Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic $p$, and with coefficients in a field of odd characteristic $?\ne p$): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We provide a quick proof, by sorting characters according to the dimension of the corresponding Springer fibre, an invariant which is directly computable from symbols.     

Nb(iPrNPMe2)3Fe–PMe3: A potential high reactivity heterobimetallic catalyst for acetylene cycloadditions

Alkynes cycloaddition reactions are powerful tools for constructing cyclic molecules with optimal atom efficiency, but these reactions cannot proceed at ambient temperature without transition-metal catalysts. In this work, a heterobimetallic complex featuring an Nb–Fe triple bond, Nb(ⁱPrNPMe₂)₃Fe–PMe₃, has been evaluated as the potential catalyst for acetylene cycloaddition, using density functional theory. The calculated results show that the singlet-state (i.e. ground-state) Nb(ⁱPrNPMe₂)₃Fe–PMe₃ can be applied to benzene synthesis, but is not suitable for cyclobutadiene. Benzene can be obtained easily at room temperature and is the unique product on the singlet potential surface. The irradiation of infrared-red light can drive the excitation of singlet Nb(ⁱPrNPMe₂)₃Fe–PMe₃ to its triplet state. Both benzene and cyclobutadiene can be formed on the triplet reaction potential surface due to their low energy barriers. Therefore, Nb(ⁱPrNPMe₂)₃Fe–PMe₃ is a potential high reactivity heterobimetallic catalyst for the cyclotrimerization of alkynes. In the reaction process, the catalytic active site of Nb(ⁱPrNPMe₂)₃Fe–PMe₃ moves from niobium to iron.     

FeOOH@Metal–Organic Framework Core–Satellite Nanocomposites for the Serum Metabolic Fingerprinting of Gynecological Cancers

High-throughput metabolic analysis is of significance in diagnostics, while tedious sample pretreatment has largely hindered its clinic application. Herein, we designed FeOOH@ZIF-8 composites with enhanced ionization efficiency and size-exclusion effect for laser desorption/ionization mass spectrometry (LDI-MS)-based metabolic diagnosis of gynecological cancers. The FeOOH@ZIF-8-assisted LDI-MS achieved rapid, sensitive, and selective metabolic fingerprints of the native serum without any enrichment or purification. Further analysis of extracted serum metabolic fingerprints successfully discriminated patients with gynecological cancers (GCs) from healthy controls and also differentiated three major subtypes of GCs. Given the low cost, high-throughput, and easy operation, our approach brings a new dimension to disease analysis and classification.     

Estimation of cosmological parameters,stability analysis and energy conditions in viable modified gravity

In the present paper, we have investigated the Friedmann-Robertson-Walker (FRW) model in viable f(R, T) gravity with f(R, T) function proposed as $f(R,T)=R+\xi T^{1/2},$ where ξ is an arbitrary constant, R is the scalar curvature and T is the trace of stress energy tensor. Defining the scale factor, the field equations are solved numerically and the energy conditions are analyzed. Further, determining Hubble parameter and deceleration parameter, their present values are estimated. Furthermore, 57 redshift data (42 redshift data from Supernova Cosmology project and 15 redshift data from Calán/ Tolono Supernova survey) are used to estimate the age of the universe and to find the best fit curves for luminosity distance and apparent magnitude.     

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.     

Ionic Conductivity of β‐Cyclodextrin–Polyethylene‐Oxide/Alkali‐Metal‐Salt Complex

Highly conductive, crystalline, polymer electrolytes, β‐cyclodextrin (β‐CD)–polyethylene oxide (PEO)/LiAsF₆ and β‐CD–PEO/NaAsF₆, were prepared through supramolecular self‐assembly of PEO, β‐CD, and LiAsF₆/NaAsF₆. The assembled β‐CDs form nanochannels in which the PEO/X⁺ (X=Li, Na) complexes are confined. The nanochannels provide a pathway for directional motion of the alkali metal ions and, at the same time, separate the cations and the anions by size exclusion.     

Integral closure of a quartic extension

Let R be a Noetherian unique factorization domain such that 2 and 3 are units,and let A=R[α]be a quartic extension over R by adding a rootαof an irreducible quartic polynomial p(z)=z4+az2+bz+c over R.We will compute explicitly the integral closure of A in its fraction field,which is based on a proper factorization of the coefficients and the algebraic invariants of p(z).In fact,we get the factorization by resolving the singularities of a plane curve defined by z4+a(x)z2+b(x)z+c(x)=0.The integral closure is expressed as a syzygy module and the syzygy equations are given explicitly.We compute also the ramifications of the integral closure over R.