On a question of Zhang

When studying the conjugacy class version of the Huppert’s ρ-σ conjecture, Jiping Zhang raised a number theory question. In this article, we provide examples to show that the question raised by Zhang is not always true in general.     

Factoriality for the reductive Zassenhaus variety and quantum enveloping algebra

Let U(g)$U(\mathfrak{g})$ be the enveloping algebra of a finite dimensional reductive Lie algebra g$\mathfrak{g}$ over an algebraically closed field of prime characteristic. Let _U?,P(s:)$U_{?,P}(\mathfrak{s}:)$ be the simply connected quantum enveloping algebra at the root of unity ?  , of a complex semi-simple finite dimensional Lie algebra s:$\mathfrak{s}:$. We show, by similar proofs, that the centers of both are factorial. While the first result was established by R. Tange [32] (by different methods), the second one confirms a conjecture in [4]. We also provide a general criterion for the factoriality of the centers of enveloping algebras in prime characteristic.     

Targeting tribbles homolog 3 (TRIB3) protein against type 2 diabetes for the identification of potential inhibitors by in silico screening

Tribbles homolog 3 (TRIB3) protein is inhibiting the insulin signaling by directly binding to the Akt/PKB leading to insulin resistance in the pancreas causing type 2 diabetes mellitus. Hence, TRIB3 protein is considered as a possible drug target for the new lead identification against type 2 diabetes. In the present study, the homology model of TRIB3 protein was generated to explore its biochemical function and molecular interactions in the new lead identification. The energy minimization of TRIB3 protein was carried out and evaluated by validation protocols for structure reliability. The druggable binding site of TRIB3 protein was identified for the virtual screening and molecular docking studies. The Asinex-fragments library of 22634 small molecules was docked at TRIB3 active site using the Glide module to identify new chemical entities. A total of 9 molecules were identified as final hits from virtual screening and their potency was ranked using Glide score, Glide energies, and residues interactions. The 6 prioritized lead molecules were further optimized using AutoDock, Prime MM/GBSA, and percentage of human oral absorption for the identification of potential leads. The molecules L2, L5, and L6 are identified as lead inhibitors and are showing consistent interactions with key residues Glu194 and Lys196 of TRIB3 protein. The identified potential leads were analyzed by ADME properties for their drug likeness and HergIC50 values are predicted for the prevention of preclinical failures. The present work sheds light on the identification of the best lead molecules against TRIB3 protein and offers a route to design as novel potential drug candidates for T2DM.     

Associated Prime Submodules of Finitely Generated Modules

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.     

Consecutive numbers with the same Legendre symbol

Let be an odd prime, and be a complete set of residues . The goal of the paper is to determine all the values of such that or , where is the Legendre symbol.

     

Repunit R49081 is a probable prime

The Repunit R is a probable prime. In order to prove primality R49080 must be approximately 33.3% factored. The status of this factorization is included.

     

The number of primes is finite

For a positive integer let and let . The number of primes of the form is finite, because if , then is divisible by . The heuristic argument is given by which there exists a prime such that for all large ; a computer check however shows that this prime has to be greater than . The conjecture that the numbers are squarefree is not true because .

     

Trace formula on the p-adic upper half-plane

This article aims at showing a p-adic analogue of Selberg's trace formula, which describes a duality between the spectrum of a Hilbert-Schmidt operator and the length of prime geodesics appearing in the p-adic upper half-plane associated with a hyperbolic discontinuous subgroup of . Then we construct Markov processes on the fundamental domain relative to such subgroups, to whose transition operators the trace formula applied and a p-adic analogue of prime geodesic theorem is proved.     

Hardness and phase analysis of IN 718 deformed at high strain rate

Specimens of the nickel base alloy IN 718 deformed at high strain rate (~10 s^–1), as realized in a screw press, have higher strength than parts forged conventionally in a hydraulic press. Microstructure analyses in light and transmission electron microscopes reveal the precipitation of very small Ni₃Nb particles (-phase) to be the reason for the increased hardness. Several processing routes are discussed and analyzed in relation to the TTT-diagram of IN 718.