Maximum Number Of Elements Research Articles

REGULAR ANALYTIC INDEPENDENCE AND EXTENSIONS OF ANALYTIC SPREAD

ABSTRACT The notion of Analytic Spread can be extended to filtrations by which denotes the maximum number of elements of J to be independent of order k with respect to a filtration f . Other extensions that we give in this paper are denoted by and . We show that if f is an adic filtration and a local ring with infinite residue field, then and are equal; but if f is not adic may be different from the other extensions, even if f is noetherian and A a local ring with infinite residue field. We introduce a weak notion, called regular analytic independence to which corresponds an extension denoted by and we show that if is a noetherian filtration with rank m and is a maximal ideal containing with infinite, then for all we have: We also prove that in the general case, if J is maximal or contains then

Choosing among Maximals

In a choice situation, it is usually assumed that the agents select the maximal elements. Nevertheless, there are situations in which the number of maximal elements is large, and a selection inside the maximal set is then required. This selection may be done in a random way or, as we propose in this work, by obtaining additional information from the preference relation that may provide a way of discriminating among the maximal elements and selecting the “best” ones.

Predicting computer run-time for the Barnes–Hut multipole method for electromagnetic scattering

This paper presents a novel simple methodology for quickly predicting and optimizing computer run-time for the Barnes–Hut multipole method for boundary element electromagnetic scattering problems. The methodology is easily extended to other multipole methods (e.g. Greengard–Rokhlin) and to other physics. The novel idea is to simply count the number of element-cell interactions, number of direct element–element interactions, and the number of cell multipole expansion creations (each expansion weighted by the number of elements in the cell), and then finally combine these three results with the associated unit costs to obtain the total computer run-time to perform a single matrix–vector multiply. By counting operations instead of actually performing them, the time to predict the computer run-time is orders of magnitude smaller than the time to actually perform the associated calculations. This new approach allows for very quick optimization of parameters, such as the maximum number of elements in a final generation cell of the tree. Numerical examples are presented herein in which the rate of return (time saved over time spent finding optimal parameter values) is significantly more than two orders of magnitude.

New Upper Bounds for Finite Bh Sequences

Let Fh(N) be the maximum number of elements that can be selected from the set {1, …, N} such that all the sums a1+…+ah, a1⩽…⩽ah are different. We introduce new combinatorial and analytic ideas to prove new upper bounds for Fh(N). In particular we proveF3(N)⩽41+16/(π+2)4N1/3+o(N1/3),F4(N)⩽81+16/(π+2)4N1/4+o(N1/4).Besides, our techniques have an independent interest for further research in additive number theory.

Dense admissible sequences

A sequence of integers in an interval of length x x is called admissible if for each prime there is a residue class modulo the prime which contains no elements of the sequence. The maximum number of elements in an admissible sequence in an interval of length x x is denoted by ϱ ∗ ( x ) \varrho ^{*}(x) . Hensley and Richards showed that ϱ ∗ ( x ) > π ( x ) \varrho ^{*}(x)>\pi (x) for large enough x x . We increase the known bounds on the set of x x satisfying ϱ ∗ ( x ) ≤ π ( x ) \varrho ^{*}(x)\le \pi (x) and find smaller values of x x such that ϱ ∗ ( x ) > π ( x ) \varrho ^{*}(x)>\pi (x) . We also find values of x x satisfying ϱ ∗ ( x ) > 2 π ( x / 2 ) \varrho ^{*}(x)>2\pi (x/2) . This shows the incompatibility of the conjecture π ( x + y ) − π ( y ) ≤ 2 π ( x / 2 ) \pi (x+y)-\pi (y)\le 2\pi (x/2) with the prime k k -tuples conjecture.

Bipartite labeling of trees with maximum degree three

Let T = (V, E) be a tree with a properly 2-colored vertex set. A bipartite labeling of T is a bijection φ: V → {1, …, |V|} for which there exists a k such that whenever φ(u) ≤ k < φ(v), then u and v have different colors. The α-size α(T) of the tree T is the maximum number of elements in the sets {|φ(u) − φ(v)|; uv ∈ E}, taken over all bipartite labelings φ of T. The quantity α(n) is defined as the minimum of α(T) over all trees with n vertices. In an earlier article (J Graph Theory 19 (1995), 201–215), A. Rosa and the second author proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4; the upper bound is believed to be the asymptotically correct value of (n). In this article, we investigate the α-size of trees with maximum degree three. Let α3(n) be the smallest α-size among all trees with n vertices, each of degree at most three. We prove that α3(n) ≥ 5n/6 for all n ≥ 12, thus supporting the belief above. This result can be seen as an approximation toward the graceful tree conjecture—it shows that every tree on n ≥ 12 vertices and with maximum degree three has “gracesize” at least 5n/6. Using a computer search, we also establish that α3(n) ≥ n − 2 for all n ≤ 17. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:7–15, 1999

