Fundamental Theorem Research Articles

Back to the fundamentals: a reply to Basener and Sanford 2018.

Fisher's fundamental theorem of natural selection has haunted theoretical population genetic literature since it was proposed in 1930, leading to numerous interpretations. Most of the confusion stemmed from Fisher's own obscure presentation. By the 1970s, a clearer view of Fisher's theorem had been achieved and it was found that, regardless of its utility or significance, it represents a general theorem of evolutionary biology. Basener and Sanford (J Math Biol 76:1589-1622, 2018) writing in JOMB, however, paint a different picture of the fundamental theorem as one hindered by its assumptions and incomplete due to its failure to explicitly incorporate mutational effects. They argue that Fisher saw his theorem as a "mathematical proof of Darwinian evolution". In this reply, we show that, contrary to Basener and Sanford, Fisher's theorem is a general theorem that applies to any evolving population, and that, far from their assertion that it needed to be expanded, the theorem already implicitly incorporates ancestor-descendant variation. We also show that their numerical simulations produce unrealistic results. Lastly, we argue that Basener and Sanford's motivations were in undermining not merely Fisher's theorem, but the concept of universal common descent itself.

Operations on Complex Hesitancy Fuzzy Graph

Abstract: In this paper, Complex Hesitancy Fuzzy Graph (CHFG) is analyzed and we have discussed new concepts in CHFG such as Complete CHFG with example. Further we defined some operations on Complex Hesitancy Fuzzy Graph such as direct product, semi strong product and strong product. Also fundamental theorems and their apllications will be examined. Additionally, new ideas in the density of a CHFG and balanced complex Hesitancy fuzzy graph have been introduced in this work.

Operations on Complex Intuitionistic Fuzzy Graph

Abstract: This study introduces the complex intuitionistic fuzzy graphs and analyzes certain fundamental theorem and applications. Further, new ideas in CIFG such as complete complex intuitionistic fuzzy graphs with example. Also, we defined operations on the Direct product, Semi strong product and Strong product of CIFG. Additionally, we presented the density of CIFG and balanced complex intuitionistic fuzzy graph.

On the Tower Factorization of Integers

Under the fundamental theorem of arithmetic, any integer n > 1 can be uniquely written as a product of prime powers pa ; factoring each exponent a as a product of prime powers qb , and so on, one will obtain what is called the tower factorization of n. Here, given an integer n > 1, we study its height h(n), that is, the number of “floors” in its tower factorization. In particular, given a fixed integer k ≥ 1 , we provide a formula for the density of the set of integers n with h(n) = k. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.

Fundamental theorem of Poisson Hopf module for weak Hopf algebras

Let H be a weak Hopf algebra with a bijective antipode and A an H-comodule Poisson algebra. In this paper, we mainly generalize the fundamental theorem of Poisson Hopf modules to the case of weak Hopf algebras. Besides we will deduce the relative projectivity in the category of Poisson Hopf module.

Double Balanced Quadrilaterals

The k-centroids Gk of a polygon, for k = 0, 1, 2, are the centroids of the polygon when the mass is equally distributed respectively between the vertices, along the perimeter, or across the area. A fundamental theorem by Al-Sharif, Hajja, and Krasopoulos in [1] asserts that the quadrilaterals with either G 0 = G 1 or G 0 = G 2 are precisely all parallelograms. Our main result describes the non-parallelograms with G 1 = G 2 by providing formulas for their diagonals in terms of the sides, as well as formulas for the ratios determined on the diagonals by their intersection point. In this way, we complete a fifteen-year-old problem by these three authors on characterizing all double balanced quadrilaterals. As an application, we show how our main theorem can be used to deduce their characterizations of double balanced circumscribed and cyclic quadrilaterals.

Leonardo and hyper-Leonardo numbers via Riordan arrays

UDC 511 A generalization of the Leonardo numbers is defined and called the hyper-Leonardo numbers. Infinite lower triangular matrices, whose elements are Leonardo and hyper-Leonardo numbers are considered. Then the A - and Z -sequences of these matrices are obtained. Finally, the combinatorial identities between the hyper-Leonardo and Fibonacci numbers are obtained using the fundamental theorem of the Riordan arrays.

