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Topic Review
Birkhoff's Representation Theorem
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937. The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.
  • 671
  • 10 Oct 2022
Topic Review Peer Reviewed
Pandemic Equation and COVID-19 Evolution
The Pandemic Equation describes multiple pandemic waves and has been applied to describe the COVID-19 pandemic. Using the generalized approaches of solid-state physics, we derive the Pandemic Equation, which accounts for the effects of pandemic mitigation measures and multiple pandemic waves. The Pandemic Equation uses slow and fast time scales for “curve flattening” and describing vaccination and mitigation measures and the Scaled Fermi–Dirac distribution functions for describing transitions between pandemic waves. The Pandemic Equation parameters extracted from the pandemic curves can be used for comparing different scenarios of the pandemic evolution and for extrapolating the pandemic evolution curves for the periods of time on the order of the instantaneous Pandemic Equation characteristic time constant. The parameter extraction for multiple locations could also allow for uncertainty quantification for such pandemic evolution predictions.
  • 660
  • 19 Apr 2024
Biography
Sunil Mukhi
Sunil Mukhi is an Indian theoretical physicist working in the areas of string theory, quantum field theory and particle physics. Currently he is a physics professor at IISER Pune. He is also the dean of faculty here. He obtained a B.Sc. degree at St. Xavier's College, Mumbai and a Ph.D. in Theoretical Physics in 1981 from Stony Brook University (then called the State University of New York at
  • 651
  • 16 Nov 2022
Topic Review
Ideal Lattice Cryptography
Ideal lattices are a special class of lattices and a generalization of cyclic lattices. Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular, they have a significant place in cryptography. Micciancio defined a generalization of cyclic lattices as ideal lattices. They can be used in cryptosystems to decrease by a square root the number of parameters necessary to describe a lattice, making them more efficient. Ideal lattices are a new concept, but similar lattice classes have been used for a long time. For example cyclic lattices, a special case of ideal lattices, are used in NTRUEncrypt and NTRUSign. Ideal lattices also form the basis for quantum computer attack resistant cryptography based on the Ring Learning with Errors. These cryptosystems are provably secure under the assumption that the Shortest Vector Problem (SVP) is hard in these ideal lattices.
  • 639
  • 20 Oct 2022
Biography
Gordon L. Kane
Gordon Leon Kane (born January 19, 1937) is Victor Weisskopf Distinguished University Professor at the University of Michigan and Director Emeritus at the Leinweber Center for Theoretical Physics (LCTP), a leading center for the advancement of theoretical physics. He was director of the LCTP from 2005 to 2011 and Victor Weisskopf Collegiate Professor of Physics from 2002 - 2011. He received the
  • 632
  • 17 Nov 2022
Biography
Pierre Deligne
Pierre René, Viscount Deligne (French: [dəliɲ]; born 3 October 1944) is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre
  • 625
  • 08 Dec 2022
Topic Review Video
Object 3d Reconstruction
The present paper summarized the existing methods of 3D reconstruction of objects by the Shape-From-Focus (SFF) method. This is a method for recovering depth from an image series of the same object taken with different focus settings, referred to as a multifocal image.
  • 622
  • 08 Oct 2021
Biography
Emilio Del Giudice
Emilio Del Giudice (1 January 1940 – 31 January 2014) was an Italian theoretical physicist who worked in the field of condensed matter. Pioneer of string theory in the early 1970s, later on he became better known for his work with Giuliano Preparata at the Italian Institute for Nuclear Physics (INFN). During the 1970s, along with Sergio Fubini, Paolo Di Vecchia and Gabriele Veneziano, Del G
  • 620
  • 29 Dec 2022
Biography
Alexey Parshin
Aleksei (or Alexei) Nikolaevich Parshin (Russian: Алексей Николаевич Паршин; born 7 November 1942 in Sverdlovsk) is a Russian mathematician, specializing in number theory and algebraic geometry. Parshin graduated in 1964 from the Faculty of Mathematics and Mechanics of Moscow State University and then enrolled as a graduate student at the Steklov Institute of Mathematic
  • 614
  • 05 Dec 2022
Topic Review
Facility Location and Vehicle-Routing Problem in Reverse Logistics
The concept of reverse logistics (RL) was put forward in 1992, whose essence was to transfer end-of-life (EOL) products from the consumer to the producer for processing.