Bipartite labeling of trees with maximum degree three

Let T = (V, E) be a tree with a properly 2-colored vertex set. A bipartite labeling of T is a bijection φ: V → {1, …, |V|} for which there exists a k such that whenever φ(u) ≤ k < φ(v), then u and v have different colors. The α-size α(T) of the tree T is the maximum number of elements in the sets {|φ(u) − φ(v)|; uv ∈ E}, taken over all bipartite labelings φ of T. The quantity α(n) is defined as the minimum of α(T) over all trees with n vertices. In an earlier article (J Graph Theory 19 (1995), 201–215), A. Rosa and the second author proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4; the upper bound is believed to be the asymptotically correct value of (n). In this article, we investigate the α-size of trees with maximum degree three. Let α3(n) be the smallest α-size among all trees with n vertices, each of degree at most three. We prove that α3(n) ≥ 5n/6 for all n ≥ 12, thus supporting the belief above. This result can be seen as an approximation toward the graceful tree conjecture—it shows that every tree on n ≥ 12 vertices and with maximum degree three has “gracesize” at least 5n/6. Using a computer search, we also establish that α3(n) ≥ n − 2 for all n ≤ 17. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:7–15, 1999

Lexicographic bottleneck combinatorial problems

We study combinatorial problems (CP) with lex-bottleneck objective function, where in addition to minimizing the largest element of a feasible solution, we are also interested in minimizing the second largest element, the third largest element, and so on. We consider CP, and in particular path, assignment and general matching problems, whose corresponding (zero–one) sum optimization problems can be solved as linear programs. Let t=min( k, l), where k is the number of distinct weights of elements and l the maximum number of elements in any feasible solution. We propose an approach which solves the lex-bottleneck optimization problem by solving bottleneck and zero–one sum optimizations for at most t iterations and reducing the problem size in each iteration. We also propose another approach which is similar to the above approach but solves only zero–one sum optimizations for at most k iterations. For a graph with n nodes and m arcs, the first approach solves the lex-bottleneck path problem in O( nm) time, and the second approach solves the lex-bottleneck assignment problem in O( m 2) time. For lex-bottleneck general matching problem, a time bound of O ( min(m,n log n)n(m+n log n)) is given.

Parallelization of 3-D Radiative Heat Transfer Analysis in Non-gray Gas.

A parallel computational method is developed for the Monte Carlo analysis of radiative heat transfer in three-dimensional nongray gas enclosed by gray walls. The highly parallel computer AP1000 is used for the analysis. For the nongray gas, a mixture of water vapor and carbon dioxide is chosen as the absorbing-emitting medium. Three types of parallelization are studied: type (1), event parallelization; type (2), combination of event parallelization and algorithm parallelization, and type (3), memory-saving algorithm of type (2). When 512 cells are used, values of the speed-up ratios of 395, 430, and 370 are obtained for types (1), (2), and (3), respectively. The maximum number of elements which can be treated is increased from 1200 for type (1) to 140000 for type (3).

Some optimal algorithms for decomposed partially ordered sets

We describe two problems and their optimal solutions for partially ordered sets. We first describe an optimal algorithm for computing the largest anti-chain of a partially ordered set given its decomposition into its chains. Our algorithm requires O( n 2 m) comparisons where n is the number of chains and m is the maximum number of elements in any chain. We also give an adversary argument to prove that this is a lower bound. Our second problem requires us to find if the given poset is a total order. Our optimal algorithm requires O( mn log n) comparisons. These algorithms have applications in distributed debugging and recovery in distributed systems.

Classification of Pk2

The set of functions of P k2 (mapping the set {0,1,…, k−1} n into {0,1}, n = 1,2,…) is divided into equivalence classes so that two functions are in the same class if their membership in the maximal subclones of P k2 coincides. This also leads to a natural classification of the set of bases (i.e. irredundant complete subsets) of P k2 . We determine all nonempty classes of functions of P k2 and show that their number is 13 B( k) − 11 B( k − 1), where B( k) is the number of equivalence relations on the set of k elements (Bell's number). The maximal number of elements in a base of P k2 is proved to be k + 2. Computational results for the numbers of classes of bases are also presented for k=3 and k=4.

On the Number of Maximal Elements in a Partially Ordered Set

AbstractLet P be a partially ordered set. For an element x ∊ P, a subset C of P is called a cutset for x in P if every element of C is noncomparable to x and every maximal chain in P meets {x} ∪ C. The following result is established: if every element of P has a cutset having n or fewer elements, then P has at most 2n maximal elements. It follows that, if some element of P covers k elements of P then there is an element x ∊ P such that every cutset for x in P has at least log2k elements.

