107 research outputs found
Discrete Midpoint Convexity
For a function defined on a convex set in a Euclidean space, midpoint
convexity is the property requiring that the value of the function at the
midpoint of any line segment is not greater than the average of its values at
the endpoints of the line segment. Midpoint convexity is a well-known
characterization of ordinary convexity under very mild assumptions. For a
function defined on the integer lattice, we consider the analogous notion of
discrete midpoint convexity, a discrete version of midpoint convexity where the
value of the function at the (possibly noninteger) midpoint is replaced by the
average of the function values at the integer round-up and round-down of the
midpoint. It is known that discrete midpoint convexity on all line segments
with integer endpoints characterizes L-convexity, and that it
characterizes submodularity if we restrict the endpoints of the line segments
to be at -distance one. By considering discrete midpoint convexity
for all pairs at -distance equal to two or not smaller than two,
we identify new classes of discrete convex functions, called local and global
discrete midpoint convex functions, which are strictly between the classes of
L-convex and integrally convex functions, and are shown to be
stable under scaling and addition. Furthermore, a proximity theorem, with the
same small proximity bound as that for L-convex functions, is
established for discrete midpoint convex functions. Relevant examples of
classes of local and global discrete midpoint convex functions are provided.Comment: 39 pages, 6 figures, to appear in Mathematics of Operations Researc
Recent Progress on Integrally Convex Functions
Integrally convex functions constitute a fundamental function class in
discrete convex analysis, including M-convex functions, L-convex functions, and
many others. This paper aims at a rather comprehensive survey of recent results
on integrally convex functions with some new technical results. Topics covered
in this paper include characterizations of integral convex sets and functions,
operations on integral convex sets and functions, optimality criteria for
minimization with a proximity-scaling algorithm, integral biconjugacy, and the
discrete Fenchel duality. While the theory of M-convex and L-convex functions
has been built upon fundamental results on matroids and submodular functions,
developing the theory of integrally convex functions requires more general and
basic tools such as the Fourier-Motzkin elimination.Comment: 50 page
Shapley-Folkman-type Theorem for Integrally Convex Sets
The Shapley-Folkman theorem is a statement about the Minkowski sum of
(non-convex) sets, expressing the closeness of the Minkowski sum to convexity
in a quantitative manner. This paper establishes similar theorems for
integrally convex sets and M-natural-convex sets, which are major classes of
discrete convex sets in discrete convex analysis.Comment: 13 page
Bifurcation analysis of a simple 3D oscillator and chaos synchronization of its coupled systems
Tamaševičius et al. proposed a simple 3d chaotic oscillator for educational purpose. In fact the oscillator can be implemented very easily and it shows typical bifurcation scenario so that it is a suitable training object for introductory education for students. However, as far as we know, no concrete studies on bifurcations or applications on this oscillator have been investigated. In this paper, we make a thorough investigation on local bifurcations of periodic solutions in this oscillator by using a shooting method. Based on results of the analysis, we study chaos synchronization phenomena in diffusively coupled oscillators. Both bifurcation sets of periodic solutions and parameter regions of in-phase synchronized solutions are revealed. An experimental laboratory of chaos synchronization is also demonstrated
A pilot study to search possible mechanisms of ultralong gonadotropin-releasing hormone agonist therapy in IVF-ET patients with endometriosis
Towards Optimal Subsidy Bounds for Envy-freeable Allocations
We study the fair division of indivisible items with subsidies among
agents, where the absolute marginal valuation of each item is at most one.
Under monotone valuations (where each item is a good), Brustle et al. (2020)
demonstrated that a maximum subsidy of and a total subsidy of
are sufficient to guarantee the existence of an envy-freeable
allocation. In this paper, we improve upon these bounds, even in a wider model.
Namely, we show that, given an EF1 allocation, we can compute in polynomial
time an envy-free allocation with a subsidy of at most per agent and a
total subsidy of at most . Moreover, we present further improved
bounds for monotone valuations.Comment: 14page
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