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$k$.     

Structural instability of cookie-cutter sets

It is proved that the cookie-cutter set in R$\mathbb{R}$ is structurally instable in C¹$C^1$ topology, that means for the invariant set E$E$ of the IFS _{fi}i${\left\{f_i\right\}}_i$, we can always perturb _{fi}i${\left\{f_i\right\}}_i$ arbitrarily small in C¹$C^1$ topology to provide an IFS _{gi}i${\left\{g_i\right\}}_i$ with its invariant set F$F$, such that _dimHE=_dimHF${dim}_{H}E={dim}_{H}F$ and E,F$E,F$ are not Lipschitz equivalent.     

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.     

Randomly weighted self-normalized Lévy processes

Let (_Ut,_Vt)$(U_t,V_t)$ be a bivariate Lévy process, where _Vt$V_t$ is a subordinator and _Ut$U_t$ is a Lévy process formed by randomly weighting each jump of _Vt$V_t$ by an independent random variable _Xt$X_t$ having cdf F$F$. We investigate the asymptotic distribution of the self-normalized Lévy process _Ut/_Vt$U_t/V_t$ at 0 and at ∞$\infty$. We show that all subsequential limits of this ratio at 0 (∞$\infty$) are continuous for any nondegenerate F$F$ with finite expectation if and only if _Vt$V_t$ belongs to the centered Feller class at 0 (∞$\infty$). We also characterize when _Ut/_Vt$U_t/V_t$ has a non-degenerate limit distribution at 0 and ∞$\infty$.     

Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {_Zs≥ζ(s),s∈S}$\{Z_s\ge \zeta \left(s\right),s\in S\}$, where Z$Z$ is a continuous Gaussian process with stationary increments, ζ$\zeta$ is a function that belongs to the reproducing kernel Hilbert space R$R$ of process Z$Z$, and S⊂R$S\subset \mathbb{R}$ is compact. The main problem considered in this paper is identifying the function β^∗∈R${\beta }^{\ast }\in R$ satisfying β^∗(s)≥ζ(s)${\beta }^{\ast }\left(s\right)\ge \zeta \left(s\right)$ on S$S$ and having minimal R$R$-norm. The smoothness (mean square differentiability) of Z$Z$ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s$\zeta \left(s\right)=s$ for s∈[0,1]$s\in [0,1]$ and Z$Z$ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.     

New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems

Let T:D⊂X→X$T:D\subset X\to X$ be an iteration function in a complete metric space X$X$. In this paper we present some new general complete convergence theorems for the Picard iteration _xn+1=T_xn$x_{n+1}=T{x}_n$ with order of convergence at least r≥1$r\ge 1$. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions   of T$T$ and a convergence function   of T$T$. We study the convergence of the Picard iteration associated to T$T$ with respect to a function of initial conditions E:D→X$E:D\to X$. The initial conditions in our convergence results utilize only information at the starting point _x0$x_0$. More precisely, the initial conditions are given in the form E(_x0)∈J$E\left(x_0\right)\in J$, where J$J$ is an interval on _R+${\mathbb{R}}_{+}$ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete ω$\omega$-versions of the famous semilocal Newton–Kantorovich theorem as well as a complete version of the famous semilocal α$\alpha$-theorem of Smale for analytic functions.     

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.     

Selections and their absolutely continuous invariant measures

Let I=[0,1]$I=[0,1]$ and let P be a partition of I   into a finite number of intervals. Let _τ1${\tau }_1$, _τ2${\tau }_2$; I→I$I\to I$ be two piecewise expanding maps on P  . Let G⊂I×I$G\subset I\times I$ be the region between the boundaries of the graphs of _τ1${\tau }_1$ and _τ2${\tau }_2$. Any map τ:I→I$\tau :I\to I$ that takes values in G is called a selection of the multivalued map defined by G  . There are many results devoted to the study of the existence of selections with specified topological properties. However, there are no results concerning the existence of selection with measure-theoretic properties. In this paper we prove the existence of selections which have absolutely continuous invariant measures (acim). By our assumptions we know that _τ1${\tau }_1$ and _τ2${\tau }_2$ possess acims preserving the distribution functions F⁽¹⁾$F^{(1)}$ and F⁽²⁾$F^{(2)}$. The main result shows that for any convex combination F   of F⁽¹⁾$F^{(1)}$ and F⁽²⁾$F^{(2)}$ we can find a map η   with values between the graphs of _τ1${\tau }_1$ and _τ2${\tau }_2$ (that is, a selection) such that F is the η-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of our multivalued maps to random maps.     

Modules satisfying the weak Nakayama property

Let R$R$ be a commutative ring with identity. We will say that an R$R$-module M$M$ satisfies the weak Nakayama property, if IM=M$IM=M$, where I$I$ is an ideal of R$R$, implies that for any x∈M$x\in M$ there exists a∈I$a\in I$ such that (a−1)x=0$(a-1)x=0$. In this paper, we will study modules satisfying the weak Nakayama property. It is proved that if R$R$ is a local ring, then R$R$ is a Max ring if and only if J(R)$J\left(R\right)$, the Jacobson radical of R$R$, is T$T$-nilpotent if and only if every R$R$-module satisfies the weak Nakayama property.     

Factorization properties of paratopological groups

In this article we continue the study of R$\mathbb{R}$-factorizability in paratopological groups. It is shown that: (1) all concepts of R$\mathbb{R}$-factorizability in paratopological groups coincide; (2) a Tychonoff paratopological group G   is R$\mathbb{R}$-factorizable if and only if it is totally ω  -narrow and has property ω-QU$\omega \text{-}\mathit{QU}$; (3) every subgroup of a _T1$T_1$ paratopological group G   is R$\mathbb{R}$-factorizable provided that the topological group G^?$G^{?}$ associated to G is a Lindelöf Σ-space, i.e., G is a totally Lindelöf Σ-space  ; (4) if Π=_∏i∈IGi$\Pi ={\prod }_{i\in I}{G}_i$ is a product of _T1$T_1$ paratopological groups which are totally Lindelöf Σ-spaces, then each dense subgroup of Π   is R$\mathbb{R}$-factorizable. These results answer in the affirmative several questions posed earlier by M. Sanchis and M. Tkachenko and by S. Lin and L.-H. Xie.     

Mosco convergence of the sets of fixed points for one-parameter nonexpansive semigroups

We consider the Mosco convergence of the sets of fixed points for one-parameter strongly continuous semigroups of nonexpansive mappings. One of our main results is the following: Let C$C$ be a closed convex subset of a Hilbert space E$E$. Let {T(t):t≥0}$\{T\left(t\right):t\ge 0\}$ be a strongly continuous semigroup of nonexpansive mappings on C$C$. The set of all fixed points of T(t)$T\left(t\right)$ is denoted by F(T(t))$F\left(T\left(t\right)\right)$ for each t≥0$t\ge 0$. Let τ$\tau$ be a nonnegative real number and let {_tn}$\left\{t_n\right\}$ be a sequence in R$\mathbb{R}$ satisfying τ+_tn≥0$\tau +t_n\ge 0$ and _tn≠0$t_n\ne 0$ for n∈N$n\in \mathbb{N}$, and _limntn=0${lim}_n{t}_n=0$. Then {F(T(τ+_tn))}$\left\{F\left(T(\tau +t_n)\right)\right\}$ converges to _?t≥0F(T(t))${?}_{t\ge 0}F\left(T\left(t\right)\right)$ in the sense of Mosco.