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Function whose composition with the logarithm is convex
In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with
Logarithmically convex function
Logarithmically_convex_function
Real function with secant line between points above the graph itself
function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function
Convex_function
Type of mathematical function
In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it
Logarithmically concave function
Logarithmically_concave_function
Negative of a convex function
concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination
Concave_function
one in terms of logarithmically convex functions, one in terms of increasing functions, and one in terms of non-negative functions. The conjecture has
Khabibullin's conjecture on integral inequalities
Khabibullin's_conjecture_on_integral_inequalities
Extension of the factorial function
is the unique interpolating function for the factorial, defined over the positive reals, which is logarithmically convex, meaning that y = log f ( x
Gamma_function
Type of mathematical functions
condition is required, which is called logarithmically convex. A Reinhardt domain D is called logarithmically convex if the image λ ( D ∗ ) {\displaystyle
Function of several complex variables
Function_of_several_complex_variables
Mathematics of convex functions and sets
Convex analysis is the branch of mathematics that studies convex sets, convex functions, and their applications to optimization, functional analysis,
Convex_analysis
Mathematical function with convex lower level sets
In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the
Quasiconvex_function
measures is log-concave. Convex measure, a generalisation of this concept Logarithmically concave function Prékopa, A. (1980). "Logarithmic concave measures and
Logarithmically concave measure
Logarithmically_concave_measure
Continuous function whose value increases to infinity
of barrier functions are inverse barrier functions and logarithmic barrier functions. Resumption of interest in logarithmic barrier functions was motivated
Barrier_function
be logarithmically convex, if M k + 1 / M k {\displaystyle M_{k+1}/M_{k}} is increasing. When M k {\displaystyle M_{k}} is logarithmically convex, then
Quasi-analytic_function
} Thus, a measure being 0-convex is the same thing as it being a logarithmically concave measure. The classes of s-convex measures form a nested increasing
Convex_measure
Complex-differentiable (mathematical) function
power series is not necessarily an open ball; these regions are logarithmically convex Reinhardt domains, the simplest example of which is a polydisk.
Holomorphic_function
self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set. Self-concordant barriers are important
Self-concordant_function
Theorem in complex analysis
writing in 1896; Hadamard published no proof. Maximum principle Logarithmically convex function Hardy's theorem Hadamard three-line theorem Borel–Carathéodory
Hadamard_three-circle_theorem
Smallest convex set containing a given set
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined
Convex_hull
_{i}(\mathbf {x} ,\mathbf {y} ))} is the partition function. This density is a logarithmically convex function, and thus the common inference task in PSL of
Probabilistic_soft_logic
Mathematical function having a characteristic S-shaped curve or sigmoid curve
asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function is convex for values less than a particular point, and it is concave for values
Sigmoid_function
completely monotonic function, logarithmically completely monotonic function, strongly logarithmically completely monotonic function, strongly completely
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Smooth approximation to the maximum function
this formula internally. LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function by adding an extra argument set
LogSumExp
Ensemble learning method
which shows that boosting performs gradient descent in a function space using a convex cost function. Given images containing various known objects in the
Boosting_(machine_learning)
Class of inequalities
In mathematics, logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f {\displaystyle f} , its logarithm, and
Logarithmic Sobolev inequalities
Logarithmic_Sobolev_inequalities
Measure of quantum entanglement in quantum mechanics
λ i {\displaystyle \lambda _{i}} are all of the eigenvalues. Is a convex function of ρ {\displaystyle \rho } : N ( ∑ i p i ρ i ) ≤ ∑ i p i N ( ρ i )
Negativity (quantum mechanics)
Negativity_(quantum_mechanics)
Measure for evaluating probabilistic forecasts
and a convex class F {\displaystyle {\mathcal {F}}} of probability measures on ( Ω , A ) {\displaystyle (\Omega ,{\mathcal {A}})} . A function defined
Scoring_rule
Probability distribution
function of the shape parameters α and β. § Moments of logarithmically transformed random variables contains formulas for moments of logarithmically transformed
Beta_distribution
Algorithms for solving convex optimization problems
