ESSENTIAL APPROXIMATE POINT SPECTRA FOR UPPER TRIANGULAR MATRIX OF LINEAR RELATIONS

When A E ∈LR（H） and B E ∈LR（K） are given, for C E∈LR（K, H） we denoteby Mc the linear relation acting on the infinite dimensional separable Hilbert space H Kof the formIn this paper, we give the necessary and sufficient conditionson A and B for wh{ch Mc is upper semi-Fredholm with negative index or Weyl for some C C ∈LR（K, H）.

LEFT-RIGHT BROWDER LINEAR RELATIONS AND RIESZ PERTURBATIONS

A closed linear relation T in a Banach space X is called left(resp. right) Fredholm if it is upper(resp. lower) semi Fredholm and its range(resp. null space) is topologically complemented in X. We say that T is left(resp. right) Browder if it is left(resp. right)Fredholm and has a finite ascent(resp. descent). In this paper, we analyze the stability of the left(resp. right) Fredholm and the left(resp. right) Browder linear relations under commuting Riesz operator perturbations. Recent results of Zivkovic et al. to the case of bounded operators are covered.

SPECTRAL MAPPING THEOREM FOR ASCENT,ESSENTIAL ASCENT, DESCENT AND ESSENTIAL DESCENT SPECTRUM OF LINEAR RELATIONS

In [7], Cross showed that the spectrum of a linear relation T on a normed space satisfies the spectral mapping theorem. In this paper, we extend the notion of essential ascent and descent for an operator acting on a vector space to linear relations acting on Banach spaces. We focus to define and study the descent, essential descent, ascent and essential ascent spectrum of a linear relation everywhere defined on a Banach space X. In particular, we show that the corresponding spectrum satisfy the polynomial version of the spectral mapping theorem.

On Regular Linear Relations

For a closed linear relation in a Banach space the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semiFredholm linear relations which are regular.

Rational design of bimetallic atoms supported on C3N monolayer to break the linear relations for efficient electrochemical nitrogen reduction

Linear relations between the adsorption free energies of nitrogen reduction reaction(NRR)intermediates limit the catalytic activity of single atom catalysts(SACs)to reach the optimal region.Significant improvements in NRR activity require the balance of binding strength of reaction intermediates.Herein,we have investigated the C_(3)N-supported monometallic(M/C_(3)N)and bimetallic(M_(1)M_(2)/C_(3)N)atoms for the electrochemical NRR by using density functional theory(DFT)calculations.The results show that this linear relation does exist for SACs because all the intermediates bind to the same site on M/C_(3)N.But the synergistic effect of the two atoms in M_(1)M_(2)/C_(3)N can create a more flexible adsorption site for intermediates,which results in the decoupling of adsorption free energies of key intermediates.Subsequently,the fundamental limitation of scaling relations on limiting potentials is broken through.Most notably,the optimal limiting potential is increased from−0.63 V for M/C_(3)N to−0.20 V for M_(1)M_(2)/C_(3)N.In addition,the presence of bimetallic atoms can also effectively inhibit the hydrogen evolution reaction(HER)as well as improve the stability of the catalysts.This study proposes that the introduction of bimetallic atoms into C_(3)N is beneficial to break the linear relations and develop efficient NRR electrocatalysts.

Seasonal changes in the relationship between species richness and community biomass in grassland under grazing and exclosure, Horqin Sandy Land, northern China

How species diversityroductivity relationships respond to temporal dynamics and land use is still not clear in semi-arid grassland ecosystems. We analyzed seasonal changes of the relationships between vegetation cover, plant density, species richness, and above- ground biomass in grasslands under grazing and exclosure in the Horqin Sandy Land of northem China. Our results showed that in grazed and fenced grassland, vegetation cover, richness, and biomass were lower in April than in August, whereas plant density showed a reverse trend. Vegetation cover during the growing season and biomass in June and August were higher in fenced grassland than in grazed grassland, whereas plant density in April and June was lower in fenced grassland than in grazed grassland. A negative relationship between species richness and biomass was found in August in fenced grassland, and in grazed grassland the relationship between plant density and biomass changed from positive in April to negative in August. The relationship between the density of the dominant plant species and the total biomass also varied with seasonal changes and land use （grazing and exclosure）. These results suggest that long-term grazing, seasonal changes, and their interaction significantly influence vegetation cover, plant density, and bio- mass in grasslands. Plant species competition in fenced grassland results in seasonal changes of the relationship between species rich- ness and biomass. Long-term grazing also affects seasonal changes of the density and biomass of dominant plant species, which fur- tiler affects the seasonal relationship between plant density and biomass in grasslands. Our study demonstrates the importance of tem- poral dynamics and land use in understanding the relationship between species richness and ecosystem fianction.

