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All Topic Review Biography Peer Reviewed Entry Video Entry
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.
  • 996
  • 08 Oct 2021
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.
  • 995
  • 19 Apr 2024
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.
  • 991
  • 20 Oct 2022
Biography
Derrick Henry Lehmer
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991) was an American mathematician who refined Édouard Lucas' work in the 1930s and devised the Lucas–Lehmer test for Mersenne primes. Lehmer's peripatetic career as a number theorist, with him and his wife taking numerous types of work in the United States and abroad to support themselves during the Great Depression, fortuitously br
  • 985
  • 24 Nov 2022
Biography
Nick Katz
Nicholas Michael Katz (born December 7, 1943) is an United States mathematician, working in the fields of algebraic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. He is currently a professor in the Mathematics Department at Princeton University in Princeton, New Jersey and an editor of the journal Annals of Mathematics.[1] Katz graduated from Johns
  • 973
  • 11 Nov 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.
  • 948
  • 04 Nov 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
  • 945
  • 15 Dec 2022
Topic Review
Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions f(z1, z2, ... zn) on the space Cn of n-tuples of complex numbers. As in complex analysis, which is the case n = 1 but of a distinct character, these are not just any functions: they are supposed to be holomorphic or complex analytic, so that locally speaking they are power series in the variables zi. Equivalently, as it turns out, they are locally uniform limits of polynomials; or local solutions to the n-dimensional Cauchy–Riemann equations.
  • 898
  • 27 Oct 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).
  • 859
  • 08 Oct 2021
Topic Review
Finite Promise Games and Greedy Clique Sequences
The finite promise games are a collection of mathematical games developed by American mathematician Harvey Friedman in 2009 which are used to develop a family of fast-growing functions [math]\displaystyle{ FPLCI(k) }[/math], [math]\displaystyle{ FPCI(k) }[/math] and [math]\displaystyle{ FLCI(k) }[/math]. The greedy clique sequence is a graph theory concept, also developed by Friedman in 2010, which are used to develop fast-growing functions [math]\displaystyle{ USGCS(k) }[/math], [math]\displaystyle{ USGDCS(k) }[/math] and [math]\displaystyle{ USGDCS_2(k) }[/math]. [math]\displaystyle{ \mathsf{SMAH} }[/math] represents the theory of ZFC plus, for each [math]\displaystyle{ k }[/math], "there is a strongly [math]\displaystyle{ k }[/math]-Mahlo cardinal", and [math]\displaystyle{ \mathsf{SMAH^+} }[/math] represents the theory of ZFC plus "for each [math]\displaystyle{ k }[/math], there is a strongly [math]\displaystyle{ k }[/math]-Mahlo cardinal". [math]\displaystyle{ \mathsf{SRP} }[/math] represents the theory of ZFC plus, for each [math]\displaystyle{ k }[/math], "there is a [math]\displaystyle{ k }[/math]-stationary Ramsey cardinal", and [math]\displaystyle{ \mathsf{SRP^+} }[/math] represents the theory of ZFC plus "for each [math]\displaystyle{ k }[/math], there is a strongly [math]\displaystyle{ k }[/math]-stationary Ramsey cardinal". [math]\displaystyle{ \mathsf{HUGE} }[/math] represents the theory of ZFC plus, for each [math]\displaystyle{ k }[/math], "there is a [math]\displaystyle{ k }[/math]-huge cardinal", and [math]\displaystyle{ \mathsf{HUGE^+} }[/math] represents the theory of ZFC plus "for each [math]\displaystyle{ k }[/math], there is a strongly [math]\displaystyle{ k }[/math]-huge cardinal".
  • 859
  • 17 Oct 2022
Biography
Cynthia Clark
Cynthia Zang Facer Clark FRSS (born April 1, 1942)[1] is an American statistician known for her work improving the quality of data in the Federal Statistical System of the United States, and especially in the National Agricultural Statistics Service.[2][3] She has also served as the president of the Caucus for Women in Statistics[4] and the Washington Statistical Society.[5] (As of 2018) she is
  • 859
  • 09 Dec 2022
Topic Review
Non-Standard Calculus
In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to infinitesimal calculus. It provides a rigorous justification that were previously considered merely heuristic. Nonrigourous calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians like Richard Courant viewed infinitesimals as being naive and vague or meaningless. Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."
