For x real, let {x} be the fractional part of x (i.e. {x} = x − bxc). In this paper we prove the k = 5 case of the following conjecture (the lonely runner conjecture): for any k positive reals v1, . . . , vk there exists a real number t such that 1/(k + 1) ≤ {vit} ≤ k/(k + 1) for i = 1, . . . , k.

We prove that every set system of bounded VC-dimension has a fractional Helly property. More precisely, if the dual shatter function of a set system F is bounded by o(m k), then F has fractional Helly number k. This means that for every > 0 there exists a > 0 such that if F 1 ; F 2 ; : : : ; F n 2 F are sets with T i2I F i 6 = ; for at least ? n k sets I f1; 2; : : :; ng of size k, then there e...

in this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the lipschitz type condition. moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

This is a generalization of a fractional programming problem (p = 1) which has been investigated quite actively in the last two decades [21 ]. In [20] many of the results in fractional programming are reviewed and extended. An extensive bibliography is given in [22]. An early application of generalized fractional programming (p > 1) is von Neumann's model of an expanding economy [25]. Here the ...

We extend the study of the algebraic structure of the V ARd,b(k) model defined from the fractional lag operator in Johansen (2005) and derive conditions under which its solution is fractional of order d and displays linear combinations that are fractional of order d− b, d− 2b, · · · , d− cb ≥ 0, for integer c. We then find the corresponding moving average representation and the cofractional rel...

We study restricted computation models related to the tree evaluation problem. The TEP was introduced in earlier work as a simple candidate for the (very) long term goal of separating L and LogDCFL. The input to the problem is a rooted, balanced binary tree of height h, whose internal nodes are labeled with binary functions on [k] = {1, . . . , k} (each given simply as a list of k2 elements of ...

Fractional regression hot deck imputation (FRHDI), suggested by J. K. Kim, imputes multiple values for each instance of a missing dependent variable. The imputed values are equal to the predicted value based on the fully observed cases plus multiple random residuals chosen from the set of empirical residuals. Fractional weights are chosen to enable variance estimation and to preserve the correl...

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair [Formula: see text] and probability density function. Com...

Based on the Riemann fractional derivative the Casimir operators and multipletts for the fractional extension of the rotation group SO(n) are calculated algebraically. The spectrum of the corresponding fractional rigid rotator is discussed. It is shown, that the rotational, vibrational and γ unstable limits of the standard geometric collective models are particular limits of this spectrum. A co...

Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. For the fractional harmonic oscillator, the corresponding q-number is derived. It is shown, that the resulting energy spectrum is an appropriate tool e.g. to describe the ground state spectra of even-even nuclei. In addition, the equiva...