More than sixty years ago, Sylvester (13) proposed the following problem: Let n given points have the property that the straight line joining any two of them passes through a …
More than sixty years ago, Sylvester (13) proposed the following problem: Let n given points have the property that the straight line joining any two of them passes through a third point of the set. Must the n points all lie on one line? An alleged solution (not by Sylvester) advanced at the time proved to be fallacious and the problem remained unsolved until about 1933 when it was revived by Erdös (7) and others.
Two permutations (displayed in the two rows) of the integers 1, 2, … , n are called discordant if a i ≠ b i , i = 1, 2, …, …
Two permutations (displayed in the two rows) of the integers 1, 2, … , n are called discordant if a i ≠ b i , i = 1, 2, …, n . Let v(4, n ), n ⩾ 4, be the number of permutations discordant with the three Permutations
The purpose of this note is to give an elementary proof of a special case of a theorem suggested by Th. Bang (2; 3) and proved by Lee et al …
The purpose of this note is to give an elementary proof of a special case of a theorem suggested by Th. Bang (2; 3) and proved by Lee et al (5; see also 1; 4; 6; 7; 8).
A parallelohedron is a convex polyhedron, in real affine three-dimensional space, which can be repeated by translation to fill the whole space without interstices. It has centrally symmetrical faces [4, …
A parallelohedron is a convex polyhedron, in real affine three-dimensional space, which can be repeated by translation to fill the whole space without interstices. It has centrally symmetrical faces [4, p. 120] and hence is centrally symmetrical. Let F i denote the number of faces each having exactly i edges, V i denote the number of vertices each incident with exactly i edges, E denote the number of edges, n denote the number of sets of parallel edges, F denote the total number of faces, V denote the total number of vertices.
We call k integers x 1 < x 2 … < x k chosen from 1, 2, …, n} a k-choice (combination) from n. With 1, 2, …, n arranged …
We call k integers x 1 < x 2 … < x k chosen from 1, 2, …, n} a k-choice (combination) from n. With 1, 2, …, n arranged in a circle, so that 1 and n are consecutive, we have a circular k-choice from n. A part of a k-choice from n is a sequence of consecutive integers not contained in a longer one. Let denote the number of circular k-choices from n with exactly r parts all ≤ w.
Abstract We give a particularly elementary solution to the following well-known problem. What is the number of k -subsets X ⊆ I n = {1, 2, 3, … , n …
Abstract We give a particularly elementary solution to the following well-known problem. What is the number of k -subsets X ⊆ I n = {1, 2, 3, … , n } satisfying “no two elements of X are adjacent in the circular display of I n ”? Then we investigate a new generalization (multiple cyclic choices without adjacencies) and apply it to enumerating a class of 3-line latin rectangles.
Abstract Let (n|k) denote the number of k-choices 1≤x 1 <x 2 <…<x k ≤n satisfying x i -x i-1 ≥2, i = 2,…, k, n + x 1 -x …
Abstract Let (n|k) denote the number of k-choices 1≤x 1 <x 2 <…<x k ≤n satisfying x i -x i-1 ≥2, i = 2,…, k, n + x 1 -x k ≥2; let (m, n | k) = Σ i+j=k (m | i)(n | j). Several elementary proofs of the new identity (m, n|k) = (m + n | k) if 0≤k<m≤n. and if 0≤m≤n, m≤k, are given. Generalizations and applications are considered.
Abstract We show that is an integer. Special cases include the theorems of Wilson and Fermat.
Abstract We show that is an integer. Special cases include the theorems of Wilson and Fermat.
There are several statements in [3] which require clarification. Theorem 1 [3, p. 246] states that U 3 = U 2 U 4 U 2 -1 U 4 U 2 …
There are several statements in [3] which require clarification. Theorem 1 [3, p. 246] states that U 3 = U 2 U 4 U 2 -1 U 4 U 2 U 4 . In fact this is (essentially) the relation O = PUPU -1 PU given in [1, p. 91, (7.35)]. To see this we note that 0 = U 3 , P = U 4 , U = U 4 U 2 U 4 (as explained in [1, p. 88]); since U 4 2, = E,
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ …
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
Geometric transformations, II, by I. M. Yaglom. 189 pages. Translated from the Russian by Allen Shields. New Mathematical Library No. 21, Random House, New York, 1969. Paper U.S. $1.95. - …
Geometric transformations, II, by I. M. Yaglom. 189 pages. Translated from the Russian by Allen Shields. New Mathematical Library No. 21, Random House, New York, 1969. Paper U.S. $1.95. - Volume 15 Issue 1
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ …
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
Abstract We give a particularly elementary solution to the following well-known problem. What is the number of k -subsets X ⊆ I n = {1, 2, 3, … , n …
Abstract We give a particularly elementary solution to the following well-known problem. What is the number of k -subsets X ⊆ I n = {1, 2, 3, … , n } satisfying “no two elements of X are adjacent in the circular display of I n ”? Then we investigate a new generalization (multiple cyclic choices without adjacencies) and apply it to enumerating a class of 3-line latin rectangles.
