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 that arises in modeling flexural-torsional vibrations of an elastic rod. For the function on the right-hand side of the system, we suggest necessary conditions for existence of a solution of the Cauchy problem in the Sobolev space with exponential weight.
Within the framework of inverse problems of photometry, we study questions on reconstruction of the spatial location and luminosity of a Lambertian optical surface from its images obtained with the use of a small number of optical systems. We study causes of ambiguity in reconstruction of the location of such a surface. We suggest criteria for existence of a unique solution of the inverse problem on reconstruction of a luminous surface from three images for general weight functions and apply the results to specific classes of weight functions that model the degree of transparency of the medium (including its absorption or scattering).
We continue to study the holomorphy problem for functions whose contour integrals over circles 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 these conditions cannot be weakened in the general case.
The main aim of the present article is to survey results on the structure of the normalizers of maximal toruses in Lie-type groups. In particular, we present results on splitting of the normalizer of a maximal torus and, in the case of exceptional groups, on the minimal order of a lift in the corresponding normalizer for elements of the Weyl group.
We study the Prony identification problem for coefficients of an autonomous difference equation by observations of noisy solutions with unknown additive perturbations from an arbitrary linear manifold. We establish a “projectivity” property of the variational objective function. For two 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 structure of finite-order moving averages is studied. The random component of this sequence is formed using a heterogeneous process in discrete time, while the non-random component is formed using a regularly varying function at infinity. The heterogeneous process with discrete time is defined as a power transform of partial sums of a certain stationary sequence. An approximation of the random processes from the above-mentioned class is studied by random processes defined as the convolution of a power transform of the fractional Brownian motion with a power function. Sufficient conditions for (C)-convergence in the Donsker invariance principle are obtained.
While modern computers are fast, there are still many practical problems that require even faster computers. It turns out that on the fundamental level, one of the main factors limiting computation speed is the fact that, according to modern physics, the speed of all processes is limited by the speed of light. Good news is that while the corresponding limitation is very severe in 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 Euclidean character 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 distance values – corresponding to all possible paths – from the values that we actually measure – which correspond only to macro-paths and thus, provide only the upper bound for the distance. In our previous papers – including our joint paper with Victor Selivanov – we provided an explicit formula for such a reconstruction. But for this formula to be useful, we need to analyze how algorithmic is this reconstructions. In this paper, we show that while in general, no reconstruction algorithm is possible, an algorithm is possible if we impose a lower limit on the distances between steps in a path. So, hopefully, this can help to eventually come up with faster computations.
We consider the variational Prony problem on approximating observations (x ) by the sum of exponentials. We find critical points and the second derivatives of the implicit function (theta ) that relates perturbation in (x ) with the corresponding exponents. We suggest upper bounds for the second order increments and describe the domain, where the accuracy of a 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 with upper 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 theorems for different dimensions (i.e., theorems on the traces) for functions in the Sobolev multianisotropic space ( W^{mathfrak {N}}_2(mathbb {R}^3)).
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