340 research outputs found
Intrinsic volumes of inscribed random polytopes in smooth convex bodies
Let be a dimensional convex body with a twice continuously
differentiable boundary and everywhere positive Gauss-Kronecker curvature.
Denote by the convex hull of points chosen randomly and independently
from according to the uniform distribution. Matching lower and upper bounds
are obtained for the orders of magnitude of the variances of the -th
intrinsic volumes of for . Furthermore,
strong laws of large numbers are proved for the intrinsic volumes of . The
essential tools are the Economic Cap Covering Theorem of B\'ar\'any and Larman,
and the Efron-Stein jackknife inequality
ON A QUESTION OF V. I. ARNOLâD
Abstract. We show by a construction that there are at least exp {cV (dâ1)/(d+1) } convex lattice polytopes in R d of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation. 1. Introduction an
Block partitions: an extended view
Given a sequence , a block of is a
subsequence . The size of a block is the sum
of its elements. It is proved in [1] that for each positive integer , there
is a partition of into blocks with for every . In this paper, we consider a generalization of the problem
in higher dimensions
Notes about the Caratheodory number
In this paper we give sufficient conditions for a compactum in
to have Carath\'{e}odory number less than , generalizing an old result of
Fenchel. Then we prove the corresponding versions of the colorful
Carath\'{e}odory theorem and give a Tverberg type theorem for families of
convex compacta
How (Not) to Cut Your Cheese
It is well known that a line can intersect at most 2nâ1 unit squares of the nâĂân chessboard. Here we consider the three-dimensional version: how many unit cubes of the 3-dimensional cube [0,n]3 can a hyperplane intersect
Tverberg's Theorem at 50: Extensions and Counterexamples
We describe how a powerful new âconstraint
methodâ yields many different extensions of the topological
version of Tverbergâs 1966 Theorem in the prime power caseâ
and how the same method also was instrumental in the recent
spectacular construction of counterexamples for the general
case
Fairly Allocating Contiguous Blocks of Indivisible Items
In this paper, we study the classic problem of fairly allocating indivisible
items with the extra feature that the items lie on a line. Our goal is to find
a fair allocation that is contiguous, meaning that the bundle of each agent
forms a contiguous block on the line. While allocations satisfying the
classical fairness notions of proportionality, envy-freeness, and equitability
are not guaranteed to exist even without the contiguity requirement, we show
the existence of contiguous allocations satisfying approximate versions of
these notions that do not degrade as the number of agents or items increases.
We also study the efficiency loss of contiguous allocations due to fairness
constraints.Comment: Appears in the 10th International Symposium on Algorithmic Game
Theory (SAGT), 201
A geoökológia és a geoökológiai térképezés néhåny elvi és gyakorlati kérdése
The geographical environment can be investigated from several aspects:
- in the biological (ecological) approach emphasis is put on the biotic factors of the environment or
on the structure itself;
- in the geographical approach research concentrates on the abiotic factors and functions; and
- the technological or planning trend focuses the analysis on the economical-technical background of
impacts.
To distinguish between the first two trends and the related disciplines, the terms (bio) ecology and
geoecology are in use. The two concepts differ in handling the role of abiogenic and biogenic factors. In
the past decade there was an intension to define geoecology as the study of abiotic factors and of issues
concerning the functioning of the physical environment, while landscape ecology investigates the
biogenic factors and problems of spatial organisation, structure. Several authors, however, use these
concepts interchangeably.
The problem is more complicated than that. On the other hand, the concept landscape is narrower or
different from that covered by landscape ecology. The latter studies the arrangement of the ecosystem
and the flows of matter and energy between its componensts. Here the question is not simply whether or
to what extent man-made elements are included in landscape functioning. On the other hand, there is a
significant difference between the landscape and the (physical) geographical environment â the true
carrier of system properties. This difference of contents was clarified by S. Marosi (1981). In his
opinion, the landscape consists of geotopes (naturally including biotopes), while the (geographical)
environment is built up of ecotopes and â as a spatial unit â from ecochores. It is the activity of the
society related to the socio- or econotopes that makes the geotopes exotopes. In the Marosi model the
relationship between landscape and environment is clearly defined. No similar is applied in either the
German or in the English-language literature. At the same time, the often used term landscape ecology is
difficult to interpret from this standpoint, since they are almost mutually exclusive categories. Spatial
pattern is often emphasised in the investigation of the landscape, of the concrete environment and the
implications for functioning are neglected, the various âtopesâ are not regarded as aspects of functioning.
