Nonoscillation theory for second order half-linear differential equations in the framework of regular variation

Criteria are established for nonoscillation of all solutions of the second order half-linear differential equation A $(\mid y^\prime \mid^{\alpha-1}y^\prime)^\prime + q(t)\mid y \mid^{\alpha -1}y = 0,\ \ \ t \geq 0,$ where α > 0 is a constant and q: [0, ∞) → ? is continuous. The criteria are designed to exhibit the role played by the integral of q(t) in guaranteeing the existence of nonoscillatory solutions of (A) in specific classes of regularly varying functions in the sense of Karamata.     

On the convergence to infinity of Fourier series along dense subsequences of numbers

В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H _k } _{k = 1} ^∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {n_k(x)} _k=1 ^∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ .     

On minor forms of Desargues and Pappus

Let Π be a projective plane coordinatized by a ternary ring (R, F). In addition to the two operations + and ·, defined bya+b =F(a,1,b and \(a \cdot b = F(a,b,0)\) , a third operation * is defined by \(a * b = F(1,a,b),\forall a,b \in R\) Several minor forms of the propositions of Desargues and Pappus are introduced in Π and their geometrical properties are developed. Several algebraic results are obtained in connection with these minor forms. For example, the first minor form of DesarguesD ₁ is proved to be equivalent to each of the following algebraic identities in every (R, F): (1) $$a \cdot c = c \cdot a \Rightarrow F(a,c,b) = F(c,a,b),$$ (2) $$a \cdot (1 + b) = a + a \cdot b,$$ (3) $$a * b = a + b$$ (4) $$F(x,m,k) = (x \cdot m) * k,\forall a,b,c,k,m,x \in R.$$ Some more algebraic identities are characterized byD ₂ andD ₃.     

Regular sets of matrices and applications

SupposeA ₁,...,A _s are (1, - 1) matrices of order m satisfying 1 $$A_i A_j = J, i,j \in \left\{ {1,...s} \right\}$$ 2 $$A_i^T A_j = A_j^T A_i = J, i \ne j, i,j \in \left\{ {1,...,s} \right\}$$ 3 $$\sum\limits_{i = 1}^s {(A_i A_i^T = A_i^T A_i ) = 2smI_m } $$ 4 $$JA_i = A_i J = aJ, i \in \left\{ {1,...,s} \right\}, a constant$$ Call A₁,…,A _s ,a regular s- set of matrices of order m if Eq. 1-3 are satisfied and a regular s-set of regular matrices if Eq. 4 is also satisfied, these matrices were first discovered by J. Seberry and A.L. Whiteman in “New Hadamard matrices and conference matrices obtained via Mathon’s construction”, Graphs and Combinatorics, 4(1988), 355-377. In this paper, we prove that

1. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn whent =sm
2. if there exist a regular s-set of order m and a regulart-set of order n there exists a regulars-set of ordermn when 2t = sm (m is odd)
3. if there exist a regulars-set of order m and a regulart-set of ordern there exists a regular 2s-set of ordermn whent = 2sm As applications, we prove that if there exist a regulars-set of order m there exists
4. an Hadamard matrices of order4hm whenever there exists an Hadamard matrix of order4h ands =2h
5. Williamson type matrices of ordernm whenever there exists Williamson type matrices of ordern and s = 2n
6. anOD(4mp;ms₁,…,ms_u whenever anOD (4p;s₁,…,s_u)exists and s = 2p
7. a complex Hadamard matrix of order 2cm whenever there exists a complex Hadamard matrix of order 2c ands = 2c

This paper extends and improves results of Seberry and Whiteman giving new classes of Hadamard matrices, Williamson type matrices, orthogonal designs and complex Hadamard matrices.     

Boundedness of the solution of a degenerate second order differential equation system

We consider a linear system of second order differential equations (1) $$A x(t) = B x (t),$$ whereA andB are singularn×n matrices,x is ann-dimensional vector with coordinates twice differentiable int. Using the Drazin inverse matrix theory we find a solution of the Cauchy problem that is bounded on the whole axis.     

