Search

Let K denote a nonsingular conic in the projective plane. Pascal's theorem says that, given six distinct points A, B, C, D, E, F on K , the three intersection points A E ∩ B F , A D ∩ C F , B D ∩ C E are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal's hexagrammum mysticum. [ABSTRACT FROM AUTHOR]

Pascal's wager faces serious criticisms and is generally considered unconvincing. We argue that it can make a comeback powered by an unlikely ally: postmodernism. If one denies the existence of objective facts (e.g. about God or His relation to the world), then various non-theological considerations should come to the fore when considering the rationality of religious commitment and the choice of education for one's children. In fact, we shall argue that, if one genuinely cares about one's children, then – in many Western countries – one cannot consistently be both secular and post-modernist. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Veronesean arc, the non-Veronesean arc by converting a point expressed in Cartesian coordinates to homogeneous coordinates in left nearfield plane of order 9 where a k--arc in a finite projective or affine plane is a set of k points no three of which are collinear. And also, we examine that whether founded complete (7,2)-Non-Veronesean arc satisfy Pascal's Theorem in the left nearfield projective plane of order 9. Six of (7,2)-Non-Veronesean arc's all points which are {1,4,11,21,35,38} points line on same conic. But it is determined that (7,2)-Non-Veronesean arc does not satisfy Pascal's Theorem. [ABSTRACT FROM AUTHOR]

Utilizing first principles calculations within PW91 exchange-correlation method, we investigated a boron sheet that exhibits related electronic properties. The 2- dimensional boron sheet is flattened and has an atomic structure where the pair cores of every three ordered hexagons within the hexagonal network are loaded up by extra atoms, which saves the triangular lattice symmetry. The boron sheet takes possession of intrinsic metal properties and the electronic bands are comparable to the bands of the graphene that are close to the Fermi level. The real and imaginary parts of the dielectric function show a metallic or semiconductor behaviour, depending on the electric field direction. [ABSTRACT FROM AUTHOR]

Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an involution point that is required to be one of the base points of the pencil. We generalise this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable. In the quartic construction the two involution points are required to be base points of the pencil of multiplicity 2. On the other hand, for the quadratic pencils the involution points can be any two distinct points in the plane (except for base points). We employ Pascal's theorem to show that the maps that preserve a quadratic pencil admit infinitely many symmetries. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where the two involution points go to infinity. We show by construction that each generalised Manin transformation can be brought to QRT form by a fractional affine transformation. We also specify classes of generalised Manin transformations which admit a root. [ABSTRACT FROM AUTHOR]

Let K denote a nonsingular conic in the complex projectiveplane. Given six distinct points a,b,c,d,e,f on K, Pascal's theorem saysthat the intersection points ab ∩ de, bc ∩ ef, cd ∩ af are collinear. The linecontaining them is called the Pascal line of the hexagonabcdef. We get sixtysuch lines by permuting the vertices. These lines satisfy some incidencetheorems, which lead to an extremely rich combinatorial structure calledthehexagrammum mysticum. In 1877, Veronese discovered a procedure tocreate an infinite sequence of such structures via successive mutations; thissequence is called themultimysticum. In this paper we prove a strong rigiditytheorem which shows that the multimysticum contains 300 projective rangesthat are absolutely invariant. The two interspersed sequences of cross-ratioswhich encode this invariance turn out to be the alternate convergents in thecontinued fraction expansions of 1±1/√3. [ABSTRACT FROM AUTHOR]

Problem 3845 in Crux Mathematicorum is a nice configuration about eight circles with many special cases. We sketch a proof of this problem and rename it as eight circles theorem. Using Miquel's six circles theorem, the bundle theorem and the eight circles theorem, we give a proof of the dual of the eight circles theorem and its converse. [ABSTRACT FROM AUTHOR]

As one of the classical theories of projective geometry, Pascal’s theorem is closely related to the conic, which is the image of a circle. A circle is the geometric element typically used as a template for camera calibration methods. In this work, we propose a calibration method based on Pascal’s theorem and its inference, in which a circle is used as the calibration template. The proposed calibration method can be applied to solve the image of the circular points via the Newton iteration method. The intrinsic parameters of the camera can then be determined based on the constraints of the images of the circular points and the image of the absolute conic. The results of simulations and experiments conducted verify the validity and feasibility of the proposed calibration method. [ABSTRACT FROM AUTHOR]

