Šindel Sequence

	Šindel Sequence In additive combinatorics, a Šindel sequence is a periodic sequence of integers with the property that its partial sums include all of the triangular numbers. For instance, the sequence that begins 1, 2, 3, 4, 3, 2 is a Šindel sequence, with the triangular partial sums etc. Another way of describing such a sequence is that it can be partitioned into contiguous subsequences whose sums are the consecutive integers: This particular example is used in the gearing of the Prague astronomical clock, as part of a mechanism for chiming the clock's bells the correct number of times at each hour. The Šindel sequences are named after Jan Šindel, a Czech scientist in the 14th and 15th centuries whose calculations were used in the design of the Prague clock. The definition and name of these sequences were given by Michal Křížek, Alena Šolcová, and Lawrence Somer, in their work analyzing the mathematics of the Prague clock. If s denotes the sum of the numbers within a single peri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Additive Combinatorics Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are ''inverse problems'': given the size of the sumset ''A'' + ''B'' is small, what can we say about the structures of A and B? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is to find a lower bound for , A + B, in terms of , A, and , B, . This can be viewed as an inverse problem with the given information that , A+B, is sufficiently small and the structural conclusion is then of the form that either A or B is the empty set; however, in literature, such problems are sometimes considered to be direct problems as well. Examples of this type include the Erdős–Heilbronn Conjecture (for a restricted sumset) and the Cauchy–Davenport Theorem. The methods used for tackling such questions often come from many different fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Partial Sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Triangular Number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a subsequence of \langle A,B,C,D,E,F \rangle obtained after removal of elements C, E, and F. The relation of one sequence being the subsequence of another is a preorder. Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as \langle B,C,D \rangle, from \langle A,B,C,D,E,F \rangle, is a substring. The substring is a refinement of the subsequence. The list of all subsequences for the word "apple" would be "''a''", "''ap''", "''al''", "''ae''", "''app''", "''apl''", "''ape''", "''ale''", "''appl''", "''appe''", "''aple''", "''apple''", "''p''", "''pp''", "''pl''", "''pe''", "''ppl''", "''ppe''", "''ple' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Jan Šindel Jan Šindel (1370s – between 1455 and 1457), also known as Jan Ondřejův ( or ''Joannes de Praga''), was a Czechs, Czech medieval scientist and Catholic priest. He was a professor at Charles University in Prague and became the Rector (academia), rector of the university in 1410. He lectured on mathematics and astronomy and was also a personal astrologer and physician of kings Wenceslaus IV of Bohemia and his brother Holy Roman Emperor Sigismund, Holy Roman Emperor, Sigismund. Life Jan Šindel was born in Hradec Králové probably in the 1370s. As a young man he came to Prague to study at Charles University. In 1395 or 1399 he became the Master of Arts at Prague University. In 1406 he worked at the parish school of the St. Nicholas Church (Malá Strana), St. Nicolas Church in Malá Strana in Prague. Later he worked as a teacher of mathematics in Vienna, where he also studied medicine. Then he came back to Prague and became the professor of astronomy at Charles University, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Alena Šolcová Alena Šolcová (born 26 March 1950) is a Czech mathematician and science historian. She is the founder of the Kepler Museum, an astronomy museum in Prague. Life and work Between 1968 and 1973, Šolcová studied mathematics at the Faculty of Mathematics and Physics and Philosophy at Charles University. Between 2002 and 2005, she completed her doctoral studies in mathematics in Civil Engineering with the doctoral thesis titled '' Fermat's Ideas Revived in Mathematics Applied in Engineering,'' and in 2009 she completed her habilitation at the Czech Technical University in Prague, and was appointed associate professor in the field of applied mathematics. Šolcová works at the Faculty of Information Technologies of the Czech Technical University in Prague, where she teaches mathematical logic and the history of mathematics and computer science. She also deals with logic, number theory, some numerical methods and the history of mathematics, computer science and astronomy. Since 1992 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Power Of Two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. The first ten powers of 2 for non-negative values of are: : 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system. Computer science Two to the exponent of , written as , is the number of ways the bits in a binary word of length can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 () to () inclusively. Corresponding signed integer values can be positive, negative and zero; see signed n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Sparse Ruler A sparse ruler is a ruler in which some of the distance marks may be missing. More abstractly, a sparse ruler of length L with m marks is a sequence of integers a_1, a_2, ..., a_m where 0 = a_1 < a_2 < ... < a_m = L. The marks $a_1$ and $a_m$ correspond to the ends of the ruler. In order to measure the distance $K$ , with $0\le K\le L$ there must be marks $a_i$ and $a_j$ such that $a_j-a_i=K$ . A ''complete'' sparse ruler allows one to measure any integer distance up to its full length. A complete sparse ruler is called ''minimal'' if there is no complete sparse ruler of length $L$ with $m-1$ marks. In other words, if any of the marks is removed one can no longer measure all of the distances, even if the marks could be rearranged. A complete sparse ruler is called ''maximal'' if there is no complete sparse ruler of length $L+1$ with $m$ marks. A sparse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]