Threshold Phenomena in Intense Electromagnetic Fields

The influence of intense electromagnetic fields on the formation and decay of quasistationary states of different quantum systems is investigated based on exact solutions of quantum equations for charged particle motion. The method allows examining systems where a spontaneous decay may occur as well as phenomena that occur only under the action of the field. Different values of the total magnetic moment of the system are taken into account in this consideration. A consistent use of the analytic continuation method allows obtaining nonlinear equations that determine complex energies in an external field. The asymptotic expansions for real and imaginary energy values under the action of weak and strong electromagnetic fields are investigated. The developed approach allows establishing the characteristic values for the length parameters that determine the formation of the processes in superstrong fields. We note that a significant decrease of distances in strong fields may lead to effects with a new characteristic length scale, characterizing a modified quantum electrodynamics (QED) formalism, namely, the QED with the fundamental mass formalism.     

The Resolution Method for One Reducible Class of Formulas of the First-Order Modal Logic S4

In the paper, we describe a resolution method for the family of the formulas of the form *ⁱ**(L ₁ L _s ), where i = 0 1 and L _j are modal literals. Negations here stand directly before classical atomic formulas, the formulas may contain constants. We also present the absorption tactics for a set of such formulas.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 481–492, October–December, 2004.     

Nonlinear Effects Induced by Vacuum Polarization by a Strong Electromagnetic Field in a Weak Static Electromagnetic Field

In the one-loop approximation of quantum electrodynamics, we study the Faraday rotation effect in the propagation of a weak linearly polarized wave through the field of a strong plane electromagnetic wave and also processes induced by a strong wave: the emission of a photon whose energy is an integer multiple of the strong-wave quantum and photon absorption with the emission of several photons identical (coherent) to the strong-wave quanta. The induced processes should occur in a vacuum in a superposition of external fields, namely, the field of a strong plane electromagnetic wave and a static spatially inhomogeneous electromagnetic field. We show that the induced photon emission in the field of a strong electromagnetic wave with circular polarization and in a static inhomogeneous field is equivalent to the merging of two strong-wave quanta with the production of one photon and that the induced photon absorption in the same combination of fields is equivalent to the decay (splitting) of a photon into two photons completely identical to the strong-wave quanta. All these effects are induced by the phenomenon of the vacuum polarization by a strong external time-dependent electromagnetic field. The probabilities of the induced processes are nontrivial nonlinear functions of the squared strength of the time-dependent field also depending on other parameters.     

Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

Transferring estimates for zonal convolution operators

_p - , . Ad- .     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

On the distribution of points of maximal deviation in complex Čebyšev approximation

f — , . p _n (f) f . , n+2 , f–p _n (f) . , n . , .

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On the distribution of points of maximal deviation in complex ebyev approximation

     

A generalized integral with applications to trigonometric series

, . . .

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The authors wish to thank the referee whose comments improved the presentation of the paper. In fact, the present form of Lemma 2, which was originally very long, is due to the referee.     

On the exactness of a theorem of G.A. Fomin

, . . [1] - . .     

Converse results on equiconvergence of interpolating polynomials

, n- n- 1 , . , , ( « ») f .     

Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

Remarks on multiple Fourier series

: (1) ( , , ), (2) ( —, , ). , .     

Асимптоический Анализ Распределения Статистики Диксона

. u 60-u u     

On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

On a test of Jordan type for double Fourier series with respect to multiplicative systems

. , , .     

A generalization of Pal-type interpolation

. .     

О решеткахИ-Радикалов, строгих слева Радикалов, наследственных слева радикалов

.     

L 2(R n )-extremal problems on the basis of incomplete information

. ( ), R ⁿ L ₂(R ²).

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The author is supported by the National Natural Science Found of China.     

Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).     

On the admissibility of function spaces of typeB p,q s andF p,q s,and boundary value problems for non-linear partial differential equations

u=f(x)+S(u), S — , u-G(u), G — . B _p,q ^s () -F _p,q ^s (). R _n — . — . _p,q ^s F _p,q ^s .