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 \(\int\limits_a^{ + \infty } {x^{r - 1} f^{(r)} (x)dx}\)      

Existence of homoclinic solutions for second order Hamiltonian systems with general potentials

In this paper we are concerned with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian systems HS $$ \ddot{q}-L(t)q+W_{q}(t,q)=0, $$ where $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$ and $L\in C(\mathbb{R},\mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in\mathbb{R}$ . Assuming that the potential W satisfies some weaken global Ambrosetti-Rabinowitz conditions and L meets the coercive condition, we show that (HS) has at least one nontrivial homoclinic solution via using the Mountain Pass Theorem. Some recent results in the literature are generalized and significantly improved.     

Conditionally oscillatory half-linear differential equations

We consider a nonoscillatory half-linear second order differential equation (*) $$ (r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x) = \left| x \right|^{p - 2} x,p > 1, $$ and suppose that we know its solution h. Using this solution we construct a function d such that the equation (**) $$ (r(t)\Phi (x'))' + [c(t) + \lambda d(t)]\Phi (x) = 0 $$ is conditionally oscillatory. Then we study oscillations of the perturbed equation (**). The obtained (non)oscillation criteria extend existing results for perturbed half-linear Euler and Euler-Weber equations.     

Diophantine properties of IETs and general systems: quantitative proximality and connectivity

Three properties of dynamical systems (recurrence, connectivity and proximality) are quantified by introducing and studying the gauges (measurable functions) corresponding to each of these properties. The properties of the proximality gauge are related to the results in the active field of shrinking targets. The emphasis in the present paper is on the IETs (interval exchange transformations) $( \mathcal {I},T)$ , $\mathcal {I}=[0,1)$ . In particular, we prove that if an IET T is ergodic (relative to the Lebesgue measure λ), then the equality A1 $$ \liminf_{n\to\infty} \, n\, \bigl|T^n(x)-y \bigr|=0 $$ holds for λ×λ-a.a. $(x,y)\in \mathcal {I}^{2}$ . The ergodicity assumption is essential: the result does not extend to all minimal IETs. Also, the factor? n? in (A1) is optimal (e.g., it cannot be replaced by n?ln(ln(lnn))). On the other hand, for Lebesgue almost all 3-IETs $( \mathcal {I},T)$ we prove that for all ?>0 A2 $$ \liminf_{n\to\infty} \, n^ \epsilon \bigl |T^n(x)-T^n(y)\bigr| = \infty,\quad\text{for Lebesgue a.a.} \ (x,y)\in \mathcal {I}^2. $$ This should be contrasted with the equality lim?inf _n→∞?|T ⁿ (x)?T ⁿ (y)|=0, for a.a. $(x,y)\in \mathcal {I}^{2}$ , which holds since $( \mathcal {I}^{2}, T\times T)$ is ergodic (because generic 3-IETs $( \mathcal {I},T)$ are weakly mixing). We introduce the notion of τ-entropy of an IET which is related to obtaining estimates of type (A2). We also prove that no 3-IET is strongly topologically mixing.     

On 1-isometries of affine quadrics over finite fields

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume $4 \leqslant dim V \leqslant \infty \wedge |\mathbb{F}| \in \mathbb{N}$ . A 1-isometry of the central quadric $\mathcal{F}: = \{ x \in V|q(x) = 1\}$ is a permutation ? of $\mathcal{F}$ such that (*) $$q(x - y) = \nu \Leftrightarrow q(x^\varphi - y^\varphi ) = \nu \forall x,y \in \mathcal{F}$$ holds true for a fixed element ν of $\mathbb{F}$ . For arbitraryν ∈ $\mathbb{F}$ we prove that? is induced (in a certain sense) by a semi-linear bijection $(\sigma ,\varrho ):(V,\mathbb{F}) \to (V,\mathbb{F})$ such thatq oσ =? oq, provided $\mathcal{F}$ contains lines and the exceptional case $(\nu = 2 \Lambda |\mathbb{F}| = 3 \Lambda \dim V = 4 \Lambda |\mathcal{F}| = 24)$ is excluded. In the exceptional case and as well in case of dim V = 3 there are counterexamples. The casesν ≠ 2 and v=2 require different techniques.     

