivo, but also showed that endothelization strongly improves their osteogenic commitment. In the end, genetic analyses confirmed that no genomical alteration in long-term in vitro culture of ASC in 3D scaffolds occurs

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

Dietary soy and meat proteins induce distinct physiological and gene expression changes in rats

This study reports on a comprehensive comparison of the effects of soy and meat proteins given at the recommended level on physiological markers of metabolic syndrome and the hepatic transcriptome. Male rats were fed semi-synthetic diets for 1 wk that differed only regarding protein source, with casein serving as reference. Body weight gain and adipose tissue mass were significantly reduced by soy but not meat proteins. The insulin resistance index was improved by soy, and to a lesser extent by meat proteins. Liver triacylglycerol contents were reduced by both protein sources, which coincided with increased plasma triacylglycerol concentrations. Both soy and meat proteins changed plasma amino acid patterns. The expression of 1571 and 1369 genes were altered by soy and meat proteins respectively. Functional classification revealed that lipid, energy and amino acid metabolic pathways, as well as insulin signaling pathways were regulated differently by soy and meat proteins. Several transcriptional regulators, including NFE2L2, ATF4, Srebf1 and Rictor were identified as potential key upstream regulators. These results suggest that soy and meat proteins induce distinct physiological and gene expression responses in rats and provide novel evidence and suggestions for the health effects of different protein sources in human diets