On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α$\alpha$ is in (0,2)$(0,2)$, equal to 2, and in (2,∞)$(2,\infty )$, respectively. The partial sum weakly converges to a functional of α$\alpha$-stable process when α<2$\alpha <2$ and converges to a functional of Brownian motion when α≥2$\alpha \ge 2$. When the process is of short-memory and α<4$\alpha <4$, the autocovariances converge to functionals of α/2$\alpha /2$-stable processes; and if α≥4$\alpha \ge 4$, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α$\alpha$ and β$\beta$ (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2$\alpha /2$-stable processes; (ii) Rosenblatt processes (indexed by β$\beta$, 1/2<β<3/4$1/2<\beta <3/4$); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α$\alpha$ and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1]$[0,1]$ with either (i) the _J1$J_1$ or the _M1$M_1$ topology (Skorokhod, 1956); or (ii) the weaker form S$S$ topology (Jakubowski, 1997). Some statistical applications are also discussed.     

On a family of differential operators with the coupling parameter in the boundary condition

We study a family of differential operators _Lα${\mathbf{L}}_{\alpha }$ in two variables, depending on the coupling parameter α?0$\alpha ?0$ that appears only in the boundary conditions. Our main concern is the spectral properties of _Lα${\mathbf{L}}_{\alpha }$, which turn out to be quite different for α<1$\alpha <1$ and for α>1$\alpha >1$. In particular, _Lα${\mathbf{L}}_{\alpha }$ has a unique self-adjoint realization for α<1$\alpha <1$ and many such realizations for α>1$\alpha >1$. In the more difficult case α>1$\alpha >1$ an analysis of non-elliptic pseudodifferential operators in dimension one is involved.     

Parametric inference for discretely observed multidimensional diffusions with small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(α,_Xt)$b(\alpha ,X_t)$ and diffusion coefficient ?σ(β,_Xt)$?\sigma (\beta ,X_t)$. The diffusion sample path is discretely observed at times _tk=kΔ$t_k=k\Delta$ for k=1…n$k=1\dots n$ on a fixed interval [0,T]$[0,T]$. We study minimum contrast estimators derived from the Gaussian process approximating X$X$ for small ?$?$. We obtain consistent and asymptotically normal estimators of α$\alpha$ for fixed Δ$\Delta$ and ?→0$?\to 0$ and of (α,β)$(\alpha ,\beta )$ for Δ→0$\Delta \to 0$ and ?→0$?\to 0$ without any condition linking ?$?$ and Δ$\Delta$. We compare the estimators obtained with various methods and for various magnitudes of Δ$\Delta$ and ?$?$ based on simulation studies. Finally, we investigate the interest of using such methods in an epidemiological framework.     

A note on distance-regular graphs with a small number of vertices compared to the valency

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2$\alpha >2$, there are finitely many distance-regular graphs Γ$\Gamma$ with valency k$k$, diameter D≥3$D\ge 3$ and v$v$ vertices satisfying v≤αk$v\le \alpha k$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3$k\ge 3$, diameter D≥3$D\ge 3$ and _c2≥εk$c_2\ge \epsilon k$ for a given 0<ε<1$0<\epsilon <1$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite).     

Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions

Let C$C$ be a closed convex subset of a real Hilbert space H$H$ and assume that T$T$ is an asymptotically κ$\kappa$-strict pseudo-contraction on C$C$ with a fixed point, for some 0≤κ<1$0\le \kappa <1$. Given an initial guess _x0∈C$x_0\in C$ and given also a real sequence {_αn}$\left\{{\alpha }_n\right\}$ in (0, 1), the modified Mann’s algorithm generates a sequence {_xn}$\left\{x_n\right\}$ via the formula: _xn+1=_αnxn+(1−_αn)Tⁿ_xn$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T^n{x}_n$, n≥0$n\ge 0$. It is proved that if the control sequence {_αn}$\left\{{\alpha }_n\right\}$ is chosen so that κ+δ<_αn<1−δ$\kappa +\delta <{\alpha }_n<1-\delta$ for some δ∈(0,1)$\delta \in (0,1)$, then {_xn}$\left\{x_n\right\}$ converges weakly to a fixed point of T$T$. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.     

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.     

Blow-up and propagation of disturbances for fast diffusion equations

This paper is concerned with the Cauchy problem for the fast diffusion equation _ut−Δu^m=αu^_p1$u_t-\Delta u^m=\alpha u^{p_1}$ in R^N${\mathbb{R}}^N$ (N≥1$N\ge 1$), where m∈(0,1)$m\in (0,1)$, _p1>1$p_1>1$ and α>0$\alpha >0$. The initial condition _u0$u_0$ is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value _u0$u_0$ so that u(t,x)$u(t,x)$ blows up in finite time, and we show how to get estimates on the profile of u(t,x)$u(t,x)$ for small enough values of t>0$t>0$.     

