Existence of Non-degenerate Continua of Singular Radial Solutions for Several Classes of Semilinear Elliptic Problems

We establish the existence of countably many branches of uncountably many solutions to elliptic boundary value problems with subcritical, and subsuper critical growth. We also prove the existence of two branches of uncountably many solutions to a problem with jumping nonlinearities. This case is remarkable since, generically, this problem has only finitely many regular solutions.     

Direct Visualization of a Mechanochemically Induced Molecular Rearrangement

Recent progress in the field of mechanochemistry has expanded the discovery of mechanically induced chemical transformations to several areas of science. However, a general fundamental understanding of how mechanochemical reactions by ball milling occur has remained unreached. For this, we have now implemented in situ monitoring of a mechanochemically induced molecular rearrangement by synchrotron X‐ray powder diffraction, Raman spectroscopy, and real‐time temperature sensing. The results of this study demonstrate that molecular rearrangements can be accomplished in the solid state by ball milling and how in situ monitoring techniques enable the visualization of changes occurring at the exact instant of a molecular migration. The mechanochemical benzil–benzilic acid rearrangement is the focal point of the study.     

Excited-state absorption and 1064-nm end-pumped laser emission of Nd:YVO4 single-crystal fiber grown by laser-heated pedestal growth

We present an investigation of the excited-state absorption and laser emission of a 1.0-at. %-Nd3+-doped YVO4 single-crystal fiber grown by the low-cost and versatile laser-heated pedestal growth technique. Efficient laser emission at 1064 nm was achieved when the fiber was pumped, in an end-pump cavity, by a Ti:sapphire laser at 808 nm. A continuous-wave threshold of 10 mW was observed with an efficiency of 42% with respect to the absorbed pump power and the maximum output power of 200 mW. These results are excellent when compared with those of a commercial bulk crystal adapted to the same cavity (48% efficiency, 250-mW maximum output power). Thus the fibers are characterized as strong candidates for the construction of compact lasers that can also be pumped by low-cost diode lasers.     

Visual spatial memory is enhanced in female rats (but inhibited in males) by dietary soy phytoestrogens

Background  

In learning and memory tasks, requiring visual spatial memory (VSM), males exhibit superior performance to females (a difference attributed to the hormonal influence of estrogen). This study examined the influence of phytoestrogens (estrogen-like plant compounds) on VSM, utilizing radial arm-maze methods to examine varying aspects of memory. Additionally, brain phytoestrogen, calbindin (CALB), and cyclooxygenase-2 (COX-2) levels were determined.     

Universal polynomials for Severi degrees of toric surfaces

The Severi variety parameterizes plane curves of degree d$d$ with δ$\delta$ nodes. Its degree is called the Severi degree. For large enough d$d$, the Severi degrees coincide with the Gromov–Witten invariants of CP²$\mathbb{C}{\mathbb{P}}^2$. Fomin and Mikhalkin (2010) [10] proved the 1995 conjecture that for fixed δ$\delta$, Severi degrees are eventually polynomial in d$d$.     

Lifted generalized permutahedra and composition polynomials

Generalized permutahedra are the polytopes obtained from the permutahedron by changing the edge lengths while preserving the edge directions, possibly identifying vertices along the way. We introduce a “lifting” construction for these polytopes, which turns an n  -dimensional generalized permutahedron into an (n+1)$(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff ?s multiplihedron from homotopy theory, and to the more general “nestomultiplihedra”, answering two questions of Devadoss and Forcey.     

Staircase skew Schur functions are Schur P-positive

We prove Stanley??s conjecture that, if ?? _n is the staircase shape, then the skew Schur functions $s_{\delta_{n} / \mu}$ are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function $s_{\delta_{n} / \delta _{n-2}}$ , we discuss connections with Eulerian numbers and alternating permutations.