О стРУктУРЕ цЕлОИ ФУН кцИИ, ОБРАЩАУЩЕИсь В НУль Н А жАДАННОИ пОслЕДОВАтЕльНОстИ тОЧЕк

The following result is proved. Theorem.Let λ _n ,0<λ _n ↑∞, be a sequence of positive numbers with finite density $$\sigma = \mathop {\lim }\limits_{n \to \infty } \frac{n}{{\lambda _n }}$$ and let a compact set K has the following property: it intersects the real axis along the interval [a, b], where a is the very left point of K, B is the very right point of K; furthermore, K intersects every vertical straight line Re z=α, a≤α≤b, along an interval. If 1) $$F(z) \in [1,S_{ - \pi \sigma }^{\pi \sigma } \cup K(\alpha + i\pi \sigma ) \cup K(\alpha - i\pi \sigma )], \alpha \in R;$$ 2) 2) $$F( \pm \lambda _n ) = 0, n = 1,2,...,$$ then $$F(z) = A(z)e^{\alpha z} \alpha (z),$$ where $$A(z) \in [1,K], \alpha (z) = \prod\limits_1^\pi {\left( {1 - \frac{{z^2 }}{{\lambda _n^2 }}} \right)}$$ . This result generalizes the theorem of Kaz'min [3]. Three corollaries are also proved, which generalize the theorems ofBoas [1] andPólya [6]. In the theorems of Boas and Pólya, we haveF(n)=0, ?n ε Z. In our case $$F( \pm \lambda _n ) = 0, 0< \lambda _n \uparrow \infty , \sigma = \mathop {\lim }\limits_{n \to \infty } \frac{n}{{\lambda _n }}$$ .     

On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation

We construct blow-up patterns for the quasilinear heat equation (QHE) $$u_t = \nabla \cdot (k(u)\nabla u) + Q(u)$$ in Ω×(0,T), Ω being a bounded open convex set in ? ^N with smooth boundary, with zero Dirichet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreoverk(u) andQ(u)/u ^p with a fixedp>1 are of slow variation asu→∞, so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation (SHE) $$u_t = \nabla u) + u^p .$$ We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption $$\smallint ^\infty k(f(e^s ))ds = \infty ,$$ wheref(v) is a monotone solution of the ODEf′(v)=Q(f(v))/v ^p defined for allv?1. If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.     

Non-Uniformity and Generalised Sacks Splitting

We show that there do not exist computable functions f ₁(e, i), f ₂(e, i), g ₁(e, i), g ₂(e, i) such that for all e, i ∈ ω, (1) $ {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (2) $ {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)} \leqslant _{{\rm T}} {\left( {W_{e} - W_{i} } \right)}; $ (3) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)} \oplus {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}; $ (4) $ {\left( {W_{e} - W_{i} } \right)} \not\leqslant _{{\rm T}} {\left( {W_{{f_{1} {\left( {e,i} \right)}}} - W_{{f_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset};{\text{and}} $ (5) $ {\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\left( {W_{{g_{1} {\left( {e,i} \right)}}} - W_{{g_{2} {\left( {e,i} \right)}}} } \right)}{\text{unless}}{\left( {W_{e} - W_{i} } \right)} \leqslant _{{\rm T}} {\emptyset}. $ It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.     

