A Construction of the Level 3 Modules for the Affine Lie Algebra and a New Combinatorial Identity of the Rogers-Ramanujan Type

We obtain a vertex operator construction of level 3 standard representations for the affine Lie algebra . As a corollary, we also get new conbinatorial identities.

     

On the extreme spectral properties of Toeplitz matrices generated byL 1 functions with several minima/maxima

In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue ₁ ⁽ⁿ⁾ of symmetric (Hermitian)n ×n Toeplitz matricesT _n (f) generated by an integrable functionf defined in [–, ]. In [7, 8, 11] it is shown that ₁ ⁽ⁿ⁾ tends to essinff =m _f in the following way: ₁ ⁽ⁿ⁾ –m _f 1/n ^2k . These authors use three assumptions:A1)f –m _f has a zero inx =x ₀ of order 2k.A2)f is continuous and at leastC ^2k in a neighborhood ofx ₀.A3)x =x ₀ is the unique global minimum off in [–, ]. In [10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf L ¹[–, ]. In this paper we further extend this theory to the case of a functionf L ¹[–, ] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off –m _f is the only parameter which characterizes the rate of convergence of ₁ ⁽ⁿ⁾ tom _f .     

Minimum blocking configurations

Some results on configurations realizing minimum blocking sets of a finite projective plane are obtained by introducing a suitable attraction property.Dedicated to Professor M. Scafati Tallini on the occasion of her 65th birthday     

Formation and stability of pyrophosphate complexes with aliphatic amines in aqueous solution

The complex formation between pyrophosphate (P(2)O(7)(4-)) and protonated methylamine, ethylenediamine, diethylenetriamine, triethylenetetramine, tetraethylenepentamine and pentaethylenehexamine, has been studied potentiometrically, in aqueous solution, at 25 degrees C. It was found that the species ALH(q)(A = amine, L = P(2)O(7)(4-)) are formed with q = 1 ... n(n = 3, 5, 6, 7, 7 and 8 for the above amines respectively). Mono- and di-amines form species A(p)LH(q) too. The stability of these species is quite high [e.g. H(4)A(4+) + HL(3-)ALH(5)(+): log K = 8.1 (A = pentaethylenehexamine)] and depends strictly on the charges involved in the formation reaction. Charges of reactants being equal, the stability trend is penten > tetren > trien > dien > en > meta and cationic mixed species are more stable than anionic ones. These results are discussed in relation to speciation problems in natural and biological fluids.     

Optimal life strategies in organisms exposed to recurrent critical events

The optimal partition of energy between survival and reproduction is considered for a population subject to recurrent and potentially lethal critical events. The best strategy is found by maximizing fitness, a functional derived from the Lotka equation. The dynamics is governed by a second-order, age-varying, nonlinear system. The energy storage and the probability of survival are the state variables, while the amounts of energy placed into and withdrawn from the storage are the controls. The optimal life strategy is shown to be as follows: build up the storage at the very beginning of life, and then progressively deplete it to resist the critical events.This work was partially supported by MURST Project Ricerche sui Fondamenti della Conservazione della Natura. The authors would like to thank C. Ricci for suggesting the problem and C. Matessi for helpful discussion.     

Copper(II) complexes of N-(phosphonomethyl)glycine in aqueous solution: a thermodynamic and spectrophotometric study

Copper(II) complexes with the herbicide N-(phosphonomethyl)glycine (glyphosate) have been investigated in aqueous solution by means of pH-metric measurements at different temperatures, 5 相似文献   

Synthesis and X-Ray Analysis of a Nickel Triphenyl-Cyclopropenyl Complex. A mo-interpretation of the geometry and bonding in the (C3H3)ML3-type of complexes

The reaction of [(C₃Ph₃)Ni(PPh₃)₂]ClO₄ with P(CH₂CH₂PPh₂)₃(pp₃) and NaBPh₄ yields the [(C₃Ph₃)Ni(pp₃)]BPh₄-complex. After long exposure of the solution of this compound in acetone/butanol to the air a new derivative [(C₃Ph₃)-Ni(pp₂po)]BPh₄· 0.5 C₄H₉OH, where pp₂po is (Ph₂PCH₂CH₂)₂P(CH₂ CH₂POPh₂), is obtained. Complete X-ray analysis has been carried out for the latter complex: a=18.303 (5); b=29.445 (6), c=13.305 (5) Å, β=112.70 (9)°; space group monoclinic, P2₁/a, Z=4. Disorder problems were encountered in the refinement of the structure. The best R is 0.093. One of the arms of the parent pp₃-molecule, not coordinated to the metal, undergoes oxidation. The Ni-atom, coordinated by the three remaining P-atoms of the ligand, is also linked in a roughly η³-mode to the cyclopropenium ligand. The geometry of the molecule is examined in detail. Extended HMO-calculations were performed to interpret how the variation of P? Ni? P angles affects the bonding between the NiP₃- and C₃H₃-fragments. The conclusion is that the overall energy of the complex may be lowered in spite of a weakening of the Ni-cyclopropenium linkage. Extensions are made to other systems containing a linkage between a metal and a X₃-ring (X=P,As).