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Abstract We show that is an integer. Special cases include the theorems of Wilson and Fermat.
Abstract We show that is an integer. Special cases include the theorems of Wilson and Fermat.
Abstract Let (n|k) denote the number of k-choices 1≤x 1 <x 2 <…<x k ≤n satisfying x i -x i-1 ≥2, i = 2,…, k, n + x 1 -x …
Abstract Let (n|k) denote the number of k-choices 1≤x 1 <x 2 <…<x k ≤n satisfying x i -x i-1 ≥2, i = 2,…, k, n + x 1 -x k ≥2; let (m, n | k) = Σ i+j=k (m | i)(n | j). Several elementary proofs of the new identity (m, n|k) = (m + n | k) if 0≤k<m≤n. and if 0≤m≤n, m≤k, are given. Generalizations and applications are considered.
Geometric transformations, II, by I. M. Yaglom. 189 pages. Translated from the Russian by Allen Shields. New Mathematical Library No. 21, Random House, New York, 1969. Paper U.S. $1.95. - …
Geometric transformations, II, by I. M. Yaglom. 189 pages. Translated from the Russian by Allen Shields. New Mathematical Library No. 21, Random House, New York, 1969. Paper U.S. $1.95. - Volume 15 Issue 1
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ …
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ …
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
Two permutations (displayed in the two rows) of the integers 1, 2, … , n are called discordant if a i ≠ b i , i = 1, 2, …, …
Two permutations (displayed in the two rows) of the integers 1, 2, … , n are called discordant if a i ≠ b i , i = 1, 2, …, n . Let v(4, n ), n ⩾ 4, be the number of permutations discordant with the three Permutations
We call k integers x 1 &lt; x 2 … &lt; x k chosen from 1, 2, …, n} a k-choice (combination) from n. With 1, 2, …, n arranged …
We call k integers x 1 &lt; x 2 … &lt; x k chosen from 1, 2, …, n} a k-choice (combination) from n. With 1, 2, …, n arranged in a circle, so that 1 and n are consecutive, we have a circular k-choice from n. A part of a k-choice from n is a sequence of consecutive integers not contained in a longer one. Let denote the number of circular k-choices from n with exactly r parts all ≤ w.
There are several statements in [3] which require clarification. Theorem 1 [3, p. 246] states that U 3 = U 2 U 4 U 2 -1 U 4 U 2 …
There are several statements in [3] which require clarification. Theorem 1 [3, p. 246] states that U 3 = U 2 U 4 U 2 -1 U 4 U 2 U 4 . In fact this is (essentially) the relation O = PUPU -1 PU given in [1, p. 91, (7.35)]. To see this we note that 0 = U 3 , P = U 4 , U = U 4 U 2 U 4 (as explained in [1, p. 88]); since U 4 2, = E,
A parallelohedron is a convex polyhedron, in real affine three-dimensional space, which can be repeated by translation to fill the whole space without interstices. It has centrally symmetrical faces [4, …
A parallelohedron is a convex polyhedron, in real affine three-dimensional space, which can be repeated by translation to fill the whole space without interstices. It has centrally symmetrical faces [4, p. 120] and hence is centrally symmetrical. Let F i denote the number of faces each having exactly i edges, V i denote the number of vertices each incident with exactly i edges, E denote the number of edges, n denote the number of sets of parallel edges, F denote the total number of faces, V denote the total number of vertices.
The purpose of this note is to give an elementary proof of a special case of a theorem suggested by Th. Bang (2; 3) and proved by Lee et al …
The purpose of this note is to give an elementary proof of a special case of a theorem suggested by Th. Bang (2; 3) and proved by Lee et al (5; see also 1; 4; 6; 7; 8).
More than sixty years ago, Sylvester (13) proposed the following problem: Let n given points have the property that the straight line joining any two of them passes through a …
More than sixty years ago, Sylvester (13) proposed the following problem: Let n given points have the property that the straight line joining any two of them passes through a third point of the set. Must the n points all lie on one line? An alleged solution (not by Sylvester) advanced at the time proved to be fallacious and the problem remained unsolved until about 1933 when it was revived by Erdös (7) and others.
(1946). On Sets of Distances of n Points. The American Mathematical Monthly: Vol. 53, No. 5, pp. 248-250.
(1946). On Sets of Distances of n Points. The American Mathematical Monthly: Vol. 53, No. 5, pp. 248-250.
