Hopf algebras and the logarithm of the S-transform in free probability

Let k be a positive integer and let G_k denote the set of non-commutative k-variable distributions \mu such that \mu (X_1) = ... = \mu (X_k) = 1. G_k is a group under the operation of free multiplicative convolution. We identify G_k as the group of characters of a certain Hopf algebra Y_k. Then, by using the log map from characters to infinitesimal characters of Y_k, we introduce a transform LS_{\mu} for distributions \mu in G_k. The main property of the LS-transform is that it linearizes commuting products in G_k. For \mu in G_k, the transform LS_{\mu} is a power series in k non-commuting indeterminates; its coefficients can be computed from the coefficients of the R-transform of \mu by using summations over chains in the lattices NC(n) of non-crossing partitions. In the particular case k=1 one has that Y_1 is naturally isomorphic to the Hopf algebra Sym of symmetric functions, and that the LS-transform is very closely related to the logarithm of the S-transform of Voiculescu, by the formula LS(z) = - z log S(z). In this case the group G_1 can be identified as the group of characters of Sym, in such a way that the S-transform, its reciprocal 1/S and its logarithm log S relate in a natural sense to the sequences of complete, elementary and respectively power sum symmetric functions.Comment: Version with minor revisions, to appear in Transactions of the American Mathematical Society. 39 pages, no figure

Anatomical and Physiological Plasticity in Leymus chinensis (Poaceae) along Large-Scale Longitudinal Gradient in Northeast China

Although it has been widely accepted that global changes will pose the most important constrains to plant survival and distribution, our knowledge of the adaptive mechanism for plant with large-scale environmental changes (e.g. drought and high temperature) remains limited.An experiment was conducted to examine anatomical and physiological plasticity in Leymus chinensis along a large-scale geographical gradient from 115° to 124°E in northeast China. Ten sites selected for plant sampling at the gradient have approximately theoretical radiation, but differ in precipitation and elevation. The significantly increasing in leaf thickness, leaf mass per area, vessel and vascular diameters, and decreasing in stoma density and stoma index exhibited more obvious xerophil-liked traits for the species from the moist meadow grassland sites in contrast to that from the dry steppe and desert sites. Significant increase in proline and soluble sugar accumulation, K(+)/Na(+) for the species with the increasing of stresses along the gradient showed that osmotic adjustment was enhanced.Obvious xerophytic anatomical traits and stronger osmotic adjustment in stress conditions suggested that the plants have much more anatomical and physiological flexibilities than those in non-stress habitats along the large-scale gradient

Cracking Behavior of Atmospheric Plasma-Sprayed 8YSZ Thermal Barrier Coatings during Thermal Shock Test

The failure of plasma-sprayed thermal barrier coatings (TBCs) during service is usually related to the cracking behavior. In this study, plasma-sprayed TBCs were prepared with two kinds of agglomerated sintered yttria-stabilized zirconia (YSZ) powders with different particle sizes. The evolution of mechanical properties and crack propagation behavior of the coatings during the whole life stage were studied by a thermal shock test. The effect of powder particle size on the cracking behavior of the TBCs during thermal shock was analyzed from the aspect of pore structure, mechanical properties, and stress state of the coatings. The crack propagation and coalescence in the direction parallel to the substrate in the coating is the main factor leading to the spalling failure of the coating during thermal shock. Although the coating prepared by fine YSZ has higher fracture toughness, the lower strain tolerance will increase the cracking driving force on the crack tip of the coating during thermal shock, and the cracks in the coating propagate merge at a faster rate during thermal shock. The larger porosity and pore size of the coating prepared by coarse YSZ help the coating suffer less thermal stress during thermal shock. Although the existence of pores reduces the fracture toughness of the coating to a certain extent, the increase of strain tolerance reduces the crack growth rate in the coating, so the coating has a longer life

The Sufficient Conditions for Orthogonal Matching Pursuit to Exactly Reconstruct Sparse Polynomials

Orthogonal matching pursuit (OMP for short) is a classical method for sparse signal recovery in compressed sensing. In this paper, we consider the application of OMP to reconstruct sparse polynomials generated by uniformly bounded orthonormal systems, which is an extension of the work on OMP to reconstruct sparse trigonometric polynomials. Firstly, in both cases of sampled data with and without noise, sufficient conditions for OMP to recover the coefficient vector of a sparse polynomial are given, which are more loose than the existing results. Then, based on a more accurate estimation of the mutual coherence of a structured random matrix, the recovery guarantees and success probabilities for OMP to reconstruct sparse polynomials are obtained with the help of those sufficient conditions. In addition, the error estimation for the recovered coefficient vector is gained when the sampled data contain noise. Finally, the validity and correctness of the theoretical conclusions are verified by numerical experiments

The Sufficient Conditions for Orthogonal Matching Pursuit to Exactly Reconstruct Sparse Polynomials

Orthogonal matching pursuit (OMP for short) is a classical method for sparse signal recovery in compressed sensing. In this paper, we consider the application of OMP to reconstruct sparse polynomials generated by uniformly bounded orthonormal systems, which is an extension of the work on OMP to reconstruct sparse trigonometric polynomials. Firstly, in both cases of sampled data with and without noise, sufficient conditions for OMP to recover the coefficient vector of a sparse polynomial are given, which are more loose than the existing results. Then, based on a more accurate estimation of the mutual coherence of a structured random matrix, the recovery guarantees and success probabilities for OMP to reconstruct sparse polynomials are obtained with the help of those sufficient conditions. In addition, the error estimation for the recovered coefficient vector is gained when the sampled data contain noise. Finally, the validity and correctness of the theoretical conclusions are verified by numerical experiments

