For the Riemannian spaces V ¯ ¯ ¯ n and V n assuming non-trivial geodesic mapping γ, we have constructed the spaces of the second-order approximation V ¯ ¯ ¯ ˜ 2 n and V ˜ 2 n. We have researched the mapping γ ˜, which is induced by the mapping γ. Also we have obtained the deformation tensor of mapping γ ˜ in an explicit form.

We find natural sufficient conditions on a subset $D$ of a vector lattice $E$ under which every orthogonally additive operator $T_0\colon D \to X$, where $X$ is a vector space, can be extended to an orthogonally additive operator $T\colon E \to X$. Two theorems on the extension from lateral ideals and lateral bands, respectively, are obtained.

Let K∈N,K≥3. Let C K= R K 0 be the Cartesian product of K copies of R 0, where R 0 denotes the set of all nonnegative real numbers. We equip this set with arithmetic operations and show that under the condition of the so-called Cancelation Law, the space C K is arithmetically isomorphic with the standard field C of complex numbers. Distinct K∈N determine distinct lexicographical orderings on C.

We consider tilings of limit spaces of self-similar groups and discuss the following problem: when does the tile of a self-similar group admit a tiling of the limit space under the action of a (self-similar) subgroup?

If a proper subharmonic function of infinite order has the full measure at the finite system of rays in the upper half-plane, then its lower order also equals infinity.

An asymptotic representations of some solutions for one class of nonlinear non-autonomous differential equations of the second order are established. We give necessary and also sufficient conditions of existence for these solutions.

With the aid of weighted sharing of sets we study the uniqueness problem of meromorphic functions having the same pole sharing of a finite set. The result of the paper improves, generalizes and extends a recent result of S. S. Bhoosnurmath and R. Davanal ([1]).

We consider some reformulation of the Riemann Hypothesis in terms of cyclic functions in a weighted Hardy space with an exponential weight. We indicate a distinctive feature of the weighted case from the classical one. Also we consider connection of the Riemann Hypothesis with a convolution type equation in the semistrip.

For any semigroup (S;⋅) let (S;∘) be a semigroup defined on the same set. Semigroup (S;∘) is called an interassociate of (S;⋅) if the following identities hold x⋅(y∘z)=(x⋅y)∘z and x∘(y⋅z)=(x∘y)⋅z. All interassociates of the free semigroup over the two-element alphabet are described.

For {a} real functions $f\in\mathcal{C}^n[a,b]$ such that $|L_nf(x)|\ge\delta$ on $[a,b]$, where $L_n$ is a differential expression, namely $L_n=(d/dx+\l_1)\dotsb(d/dx+\l_n)$ with real $\l_1,\dotsc,\l_n$, {we find a} Lebesgue measure estimate for the level set $G_{L_n}(\varepsilon,\delta;f)=\{x\in[a,b]\colon|f(x)|<\e\}$. {In particular, we establish } the inequality $\meas G_{L_n}(\e,\delta;f)\le\min\{b-a,n\sqrt{2^{n+1}}\sqrt[n]{q_n\e/\delta}\},$ where $q_n=\prod_{r=1}^n \frac{|\l_r|(b-a)/2}{\th|\l_r|(b-a)/2}$.

It is proved that the Gauss equation $z(z-1)w''+((\alpha+\beta+1)z-\gamma)w'+\alpha\beta w=0$ apart from the hypergeometric function has a power solution with negative exponents. Its possible growth, close-to-convexity and $l$-index boundedness are investigated.

We obtain new integral test for convergence of series.

The theorems on construction of rational approximation by means of the method of generalized moment representations are generalized. Under conditions of the generalized theorems the formulas for the errors of approximation are established. The two-dimensional Pade approximants for some analytic two-variable functions are constructed.