a convex function and G is a convex set. Without loss of generality, we can assume that the objective f is a linear function. Usually, the convex set
Interior-point_method
Theorem in complex analysis
x f (x) for x > 0 and f is logarithmically convex. A treatment of this theorem is in Artin's book The Gamma Function, which has been reprinted by the
Bohr–Mollerup_theorem
Meromorphic function
) {\displaystyle \ln \Gamma (x)} is strictly convex. For m = 0 {\displaystyle m=0} , the digamma function, ψ ( x ) = ψ ( 0 ) ( x ) {\displaystyle \psi
Polygamma_function
Mathematical function
As with the gamma function that extends the ordinary factorial function, this double factorial function is logarithmically convex in the sense of the
Double_factorial
Type of market maker
roundtrip arbitrage in a CFMM implies that the level function φ {\displaystyle \varphi } must be convex. Execution costs in the CFMM are defined as the difference
Constant function market maker
Constant_function_market_maker
Equation whose unknown is a function
functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically
Functional_equation
caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Measure of the decline from a historical peak
{\displaystyle \mu } : μ > 0 {\displaystyle \mu >0} implies that the MDD grows logarithmically with time μ = 0 {\displaystyle \mu =0} implies that the MDD grows as
Drawdown_(economics)
2D graphic with logarithmic scales on both axes
two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y = a x k
Log–log_plot
Mathematical concept
-Lipschitz gradient. When every f i {\displaystyle f_{i}} is convex the function is convex, and an ε {\displaystyle \varepsilon } -optimal point is reachable
Multi-objective_optimization
Mathematical function often applied to matrices
ellipticity in differential operators on function spaces, subject to specific boundary conditions. The logarithmic norm has a wide range of applications
Logarithmic_norm
Extension of superfactorials to the complex numbers
(2022). A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions (PDF). Developments in Mathematics. Vol. 70. Springer. p. 218. doi:10
Barnes_G-function
Geometric relation between the roots of a polynomial and those of its derivative
polynomial P (quartic function) with four distinct zeros forming a concave quadrilateral, one of the zeros of P lies within the convex hull of the other three;
Gauss–Lucas_theorem
Cartesian product of discs
the term bidisc is sometimes used. A polydisc is an example of logarithmically convex Reinhardt domain. Poincare, H, Les fonctions analytiques de deux
Polydisc
Matrix of second derivatives
Hessian determinant is a polynomial of degree 3. The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test
Hessian_matrix
In Euclidean space, a measure of that set's "size"
u|^{2}\mathrm {d} x} This is often called the logarithmic capacity, the term logarithmic arises, as the potential function goes from being an inverse power to a
Capacity_of_a_set
Product of numbers from 1 to n
Bohr–Mollerup theorem, which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates
Factorial
Exponentially decreasing bounds on tail distributions of random variables
{\displaystyle I(a)=\sup _{t}at-K(t)} The moment generating function is log-convex, so by a property of the convex conjugate, the Chernoff bound must be log-concave
Chernoff_bound
Algorithm for computing convex hulls in a set of points
to sort dominates the time to actually compute the convex hull. The pseudocode below uses a function ccw: ccw > 0 if three points make a counter-clockwise
Graham_scan
Family of optimization algorithms
\left({\frac {1}{\epsilon }}\right)\right).} The number of steps depends only logarithmically on the level of accuracy required, in contrast to the stochastic approximation
Stochastic_variance_reduction
Count of the possible partitions of a set
descent. The Bell numbers form a logarithmically convex sequence. Dividing them by the factorials, Bn/n!, gives a logarithmically concave sequence. Several asymptotic
Bell_number
Quantum
is the largest function that is convex and that equals four times the variance for pure states. That is, it equals four times the convex roof of the variance
Quantum_Fisher_information
Number, approximately 3.14
(optimal) upper bound on the volume of a convex body containing only one integer lattice point. The Riemann zeta function ζ(s) is used in many areas of mathematics
Pi
Graph showing survival against age
regardless of age. Some birds and some lizards follow this pattern. Type III or convex curves have the greatest mortality (lowest age-specific survival) early
Survivorship_curve
Measure of ranking quality
DCG in both formulations. Convex and smooth approximations to DCG have also been developed, for use as an objective function in gradient based learning
Discounted_cumulative_gain
(unlike posynomials) signomials cannot necessarily be made convex by applying a logarithmic change of variables. Nevertheless, signomial optimization problems
Signomial
Length of a line segment