Given six points on a conic, Pascal's theorem gives rise to a configuration called the hexagrammum mysticum. It contains 20 Steiner points and 20 Cayley-Salmon lines. By a classical theorem due to von Staudt, the Steiner points fall into 10 conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. We show that the C-S lines are pairwise conjugate precisely when the original sextuple is tri-symmetric. The variety of tri-symmetric sextuples turns out to be arithmetically Cohen-Macaulay of codimension two. We determine its SL2-equivariant minimal resolution. [ABSTRACT FROM AUTHOR]

In this article we check, with the Mizar system [2], Pascal's theorem in the real projective plane (in projective geometry Pascal's theorem is also known as the Hexagrammum Mysticum Theorem). Pappus' theorem is a special case of a degenerate conic of two lines. For proving Pascal's theorem, we use the techniques developed in the section "Projective Proofs of Pappus' Theorem" in the chapter "Pappus' Theorem: Nine proofs and three variations" [11]. We also follow some ideas from Harrison's work. With HOL Light, he has the proof of Pascal's theorem. For a lemma, we use PROVER9 and OTT2MIZ by Josef Urban [12, 6, 7]. We note, that we don't use Skolem/Herbrand functions (see "Skolemization" in [1]). [ABSTRACT FROM AUTHOR]

Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a suitable first order initial value problem. The paper aims to confirm that the unifying character of this approach can also be applied to the construction of homogeneous Appell polynomials that are solutions of a generalized Cauchy–Riemann system in Euclidean spaces of arbitrary dimension. The result contributes to the development of techniques for polynomial approximation and interpolation in non-commutative Hypercomplex Function Theories with Clifford algebras. [ABSTRACT FROM AUTHOR]

Purpose Cardinal points are used for ray tracing through Gaussian systems. Anti-principal and anti-nodal points (which we shall refer to as the anti-cardinal points), along with the six familiar cardinal points, belong to a much larger set of special points. The purpose of this paper is to obtain a set of relationships and resulting equalities among the cardinal and anti-cardinal points and to illustrate them using Pascal's ring. Methods The methodology used relies on Gaussian optics and the transference T. We make use of two equations, obtained via the transference, which give the locations of the six cardinal and four anti-cardinal points with respect to the system. We obtain equalities among the cardinal and anti-cardinal points. We utilise Pascal's ring to illustrate which points depend on frequency and their displacement with change in frequency. Results Pascal described a memory schema in the shape of a hexagon for remembering equalities among the points and illustrating shifts in these points when an aspect of the system changes. We modify and extend Pascal's ring to include the anti-cardinal points. We make use of Pascal's ring extended to illustrate which points are dependent on the frequency of light and the direction of shift of the equalities with change in frequency. For the reduced eye the principal and nodal points are independent of frequency, but the focal points and the anti-cardinal points depend on frequency. For Le Grand's four-surface model eye all six cardinal and four anti-cardinal points depend on frequency. This has implications for definitions, particularly of chromatic aberrations of the eye, that make use of cardinal points and that themselves depend on frequency. Conclusions Pascal's ring and Pascal's ring extended are novel memory schema for remembering the equalities among the cardinal and anti-cardinal points. The rings are useful for illustrating changes among the equalities and direction of shift of points when an aspect of a system changes. Care should be taken when defining concepts that rely on cardinal points that depend on frequency. [ABSTRACT FROM AUTHOR]

Existence of non-abelian representations of the near hexagon Q(5, 2) ⊗ Q(5, 2)In [5], a new combinatorial model with four types of points and nine types of lines of the slim dense near hexagon Q(5, 2)⊗Q(5, 2) was provided and it was then used to construct a non-abelain representation of Q(5, 2) ⊗ Q(5, 2) in the extraspecial 2-group 21+18-. In this paper, we give a direct proof for the existence of a non-abelian representation of Q(5, 2) ⊗ Q(5, 2) in 21+18-. [ABSTRACT FROM AUTHOR]

