On averages of Fourier coefficients of Maass cusp forms

Let φ be a primitive Maass cusp form and t _φ (n) be its nth Fourier coefficient at the cusp infinity. In this short note, we are interested in the estimation of the sums ${\sum_{n \leq x}t_{\varphi}(n)}$ and ${\sum_{n \leq x}t_{\varphi}(n^2)}$ . We are able to improve the previous results by showing that for any ${\varepsilon > 0}$ $$\sum_{n \leq x}t_{\varphi}(n) \ll\, _{\varphi, \varepsilon} x^{\frac{1027}{2827} + \varepsilon} \quad {and}\quad\sum_{n \leq x}t_{\varphi}(n^2) \ll\,_{\varphi, \varepsilon} x^{\frac{489}{861} + \varepsilon}.$$      

Generalization of the steinhaus problem on the convergence of series with random signs

For series of random variables $\sum\limits_{k = 1}^\infty {a_k x_k }$ ,a _K ∈R ¹, {X _K } _K=1 ^∞ being an Ising system, i.e., for each n ≥ 2 the joint distribution of {X _K } _K=1 ⁿ has the form $$P_n (t_1 ,...,t_n ) = ch^{ - (n - 1)} J \cdot exp(J\sum\limits_{k - 1}^{n - 1} {t_k t_{k + 1} )\prod\limits_{k = 1}^n {\frac{1}{2}\delta (t_{k^{ - 1} }^2 ),J > 0} }$$ one obtains a criterion for almost everywhere convergence: $\sum\limits_{k = 1}^\infty {a_k^2< \infty }$ . The relation between the asymptotic behavior of large deviations of the sum and the rate of decrease of the sequence {a_k} of the coefficients is investigated.     

On the asymptotic behavior of solutions of Emden–Fowler equations on time scales

Consider the Emden-Fowler dynamic equation $$ x^{\Delta\Delta}(t)+p(t)x^\alpha(t)=0,\:\:\alpha >0 , \qquad \qquad \qquad \qquad (0.1) $$ where ${p\in C_{rd}([t_0,\infty)_{\mathbb{T}},\mathbb{R}), \alpha}$ is the quotient of odd positive integers, and ${\mathbb{T}}$ denotes a time scale which is unbounded above and satisfies an additional condition (C) given below. We prove that if ${\int^\infty_{t_0}t^\alpha |p(t)|\Delta t<\infty}$ (and when ???=?1 we also assume lim _t???? tp(t)??(t)?=?0), then (0.1) has a solution x(t) with the property that $$ \lim_{t\rightarrow\infty} \frac{x(t)}{t}=A\neq 0.$$      

A mean-square bound concerning the lattice discrepancy of a torus in ℝ3

For positive constants a > b > 0, let P _T (t) denote the lattice point discrepancy of the body tT _a,b , where t is a large real parameter and T = T _a,b is bounded by the surface $$ \partial \tau _{a,b} :\left( {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {(a + b\cos \alpha )\cos \beta } \\ {(a + b\cos \alpha )\sin \beta } \\ {b\sin \alpha } \\ \end{array} } \right), 0 \leqq \alpha ,\beta < 2\pi . $$ In a previous paper [12] it has been proved that $$ P_\tau (t) = \mathcal{F}_{a,b} (t)t^{3/2} + \Delta _\tau (t), $$ where F _a,b (t) is an explicit continuous periodic function, and the remainder satisfies the (“pointwise”) estimate Δ _T (t) ? t ^11/8+? . Here it will be shown that this error term is only ? t ^1+? in mean-square, i.e., that $$ \int\limits_0^T {(\Delta _\tau (t))^2 dt} \ll T^{3 + \varepsilon } $$ for any ? > 0.     

