Indagationes Mathematicae-New Series最新文献

We deal with the finite family $F$ of continuous maps on the Hausdorff space $X$. A nonempty compact subset $A$ of such space is called a strict attractor if it has an open neighborhood $U$ such that $A={lim}_{n\to \infty }{F}^n\left(S\right)$ for every nonempty compact $S\subset U$. Every strict attractor is a pointwise attractor, which means that the set $\{x\in X;{lim}_{n\to \infty }{F}^n\left(x\right)=A\}$ contains $A$ in its interior.

We present a class of examples of pointwise attractors – from the finite set to the Sierpiński carpet – which are not strict when we add to the system one nonexpansive map.

We give a sufficient condition for a pair of Banach spaces $(X,Y)$ to have the following property: whenever $W_1\subseteq X$ and $W_2\subseteq Y$ are sets such that $\{x\otimes y:x\in W_1,y\in W_2\}$ is weakly precompact in the projective tensor product $X{\hat{\otimes }}_{\pi }Y$, then either $W_1$ or $W_2$ is relatively norm compact. For instance, such a property holds for the pair $({\ell }_p,{\ell }_q)$ if $1<p,q<\infty$ satisfy $1/p+1/q\ge 1$. Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ${\ell }_1$ into $X{\hat{\otimes }}_{\pi }Y$ for arbitrary Banach spaces $X$ and $Y$, in connection with the compactness of all operators from $X$ to $Y^{\ast }$.

Many results about the geometry of the trianguline variety have been obtained by Breuil–Hellmann–Schraen. Among them, using geometric methods, they have computed a formula for the dimension of the tangent space of the trianguline variety at dominant crystalline generic points, which has a conjectural generalisation to companion (i.e. non-dominant) points. In an earlier work, they proved a weaker form of this formula under the assumption of modularity using arithmetic methods. We prove a generalisation of a result of Bellaïche–Chenevier in $p$-adic Hodge theory and use it to extend the arithmetic methods of Breuil–Hellmann–Schraen to a wide class of companion points.

This paper is a sequel to a previous paper of the authors in which the cohomology of semi-simple Leibniz algebras was computed by using spectral sequences. In the present paper we generalize the vanishing theorems of Dixmier and Barnes for nilpotent and (super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the cohomology of the one-dimensional Lie algebra with values in an arbitrary Leibniz bimodule and show that it is periodic with period two. As a consequence, we establish the Leibniz analogue of a non-vanishing theorem of Dixmier for nilpotent Leibniz algebras. In addition, we prove a Fitting lemma for Leibniz bimodules

With this article, we hope to launch the investigation of what we call the Real Zero Amalgamation Problem. Whenever a polynomial arises from another polynomial by substituting zero for some of its variables, we call the second polynomial an extension of the first one. The Real Zero Amalgamation Problem asks when two (multivariate real) polynomials have a common extension (called amalgam) that is a real zero polynomial. We show that the obvious necessary conditions are not sufficient. Our counterexample is derived in several steps from a counterexample to amalgamation of matroids by Poljak and Turzík. On the positive side, we show that even a degree-preserving amalgamation is possible in three very special cases with three completely different techniques. Finally, we conjecture that amalgamation is always possible in the case of two shared variables. The analogue in matroid theory is true by another work of Poljak and Turzík. This would imply a very weak form of the Generalized Lax Conjecture.

Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the $q$-aspect for the logarithmic derivative $\left(L^{\prime }/L\right)\left(\sigma ,\chi \right)$ of Dirichlet $L$-functions, where $\chi$ is a primitive character modulo $q\ge 10^{30}$ and $1/2+1/loglogq\le \sigma \le 1-1/loglogq$. In addition, for $\sigma =1$ we improve upon the result by Ihara, Murty and Shimura (2009). Similar results for the logarithmic derivative of the Riemann zeta-function are given.

We consider the impulse control of Lévy processes under the infinite horizon, discounted cost criterion. Our motivating example is the cash management problem in which a controller is charged a fixed plus proportional cost for adding to or withdrawing from his/her reserve, plus an opportunity cost for keeping any cash on hand. Our main result is to provide a verification theorem for the optimality of control band policies in this scenario. We also analyze the transient and steady-state behavior of the controlled process under control band policies and explicitly solve for an optimal policy in the case in which the Lévy process to be controlled is the sum of a Brownian motion with drift and a compound Poisson process with exponentially distributed jump sizes.

Heavy-traffic limit theory is concerned with queues that operate close to criticality and face severe queueing times. Let $W$ denote the steady-state waiting time in the $\mathrm{GI}/G/1$ queue. Kingman (1961) showed that $W$, when appropriately scaled, converges in distribution to an exponential random variable as the system’s load approaches 1. The original proof of this famous result uses the transform method. Starting from the Laplace transform of the pdf of $W$ (Pollaczek’s contour integral representation), Kingman showed convergence of transforms and hence weak convergence of the involved random variables. We apply and extend this transform method to obtain convergence of moments with error assessment. We also demonstrate how the transform method can be applied to so-called nearly deterministic queues in a Kingman-type and a Gaussian heavy-traffic regime. We demonstrate numerically the accuracy of the various heavy-traffic approximations.

Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider.

The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the stationary distribution can indeed be found by solving a boundary value problem of the same kind as the one encountered in the quarter plane, up to various dualities and symmetries. The main idea is to start from Fourier (and not Laplace) transforms, allowing to get a functional equation for a single function of two complex variables.