strictly convex function of the two points, unlike the distance, which is non-smooth (near pairs of equal points) and convex but not strictly convex. The
Euclidean_distance
Abstract data type
These include: Graham scan, an algorithm for the convex hull of a two-dimensional system of points. A convex hull of a subset of the input is maintained in
Stack_(abstract_data_type)
Probability distribution
Pedersen also proved many properties of the median, showing that it is a convex function of α, and that the asymptotic behavior near α = 0 {\displaystyle \alpha
Gamma_distribution
Theorem in mathematics
complex-valued functions. Instead, a generalization of the theorem is stated such: Let f : Ω → C be a holomorphic function on the open convex set Ω, and let
Mean_value_theorem
List of laws List of lemmas List of limits List of logarithmic identities List of mathematical functions List of mathematical identities List of mathematical
List_of_theorems
a polytope, but a diagram showing how the elements meet. The classical convex polytopes may be considered tessellations, or tilings, of spherical space
List_of_mathematical_shapes
Measure of complexity of real-valued functions
complexity of the convex hull of A {\displaystyle A} equals Rad(A). (Massart Lemma) The Rademacher complexity of a finite set grows logarithmically with the set
Rademacher_complexity
Visualization method for regularization
to the size of its residual as a function of a regularization parameter. When the points are plotted in logarithmic scale, the curve typically has a characteristic
L-curve
Description of a quantum-mechanical system
Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant
Schrödinger_equation
Algorithm used for pathfinding and graph traversal
of the least cost path in the graph. Convex Upward/Downward Parabola (XUP/XDP). Modification to the cost function in weighted A* to push optimality toward
A*_search_algorithm
the function is convex. Well-known examples of convex functions include the quadratic function x 2 {\displaystyle x^{2}} and the exponential function e
Glossary_of_calculus
coefficient c0 is called the conformal center of D. It can be shown to lie in the convex hull of D; moreover, D ⊆ { z : | z − c 0 | ≤ 2 c 1 } , {\displaystyle D\subseteq
Conformal_radius
Foundational principle in quantum physics
quantum Fisher information is the convex roof of the variance times four. A simpler inequality follows without a convex roof σ A 2 F Q [ ϱ , B ] ≥ | ⟨ i
Uncertainty_principle
Number divisible only by 1 and itself
2^{a}3^{b}+1} . It is possible to partition any convex polygon into n {\displaystyle n} smaller convex polygons of equal area and equal perimeter, when
Prime_number
Method for constructing existence proofs and calculating solutions in variational calculus
{\displaystyle \mathbb {R} ^{mn}} ). If the function A ↦ F ( x , y , A ) {\displaystyle A\mapsto F(x,y,A)} is convex for almost every x ∈ Ω {\displaystyle x\in
Direct method in the calculus of variations
Direct_method_in_the_calculus_of_variations
Rectangle with side lengths in the golden ratio
the golden ratio, so this triangle forms half of a golden rectangle. The convex hull of two opposite edges of a regular icosahedron forms a golden rectangle
Golden_rectangle
Generalization of the concept of directional derivative
differential calculus. Named after René Gateaux, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet
Gateaux_derivative
Continuous stochastic process
of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options. If instead we assume that
Geometric_Brownian_motion
w} is the indicator function of a given shape, this is the same as the axiality. Lassak, Marek (2002), "Approximation of convex bodies by axially symmetric
Axiality_(geometry)
Mathematical operation
convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function. Similarly, a function whose
Second_derivative
Generalization of means
partitioning property of the mean. Consider a Legendre-type strictly convex function F {\displaystyle F} . Then the gradient map ∇ F {\displaystyle \nabla
Quasi-arithmetic_mean
Swiss mathematician (1707–1783)
logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative
Leonhard_Euler
Risk measure estimating the average loss in the worst tail of the distribution
incomplete gamma function, l i ( x ) = ∫ d x ln x {\displaystyle \mathrm {li} (x)=\int {\frac {dx}{\ln x}}} is the logarithmic integral function. If the loss
Expected_shortfall
Polygon intersected up to twice by lines orthogonal to a given line
name. A convex polygon is monotone with respect to any straight line and a polygon which is monotone with respect to every straight line is convex. A linear
Monotone_polygon
Mathematical space with a notion of distance
distance between its points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many
Metric_space
Norm on a vector space of matrices
The nuclear norm ‖ A ‖ ∗ {\displaystyle \|A\|_{*}} is a convex envelope of the rank function rank ( A ) {\displaystyle {\text{rank}}(A)} , so it is often
Matrix_norm
Fundamental construction of differential calculus
