It is demonstrated that every (0, 1)-matrix of size n×m having Boolean rank n contains a column with at least √n/2 − 1 zero entries. This bound is shown to be asymptotically optimal. As a corollary, it is established that the size of a full-rank Boolean matrix is bounded from above by a function of its tropical and determinantal ranks.

We consider the Paley--Wiener spaces of L2-functions whose Fourier transform has a bounded support. We show that every continuous mapping that generates a superposition operator acting on these spaces is affine and injective.

The Barth-Van de Ven-Tyurin-Sato Theorem states that any finite-rank vector bundle on the complex projective ind-space P∞ is isomorphic to a direct sum of line bundles. We establish sufficient conditions on a locally complete linear ind-variety X which ensure that the same result holds on X. We then exhibit natural classes of locally complete linear ind-varieties which satisfy these sufficient conditions. © 2015 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

Let R be a commutative Noetherian ring such that X=SpecR is connected. We prove that the category D^b(cohX) contains no proper full triangulated subcategories which are strongly generated. We also bound below the Rouquier dimension of a triangulated category T, if there exists a triangulated functor T→D^b(cohX) with certain properties. Applications are given to the cohomological annihilator of R and to point-like objects in T.

We study singular Fano threefolds of type $ V_{22}$.

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions.

In this paper a six-valued two-dimensional formal group with ring of coefficients Λ2, lying in ΩU[1/2], is constructed. It is proved that the ring Λ2[1/2] coincides with the image of the ring ΩSU[1/2] in the ring ΩU[1/2]. 

In this work we explicitly calculate the syzygies of the quadratic Veronese embedding ℙ (V) ⊂ ℙ (Sym2V) as representations of the group GL(V ). Resolutions of the sheaves Oℙ (V)(i) are also constructed in the category D(ℙ (Sym2 V)). © 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

The Pontryagin-van Kampen duality for locally compact Abelian groups can be generalized in two ways to wider classes of topological Abelian groups: in the first approach the dual group X• is endowed with the topology of uniform convergence on compact subsets of X and in the second, with the topology of uniform convergence on totally bounded subsets of X. The corresponding two classes of groups "reflexive in the sense of Pontryagin-van Kampen" are very wide and are so close to each other that it was unclear until recently whether they coincide or not. A series of counterexamples constructed in this paper shows that these classes do not coincide and also answer several other questions arising in this theory. The results of the paper can be interpreted as evidence that the second approach to the generalization of the Pontryagin duality is more natural.

The zero-one law for the model of random distance graphs with vertices in Z^n is studied. Sufficient conditions for a sequence of random distance graphs to obey the zero-one law are derived, as well as conditions under which it contains a subsequence obeying the zero-one law.

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies a simple condition.