Coalescence in the recent past in rapidly growing populations

In a rapidly growing population one expects that two individuals chosen at random from the n$n$th generation are unlikely to be closely related if n$n$ is large. In this paper it is shown that for a broad class of rapidly growing populations this is not the case. For a Galton–Watson branching process with an offspring distribution {_pj}$\left\{p_j\right\}$ such that _p0=0$p_0=0$ and ψ(x)=_{∑jpjI{j≥x}}$\psi \left(x\right)={\sum }_j{p}_j{I}_{\{j\ge x\}}$ is asymptotic to x^−αL(x)$x^{-\alpha }L\left(x\right)$ as x→∞$x\to \infty$ where L(⋅)$L(\cdot )$ is slowly varying at ∞$\infty$ and 0<α<1$0<\alpha <1$ (and hence the mean m=∑j_pj=∞$m=\sum j{p}_j=\infty$) it is shown that if _Xn$X_n$ is the generation number of the coalescence of the lines of descent backwards in time of two randomly chosen individuals from the n$n$th generation then n−_Xn$n-X_n$ converges in distribution to a proper distribution supported by N={1,2,3,…}$\mathbb{N}=\{1,2,3,\dots \}$. That is, in such a rapidly growing population coalescence occurs in the recent past rather than the remote past. We do show that if the offspring mean m$m$ satisfies 1<m≡∑j_pj<∞$1<m\equiv \sum j{p}_j<\infty$ and _p0=0$p_0=0$ then coalescence time _Xn$X_n$ does converge to a proper distribution as n→∞$n\to \infty$, i.e., coalescence does take place in the remote past.     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

Crossing number additivity over edge cuts

Consider a graph G$G$ with a minimal edge cut F$F$ and let _G1$G_1$, _G2$G_2$ be the two (augmented) components of G−F$G-F$. A long-open question asks under which conditions the crossing number of G$G$ is (greater than or) equal to the sum of the crossing numbers of _G1$G_1$ and _G2$G_2$—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2}$\left|F\right|\in \{0,1,2\}$ and that there exist graphs violating this property with |F|≥4$\left|F\right|\ge 4$. In this paper, we show that crossing number is additive for |F|=3$\left|F\right|=3$, thus closing the final gap in the question.     

The relaxed Newton method derivative: Its dynamics and non-linear properties

The dynamic behaviour of the one-dimensional family of maps f(x)=_c2[(a−1)x+_c1]^{−λ/(α−1)}$f\left(x\right)=c_2{[(a-1)x+c_1]}^{-\lambda /(\alpha -1)}$ is examined, for representative values of the control parameters a,_c1$a,c_1$, _c2$c_2$ and λ$\lambda$. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a$a$. The maps f(x)$f\left(x\right)$ are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an _xn$x_n$ versus λ$\lambda$ plot, an initial exponential decay followed by a bifurcation. The value of λ$\lambda$ at which this bifurcation takes place depends on the values of the parameters a,_c1$a,c_1$ and _c2$c_2$. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x)$f\left(x\right)$ undergoing a period doubling. For values of a$a$ higher than 1 and at higher values of λ$\lambda$ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter _c1$c_1$ between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Characterizations of hypersurfaces of a Danielewski type

Let k$k$ be a field of characteristic zero and R$R$ a factorial affine k$k$-domain. Let B$B$ be an affineR$R$-domain. In terms of locally nilpotent derivations, we give criteria for B$B$ to be R$R$-isomorphic to the residue ring of a polynomial ring R[_X1,_X2,Y]$R[X_1,X_2,Y]$ over R$R$ by the ideal (_X1X2−φ(Y))$(X_1{X}_2-\phi \left(Y\right))$ for φ(Y)∈R[Y]?R$\phi \left(Y\right)\in R\left[Y\right]?R$.     

Frobenius groups and retract rationality

Let k$k$ be any field, G$G$ be a finite group acting on the rational function field k(_xg:g∈G)$k(x_g:g\in G)$ by h⋅_xg=_xhg$h\cdot x_g=x_{hg}$ for any h,g∈G$h,g\in G$. Define k(G)=k(_xg:g∈G)^G$k\left(G\right)=k{(x_g:g\in G)}^G$. Noether’s problem asks whether k(G)$k\left(G\right)$ is rational (= purely transcendental) over k$k$. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if G$G$ is a Frobenius group with abelian Frobenius kernel, then k(G)$k\left(G\right)$ is retract k$k$-rational for any field k$k$ satisfying some mild conditions. As an application, we show that, for any algebraic number field k$k$, for any Frobenius group G$G$ with Frobenius complement isomorphic to S_L2(_F5)$S{L}_2\left({\mathbb{F}}_5\right)$, there is a Galois extension field K$K$ over k$k$ whose Galois group is isomorphic to G$G$, i.e. the inverse Galois problem is valid for the pair (G,k)$(G,k)$. The same result is true for any non-solvable Frobenius group if k(_ζ8)$k\left({\zeta }_8\right)$ is a cyclic extension of k$k$.     

Geodesic continued fractions and LLL

We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to d$d$ real numbers _α1,…,_αd${\alpha }_1,\dots ,{\alpha }_d$. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter t$t$ as t↓0$t\downarrow 0$. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in t$t$.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.