First integrals for Finsler metrics with vanishing $$\chi $$ -curvature

Annals of Global Analysis and Geometry - We prove that in a Finsler manifold with vanishing $$\chi $$ -curvature (in particular with constant flag curvature) some non-Riemannian geometric...     

The Forgotten Pyrazines: Exploring the Dakin–West Reaction

Pyrazines are an underreported class of N-heterocycles available from nitrogen-rich biomass presenting an interesting functional alternative for current aromatics. In this work, access to pyrazines obtained from amino acids by using the 90 year old Dakin–West reaction was explored. After a qualitative screening several functional proteinogenic amino acids proved good substrates for this reaction, which were successfully scaled to multigram scale synthesis of the corresponding intermediate α-acetamido ketones. Subsequently, the conditions towards pyrazine formation using δ-amino-levulinic acid were optimized, and these were employed to synthesize a relevant set of five functional dimethylpyrazines in high purity. These pyrazines can be considered a versatile toolbox of aromatic building blocks for a wide range of applications, such as in the synthesis of polymers or metal–organic frameworks.     

Engineering Multifunctional Gold Decorated Dendritic Mesoporous Silica/Tantalum Oxide Nanoparticles for Intraperitoneal Tumor‐Specific Delivery

Nonspecific high‐energy radiation for treatment of metastatic ovarian cancer is limited by damage to healthy organs, which can be mitigated by the use of radiosensitizers and image‐guided radiotherapy. Gold (Au) and tantalum oxide (TaO_x) nanoparticles (NPs), by virtue of their high atomic numbers, find utility in the design of bimetallic NP systems capable of high‐contrast computed tomography (CT) imaging as well as a potential radiosensitizing effect. These two radio‐dense metals are integrated into dendritic mesoporous silica NPs (dMSNs) with radial porous channels for high surface‐area loading of therapeutic agents. This approach results in stable, monodispersed dMSNs with a uniform distribution of Au on the surface and TaO_x in the core that exhibits CT attenuation up to seven times greater than iodine or monometallic dMSNs without either TaO_x or Au. Tumor targeting is assessed in a metastatic ovarian cancer mouse model. Ex vivo micro‐CT imaging of collected tumors shows that these NPs not only accumulate at tumor sites but also penetrate inside tumor tissues. This study demonstrates that after intraperitoneal administration, rationally designed bimetallic NPs can simultaneously serve as targeted contrast agents for imaging tumors and to enhance radiation therapy in metastatic ovarian cancer.     

Low- and high-energy solutions of nonlinear elliptic oscillatory problems

In this Note, we study the existence of low- or high-energy solutions for a class of elliptic problems containing a nonlinear term that oscillates either near the origin or at infinity. We point out the competition effect between the oscillatory nonlinearity, a polynomial growth term, and the values of a real parameter. The proofs combine related variational methods.     

Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition

We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally Lipschitz integrand. We assume that asymptotically this term is resonant with respect the principal eigenvalue (from the left). We prove the existence of three nontrivial smooth solutions, two of constant sign and the third nodal. We also show the existence of extremal constant sign solutions. The tools come from nonsmooth critical point theory and from global optimization (direct method).     

A sharp eigenvalue theorem for fractional elliptic equations

The invisibility graph I(X) of a set X ? R ^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number ω(I(X)), the chromatic number χ(I(X)) and the convexity number γ(X), which is the minimum number of convex subsets of X that cover X.We settle a conjecture of Matou?ek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also find sets X in R⁵ with χ(X) = 2, but γ(X) arbitrarily large.     

Connections on the Total Space of a Holomorphic Lie Algebroid

The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle \(T_{\mathbb {C}}E=T'E\oplus T''E\) and its link, via the tangent anchor map, with the complexified tangent bundle \(T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)\). A holomorphic Lie algebroid structure is emphasized on \(T'E\). A special study is made for integral curves of a spray on \(T'E\). Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on \(T'E\). In the second part of section two, we study the holomorphic prolongation \(\mathcal {T}'E\) of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on \(T'M\) induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from \(T'M\).