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Optimization performance test
Sphere function of two variables In mathematical optimization, the sphere function is a convex function used as a performance test problem for optimization
Sphere_function
Model of the extended complex plane plus a point at infinity
any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical
Riemann_sphere
Special mathematical functions defined on the surface of a sphere
and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential
Spherical_harmonics
Set of points equidistant from a center
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'ball') is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that
Sphere
Generalized sphere of dimension n (mathematics)
In mathematics, an n-sphere or hypersphere is an n {\displaystyle n} -dimensional generalization of the 1 {\displaystyle 1} -dimensional circle and
N-sphere
Theorem in differential topology
even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in ℝ3 to every point p on a sphere such that
Hairy_ball_theorem
Class of mathematical function
a meromorphic function is the same as a holomorphic function that maps to the Riemann sphere and which is not the constant function equal to ∞. The
Meromorphic_function
Counterintuitive mathematical object
Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function is
Pathological_(mathematics)
Hypothetical megastructure around a star
A Dyson sphere is a hypothetical megastructure that encompasses a star and captures a large percentage of its power output. The concept is a thought experiment
Dyson_sphere
Geometric representation of the complex numbers
surface of a sphere. Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides
Complex_plane
Mathematical object
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space
3-sphere
Volume space bounded by a sphere
bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball
Ball_(mathematics)
One-dimensional complex manifold
can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include graphs of multivalued functions such as z {\displaystyle
Riemann_surface
Concept in complex analysis
point at infinity is called the Riemann sphere. If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros
Zeros_and_poles
Optical component
An integrating sphere (also known as an Ulbricht sphere) is an optical component consisting of a hollow spherical cavity with its interior covered with
Integrating_sphere
3D computer graphics rendering method
some function at each step. For example, in volume ray casting the function would access data points from a 3D scan. In Sphere tracing, the function estimates
Ray_marching
World War II era Pan-Asian union under the Empire of Japan
The Greater East Asia Co-Prosperity Sphere (Japanese: 大東亜共栄圏, Hepburn: Dai Tōa Kyōeiken), also known as the GEACPS, was a pan-Asian union that the Empire
Greater East Asia Co-Prosperity Sphere
Greater_East_Asia_Co-Prosperity_Sphere
Sphere was a blog search engine. The Sphere search engine delivered blog posts based on algorithms that combine semantic matching with authority factors
Sphere_(website)
Functions used to evaluate optimization algorithms
In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as convergence
Test functions for optimization
Test_functions_for_optimization
Topics referred to by the same term
three-dimensional three spheres inequality, a bound of a harmonic function on a sphere Gankyil § Three Spheres, a concept in some schools of Buddhism Trailokya, a cosmological
Three_spheres
Geometrical structure
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical
Sphere_packing
Formula for the great-circle distance between two points on a sphere
. The haversine function computes half a versine of the angle θ, or the squares of half chord of the angle on a unit circle (sphere). It is related to
Haversine_formula
Extension of the factorial function
complex-valued gamma function is undefined for non-positive integers, but in these cases the value can be defined in the Riemann sphere as ∞ {\displaystyle
Gamma_function
Topological space that locally resembles Euclidean space
Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The
Manifold
the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting
Fuzzy_sphere
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Elements of some cosmological models
The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus
Celestial_spheres
Equation for the velocity of a body in viscous fluid
numbers of the Navier–Stokes equations. The force of viscosity on a small sphere moving through a viscous fluid is given by: F → d = − 6 π μ R v → {\displaystyle
Stokes's_law
Scattering of an electromagnetic plane wave by a sphere
describes the scattering of an electromagnetic plane wave by a homogeneous sphere. The solution takes the form of an infinite series of spherical multipole
Mie_scattering
Mathematical technique
the double Fourier sphere (DFS) method is a technique that transforms a function defined on the surface of the sphere to a function defined on a rectangular
Double_Fourier_sphere_method
mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point
Spherical_mean
Mathematical concept
_{i=1}^{n}x_{i}=1} is a hyperbolic affine sphere centered at the origin, even though it is not a quadric. If ƒ is a smooth function on the plane and the determinant
Affine_sphere
Ratio of inertial to viscous forces acting on a liquid
describe spheres or circles, but one is chosen by convention. For aircraft or ships, the length or width can be used. For flow in a pipe, or for a sphere moving
Reynolds_number
Retarding force on a body moving in a fluid
is, in general, a function of the orientation of the flow with respect to the object (apart from symmetrical objects like a sphere). Under the assumption
Drag_(physics)
Area in social life with political ramifications
The public sphere (German: Öffentlichkeit) is an area in social life where individuals can come together to freely discuss and identify societal problems
Public_sphere
Type of function in mathematics
complement in the Riemann sphere of any simple arc connecting 0 {\displaystyle 0} to ∞ {\displaystyle \infty } . Power functions are analytic everywhere
Analytic_function
Concept in neo-Calvinism
neo-Calvinism, sphere sovereignty (Dutch: soevereiniteit in eigen kring), also known as differentiated responsibility, is the concept that each sphere (or sector)
Sphere_sovereignty
Model particles in statistical mechanics
the value of the radial distribution function, g ( r ) {\displaystyle g(r)} , evaluated at the surface of a sphere, g ( σ ) = 1 − 1 2 η ( 1 − η ) 3 . {\displaystyle
Hard_spheres
Statement on the gravitational attraction of spherical bodies
provided the density at any point inside the sphere is a function only of its distance from the center of the sphere. Although the following are completely
Shell_theorem
spheres are transformed into spheres, and the exterior of a sphere is transformed to the interior, and vice versa. The Kelvin transform of a function
Kelvin_transform
Sphere with radius one, usually centered on the origin of the space
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space.
Unit_sphere
Ratio of polynomial functions
Q(z)\neq 0} . Every rational function can be naturally extended to a function whose domain and range are the whole Riemann sphere, i.e., a rational mapping
Rational_function
Dimensionless parameter to quantify fluid resistance
area, depending on where the cross-section is measured. For example, for a sphere A = π r 2 {\displaystyle A=\pi r^{2}} (note this is not the surface area
Drag_coefficient
Mathematical description of quantum state
orthonormal set). The square integrable functions on the unit sphere S2 is a Hilbert space. The basis functions in this case are the spherical harmonics
Wave_function
Operator generalizing the Laplacian in differential geometry
sphere as isometrically embedded into Rn as the unit sphere centred at the origin. Then for a function f on Sn−1, the spherical Laplacian is defined by Δ
Laplace–Beltrami_operator
Study of angle-preserving transformations of a geometric space
a mapping into N+. The function κ is an arbitrary choice of conformal scale. A representative Riemannian metric on the sphere is a metric that is proportional
Conformal_geometry
Analytic function in mathematics
complex infinity on the Riemann sphere the zeta function has an essential singularity. For sums involving the zeta function at integer and half-integer values
Riemann_zeta_function
Areas historically influenced by Chinese culture
scripts. The Sinosphere, also known as the Chinese cultural sphere, East Asian cultural sphere, or the Sinic world, encompasses multiple countries in East
Sinosphere
Mapping which preserves all topological properties of a given space
or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are
Homeomorphism
Mathematical concept in algebraic geometry
operations of addition and multiplication of functions, this is a field in the sense of algebra. For the Riemann sphere, which is the variety P 1 {\displaystyle
Function field of an algebraic variety
Function_field_of_an_algebraic_variety
Assignment of a vector to each point in a subset of Euclidean space
(homeomorphic to the (n-1)-sphere) S around the zero, so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimension n − 1
Vector_field
Spherical geometry analog of a straight line
intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles
Great_circle
Particular mapping that projects a sphere onto a plane
smooth, bijective function from the entire sphere except the center of projection to the entire plane. It maps circles on the sphere to circles or lines
Stereographic_projection
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from
Exotic_sphere
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Curve whose range contains the unit square
endpoints) is a continuous function whose domain is the unit interval [0, 1]. In the most general form, the range of such a function may lie in an arbitrary
Space-filling_curve
Shape formed from points common to other shapes
