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Negative of a convex function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to
Concave_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
Real function with secant line between points above the graph itself
like a linear function), while a concave function's graph is shaped like a cap ∩ {\displaystyle \cap } . A twice-differentiable function of a single variable
Convex_function
Mathematical function with convex lower level sets
Gaussian distribution are common examples of quasi-concave functions that are not concave. A function that is both quasiconvex and quasiconcave is quasilinear
Quasiconvex_function
Function in mathematical analysis
Schur-convex functions are used in the study of majorization. A function f is 'Schur-concave' if its negative, −f, is Schur-convex. Every function that is
Schur-convex_function
Topics referred to by the same term
Look up concave or concavity in Wiktionary, the free dictionary. Concave or concavity may refer to: Concave lens Concave mirror Concave function, the negative
Concave
Subfield of mathematical optimization
studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex
Convex_optimization
Set-to-real map with diminishing returns
optimizing a convex or concave function can also be described as the problem of maximizing or minimizing a submodular function subject to some constraints
Submodular_set_function
Concept in convex analysis
-\infty .} Convex functions that are not proper are called improper convex functions. A proper concave function is by definition, any function g : X → [ − ∞
Proper_convex_function
converting a non-concave function to a concave function. A related concept is convexification – converting a non-convex function to a convex function. It is especially
Concavification
In geometry, set whose intersection with every line is a single line segment
this usage. The complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context
Convex_set
Representation of a mathematical function
function: f ( x , y ) = − ( cos ( x 2 ) + cos ( y 2 ) ) 2 . {\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}.} Asymptote Chart Plot Concave function
Graph_of_a_function
Mathematical operation
function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite
Second_derivative
Fundamental trigonometric functions
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle:
Sine_and_cosine
Type of sequence of numbers
interval of Z. These conditions mirror the ones required for log-concave functions. Sequences that fulfill the three conditions are also called Pólya
Logarithmically concave sequence
Logarithmically_concave_sequence
Rounding of an interior or exterior corner
an interior corner is a line of concave function, whereas a fillet on an exterior corner is a line of convex function (in these cases, fillets are typically
Fillet_(mechanics)
Topics referred to by the same term
Log-concave may refer to: Logarithmically concave function Logarithmically concave measure Logarithmically concave sequence This disambiguation page lists
Log-concave
Mathematical result in convex functions theory
{\displaystyle \mathbb {R} ^{n}} and let g {\displaystyle g} be a proper concave function on R n {\displaystyle \mathbb {R} ^{n}} . Then, if regularity conditions
Fenchel's_duality_theorem
Point where the curvature of a curve changes sign
case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa
Inflection_point
Gives conditions that guarantee the max–min inequality holds with equality
} is a continuous function that is concave-convex, i.e. f ( ⋅ , y ) : X → R {\displaystyle f(\cdot ,y):X\to \mathbb {R} } is concave for every fixed y
Minimax_theorem
Disproven hypothesis
been called the "concave" Hollow Earth hypothesis or skycentrism. Cyrus Teed, a doctor from upstate New York, proposed such a concave Hollow Earth in 1869
Hollow_Earth
Function whose composition with the logarithm is convex
characterize Euler's gamma function among the possible extensions of the factorial function to real arguments. Logarithmically concave function Kingman, J.F.C. 1961
Logarithmically convex function
Logarithmically_convex_function
Type of chart
Because the values are in decreasing order, the cumulative function is a concave function. To take the example below, in order to lower the amount of
Pareto_chart
Generalization of the normal-form game
the constraint functions are linear functions defining the simpices of the players; in a concave game, the hj can be any concave function of x. For the
Concave_game
function. Thus, any Gaussian measure is log-concave. The Prékopa–Leindler inequality shows that a convolution of log-concave measures is log-concave.
