On the recursive sequencex_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the positive solutions of the difference equation of the form $$x_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}, n = 0,1,...$$ is investigated, where all the coefficients are nonnegative real numbers.     

On the rational recursive sequencex_{n + 1} = \frac{{\alpha x_n + \beta x_{n - 1} + \gamma x_{n - 2} + \delta x_{n - 3} }}{{Ax_n + Bx_{n - 1} + Cx_{n - 2} + Dx_{n - 3} }}

The main objective of this paper is to study the boundedness character, the periodic character and the global stability of the positive solutions of the following difference equation $x_{n + 1} = \frac{{\alpha x_n + \beta x_{n - 1} + \gamma x_{n - 2} + \delta x_{n - 3} }}{{Ax_n + Bx_{n - 1} + Cx_{n - 2} + Dx_{n - 3} }},n = 0,1,2.....$ where the coefficientsA, B, C, D, α, β, γ, δ, and the initial conditionsx _-3,x _-2,x _-1,x ₀ are arbitrary positive real numbers.     

On asymptotic behaviour of the difference equationX_{N + 1} = \alpha + \frac{{X_{N - 1} ^P }}{{X_N ^P }}

In this paper, we investigate local stability, oscillation and boundeness character of positive solutions of the difference equation $$x_{n + 1} = \alpha + \frac{{x_{n - 1} ^p }}{{x_n ^p }},n = 0,1,...$$ under specified conditions.     

On the recursive sequencex_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation $$x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}, n = 0, 1,...$$ wherem, k ∈ N, 2k > 2m?1,a, b, c are nonnegative real numbers andp < 1, are investigated.     

On the difference equationx_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}}

In this paper we consider the difference equation $$x_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}},$$ wherea, b, c are nonegative real numbers,k, l, m are nonnegative integers andg(x) is a nonegative real function. The oscillatory and periodic character, the boundedness and the stability of positive solutions of the equation is investigated. The existence and nonexistence of two-period positive solutions are investigated in details. In the last section of the paper we consider a generalization of the equation.     

Unbounded solutions of the max-type difference equation x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\}

We investigate the boundedness nature of positive solutions of the difference equation $$ x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\},n = 0,1,..., $$ where {A _n } _n=0 ^∞ and {B _n } _n=0 ^∞ are periodic sequences of positive real numbers.     

Subalgebras of free products of algebras of the variety\mathfrak{u}_{m,n}

The variety \(\mathfrak{u}_{m,n} \) is defined by the system of n-ary operations ω_i,..., ω_m, the system of m-ary operations ?_i,..., ?_n, 1≤ m ≤ n, and the system of identities $$\begin{gathered} x_1 ...x_n \omega _1 ...x_1 ...x_n \omega _m \varphi _i = x_i (i = 1,...,n), \hfill \\ x_1 ...x_m \varphi _1 ...x_1 ...x_m \varphi _n \omega _j = x_j (i = 1,...,m), \hfill \\ \end{gathered} $$ It is proved in this paper that the subalgebra U of the free product \(\Pi _{i \in I}^* A_i \) of the algebras A_i (i ε I) can be expanded as the free product of nonempty intersections U ∩ A_i (i ε I) and a free algebra.     

Sensitivity of best recovery in the Sobolev spaces W_\infty ^{r,d} , \widetilde W_\infty ^{r,d}for perturbed sampling

Let $I^d $ be the d‐dimensional cube, $I^d = [0,1]^d $ , and let $F \ni f \mapsto Sf \in L_\infty (I^d ) $ be a linear operator acting on the Sobolev space F, where Fis either $$$$ or $$$$ where $$\left\| f \right\|_F = \sum\limits_{\left| m \right| = r} {\mathop {{\text{esssup}}}\limits_{x \in I^d } \left| {\frac{{\partial f^{\left| m \right|} }} {{\partial x_1^{m_1 } \partial x_2^{m_2 } \cdot \cdot \cdot \partial x_d^{m_d } }}(x)} \right|.} $$ We assume that the problem elements fsatisfy the condition $\sum\nolimits_{\left| m \right| = r} {{\text{esssup}}} _{x \in I^d } \left| {f^{(m)} (x)} \right| \leqslant 1 $ and that Sis continuous with respect to the supremum norm. We study sensitivity of optimal recovery of Sfrom inexact samples of ftaken at npoints forming a uniform grid on $I^d $ . We assume that the inaccuracy in reading the sample vector is measured in the pth norm and bounded by a nonnegative number δ. The sensitivity is defined by the difference between the optimal errors corresponding to the exact and perturbed readings, respectively. Our main result is that this difference is bounded by $\mathcal{A}\delta $ , where $\mathcal{A} $ is a positive constant independent of the number of samples. This indicates that the curse of dimension, which badly affects the optimal errors, does not extend to sensitivity.     