Functional regular variation of Lévy-driven multivariate mixed moving average processes

We consider the functional regular variation in the space $\mathbb {D}$ of càdlàg functions of multivariate mixed moving average (MMA) processes of the type $X_t = \int \int f(A, t - s) \Lambda (d A, d s)$ . We give sufficient conditions for an MMA process $(X_t)$ to have càdlàg sample paths. As our main result, we prove that $(X_t)$ is regularly varying in $\mathbb {D}$ if the driving Lévy basis is regularly varying and the kernel function f satisfies certain natural (continuity) conditions. Finally, the special case of supOU processes, which are used, e.g., in applications in finance, is considered in detail.     

On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over ℝ̄ + m

We investigate the regular convergence of the m-multiple series (*) $$\sum\limits_{j_1 = 0}^\infty {\sum\limits_{j_2 = 0}^\infty \cdots \sum\limits_{j_m = 0}^\infty {c_{j_1 ,j_2 } , \ldots j_m } }$$ of complex numbers, where m ≥ 2 is a fixed integer. We prove Fubini’s theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim’s sense can also be computed by successive summation. We introduce and investigate the regular convergence of the m-multiple integral (**) $$\int_0^\infty {\int_0^\infty { \cdots \int_0^\infty {f\left( {t_1 ,t_2 , \ldots ,t_m } \right)dt_1 } } } dt_2 \cdots dt_m ,$$ where f : ?? ₊ ^m → ? is a locally integrable function in Lebesgue’s sense over the closed nonnegative octant ?? ₊ ^m := [0,∞) ^m . Our main result is a generalized version of Fubini’s theorem on successive integration formulated in Theorem 4.1 as follows. If f ∈ L _loc ¹ (?? ₊ ^m ), the multiple integral (**) converges regularly, and m = p + q, where p and q are positive integers, then the finite limit $$\mathop {\lim }\limits_{v_{_{p + 1} } , \cdots ,v_m \to \infty } \int_{u_1 }^{v_1 } {\int_{u_2 }^{v_2 } { \cdots \int_0^{v_{p + 1} } { \cdots \int_0^{v_m } {f\left( {t_1 ,t_2 , \ldots t_m } \right)dt_1 dt_2 } \cdots dt_m = :J\left( {u_1 ,v_1 ;u_2 v_2 ; \ldots ;u_p ,v_p } \right)} , 0 \leqslant u_k \leqslant v_k < \infty } ,k = 1,2, \ldots p,}$$ exists uniformly in each of its variables, and the finite limit $$\mathop {\lim }\limits_{v_1 ,v_2 \cdots ,v_p \to \infty } J\left( {0,v_1 ;0,v_2 ; \ldots ;0,v_p } \right) = I$$ also exists, where I is the limit of the multiple integral (**) in Pringsheim’s sense. The main results of this paper were announced without proofs in the Comptes Rendus Sci. Paris (see [8] in the References).     

Existence and uniqueness of a regular solution of the Cauchy-Dirichlet problem for a class of doubly nonlinear parabolic equations

The class of equations of the type (1) $$\partial u/\partial t - div\overrightarrow a (u,\nabla u) = f,$$ such that (2) $$\begin{array}{l} \overrightarrow a (u,p) \cdot p \ge v_0 |u|^l |p|^m - \Phi _0 (u), \\ |\overrightarrow a (u,p)| \le \mu _1 |u|^l |p|^{m - 1} + \Phi _1 (u) \\ \end{array}$$ with some m ∈ (1,2), l≥0, and Φ _i (u)≥0 is studied. Similar equations arise in the study of turbulent filtration of gas or liquid through porous media. Existence and uniqueness in some class of Hölder continuous generalized solutions of the Cauchy-Dirichlet problem for equations of the type (1), (2), is proved. Bibliography: 9 titles.     