subderivative and subgradient are generalizations of the derivative to convex functions used in convex analysis. In commutative algebra, Kähler differentials are
Generalizations of the derivative
Generalizations_of_the_derivative
Concept in integral mathematics
relaxations: a linear programming relaxation can be viewed geometrically, as a convex polytope that includes all feasible solutions and excludes all other 0–1
Linear_programming_relaxation
a function, and introduced the use of the exponential function and logarithms in analytic proofs. Euler frequently used the logarithmic functions as
Contributions of Leonhard Euler to mathematics
Contributions_of_Leonhard_Euler_to_mathematics
of a matrix. conv – convex hull of a set. Cor – corollary. corr – correlation. cos – cosine function. cosec – cosecant function. (Also written as csc
List of mathematical abbreviations
List_of_mathematical_abbreviations
On tangency patterns of circles
Verdière proved the existence of the circle packing as a minimizer of a convex function on a certain configuration space. Bennett Chow and Feng Luo found another
Circle_packing_theorem
Economical computational problem
with linear utility functions. Their algorithm uses the primal–dual paradigm in the enhanced setting of KKT conditions and convex programs. Their algorithm
Market equilibrium computation
Market_equilibrium_computation
Measure of value difference between best possible decision and made decision
cannot be achieved in the latter case. In this case, the solution of a convex optimization problem gives the optimal, minimax regret-minimizing linear
Regret_(decision_theory)
Mathematician (1912–1999)
convex curves)" (PDF). Matematikai Lapok (9): 19–36. George Csordás; István Vincze (1992). "Convexity properties of power series with logarithmically
István_Vincze_(mathematician)
Differential calculus on function spaces
which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals
Calculus_of_variations
Arrangement of points on a sphere
Smale's problem the function to minimise is not the electrostatic potential 1 r i j {\displaystyle 1 \over r_{ij}} but a logarithmic potential given by
Thomson_problem
Probability distribution and special case of gamma distribution
function for χ 2 ( k ) {\displaystyle \chi ^{2}(k)} is K ( t ) = − k 2 ln ( 1 − 2 t ) {\textstyle K(t)=-{\frac {k}{2}}\ln(1-2t)} , and its convex dual
Chi-squared_distribution
Long, BY. Optimal bounds for Neuman–Sándor mean in terms of the geometric convex combination of two Seiffert means. J Inequal Appl (2016) 2016: 14. https://doi
Neuman–Sándor_mean
Physical simulation to visualize graphs
in the plane with all faces convex by fixing the vertices of the outer face of a planar embedding of the graph into convex position, placing a spring-like
Force-directed_graph_drawing
Partition of space by hyperplanes
is either a bounded region that is (the interior of) a convex polytope, or an unbounded convex polyhedral region. Each flat of A is also divided into
Arrangement_of_hyperplanes
Area of geometry, about angles and lengths
angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths
Trigonometry
Algorithm that employs a degree of randomness as part of its logic or procedure
computational geometry, a standard technique to build a structure like a convex hull or Delaunay triangulation is to randomly permute the input points and
Randomized_algorithm
Number, approximately 1.618
36^{\circ }\!} , one of 72 ∘ {\displaystyle 72^{\circ }\!} , and one non-convex angle of 216 ∘ {\displaystyle 216^{\circ }\!} . Special matching rules
Golden_ratio
Mathematical optimization algorithm
However, an interesting case appears when the eigenvalues are spaced logarithmically for a large symmetric matrix. For example, let A = Q D Q T {\displaystyle
Conjugate_gradient_method
Economical computational problem
ΣP2-complete. However, when the utility function for each player depends only on the actions of a logarithmically small number of other players (that is
Nash_equilibrium_computation
Generalization of the standard Boltzmann–Gibbs entropy
{\displaystyle [z]_{+}:=\max\{z,0\}} . These functions recover the standard exponential and logarithmic functions in the limit q → 1 {\displaystyle q\to 1}
Tsallis_entropy
Method of DNA analysis
gaps. The five main types of gap penalties are constant, linear, affine, convex, and profile-based. Genetic sequence alignment - In bioinformatics, gaps
Gap_penalty
Geometry problem on grid points
finds points forming convex polygons, which satisfy the requirement of having no three in line, but are too small. The largest convex polygons with vertices
No-three-in-line_problem
LOGARITHMICALLY CONVEX-FUNCTION
LOGARITHMICALLY CONVEX-FUNCTION
Male
English
Variant spelling of English Connor, CONNER means "hound-lover."
Boy/Male
Irish American
Hound lover. Full of desire; much desire.
Boy/Male
Irish American
Strong willed or wise. Also a : Hero.
Surname or Lastname
English
English : habitational name from a place named Cove, examples of which are found in Devon, Hampshire, and Suffolk, from Old English cofa ‘cove’, ‘bay’, ‘inlet’, also ‘shelter’, ‘hut’, or a topographic name with the same meaning.
