Whitehead test modules

A (right -) module is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module , implies is projective. Dually, i-test modules are defined. For example, is a p-test abelian group iff each Whitehead group is free. Our first main result says that if is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring , there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring.

A non-semisimple ring is said to be fully saturated (-saturated) provided that all non-projective (-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, . The four parameters involved here are skew-fields and , and natural numbers . For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of -modules.

     

Relations between the wave solutions and the jost solutions off the energy shell for local central potentials

Using the Green function techniques we express the wave solutions of the radial inhomogeneous Schrödinger equation by means of the on-shell Jost and regular solutions. Making use of their boundary behaviour atr = andr = 0 we reexpress them alternatively in terms of the off-shell Jost and regular solutions. Relations among the different generalized (fully off the energy shell) Jost functions are derived and the radial matrix elements of the transition and reaction (reactance) operators are given in terms of these Jost functions. The relations reflect the principle of detailed balance.     

Tilting theory and the finitistic dimension conjectures

Let be a right noetherian ring and let be the class of all finitely presented modules of finite projective dimension. We prove that findim iff there is an (infinitely generated) tilting module such that pd and . If is an artin algebra, then can be taken to be finitely generated iff is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.

     

Non-radiative decay of excitons on ionized donors and acceptors in semi-conducting and dielectric crystals

The possible processes of non-radiative decay of excitons with a large radius on ionized donors and acceptors in semi-conducting and dielectric crystals are theoretically investigated in the first approximation by means of time perturbation calculus of quantum mechanics using the method of second quantization. Theoretical relations are found for the probability per sec of non-radiative decay of excitons on ionized acceptors in Cu₂O crystals and on electron trapping levels in CdS crystals.     

Double quantum transitions in free radicals

The observation of double quantum transitions on three kinds of free radicals (DPPH, DPANO, ultramarine) at temperatures of liquid nitrogen is described. Signals having an absorption and emission character were observed in the 12-5 MHz–18 MHz frequency band in the direction parallel to the external magnetic fieldH ₀ and their dependence on different parameters was studied.Bloch equations were solved for two mutually perpendicular high-frequency fields and expressions were found for the components of dynamic susceptibility in theH ₀ direction. The experiments agree with theory.The possibility of using double quantum transitions for measuring relaxation times is discussed.

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(DPPH, DPANO, ) . 12,5MHz–18MHz H ₀, ; . H ₀. . .

     

An alternative derivation of the pulse solutions of the KdV hierarchy

An alternative approach issues from the Appelle transformation of the Schrödinger equation. One solves the inverse problem for the transformed equation, a general solution of which is a quadratic form of two independent solutions of the primary Schrödinger equation. If the potential in the Schrödinger equation obeys one equation of the KdV hierarchy, the time derivative of this form is a linear combination of the form and its space derivative. The coefficients in the combination depend on the potential and the energy parameter of the Schrödinger equation only. This relation also determines the time dependence of the spectral data which along with the solution of the inverse problem gives the solution of the KdV equations as usual.     

Non-radiative transformation of a singlet exciton into a triplet one in organic molecular crystals

The probability of non-radiative transformation of a singlet exciton into a triplet one in organic molecular crystals is calculated. Mechanisms caused by intermolecular interactions apparent at grouping molecules into a crystal together with spin-orbital molecular interactions are considered as mechanisms of non-radiative transformation. Dependence of the calculated probability on the temperature and the character of an energetic spectrum of a singlet and a triplet exciton is discussed. On the basis of the calculated probabilities for various specific mechanisms the influence of intermolecular interactions, when arranging molecules into a crystal, on the intensity of non-radiative transitions between singlet and triplet excited states of molecules in the crystal is analyzed.Na Slovance 2, Praha 8, Czechoslovakia.     

Nonradiative heterofission of a singlet host exciton into a pair of triplet electronic excitations in doped aromatic hydrocarbon molecular crystals

The nonradiative heterofission process of a singlet host exciton into one triplet host exciton and one triplet guest molecule electronic excitation in some aromatic hydrocarbon molecular crystals is investigated theoretically. The formulae for the heterofission rate constants are derived and discussed and used for the numerical estimates. The conditions for the occurence of the heterofission of a singlet host exciton in the investigated crystals are discussed.     

Nonradiative heterofusion of excitons in doped organic molecular crystals

The nonradiative heterofusion of a host exciton with a guest molecule electronic excitation resulting in a higher electronic neutral single excited state is examined theoretically in the doped organic molecular crystals. The heterofusion process is treated as a nonradiative spontaneous multi-lattice phonon transition of the crystal from the electronic double excited initial state to the electronic single excited final state under the action of the interaction between the host molecules and the guest molecule. Two heterofusion channels are considered: the first one resulting in a higher electronic excited state of the guest molecule and the second one resulting in a higher host exciton state. The corresponding rate constants are calculated and discussed within the framework of the first order time dependent perturbation theory.