Oscillatory Properties of Solutions of Impulsive Differential Equations With Several Retarded Arguments

The impulsive differential equation $\begin{gathered} x\prime (t) + \sum\limits_{i = 1}^m {p_i (t)x(t - \tau _i ) = 0,} {\text{ }}t \ne \xi _k , \\ \Delta x(\xi _k ) = b_k x(\xi _k ) \\ \end{gathered} $ with several retarded arguments is considered, where p _i(t) ≥ 0, 1 + b _k > 0 for i = 1, ..., m, t ≥ 0, $k \in \mathbb{N}$ . Sufficient conditions for the oscillation of all solutions of this equation are found.     

On the dynamics of certain recurrence relations

In recent analyses [3, 4] the remarkable AGM continued fraction of Ramanujan—denoted ${\cal R}_1$ (a, b)—was proven to converge for almost all complex parameter pairs (a, b). It was conjectured that ${\cal R}_1$ diverges if and only if (0≠ a = be ^{i φ} with cos ²φ ≠ 1) or (a ² = b ²? (?∞, 0)). In the present treatment we resolve this conjecture to the positive, thus establishing the precise convergence domain for ${\cal R}_1$ . This is accomplished by analyzing, using various special functions, the dynamics of sequences such as (t _n ) satisfying a recurrence $$ t_n = (t_{n-1} + (n-1) \kappa_{n-1}t_{n-2})/n, $$ where κ _n ? a ², b ² as n be even, odd respectively. As a byproduct, we are able to give, in some cases, exact expressions for the n-th convergent to the fraction ${\cal R}_1$ , thus establishing some precise convergence rates. It is of interest that this final resolution of convergence depends on rather intricate theorems for complex-matrix products, which theorems evidently being extensible to more general continued fractions.     

A certain generalization of the law of the iterated logarithm

Let X₁ ,..., X_n be independent random variables and let \(S_n = \sum\limits_{i = 1}^n {X_i }\) . For the sequence of random variables $$T_n = \sum\limits_1^p {(S_{t_j } - S_{t_{j - 1} } )} ,$$ where t₀=01<...p=n, p?1, under certain conditions on t_i, \(i = \overline {1,n}\) , one proves a series of general theorems of the type of the iterated logarithm laws.     

On Asymptotic and Strict Monotonicity of a Sharper Lower Bound for Student’s t Percentiles

Let z _α and t _ν,α denote the upper 100α% points of a standard normal and a Student’s t _ν distributions respectively. It is well-known that for every fixed $0<\alpha <\frac{1}{2}$ and degree of freedom ν, one has t _ν,α ?>?z _α and that t _ν,α monotonically decreases to z _α as ν increases. Recently, Mukhopadhyay (Methodol Comput Appl Probab, 2009) found a new and explicit expression b _ν (?>?1) such that t _ν,α ?>?b _ν z _α for every fixed $0<\alpha <\frac{1}{2}$ and ν. He also showed that b _ν converges to 1 as ν increases. In this short note, we prove three key results: (i) $\log(b_{\nu+1}/b_{\nu})\sim -\frac{1}{4}\nu^{-2}$ for large enough ν, (ii) b _ν strictly decreases as ν increases, and (iii) $b_{\nu}\sim 1+\frac14\nu^{-1}+\frac1{32}\nu^{-2}$ for large enough ν.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Positive Periodic Solutions of Systems of First Order Ordinary Differential Equations

Consider the n-dimensional nonautonomous system ?(t) = A(t)G(x(t)) ? B(t)F(x(t ? τ(t))) Let u = (u ₁,…,u _n ), $f^{i}_{0}={\rm lim}_{\|{\rm u}\|\rightarrow 0}{f^{i}(\rm u)\over \|u\|}$ , $f^{i}_{\infty}={\rm lim}_{\|{\rm u}\|\rightarrow \infty}{f^{i}(\rm u)\over \|u\|}$ , i = l,…,n, F = (f ¹…,f ⁿ ), ${\rm F_{0}}={\rm max}_{i=1,\ldots,n}{f^{i}_{0}}$ and ${\rm F_{\infty}}={\rm max}_{i=1,\ldots,n}{f^{i}_{\infty}}$ . Under some quite general conditions, we prove that either F₀ = 0 and F_∞ = ∞, or F₀ = ∞ and F_∞ = 0, guarantee the existence of positive periodic solutions for the system for all λ > 0. Furthermore, we show that F₀ = F_∞ = 0, or F_∞ = F_∞ = ∞ guarantee the multiplicity of positive periodic solutions for the system for sufficiently large, or small λ, respectively. We also establish the nonexistence of the system when either F₀ and F_∞ > 0, or F₀ and F_∞, < for sufficiently large, or small λ, respectively. We shall use fixed point theorems in a cone.     