The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem, poles and polars. The axioms used for the synthetic treatment are constructive versions of the traditional axioms. The analytic construction is used to verify the consistency of the axiom system; it is based on the usual model in three-dimensional Euclidean space, using only constructive properties of the real numbers. The methods of strict constructivism, following principles put forward by Errett Bishop, reveal the hidden constructive content of a portion of classical geometry. A number of open problems remain for future studies. [ABSTRACT FROM AUTHOR]

In Sturmfels and Whiteley (J Symb Comput 11(5):439-453, ), it is proven that any multihomogenous bracket polynomial with integer coefficients can be interpreted by a ruler construction when introducing simple non-degeneracy conditions. This problem is called (generalized) Cayley factorization. This allows for geometrically interpreting many projective invariant properties. We reprove the above statement in rank 3 giving a better bound on the size of the non-degeneracy conditions. The constant factor in the bound is essentially reduced from 105 to 9. The algorithm described is concise enough to be implemented on a computer. With this algorithm, interpreting a common condition for six points to lie on a common conic is very close to Pascal's construction (see Apel, in: The geometry of brackets and the area principle, Dissertation, TU München, ). [ABSTRACT FROM AUTHOR]

The article discusses proof of Poncelet's theorem using real projective plane based on Pascal's theorem. Topics discussed include Poncelet's theorem related to n-sided polygon in nondegenerate conics and vertices, intersections of circles, Abelian integrals theory, interpretation of elliptic billiards, Poncelet's theorem being consequence of Pascal's theorem and a theorem proving sides of triangles being tangent if triangles are inscribed in a conic.

Projective invariants are not only important objects in mathematics especially in geometry, but also widely used in many practical applications such as in computer vision and object recognition. In this work, we show a projective invariant named as characteristic number, from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property, two high-dimensional generalizations of Pascal's theorem are given, one establishing the connection of hypersurfaces of distinct degrees, and the other concerned with the intersections of a hypersurface and a simplex. [ABSTRACT FROM AUTHOR]

The article offers information on a generalization of both Pascal's Theorem and its converse Braikenridge-Maclaudn Theorem. It mentions that a simple dimension count was conducted which revealed that approximately all curves of high degree are not d-degree constructible, and all curves of degree three or less are d-constructible. It also states that using the group law on elliptic curves allows to show that d-construction is not dense in high degrees.

Abstract: A result on the ill-conditioning of Pascal matrices is proved. However, it is shown that bidiagonal factorizations of Pascal matrices can be applied to perform computations with high relative accuracy. [Copyright &y& Elsevier]

The article focuses on the concept of classical mysticum by French mathematician Blaise Pascal. It states that the Pascal lines are defined by the classical mysticum which starts from the 45 meeting points. It mentions that the Polar Pascal theorem is hard to prove because the standard method of applying Desargues’s theorem to earlier lines and points does not work.

We solve the currently smallest open case in the 1976 problem of Molluzzo on , namely the case . This amounts to constructing, for all positive integers n congruent to 0 or 7 mod 8, a sequence of integers modulo 4 of length n generating, by Pascal's rule, a Steinhaus triangle containing 0, 1, 2, 3 with equal multiplicities. [ABSTRACT FROM AUTHOR]

By combining the extended Cauchy double alternant with a recent theorem due to Chu (2009), we review some determinantal evaluations for Pascal-like matrices due to Krattenthaler (2002). [ABSTRACT FROM AUTHOR]

The article presents a thought experiment that aims to discover Pascal's Hexagon Theorem, in response to the complaint of German philosopher G. W. F. Hegel regarding the ad hoc quality of geometrical proofs. It focuses on elementary considerations that lead to the formation of the theorem through natural questions and plausible steps. It suggests that the thought experiment does not give any new result, but fills the steps of a thought sequence of discovering the theorem.

The paper proposes a formal derivation of recurrent equations describing the occupancy distribution in the full-availability group with multirate Binomial-Poisson-Pascal (BPP) traffic. The paper presents an effective algorithm for determining the occupancy distribution on the basis of derived recurrent equations and for the determination of the blocking probability as well as the loss probability of calls of particular classes of traffic offered to the system. A proof of the convergence of the iterative process of estimating the average number of busy traffic sources of particular classes is also given in the paper. [ABSTRACT FROM AUTHOR]

In this article we present a simple and elegant algebraic proof of Pascal's hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic. [ABSTRACT FROM AUTHOR]

The article discusses some apparently new theorems in projective geometry that are similar in spirit to Pascal's Theorem and Briancon's Theorem. It is informed that Pappus's Theorem goes back about 1700 years. In 1639, Blaise Pascal discovered a generalization in which there was six points that lie on a conic section.

