Global and local behavior of zeros of nonpositive type

A generalized Nevanlinna function Q(z)$Q(z)$ with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by _Qτ(z)=(Q(z)−τ)/(1+τQ(z))$Q_{\tau }(z)=(Q(z)-\tau )/(1+\tau Q(z))$, τ∈R∪{∞}$\tau \in \mathbb{R}\cup \{\infty \}$, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ)$\alpha (\tau )$ as a function of τ is being studied. In particular, it is shown that it is continuous and its behavior in the points where the function extends through the real line is investigated.     

Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory

In this paper, we introduce the concept of a Q$Q$-function defined on a quasi-metric space which generalizes the notion of a τ$\tau$-function and a w$w$-distance. We establish Ekeland-type variational principles in the setting of quasi-metric spaces with a Q$Q$-function. We also present an equilibrium version of the Ekeland-type variational principle in the setting of quasi-metric spaces with a Q$Q$-function. We prove some equivalences of our variational principles with Caristi–Kirk type fixed point theorems for multivalued maps, the Takahashi minimization theorem and some other related results. As applications of our results, we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps. We also extend the Nadler’s fixed point theorem for multivalued maps to a Q$Q$-function and in the setting of complete quasi-metric spaces. As a consequence, we prove the Banach contraction theorem for a Q$Q$-function and in the setting of complete quasi-metric spaces. The results of this paper extend and generalize many results appearing recently in the literature.     

Measure preserving homeomorphism of the Hilbert cube embedding the pseudo-boundary into the pseudo-interior

We prove that the pseudo-boundary B(Q)$B(Q)$ of the Hilbert cube Q can be embedded into the pseudo-interior s of Q by a homeomorphism of Q that preserves the product Lebesgue measure.     

Iterative construction of fixed points of nonself-mappings in Banach spaces

The purpose of this paper is to study the iterative methods for constructing fixed points of nonself-mappings in Banach spaces. The concept of the class of asymptotically _QG$Q_G$-weakly contractive nonself-mappings is introduced and a new iterative algorithm for finding fixed points of this class of mappings is studied. Several strong convergence results on this algorithm are established under different conditions.     

A stabilized finite element method based on two local Gauss integrations for the Stokes equations

This paper considers a stabilized method based on the difference between a consistent and an under-integrated mass matrix of the pressure for the Stokes equations approximated by the lowest equal-order finite element pairs (i.e., the _P1$P_1$–_P1$P_1$ and _Q1$Q_1$–_Q1$Q_1$ pairs). This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf–sup condition. Optimal error estimates are obtained by applying the standard Galerkin technique. Finally, the numerical illustrations agree completely with the theoretical expectations.     

Constant Q-curvature metrics near the hyperbolic metric

Let (M,g)$(M,g)$ be a Poincaré–Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant Q-curvature in the conformal class of an asymptotically hyperbolic metric close enough to g. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.     

Fixed point theorems and the Ulam–Hyers stability in non-Archimedean cone metric spaces

Let (X,p)$(X,p)$ be a metric space with a K-valued non-Archimedean metric p  . In this paper, we prove the existence and approximation of a fixed point for operators F:X→X$F:X\to X$ satisfying the contractive condition in the form p(F(x),F(y))?Q[p(x,y)]$p(F(x),F(y))?Q[p(x,y)]$, where Q:K→K$Q:K\to K$ is an increasing operator. Then, we study the generalized Ulam–Hyers stability of fixed point equations. We next obtain an extension of the Krasnoselskii fixed point theorem for the sum of two operators. Finally, an application to functional equations is given.     

Test ideals of non-principal ideals: Computations,jumping numbers,alterations and division theorems

Given an ideal a⊆R$\mathfrak{a}\subseteq R$ in a (log) Q$\mathbb{Q}$-Gorenstein F  -finite ring of characteristic p>0$p>0$, we study and provide a new perspective on the test ideal τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ for a real number t>0$t>0$. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F  -jumping numbers of τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.     

Density of rational points on del Pezzo surfaces of degree one

We state conditions under which the set S(k)$S(k)$ of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k   of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration S?P¹$S?{\mathbb{P}}^1$ induced by the anticanonical map has a nodal fiber over a k  -rational point of P¹${\mathbb{P}}^1$. It also suffices to require the existence of a point in S(k)$S(k)$ that does not lie on six exceptional curves of S   and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over R$\mathbb{R}$, the set of surfaces S   defined over Q$\mathbb{Q}$ for which the set S(Q)$S(\mathbb{Q})$ is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them.     

Minimal entropy preserves the Lévy property: how and why

Let L be a multidimensional Lévy process under P in its own filtration and consider all probability measures Q turning L into a local martingale. The minimal entropy martingale measure $Q^E$ is the unique Q which minimizes the relative entropy with respect to P. We prove that L is still a Lévy process under $Q^E$ and explain precisely how and why this preservation of the Lévy property occurs.     

An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces

In this paper, a new monotonicity, M$M$-monotonicity, is introduced, and the resolvent operator of an M$M$-monotone operator is proved to be single valued and Lipschitz continuous. With the help of the resolvent operator, an equivalence between the variational inequality VI(C,F+G)$(C,F+G)$ and the fixed point problem of a nonexpansive mapping is established. A proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F$F$ in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method, which is based on the assumption that the projection mapping _∏C(⋅)${\prod }_C(\cdot )$ is semismooth, is given for calculating ε$\epsilon$-solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.     

Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

We give a definition, in the ring language, of _Zp${\mathbb{Z}}_p$ inside _Qp${\mathbb{Q}}_p$ and of _Fp[[t]]${\mathbb{F}}_p[[t]]$ inside _Fp((t))${\mathbb{F}}_p((t))$, which works uniformly for all p   and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining _Zp${\mathbb{Z}}_p$ inside _Qp${\mathbb{Q}}_p$ uniformly for all p  . For any fixed finite extension of _Qp${\mathbb{Q}}_p$, we give an existential formula and a universal formula in the ring language which define the valuation ring.     

Extendability of quadratic modules over a polynomial extension of an equicharacteristic regular local ring

We prove that a quadratic A[T]$A[T]$-module Q   with Witt index (Q/TQ)?d$(Q/TQ)?d$, where d is the dimension of the equicharacteristic regular local ring A, is extended from A. This improves a theorem of the second named author who showed it when A is the local ring at a smooth point of an affine variety over an infinite field. To establish our result, we need to establish a local–global principle (of Quillen) for the Dickson–Siegel–Eichler–Roy (DSER) elementary orthogonal transformations.     

Probabilistic properties of number fields

In general a bound on number theoretic invariants under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta function of a number field K   is much stronger than an unconditional one. In this article, we consider three invariants; the residue of _ζK(s)${\zeta }_K(s)$ at s=1$s=1$, the logarithmic derivative of Artin L-function attached to K   at s=1$s=1$, and the smallest prime which does not split completely in K. We obtain bounds on them just as good as the bounds under GRH except for a density zero set of number fields.