Family Of Rectangles Research Articles

Mean oscillation bounds on rearrangements

We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on R n {\mathbb {R}^n} . For the decreasing rearrangement f ∗ f^* of a rearrangeable function f f of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett–DeVore–Sharpley, ‖ f ∗ ‖ BMO ( R + ) ≤ C n ‖ f ‖ BMO ( R n ) \|f^*\|_{{\operatorname {BMO}}(\mathbb {R}_+)}\leq C_n \|f\|_{{\operatorname {BMO}}(\mathbb {R}^n)} , by showing the growth of C n C_n in the dimension n n is not exponential but at most of the order of n \sqrt {n} . This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calderón–Zygmund decomposition. By comparing cubes to a family of polar rectangles, we provide a first proof that an analogous inequality holds for the symmetric decreasing rearrangement, S f Sf .

On Wegner’s inequality for axis-parallel rectangles

According to an old conjecture of Wegner, the piercing number of a set of axis-parallel rectangles in the plane is at most twice the independence number (or matching number) minus 1, that is, τ(F)≤2ν(F)−1. On the other hand, the current best upper bound, due to Corea et al. (2015), is a Ologlogν(F)2 factor away from the current best lower bound. From the other direction, lower bound constructions with τ(F)≥2ν(F)−4 are known. Here we exhibit families of rectangles with τ=7 and ν=4 and thereby show that Wegner’s inequality, if true, cannot be improved for ν=4. The analogous result for ν=3, due to Wegner, dates back to 1968.A key element in our proof is establishing a connection with the Maximum Empty Box problem: Given a set P of n points inside an axis-parallel box U in Rd, find a maximum-volume axis-parallel box that is contained in U but contains no points of P in its interior.Whereas our construction can be extended to any larger independence number (ν=5,6,…), its analysis remains open.

On the Systems of Finite Weights on the Algebra of Bounded Operators and Corresponding Translation Invariant Measures

We describe the class of translation invariant measures on the algebra ℬ(ℋ) of bounded linear operators on a Hilbert space ℋ and some of its subalgebras. In order to achieve this we apply two steps. First we show that a total minimal system of finite weights on the operator algebra defines a family of rectangles in this algebra through construction of operator intervals. The second step is construction of a translation invariant measure on some subalgebras of algebra ℬ(ℋ) by the family of rectangles. The operator intervals in the Jordan algebra ℬ(ℋ)sa is investigated. We also obtain some new operator inequalities.

Numerical Preservation of Velocity Induced Invariant Regions for Reaction\u2013Diffusion Systems on Evolving Surfaces

We propose and analyse a finite element method with mass lumping (LESFEM) for the numerical approximation of reaction–diffusion systems (RDSs) on surfaces in {mathbb {R}}^3 that evolve under a given velocity field. A fully-discrete method based on the implicit–explicit (IMEX) Euler time-discretisation is formulated and dilation rates which act as indicators of the surface evolution are introduced. Under the assumption that the mesh preserves the Delaunay regularity under evolution, we prove a sufficient condition, that depends on the dilation rates, for the existence of invariant regions (i) at the spatially discrete level with no restriction on the mesh size and (ii) at the fully-discrete level under a timestep restriction that depends on the kinetics, only. In the specific case of the linear heat equation, we prove a semi- and a fully-discrete maximum principle. For the well-known activator-depleted and Thomas reaction–diffusion models we prove the existence of a family of rectangles in the phase space that are invariant only under specific growth laws. Two numerical examples are provided to computationally demonstrate (i) the discrete maximum principle and optimal convergence for the heat equation on a linearly growing sphere and (ii) the existence of an invariant region for the LESFEM–IMEX Euler discretisation of a RDS on a logistically growing surface.