Solutions of the Allen-Cahn equation which are invariant under screw-motion

Let ${\phi(x)}$ be a rational function of degree >?1 defined over a number field K and let ${\Phi_{n}(x,t) = \phi^{(n)}(x)-t \in K(x,t)}$ where ${\phi^{(n)}(x)}$ is the nth iterate of ${\phi(x)}$ . We give a formula for the discriminant of the numerator of Φ _n (x, t) and show that, if ${\phi(x)}$ is postcritically finite, for each specialization t ₀ of t to K, there exists a finite set ${S_{t_0}}$ of primes of K such that for all n, the primes dividing the discriminant are contained in ${S_{t_0}}$ .     

Extinction profile of the logarithmic diffusion equation

Let ${N \geq 3}$ and u be the solution of u _t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u ₀ satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k ₁ > k ₂ > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k ₀ > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L ¹ space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ .     

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

An interstice relationship for flowers with four petals

Given three mutually tangent circles with bends (related to the reciprocal of the radius) a, b and c respectively, an important quantity associated with the triple is the value ${\langle a,b,c \rangle:=ab+ac+bc}$ . In this note we show in the case when a central circle with bend b ₀ is “surrounded” by four circles, i.e., a flower with four petals, with bends b ₁, b ₂, b ₃,b ₄ that either $$\sqrt{\langle b_{0},b_{1},b_{2} \rangle}+\sqrt{\langle b_{0},b_{3},b_{4} \rangle}=\sqrt{\langle b_{0},b_{2},b_{3} \rangle}+\sqrt{\langle b_{0},b_{4},b_{1} \rangle}$$ or $$\sqrt{\langle b_{0},b_{1},b_{2} \rangle}=\sqrt{\langle b_{0},b_{2},b_{3} \rangle}+\sqrt{\langle b_{0},b_{3},b_{4} \rangle}+\sqrt{\langle b_{0},b_{4},b_{1} \rangle}$$ (where ${\langle b_{0},b_{1},b_{2} \rangle}$ is chosen to be maximal). As an application we give a sufficient condition for the alternating sum of the ${\sqrt{\langle a,b,c\rangle}}$ of a packing in standard position to be 0. (A packing is in standard position when we have two circles with bend 0, i.e., parallel lines, and the remaining circles are packed in between.)     

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

Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

We consider evolution inclusions, in a separable and reflexive Banach space ${\mathbb{E}}$ , of the form ${(\ast) x'(t) \in Ax(t) + F(t, x(t)), x(t_0) = c}$ and ${(**) x'(t) \in Ax(t) + {\rm ext} F(t,x(t)), x(t_0) = c}$ , where A is the infinitesimal generator of a C ₀-semigroup, F is a continuous and bounded multifunction defined on ${[t_0, t_1] \times \mathbb{E}}$ with values F(t, x) in the space of all closed convex and bounded subsets of ${\mathbb{E}}$ with nonempty interior, and ext F(t, x(t)) denotes the set of the extreme points of F(t, x(t)). For (*) and (**) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of (**) contains the set of all internal solutions of (*). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C ₀-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values ${F(t, x) \subset \mathbb{E}}$ , the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of (**) is equal to the set of the mild solutions of (*).     