Other types of geometric intersection include: Line–plane intersection Line–sphere intersection Intersection of a polyhedron with a line Line segment intersection
Intersection_(geometry)
Seven mathematical problems with a US$1 million prize for each solution
its name to them". In the field of geometric topology, a two-dimensional sphere is characterized by the fact that it is the only closed and simply-connected
Millennium_Prize_Problems
Fourier extension operator is an operator that takes a function defined on the surface of the unit sphere in R n {\displaystyle \mathbb {R} ^{n}} and applies
Fourier_extension_operator
American–Canadian animated television series
Cities PBS (TPT) and Sphere Media. The show is set in a gaming environment and is designed to help children develop executive function skills and learn about
Skillsville
Second-order partial differential equation
{a^{2}}{\rho }}.\,} Note that if P is inside the sphere, then P′ will be outside the sphere. The Green's function is then given by 1 4 π R − a 4 π ρ R ′ , {\displaystyle
Laplace's_equation
Geometrical concept relating area and volume
method resembling Cavalieri's principle, was able to find the volume of a sphere given the volumes of a cone and cylinder in his work The Method of Mechanical
Cavalieri's_principle
Point where function's value is zero
unit m {\displaystyle m} -sphere in R m + 1 {\displaystyle \mathbb {R} ^{m+1}} is the zero set of the real-valued function f ( x ) = ‖ x ‖ 2 − 1 {\displaystyle
Zero_of_a_function
mathematics, the three spheres inequality bounds the L 2 {\displaystyle L^{2}} norm of an harmonic function on a given sphere in terms of the L 2 {\displaystyle
Three_spheres_inequality
Size of a mathematical ball
or double factorial function. The volume can also be expressed in terms of A n {\displaystyle A_{n}} , the area of the unit n-sphere. The first volumes
Volume_of_an_n-ball
Mathematical algorithm
in some geometries by using Green's Functions First Passage method: one can change the geometry of the "spheres" when close enough to the border, so
Walk-on-spheres_method
Analyzes the topology of a manifold by studying differentiable functions on that manifold
studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will
Morse_theory
Distance from origin of tangent hyperplanes
valued function is the (convex) indicator function of a compact convex set. Many authors restrict the support function to the Euclidean unit sphere and consider
Support_function
Cylindrical conformal map projection
sphere, reaching a minimum at the contact circle. This is sometimes visualized as a projection onto a cylinder which is secant to (cuts) the sphere,
Mercator_projection
Complex exponential in terms of sine and cosine
fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x
Euler's_formula
Function used in computer graphics
interpolation, commonly abbreviated slerp, is a function which interpolates between two points on a sphere, such that spherical distance from the starting
Spherical linear interpolation
Spherical_linear_interpolation
How spheres of various dimensions can wrap around each other
mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples
Homotopy_groups_of_spheres
Branch of mathematics
tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind of
Topology
Section of a sphere
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i
Spherical_cap
Numerical optimization algorithm
(GECCO). T. Schaul (2012). Natural Evolution Strategies Converge on Sphere Functions. Genetic and Evolutionary Computation Conference (GECCO). Collection
Natural_evolution_strategy
Open source software suite
defined functions (UDFs) instead of the map and reduce functions. A UDF can be either a map function or a reduce function, or even others. Sphere can manipulate
Sector/Sphere
Special functions
spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics
Spin-weighted spherical harmonics
Spin-weighted_spherical_harmonics
Methods of calculating definite integrals
integration rules for a variety of weighting functions are given in the monograph by Stroud. Integration on the sphere has been reviewed by Hesse et al. (2015)
Numerical_integration
Strength of an object's radar echo
reflecting sphere that would produce the same strength reflection as would the object in question. (Bigger sizes of this imaginary sphere would produce
Radar_cross_section
Continuous surjection satisfying a local triviality condition
given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle. The theory of fibered spaces, of which
Fiber_bundle
Software
WebSphere Application Server (WAS) is a software product that performs the role of a web application server. More specifically, it is a software framework
IBM WebSphere Application Server
IBM_WebSphere_Application_Server
Function with a repeating pattern
Easily discernible pitch Double Fourier sphere method – Mathematical technique Doubly periodic function – Function with two complex number "periods" Fourier
Periodic_function
Function that is holomorphic on the whole complex plane
bounded entire function must be constant. As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere is constant.