Atomic Dispersed Hetero‑Pairs for Enhanced Electrocatalytic CO_(2)Reduction

Electrochemical carbon dioxide reduction reaction(CO_(2)RR)involves a variety of intermediates with highly correlated reaction and ad-desorption energies,hindering optimization of the catalytic activity.For example,increasing the binding of the*COOH to the active site will generally increase the*CO desorption energy.Breaking this relationship may be expected to dramatically improve the intrinsic activity of CO_(2)RR,but remains an unsolved challenge.Herein,we addressed this conundrum by constructing a unique atomic dispersed hetero-pair consisting of Mo-Fe di-atoms anchored on N-doped carbon carrier.This system shows an unprecedented CO_(2)RR intrinsic activity with TOF of 3336 h−1,high selectivity toward CO production,Faradaic efficiency of 95.96%at−0.60 V and excellent stability.Theoretical calculations show that the Mo-Fe diatomic sites increased the*COOH intermediate adsorption energy by bridging adsorption of*COOH intermediates.At the same time,d-d orbital coupling in the Mo-Fe di-atom results in electron delocalization and facilitates desorption of*CO intermediates.Thus,the undesirable correlation between these steps is broken.This work provides a promising approach,specifically the use of di-atoms,for breaking unfavorable relationships based on understanding of the catalytic mechanisms at the atomic scale.

PDRK:A General Kinetic Dispersion Relation Solver for Magnetized Plasma

A general,fast,and effective approach is developed for numerical calculation of kinetic plasma linear dispersion relations.The plasma dispersion function is approximated by J-pole expansion.Subsequently,the dispersion relation is transformed to a standard matrix eigenvalue problem of an equivalent linear system.Numerical solutions for the least damped or fastest growing modes using an 8-pole expansion are generally accurate;more strongly damped modes are less accurate,but are less likely to be of physical interest.In contrast to conventional approaches,such as Newton＇s iterative method,this approach can give either all the solutions in the system or a few solutions around the initial guess.It is also free from convergence problems.The approach is demonstrated for electrostatic dispersion equations with one-dimensional and twodimensional wavevectors,and for electromagnetic kinetic magnetized plasma dispersion relation for bi-Maxwellian distribution with relative parallel velocity flows between species.

Estimation of peak relative velocity and peak absolute acceleration of linear SDOF systems

We have found some mistakes in the article by Jianwei Song et al. （2007）. The revisions are given below：

HARDY-LITTLEWOOD-POLYA INEQUALITY FOR A LINEAR DIFFERENTIAL OPERATOR AND SOME RELATED OPTIMAL PROBLEMS

In this paper a generalized version of the classical Hardy-Littlewood-Polya inequality is given.Furthermore,the Stechkin's problem for a linear differential operator is solved in L_2(R), and the optimal recovery problem for such differential operator is considered.

线性热力学对大气系统的应用（Ⅱ）大气系统中线性唯象关系的例证

The linear phenomenological relations in the atmospheric boundary layer are proved indirectly using observational facts to combine linear thermodynamic theory and similarity theory in the boundary layer. Furthermore, it is proved that Ihe turbulent transport coefficients are in proportion to Ihe corresponding linear phenomenological coefficients. But the experimental facts show that the linear phenomenological relations are not (tenable in the atmospheric mixing layer because the turbulenl transport process is an intense non-linear process in the mixing layer. Hence the convection boundary layer is a thermodynamic stale in a non-linear region far from the equilibrium state. The geostrophic wind is a special cross-coupling phenomenon between the dynamic process and the thermodynamic process in the atmospheric system. It is a practical exemplification of a cross-coupling phenomenon in the atmospheric system.

Fractional-Linear Transformations of Operator Balls;Applications to Dynamical Systems

The operator sets, which are the subject of this paper, have been studied in many papers where, under different restrictions on the generating operators, convexity, compactness in the weak operator topology, and nonemptiness were proved for sets of different classes under study. Then the results obtained were used in these papers to solve several applied problems. Namely, they played the key role in establishing the dichotomy of nonautonomous dynamical systems, with either continuous or discrete time. In the present paper, we generalize and sharpen the already known criteria and obtain several new criteria for convexity, compactness, and nonemptiness of several special operator sets. Then, using the assertions obtained, we construct examples of sets of the form under study which are nonconvex, noncompact in the weak operator topology, as well as empty, and are generated by ＂smooth＂ operators of a special class. The existence problem for such sets remained open until the authors of this paper announced some of its results.