|
Jacob's Ladder (manifold)
In mathematics, Jacob's ladder is a surface with infinite genus and two ends. It was named after Jacob's ladder by Étienne , because the surface can be constructed as the boundary of a ladder that is infinitely long in both directions. See also *Cantor tree surface In dynamical systems, the Cantor tree is an infinite-genus surface homeomorphic to a sphere with a Cantor set removed. The blooming Cantor tree is a Cantor tree with an infinite number of handles added in such a way that every end is a limit of ha ... * Loch Ness monster surface References * * Surfaces {{Topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob S Ladder Surface
Jacob (; ; ar, يَعْقُوب, Jacob in Islam, Yaʿqūb; gr, Ἰακώβ, Iakṓb), later given the name Israel (name), Israel, is regarded as a Patriarchs (Bible), patriarch of the Israelites and is an important figure in Abrahamic religions, such as Judaism, Christianity, and Islam. Jacob first appears in the Book of Genesis, where he is described as the son of Isaac and Rebecca, and the grandson of Abraham, Sarah, and Bethuel. According to the biblical account, he was the second-born of Isaac's children, the elder being Jacob's fraternal twin brother, Esau. Jacob is said to have bought Esau's Primogeniture, birthright and, with his mother's help, deceived his aging father to bless him instead of Esau. Later in the narrative, following a severe drought in his homeland of Canaan, Jacob and his descendants, with the help of his son Joseph (Genesis), Joseph (who had become a confidant of the pharaoh), moved to Biblical Egypt, Egypt where Jacob died at the age of 147. He is su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. CT ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic ''χ'', via the relationship ''χ'' = 2 − 2''g'' for closed surfaces, where ''g'' is the genus. For surfaces with ''b'' boundary components, the equation reads ''χ'' = 2 − 2''g'' − ''b''. In layman's terms, it's the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
End (topology)
In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification. The notion of an end of a topological space was introduced by . Definition Let ''X'' be a topological space, and suppose that :K_1 \subseteq K_2 \subseteq K_3 \subseteq \cdots is an ascending sequence of compact subsets of ''X'' whose interiors cover ''X''. Then ''X'' has one end for every sequence :U_1 \supseteq U_2 \supseteq U_3 \supseteq \cdots, where each ''U''''n'' is a connected component of ''X'' \ ''K''''n''. The number of ends does not depend on the specific sequence of compact sets; there is a natural bijection between the sets of ends associated with any two such sequences. Using this definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob's Ladder
Jacob's Ladder ( he, סֻלָּם יַעֲקֹב ) is a ladder leading to heaven that was featured in a dream the biblical Patriarch Jacob had during his flight from his brother Esau in the Book of Genesis (chapter 28). The significance of the dream has been debated, but most interpretations agree that it identified Jacob with the obligations and inheritance of the people chosen by God, as understood in Abrahamic religions. Biblical narrative The description of Jacob's Ladder appears in : Judaism The classic Torah commentaries offer several interpretations of Jacob's Ladder. According to the Midrash Genesis Rabbah, the ladder signified the exiles which the Jewish people would suffer before the coming of the Jewish messiah. First, the angel representing the 70-year exile of Babylonia climbed "up" 70 rungs, and then fell "down". Then the angel representing the exile of Persia went up a number of steps, and fell, as did the angel representing the exile of Greece. Only the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Tree Surface
In dynamical systems, the Cantor tree is an infinite-genus surface homeomorphic to a sphere with a Cantor set removed. The blooming Cantor tree is a Cantor tree with an infinite number of handles added in such a way that every end is a limit of handles. See also *Jacob's ladder surface *Loch Ness monster surface In mathematics, the Loch Ness monster is a surface with infinite genus but only one end. It appeared named this way already in a 1981 article by . The surface can be constructed by starting with a plane (which can be thought of as the surface ... References * *{{Citation , last1=Walczak , first1=Paweł , title=Dynamics of foliations, groups and pseudogroups , url=https://books.google.com/books?id=Tl4WkcHzhIAC , publisher=Birkhäuser Verlag , series=Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) athematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series), isbn=978-3-7643-7091-6 , mr=20563 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loch Ness Monster Surface
In mathematics, the Loch Ness monster is a surface with infinite genus but only one end. It appeared named this way already in a 1981 article by . The surface can be constructed by starting with a plane (which can be thought of as the surface of Loch Ness) and adding an infinite number of handles (which can be thought of as loops of the Loch Ness monster). See also * Cantor tree surface In dynamical systems, the Cantor tree is an infinite-genus surface homeomorphic to a sphere with a Cantor set removed. The blooming Cantor tree is a Cantor tree with an infinite number of handles added in such a way that every end is a limit of ha ... * Jacob's ladder surface References * * * *{{Citation , last1=Arredondo , first1=John A. , last2=Ramírez-Maluendas , first2= Camilo, title=On the Infinite Loch Ness monster , url=http://cmuc.karlin.mff.cuni.cz/cmuc1704/abs/arredo.htm , doi=10.14712/1213-7243.2015.227 , year=2017 , journal= Commentationes Mathematicae Universitatis C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