On integral graphs which belong to the class\overline {\alpha K_a \cup \beta K_b }

LetG be a simple graph and let $\bar G$ denotes its complement. We say thatG is integral if its spectrum consists entirely of integers. If $\overline {\alpha K_a \cup \beta K_b } $ is integral we show that it belongs to the class of integral graphs $$\overline {[\frac{{kt}}{\tau }x_o + \frac{{mt}}{\tau }z]K(t + \ell n)k + \ell m \cup [\frac{{kt}}{\tau }y_o + \frac{{(t + \ell n)k + \ell m}}{\tau }z]nK\ell m,} $$ where (i) t, k, l, m, n ∈ ? such that (m, n) =1, (n, t) =1 and (l, t)=1; (ii) τ=((t+ln)k+lm, mt) such that τ| kt; (iii) (x₀, y₀) is aparticular solution of the linear Diophantine equation ((t+ln)k+lm)x-(mt)y=τ and (iv) z≥z₀ where z₀ is the least integer such that $(\frac{{kt}}{\tau }x_0 + \frac{{mt}}{\tau }z_0 ) \geqslant 1$ and $(\frac{{kt}}{\tau }y_0 + \frac{{(t + \ell n)k + \ell m}}{\tau }z_0 ) \geqslant 1$ .     

On the Diophantine equation \dfrac{ax^{n+2l}+c}{abt^{2}x^{n}+c}{\displaystyle\frac{ax^{n+2l}+c}{abt^{2}x^{n}+c}}=by^{2}

We obtain all solutions of the equation $\frac{ax^{n+2l}+c}{abt^{2}x^{n}+c} = by^{2}$ with c??{??1,??2,??4}.     

Asymptotic behavior of multiperiodic functionsG(x) = \prod\limits_{n = 1}^\infty {g(x/2^n )}

Let 0≤g be a dyadic Hölder continuous function with period 1 and g(0)=1, and let $G(x) = \prod\nolimits_{n = 0}^\infty {g(x/{\text{2}}^n )} $ . In this article we investigate the asymptotic behavior of $\smallint _0^{\rm T} \left| {G(x)} \right|^q dx$ and $\frac{1}{n}\sum\nolimits_{k = 0}^n {\log g(2^k x)} $ using the dynamical system techniques: the pressure function and the variational principle. An algorithm to calculate the pressure is presented. The results are applied to study the regulatiry of wavelets and Bernoulli convolutions.     

On the recursive sequencex n +1=α-(x n /x n −1)

We study the global asymptotic stability, global attractivity, boundedness character, and periodic nature of all positive solutions and all negative solutions of the difference equation $$x_{n + 1} = \alpha - \frac{{x_n }}{{x_{n - 1} }}, n = 0,1,...,$$ where α∈R is a real number, and the initial conditionsx?1,x ₀ are arbitrary real numbers.     

On positive solutions of a reciprocal difference equation with minimum

In this paper we consider positive solutions of the following difference equation $$x_{n + 1} = \min \left\{ {\frac{A}{{x_n }},\frac{B}{{x_{n - 2} }}} \right\}, A, B > 0.$$ We prove that every positive solution is eventually periodic. Also, we present here some results concerning positive solutions of the difference equation $$x_{n + 1} = \min \left\{ {\frac{A}{{x_n x_{n - 1} ...x_{n - k} }},\frac{B}{{x_{n - (k + 2)} ...x_{n - (2k + 2)} }}} \right\}, A, B > 0.$$      

Global attractivity of the recursive sequencex_{n + 1} = \frac{{\alpha - \beta x_{n - k} }}{{\gamma + x_n }}

Our aim in this paper is to investigate the global attractivity of the recursive sequence $$x_{n + 1} = \frac{{\alpha - \beta x_{n - k} }}{{\gamma + x_n }},$$ where α, β, γ >0 andk=1,2,… We show that the positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients.     

\mathfrak{E} of topological spaces II

In this paper, we obtain analogues, in the situation of \(\mathfrak{E}\) -extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any \(\mathfrak{E}\) -regular spaceX, every Hausdorff quotient of \(\beta _\mathfrak{E} X\) which is Urysohn on \(\beta _\mathfrak{E} X - X\) (respectively which is finitary on \(\beta _\mathfrak{E} X - X\) ) and which is identity onX, has \(\mathfrak{E}\) . We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when \(\mathfrak{E}\) is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary \(\mathfrak{E}\) -extensions of two spacesX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. Further if \(\mathfrak{E}\) is admissive, then the lattices of Urysohn \(\mathfrak{E}\) -extensions ofX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic.     

Bases formed of successive primitives

Necessary and sufficient conditions are found in order for the system of successive primitives $$\left\{ {F_n (z) = \sum\nolimits_{k = 0}^\infty {\frac{{a_{k - n} }}{{k!}}z^k } } \right\}, n = 0,1,2, ...,$$ generated by the integer-valued function \(F_n (z) = \sum\nolimits_{k = 0}^\infty {\frac{{a_k }}{{k!}}zk} \) of growth no higher than first order of the normal typeσ(F₀(z) ε [1;σ] to form a quasi-power basis in the class [1; σ].     