Entire_function
set, and not simply a sphere as in the case of the spherical contact distribution function. Spherical contact distribution functions are used in the study
Spherical contact distribution function
Spherical_contact_distribution_function
Environment mapping technique
reflection of a mirrored sphere onto a plane using an orthographic projection. The resulting 2D texture encodes incident light as a function of direction and
Sphere_mapping
Measure in 3-dimensional geometry
One steradian corresponds to one unit of area (of any shape) on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would
Solid_angle
Integral transform
transform) is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1911, based on the work
Funk_transform
Function defined by a hypergeometric series
characterised (on the Riemann sphere) by its three regular singularities. The cases where the solutions are algebraic functions were found by Hermann Schwarz
Hypergeometric_function
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
many-to-one continuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped from a distinct
Hopf_fibration
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
Type of fluid flow
well-known methods for linear differential equations. The primary Green's function of Stokes flow is the Stokeslet, which is associated with a singular point
Stokes_flow
Physical law relating heat loss to temperature difference
gradients within the sphere become important, even though the sphere material is a good conductor. Equivalently, if the sphere is made of a thermally
Newton's_law_of_cooling
Ukrainian mathematician (born 1984)
born 2 December 1984) is a Ukrainian mathematician known for her work in sphere packing. She is a full professor and Chair of Number Theory at the Institute
Maryna_Viazovska
Stochastic process generalizing Brownian motion
\lfloor nt\rfloor }\xi _{k},\qquad t\in [0,1].} This is a random step function. Increments of Wn are independent because the ξ k {\textstyle \xi _{k}}
Wiener_process
Real function on a Euclidean space whose value depends only on distance from the origin
for every test function φ and rotation ρ. Given any (locally integrable) function f, its radial part is given by averaging over spheres centered at the
Radial_function
Calculation technique for classical electrostatics
)}{p^{2}}}\right]} The method of images for a sphere leads directly to the method of inversion. If we have a harmonic function of position Φ ( r , θ , ϕ ) {\displaystyle
Method_of_image_charges
Rational function of the form (az + b)/(cz + d)
transformation is always a bijective holomorphic function from the Riemann sphere to the Riemann sphere. The set of all Möbius transformations forms a group
Möbius_transformation
Solutions of Legendre's differential equation
mentioned. Ferrers function Creasey, Peter E.; Lang, Annika (2018). "Fast generation of isotropic Gaussian random fields on the sphere". Monte Carlo Methods
Legendre_function
Marked objects for finding random numbers
dice and intransitive dice. A die can be constructed in the shape of a sphere, with the addition of an internal cavity in the shape of the dual polyhedron
Dice
scattering by spheres - this article list codes for electromagnetic scattering by a homogeneous sphere, layered sphere, and cluster of spheres. Majority of
Codes for electromagnetic scattering by spheres
Codes_for_electromagnetic_scattering_by_spheres
SPHERE FUNCTION
SPHERE FUNCTION
Male
Hebrew
(עֵפֶר) Hebrew name EPHER means "calf" or "gazelle." In the bible, this is the name of several characters, including a son of Ezra.
Boy/Male
American, British, English
Spear
Boy/Male
Australian, French, Portuguese
Stern; Severe
Girl/Female
American, Christian, French, German, Hebrew
Darling; Little and Womanly; Beloved; The Plain
Female
English
Variant spelling of English Sherry, SHEREE means "darling."
Girl/Female
French, German, Hebrew
Little and Womanly; Dear; Man; The Plain
Female
English
English variant spelling of Greek Phoebe, PHEBE means "shining one."
