A Poset Fiber Theorem for Doubly Cohen-Macaulay Posets and Its Applications

This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen- Macaulayness of a poset. Applications to complexes of injective words are also included.     

Bijections between generalized Catalan families of types A and C

Motivated by the relation ${\mathsf{N}}^m\left(C_n\right)=(mn+1){\mathsf{N}}^m\left(A_{n?1}\right)$, holding for the $m$-generalized Catalan numbers of type $A$ and $C$, the connection between dominant regions of the $m$-Shi arrangement of type $A_{n?1}$ and $C_n$ is investigated. More precisely, it is explicitly shown how $mn+1$ copies of the set of dominant regions of the $m$-Shi arrangement of type $A_{n?1}$, biject onto the set of type $C_n$ such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between $mn+1$ copies of the set of $m$-Dyck paths of height $n$ and the set of $N?E$ lattice paths inside an $n\times mn$ rectangle is provided.     

Porphyrin Containing Polymersomes with Enhanced ROS Generation Efficiency: In Vitro Evaluation

Porphyrins are molecules possessing unique photophysical properties making them suitable for application in photodynamic therapy. The incorporation of porphyrins into natural or synthetic nano‐assemblies such as polymersomes is a strategy to improve and prolong their therapeutic capacities and to overcome their limitations as therapeutic and diagnostic agents. Here, 5,10,15,20‐tetrakis(1‐(6‐ethoxy‐6‐oxohexyl)‐4‐pyridin‐1‐io)‐21H,23H‐porphyrin tetrabromide porphyrin is inserted into polymersomes in order to demonstrate that the encapsulation enhances its ability to generate highly reactive singlet oxygen (¹O₂) upon irradiation in vitro. The photoactivation of the free and polymersome‐encapsulated porphyrin is evaluated by electron spin resonance and cell viability assays on three different mammalian cell lines. The results indicate that by encapsulating the porphyrin, a controlled ROS delivery within the cells is achieved, at the same time avoiding side effects such as dark toxicity, non‐specific porphyrin release and over time decreased activity in vitro. This work focuses on showing a not‐toxic model system for modern therapeutic nanomedicine, which works under mild irradiation and dosage conditions.     

Comparative Study of a Series of 99mTc(CO)3 Mannosylated Dextran Derivatives for Sentinel Lymph Node Detection

Sentinel lymph node detection (SLND) is rapidly entering common practice in the management of patients with tumors. The introduction of mannose molecules to ^99mTc-labeled dextrans, so far, showed that the sentinel node could trap these agents due to their recognition by the mannose receptors of lymph node macrophages. The current study aimed to synthesize, characterize, and biologically evaluate a series of mannosylated dextran derivatives labeled with ^99mTc for potential use in SLND. The compounds were designed to have a dextran with a molecular weight of 10–500 kDa as a backbone, S-derivatized cysteines, efficient SNO chelators, and mannose moieties for binding to mannose receptors. They were successfully synthesized, thoroughly characterized using NMR techniques, and labeled with the fac-[^99mTc(CO)₃]⁺ synthon. Labeling with high yields and radiochemical purities was achieved with all derivatives. In vivo biodistribution and imaging studies demonstrated high uptake in the first lymph node and low uptakes in the following node and confirmed the ability to visualize the SLN. Among the compounds studied, ^99mTc-D75CM demonstrated the most attractive biological features, and in combination with the high radiochemical yield and stability of the compound, its further evaluation as a new radiopharmaceutical for sentinel lymph node detection was justified.     

Ostrowski-type theorems for harmonic functions

Ostrowski showed that there are intimate connections between the gap structure of a Taylor series and the behaviour of its partial sums outside the disk of convergence. This paper investigates the corresponding problem for the homogeneous polynomial expansion of a harmonic function. The results for harmonic functions display new features in the case of higher dimensions.     

Universal Polynomial Expansions of Harmonic Functions

Let Ω be a domain in ? ^N such that $\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega$ is connected. It is known that, for each w?∈?Ω, there exist harmonic functions on Ω that are universal at w, in the sense that partial sums of their homogeneous polynomial expansion about w approximate all plausibly approximable functions in the complement of Ω. Under the assumption that Ω omits an infinite cone, it is shown that the connectedness hypothesis on $\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega$ is essential, and that a harmonic function which is universal at one point is actually universal at all points of Ω.     

Properties of the solutions of the Ginzburg-Landau equation on the bifurcation branch

Generalizing previous results of M. Comte and P. Mironescu, it is shown that for degree d large enough (such that ), there is a bifurcation branch in the set of the solutions of the Ginzburg-Landau equation, emanating from the branch of radial solutions at the critical value _d of the parameter. Moreover, the solutions on the bifurcation branch admit exactly d zeroes, and the energy on the bifurcation branch is strictly smaller than the energy on the radial branch.     

Patterns in Shi Tableaux and Dyck Paths

Order - Shi tableaux are special binary fillings of certain Young diagrams which arise in the study of Shi hyperplane arrangements related to classical root systems. For type A, the set $\mathcal...     

The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes

The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.