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In particular, it is proved that if G is a bounded set satisfying thel horn condition, then there holds the imbedding[Figure not available: see fulltext.] where
      
The quantities[Figure not available: see fulltext.] are calculated, where[Figure not available: see fulltext.] is the best approximation from above of the function f by trigonometric polynomials of order ?n-1 in the metric of L1.
      
In this paper we solve the problem of finding the Borsuk number of any bounded set in an arbitrary two-dimensional normed space (the solution is given in terms of the enlargement of a set to a figure of constant width).
      
Estimates for[Figure not available: see fulltext.] and moduli of continuity of the operators of best approximation on WrHw are established.
      
For example, if ω(δ)=δα, then[Figure not available: see fulltext.]
      
Among others, we establish the following results: If F(z)εs and 0 >amp;lt; α >amp;lt; 1, then[Figure not available: see fulltext.] IfF (z) ε s (0 >amp;lt; a >amp;lt; 1) and
      
The functionγ? is assumed to belong to the Dirichlet space, i.e., it satisfies[Figure not available: see fulltext.], wheredv(ξ) is the area element.
      
It is shown that forγ? to satisfy the above conditions, it is necessary and sufficient to have[Figure not available: see fulltext.] and the growth indicatrixh? of? satisfies the relation 0>amp;lt;m≤h″(?)+h(?)≤M>amp;lt;∞.
      
The polysaccharide has the following structure, which is unique among the known bacterial polysaccharides: [Figure not available: see fulltext.]
      
The polysaccharide had the following structure: [Figure not available: see fulltext.]
      
The Pd-catalyzed alkylation of 1,3-diphenylallylacetate with dimethyl malonate results in a 87% enantiomer excess (ee), while in sulfonylation of 1,3-diphenylallylacetate with sodium para-toluenesulfinite, this figure amounts to 67%.
      
A new structured-laser-based system for measuring the 3D inner-contour of pipe figure components
      
In order to measure the 3D inner-contour parameter of pipe figure components, a new measuring system is designed and, correspondingly, a real-time mathematical model is developed.
      
The measuring system and mathematical model are able to complete the measurement of 3D inner-contour parameters of pipe figure components rapidly and efficiently during experiments.
      
Estimates of the potassium-argon age (gas-retention age) yielded the figure of 560 million years, which is within the range of values obtained for SNC meteorites by other authors, who used the rubidium-strontium and the potassium-argon technique.
      
Finally, we calculated parameters of the equilibrium figure of Mars for the M13 model: J20 = 1.82 × 10-3, J40 = -7.79 × 10-6, ec-mD = 1/242.3 (the dynamical flattening of the core-mantle boundary).
      
In some cases, we found transitions between motions along the figure of eight in its neighborhood and motions in the vicinity of a periodic Broucke orbit in the isosceles three-body problem.
      
A theory of the equilibrium figure and gravitational field of the Galilean satellite Io: The second approximation
      
We construct a theory of the equilibrium figure and gravitational field of the Galilean satellite Io to within terms of the second order in the small parameter α.
      
We show that to describe all effects of the second approximation, the equation for the figure of the satellite must contain not only the components of the second spherical function, but also the components of the third and fourth spherical functions.
      
 

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