Logarithmically concave measure
Logarithmically_concave_measure
Measure of variation in statistics
downward bias, by Jensen's inequality, due to the square root's being a concave function. The bias in the variance is easily corrected, but the bias from the
Standard_deviation
Mathematical relation assigning a probability event to a cost
ISBN 978-3-540-42669-1. Tangian, Andranik (2002). "Constructing a quasi-concave quadratic objective function from interviewing a decision maker". European Journal of
Loss_function
Method to solve optimization problems
linear function is a convex function, which implies that every local minimum is a global minimum; similarly, a linear function is a concave function, which
Linear_programming
Optical device which transmits and refracts light
lens with two concave surfaces is biconcave (or just concave). If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the
Lens
p-adic function: a function whose domain is p-adic. Convex function: line segment between any two points on the graph lies above the graph. Also concave function
List_of_types_of_functions
Mathematical function
chemistry to form basis sets. Gaussian functions arise by composing the exponential function with a concave quadratic function: f ( x ) = exp ( α x 2 + β x
Gaussian_function
equilibrium computation. The function f is called convex-concave if it is a convex function of x for any fixed y, and a concave function of y for any fixed x
Min-max_optimization
Solution concept of a non-cooperative game
utility function of each player is continuous in all strategies and a concave function of the player's own strategy, then a GNE exists; see concave games
Nash_equilibrium
Mathematical and computational problem
more general cost and load functions: Anily, Bramel and Simchi-Levi study a setting where the cost of a bin is a concave function of the number of items in
Bin_packing_problem
Probability distribution
the AM–GM inequality and is a consequence of the logarithm being a concave function. In fact, E [ X ] = e μ + 1 2 σ 2 = e μ ⋅ e σ 2 = GM [ X ] ⋅ GVar
Log-normal_distribution
Concept in machine learning
strictly concave function such that C ( η ) = C ( 1 − η ) {\displaystyle C(\eta )=C(1-\eta )} . Table-I shows the generated Bayes consistent loss functions for
Loss functions for classification
Loss_functions_for_classification
Scalar physical quantities representing system states
V^{2}}}{\biggr )}_{T,N}\geq 0} Where Helmholtz energy is a concave function of temperature and convex function of volume. ( ∂ 2 H ∂ P 2 ) S , N ≤ 0 {\displaystyle
Thermodynamic_potential
Model in probability theory
n 2 − n {\displaystyle X_{n}^{2}-n} is a martingale). Similarly, a concave function of a martingale is a supermartingale. A stopping time with respect
Martingale (probability theory)
Martingale_(probability_theory)
Mathematical function having a characteristic S-shaped curve or sigmoid curve
x\rightarrow \pm \infty } . A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point:
Sigmoid_function
Pentagon with all sides equal but the angles may not be equal
themselves are called simple, and they can be classified as either convex or concave. We here use the term "stellated" to refer to the ones that intersect themselves
Equilateral_pentagon
Integral inequality
with log-concave distribution. Since the product of two log-concave functions is log-concave, the joint distribution of (X,Y) is also log-concave. Log-concavity
Prékopa–Leindler_inequality
Mathematical relationships
of a concave function of an arithmetic mean is greater than or equal to the arithmetic mean of the function's values. Since the logarithm function is concave
QM–AM–GM–HM_inequalities
Mathematical space with a notion of distance
found many applications. Given a metric space (X, d) and an increasing concave function f : [ 0 , ∞ ) → [ 0 , ∞ ) {\displaystyle f\colon [0,\infty )\to [0
Metric_space
Inequality between integrals in Lp spaces
another proof as part of a work developing the concept of convex and concave functions and introducing Jensen's inequality, which was in turn named for work
Hölder's_inequality
Mathematical function
(z)}}.} It is the first of the polygamma functions. This function is strictly increasing and strictly concave on ( 0 , ∞ ) {\displaystyle (0,\infty )}
Digamma_function
Probabilistic optimal control
for example, a concave function of a state variable over a horizon from time zero (the present) to a terminal time T, or a concave function of a state variable
Stochastic_control
Concept in mathematical optimization
optimality if the objective function f {\displaystyle f} of a maximization problem is a differentiable concave function, the inequality constraints g
Karush–Kuhn–Tucker_conditions
Statistical measure of how far values spread from their average
standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality), which depends
Variance
Function related to statistics and probability theory
distributions—notably the exponential family—are only logarithmically concave, and concavity of the objective function plays a key role in the maximization. Given the independence
Likelihood_function
Algorithm for finding zeros of functions
_{n+1}\vert \leq M\cdot \varepsilon _{n}^{2}\,.} Suppose that f(x) is a concave function on an interval, which is strictly increasing. If it is negative at
Newton's_method
Maximum size of an independent set of the matroid
does not necessarily maximize a concave function), and a polynomial-time algorithm that maximizes a concave function for the special case of MRFs based
Matroid_rank
Physics phenomenon
S2CID 118952931. Huang, Yichen (29 July 2010). "Entanglement criteria via concave-function uncertainty relations". Physical Review A. 82 (1) 012335. Bibcode:2010PhRvA
Quantum_entanglement
In mathematics, invariant of square matrices
Brunn–Minkowski theorem implies that the nth root of determinant is a concave function, when restricted to Hermitian positive-definite n × n {\displaystyle
Determinant
Correction for sample variance bias
of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality. There is no general
Bessel's_correction
relationship between the expenditure function and the utility function. If given a specific regular quasi-concave utility function, the corresponding price is
Expenditure_function
