Rectangular Well as Perturbation

We discuss a finite rectangular well as a perturbation for the infinite one with a depth $\lambda^2$ of the former as a perturbation parameter. In particular we consider a behaviour of energy levels in the well as functions of complex $\lambda$. It is found that all the levels of the same parity are defined on infinitely sheeted Riemann surfaces which topological structures are described in details. These structures differ considerably from those found in models investigated earlier. It is shown that perturbation series for all the levels converge what is in contrast with the known results of Bender and Wu. The last property is shown to hold also for the finite rectangular well with Dirac delta barier as a perturbation considered earlier by Ushveridze.Comment: 19 pages, 5 Postscript figures, uses psfig.st

Fundamental solution method applied to time evolution of two energy level systems: exact and adiabatic limit results

A method of fundamental solutions has been used to investigate transitions in two energy level systems with no level crossing in a real time. Compact formulas for transition probabilities have been found in their exact form as well as in their adiabatic limit. No interference effects resulting from many level complex crossings as announced by Joye, Mileti and Pfister (Phys. Rev. {\bf A44} 4280 (1991)) have been detected in either case. It is argued that these results of this work are incorrect. However, some effects of Berry's phases are confirmed.Comment: LaTeX2e, 23 pages, 8 EPS figures. Style correcte

Irreducible Representations of an Algebra underlying Hidden Symmetries of a class of Quasi Exactly Solvable Systems of Equations

The set of linear, differential operators preserving the vector space of couples of polynomials of degrees n and n-2 in one real variable leads to an abstract associative graded algebra A(2). The irreducible, finite dimensional representations of this algebra are classified into five infinite discrete sets and one exceptional case. Their matrix elements are given explicitely. The results are related to the theory of quasi exactly solvable equations.Comment: 38 pages, late

The Quantum Galilei Group

The quantum Galilei group $G_{\varkappa}$ is defined. The bicrossproduct structure of $G_{\varkappa}$ and the corresponding Lie algebra is revealed. The projective representations for the two-dimensional quantum Galilei group are constructed.Comment: AMSTe

Fractal properties of quantum spacetime

We show that in general a spacetime having a quantum group symmetry has also a scale dependent fractal dimension which deviates from its classical value at short scales, a phenomenon that resembles what observed in some approaches to quantum gravity. In particular we analyze the cases of a quantum sphere and of \k-Minkowski, the latter being relevant in the context of quantum gravity.Comment: 4 pages, 2 figures; some minor corrections; reference adde

κ\kappa-Deformed Statistics and Classical Fourmomentum Addition Law

We consider $\kappa$-deformed relativistic symmetries described algebraically by modified Majid-Ruegg bicrossproduct basis and investigate the quantization of field oscillators for the $\kappa$-deformed free scalar fields on $\kappa$-Minkowski space. By modification of standard multiplication rule, we postulate the $\kappa$-deformed algebra of bosonic creation and annihilation operators. Our algebra permits to define the n-particle states with classical addition law for the fourmomenta in a way which is not in contradiction with the nonsymmetric quantum fourmomentum coproduct. We introduce $\kappa$-deformed Fock space generated by our $\kappa$-deformed oscillators which satisfy the standard algebraic relations with modified $\kappa$-multiplication rule. We show that such a $\kappa$-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of $\kappa$-deformed algebra of oscillators in field-theoretic noncommutative framework.Comment: LaTeX, 12 pages. V2: second part of chapter 4 changed, new references and comments added. V3: formula (14) corrected. Some additional explanations added. V4: further comments about algebraic structure are adde

Multiple benefits of manure: the key to maintenance of soil fertility and restoration of depleted sandy soils on African smallholder farms

Manure is a key nutrient resource on smallholder farms in the tropics, especially on poorly buffered sandy soils, due to its multiple benefits for soil fertility. Farmers preferentially apply manure to fields closest to homesteads (homefields), which are more fertile than fields further away (outfields). A three-year experiment was established on homefields and outfields on sandy and clayey soils to assess the effects of mineral nitrogen (N) fertilizer application in combination with manure or mineral phosphorus (P) on maize yields and soil chemical properties. Significant maize responses to application of N and manure were observed on all fields except the depleted sandy outfield. Large amounts of manure (17 t ha¿1 year¿1) were required to significantly increase soil organic carbon (SOC), pH, available P, and base saturation, and restore productivity of the depleted sandy outfield. Sole N as ammonium nitrate (100 kg N ha¿1) or in combination with single superphosphate led to acidification of the sandy soils, with a decrease of up to 0.8 pH units after three seasons. In a greenhouse experiment, N and calcium (Ca) were identified as deficient in the sandy homefield, while N, P, Ca, and zinc (Zn) were deficient or low on the sandy outfield. The deficiencies of Ca and Zn were alleviated by the addition of manure. This study highlights the essential role of manure in sustaining and replenishing soil fertility on smallholder farms through its multiple effects, although it should be used in combination with N mineral fertilizers due to its low capacity to supply N

Effect of farmer management strategies on spatial variability of soil fertility and crop nutrient uptake in contrasting agro-ecological zones in Zimbabwe

Variability of soil fertility within, and across farms, poses a major challenge for increasing crop productivity in smallholder systems of sub-Saharan Africa. This study assessed the effect of farmers’ resource endowment and nutrient management strategies on variability in soil fertility and plant nutrient uptake between different fields in Gokwe South (ave. rainfall ~650 mm year-1; 16.3 persons km-2) and Murewa (ave. rainfall ~850 mm year-1; 44.1 persons km-2) districts, Zimbabwe. In Murewa, resource-endowed farmers applied manure (>3.5 t ha-1 year-1) on fields closest to their homesteads (homefields) and none to fields further away (outfields). In Gokwe the manure was not targeted to any particular field, and farmers quickly abandoned outfields and opened up new fields further way from the homestead once fertility had declined, but homefields were continually cultivated. Soil available P was higher in homefields (8–13 mg kg-1) of resource-endowed farmers than on outfields and all fields on resource constrained farms (2–6 mg kg-1) in Murewa. Soil fertility decreased with increasing distance from the homestead in Murewa while the reverse trend occurred in Gokwe South, indicating the impact of different soil fertility management strategies on spatial soil fertility gradients. In both districts, maize showed deficiency of N and P, implying that these were the most limiting nutrients. It was concluded that besides farmers’ access to resources, the direction of soil fertility gradients also depends on agro-ecological conditions which influence resource management strategie