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With distributed computing and mobile applications becoming ever more prevalent, synchronizing diverging replicas of the same data is a common problem. Reconciliation – bringing two replicas of the same data structure as close as possible without overriding local changes – is investigated in an algebraic model. Our approach is to consider two sequences of simple commands that describe the changes in the replicas compared to the original structure, and then determine the maximal subsequences of each that can be propagated to the other. The proposed command set is shown to be functionally complete, and an update detection algorithm is presented which produces a command sequence transforming the original data structure into the replica while traversing both simultaneously. Syntactical characterization is provided in terms of a rewriting system for semantically equivalent command sequences. Algebraic properties of sequence pairs that are applicable to the same data structure are investigated. Based on these results the reconciliation problem is shown to have a unique maximal solution. In addition, syntactical properties of the maximal solution allow for an efficient algorithm that produces it.

This manuscript deals with the global existence and asymptotic behavior of solutions for a Kirchhoff beam equation with internal damping. The existence of solutions is obtained by using the Faedo-Galerkin method. Exponential stability is proved by applying Nakao’s theorem.

We consider hypersphere x = x(*u, v, w*) in the four dimensional Euclidean space. We calculate the Gauss map, and the curvatures of it. Moreover, we compute the second Laplace-Beltrami operator the hypersphere satisfying Δ^{II}x = *A*x, where *A* ϵ *Mat* (4,4).

In this paper, we show a Marcinkiewicz type interpolation theorem for Orlicz spaces. As an application, we obtain an existence result for a parabolic equation in divergence form.

Extending Blaschke and Lebesgue’s classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width *D*. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.

Let P be a set of n points in general position in the plane. Let *R* be a set of points disjoint from P such that for every *x, y € P* the line through *x* and *y* contains a point in *R*. We show that if *c* in the plane, then *P* has a special property with respect to the natural group structure on *c*. That is, *P* is contained in a coset of a subgroup *H* of c of cardinality at most |*R*|.

We use the same approach to show a similar result in the case where each of *B* and *G* is a set of n points in general position in the plane and every line through a point in *B* and a point in *G* passes through a point in *R*. This provides a partial answer to a problem of Karasev.

The bound

Let *E, G* be Fréchet spaces and *F* be a complete locally convex space. It is observed that the existence of a continuous linear not almost bounded operator *T* on *E* into *F* factoring through *G* causes the existence of a common nuclear Köthe subspace of the triple (*E, G, F*). If, in addition, *F* has the property (*y*), then (*E, G, F*) has a common nuclear Köthe quotient.

In this paper we study the sum *n*, and {*n _{p}
*} is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is

In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.

The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.