An Almost-Markov-Type Mixing Condition and Large Deviations for Boolean Models on the Line

We consider a not necessarily stationary one-dimensional Boolean model Ξ=∪ _i≥1(Ξ _i +X _i ) defined by a Poisson process $\Psi=\sum_{i\ge 1}\delta_{X_{i}}$ with bounded intensity function λ(t)≤λ ₀ and a sequence of independent copies Ξ ₁,Ξ ₂,… of a random compact subset Ξ ₀ of the real line ?¹ whose diameter ‖Ξ ₀‖ possesses a finite exponential moment $\mathsf{E}\exp\{a\|\Xi_{0}\|\}$ . We first study the higher-order covariance functions $\mathop{\mathsf{E}}\limits^{\frown}\xi(t_{1})\xi(t_{2})\cdots \xi(t_{k})$ of the {0,1}-valued stochastic process $\xi(t)=\mathbf{1}_{\Xi^{c}(t)},\ t\in \mathbb{R}^{1}$ , and derive exponential estimates of them as well as of the mixed cumulants Cum _k (ξ(t ₁),ξ(t ₂),…,ξ(t _k )). From this, we derive Cramér-type large deviations relations and a Berry–Esseen bound for the distribution of empirical total length meas(Ξ∩[0,T]) of Ξ within [0,T] as T grows large. Second, we prove that the family of events {ξ(t)=1}={t ? Ξ}, t∈?¹, satisfies an almost-Markov-type mixing condition with an exponentially decaying mixing rate. In case of a stationary Boolean model, i.e. λ(t)≡λ ₀, these properties enable us to show the existence and analyticity of the thermodynamic limit $$L(z)=\lim_{T\to \infty}\frac{1}{T}\log \mathsf{E}\exp\bigl\{z\mathop{\mathrm{meas}}\bigl(\Xi\cap [0,T]\bigr)\bigr\}\quad \hbox{for}\ |z|<\varepsilon(a,\lambda_{0}).$$      

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Convergence of symmetric Markov chains on {\mathbb{Z}^d}

For each n let ${Y^{(n)}_t}$ be a continuous time symmetric Markov chain with state space ${n^{-1} \mathbb{Z}^d}$ . Conditions in terms of the conductances are given for the convergence of the ${Y^{(n)}_t}$ to a symmetric Markov process Y _t on ${\mathbb{R}^d}$ . We have weak convergence of $\{{Y^{(n)}_t: t \leq t_0\}}$ for every t ₀ and every starting point. The limit process Y has a continuous part and may also have jumps.     

Isomorphisms of Brin-Higman-Thompson groups

Let m, m′, r, r′, t, t′ be positive integers with r, r′ ? 2. Let \(\mathbb{L}_r \) denote the ring that is universal with an invertible 1×r matrix. Let \(M_m (\mathbb{L}_r^{ \otimes t} )\) denote the ring of m × m matrices over the tensor product of t copies of \(\mathbb{L}_r \) . In a natural way, \(M_m (\mathbb{L}_r^{ \otimes t} )\) is a partially ordered ring with involution. Let \(PU_m (\mathbb{L}_r^{ \otimes t} )\) denote the group of positive unitary elements. We show that \(PU_m (\mathbb{L}_r^{ \otimes t} )\) is isomorphic to the Brin-Higman-Thompson group tV _r,m ; the case t=1 was found by Pardo, that is, \(PU_m (\mathbb{L}_r )\) is isomorphic to the Higman-Thompson group V _r,m . We survey arguments of Abrams, Ánh, Bleak, Brin, Higman, Lanoue, Pardo and Thompson that prove that t′V _r′,m′ ≌ tV _r,m if and only if r′ =r, t′ =t and gcd(m′, r′?1) = gcd(m, r?1) (if and only if \(M_{m'} (\mathbb{L}_{r'}^{ \otimes t'} )\) and \(M_m (\mathbb{L}_r^{ \otimes t} )\) are isomorphic as partially ordered rings with involution).