Consider the restriction of directed landscape $${\mathcal {L}}(x, s; y, t)$$ to a set form $$\{x_1, \dots , x_k\} \times \{s_0\} {\mathbb {R}}\times \{t_0\}$$ . We show that on any such set, is given by last passage problem across k locally Brownian functions. The functions in this isometry are built from certain marginals extended landscape. As applications construction, we Airy difference pr...

We give a new proof of a version of Klein's theorem on the existence of absolutely continuous spectrum for the Anderson model on the Bethe Lattice at weak disorder. It is widely believed that the Anderson model [An] should exhibit absolutely continuous spectrum at weak disorder in dimensions three and higher. But it is only for the Bethe lattice B, or Cayley tree, that this has been established...

Abstract. We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum ma...

to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our consists reducing problem (nondynamic) optimization normed space and then applying the results recently obtained latter class by using generalized differentiation. In this way, we derive nonconvex calculus ...

this paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

In [9] it is proved that there are subsets M of the complex plane such that for any two entire functions f and g if f [M ] = g[M ] then f = g. In [3] it was shown that the continuum hypothesis (CH) implies the existence of a similar set M ⊂ R for the class Cn(R) of continuous nowhere constant functions from R to R, while it follows from the results in [5] and [7] that the existence of such a se...