In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy--Littlewood constants for $2$-homogeneous polynomials on $ell_p^2$ spaces, $2<pleqinfty$ and lower estimates for polynomials of higher degrees.
In recent years, the topic of car-following has experimented an increased importance in traffic engineering and safety research. This has become a very interesting topic because of the development of driverless cars cite{google_driverless_cars}. Driv
ing models which describe the interaction between adjacent vehicles in the same lane have a big interest in simulation modeling, such as the Quick-Thinking-Driver model. A non-linear version of it can be given using the logistic map, and then chaos appears. We show that an infinite-dimensional version of the linear model presents a chaotic behavior using the same approach as for studying chaos of death models of cell growth.
A number of sharp inequalities are proved for the space ${mathcal P}left(^2Dleft(frac{pi}{4}right)right)$ of 2-homogeneous polynomials on ${mathbb R}^2$ endowed with the supremum norm on the sector $Dleft(frac{pi}{4}right):=left{e^{itheta}:thetain le
ft[0,frac{pi}{4}right]right}$. Among the main results we can find sharp Bernstein and Markov inequalities and the calculation of the polarization constant and the unconditional constant of the canonical basis of the space ${mathcal P}left(^2Dleft(frac{pi}{4}right)right)$.
In this paper we prove that the complex polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${mathbb C}^2$ is exactly $sqrt[4]{frac{3}{2}}$. We also give the exact value of the real polynomial Bohnenblust-Hille constant for $2$-h
omogeneous polynomials in ${mathbb R}^2$. Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille constant for polynomials in ${mathbb R}^2$ of higher degrees.
In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that
in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust--Hille type inequalities.
We show that a recent interpolative new proof of the Bohnenblust--Hille inequality, when suitably handled, recovers its best known constants. This seems to be unexpectedly surprising since the known interpolative approaches only provide constants hav
ing exponential growth. This preprint is no longer an independent submission, it is now contained in the preprint arXiv 1310.2834.
We introduce the concept of {em maximal lineability cardinal number}, $mL(M)$, of a subset $M$ of a topological vector space and study its relation to the cardinal numbers known as: additivity $A(M)$, homogeneous lineability $HL(M)$, and lineability
$LL(M)$ of $M$. In particular, we will describe, in terms of $LL$, the lineability and spaceability of the families of the following Darboux-like functions on $real^n$, $nge 1$: extendable, Jones, and almost continuous functions.
Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p in [1,infty]$) $Z(ell_p)$ does
not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(ell_infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X setminus Z(X)$ and its algebraic genericity.