ANGLE BETWEEN INTERSECTING CIRCLES

The article demonstrates the concept of orthogonal circles and solves fundamental theorem problems involving right angles between circles. Special attention is paid to the orthogonality of two circles as in a right triangle, as well as the perpendicularity of the Apollonian circle to the respective circles. These problems can be considered both in math olympiads and in the study of geometry courses.

Sustainability-valued discrete option pricing in complete markets

ABSTRACT We consider option pricing using recombining binomial trees, with a two-fold purpose. The first is to introduce environmental, social and governance (ESG) sustainability valuation into option pricing. We explore this in a number of scenarios, including enhancement of yield due to trader information and the impact of the past history of a market driver. The second is to emphasize the use of discrete dynamic pricing, rather than continuum models, as the natural model that governs actual market practice. Furthermore, use of continuous, rather than discrete, compounding (e.g. of the risk-free rate) in discrete option pricing models violates conditions of the fundamental theorem of asset pricing and no-arbitrage. By considering models based upon both compounding methods, we show explicitly where this violation occurs and demonstrate computational differences that arise as a result.

Algebraic Representation of Primes by Hybrid Factorization

The representation of integers by prime factorization, proved by Euclid in the Fundamental Theorem of Arithmetic −also referred to as the Prime Factorization Theorem− although universal in scope, does not provide insight into the algebraic structure of primes themselves. No such insight is gained by summative prime factorization either, where a number can be represented as a sum of up to three primes, assuming Goldbach’s conjecture is true. In this paper, a third type of factorization is introduced, called hybrid prime factorization, defined as the representation of a number as sum −or difference− of two products of primes with no common factors between them. By using hybrid factorization, primes are expressed as algebraic functions of other primes, and primality is established by a single algebraic condition. Following a hybrid factorization approach, sufficient conditions for the existence of Goldbach pairs are derived, and their values are algebraically evaluated, based on the symmetry exhibited by Goldbach primes around their midpoint. Hybrid prime factorization is an effective way to represent, predict, compute, and analyze primes, expressed as algebraic functions. It is shown that the sequence of primes can be generated through an algebraic process with evolutionary properties. Since prime numbers do not follow any predetermined pattern, proving that they can be represented, computed and analyzed algebraically has important practical and theoretical ramifications.

Adjacency preserving maps on symmetric tensors

Let r and s be positive integers such that r⩾2. Let U and V be vector spaces over fields F and K, respectively, such that dim⁡U⩾3 and F has at least r+1 elements. In this paper, we characterize surjective maps ψ:SrU→SsV preserving adjacency in both directions on symmetric tensors of finite order, which generalizes Hua's fundamental theorem of geometry of symmetric matrices. We give examples showing the indispensability of the assumptions dim⁡U⩾3 and the cardinality |F|⩾r+1 in our result.

3-Uniform hypergraphs from vector spaces

The Fundamental Theorem of Projective Geometry states that, in a vector space, a permutation of vector lines preserving triples that span a vector plane is induced by a semi-linear automorphism. We consider a generalisation to triples of subspaces, not necessarily of the same dimension, spanning, or being contained in a subspace of fixed dimension. We determine all cases in which the permutation is necessarily induced by a semi-linear automorphism.

Fundamental Theorem of Asset Pricing under fixed and proportional costs in multi-asset setting and finite probability space

The Fundamental Theorem of Asset Pricing is extended to a market model over a finite probability space with many assets that can be exchanged into one another under combined fixed and proportional transaction costs. The absence of arbitrage in this setting is shown to be equivalent to the existence of a family of absolutely continuous single-step probability measures and a multi-dimensional martingale with respect to such a family.

The Fundamental Theorem of Arithmetic and q-series

We connect the fundamental theorem of arithmetic to the Euler-Nichole identity and discuss various useful q-identities.

