scholarly journals Generalization of Heron’s and Brahmagupta’s equalities to any cyclic polygon

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Enrico Laeng
Keyword(s):  
Explicit Formula ◽  
Symmetric Function ◽  
Explicit Solutions ◽  
Elementary Approach ◽  
The Real ◽  
Simple Geometry ◽  
Special Cases ◽  
Cyclic Polygon

AbstractIt is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having $$n>4$$ n > 4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality.

scholarly journals Explicit Solutions of the Extended Skorokhod Problems in Affine Transformations of Time-Dependent Strata

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Marek Slaby
Keyword(s):  
Coordinate System ◽  
Explicit Formula ◽  
Special Class ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Affine Transformations ◽  
Skorokhod Problem ◽  
Special Cases ◽  
Lipschitz Constants ◽  
Lipschitz Properties

The goal of this paper is to expand the explicit formula for the solutions of the Extended Skorokhod Problem developed earlier for a special class of constraining domains in ℝ n with orthogonal reflection fields. We examine how affine transformations convert solutions of the Extended Skorokhod Problem into solutions of the new problem for the transformed constraining system. We obtain an explicit formula for the solutions of the Extended Skorokhod Problem for any ℝ n - valued càdlàg function with the constraining set that changes in time and the reflection field naturally defined by any basis. The evolving constraining set is a region sandwiched between two graphs in the coordinate system generating the reflection field. We discuss the Lipschitz properties of the extended Skorokhod map and derive Lipschitz constants in special cases of constraining sets of this type.

Topic: Rare and Special Cases, The Real “Strange Cases”

Hernia ◽  
2015 ◽  
Vol 19 (S1) ◽  
pp. S317-S327 ◽  
Author(s):  
E. Zarrinkhoo ◽  
J. Miller ◽  
A. Walker ◽  
M. Weisman ◽  
S. Towfigh ◽  
...  
Keyword(s):  
The Real ◽  
Special Cases

scholarly journals Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

10.37236/734 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Combinatorial Nullstellensatz ◽  
Algebraic Solution ◽  
Explicit Solutions ◽  
Polynomial Map ◽  
Combinatorial Number Theory ◽  
Special Cases ◽  
The Matrix ◽  
Grid Points ◽  
Algebraic Solutions ◽  
Solvable Problems

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

The postulation of a multiple variety

1940 ◽  
Vol 36 (1) ◽  
pp. 27-33 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Explicit Formula ◽  
Complete Intersection ◽  
Trivial Case ◽  
Multiple Points ◽  
Ordinary Space ◽  
Large Order ◽  
Special Cases

1. The determination of an explicit formula for the postulation of a multiple variety of given characters for primals of sufficiently large order is one which seems to have received attention only in special cases. The postulation of a multiple curve for surfaces in ordinary space is known*, and Roth† has obtained the postulation of a multiple surface in [4], free from singularities, for primals in that space. Apart from these results, and the trivial case of isolated multiple points, the only general results seem to be those of Torelli‡ and Giambelli§ for the case in which the multiple variety is a Vh which is the complete intersection of r − h primals in [r].

scholarly journals (2n-1)-Point Ternary Approximating and Interpolating Subdivision Schemes

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Muhammad Aslam ◽  
Ghulam Mustafa ◽  
Abdul Ghaffar
Keyword(s):  
Explicit Formula ◽  
Error Bounds ◽  
Computational Cost ◽  
Subdivision Schemes ◽  
Special Cases ◽  
Better Than

We present an explicit formula which unifies the mask of(2n-1)-point ternary interpolating as well as approximating subdivision schemes. We observe that the odd point ternary interpolating and approximating schemes introduced by Lian (2009), Siddiqi and Rehan (2010, 2009) and Hassan and Dodgson (2003) are special cases of our proposed masks/schemes. Moreover, schemes introduced by Zheng et al. (2009) can easily be generated by our proposed masks. It is also proved from comparison that(2n-1)-point schemes are better than2n-scheme in the sense of computational cost, support and error bounds.

scholarly journals O cuidado moral e pastoral de nossas famílias em situações “especiais”

2019 ◽  
Vol 68 (271) ◽  
pp. 614
Author(s):  
Nilo Agostini
Keyword(s):  
Pastoral Care ◽  
Real Situation ◽  
The Real ◽  
Special Cases ◽  
The Family ◽  
A Priority ◽  
The Church

