scholarly journals The Hodge conjecture: The complications of understanding the shape of geometric spaces

2018 ◽  
pp. 51
Author(s):  
Vicente Muñoz Velázquez
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Natural Condition ◽  
Mathematical Physics ◽  
Complex Numbers ◽  
Hodge Conjecture ◽  
Real Numbers ◽  
Geometric Objects ◽  
Complex Submanifolds ◽  
Geometry Analysis

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

scholarly journals THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

2018 ◽  
Vol 83 (04) ◽  
pp. 1667-1679
Author(s):  
MATÍAS MENNI
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Algebraic Topology ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Double Negation ◽  
Synthetic Differential Geometry ◽  
Simplicial Sets ◽  
Image Position

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

Encircling Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Nineteenth Century ◽  
Real Number ◽  
Traveling Salesman Problem ◽  
Complex Number ◽  
Traveling Salesman ◽  
Complex Numbers ◽  
Real Numbers ◽  
Hamiltonian Graphs ◽  
The World ◽  
The Traveling Salesman Problem

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

Unbounded Positive Definite Functions

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart
Keyword(s):  
Abelian Group ◽  
Positive Measure ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Definite Function ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

scholarly journals Non-standard Aspects of Fibonacci Type Sequences

KoG ◽  
2017 ◽  
pp. 26-34
Author(s):  
Gunter Weiss
Keyword(s):  
Number Field ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
General Knowledge ◽  
Real Numbers ◽  
Number Series ◽  
Golden Mean ◽  
Quadratic Equations ◽  
Geometric Objects ◽  
Art And Architecture

Fibonacci sequence and the limit of the quotient of adjacent Fibonacci numbers, namely the Golden Mean, belong to general knowledge of almost anybody, not only of mathematicians and geometers. There were several attempts to generalize these fundamental concepts which also found applications in art and architecture, as e.g. number series and quadratic equations leading to the so-called ``Metallic means'' by V. de Spinadel [8] or the cubic ``plastic number'' by van der Laan [5] resp. the ``cubi ratio'' by L. Rosenbusch [7]. The mentioned generalisations consider series of integers or real numbers. ``Non-standard aspects'' now mean generalisations with respect to a given number field or ring as well as visualisations of the resulting geometric objects. Another aspect concerns Fibonacci type resp. Padovan type combinations of given start objects. Here it turns out that the concept ``Golden Mean'' or ``van der Laan Mean'' also makes sense for vectors, matrices and mappings.

scholarly journals BRACKETS

2013 ◽  
Vol 10 (08) ◽  
pp. 1360001 ◽  
Author(s):  
JANUSZ GRABOWSKI
Keyword(s):  
Differential Geometry ◽  
Mathematical Physics ◽  
New Concepts

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the classical, as well as in the supergeometric setting. The review is supplemented by some new concepts and examples.

scholarly journals Handling of Complex Numbers in the CHProgramming Language

1993 ◽  
Vol 2 (3) ◽  
pp. 77-106 ◽  
Author(s):  
Harry H. Cheng
Keyword(s):  
Variable Number ◽  
Language Design ◽  
Complex Numbers ◽  
Computer Language ◽  
Real Numbers ◽  
Mathematical Functions ◽  
Complex Arithmetic ◽  
Complex Functions ◽  
Complex Features ◽  
Branch Cuts

The handling of complex numbers in the CHprogramming language will be described in this paper. Complex is a built-in data type in CH. The I/O, arithmetic and relational operations, and built-in mathematical functions are defined for both regular complex numbers and complex metanumbers of ComplexZero, Complexlnf, and ComplexNaN. Due to polymorphism, the syntax of complex arithmetic and relational operations and built-in mathematical functions are the same as those for real numbers. Besides polymorphism, the built-in mathematical functions are implemented with a variable number of arguments that greatly simplify computations of different branches of multiple-valued complex functions. The valid lvalues related to complex numbers are defined. Rationales for the design of complex features in CHare discussed from language design, implementation, and application points of views. Sample CHprograms show that a computer language that does not distinguish the sign of zeros in complex numbers can also handle the branch cuts of multiple-valued complex functions effectively so long as it is appropriately designed and implemented.

ITERATIVE GEOMETRIC STRUCTURES

2010 ◽  
Vol 07 (07) ◽  
pp. 1103-1114 ◽  
Author(s):  
GABRIEL BERCU ◽  
CLAUDIU CORCODEL ◽  
MIHAI POSTOLACHE
Keyword(s):  
Differential Geometry ◽  
Special Functions ◽  
Riemannian Metrics ◽  
General Setting ◽  
Mathematical Physics ◽  
Geometric Structures ◽  
Geometric Characteristics ◽  
Geometrical Characteristics ◽  
Metric Connection

In this work, we propose a study of geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry. Then we find geometrical characteristics of some ODE or PDE of Mathematical Physics. While Sec. 1 contains the general setting, Secs. 2–5 contain our results. In Sec. 2, we introduce a Hessian structure having the same connection as the initial metric. In Sec. 3, we initiate a study on iterative 2D Hessian structures. In Sec. 4, we find pairs (metric, connection) generated by special functions. In Sec. 5, we find geometric characteristics of a PDE.

Didactic Features of Studying the Real Numbers in the School Course of Mathematics

2017 ◽  
Vol 6 (1) ◽  
pp. 65-70
Author(s):  
Алексеенко ◽  
A. Alekseenko ◽  
Лихачева ◽  
M. Likhacheva
Keyword(s):  
Real Number ◽  
Complex Numbers ◽  
Algebraic Structures ◽  
Real Numbers ◽  
Irrational Numbers ◽  
The Real ◽  
Clear Idea

The article is devoted to the study of the peculiarities of real numbers in the discipline "Algebra and analysis" in the secondary school. The theme of "Real numbers" is not easy to understand and often causes difficulties for students. However, the study of this topic is now being given enough attention and time. The consequence is a lack of understanding of students and school-leavers, what constitutes the real numbers, irrational numbers. At the same time the notion of a real number is required for further successful study of mathematics. To improve the efficiency of studying the topic and form a clear idea about the different numbers offered to add significantly to the material of modern textbooks, increase the number of hours in the study of real numbers, as well as to include in the school course of algebra topics "Complex numbers" and "Algebraic structures".

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