Kleiner's theorem for unitary representations of posets

A subspace representation of a poset $\mathcal S=\{s_1,...,s_t\}$ is given by a system $(V;V_1,...,V_t)$ consisting of a vector space $V$ and its subspaces $V_i$ such that $V_i\subseteq V_j$ if $s_i \prec s_j$. For each real-valued vector $\chi=(\chi_1,...,\chi_t)$ with positive components, we define a unitary $\chi$-representation of $\mathcal S$ as a system $(U;U_1,...,U_t)$ that consists of a unitary space $U$ and its subspaces $U_i$ such that $U_i\subseteq U_j$ if $s_i\prec s_j$ and satisfies $\chi_1 P_1+...+\chi_t P_t= \mathbb 1$, in which $P_i$ is the orthogonal projection onto $U_i$. We prove that $\mathcal S$ has a finite number of unitarily nonequivalent indecomposable $\chi$-representations for each weight $\chi$ if and only if $\mathcal S$ has a finite number of nonequivalent indecomposable subspace representations; that is, if and only if $\mathcal S$ contains any of Kleiner's critical posets.Comment: 12 pages, paper reorganized and rewritten. some statements were adde

In operando Synchrotron XRD/XAS Investigation of Sodium Insertion into the Prussian Blue Analogue Cathode Material Na 1.32 Mn[Fe(CN) 6 ] 0.83 · z H 2 O

Prussian Blue Analogues (PBAs) with general formula NaxMA[MB(CN)6]y·z H2O (MA, MB = transition metal) are promising low cost, high rate and high capacity cathodes for sodium ion battery (SIB) technology. Here, we have studied the PBA Na1.32Mn[Fe(CN)6]0.83·z H2O (z = 3.0 and 2.2) with varying structural modifications (monoclinic and cubic) using in operando quasi-simultaneous X-ray diffraction (XRD) and X-ray absorption spectroscopy (XAS). We observed a series of reversible structural phase transitions which accompany Na insertion/extraction during electrochemical cycling. The samples show pronounced differences in their galvanostatic charge and discharge profiles which could be linked to structural and electronic response. Different desodiation and sodiation mechanisms were identified. The influence of [Fe(CN)6] vacancies and water content on the electrochemical performance was investigated

Bis(1,10-phenanthrolin-1-ium) hexa­bromidoplatinate(IV) dihydrate

The asymmetric unit of the title compound, (C12H9N2)2[PtBr6]·2H2O, contains a protonated 1,10-phenanthroline cation (H-phen), one half of a [PtBr6]2− anionic complex and a solvent water mol­ecule. The PtIV ion is located on an inversion centre and is coordinated in an octa­hedral environment by six Br atoms. The crystal structure displays numerous inter­molecular π–π inter­actions between six-membered rings of H-phen, with a shortest centroid–centroid distance of 3.670 (5) Å, and inter­molecular N—H⋯O, O—H⋯Br and O—H⋯N hydrogen bonds

High-pressure high-temperature stability of hcp-IrxOs1−x (x = 0.50 and 0.55) alloys

An in situ powder X-ray diffraction has been used for a monitoring a formation of hcp-Ir0.55Os0.45 alloy from (NH4)2[Ir0.55Os0.45Cl6] precursor. A crystalline intermediate compound and nanodimentional metallic particles with a large concentration of defects has been found as key intermediates in the thermal decomposition process in hydrogen flow. High-temperature stability of titled hcp-structured alloys has been investigated upon compression up to 11 GPa using a multi-anvil press and up to 80 GPa using laser-heated diamond-anvil cells to obtain a phase separation into fcc + hcp mixture. Obtained high-pressure high-temperature data allowed us to construct the first model for pressure-dependent Ir─Os phase diagram

Systems of subspaces of a unitary space

For a given poset, we consider its representations by systems of subspaces of a unitary space ordered by inclusion. We classify such systems for all posets for which an explicit classification is possible.Comment: 20 page

First hexagonal close packed high-entropy alloy with outstanding stability under extreme conditions and electrocatalytic activity for methanol oxidation

High-entropy alloys containing 5 and 6 platinum group metals have been prepared by thermal decomposition of single-source precursors non requiring high temperature or mechanical alloying. The prepared Ir0.19Os0.22Re0.21Rh0.20Ru0.19 alloy is the first example of a single-phase hexagonal high-entropy alloy. Heat treatment up to 1500 K and compression up to 45 GPa do not result in phase changes, a record temperature and pressure stability for a single-phase high-entropy alloy. The alloys show pronounced electrocatalytic activity in methanol oxidation, which opens a route for the use of high-entropy alloys as materials for sustainable energy conversion

High-pressure high-temperature tailoring of High Entropy Alloys for extreme environments

The exceptional performance of some High Entropy Alloys (HEAs) under extreme conditions holds out the possibility of new and exciting materials for engineers to exploit in future applications. In this work, instead of focusing solely on the effects of high temperature on HEAs, the effects of combined high temperature and high pressure were observed. Phase transformations occurring in a pristine HEA, the as-cast bcc–Al2CoCrFeNi, are heavily influenced by temperature, pressure, and by scandium additions. As-cast bcc–Al2CoCrFeNi and fcc–Al0.3CoCrFeNi HEAs are structurally stable below 60 GPa and do not undergo phase transitions. Addition of scandium to bcc–Al2CoCrFeNi results in the precipitation of hexagonal AlScM intermetallic (W-phase), which dissolves in the matrix after high-pressure high-temperature treatment. Addition of scandium and high-pressure sintering improve hardness and thermal stability of well-investigated fcc- and bcc- HEAs. The dissolution of the intermetallic in the main phase at high pressure suggests a new strategy in the design and optimization of HEAs