Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Mathematical beautyMathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful or describe mathematics as an art form, (a position taken by G. H. Hardy) or, at a minimum, as a creative activity. Comparisons are made with music and poetry. Mathematicians describe an especially pleasing method of proof as elegant.
Residue theoremIn complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof.
Residue (complex analysis)In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.
Philosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics makes this branch of philosophy broad and unique. The philosophy of mathematics has two major themes: mathematical realism and mathematical anti-realism. The origin of mathematics is of arguments and disagreements.
BambooBamboos are a diverse group of mostly evergreen perennial flowering plants making up the subfamily Bambusoideae of the grass family Poaceae. Giant bamboos are the largest members of the grass family, in the case of Dendrocalamus sinicus individual culms reaching a length of 46 meters, up to 36 centimeters in thickness and a weight of up to 450 kilograms. The internodes of bamboos can also be of great length. Kinabaluchloa wrayi has internodes up to 2.5 meters in length.
Contour integrationIn the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. Contour integration methods include: direct integration of a complex-valued function along a curve in the complex plane; application of the Cauchy integral formula; and application of the residue theorem.
Toilet paperToilet paper (sometimes called toilet tissue or bathroom tissue) is a tissue paper product primarily used to clean the anus and surrounding region of feces (after defecation), and to clean the external genitalia and perineal area of urine (after urination). It is usually supplied as a long strip of perforated paper wrapped around a paperboard core for storage in a dispenser near a toilet. The bundle, or roll of toilet paper, is known as a toilet roll, or loo roll or bog roll in Britain.
Well-formed formulaIn mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic.
History of paperPaper is a thin nonwoven material traditionally made from a combination of milled plant and textile fibres. The first paper-like plant-based writing sheet was papyrus in Egypt, but the first true papermaking process was documented in China during the Eastern Han period (25–220 AD), traditionally attributed to the court official Cai Lun. This plant-puree conglomerate produced by pulp mills and paper mills was used for writing, drawing, and money. During the 8th century, Chinese paper making spread to the Islamic world, replacing papyrus.
Mellin transformIn mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.
Laplace transformIn mathematics, the 'Laplace transform, named after its discoverer Pierre-Simon Laplace (ləˈplɑ:s), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex frequency domain, also known as s-domain', or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication.
Nachbin's theoremIn mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated.
ZX SpectrumThe ZX Spectrum (zɛd_ɛks) is an 8-bit home computer that was developed by Sinclair Research. It was released in the United Kingdom on 23 April 1982, and became Britain's best-selling microcomputer. Referred to during development as the ZX81 Colour and ZX82, it was launched as the ZX Spectrum to highlight the machine's colour display, which differed from the black and white display of its predecessor, the ZX81.
ZX80The Sinclair ZX80 is a home computer launched on 29 January 1980 by Science of Cambridge Ltd. (later to be better known as Sinclair Research). It is notable for being one of the first computers available in the United Kingdom for less than a hundred pounds. It was available in kit form for £79.95, where purchasers had to assemble and solder it together, and as a ready-built version at £99.95. The ZX80 was advertised as the first personal computer for under £100 and received praise for its value and documentation.
ZX81The ZX81 is a home computer that was produced by Sinclair Research and manufactured in Dundee, Scotland, by Timex Corporation. It was launched in the United Kingdom in March 1981 as the successor to Sinclair's ZX80 and designed to be a low-cost introduction to home computing for the general public. It was hugely successful; more than 1.5 million units were sold. In the United States it was initially sold as the ZX-81 under licence by Timex. Timex later produced its own versions of the ZX81: the Timex Sinclair 1000 and Timex Sinclair 1500.