Abstract: This article deals with an unknown mathematical manuscript to be found in the Recueil Original of the manuscript of the Pensées. I give a technical description of the original manuscript, followed by a mathematical analysis of the proposition proved by Pascal. The interest of this study lies in the fact that this theorem is the only handwritten draft remaining in Pascal’s mathematical works. [ABSTRACT FROM AUTHOR]

Let L(n)p be the Pascal lower triangular matrix with coefficients (i j) (mod p); 0 ≤ i; j < n. In this paper, we found the inverse of L(n)p modulo p. In fact, we generalize a result due to David Callan [4]. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the notion of the Catalan matrix whose non-zero elements are expressions which contain the Catalan numbers arranged into a lower triangular Toeplitz matrix. Inverse of the Catalan matrix is derived. Correlations between the matrix and the generalized Pascal matrix are considered. Some combinatorial identities involving Catalan numbers, binomial coefficients and the generalized hypergeometric function are derived using these correlations. Moreover, an additional explicit representation of the Catalan number, as well as an explicit representation of the sum of the first m Catalan numbers are given. [Copyright &y& Elsevier]

`Riders' in geometry are always a pleasure to tackle , and this pleasure is doubled when on e finds connections between plane geometry and algebra . This three -part article is about such connection s. In Parts 1 and 2 , we explore some connections between plane geometry and the algebra of conics and cubics; in P art 1 we give algebraic proofs of results such as Pascal's Theorem and the Butterfly. Theorem , and in Part 2 we study some group theoretic and number theoretic aspects of cubic curves. In Part 3 we look at the role of mappings and transformation groups in plane geometry. [ABSTRACT FROM AUTHOR]

Abstract   Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon. [ABSTRACT FROM AUTHOR]

We construct pairs of reciprocal relations which contain as the coefficients three-index combinatorial numbers. We introduce three-index generalisations of a series of known combinatorial numbers. [ABSTRACT FROM AUTHOR]

The article discusses the mathematical principles and applications concerning the families of plane algebraic curves that contain a special set of points. The case of curves in cages is associated with the elementary methods that presume the familiarity with algebraic geometry and geometrical applications. These applications can be considered as natural generalizations of classical theorems in projective geometry and are associated with familiarity with Pascal's theorem and its comparison with the classical Pascal diagram.

Abstract: We determine all dense near -gons, , in which each hex is isomorphic to either the dual polar space , the product near hexagon or a glued near hexagon of type . [Copyright &y& Elsevier]

Abstract: In this paper, we present a fast algorithm of the complexity O(n log n) for solving the linear systems with coefficient matrix of the Pascal type, which is faster than the algorithm of the complexity O(n 2) given in the paper [M.E.A. El-Mikkawy, On solving linear systems of the Pascal type. Applied mathematics and computation 136 (2003) 195–202]. An application in solving non-homogeneous differential equation with constant coefficients is given. [Copyright &y& Elsevier]

In 1963 Fisher and Stephenson (Phys. Rev. (2) 132 (1963) 1411) conjectured that the monomer–monomer correlation on the square lattice is rotationally invariant. In this paper we prove a closely related statement on the hexagonal lattice. Namely, we consider correlations of two quadromers (four-vertex subgraphs consisting of a monomer and its three neighbors) and show that they are rotationally invariant. [Copyright &y& Elsevier]

Focuses on the inversion of the Pascal matrix plus one. Introduction to the Pascal matrix; Inversion of Pascal's matrix; Use of polylogarithm functions.

By a transfer principle, Pascal's Theorem is equivalent to a theorem about point pairs on the real line. It appears that Pascal's Theorem is equivalent to the vanishing of a common invariant of six quadratic forms. Using the q-deformed invariant theory of Leitenberger ( J. Algebra 222 (1999), 82), we construct corresponding quantum invariants by a computer calculation. [ABSTRACT FROM AUTHOR]