The Optimal Shape Parameter for the Least Squares Approximation Based on the Radial Basis Function

The radial basis function (RBF) is a class of approximation functions commonly used in interpolation and least squares. The RBF is especially suitable for scattered data approximation and high dimensional function approximation. The smoothness and approximation accuracy of the RBF are affected by its shape parameter. There has been some research on the shape parameter, but the research on the optimal shape parameter of the least squares based on the RBF is scarce. This paper proposes a way for the measurement of the optimal shape parameter of the least squares approximation based on the RBF and an algorithm to solve the corresponding optimal parameter. The method consists of considering the shape parameter as an optimization variable of the least squares problem, such that the linear least squares problem becomes nonlinear. A dimensionality reduction is applied to the nonlinear least squares problem in order to simplify the objective function. To solve the optimization problem efficiently after the dimensional reduction, the derivative-free optimization is adopted. The numerical experiments indicate that the proposed method is efficient and reliable. Multiple kinds of RBFs are tested for their effects and compared. It is found through the experiments that the RBF least squares with the optimal shape parameter is much better than the polynomial least squares. The method is successfully applied to the fitting of real data

The Transport Performance of Feedback Coupled Brownian Ratchets

The directed transport properties of feedback coupled Brownian ratchets under the effect of external periodic force and load force are investigated. The influence of the coupling strength on the transport properties of coupled system is discussed in detail. It is found that the centre-of-mass mean velocity, the average diffusion coefficient and the energy conversion efficiency of the feedback system can achieve the maximum for the different optimal coupling strength, and the maximum move to right with the increase of the natural length. It implies that the optimal coupling strength and natural length can promote the directed transport and energy conversion efficiency of the feedback ratchets

The Transport Performance of Feedback Coupled Brownian Ratchets

The directed transport properties of feedback coupled Brownian ratchets under the effect of external periodic force and load force are investigated. The influence of the coupling strength on the transport properties of coupled system is discussed in detail. It is found that the centre-of-mass mean velocity, the average diffusion coefficient and the energy conversion efficiency of the feedback system can achieve the maximum for the different optimal coupling strength, and the maximum move to right with the increase of the natural length. It implies that the optimal coupling strength and natural length can promote the directed transport and energy conversion efficiency of the feedback ratchets

Evaluation of the industrial cooperation of the Guangdong Hong Kong Macao Greater Bay Area: Based on the Origin-Destination pairs of industrial parks and coupling model efficiency

The industrial cooperation of Guangdong Hong Kong Macao Greater Bay Area (GBA) is one of the leading regional development strategies of this world-class urban agglomeration. This study constructed the industrial cooperation network based on the travel Origin-Destination (OD) connections among industrial parks. A multi-dimensional industrial cooperation and industrial development calculation index system were also set up to measure the nonlinear interaction relationship between them. The research found that an industrial collaboration network has been basically formed in the GBA, particularly presented by major cities. Some undeveloped cities may receive more benefits in the industrial collaboration network. The Covid-19 pandemic has had an impact in terms of within city connection instead of cross-city industrial cooperation. In addition, the degree of coupling between urban industrial coordination and urban industrial development has improved significantly over one decade and taking industrial collaboration as the input variables, industrial collaboration efficiently leads to industrial development outputs in almost every city in the GBA. Practically, decision makers should encourage and support intercity industrial collaboration, particularly between cities with closer geographic proximity, as it has been found to result in stronger cooperation and better economic enhancement. In addition, although industrial collaboration does not guarantee industrial development, when the collaboration systems and policies are enhanced, the synergy and coordination between them gradually improve. This highlights the potential benefits of continued investment in industrial collaboration for economic development

Soil Microbial Properties and Plant Growth Responses to Carbon and Water Addition in a Temperate Steppe: The Importance of Nutrient Availability

Background: Global climatic change is generally expected to stimulate net primary production, and consequently increase soil carbon (C) input. The enhanced C input together with potentially increased precipitation may affect soil microbial processes and plant growth. Methodology/Principal Findings: To examine the effects of C and water additions on soil microbial properties and plant growth, we conducted an experiment lasting two years in a temperate steppe of northeastern China. We found that soil C and water additions significantly affected microbial properties and stimulated plant growth. Carbon addition significantly increased soil microbial biomass and activity but had a limited effect on microbial community structure. Water addition significantly increased soil microbial activity in the first year but the response to water decreased in the second year. The water-induced changes of microbial activity could be ascribed to decreased soil nitrogen (N) availability and to the shift in soil microbial community structure. However, no water effect on soil microbial activity was visible under C addition during the two years, likely because C addition alleviated nutrient limitation of soil microbes. In addition, C and water additions interacted to affect plant functional group composition. Water addition significantly increased the ratio of grass to forb biomass in C addition plots but showed only minor effects under ambient C levels. Our results suggest that soil microbial activity and plant growth are limited by nutrient (C and N) and water availability, and highlight the importance of nutrien