Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComparison between the non-crossing and the non-crossing on lines properties
262
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In the recent paper [2], it was proved that the closure of the planar diffeomorphisms in the Sobolev norm consists of the functions which are non-crossing (NC), i.e., the functions which can be uniformly approximated by continuous one-to-one functions on the grids. A deep simplification of this property is to consider curves instead of grids, so considering functions which are non-crossing on lines (NCL). Since the NCL property is way easier to check, it would be extremely positive if they actually coincide, while it is only obvious that NC implies NCL. We show that in general NCL does not imply NC, but the implication becomes true with the additional assumption that $det(Du)>0$ a.e., which is a very common assumption in nonlinear elasticity.
rate research
Read More
A comparative study of the numerical renormalization group and non-crossing approximation results for the spectral functions of the $U=infty$ Anderson impurity model is carried out. The non-crossing approximation is the simplest conserving approximation and has led to useful insights into strongly correlated models of magnetic impurities. At low energies and temperatures the method is known to be inaccurate for dynamical properties due to the appearance of singularities in the physical Greens functions. The problems in developing alternative reliable theories for dynamical properties have made it difficult to quantify these inaccuracies. As a first step in obtaining a theory which is valid also in the low energy regime, we identify the origin of the problems within the NCA. We show, by comparison with close to exact NRG calculations for the auxiliary and physical particle spectral functions, that the main source of error in the NCA is in the lack of vertex corrections in the convolution formulae for physical Greens functions. We show that the dynamics of the auxiliary particles within NCA is essentially correct for a large parameter region, including the physically interesting Kondo regime, for all energy scales down to $T_{0}$, the low energy scale of the model, and often well below this scale. Despite the satisfactory description of the auxiliary particle dynamics, the physical spectral functions are not obtained accurately on scales $sim T_{0}$. Our results suggest that self--consistent conserving approximations which include vertex terms may provide a highly accurate way of dealing with strongly correlated systems at low temperatures.
Let $P$ be a set of $2n$ points in the plane, and let $M_{rm C}$ (resp., $M_{rm NC}$) denote a bottleneck matching (resp., a bottleneck non-crossing matching) of $P$. We study the problem of computing $M_{rm NC}$. We first prove that the problem is NP-hard and does not admit a PTAS. Then, we present an $O(n^{1.5}log^{0.5} n)$-time algorithm that computes a non-crossing matching $M$ of $P$, such that $bn(M) le 2sqrt{10} cdot bn(M_{rm NC})$, where $bn(M)$ is the length of a longest edge in $M$. An interesting implication of our construction is that $bn(M_{rm NC})/bn(M_{rm C}) le 2sqrt{10}$.
A non-crossing pairing on a bitstring matches 1s and 0s in a manner such that the pairing diagram is nonintersecting. By considering such pairings on arbitrary bitstrings $1^{n_1} 0^{m_1} ... 1^{n_r} 0^{m_r}$, we generalize classical problems from the theory of Catalan structures. In particular, it is very difficult to find useful explicit formulas for the enumeration function $phi(n_1, m_1, ..., n_r, m_r)$, which counts the number of pairings as a function of the underlying bitstring. We determine explicit formulas for $phi$, and also prove general upper bounds in terms of Fuss-Catalan numbers by relating non-crossing pairings to other generalized Catalan structures (that are in some sense more natural). This enumeration problem arises in the theory of random matrices and free probability.
We consider $m$-divisible non-crossing partitions of ${1,2,ldots,mn}$ with the property that for some $tleq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapotons $M$-triangle in this setting and conjecture a combinatorial interpretation for the $H$-triangle. This conjecture is proved for $m=1$.
In this paper, the problem of pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding patterns of length one and two are obtained. Lagrange inversion formula is used to obtain the explicit formulas for some special cases. Bijection is also established between generalized non-crossing trees with special pattern avoidance and the little Schr{o}der paths.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
