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Doubly Triangular Number
In mathematics, the doubly triangular numbers are the numbers that appear within the sequence of triangular numbers, in positions that are also triangular numbers. That is, if T_n=n(n+1)/2 denotes the nth triangular number, then the doubly triangular numbers are the numbers of the form T_. Sequence and formula The doubly triangular numbers form the sequence :0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, ... The nth doubly triangular number is given by the quartic formula T_ = \frac. The sums of row sums of Floyd's triangle give the doubly triangular numbers. Another way of expressing this fact is that the sum of all of the numbers in the first n rows of Floyd's triangle is the nth doubly triangular number. In combinatorial enumeration Doubly triangular numbers arise naturally as numbers of of objects, including pairs where both objects are the same: *An example from mathematical chemistry is given by the numbers of overlap integrals between Slater-type orbitals. *An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square 3-colorings
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Of The Beast
The number of the beast ( grc-koi, Ἀριθμὸς τοῦ θηρίου, ) is associated with the Beast of Revelation in chapter 13, verse 18 of the Book of Revelation. In most manuscripts of the New Testament and in English translations of the Bible, the number of the beast is six hundred sixty-six or (in Greek numerals, represents 600, represents 60 and represents 6). Papyrus 115 (which is the oldest preserved manuscript of the ''Revelation'' ), as well as other ancient sources like ''Codex Ephraemi Rescriptus'', give the number of the beast as χιϛ or χιϲ, transliterable in Arabic numerals as 616 (), not 666; critical editions of the Greek text, such as the '' Novum Testamentum Graece'', note χιϛ as a variant. In the Bible χξϛ The number of the beast is described in Revelation 13:15–18. Several translations have been interpreted for the meaning of the phrase "Here is Wisdom. Let him that hath understanding count the number of the beast..." where the pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
666 (number)
666 (six hundred ndsixty-six) is the natural number following 665 and preceding 667. In Christianity, 666 is called the " number of the beast" in (most manuscripts of) chapter 13 of the Book of Revelation of the New Testament.Beale, Gregory K. (1999). The Book of Revelation: A Commentary on the Greek Text. Grand Rapids, Michigan: Wm. B. Eerdmans Publishing. p. 718. . Retrieved 9 July 2012. In Mathematics 666 is the sum of the first 36 natural numbers (\sum_^ i, i.e. ), and thus it is a triangular number. Because 36 is also triangular, 666 is a doubly triangular number. Also, ; 15 and 21 are also triangular numbers, and . In base 10, 666 is a repdigit (and therefore a palindromic number) and a Smith number. A prime reciprocal magic square based on 1/149 in base 10 has a magic total of 666. The prime factorization of 666 is 2 ⋅ 32 ⋅ 37. Also, 666 is the sum of the squares of the first seven primes: 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 The number of integers whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biblical Studies
Biblical studies is the academic application of a set of diverse disciplines to the study of the Bible (the Old Testament and New Testament).''Introduction to Biblical Studies, Second Edition'' by Steve Moyise (Oct 27, 2004) pages 11–12 For its theory and methods, the field draws on disciplines ranging from ancient history, historical criticism, philology, textual criticism, literary criticism, historical backgrounds, mythology, and comparative religion. Many secular as well as religious universities and colleges offer courses in biblical studies, usually in departments of religious studies, theology, Judaic studies, history, or comparative literature. Biblical scholars do not necessarily have a faith commitment to the texts they study, but many do. Definition The ''Oxford Handbook of Biblical Studies'' defines the field as a set of various, and in some cases independent disciplines for the study of the collection of ancient texts generally known as the Bible.''The Oxf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerology
Numerology (also known as arithmancy) is the belief in an occult, divine or mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value, via an alphanumeric system, of the letters in words and names. When numerology is applied to a person's name, it is a form of onomancy. It is often associated with the paranormal, alongside astrology and similar to divinatory arts. Despite the long history of numerological ideas, the word "numerology" is not recorded in English before c. 1907. The term numerologist can be used for those who place faith in numerical patterns and draw inferences from them, even if those people do not practice traditional numerology. For example, in his 1997 book ''Numerology: Or What Pythagoras Wrought'' (), mathematician Underwood Dudley uses the term to discuss practitioners of the Elliott wave principle of stock market analysis. History The practice of gematria, assigning numerical values to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multigraph
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. There are two distinct notions of multiple edges: * ''Edges without own identity'': The identity of an edge is defined solely by the two nodes it connects. In this case, the term "multiple edges" means that the same edge can occur several times between these two nodes. * ''Edges with own identity'': Edges are primitive entities just like nodes. When multiple edges connect two nodes, these are different edges. A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not just two. For some authors, the terms ''pseudograph'' and ''multigraph'' are synonymous. For others, a pseudograph is a multigraph that is permitted to have loops. Undirected multigraph (edges without ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slater-type Orbital
Slater-type orbitals (STOs) are functions used as atomic orbitals in the linear combination of atomic orbitals molecular orbital method. They are named after the physicist John C. Slater, who introduced them in 1930. They possess exponential decay at long range and Kato's cusp condition at short range (when combined as hydrogen-like atom functions, i.e. the analytical solutions of the stationary Schrödinger equation for one electron atoms). Unlike the hydrogen-like ("hydrogenic") Schrödinger orbitals, STOs have no radial nodes (neither do Gaussian-type orbitals). Definition STOs have the following radial part: : R(r) = N r^ e^\, where * is a natural number that plays the role of principal quantum number, = 1,2,..., * is a normalizing constant, * is the distance of the electron from the atomic nucleus, and * \zeta is a constant related to the effective charge of the nucleus, the nuclear charge being partly shielded by electrons. Historically, the effective nuclear charge was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: *Abundant numbers *Baum–Sweet sequence *Bell numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbital Overlap
In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. Linus Pauling explained the importance of orbital overlap in the molecular bond angles observed through experimentation; it is the basis for orbital hybridization. As ''s'' orbitals are spherical (and have no directionality) and ''p'' orbitals are oriented 90° to each other, a theory was needed to explain why molecules such as methane (CH4) had observed bond angles of 109.5°. Pauling proposed that s and p orbitals on the carbon atom can combine to form hybrids (sp3 in the case of methane) which are directed toward the hydrogen atoms. The carbon hybrid orbitals have greater overlap with the hydrogen orbitals, and can therefore form stronger C–H bonds.Pauling, Linus. (1960). ''The Nature Of The Chemical Bond''. Cornell University Press. A quantitative measure of the overlap of two atomic orbitals ΨA and ΨB o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Chemistry
Mathematical chemistry is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena. Mathematical chemistry has also sometimes been called computer chemistry, but should not be confused with computational chemistry. Major areas of research in mathematical chemistry include chemical graph theory, which deals with topology such as the mathematical study of isomerism and the development of topological descriptors or indices which find application in quantitative structure-property relationships; and chemical aspects of group theory, which finds applications in stereochemistry and quantum chemistry. Another important area is molecular knot theory and circuit topology that describe the topology of folded linear molecules such as proteins and Nucleic Acids. The history of the approach may be traced back to the 19th century. Georg Helm published a treatise titled "The Principles of M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
