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Lah Numbers
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., the Lah numbers, discovered by Ivo Lah in 1954, are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the nth derivatives of e^. Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a Set (mathematics), set of ''n'' elements can be Partition of a set, partitioned into ''k'' nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers. Unsigned Lah numbers : : L(n,k) = \frac. Signed Lah numbers : : L'(n,k) = (-1)^n \frac. ''L''(''n'', 1) is always ''n''!; in the interpretation above, the only partition of into 1 set can have its set ordered in 6 ways: :, , , , or ''L''(3, 2) corresponds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lah Numbers
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., the Lah numbers, discovered by Ivo Lah in 1954, are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the nth derivatives of e^. Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a Set (mathematics), set of ''n'' elements can be Partition of a set, partitioned into ''k'' nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers. Unsigned Lah numbers : : L(n,k) = \frac. Signed Lah numbers : : L'(n,k) = (-1)^n \frac. ''L''(''n'', 1) is always ''n''!; in the interpretation above, the only partition of into 1 set can have its set ordered in 6 ways: :, , , , or ''L''(3, 2) corresponds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Numbers Of The First Kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers in general. Definitions The original definition of Stirling numbers of the first kind was algebraic: they are the coefficients s(n,k) in the expansion of the falling factorial :(x)_n = x(x-1)(x-2)\cdots(x-n+1) into powers of the variable x: :(x)_n = \sum_^n s(n,k) x^k, For example, (x)_3 = x(x-1)(x - 2) = 1x^3 - 3x^2 + 2x, leading to the values s(3, 3) = 1, s(3, 2) = -3, and s(3, 1) = 2. Subsequently, it was discovered that th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial And Binomial Topics
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Matrix
In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × 5 matrices are: L_5 = \begin 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 \\ 1 & 3 & 3 & 1 & 0 \\ 1 & 4 & 6 & 4 & 1 \end\,\,\,U_5 = \begin 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 3 & 6 \\ 0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 0 & 1 \end\,\,\,S_5 = \begin 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 6 & 10 & 15 \\ 1 & 4 & 10 & 20 & 35 \\ 1 & 5 & 15 & 35 & 70 \end=L_5 \times U_5 There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity. Definition The non-zero elements of a Pascal matrix are given by the bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optics Express
''Optics Express'' is a biweekly peer-reviewed scientific journal published by Optica. It was established in 1997. The journal reports on scientific and technology innovations in all aspects of optics and photonics. The Energy Express supplement reports research on the science and engineering of light and its impact on sustainable energy development, the environment, and green technologies. The editor-in-chief is James Leger (University of Minnesota). According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 3.833, ranking it 28th out of 101 journals in the category "Optics". References External links * Open access journals Optics journals Optica (society) academic journals Publications estab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Dispersion
In optics, and by analogy other branches of physics dealing with wave propagation, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to optics in particular. A medium having this common property may be termed a dispersive medium (plural ''dispersive media''). Although the term is used in the field of optics to describe light and other electromagnetic waves, dispersion in the same sense can apply to any sort of wave motion such as acoustic dispersion in the case of sound and seismic waves, and in gravity waves (ocean waves). Within optics, dispersion is a property of telecommunication signals along transmission lines (such as microwaves in coaxial cable) or the pulses of light in optical fiber. Physically, dispersion translates in a loss of kinetic energy through absorption. In optics, one important and familiar consequence of dispersion is the change in the angle of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Wavelet Transform
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency ''and'' location information (location in time). Examples Haar wavelets The first DWT was invented by Hungarian mathematician Alfréd Haar. For an input represented by a list of 2^n numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the sum. This process is repeated recursively, pairing up the sums to prove the next scale, which leads to 2^n-1 differences and a final sum. Daubechies wavelets The most commonly used set of discrete wavelet transforms was formulated by the Belgian mathematician Ingrid Daubechies in 1988. This formulation is based on the use of recurrence relations to generate progressively finer discrete samplings o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. The DFT is the most important discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steganography
Steganography ( ) is the practice of representing information within another message or physical object, in such a manner that the presence of the information is not evident to human inspection. In computing/electronic contexts, a computer file, message, image, or video is concealed within another file, message, image, or video. The word ''steganography'' comes from Greek ''steganographia'', which combines the words ''steganós'' (), meaning "covered or concealed", and ''-graphia'' () meaning "writing". The first recorded use of the term was in 1499 by Johannes Trithemius in his '' Steganographia'', a treatise on cryptography and steganography, disguised as a book on magic. Generally, the hidden messages appear to be (or to be part of) something else: images, articles, shopping lists, or some other cover text. For example, the hidden message may be in invisible ink between the visible lines of a private letter. Some implementations of steganography that lack a shared secret are f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Numbers Of The Second Kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \left\. Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. Stirling numbers of the second kind are one of two kinds of Stirling numbers, the other kind being called Stirling numbers of the first kind (or Stirling cycle numbers). Mutually inverse (finite or infinite) triangular matrices can be formed from the Stirling numbers of each kind according to the parameters ''n'', ''k''. Definition The Stirling numbers of the second kind, written S(n,k) or \lbrace\textstyle\rbrace or with other notations, count the number of ways to partition a set of n labelled objects into k nonempty unlabelled subsets. Equivalently, they count the number of different equivalence relations with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