About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets

LetP be a finite partially ordered set. The lengthl(x) of an elementx ofP is defined by the maximal number of elements, which lie in a chain withx at the top, reduced by one. Letw(P) (d(P)) be the maximal number of elements ofP which have the same length (which form an antichain). Further let\(p^n : = \underbrace {PX...XP}_{n - times}\). The numbers\(r_k : = \mathop {\max }\limits_{P:|P| = k} \frac{{d(P)}}{{w(P)}}\) and\(s_k : = \mathop {\max }\limits_{P:|P| = k} \mathop {\lim }\limits_{n \to \infty } \frac{{d(P^n )}}{{w(P^n )}}\) as well as all partially ordered sets for which these maxima are attained are determined.

The Use of Micro- and Supercomputers for Simulation of Metal Forming Processes

A 16 bit microcomputer and a supercomputer are used to simulate the plastic deformation of work-piece in metal forming by the rigid-plastic finite element method. Their limits of application are compared with those of the large- and minicomputers. With the use of FORTRAN language and a large memory of about 600k bites the finite element calculation is possible by a microcomputer. It is experienced that the program development is conveniently carried out by the aids of screen editing and graphic functions. The time for execution of benchmark program of forward extrusion is 17 - 300 times longer than those of large computers and 4 times longer than a 32 bit minicomputer. The time necessary for computation by the microcomputer increases in proportion to N 1.4 where N is the number of nodal velocities, and thus, the number of elements is limited by the computing time in practical applications. The maximum number of elements is also limited by the sector of 64k bites set on the 18086 processor. Although a supercomputer is claimed to be 10 - 20 times faster than the existing large computers, it is found to be about 4 times faster in calculating a benchmark program of three dimensional compression of block. This is because considerable part of the program cannot be vectorized. The speed and size of memory is again a limit of calculation for three-dimensional problems.

It’s Hard to Color Antirectangles

The maximum number of elements in an antirectangle of a convex board equals the minimum number of rectangles it takes to cover that board. It is shown here that the dual of this theorem, that the minimum number of antirectangles needed to cover a convex board equals the maximum size of any rectangle of the board, is not generally true.

Independent elements, integrally closed ideals, and quasi-unmixedness

Let I a denote the integral closure of an ideal I in a Noetherian ring A. Then it is shown that sup a ∞ I, the maximum number of elements which are ( I n ) a -independent as n → ∞, exists and is equal to min{height (I(A P) ∗ + z) z : P is a prime divisor of ( I n ) a for all large n and z is a minimal prime ideal in the completion (A p) ∗ of A p} . From this it follows that A is locally quasi-unmixed if and only if sup a ∞ I = height I for all ideals I in A.

Matroid intersection algorithms

LetM 1 = (E, 91),M 2 = (E, 92) be two matroids over the same set of elementsE, and with families of independent sets 91, 92. A setI ∈ 91 ∩ 92 is said to be anintersection of the matroidsM 1,M 2. An important problem of combinatorial optimization is that of finding an optimal intersection ofM 1,M 2. In this paper three matroid intersection algorithms are presented. One algorithm computes an intersection containing a maximum number of elements. The other two algorithms compute intersections which are of maximum total weight, for a given weighting of the elements inE. One of these algorithms is “primal-dual”, being based on duality considerations of linear programming, and the other is “primal”. All three algorithms are based on the computation of an “augmenting sequence” of elements, a generalization of the notion of an augmenting path from network flow theory and matching theory. The running time of each algorithm is polynomial inm, the number of elements inE, and in the running times of subroutines for independence testing inM 1,M 2. The algorithms provide constructive proofs of various important theorems of matroid theory, such as the Matroid Intersection Duality Theorem and Edmonds' Matroid Polyhedral Intersection Theorem.

ON THE INDEPENDENCE NUMBERS OF COMPLEMENTARY GRAPHS

AbstractA set of vertices (edges) is called independent if no two vertices (edges) in the set are adjacent. The independence number β(G) (edge independence number β1(G)) is the maximum number of elements in an independent set of vertices (edges) of G. It is shown for any graph G of order p ≥ 3, having complement Ḡ, that [p/2] ≤ β1(G) + β1(Ḡ) ≤ 2[p/2]. From this, upper and lower bounds for β1(G) · β1(Ḡ) are determined. All four bounds are shown to be best possible. It is verified that for every graph G of order p, the bounds β(G) + β(Ḡ) ≤ p + 1 and β(G) · β(Ḡ) ≤ [(p + 1)/2] · {(p + 1)/2} hold and are best possible. Lower bounds for β(G) + β(Ḡ) and β(G) · β(Ḡ) are considered and are shown to be related to Ramsey numbers.

A proof of dilworth’s decomposition theorem for partially ordered sets

A short proof of the following theorem is given: LetP be a finite partially ordered set. If the maximal number of elements in an independent subset ofP isk, thenP is the union ofk chains.