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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.
  • 312
  • 13 Oct 2022
Biography
Ezra A. (Bud) Brown
Ezra A. Brown (born January 22, 1944 in Reading, PA) is an American mathematician active in combinatorics, algebraic number theory, elliptic curves, graph theory, expository mathematics and cryptography. He spent most of his career at Virginia Tech where he is now Alumni Distinguished Professor Emeritus of Mathematics.[1] Brown earned a B.A. at Rice University in 1965.[2] He then studied math
  • 545
  • 15 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.
  • 479
  • 13 Jan 2023
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".
  • 341
  • 17 Oct 2022
Topic Review
First-Order Arithmetic
In set theory and mathematical logic, first-order arithmetic is a collection of axiomatic systems formalising natural and subsets of the natural numbers. It is a choice for axiomatic theory as a basis for many mathematics, but not all. The primary first-order axiom is Peano arithmetic, created by Giuseppe Peano: Peano arithmetic has a proof-theoretic ordinal of [math]\displaystyle{ \varepsilon_0 = \varphi(1, 0) = \psi_0(\Omega) }[/math].
  • 1.5K
  • 10 Oct 2022
Topic Review
Fresh Fruit Supply Chain Optimization
The fresh fruit chain has been recognized as a very important and strategic part of the economic development of many countries. The planning framework for production and distribution is highly complex as a result. Mathematical models have been developed over the decades to deal with this complexity. This review focuses on the recent progress in mathematically based decision making to account for uncertainties in the fresh fruit supply chain
  • 4.5K
  • 27 Jul 2021
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.
  • 104
  • 22 Nov 2023
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
  • 364
  • 17 Nov 2022
Topic Review
Hybrid Number
A hybrid number is a generalization of complex numbers [math]\displaystyle{ \left(a+\mathbf{i}b, \mathbf{i}^{2}=-1\right) }[/math], split-complex numbers (or "hyperbolic number") [math]\displaystyle{ \left(a+\mathbf{h}b, \mathbf{h}^2=1\right) }[/math] and dual numbers [math]\displaystyle{ \left(a+\mathbf{\varepsilon} b, \mathbf{\varepsilon}^2 = 0\right) }[/math]. Hybrid numbers form a noncommutative ring. Complex, hyperbolic and dual numbers are well known two-dimensional number systems. It is well known that, the set of complex numbers, hyperbolic numbers and dual numbers are respectively. The algebra of hybrid numbers is a noncommutative algebra which unifies all three number systems calls them hybrid numbers., , . A hybrid number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation Because these numbers are a composition of dual, complex and hyperbolic numbers, Ozdemir calls them hybrid numbers . A commutative two-dimensional unital algebra generated by a 2 by 2 matrix is isomorphic to either complex, dual or hyperbolic numbers . Due to the set of hybrid numbers is a two-dimensional commutative algebra spanned by 1 and [math]\displaystyle{ \mathbf{i}b+c\mathbf{\varepsilon }+d\mathbf{h} }[/math], it is isomorphic to one of the complex, dual or hyperbolic numbers. Especially in the last century, a lot of researchers deal with the geometric and physical applications of these numbers. Just as the geometry of the Euclidean plane can be described with complex numbers, the geometry of the Minkowski plane and Galilean plane can be described with hyperbolic numbers. The group of Euclidean rotations SO(2) is isomorphic to the group U(1) of unit complex numbers. The geometrical meaning of multiplying by [math]\displaystyle{ e^{\mathbf{i}\theta}=\cos \theta +\mathbf{i}\sin \theta }[/math] means a rotation of the plane. , . The group of Lorentzian rotations [math]\displaystyle{ SO(1,1) }[/math] is isomorphic to the group of unit spacelike hyperbolic numbers. This rotation can be viewed as hyperbolic rotation. Thus, multiplying by [math]\displaystyle{ e^{\mathbf{h}\theta }=\cosh \theta +\mathbf{h} \sinh \theta }[/math] means a map of hyperbolic numbers into itself which preserves the Lorentzian metric. , , , The Galilean rotations can be interpreted with dual numbers. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since [math]\displaystyle{ \left( 1+x\mathbf{\varepsilon }\right) \left( 1+y\mathbf{\varepsilon }\right) =1+\left( x+y\right) \mathbf{\varepsilon } }[/math]. The Euler formula for dual numbers is [math]\displaystyle{ e^{\mathbf{\varepsilon }\theta }=1+\mathbf{\varepsilon }\theta }[/math]. Multiplying by [math]\displaystyle{ e^{\mathbf{\varepsilon \theta }} }[/math] is a map of dual numbers into itself which preserves the Galilean metric. This rotation can be named as parabolic rotation , , , , , . File:Planar rotations.tif In abstract algebra, the complex, the hyperbolic and the dual numbers can be described as the quotient of the polynomial ring [math]\displaystyle{ \mathbb{R}[x] }[/math] by the ideal generated by the polynomials [math]\displaystyle{ x^2+1, }[/math], [math]\displaystyle{ x^2-1 }[/math] and [math]\displaystyle{ x^{2} }[/math] respectively. That is, Matrix represantations of the units [math]\displaystyle{ \mathbf{i} }[/math], [math]\displaystyle{ \mathbf{\varepsilon } }[/math], [math]\displaystyle{ \mathbf{h} }[/math] are respectively.
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  • 08 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.
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  • 20 Oct 2022
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