An Information-Spectrum Approach to the Capacity Region of the Interference Channel

In this paper, a general formula for the capacity region of a general interference channel with two pairs of users is derived, which reveals that the capacity region is the union of a family of rectangles. In the region, each rectangle is determined by a pair of spectral inf-mutual information rates. The presented formula provides us with useful insights into the interference channels in spite of the difficulty of computing it. Specially, when the inputs are discrete, ergodic Markov processes and the channel is stationary memoryless, the formula can be evaluated by the BCJR (Bahl-Cocke-Jelinek-Raviv) algorithm. Also the formula suggests that considering the structure of the interference processes contributes to obtaining tighter inner bounds than the simplest one (obtained by treating the interference as noise). This is verified numerically by calculating the mutual information rates for Gaussian interference channels with embedded convolutional codes. Moreover, we present a coding scheme to approach the theoretical achievable rate pairs. Numerical results show that the decoding gains can be achieved by considering the structure of the interference.

Tiling Rectangles with Gaps by Ribbon Right Trominoes

We show that the least number of cells (the gap number) one needs to take out from a rectangle with integer sides of length at least 2 in order to be tiled by ribbon right trominoes is less than or equal to 4. If the sides of the rectangle are of length at least 5, then the gap number is less than or equal to 3. We also show that for the family of rectangles that have nontrivial minimal number of gaps, with probability 1, the only obstructions to tiling appear from coloring invariants. This is in contrast to what happens for simply connected regions. For that class of regions Conway and Lagarias found a tiling invariant that does not follow from coloring.

A new algorithm for solving planar multiobjective location problems involving the Manhattan norm

This paper is devoted to the study of unconstrained planar multiobjective location problems, where distances between points are defined by means of the Manhattan norm. We characterize the nonessential objectives and, by eliminating them, we develop an effective algorithm for generating the whole set of efficient solutions as the union of a special family of rectangles and line segments. We prove the correctness of this algorithm, analyze its complexity, and present illustrative computational results obtained by a MATLAB-based implementation.

Taylor series of conformal mappings onto symmetric quadrilaterals

We discuss the positivity (or lack thereof) of the non-vanishing coefficients of the Taylor series expansion about for the Riemann map of a family of rectangles and rhombi with defined symmetries.

Minimum vertex cover in rectangle graphs

We consider the Minimum Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families R where R 1 ∖ R 2 is connected for every pair of rectangles R 1 , R 2 ∈ R . This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of ( 1.5 + ε ) in general rectangle families, for any fixed ε > 0 , and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a novel way.

Covering a polygonal region by rectangles

The problem of covering a compact polygonal region, called target region, with a finite family of rectangles is considered. Tools for mathematical modeling of the problem are provided. Especially, a function, called Γ-function, is introduced which indicates whether the rectangles with respect to their configuration form a cover of the target region or not. The construction of the Γ-function is similar to that of ?-functions which have been proved to be an efficient tool for packing problems. A mathematical model of the covering problem based on the Γ-function is proposed as well as a solution strategy. The approach is illustrated by an example and some computational results are presented.

The Simple Geometry of Cournot-Nash Equilibrium with Application to Commons and Anti-Commons

The geometric analysis of Cournot-Nash (CN) equilibrium has been developed mainly by using reaction curves (see, e.g. Tirole (1988), Cabral (2000)). Since they are defined in the quantity space, the use of such graphical approach yields a discontinuity with respect to the perfectly competitive, the competitive monopolistic and the monopolistic cases, whose graphical representation is carried out in the price-quantity space. Recently, Fulton (1997), using the graphical version of the first order condition, shows how reactions curves are obtained from the price-quantity space. More recently, Sarkar, Gupta and Pal (1998) and Dufwenberg (2001) have provided graphical analyses of CN equilibrium using directly diagrams in the price-quantity space. While the former work's geometrical approach is obtained from mathematical elaboration of the first-order condition for profit maximization, the latter bases the geometrical conditions of equilibrium exploiting an interesting property of a family of rectangles. In either cases, however, the geometric method employed lacks of immediate economic interpretation. The aim of this paper is to develop a simple geometrical analysis of CN equilibrium only in the price-quantity space by exploiting the economic content of the first-order condition. Apart its conceptual simplicity, the advantage of our approach is threefold: first, the first-order condition is used only in terms of its economic content, rather than for its mathematical properties; this ensures that our graphical approach maintains a strong economic interpretation. Moreover, as we shall point out later, our approach yields, as particular case, Dufwenberg's rectangle method; thus, it enriches the latter method with economic content and with a geometric approach. Second, our approach makes it crystal-clear the economic nature of the externality existing among oligopolists. Two immediate implications of this fact are: (i) that our approach provides an economically grounded explanation of the origin of cartel instability; (ii) it allows us also to immediately extend oligopoly within the more general commons and anti-commons framework, as recently considered by Buchanan and Yoon (2000). Third, the approach here proposed seems to be particularly suitable for generalizations, further applications and very fine-grained analysis.

