We explore thermal fluctuations of thin planar membranes with a frozen spatially varying background metric and a shear modulus. We focus on a special class of D-dimensional "warped membranes" embedded in a d-dimensional space with d ≥ D + 1 and a preferred height profile characterized by quenched random Gaussian variables {h(α)(q)}, α = D + 1,...,d, in Fourier space with zero mean and a power-l...

Let p be a prime and let q be a power of p. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distanceseparable (MDS) codes with parameters [[tq, tq − 2d+ 2, d]]q for any 1 ≤ t ≤ q, 2 ≤ d ≤ ⌊ tq+q−1 q+1 ⌋+ 1, and [[t(q + 1) + 2, t(q + 1)− 2d+ 4, d]]q for any 1 ≤ t ≤ q− 1, 2 ≤ d ≤ t+2 with (p, t, d) 6= (...

New polarized fragmentation functions are introduced and justified, in addition to those conventional ones assumed to be independent of the helicity of the parent parton. It is demonstrated that due to our present ignorance concerning these new parton–spin dependent leading–twist fragmentation functions, it is impossible to utilize current experiments on spin–dependent semi–inclusive deep inela...

The quadrance between two points A 1 = (x 1 , y 1) and A 2 = (x 2 , y 2) is the number Q(A 1 , A 2) = (x 1 − x 2) 2 + (y 1 − y 2) 2. Let q be an odd prime power and F q be the finite field with q elements. The unit-quadrance graph D q has the vertex set F 2 q , and X, Y ∈ F 2 q are adjacent if and only if Q(A 1 , A 2) = 1. Let χ(F 2 q) be the chromatic number of graph D q. In this note, we will...

‎Hensel [K‎. ‎Hensel‎, ‎Deutsch‎. ‎Math‎. ‎Verein‎, ‎{6} (1897), ‎83-88.] discovered the $p$-adic number as a‎ ‎number theoretical analogue of power series in complex analysis‎. ‎Fix ‎a prime number $p$‎. ‎for any nonzero rational number $x$‎, ‎there‎ ‎exists a unique integer $n_x inmathbb{Z}$ such that $x = ‎frac{a}{b}p^{n_x}$‎, ‎where $a$ and $b$ are integers not divisible by ‎$p$‎. ‎Then $|x...

Let $$(R,\mathfrak {m})$$ be a Noetherian local ring of prime characteristic p and Q an $$\mathfrak {m}$$ -primary parameter ideal. We give criteria for F-rationality R using the tight Hilbert function $$H^*_Q(n)=\ell (R/(Q^n)^*)$$ coefficient $$e_1^*(Q)$$ polynomial $$P^*_Q(n)=\sum _{i=0}^d(-1)^ie_i^*(Q)\left( {\begin{array}{c}n+d-1-i\\ d-i\end{array}}\right) .$$ obtain lower bound equidimensi...

An M-space is a metric space (X, d) having the property that for each pair of points p, q ∈ X with d(p, q) = λ and for each real number α ∈ [0, λ], there is a unique rα ∈ X such that d(p, rα) = α and d(rα, q) = λ − α. In an M-space (X, d), we say that metric segments have unique prolongations if points p, q, r, s satisfy d(p, q) + d(q, r) = d(p, r), d(p, q) + d(q, s) = d(p, s) and d(q, r) = d(q...

For m > n ≥ 0 and 1 ≤ d ≤ m, it is shown that the q-Euler number E 2m (q) is congruent to q m−n E 2n (q) mod (1 + q d) if and only if m ≡ n mod d. The q-Salié number S 2n (q) is shown to be divisible by (1 + q 2r+1) ⌊ n 2r+1 ⌋ for any r ≥ 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.

The dilute-to-dense transition of granular flow of particle size d(0) is studied experimentally in a two-dimensional channel (width D) with confined exit (width d). Our results show that with fixed d and D there is a maximum inflow rate Q(c) above which the flow changes from dilute to dense and the outflow rate drops abruptly from Q(c) to a dense rate Q(d). A rescaled critical rate q(c) ...