  • 840
  • 27 Oct 2022
Biography
Margarete Kahn
Margarete Kahn (known as Grete Kahn, born 27 August 1880, missing after deportation to Piaski, Poland on 28 March 1942) was a German mathematician and Holocaust victim.[1] She was among the first women to obtain a doctorate in Germany . Her doctoral work was on the topology of algebraic curves. Margarete Kahn was the daughter of Eschwege merchant and flannel factory owner Albert Kahn (1853–
  • 814
  • 26 Dec 2022
Topic Review
Explicit Mathematics
In the field of logic and set theory, American mathematician and philosopher Solomon Feferman developed explicit mathematics, which is a collection of axiomatic systems related to constructive mathematics. It refers to a set of fundamental logical systems based on mathematical logic. A second type of class is added to a first-order logic of partial terms axiomatizing combinatory algebra. Alternative names: classifications, or types are sometimes used. The aim of explicit mathmetics was to have a straightforward and principled transfer of the notions of indescribable cardinals from set theory to admissible ordinals, yet the approach leaves open the question as to what is the proper analogue for admissible ordinals — if any — of a cardinal κ being [math]\displaystyle{ \Pi^m_n }[/math]-indescribable for m > 1.
  • 796
  • 13 Oct 2022
Biography
Valentina Mikhailovna Borok
Valentina Mikhailovna Borok (9 July 1931, Kharkiv, Ukraine , USSR – 4 February 2004, Haifa, Israel) was a Soviet Ukraine mathematician. She is mainly known for her work on partial differential equations.[1] Borok was born on July 9, 1931 in Kharkiv in Ukraine (then USSR), into a Jewish family.[2] Her father, Michail Borok, was a chemist, scientist and an expert in material science. Her moth
  • 754
  • 23 Dec 2022
Topic Review
Matrix Function
In mathematics, a matrix function is a function which maps a matrix to another matrix.
  • 741
  • 24 Oct 2022
Topic Review
Synthetic Datasets
With the consistent growth in the importance of machine learning and big data analysis, feature selection stands to be one of the most relevant techniques in the field. Extending into many disciplines, the use of feature selection in medical applications, cybersecurity, DNA micro-array data, and many more areas is witnessed. Machine learning models can significantly benefit from the accurate selection of feature subsets to increase the speed of learning and also to generalize the results. Feature selection can considerably simplify a dataset, such that the training models using the dataset can be “faster” and can reduce overfitting. Synthetic datasets were presented as a valuable benchmarking technique for the evaluation of feature selection algorithms.
  • 712
  • 20 Mar 2024
Topic Review
Role of Visual Tools in Understanding Mathematical Culture
The term mathematical culture’ cannot be naturally defined; we will understand it in the same way as ‘good mathematics’ is understood by, i.e., good ways of solving mathematical issues, good mathematical techniques, good mathematical applications and cultivating of mathematical insight, creativity and beauty of mathematics. Cultivation of a mathematical culture means teaching how to see the roots of mathematics in reality (in nature, in society, but also in mathematics itself), getting to know the world of mathematical concepts, understanding this world and being able to apply it in a cultivated and correct way when solving various problems.
  • 556
  • 25 Jul 2023
Topic Review
Fuzzy Time Series for Electricity Demand
A time series is a succession of data ordered chronologically in defined time intervals. The data may be evenly spaced, such as the record of daily solar generation from a photovoltaic plant, or it may be different, such as the number of annual earthquakes in a defined area. This type of representation offers many advantages because its analysis allows us to discover underlying relationships in the data, which can be from various time series or within the data itself. These can be used to extrapolate behavior in the past, during periods of data loss, and in the future.
  • 554
  • 22 Nov 2023
Topic Review Peer Reviewed
A Survey on Orthogonal Polynomials from Monomiality Principle Point of View
This survey highlights the significant role of exponential operators and the monomiality principle in the theory of special polynomials. Using operational calculus formalism, we revisited classical and current results corresponding to a broad class of special polynomials. For instance, we explore the 2D Hermite polynomials and their generalizations. We also present an integral representation of Gegenbauer polynomials in terms of Gould–Hopper polynomials, establishing connections with a simple case of Gegenbauer–Sobolev orthogonality. The monomiality principle is examined, emphasizing its utility in simplifying the algebraic and differential properties of several special polynomial families. This principle provides a powerful tool for deriving properties and applications of such polynomials. Additionally, we review advancements over the past 25 years, showcasing the evolution and extensive applicability of this operational formalism in understanding and manipulating special polynomial families.
  • 529
  • 20 Sep 2024
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