scholarly journals POINT SET LABELING WITH SPECIFIED POSITIONS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 29-66 ◽  
Author(s):  
SRINIVAS R. DODDI ◽  
MADHAV V. MARATHE ◽  
BERNARD M. MORET
Keyword(s):  
Fixed Point ◽  
Computer Graphics ◽  
Best Approximation ◽  
Maximum Size ◽  
Map Labeling ◽  
Combinatorial Properties ◽  
The Best Approximation ◽  
Point Set ◽  
Set Of Points ◽  
Aesthetic Criterion

Motivated by applications in cartography and computer graphics, we study a version of the map-labeling problem that we call the k-Position Map-Labeling Problem: given a set of points in the plane and, for each point, a set of up to k allowable positions, place uniform and non-intersecting labels of maximum size at each point in one of the allowable positions. This version combines an aesthetic criterion and a legibility criterion and comes close to actual practice while generalizing the fixed-point and slider models found in the literature. We present a general heuristic that given an ∊ > 0, runs in time O(n log n + n log (R*/ ∊) log (k)), where R* is the size of the optimal label, and guarantees a constant approximation for any regular labels. For circular labels, our technique yields a (3.6 + ∊)-approximation, improving in the case of arbitrary placement over the previous bound of approximately 19.5 obtained by Strijk and Wolff.28 We then extend our approach to arbitrary positions, obtaining an algorithm that is easy to implement and also substantially improves the best approximation bounds. Our technique combines several geometric and combinatorial properties, which may be of independent interest.

scholarly journals A global optimality result using Geraghty type contraction

2014 ◽  
Vol 4 (2) ◽  
pp. 99-104 ◽  
Author(s):  
Binayak S. Choudhury ◽  
Pranati Maity ◽  
P. Konar
Keyword(s):  
Fixed Point ◽  
Best Approximation ◽  
Approximate Solutions ◽  
Global Optimality ◽  
Approximation Theorems ◽  
The Best Approximation ◽  
Optimal Values ◽  
Type Contraction

In this paper we prove two proximity point results for finding the distance between two sets. Unlike the best approximation theorems they provide with globally optimal values. Here our approach is to reduce the problem to that of finding optimal approximate solutions of some fixed point equations. We use Geraghty type contractive inequalities in our theorem. Two illustrative examples are given.

scholarly journals Properties of fixed point set of a multivalued map

2005 ◽  
Vol 2005 (3) ◽  
pp. 323-331 ◽  
Author(s):  
Abdul Rahim Khan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Best Approximation ◽  
Convex Banach Space ◽  
Multivalued Map ◽  
Multivalued Maps ◽  
Fixed Point Set ◽  
Strictly Convex Banach Space ◽  
Point Set

Properties of the set of fixed points of some discontinuous multivalued maps in a strictly convex Banach space are studied; in particular, affirmative answers are provided to the questions related to set of fixed points and posed by Ko in 1972 and Xu and Beg in 1998. A result regarding the existence of best approximation is derived.

scholarly journals The best approximation and an extension of a fixed point theorem of F. E. Browder

1995 ◽  
Vol 18 (4) ◽  
pp. 745-748
Author(s):  
V. M. Sehgal ◽  
S. P. Singh
Keyword(s):  
Fixed Point ◽  
Continuous Function ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Best Approximation ◽  
Fixed Point Theorems ◽  
Affine Map ◽  
The Best Approximation

In this paper, theKKMprinciple has been used to obtain a theorem on the best approximation of a continuous function with respect to an affine map. The main result provides extensions of some well-known fixed point theorems.

scholarly journals String-averaging methods for best approximation to common fixed point sets of operators: the finite and infinite cases

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yair Censor ◽  
Ariel Nisenbaum
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Best Approximation ◽  
Fixed Point Problem ◽  
Feasibility Problem ◽  
Point Problem ◽  
Fixed Point Set ◽  
Averaging Methods ◽  
Point Set ◽  
The Common

AbstractString-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm in hand requires to employ the operators in a specific order. Sequential orderings are well known, and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by subsets of the index set of all operators, to apply the operators along these strings, and then to combine their end-points in some agreed manner to yield the next iterate of the algorithm. String-averaging methods were discussed and used for solving the common fixed point problem or its important special case of the convex feasibility problem. In this paper we propose and investigate string-averaging methods for the problem of best approximation to the common fixed point set of a family of operators. This problem involves finding a point in the common fixed point set of a family of operators that is closest to a given point, called an anchor point, in contrast with the common fixed point problem that seeks any point in the common fixed point set.We construct string-averaging methods for solving the best approximation problem to the common fixed points set of either finite or infinite families of firmly nonexpansive operators in a real Hilbert space. We show that the simultaneous Halpern–Lions–Wittman–Bauschke algorithm, the Halpern–Wittman algorithm, and the Combettes algorithm, which were not labeled as string-averaging methods, are actually special cases of these methods. Some of our string-averaging methods are labeled as “static” because they use a fixed pre-determined set of strings. Others are labeled as “quasi-dynamic” because they allow the choices of strings to vary, between iterations, in a specific manner and belong to a finite fixed pre-determined set of applicable strings. For the problem of best approximation to the common fixed point set of a family of operators, the full dynamic case that would allow strings to unconditionally vary between iterations remains unsolved, although it exists and is validated in the literature for the convex feasibility problem where it is called “dynamic string-averaging”.

scholarly journals Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

2021 ◽  
Author(s):  
Jiahao Qiu ◽  
Jianjie Zhao
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Minimal System ◽  
Combinatorial Properties ◽  
Return Times ◽  
Residual Set ◽  
Closed Sets ◽  
Open Set ◽  
Geometric Progressions ◽  
Set Of Points

AbstractIn this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

Intracranial volume-pressure relationship in man

1982 ◽  
Vol 56 (4) ◽  
pp. 524-528 ◽  
Author(s):  
Joseph Th. J. Tans ◽  
Dick C. J. Poortvliet
Keyword(s):  
Best Approximation ◽  
Continuous Monitoring ◽  
Fluid Pressure ◽  
Constant Term ◽  
Volume Index ◽  
Intracranial Volume ◽  
Content Type ◽  
The Best Approximation ◽  
Ventricular Fluid Pressure ◽  
Fine Print

✓ The pressure-volume index (PVI) was determined in 40 patients who underwent continuous monitoring of ventricular fluid pressure. The PVI value was calculated using different mathematical models. From the differences between these values, it is concluded that a monoexponential relationship with a constant term provides the best approximation of the PVI.

Sign in / Sign up

Export Citation Format

Share Document