In this paper, some results of the Chebyshev type integral inequality for the pseudo-integral are proven. The obtained results, are related to the measure of a level set of the maximum and the sum of two non-negative integrable functions. Finally, we applied our results  to the case of comonotone functions.

‎This paper is devoted to definition standard higher dimension shearlet group $ mathbb{S} = mathbb{R}^{+} times mathbb {R}^{n-1} times mathbb {R}^{n} $ and determination of square integrable subrepresentations of this group‎. ‎Also we give a characterisation of admissible vectors associated to the Hilbert spaces corresponding to each su brepresentations‎.

In this paper, we extend and generalize some recent results on the strong laws of large numbers (SLLN) for pairwise independent random variables [3]. No assumption is made concerning the existence of independence among the random variables (henceforth r.v.’s). Also Chandra’s result on Cesàro uniformly integrable r.v.’s is extended.

Typically in time series econometrics, for many statistics, a rescaled integrated process is replaced with Brownian motion in order to find the limit distribution. For averages of functions of a rescaled integrated process, Park and Phillips have shown that this remains true for functions with poles, as long as a sample-size dependent region around the poles is excluded from consideration and t...

In this note we describe the notion of a rectifiable scan and consider some applications [DH1], [DH2] to Plateau-type minimization problems. “Scans” were first introduced in the work [HR1] of Tristan Rivière and the second author to adequately describe certain bubbling phenomena. There, the behavior of certain W 1,3 weakly convergent sequences of smooth maps from 4 dimensional domains into S le...

In this note we define Riemann integrabillty for real valued functions defined on a compact metric space accompanied by a finite Borel measure. If the measure of each open ball equals the measure of its corresponding closed ball, then a bounded function is Riemann integrable if and only if its set of points of discontinuity has measure zero. Let denote the algebra of sets generated by the open ...

A well-known theorem proved by A. C. Lazer and S. Solimini claims that the singular equation u′′ + 1 uλ = h(t), λ > 0, has a periodic solution if and only if the mean value of the continuous external force is positive. In this paper, we show that this result cannot be extended to the case when h is an integrable function, unless the additional assumptions are introduced. In addition, for each p...

In this paper, we construct a new integrable equation which is generalization of $q$-Toda equation. Meanwhile its soliton solutions are constructed to show property. Further the Lax pairs generalized and whole hierarchy also constructed. To integrability, Bi-hamiltonian structure tau symmetry given leads function.

We construct a family of integrable equations the form vt = f(v; vx; vxx; vxxx) such that f is transcendental function in v; vxx. This related to Krichever-Novikov equation by differential substitution. Our construction and corresponding substitutions involves geometry genus two curves their Jacobians.

Solitary wave solutions to the Broer-Kaup equations and approximate long water wave equations are considered challenging by using the rst integral method.The exact solutions obtained during the present investigation are new. This method can be applied to nonintegrable equations as well as to integrable ones.