Substantiation of Optimal Design Parameters for New System Angle Headframes in State-ofthe Art Mines

The article formulates a new concept for design of the steel angle headframes of the new system for state-of-the art mines. One of the important structural elements of the steel headframe is a rig, installed above mouth of the vertical shaft and designed for equipment and passage of mine conveyances in shaft. A traditional frame rig of rectangular section is the most cumbersome and heavy-weight part of the headframe due its load-bearing. The proposed by the author headframe of multifunctional purpose, that includes a non-load bearing rig of circular cross-section, will allow reducing load on mine mouth and forgoing the sub-frame that greatly simplifies the task of finding an optimal design solution for the structure. There is an analysis of the most significant factors that significantly affect formation of the main design parameters of the sheave wheels and the rig of the examined headframe. The proposed idea of the independently operating structural units, one of which is a circular rig, significantly reduces a number of controlled variables, with which a designer can vary designing a headframe.

Improved Repeatability of Oscillatory Rheological Measurements for Polycarbosilane Melt

Dynamic Time Sweep of Rotational Rheometer has been Used to Study the Variation in Rheological Properties of Polymers with Time. Polycarbosilane (PCS) is a Solid Oligomer at Room Temperature and Becomes Melt at its Softening Temperature. Bubbles are Unavoidably Produced by Gasification of Low Molecular Weight Part of PCS, which Significantly Disturbs the Subsequent Rheological Measurement. however, the Rheological Data of PCS Melt Cannot Be Repeated on Conventional Dynamic Time Sweep Test even after Reducing the Bubble. in this Work, a Series of Oscillatory Rheological Measurements were Carried out by Temperature Control to Improve the Data Reliability. the Axial Force Data of PCS Melt were Manually Recorded and Compared before and during the Test, which Reflected the Response of PCS Melt to Temperature Change. the Results Confirmed that the Shear Stress of PCS Melt was Readily Affected by Temperature Alteration. the Data Repeatability of the Rheological Test was Evidently Improved for PCS Melt with the Temperature Control.

Serre weights for U ⁢ ( n ) {U(n)}

We study the weight part of (a generalisation of) Serre’s conjecture for mod l Galois representations associated to automorphic representations on unitary groups of rank n for odd primes l. Given a modular Galois representation, we use automorphy lifting theorems to prove that it is modular in many other weights. We make no assumptions on the ramification or inertial degrees of l. We give an explicit strengthened result when {n=3} and l splits completely in the underlying CM field.

THE WEIGHT PART OF SERRE’S CONJECTURE FOR

Let $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

SERRE WEIGHTS FOR LOCALLY REDUCIBLE TWO-DIMENSIONAL GALOIS REPRESENTATIONS

Let $F$ be a totally real field, and $v$ a place of $F$ dividing an odd prime $p$. We study the weight part of Serre’s conjecture for continuous totally odd representations $\overline{{\it\rho}}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbb{F}}_{p})$ that are reducible locally at $v$. Let $W$ be the set of predicted Serre weights for the semisimplification of $\overline{{\it\rho}}|_{G_{F_{v}}}$. We prove that, when $\overline{{\it\rho}}|_{G_{F_{v}}}$ is generic, the Serre weights in $W$ for which $\overline{{\it\rho}}$ is modular are exactly the ones that are predicted (assuming that $\overline{{\it\rho}}$ is modular). We also determine precisely which subsets of $W$ arise as predicted weights when $\overline{{\it\rho}}|_{G_{F_{v}}}$ varies with fixed generic semisimplification.

THE BREUIL–MÉZARD CONJECTURE FOR POTENTIALLY BARSOTTI–TATE REPRESENTATIONS

We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$ , $K$ a finite extension of $\mathbb{Q}_{p}$ , for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

A geometric perspective on the Breuil–Mézard conjecture

Let$p\gt 2$be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard conjecture for two-dimensional mod$p$representations of the absolute Galois group of${ \mathbb{Q} }_{p} $. We also state a conjectural generalization to$n$-dimensional representations of the absolute Galois group of an arbitrary finite extension of${ \mathbb{Q} }_{p} $, and give a conditional proof of this conjecture, subject to a certain$R= \mathbb{T} $-type theorem together with a strong version of the weight part of Serre’s conjecture for rank $n$unitary groups. We deduce an unconditional result in the case of two-dimensional potentially Barsotti–Tate representations.