Vol 105, No 5-6 (2019)

Let φ be atrace on aunital C*-algebra \(\mathcal{A}\), let \(\mathfrak{M}_\varphi\) be the ideal of definition of the trace φ, and let \(P,Q\in\mathcal{A}\) be idempotents such that QP = P. If \(Q\in\mathfrak{M}_\varphi\) then \(P\in\mathfrak{M}_\varphi\) and 0 ≤ φ(P) ≤ φ(Q). If \(Q-P\in\mathfrak{M}_\varphi\) then φ(Q − P) ∈ ℝ⁺. Let \(A,B\in\mathcal{A}\) be tripotents. If AB = B and \(A\in\mathfrak{M}_\varphi\), then \(B\in\mathfrak{M}_\varphi\) and 0 ≤ φ(B²) ≤ φ(A²) < +∞. Let \(\mathcal{A}\) be a von Neumann algebra. Then

\(\varphi(|PQ-QP|)\le {\rm{min}}\{\varphi(P),\varphi(Q),\varphi(|P-Q|)\}\)

A sharp Jackson inequality in the space L_p(ℝ^d), 1 ≤ p < 2, with Dunkl weight is proved. The best approximation is realized by entire functions of exponential spherical type. The modulus of continuity is defined by means of a generalized shift operator bounded on L_p, which was constructed earlier by the authors. In the case of the unit weight, this operator coincides with the mean-value operator on the sphere.

We prove that it is true in Sacks, Cohen, and Solovay generic extensions that any ordinal definable Borel set of reals necessarily contains an ordinal definable element. This result has previously been known only for countable sets.

In the case of approximation of functions by using linear methods of summation of their Fourier-Laplace series in the spaces S^(p,q) (σ^m−1), m ≥ 3, for classes of functions defined by transformations of their Fourier-Laplace series using multipliers, Jackson-type inequalities are established in terms of operators which are also defined by the corresponding transformations of the Fourier-Laplace series.

The equation describing a perturbed harmonic oscillator is considered. Using transformation operators, we obtain representations of solutions of this equation with conditions at infinity. Estimates for the kernels of the transformation operators are obtained.

The paper deals with the Rees algebra \(\mathcal{R}\) of a graded ideal I of a standard graded algebra A generated by a subset of generators of the maximal graded ideal of A. We compute the cohomology modules of \(\mathcal{R}\) in terms of the cohomology modules of A and depth(\(\mathcal{R}\)) in terms of depth (A).

A smooth stably complex manifold is said to be totally tangentially/normally split if its stably tangential/normal bundle is isomorphic to a sum of complex line bundles. It is proved that each class of degree greater than 2 in the graded unitary cobordism ring contains a quasitoric totally tangentially and normally split manifold.

A new combinatorial proof of the Protasov-Voynov theorem on the structure of irreducible semigroups of nonnegative matrices is proposed. The original proof was obtained by geometric methods.

We describe homogeneous locally nilpotent derivations of the algebra of regular functions for a class of affine trinomial hypersurfaces. This class comprises all nonfactorial trinomial hypersurfaces.

The relationship between the Bohr-Sommerfeld quantization condition and the integrality of the symplectic structure in Planck constant units is considered. Constructions of spherical and toric Θ-handles are proposed which allow one to obtain symplectic manifolds with contact singularities, preserve Kostant-Souriau prequantization, and expect interesting topological applications. In particular, the toric Θ-handle glues Liouville foliations, while the spherical handle generates (pre)quantized connected sums of symplectic manifolds. In this way, nonorientable manifolds may arise.

Borsuk’s celebrated conjecture, which has been disproved, can be stated as follows: in ℝⁿ, there exist no diameter graphs with chromatic number larger than n + 1. In this paper, we prove the existence of counterexamples to Borsuk’s conjecture which, in addition, have large girth. This study is in the spirit of O’Donnell and Kupavskii, who studied the existence of distance graphs with large girth. We consider both cases of strict and nonstrict diameter graphs. We also prove the existence of counterexamples with large girth to a statement of Lovász concerning distance graphs on the sphere.

Asubgroup H of G is said to be \({{\cal M}_p}\)-supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H with ∣H: T∣ = p^α. In this paper, we will investigate the structure of \({Z_{{\cal F}\Phi }}\left( G \right)\) by using \({{\cal M}_p}\)-supplemented subgroups.

Upper bounds for the norms of Hermite—Fejér interpolation operators in one-dimensional and multidimensional periodic Sobolev spaces are obtained. It is shown that, in the one-dimensional case, the norm of this operator is bounded. In the multidimensional case, the upper bound depends on the ratio of the numbers of nodes on separate coordinates.