Boy/Male
American, Christian, German, Indian
High Desire
Surname or Lastname
English
English : from Middle English cony ‘rabbit’ (a back-formation from conies, from Old French conis, plural of conil), a nickname for someone thought to resemble a rabbit in some way or a metonymic occupational name for a dealer in rabbits or rabbit skins.
Surname or Lastname
English
English : unexplained.
Surname or Lastname
English (Leicestershire)
English (Leicestershire) : variant of Culver.
Surname or Lastname
Italian
Italian : from the title of rank conte ‘count’ (from Latin comes, genitive comitis ‘companion’). Probably in this sense (and the Late Latin sense of ‘traveling companion’), it was a medieval personal name; as a title it was no doubt applied ironically as a nickname for someone with airs and graces or simply for someone who worked in the service of a count.English : variant of Count, cognate with 1.French : nickname for someone in the service of a count or for someone who behaved pretentiously, from Old French conte, cunte ‘count’ (of the same derivation as 1).French (Conté) : variant of Comté (see Comte).
Boy/Male
American, British, English
Dove
Boy/Male
British, Christian, English
Wagoner; To Convey
Boy/Male
American, British, English
Shepherd
Surname or Lastname
Irish
Irish : variant spelling of Connor, now common in Scotland.English : occupational name for an inspector of weights and measures, Middle English connere, cunnere ‘inspector’, an agent derivative of cun(nen) ‘to examine’.
Surname or Lastname
Spanish and Portuguese
Spanish and Portuguese : nickname from the title of rank conde ‘count’, a derivative of Latin comes, comitis ‘companion’.English : unexplained.
Surname or Lastname
English
English : metathesized form of the occupational name Coyner.English : possibly an occupational name for a dealer in rabbits or rabbit skins, from an agent derivative of Middle English cony ‘rabbit’ (see Coney).
Boy/Male
Irish
Hound of the plains.
Male
English
Anglicized form of Irish Gaelic Conláed, CONLEY means "purifying fire."
Boy/Male
Indian, Kannada, Tamil
God Murugan
Boy/Male
Irish
Hero.
Surname or Lastname
English
English : from Old French covine ‘fraud’, ‘deceit’, hence a derogatory nickname for a trickster.English : habitational name from a place in Staffordshire named Coven ‘(place) at the huts or shelters (Old English cofa, dative plural cofum)’.
LOGARITHMICALLY CONVEX-FUNCTION
LOGARITHMICALLY CONVEX-FUNCTION
Boy/Male
Tamil
Kamkrish | கமà¯à®•à¯à®°à¯€à®·
Boy/Male
Arabic, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Tamil, Telugu, Traditional
Dignity; Lord Shiva's Daughter
Surname or Lastname
English
English : variant spelling of Seeley.
Surname or Lastname
English
English : from a diminutive of Trick.
Boy/Male
Hindu, Indian
King of Earth
Boy/Male
Hindu
Free, From france
Boy/Male
American, British, English, Gaelic, Irish
Great
Female
English
Variant spelling of English Colleen, KOLLEEN means "girl."
Boy/Male
Tamil
Winter
Boy/Male
Tamil
Unassuming
LOGARITHMICALLY CONVEX-FUNCTION
LOGARITHMICALLY CONVEX-FUNCTION
LOGARITHMICALLY CONVEX-FUNCTION
LOGARITHMICALLY CONVEX-FUNCTION
LOGARITHMICALLY CONVEX-FUNCTION
dv.
In a convex form; convexly.
a.
Concave on one side and convex on the other, as an eggshell or a crescent.
v. t.
To impart or communicate; as, to convey an impression; to convey information.
a.
Convex on one side, and flat on the other; plano-convex.
adv.
In a convex form; as, a body convexly shaped.
a.
Alt. of Logarithmical
v. t.
To context.
a.
Specifically, having such a combination of concave and convex sides as makes the focal axis the shortest line between them. See Illust. under Lens.
n.
The conger eel; -- called also congeree.
n.
A convex body or surface.
v. t.
To cause to pass from one place or person to another; to serve as a medium in carrying (anything) from one place or person to another; to transmit; as, air conveys sound; words convey ideas.
adv.
Logarithmically.
a.
Convex on both sides; double convex. See under Convex, a.
a.
Made convex; protuberant in a spherical form.
v. t.
To accompany; to convoy.
a.
Convex on one side, and concave on the other. The curves of the convex and concave sides may be alike or may be different. See Meniscus.
n. & v.
See Conge, Conge.
a.
Plane or flat on one side, and convex on the other; as, a plano-convex lens. See Convex, and Lens.
adv.
By the use of logarithms.
a.
Convex on both sides; as, a biconvex lens.