Surname or Lastname
English
English : variant spelling of Shear 1.Indian (Maharashtra); pronounced as two syllables : Hindu (Vani) name, probably from Marathi šera ‘rate’.
Girl/Female
Indian, Telugu
Veda means Vedham and Shree means Sriman Narayana
Boy/Male
British, English
Spear-man
Surname or Lastname
English
English : topographic name for someone who lived by the seashore, Middle English schore.English : topographic name for someone who lived on or by a bank or steep slope, Old English scora. There are minor places named with this word in Lancashire and West Yorkshire, and the surname may also be a habitational name from these.Americanized spelling of Ashkenazic Jewish S(c)hor(r) or Szor, variants of Schauer.
Surname or Lastname
English and Irish (County Limerick; of English origin)
English and Irish (County Limerick; of English origin) : from Old English scīr, Middle English s(c)hire ‘shire’, perhaps a topographic name for someone who lived by the meeting place of a shire.
Girl/Female
French, German, Hebrew
Beloved; A Man; The Plain
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Surname or Lastname
English
English : variant of Sherrin.
Surname or Lastname
English
English : variant of Shear 1.Jewish (eastern Ashkenazic) : variant spelling of Scher.
Male
English
Variant spelling of English Ophir, OPHER means "gold" or "reducing to ashes."
Surname or Lastname
English
English : variant of Spear.
Female
English
Variant spelling of English Sherry, SHERIE means "darling."
Female
English
Variant spelling of English Sherry, SHERI means "darling."
SPHERE FUNCTION
SPHERE FUNCTION
Girl/Female
Tamil
Akarshana | அகரà¯à®·à®¾à®¨à®¾
Attraction
Boy/Male
French
Red haired.
Girl/Female
Indian
Beauty, Splendor, Brilliance, Fashion, Form, Figure
Surname or Lastname
English
English : unexplained.
Male
Egyptian
, a hawk-headed deity.
Girl/Female
Tamil
Yokshita | யோகà¯à®·à¯€à®¤à®¾
Male
Italian
Italian form of Latin Americus, AMERIGO means "work-power."Â
Girl/Female
Arabic, Muslim
Forest Flower
Girl/Female
Tamil
A name of Goddess Lakshmi
Female
Hebrew
(לָבִי×) Hebrew unisex name LAVI means "lion."
SPHERE FUNCTION
SPHERE FUNCTION
SPHERE FUNCTION
SPHERE FUNCTION
SPHERE FUNCTION
v. t.
To form into roundness; to make spherical, or spheral; to perfect.
n.
The apparent surface of the heavens, which is assumed to be spherical and everywhere equally distant, in which the heavenly bodies appear to have their places, and on which the various astronomical circles, as of right ascension and declination, the equator, ecliptic, etc., are conceived to be drawn; an ideal geometrical sphere, with the astronomical and geographical circles in their proper positions on it.
v. t.
To place in a sphere; to envelop.
a.
Of or pertaining to the spheres.
n.
A sphere or scheme of operation.
superl.
Sharp; afflictive; distressing; violent; extreme; as, severe pain, anguish, fortune; severe cold.
a.
Of or pertaining to a sphere or the spheres.
a.
Rounded like a sphere; sphere-shaped; hence, symmetrical; complete; perfect.
v. t.
To remove, as a planet, from its sphere or orb.
v. t.
To place in, or as in, an orb a sphere. Cf. Ensphere.
n.
A sphere.
v. i.
To form a scheme or schemes.
adv.
In this place; in the place where the speaker is; -- opposed to there.
a.
Having the form of a sphere; like a sphere; globular; orbicular; as, a spherical body.
v. t.
To place in a sphere, or among the spheres; to insphere.
a.
Of or pertaining to the heavenly orbs, or to the sphere or spheres in which, according to ancient astronomy and astrology, they were set.
v. t.
To form into a sphere.
n.
A sphere.
a.
Of or pertaining to a sphere.
imp. & p. p.
of Sphere