Tiling Parity Results and the Holey Square Solution

We prove combinatorially that the parity of the number of domino tilings of a region is equal to the parity of the number of domino tilings of a particular subregion. Using this result we can resolve the holey square conjecture. We furthermore give combinatorial proofs of several other tiling parity results, including that the number of domino tilings of a particular family of rectangles is always odd.

Complexity of perceptron recognition for a class of geometric patterns

In this paper, we study the recognition complexity of discrete geometric figures (rectangles, squares, circles, ellipses) on a retina by diameter-limited and order-restricted perceptrons. We construct a diameter-limited recognition perceptron for the family of rectangles, beginning with local configurations, which is different from the one shown by Minsky et al. (Perceptrons: An Introduction to Computational Geometry, extended edition, MIT Press, Cambridge, MA, 1988). In addition, we demonstrate the nonexistence of diameter-limited recognition perceptrons for squares, circles and ellipses. Finally, for squares and ellipses we construct an order-restricted perceptron with constant coefficients, using an original technique which decomposes the characterization of the figures into local and global features.

An Unexpected Maximum in a Family of Rectangles

An Unexpected Maximum in a Family of Rectangles

An Unexpected Maximum in a Family of Rectangles

The reader is encouraged to try to solve the problem now, before proceeding. In our solution to this problem we realized that A1 A2 A3A4 need not be a square. This observation led to the study of families of rectangles that satisfy hypotheses like those of Problem 749, and ultimately to this paper. A quick solution to Problem 749 is provided by Theorem 1, which is a special case of Feuerbach's Relation (see [4] or [6]).

A theorem on independence

Suppose we are given a family of sets C = {S(j), j∈J} , where S( j) = ∩ k i=1 H i ( j), and suppose each collection of sets H i ( j 1),…, H i ( j k+1 ) has a lower bound under the partial ordering defined by inclusion, then the maximal size of an independent subcollection of C is k. For example, for a fixed collection of half-spaces H 1,…, H k in R d , we define C to be the collection of all sets of the form ∩ i=1 k χ i+H i where χ i , i=1,…, k are points in R d . Then the maximal size of an independent collection of such sets us k. This leads to a proof of the bound of 2 d due to Rényi et al. (1951) for the maximum size of an independent family of rectangles in R d with sides parallel to the coordinate axes, and to a bound of d+1 for the maximum size of an independent family of simplices in R d with sides parallel to given hyperplanes H 1,…, H d+1 .

An integral inequality for the disc multiplier

In this paper an integral inequality is proved for the Bochner-Riesz operators in dimension two. This inequality expresses how those operators are controlled by maximal functions associated to families of rectangles with an appropriate number of directions.

The structure of rectangle families dividing the plane into maximum number of atoms

The structure of rectangle families dividing the plane into maximum number of atoms

On the Hardy-Littlewood maximal function and some applications

With a monotone family F = { S α } , S α ⊂ R n F\, = \,\{ {S_\alpha }\} ,\,{S_\alpha }\, \subset \,{{\textbf {R}}^n} , we associate the Hardy-Littlewood maximal function M F f ( x ) = sup α ( 1 / | S α | ) ∫ S α + x | f | {M_F}f(x)\, = \,{\sup _\alpha }(1/\left | {{S_\alpha }} \right |)\int _{{S_\alpha }\, + \,x} {\left | f \right |} . In general, M F {M_F} is not weak type (1.1). However, if we replace in the denominator S α {S_\alpha } by S F ∗ = { x − y : x , y ∈ S α } S_F^ {\ast } \, = \,\{ x\, - \,y:\,x,\,y\, \in \,{S_\alpha }\} , and denote the resulting maximal function by M F ∗ M_F^ {\ast } , then M F ∗ M_F^ {\ast } is weak type (1, 1) with weak type constant 1.