اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA random field formulation of Hookes law in all elasticity classes
79
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
For each of the $8$ isotropy classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set $mathsf{V}$ of the corresponding class, that is isotropic with respect to the natural orthogonal representation of a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders $1$ and $2$ of such a field, and the fields spectral expansion.
قيم البحث
اقرأ أيضاً
The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the genera
lization of this law to random matrices exhibiting higher-order cyclic correlations between $k$-tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with $k$-fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable universality. We further extend our analysis to matrices and graphs with competing cycle motifs, which are described more generally by polytrochoid spectral boundaries.
Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrodinger operators into a new spectral perspective. Based on a stationary random version of a Floquet type fibration, we redu
ce the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space. We state a natural resonance conjecture for these fibered operators: in contrast with periodic and quasiperiodic settings, this would entail that Bloch waves do not exist as extended states, but rather as resonant modes, and this would justify the expected exponential decay of time correlations. Although this resonance conjecture remains open, we develop new tools for spectral analysis on the probability space, and in particular we show how ideas from Malliavin calculus lead to rigorous Mourre type results: we obtain an approximate dynamical resonance result and the first spectral proof of the decay of time correlations on the kinetic timescale. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques.
We introduce and study the following model for random resonances: we take a collection of point interactions $Upsilon_j$ generated by a simple finite point process in the 3-D space and consider the resonances of associated random Schrodinger Hamilton
ians $H_Upsilon = -Delta + ``sum mathfrak{m}(alpha) delta (x - Upsilon_j)``$. These resonances are zeroes of a random exponential polynomial, and so form a point process $Sigma (H_Upsilon)$ in the complex plane $mathbb{C}$. We show that the counting function for the set of random resonances $Sigma (H_Upsilon)$ in $mathbb{C}$-discs with growing radii possesses Weyl-type asymptotics almost surely for a uniform binomial process $Upsilon$, and obtain an explicit formula for the limiting distribution as $m to infty$ of the leading parameter of the asymptotic chain of `most narrow resonances generated by a sequence of uniform binomial processes $Upsilon^m$ with $m$ points. We also pose a general question about the limiting behavior of the point process formed by leading parameters of asymptotic sequences of resonances. Our study leads to questions about metric characteristics for the combinatorial geometry of $m$ samples of a random point in the 3-D space and related statistics of extreme values.
We introduce a simple yet powerful calculational tool useful in calculating averages of ratios and products of characteristic polynomials. The method is based on Dyson Brownian motion and Grassmann integration formula for determinants. It is intended
as an alternative to other RMT techniques applicable to general gaussian measures. Resulting formulas are exact for finite matrix size N and form integral representations convenient for large N asymptotics. Quantities obtained by the method can be interpreted as averages over matrix models with an external source. We provide several explicit and novel calculations showing a range of applications.
Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric ob
servable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
