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Å indel Sequence
In additive combinatorics, a Å indel 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 Å indel 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 Å indel sequences are named after Jan Å indel, 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 KřÞek, Alena Å olcová, and Lawrence Somer, in their work analyzing the mathematics of the Prague clock. If s denotes the sum of the numbers within a single per ... [...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 is small, what can we say about the structures of and ? 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 in terms of and . This can be viewed as an inverse problem with the given information that is sufficiently small and the structural conclusion is then of the form that either or 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 fields of mathematics, including combinatorics, ergod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Sum
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. 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 in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. The re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are Formula The triangular numbers are given by the following explicit formulas: where \textstyle is notation for 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 fact that the nth triangular number equals n(n+1)/2 can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement ... [...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 partial order. 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''", " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prague Astronomical Clock
The Prague astronomical clock or Prague Orloj ( ) is a medieval astronomical clock attached to the Old Town Hall in Prague, the capital of the Czech Republic. Description The Orloj is mounted on the southern wall of Old Town Hall in the Old Town Square. The clock mechanism has three main components – the astronomical dial, representing the position of the Sun and Moon in the sky and displaying various astronomical details; statues of various Catholic saints stand on either side of the clock; "The Walk of the Apostles", an hourly show of moving Apostle figures and other sculptures, notably a figure of a skeleton that represents Death, striking the time; and a calendar dial with medallions representing the months. According to local legend, the city will suffer if the clock is neglected and its good operation is placed in jeopardy; a ghost, mounted on the clock, was supposed to nod its head in confirmation. According to the legend, the only hope was represented by a bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Å indel
Jan Šindel (1370s – between 1455 and 1457), also known as Jan Ondřejův ( or ''Joannes de Praga''), was a Czech medieval scientist and Catholic priest. He was a professor at Charles University in Prague and became the 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. 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. 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 he became Doctor of Medicine and rector of the university in 1410. At the beginning he ... [...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] |
Striking Clock
A striking clock is a clock that sounds the hours audibly on a bell, gong, or other audible device. In 12-hour striking, used most commonly in striking clocks today, the clock strikes once at 1:00 am, twice at 2:00 am, continuing in this way up to twelve times at 12:00 mid-day, then starts again, striking once at 1:00 pm, twice at 2:00 pm, and the pattern continues up to twelve times at 12:00 midnight. The striking feature of clocks was originally more important than their clock faces; the earliest clocks struck the hours, but had no dials to enable the time to be read. The development of mechanical clocks in 12th century Europe was motivated by the need to ring bells upon the canonical hours to call the community to prayer. The earliest known mechanical clocks were large striking clocks installed in towers in monasteries or public squares, so that their bells could be heard far away. Though an early striking clock in Syria was a 12-hour clock, many early clocks struck up to 24 st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 2, two as the Base (exponentiation), base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hierarchy, is exactly equal to H_(1). Powers of two with Sign (mathematics)#Terminology for signs, non-negative exponents are integers: , , and is two multiplication, multiplied by itself times. The first ten powers of 2 for non-negative values of are: :1, 2, 4, 8, 16 (number), 16, 32 (number), 32, 64 (number), 64, 128 (number), 128, 256 (number), 256, 512 (number), 512, ... By comparison, powers of two with negative exponents are fractions: for positive integer , is one half multiplied by itself times. Thus the first few negative powers of 2 are , , , , etc. Sometimes these are called ''inverse powers of two'' because each is the multiplicative inverse of a positive power of two. Base of the binary numeral sy ... [...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 and correspond to the ends of the ruler. In order to measure the distance , with there must be marks and such that . A ''complete'' sparse ruler allows one to measure any distance up to its full length. A complete sparse ruler is called ''minimal'' if there is no complete sparse ruler of length with 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
