The k-server problem, introduced by Manasse, McGeoch and Sleator [29, 30] is a fundamental online problem where k mobile servers are required to serve a sequence of requests on a metric space, with the minimum possible movement cost. Despite the conjectured existence of a Θ(log k)-competitive randomized algorithm, the best known upper bound for arbitrary metric spaces is 2k − 1. Even for one of...

We review some basic elements on k-fermions, which are objects interpolating between bosons and fermions. In particular, we define k-fermionic coherent states and study some of their properties. The decomposition of a Q-uon into a boson and a k-fermion leads to a definition of fractional supercoherent states. Such states involve bosonic coherent states and k-fermionic coherent states. We constr...

We define the total weak discrepancy of a poset P as the minimum nonnegative integer k for which there exists a function f : V → Z satisfying (i) if a ≺ b then f(a) + 1 ≤ f(b) and (ii) ∑ |f(a)− f(b)| ≤ k, where the sum is taken over all unordered pairs {a, b} of incomparable elements. If we allow k and f to take real values, we call the minimum k the fractional total weak discrepancy of P . The...

Our main result is that every graph G on n ≥ 10r vertices with minimum degree δ(G) ≥ (1−1/10r)n has a fractional Kr-decomposition. Combining this result with recent work of Barber, Kühn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) F -decompositions for a wide class of graphs F (including large cliques). For general k-uniform hypergraphs, we give a ...

We investigate some issues relating to recently proposed fractional superstring theories with Dcritical < 10. Using the factorization approach of Gepner and Qiu, we systematically rederive the partition functions of the K = 4, 8, and 16 theories and examine their spacetime supersymmetry. Generalized GSO projection operators for the K = 4 model are found. Uniqueness of the twist field, φ K/4 K/4...

In the present paper, the notion of generalized $(r;g,s,m,varphi)$-preinvex function is applied to establish some new generalizations of Ostrowski type integral inequalities via Caputo $k$-fractional derivatives. At the end, some applications to special means are given.

We study the general (non-metric) facility-location and weighted k-medians problems, as well as the fractional facility-location and k-medians problems. We describe a natural randomized rounding scheme and use it to derive approximation algorithms for all of these problems. For facility location and weighted k-medians, the respective algorithms are polynomial-time [H∆k + d]and [(1 + ǫ)d; ln(n +...

We study the general (non-metric) facility-location and weighted k-medians problems, as well as the fractional facility-location and k-medians problems. We describe a natural randomized rounding scheme and use it to derive approximation algorithms for all of these problems. For facility location and weighted k-medians, the respective algorithms are polynomial-time [H∆k + d]and [(1 + ǫ)d; ln(n +...

The combinatorial dimension relative to an arbitrary fractional Cartesian product is defined. Relations between dimensions in certain archetypal instances are derived. Random sets with arbitrarily prescribed dimensions are produced; in particular, scales of combinatorial dimension are shown to be continuously and independently calibrated. A combinatorial concept of cylindricity is key. © 2004 W...

and Applied Analysis 3 Rn×m : {M Mij ∈ Rn×m : Mij > 0; ∀ i, j ∈ n×m} is the set of all n×m real matrices of positive entries. If M ∈ Rn×m then M 0 is used as a simpler notation for M ∈ Rn×m . The superscript T denotes the transpose, M i and Mj are, respectively, the ith row and the jth column of the matrix M. A close notation to characterize the positivity of vectors is the following: R0 : {v v...