  • 614
  • 13 Jan 2023
Topic Review
Petr–Douglas–Neumann Theorem
In geometry, the Petr–Douglas–Neumann theorem (or the PDN-theorem) is a result concerning arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same number of sides as the initial polygon. The theorem was first published by Karel Petr (1868–1950) of Prague in 1908. The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940 and also by B H Neumann (1909–2002) in 1941. The naming of the theorem as Petr–Douglas–Neumann theorem, or as the PDN-theorem for short, is due to Stephen B Gray. This theorem has also been called Douglas's theorem, the Douglas–Neumann theorem, the Napoleon–Douglas–Neumann theorem and Petr's theorem. The PDN-theorem is a generalisation of the Napoleon's theorem which is concerned about arbitrary triangles and of the van Aubel's theorem which is related to arbitrary quadrilaterals.
  • 603
  • 28 Nov 2022
Biography
Kate Claghorn
Kate Holladay Claghorn (1864–1938) was an American sociologist, economist, statistician, legal scholar, and Progressive Era activist, who became one of the founders of the National Association for the Advancement of Colored People. Claghorn was born on February 12, 1864 in Aurora, Illinois, but grew up in New York City . She earned a bachelor's degree in 1892 from Bryn Mawr College, and com
  • 594
  • 09 Dec 2022
Topic Review Video
K-Center Problem
This entry is adapted from the peer-reviewed paper 10.1109/ACCESS.2019.2933875 K-center problems are particular cases of the facility location problem, where a set of optimal centers are to be found given a set of constraints. In a nutshell, a k-center problem usually seeks a set of at most k centers that minimize the distance a client must travel to its nearest center. Namely, their objective function is often a minmax one. Naturally, these problems are well suited for modeling real location problems. Although many different problems fit within the description of a k-center problem, the most popular of these is the vertex k-center problem, where the input is a simple graph and an integer k, and the goal is to find at most k vertices whose distance to the remaining vertices is minimal.
  • 592
  • 30 Nov 2023
Biography
John Howard Redfield
John Howard Redfield (June 8, 1879 – April 17, 1944) was an United States mathematician, best known for discovery of what is now called Pólya enumeration theorem (PET) in 1927,[1] ten years ahead of similar but independent discovery made by George Pólya. Redfield was a great-grandson of William Charles Redfield, one of the founders and the first president of AAAS. Redfield's ability is ev
  • 589
  • 09 Dec 2022
Topic Review
Lebesgue Constant (Interpolation)
In mathematics, the Lebesgue constants (depending on a set of nodes and of its size) give an idea of how good the interpolant of a function (at the given nodes) is in comparison with the best polynomial approximation of the function (the degree of the polynomials are obviously fixed). The Lebesgue constant for polynomials of degree at most n and for the set of n + 1 nodes T is generally denoted by Λn(T ). These constants are named after Henri Lebesgue.
  • 584
  • 01 Dec 2022
Topic Review
Boundary Conditions, Nonequilibrium Thermodynamics Equations
In this entry, we present a systematical review on boundary conditions (BCs) for partial differential equations (PDEs) from nonequilibrium thermodynamics. From a stability point of view, such PDEs should satisfy the structural stability condition. In particular, they constitute hyperbolic systems, for which the generalized Kreiss condition (UKC) is a sufficient and essentially necessary condition for the well-posedness of the corresponding models (PDEs with BCs).
  • 581
  • 08 Oct 2021
Biography
Margaret Jarman Hagood
Margaret Loyd Jarman "Marney" Hagood (1907–1963) was an American sociologist and demographer who "helped steer sociology away from the armchair and toward the calculator".[1] She wrote the books Mothers of the South (1939) and Statistics for Sociologists (1941), and later became president of the Population Association of America and of the Rural Sociological Society. Hagood was born on Octo
  • 576
  • 08 Dec 2022
Biography
Brendan McKay
Brendan Damien McKay (born 26 October 1951 in Melbourne, Australia ) is an Emeritus Professor in the Research School of Computer Science at the Australian National University (ANU). He has published extensively in combinatorics. McKay received a Ph.D. in mathematics from the University of Melbourne in 1980, and was appointed Assistant Professor of Computer Science at Vanderbilt University, Nash
  • 555
  • 13 Dec 2022
Biography
David R. Morrison
David Robert Morrison (born July 29, 1955, in Oakland, California) is an American mathematician and theoretical physicist. He works on string theory and algebraic geometry, especially its relations to theoretical physics. Morrison studied at Princeton University with bachelor's degree in 1976 and at Harvard University with master's degree in 1977 and PhD under Phillip Griffiths in 1980 with the
  • 552
  • 15 Dec 2022
Topic Review
Criticism of Non-Standard Analysis
Non-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
  • 525
  • 04 Nov 2022
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