The Quintic Grassmannian G_{1,4,2} in PG(9, 2)

The 155 points of the Grassmannian $G_{1,4,2}$ of lines of PG (4, 2) = $\mathbb{P}V\left( {5,2} \right)$ are those points $x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X $ \subset $ PG (9, 2) will be termed odd or even according as X intersects $G_{1,4,2}$ in an odd or even number of points. Let $Q^\ddag \left( {x_1 ,...,x_5 } \right)$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X ^# of a r-flat X $ \subset $ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$ . Because $Q^\ddag$ is quinquelinear, the associate X ^# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X $ \subseteq$ X ^# while if X is an even 4-flat then X ^# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X ^# = X. An example of an even 4-flat X such that $\left( {X^\# } \right)^\#$ = X is provided by any 4-flat X which is external to $G_{1,4,2}$ . However, it appears that the two possibilities just illustrated, namely X ^# = X for an odd 4-flat and $\left( {X^\# } \right)^\#$ = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X ^# = PG (9, 2) and of even 4-flats for which ${X^{\# \# \# } }$ = X.     

Towards a classification of modular compactifications of \mathcal {M}_{g,n}

The moduli space of smooth curves admits a beautiful compactification $\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,n}$ by the moduli space of stable curves. In this paper, we undertake a systematic classification of alternate modular compactifications of $\mathcal{M}_{g,n}$ . Let $\mathcal{U}_{g,n}$ be the (non-separated) moduli stack of all n-pointed reduced, connected, complete, one-dimensional schemes of arithmetic genus g. When g=0, $\mathcal{U}_{0,n}$ is irreducible and we classify all open proper substacks of $\mathcal{U}_{0,n}$ . When g≥1, $\mathcal{U}_{g,n}$ may not be irreducible, but there is a unique irreducible component $\mathcal{V}_{g,n} \subset\mathcal{U}_{g,n}$ containing $\mathcal{M}_{g,n}$ . We classify open proper substacks of $\mathcal {V}_{g,n}$ satisfying a certain stability condition.     

Criterion of polynomial increase of a function and its derivatives

ДОкАжАНО, ЧтО Дль тОгО, ЧтОБы Дльr РАж ДИФФЕРЕНцИРУЕМОИ НА пРОМЕжУткЕ [А, + ∞) ФУНкцИИf сУЩЕстВОВА л тАкОИ МНОгОЧлЕН (1) $$P(x) = \mathop \Sigma \limits_{\kappa = 0}^{r - 1} a_k x^k ,$$ , ЧтО (2) $$\mathop {\lim }\limits_{x \to + \infty } (f(x) - P(x))^{(k)} = 0,k = 0,1,...,r - 1,$$ , НЕОБхОДИМО И ДОстАтО ЧНО, ЧтОБы схОДИлсь ИН тЕгРАл (3) $$\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{r - 1} }^{ + \infty } {f^{(r)} (t)dt.}$$ ЕслИ ЁтОт ИНтЕгРАл сх ОДИтсь, тО Дль кОЁФФИц ИЕНтОВ МНОгОЧлЕНА (1) ИМЕУт МЕс тО ФОРМУлы $$\begin{gathered} a_{r - m} = \frac{1}{{(r - m)!}}\left( {\mathop \Sigma \limits_{j = 1}^m \frac{{( - 1)^{m - j} f^{(r - j)} (x_0 )}}{{(m - j)!}}} \right.x_0^{m - j} + \hfill \\ + ( - 1)^{m - 1} \left. {\mathop \Sigma \limits_{l = 0}^{m - 1} \frac{{x_0^l }}{{l!}}\int\limits_a^{ + \infty } {dt_1 } \int\limits_{t_1 }^{ + \infty } {dt_2 ...} \int\limits_{t_{m - l - 1} }^{ + \infty } {f^{(r)} (t_{m - 1} )dt_{m - 1} } } \right),m = 1,2,...,r. \hfill \\ \end{gathered}$$ ДОстАтОЧНыМ, НО НЕ НЕОБхОДИМыМ Усл ОВИЕМ схОДИМОстИ кРА тНОгО ИНтЕгРАлА (3) ьВльЕтсь схОДИМОсть ИНтЕгРАл А \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

Some new estimates of the Fourier-Bessel transform in the space \mathbb{L}_2 \left( {\mathbb{R}_ + } \right)

The Fourier-Bessel integral transform $$g\left( x \right) = F\left[ f \right]\left( x \right) = \frac{1} {{2^p \Gamma \left( {p + 1} \right)}}\int\limits_0^{ + \infty } {t^{2p + 1} f\left( x \right)j_p \left( {xt} \right)dt}$$ is considered in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ . Here, j _p (u) = ((2 ^p Γ(p+1))/(u ^p ))J _p (u) and J _p (u) is a Bessel function of the first kind. New estimates are proved for the integral $$\delta _N^2 \left( f \right) = \int\limits_N^{ + \infty } {x^{2p + 1} g^2 \left( x \right)dx, N > 0,}$$ in $\mathbb{L}_2 \left( {\mathbb{R}_ + } \right)$ for some classes of functions characterized by a generalized modulus of continuity.