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Labs Surface
In mathematics, the Labs septic surface is a degree-7 ( septic) nodal surface with 99 nodes found by . As of 2015, it has the largest known number of nodes of a degree-7 surface, though this number is still less than the best known upper bound of 104 nodes given by .The upper bound in degree 7 given by is 106. See also * Barth surface *Endrass surface *Sarti surface In algebraic geometry, a Sarti surface is a degree-12 nodal surface with 600 nodes, found by . The maximal possible number of nodes of a degree-12 surface is not known (as of 2015), though Yoichi Miyaoka showed that it is at most 645. Sarti has ... * Togliatti surface References * * * External links * Algebraic surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Septic Equation
In algebra, a septic equation is an equation of the form :ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\, where . A septic function is a function of the form :f(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\, where . In other words, it is a polynomial of degree seven. If , then ''f'' is a sextic function (), quintic function (), etc. The equation may be obtained from the function by setting . The ''coefficients'' may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. Because they have an odd degree, septic functions appear similar to quintic or cubic function when graphed, except they may possess additional local maxima and local minima (up to three maxima and three minima). The derivative of a septic function is a sextic function. Solvable septics Some seventh degree equations can be solved by factorizing into radicals, but other septics cannot. Évariste Galois developed techniques for determining whether a given equation cou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nodal Surface
In algebraic geometry, a nodal surface is a surface in (usually complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...) projective space whose only singularities are nodes. A major problem about them is to find the maximum number of nodes of a nodal surface of given degree. The following table gives some known upper and lower bounds for the maximal number of nodes on a complex surface of given degree. In degree 7, 9, 11, and 13, the upper bound is given by , which is better than the one by . See also * Algebraic surface References * * * *{{citation , mr=3124329 , doi=10.1016/j.crma.2013.09.009 , last=Escudero , first=Juan García , title=On a family of complex algebraic surfaces of degree 3''n'' , journal=C. R. Math. Acad. Sci. Paris , volume=351 , year=2013 , i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Point Of An Algebraic Variety
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth. Definition A plane curve defined by an implicit equation :F(x,y)=0, where is a smooth function is said to be ''singular'' at a point if the Taylor series of has order at least at this point. The reason for this is that, in differential calculus, the tangent at the point of such a curve is defined by the equation :(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0)=0, whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D Model Of Real Points Of The Labs Septic
3-D, 3D, or 3d may refer to: Science, technology, and mathematics Relating to three-dimensionality * Three-dimensional space ** 3D computer graphics, computer graphics that use a three-dimensional representation of geometric data ** 3D film, a motion picture that gives the illusion of three-dimensional perception ** 3D modeling, developing a representation of any three-dimensional surface or object ** 3D printing, making a three-dimensional solid object of a shape from a digital model ** 3D display, a type of information display that conveys depth to the viewer ** 3D television, television that conveys depth perception to the viewer ** Stereoscopy, any technique capable of recording three-dimensional visual information or creating the illusion of depth in an image Other uses in science and technology or commercial products * 3D projection * 3D rendering * 3D scanning, making a digital representation of three-dimensional objects * 3D video game (other) * 3-D Secure, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barth Surface
__NOTOC__ In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points. For degree 6 surfaces in P3, showed that 65 is the maximum number of double points possible. The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum number of double points possible. Informal accounting of the 65 ordinary double points of the Barth Sextic The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes). Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly tetrahedral shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular icosidodecahedron. To these 30 icosidodeca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endrass Surface
In algebraic geometry, an Endrass surface is a nodal surface of degree 8 with 168 real nodes, found by . , it remained the record-holder for the most number of real nodes for its degree; however, the best proven upper bound, 174, does not match the lower bound given by this surface. See also *Barth surface * Sarti surface *Togliatti surface In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 nodes. The first examples were constructed by . proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to b ... References {{reflist Algebraic surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sarti Surface
In algebraic geometry, a Sarti surface is a degree-12 nodal surface with 600 nodes, found by . The maximal possible number of nodes of a degree-12 surface is not known (as of 2015), though Yoichi Miyaoka showed that it is at most 645. Sarti has also found sextic, octic and dodectic nodal surfaces with high numbers of nodes and high degrees of symmetry. File:Sarti sextic 48 A.png, Sextic with 48 node File:Sarti sextic 48 (Stabchen).png, Sextic with 48 node File:Sarti's Octic with 72.png, Octic with 72 nodes File:Sarti's octic with 144 nodes.png, Octic with 144 nodes File:Sarti dodectic 360.png, Dodectic surface with 360 nodes File:3D model of Sarti surface.stl, 3D model of Sarti surface See also *Nodal surface In algebraic geometry, a nodal surface is a surface in (usually complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex ... References * * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Togliatti Surface
In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 Node (algebraic geometry), nodes. The first examples were constructed by . proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to be optimal. See also *Barth surface *Endrass surface *Sarti surface *List of algebraic surfaces References *. *. External links * * Algebraic surfaces Complex surfaces {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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