The fundamental theorem of natural selection: the end of a story.

The direction of research in population genetics theory is currently, and correctly, retrospective, that is directed toward the past. What events in the past have led to the presently observed genetic constitution of a population? This direction is inspired, first, by the large volumes of genomic data now available and, second, by the success of the classical prospective theory in validating the Darwinian theory in terms of Mendelian genetics. However, the prospective theory should not be forgotten, and in that theory, perhaps the most interesting and certainly the most controversial, is Fisher's so-called "Fundamental Theorem of Natural Selection." This article describes the history and the current status of that theorem.

Submersions, immersions, and étale maps in diffeology

Although structural maps such as subductions and inductions appear naturally in diffeology, one of the challenges is providing suitable analogues for submersions, immersions, and étale maps (i.e., local diffeomorphisms) consistent with the classical versions of these maps between manifolds. In this paper, we consider diffeological or plotwise versions of submersions, immersions, and étale maps as an adaptation of these maps to diffeology by a nonlinear approach. We study their diffeological properties from different aspects in a systematic fashion with respect to the germs of plots.In order to characterize the considered maps from their linear behaviors, we introduce a class of diffeological spaces, so-called diffeological étale manifolds, which not only contains the usual manifolds but also includes irrational tori. We state and prove versions of the rank and implicit function theorems, as well as the fundamental theorem on flows in this class. As an application, we use the results of this work to facilitate the computations of the internal tangent spaces and diffeological dimensions in a few interesting cases.

Fundamental Theorems for Timelike Surfaces in the Minkowski 4-Space

In the present paper, we study timelike surfaces free of minimal points in the four-dimensional Minkowski space. For each such surface we introduce a geometrically determined pseudo-orthonormal frame field and writing the derivative formulas with respect to this moving frame field and using the integrability conditions, we obtain a system of six functions satisfying some natural conditions. In the general case, we prove a Fundamental Bonnet-type theorem (existence and uniqueness theorem) stating that these six functions, satisfying the natural conditions, determine the surface up to a motion. In some particular cases, we reduce the number of functions and give the fundamental theorems.

Noncommutative Poisson Boundaries, Ultraproducts, and Entropy

Abstract We construct the noncommutative Poisson boundaries of tracial von Neumann algebras through the ultraproducts of von Neumann algebras. As an application of this result, we complete the proof of Kaimanovich-Vershik’s fundamental theorems regarding noncommutative entropy. We also prove the Amenability-Trivial Boundary equivalence and Choquet-Deny-Type I equivalence for tracial von Neumann algebras.

Stabilizer Formalism for Operator Algebra Quantum Error Correction

We introduce a stabilizer formalism for the general quantum error correction framework called operator algebra quantum error correction (OAQEC), which generalizes Gottesman's formulation for traditional quantum error correcting codes (QEC) and Poulin's for operator quantum error correction and subsystem codes (OQEC). The construction generates hybrid classical-quantum stabilizer codes and we formulate a theorem that fully characterizes the Pauli errors that are correctable for a given code, generalizing the fundamental theorems for the QEC and OQEC stabilizer formalisms. We discover hybrid versions of the Bacon-Shor subsystem codes motivated by the formalism, and we apply the theorem to derive a result that gives the distance of such codes. We show how some recent hybrid subspace code constructions are captured by the formalism, and we also indicate how it extends to qudits.

Teaching Abstract Algebra Concretely via Embodiment

ABSTRACT We describe how an instructor integrated embodiment to teach the Fundamental Homomorphism Theorem (FHT) and preliminary concepts in an undergraduate abstract algebra course. The instructor’s use of embodiment reduced levels of abstraction for formal definitions, theorems, and proofs. The instructor’s simultaneous use of various forms of embodiment primed students for the formalism and symbolism, highlighted and disambiguated students’ referents, amplified students’ contributions to develop conceptual fluency, and linked students’ body form catchments to motivate the FHT. Our results offer practical implications for teaching by illustrating examples of how embodiment can assist in making abstract concepts concrete.