Realizations of regular polytopes

Let be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets ^{** *–1*} ( ^*), where ^* is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed _G , and let the subgroup ofG corresponding to ^* have a fixed space of dimensionw _G . Then the relations

Determinants of Lattices induced by Rational Subspaces

Let L and be orthogonal complementary rational linear subspaces of Eⁿ, and let = L Zⁿ and $$\stackrel{\¯}{\Lambda}$$ = Zⁿ be the sublatticesof the usual integer lattice Zⁿ induced by L and . Then the determinants of and are equal. The samerelationship holds between the determinants of the lattices and obtained by orthogonal projection of Zⁿ on to L and .     

On simple polytopes

Summary LetP be a simpled-polytope ind-dimensional euclidean space , and let (P) be the subalgebra of the polytope algebra generated by the classes of summands ofP. It is shown that the dimensions of the weight spaces _r(P) of (P) are theh-numbers ofP, which describe the Dehn-Sommerville equations between the numbers of faces ofP, and reflect the duality between _r (P) and _d-r (P). Moreover, (P) admits a Lefschetz decomposition under multiplication by the element of ₁(P) corresponding toP itself, which yields a proof of the necessity of McMullen's conditions in theg-theorem on thef-vectors of simple polytopes. The Lefschetz decomposition is closely connected with the new Hodge-Riemann-Minkowski quadratic inequalities between mixed volumes, which generalize Minkowski's second inequality; also proved are analogous generalizations of the Aleksandrov-Fenchel inequalities. A striking feature is that these are obtained without using Brunn-Minkowski theory; indeed, the Brunn-Minkowski theorem (without characterization of the cases of equality) can be deduced from them. The connexion found between (P) and the face ring of the dual simplicial polytopeP ^* enables this ring to be looked at in two ways, and a conjectured formulation of theg-theorem in terms of a Gale diagram ofP ^* is also established.Oblatum 1-X-1992     

BiBr3 initiated cyclization-addition reactions: effect of π-nucleophile on oxocarbenium ion addition and total syntheses of (+)-(S,S)-(cis-6-methyltetrahydropyran-2-yl)acetic acid and its trans-diastereomer

For BiBr₃ initiated tandem cyclization-additions of very reactive silyl ketene acetal nucleophiles with δ-silyloxy aldehydes to afford 2,6-disubstituted THP products, diastereoselectivities range from 5-6:1 (trans-/cis-). The selectivity for axial attack on the intermediate oxocarbenium ion is inversely proportional to π-nucleophilicity. We have utilized this chemistry to convert a common starting material to both cis- and trans-diastereomers of an acid first isolated from the secretions of Viverra civetta.     

Anionic Magnesium and Calcium Hydrides: Transforming CO into Unsaturated Disilyl Ethers

The synthesis, characterisation and reactivity of two isostructural anionic magnesium and calcium complexes is reported. By X-ray and neutron diffraction techniques, the anionic hydrides are shown to exist as dimers, held together by a range of interactions between the two anions and two bridging potassium cations. Unlike the vast proportion of previously reported dimeric group 2 hydrides, which have hydrides that bridge two group 2 centres, here the hydrides are shown to be “terminal”, but stabilised by interactions with the potassium cations. Both anionic hydrides were found to insert and couple CO under mild reaction conditions to give the corresponding group 2 cis-ethenediolate complexes. These cis-ethenediolate complexes were found to undergo salt elimination reactions with silyl chlorides, allowing access to small unsaturated disilyl ethers with a high percentage of their mass originating from the C₁ source CO.     

Aerodynamic shape optimization for unsteady three-dimensional flows

This paper presents an adjoint method for the optimum shape design of unsteady flows. The goal is to develop a set of discrete unsteady adjoint equations and the corresponding boundary condition for the non-linear frequency domain method. First, this paper presents the complete formulation of the time dependent optimal design problem. Second, we present the non-linear frequency domain adjoint equations for three-dimensional flows. Third, we present results that demonstrate the application of the theory to a three-dimensional wing.     

Polytopes with centrally symmetric faces

It is shown that if, for some 2≦j≦d-2, all thej-faces of ad-polytopeP are centrally symmetric, then all the faces ofP of every dimension are centrally symmetric.     

Zwitterionic Peptide Cloak Mimics Protein Surfaces for Protein Protection

Inspired by the amino acid composition of natural protein surfaces, we developed a zwitterionic cloak containing multi-layers of short alternating glutamic acid and lysine (EK) peptides as a facile, highly effective and low-immunogenicity approach for the protection and delivery of biotherapeutics. Each EK layer grafted to proteins provides multiple times of new lysine reaction sites for the growth of subsequent EK layers. This unique design allows EK peptides to achieve high coating density on proteins, overcoming the limitation of traditional conjugation strategies that rely on the number of innate lysine groups. A triple-layer EK cloak manifests to successfully eliminate the specific and non-specific interactions of protected asparaginase with biological media while prolong the drug circulation time and significantly mitigate its immunogenicity in vivo, suggesting an EK peptide cloak as a promising approach to improve the safety and efficacy of biotherapeutics.     

Molecular Targeting of Epidermal Growth Factor Receptor (EGFR) and Vascular Endothelial Growth Factor Receptor (VEGFR)

Epidermal growth factor receptor (EGFR) and vascular endothelial growth factor receptor (VEGFR) are two extensively studied membrane-bound receptor tyrosine kinase proteins that are frequently overexpressed in many cancers. As a result, these receptor families constitute attractive targets for imaging and therapeutic applications in the detection and treatment of cancer. This review explores the dynamic structure and structure-function relationships of these two growth factor receptors and their significance as it relates to theranostics of cancer, followed by some of the common inhibition modalities frequently employed to target EGFR and VEGFR, such as tyrosine kinase inhibitors (TKIs), antibodies, nanobodies, and peptides. A summary of the recent advances in molecular imaging techniques, including positron emission tomography (PET), single-photon emission computerized tomography (SPECT), computed tomography (CT), magnetic resonance imaging (MRI), and optical imaging (OI), and in particular, near-IR fluorescence imaging using tetrapyrrolic-based fluorophores, concludes this review.