In the same manner it would be a mistake to restrict the study only to the biogenic or to the abiogenic
factors or to disregard functional or system properties. In our opinion â after the scheme by H. Leser
(1984) â the German and English schools and the Hungarian views can be reconciled as shown in Fig. l.
The size of the landscape ecology frame in the figure may change with various approaches and even
it location may vary with the emphasis being on spatiality (like in the Russian literature) or on systems
approach (like in the concept of English speaking researchers). Although it contradicts rigid
delimitations, geoecology â among others for the above reasons â should cover the analysis of biotic
factors too (hence is the uncertainty of delimitation), since they reflect the joint impact of abiotic factors
and also point to human influences.
Hopefully, the series of examples in the paper call attention to the flexibility of categories. There is
communication between them, e.g. geoecology may also reveal structural properties and landscape
ecology may answer functional questions of the physical environment. In this respect, the distinction
between the two concepts may seem groundless. In our opinion, the in dependent treatment of
geoecology separate from landscape ecology, a discipline with more traditions and broader contents, can
be justified by the increasing importance of issues of environmental functioning, assessment of the partial potentials of the physical environment (i.e. landscape capacity controlled by landscape budget),
data aquisition from field measurements and other practical requirements.
The principles of geoecological mapping outlined here (Figure 2) reach beyond the 1:25,000 scale
geoecological mapping in Germany, both in methodology and in objective4s. It seemed necessary to
apply â in addition to the conventional field surveys, mapping and laboratory techniques â GIS for data
storage and processing and for the regional extension of results automated aerial photo interpretation
(with scanner) and other remote sensing methods. Although complex systems (such as the landscape) can
only be fragmented in a holistic approach, efficiency required the application of a GIS.
In the paper three examples are used to illustrate the opportunities to geoecological mapping. The first
of them concerns the reclamation or optimal utilisation of surfaces partially used for agricultural
purposes, while the second identifies areas affected by hazards, soil erosion, and the third deals with
physical loadability through recreation
Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3
In an epsilon-approximate Nash equilibrium, a player can gain at most epsilon
in expectation by unilateral deviation. An epsilon well-supported approximate
Nash equilibrium has the stronger requirement that every pure strategy used
with positive probability must have payoff within epsilon of the best response
payoff. Daskalakis, Mehta and Papadimitriou conjectured that every win-lose
bimatrix game has a 2/3-well-supported Nash equilibrium that uses supports of
cardinality at most three. Indeed, they showed that such an equilibrium will
exist subject to the correctness of a graph-theoretic conjecture. Regardless of
the correctness of this conjecture, we show that the barrier of a 2/3 payoff
guarantee cannot be broken with constant size supports; we construct win-lose
games that require supports of cardinality at least Omega((log n)^(1/3)) in any
epsilon-well supported equilibrium with epsilon < 2/3. The key tool in showing
the validity of the construction is a proof of a bipartite digraph variant of
the well-known Caccetta-Haggkvist conjecture. A probabilistic argument shows
that there exist epsilon-well-supported equilibria with supports of cardinality
O(log n/(epsilon^2)), for any epsilon> 0; thus, the polylogarithmic cardinality
bound presented cannot be greatly improved. We also show that for any delta >
0, there exist win-lose games for which no pair of strategies with support
sizes at most two is a (1-delta)-well-supported Nash equilibrium. In contrast,
every bimatrix game with payoffs in [0,1] has a 1/2-approximate Nash
equilibrium where the supports of the players have cardinality at most two.Comment: Added details on related work (footnote 7 expanded
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