Funktionalgleichungen und Charakterisierungen für die Riemannsche Funktion

Riemann's functionR=Σ _v=1 ^∞ v ^?2 sin(2πv ² x) satisfies the following infinite system of functional equations: (*) $$\sum\limits_{k = 0}^{n - 1} {R\left( {\frac{{x + k}}{n}} \right) = \frac{1}{q}R(qx)} $$      

Large deviations of the first passage time for a random walk with semiexponentially distributed jumps

Suppose that ξ, ξ(1), ξ(2), ... are independent identically distributed random variables such that ?ξ is semiexponential; i.e., $P( - \xi \geqslant t) = e^{ - t^\beta L(t)} $ is a slowly varying function as t → ∞ possessing some smoothness properties. Let E ξ = 0, D ξ = 1, and S(k) = ξ(1) + ? + ξ(k). Given d > 0, define the first upcrossing time η ₊(u) = inf{k ≥ 1: S(k) + kd > u} at nonnegative level u ≥ 0 of the walk S(k) + kd with positive drift d > 0. We prove that, under general conditions, the following relation is valid for $u = (n) \in \left[ {0, dn - N_n \sqrt n } \right]$ : 0.1 $P(\eta + (u) > n) \sim \frac{{E\eta + (u)}}{n}P(S(n) \leqslant x) as n \to \infty $ , where x = u ? nd < 0 and an arbitrary fixed sequence N _n not exceeding $d\sqrt n $ tends to ∞. The conditions under which we prove (0.1) coincide exactly with the conditions under which the asymptotic behavior of the probability P(S(n) ≤ x) for $x \leqslant - \sqrt n $ was found in [1] (for $x \in \left[ { - \sqrt n ,0} \right]$ it follows from the central limit theorem).     

A unifying approach to the regularization of Fourier polynomials

In a previous paper [4] the following problem was considered:find, in the class of Fourier polynomials of degree n, the one which minimizes the functional: (0.1) $$J^* [F_n ,\sigma ] = \left\| {f - F_n } \right\|^2 + \sum\limits_{r = 1}^\infty {\frac{{\sigma ^r }}{{r!}}} \left\| {F_n^{(r)} } \right\|^2$$ , where ∥·∥ is theL ² norm,F _n ^(r) is therth derivative of the Fourier polynomialF _n (x), andf(x) is a given function with Fourier coefficientsc _k . It was proved that the optimal polynomial has coefficientsc _k ^* given by (0.2) $$c_k^* = c_k e^{ - \sigma k^2 } ; k = 0, \pm ,..., \pm n$$ . In this paper we consider the more general functional (0.3) $$\hat J[F_n ,\sigma _r ] = \left\| {f - F_n } \right\|^2 + \sum\limits_{r = 1}^\infty {\sigma _r \left\| {F_n^{(r)} } \right\|^2 }$$ , which reduces to (0.1) forσ _r =σ ^r /r!. We will prove that the classical sigma-factor method for the regularization of Fourier polynomials may be obtained by minimizing the functional (0.3) for a particular choice of the weightsσ _r . This result will be used to propose a motivated numerical choice of the parameterσ in (0.1).     

The convex hull of consecutive pairs of observations from some time series models

We examine the asymptotic behavior of the number of vertices of the convex hull spanned by n consecutive pairs from a time series model. We consider data from three models, the moving average (MA) process with regularly varying noise, the stochastic volatility (SV) process with regularly varying noise and the GARCH process. The latter two processes are commonly used for modeling returns of financial assets. If $N_n$ denotes the number of vertices of the convex hull of n consecutive pairs of observations, we show that for a SV model, $N_n \stackrel {P}{\rightarrow } 4 $ as $n \rightarrow \infty$ , whereas for a GARCH model, $N_n \geq 5$ with positive probability. This provides another measure that distinguishes the behavior of the extremes for SV and GARCH models. Geometrically the extreme GARCH pairs fall in butterfly-like shapes away from the axes, while the SV pairs suitably scaled drift towards the coordinate axes with increasing n. MA pairs show a similar flavor as the SV pairs except that their convex hull vertices produce segments of extreme pairs that no longer align themselves exclusively along the axes, but are also distributed along other directions, determined solely by the MA coefficients. We show that the non-degenerate limiting distribution of $ N_n $ as $n \rightarrow \infty $ depends on the model parameters and limiting law of the ratio of the maximal and minimal observations.