Abstract Let (n|k) denote the number of k-choices 1≤x 1 <x 2 <…<x k ≤n satisfying x i -x i-1 ≥2, i = 2,…, k, n + x 1 -x …
Abstract Let (n|k) denote the number of k-choices 1≤x 1 <x 2 <…<x k ≤n satisfying x i -x i-1 ≥2, i = 2,…, k, n + x 1 -x k ≥2; let (m, n | k) = Σ i+j=k (m | i)(n | j). Several elementary proofs of the new identity (m, n|k) = (m + n | k) if 0≤k<m≤n. and if 0≤m≤n, m≤k, are given. Generalizations and applications are considered.
Abstract In this paper diagonals of various orders in a (strict) convex polygon P n are considered. The sums of lengths of diagonals of the same order are studied. A …
Abstract In this paper diagonals of various orders in a (strict) convex polygon P n are considered. The sums of lengths of diagonals of the same order are studied. A relationship between the number of consecutive diagonals which do not intersect a given maximal diagonal and lie on one side of it and the order of the smallest diagonal among them is established. Finally a new proof of a conjecture of P. Erdos, considered already in [1], is given.
(1971). On the Packing of Ten Equal Circles in a Square. Mathematics Magazine: Vol. 44, No. 3, pp. 139-140.
(1971). On the Packing of Ten Equal Circles in a Square. Mathematics Magazine: Vol. 44, No. 3, pp. 139-140.
This paper is concerned with the problem of placing <italic>N</italic> points on the unit sphere in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mn>3</mml:mn> </mml:msup> …
This paper is concerned with the problem of placing <italic>N</italic> points on the unit sphere in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{E^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so as to maximize the sum of their mutual distances. A necessary condition is proved which led to a computer algorithm. This in turn led to the apparent best arrangements for values of <italic>N</italic> from 5 to 10 inclusive.
How many spheres of given diameter can be packed in a cube of given size? Or: What is the maximum diameter of k identical spheres if they can be packed …
How many spheres of given diameter can be packed in a cube of given size? Or: What is the maximum diameter of k identical spheres if they can be packed in a cube of given size? These questions are obviously equivalent to the following problem: Let d(P i , P j ) denote the distance between the points P i and P j , and Γ k the set of all configurations of k points P i (1 ≤ i < j ≤ k) in a closed unit cube C.
(1973). Inequalities for Sums of Distances. The American Mathematical Monthly: Vol. 80, No. 9, pp. 1009-1017.
(1973). Inequalities for Sums of Distances. The American Mathematical Monthly: Vol. 80, No. 9, pp. 1009-1017.
found in the writings of Jacobson [9], Becker [2], Motzkin [11], and Bourbaki [3; 4]. This paper will be concerned with a natural generalization of Cayley's problem, and will show …
found in the writings of Jacobson [9], Becker [2], Motzkin [11], and Bourbaki [3; 4]. This paper will be concerned with a natural generalization of Cayley's problem, and will show that the solution to the generalized problem contains all of the combinatorial information needed to establish the well known formula of Lagrange for the reversion of power series. To describe the problem, we consider expressions which are built from operator symbols and argument symbols, using a prefix notation for operators. Weights are assigned to the symbols in an expression, an argument symbol having the weight 0 and an n-ary operator symbol having the weight n. Expressions are of various types, the type of an expression depending only on the weights of the symbols in it and on the order in which they appear. The expression (a+b) +c, for example, is written + +abc and is of the type 22000, while a+(b+c) is written +a+bc and is of the type 20200. The expression F(G(x, H(y, z), t), K(u)) is of the type 230200010. Following P. C. Rosenbloom [13], we call those finite sequences of natural numbers which designate the types of expressions words. Definitions and some special properties of these sequences are stated in ?2.
Keywords: conjecture of Leo Moser ; problem of Erdos Note: Professor Pach's number: [056] Reference DCG-ARTICLE-1989-001doi:10.2307/2325175 Record created on 2008-11-14, modified on 2017-05-12
Keywords: conjecture of Leo Moser ; problem of Erdos Note: Professor Pach's number: [056] Reference DCG-ARTICLE-1989-001doi:10.2307/2325175 Record created on 2008-11-14, modified on 2017-05-12
P. Erdös and G. Szekeres [1] proved that from any points in the plane one can always choose n + 1 of them which are the vertices of a convex …
P. Erdös and G. Szekeres [1] proved that from any points in the plane one can always choose n + 1 of them which are the vertices of a convex polygon, thus answering a question due to Miss Esther Klein (who later became Mrs. G. Szekeres).
In a recent study of generalized transfinite dimeters [4, 5] some geometric extremal problems were encountered. These form the subject matter of this note.
In a recent study of generalized transfinite dimeters [4, 5] some geometric extremal problems were encountered. These form the subject matter of this note.