Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConvergences and the Intermediate Value Property in Fermat Reals
68
0
0.0
(
0
)
Authors
Enxin Wu
Ask ChatGPT about the research
No Arabic abstract
This paper contains two topics of Fermat reals, as suggested by the title. In the first part, we study the omega-topology, the order topology and the Euclidean topology on Fermat reals, and their convergence properties, with emphasis on the relationship with the convergence of sequences of ordinary smooth functions. We show that the Euclidean topology is best for this relationship with respect to pointwise convergence, and Lebesgue dominated convergence does not hold, among all additive Hausdorff topologies on Fermat reals. In the second part, we study the intermediate value property of quasi-standard smooth functions on Fermat reals, together with some easy applications. The paper is written in the language of Fermat reals, and the idea could be extended to other similar situations.
rate research
Read More
We develop the integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano-Jordan-like integration domains.
The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy Mean Value Theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.
We establish partial semigroup property of Riemann-Liouville and Caputo fractional differential operators. Using this result we prove theorems on reduction of multi-term fractional differential systems to single-term and multi-order systems, and prove existence and uniqueness of solution to multi-term Caputo fractional differential systems
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.
Let $(X,d,mu)$ be a metric measure space satisfying a $Q$-doubling condition, $Q>1$, and an $L^2$-Poincar{e} inequality. Let $mathscr{L}=mathcal{L}+V$ be a Schrodinger operator on $X$, where $mathcal{L}$ is a non-negative operator generalized by a Dirichlet form, and $V$ is a non-negative Muckenhoupt weight that satisfies a reverse Holder condition $RH_q$ for some $qge (Q+1)/2$. We show that a solution to $(mathscr{L}-partial_t^2)u=0$ on $Xtimes mathbb{R}_+$ satisfies the Carleson condition, $$sup_{B(x_B,r_B)}frac{1}{mu(B(x_B,r_B))} int_{0}^{r_B} int_{B(x_B,r_B)} |t abla u(x,t)|^2 frac{mathrm{d}mumathrm{d} t}{t}<infty,$$ if and only if, $u$ can be represented as the Poisson integral of the Schrodinger operator $mathscr{L}$ with trace in the BMO space associated with $mathscr{L}$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
