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In this paper, we will consider an arithmetical approach to the Eratosthenes sieve and to the problem of the twin primes.

The aim of this paper is to study, by using the mathematical tools developed by Chałupnik, Touzé, and Van der Kallen, the effect of the Frobenius twist on $\operatorname{Ext}$-group in the category of strict polynomial functors. As an application, we obtain explicit formulas of cohomology of the orthogonal groups and symplectic ones.

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr\"odinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related Lp-Lq estimates on the Schr\"odinger solution. These estimates extend known results from the literature and are sharp. This result was latetly already generalized to a degenerate case (cf. [9]).

In this paper, we classify the invariant complex structures on four-dimensional, solvable, simply-connected real Lie groups with commutator of dimension three. The resulting complex surfaces corresponding to these structures are also determined. The classification is based on the determination of certain complex subalgebras of the complexifications of the corresponding real Lie algebras.

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends", and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic., Minor edits in exposition, 19 pages. The paper now uses the style file of Trans. AMS (to appear)

Accurate sky identification is one of the most important functions of an automated telescope mount. The more accurately the robotic telescope is navigated to the investigated part of the sky, the better the observations and discoveries made. In this paper, we present mathematical methods for accurate sky identification (celestial coordinates determination). They include the automatic selection of the reference stars, preliminary and full sky identification, as well as an interaction with international databases, which are a part of the astrometric calibration. All described methods help to receive accurately calculated astrometric data and use it for the positional calibration and better navigation of the automated telescope mount. The developed methods were successfully implemented in the Collection Light Technology (CoLiTec) software. Through its use, more than 1600 small solar system objects were discovered. It has been used in more than 700,000 observations and successful sky identifications, during which, five comets were discovered. Additionally, the accuracy indicators of the processing results of the CoLiTec software are provided in the paper, which shows benefits of the CoLiTec software and lower standard deviation of the sky identification in the case of low signal-to-noise ratios.

This paper extends the results from a work of Hytönen, Petermichl, and Volberg about sharp A 2 A_2 - A ∞ A_\infty estimates with matrix weights to the non-homogeneous situation.

In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T)⫋P⫋C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Zfˆ(n)ξnexp(2πint) in the general setting of dependent random variables (ξn).

Let a reductive groupG act linearly on a vectorspaceV, and letO∋V be an orbit. For each irreducible representation ω ofG, the multiplicity with which ω occurs in the ring of regular functions onO (or on its closureŌ) provides an interesting numerical invariant ofO. We introduce an algebraic notion of a “deformation” of an orbit into another one. The main goal of this paper is to give sufficient conditions on an orbit, in order that an arbitrary deformation of this orbit has to preserve all the multiplicites mentioned above.

In this paper some non existence results are given for smooth varieties of dimension bigger then 1, embedded in projective spaces with low codimension, by the pluricanonical system |mK X |, withm≥2. The only meaningful cases of codimension 2 are surfaces in P4 and threefolds in P5, whose existence is excluded. When the codimension is 3, for surfaces in P5 is proven thatm=2, while for three-folds in P6 and fourfolds in P7 only two numerical possibilities for the degree are given.