As famílias representam para a Igreja um setor prioritário, merecedor das melhores forças e de uma Pastoral Familiar realmente organizada. Uma redobrada atenção é dirigida às famílias em “situações especiais”, em favor das quais a Igreja é chamada a devotar uma solicitude pastoral ainda mais viva. O cuidado moral e pastoral começará por tomar em conta a “situação real” para, em seguida, levar a um itinerário progressivo, abrindo caminhos, apontando possibilidades, criando condições, evitando degradações, num empenho pastoral na linha do Bom Pastor. Contempla-se, neste texto, com brevidade, os casos especiais mais comuns na pastoral familiar.Abstract: Families represent a priority sector for the Church, one that deserves its best efforts and a thoroughly organized Family Pastoral. Redoubled attention is paid to families in “special situations” in whose benefit the Church is called to devote an even livelier part of its concern. Moral and pastoral care will start by taking into consideration the “real situation” and then will take a progressive path, opening new fronts, indicating possibilities, creating conditions, avoiding degradation, in the Good Shepherd’s type of pastoral effort. In this text, we will examine briefly some of the most common special cases in the family pastoral.

scholarly journals Quantum Hermite-Hadamard type inequalities for generalized strongly preinvex functions

2021 ◽  
Vol 6 (12) ◽  
pp. 13291-13310
Author(s):  
Humaira Kalsoom ◽  
◽  
Muhammad Amer Latif ◽  
Muhammad Idrees ◽  
Muhammad Arif ◽  
...  
Keyword(s):  
Real World ◽  
Higher Order ◽  
Integral Type ◽  
The Real ◽  
Quantum Calculus ◽  
Special Cases ◽  
Mathematical Problems ◽  
Preinvex Functions ◽  
Function Approximations ◽  
Integral Identities

<abstract><p>In accordance with the quantum calculus, the quantum Hermite-Hadamard type inequalities shown in recent findings provide improvements to quantum Hermite-Hadamard type inequalities. We acquire a new $ q{_{\kappa_1}} $-integral and $ q{^{\kappa_2}} $-integral identities, then employing these identities, we establish new quantum Hermite-Hadamard $ q{_{\kappa_1}} $-integral and $ q{^{\kappa_2}} $-integral type inequalities through generalized higher-order strongly preinvex and quasi-preinvex functions. The claim of our study has been graphically supported, and some special cases are provided as well. Finally, we present a comprehensive application of the newly obtained key results. Our outcomes from these new generalizations can be applied to evaluate several mathematical problems relating to applications in the real world. These new results are significant for improving integrated symmetrical function approximations or functions of some symmetry degree.</p></abstract>

Interconnected birth and death processes

1968 ◽  
Vol 5 (02) ◽  
pp. 334-349 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Limit Theorems ◽  
Probability Generating Function ◽  
Random Variables ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Transition Rates ◽  
Birth And Death Processes ◽  
Death Rates ◽  
Standard Methods ◽  
Special Cases

SummaryTwo cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f)gof the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution forg.This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends togasn → ∞in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λiandμi, i =1,2, interconnected linearly with transition rates v andδ(see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1=δ= 0, with the remaining rates time dependent and (ii) λ2=δ= 0, with the remaining rates constant, explicit solutions forgare obtained and studied.

Omitting types for stable ccc theories

1990 ◽  
Vol 55 (3) ◽  
pp. 1037-1047 ◽  
Author(s):  
Ludomir Newelski
Keyword(s):  
Real Line ◽  
Basic Knowledge ◽  
Complete Theory ◽  
The Real ◽  
Special Cases ◽  
Standard Set ◽  
Tarski Algebra ◽  
Independence Results ◽  
Omitting Types ◽  
Meager Sets

In this paper we investigate omitting types for a certain kind of stable theories which we call stable ccc theories. In Theorem 2.1 we improve Steinhorn's result from [St]. We prove also some independence results concerning omitting types. The main results presented in this paper were part of the author's Ph.D. thesis [N1].Throughout, we use the standard set-theoretic and model-theoretic notation, such as can be found for example in [Sh] or [M]. So in particular T is always a countable complete theory in the language L. We consider all models of T and all sets of parameters subsets of the monster model ℭ, which is very saturated. Ln(A) denotes the Lindenbaum-Tarski algebra of formulas with parameters from A and n free variables. We omit n in Ln(A) when n = 1 or when it is clear from the context what n is. If φ, ψ ∈ L(A) are consistent then we say that φ is below ψ if ψ⊢ψ. For a type p and a set A ⊆ ℭ, p(A) is the set of tuples of elements of A which satisfy p. Formulas are special cases of types. We say that a type p is isolated over A if, for some φ() ∈ L(A), φ() ⊢ p(x), i.e. φ isolates p. For a formula φ, [φ] denotes the class of types which contain φ. We assume that the reader is familiar with some basic knowledge of forking, as presented in [Sh, III] or [M].Throughout, we work in ZFC. and denote (countable) transitive models of ZFC. cov K is the minimal number of meager sets covering the real line R. In this paper we prove theorems showing connections between omitting types and the combinatorics of the real line. More results in this direction are presented in [N2] and [N3].

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