Abstract
We introduce generalizations of o-minimality ((lambda )-o-minimality and weak (lambda )-p.o.-lin-minimality) and study their properties.
We introduce generalizations of o-minimality ((lambda )-o-minimality and weak (lambda )-p.o.-lin-minimality) and study their properties.
In the present article, we consider the Cauchy problem for a pseudohyperbolic system thatarises in modeling flexural-torsional vibrations of an elastic rod. For the function onthe right-hand side of the system, we suggest necessary conditions for existence of a solution ofthe Cauchy problem in the Sobolev space with exponential weight.
Within the framework of inverse problems of photometry, we study questions onreconstruction of the spatial location and luminosity of a Lambertian optical surface from itsimages obtained with the use of a small number of optical systems. We study causes of ambiguityin reconstruction of the location of such a surface. We suggest criteria for existence of a uniquesolution of the inverse problem on reconstruction of a luminous surface from three images forgeneral weight functions and apply the results to specific classes of weight functions that modelthe degree of transparency of the medium (including its absorption or scattering).
We continue to study the holomorphy problem for functions whose contour integrals overcircles vanish. We consider the case in which a function (f ) is defined on a deleted ball (mathcal {D} ) in (mathbb {C}^n)(without its center) and integrate over all spheres of two fixed radii inside (mathcal {D} ). For (fin C^{infty }(mathcal {D}) ), we find conditions on the radii and size of(mathcal {D} ) implying that (f ) is a holomorphic function. We also show that theseconditions cannot be weakened in the general case.
The main aim of the present article is to survey results on the structure of the normalizersof maximal toruses in Lie-type groups. In particular, we present results on splitting ofthe normalizer of a maximal torus and, in the case of exceptional groups, on the minimal order ofa lift in the corresponding normalizer for elements of the Weyl group.
We study the Prony identification problem for coefficients of an autonomous differenceequation by observations of noisy solutions with unknown additive perturbations from an arbitrarylinear manifold. We establish a “projectivity” property of the variational objective function. Fortwo main types of equations, we obtain criteria and sufficient conditions for identifiability.
A class of partial sum processes based on a sequence of observations having the structureof finite-order moving averages is studied. The random component of this sequence is formedusing a heterogeneous process in discrete time, while the non-random component is formed using aregularly varying function at infinity. The heterogeneous process with discrete time is defined as apower transform of partial sums of a certain stationary sequence. An approximation of therandom processes from the above-mentioned class is studied by random processes defined as theconvolution of a power transform of the fractional Brownian motion with a power function.Sufficient conditions for (C)-convergence in theDonsker invariance principle are obtained.
While modern computers are fast, there are still many practical problems that requireeven faster computers. It turns out that on the fundamental level, one of the main factors limitingcomputation speed is the fact that, according to modern physics, the speed of all processes islimited by the speed of light. Good news is that while the corresponding limitation is very severein Euclidean geometry, it can be more relaxed in (at least some) non-Euclidean spaces, and,according to modern physics, the physical space is not Euclidean. The differences from Euclideancharacter are especially large on micro-level, where quantum effects need to be taken into account.To analyze how we can speed up computations, it is desirable to reconstruct the actual distancevalues – corresponding to all possible paths – from the values that we actually measure – whichcorrespond only to macro-paths and thus, provide only the upper bound for the distance. In ourprevious papers – including our joint paper with Victor Selivanov – we provided an explicitformula for such a reconstruction. But for this formula to be useful, we need to analyze howalgorithmic is this reconstructions. In this paper, we show that while in general, no reconstructionalgorithm is possible, an algorithm is possible if weimpose a lower limit on the distances between steps in a path. So, hopefully, this can help toeventually come up with faster computations.
We consider the variational Prony problem on approximating observations(x ) by the sum of exponentials. We find critical pointsand the second derivatives of the implicit function (theta ) that relates perturbation in (x ) with the corresponding exponents. We suggestupper bounds for the second order increments and describe the domain, where the accuracy ofa linear approximation of (theta ) is acceptable.We deduce lower estimates of the norm of deviation of (theta ) for small perturbations in (x ). We compare our estimates of this norm withupper bounds obtained with the use of Wilkinson’s inequality.
We consider a class of completely regular polyhedrons (mathfrak {N} ) and prove direct and inverse embedding theoremsfor different dimensions (i.e., theorems on the traces) for functions in the Sobolev multianisotropicspace ( W^{mathfrak {N}}_2(mathbb {R}^3)).
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