Type of function in linear algebra
of sublinear growth: every function f ( n ) ∈ o ( n ) {\displaystyle f(n)\in o(n)} can be upper-bounded by a concave function of sublinear growth. Asymmetric
Sublinear_function
Arithmetic mean is greater than or equal to geometric mean
of a concave function of an arithmetic mean is greater than or equal to the arithmetic mean of the function's values. Since the logarithm function is concave
AM–GM_inequality
Probability distribution
1981, states that the entropy of a Poisson binomial distribution is a concave function of the success probabilities p 1 , p 2 , … , p n {\displaystyle p_{1}
Poisson_binomial_distribution
Function linear in one argument, used in economics and consumer theory
{\displaystyle \theta _{i}} is strictly increasing and concave. A useful property of the quasilinear utility function is that the Marshallian/Walrasian demand for
Quasilinear_utility
Probability distribution
real-valued random variable. The general form of its probability density function is f ( x ) = 1 2 π σ 2 exp ( − ( x − μ ) 2 2 σ 2 ) . {\displaystyle f(x)={\frac
Normal_distribution
Mathematical finance term
logarithms of the arithmetic price changes. Because the logarithm is a concave function (it curves down), it increasingly penalizes negative arithmetic returns
Volatility_tax
Iterative optimization method
objective function or leave it unchanged. Taking the minorize-maximization version, let f ( θ ) {\displaystyle f(\theta )} be the objective concave function to
MM_algorithm
Russian mathematician (born 1966)
[PP93] In further unpublished work, Perelman studied DC functions (difference of concave functions) on Alexandrov spaces and established that the set of
Grigori_Perelman
Statistical property
inequality, a convex function as transformation will introduce positive bias, while a concave function will introduce negative bias, and a function of mixed convexity
Bias_of_an_estimator
Inverse of the average of the inverses of a set of numbers
x_{n})={\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}.} The harmonic mean is a Schur-concave function, and is greater than or equal to the minimum of its arguments: for
Harmonic_mean
Property of some mathematical functions
case of subadditive function, if a sequence is interpreted as a function on the set of natural numbers. Note that while a concave sequence is subadditive
Subadditivity
Statistical method for handling multiple comparisons
linear denominator R in the expected ratio E[V/R] with a non-decreasing concave function s(R), yielding the criterion E[V/s(R)]. This approach allows the control
False_discovery_rate
Quantum states that are not entangled
S2CID 118952931. Huang, Yichen (July 29, 2010). "Entanglement criteria via concave-function uncertainty relations". Physical Review A. 82 (1) 012335. Bibcode:2010PhRvA
Separable_state
Statement in information theory
Kullback-Leibler divergence is a form of Bregman divergence. Because log is a concave function, we have that: ∑ i p i log q i p i ≤ log ∑ i p i q i p i = log
Gibbs'_inequality
Function that maps matrices to matrices
domain of f. This is analogous to monotone function in the scalar case. A function f is called operator concave if and only if τ f ( A ) + ( 1 − τ ) f (
Analytic_function_of_a_matrix
Meromorphic function
the digamma function, ψ ( x ) = ψ ( 0 ) ( x ) {\displaystyle \psi (x)=\psi ^{(0)}(x)} , is strictly monotonic increasing and strictly concave. For m {\displaystyle
Polygamma_function
Function in mathematical analysis
continuous functions. For a function between metric spaces, it is equivalent to admit a modulus of continuity that is either concave, or subadditive, or uniformly
Modulus_of_continuity
Game class in game theory
studied potential games in which (a) the potential is a smooth and concave function, (b) the strategy sets are convex, (c) the utilities are bounded. He
Potential_game
epigraph has an SCB. Let g(t) be a 3-times continuously differentiable concave function on t>0, such that t ⋅ | g ‴ ( t ) | / | g ″ ( t ) | {\displaystyle
Self-concordant_function
Partial order on matrices
definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix
Loewner_order
Concept in auditing and accounting
profit. These ranges can also be combined into blended methods. A concave function, such as the "gauge" formula. Gauge is a measure of materiality that
Materiality_(auditing)
into account. Typically the criterion is the expected value of some concave function of the value of the portfolio after a certain number of time periods—that
Intertemporal portfolio choice
Intertemporal_portfolio_choice
Mathematical inequality
all i = 1, . . ., n. Idea: Apply Jensen's inequality to the strictly concave function f ( x ) := ln x − ln ( 1 − x ) = ln x 1 − x , x ∈ ( 0 , 1 2 ]
Ky_Fan_inequality
Algebra theorem about convex functions
inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the
Karamata's_inequality
Is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper
Glossary_of_calculus
Organ of the urinary system
kidney is a bean-shaped structure with a convex and a concave border. A recessed area on the concave border is the renal hilum, where the renal artery enters
Kidney
Concept in game design
Positive feedback processes may be limited by making capabilities some concave function of a measure of raw success. For example: In RPG (role-playing games)
Game_balance
of a concave fractional program where all functions f , g , h j , j = 1 , … , m {\displaystyle f,g,h_{j},j=1,\ldots ,m} are affine. The function q ( x
Fractional_programming
Theorem of convex functions
opposite inequality for concave transformations). Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph
Jensen's_inequality
p; Composite-concave functions f(x) = g(Wx), where W is a d × n integer matrix with d fixed, and where g is a d-variate concave function; Certain (in)-definite
Graver_basis
Reduced quality of service due to high network traffic
matrix. Let U ( x ) {\displaystyle U(x)} be an increasing, strictly concave function, called the utility, which measures how much benefit a user obtains
Network_congestion
Concept in economics and decision theory
(indirect) utility function for money is a nonlinear function that is bounded and asymmetric about the origin. The utility function is concave in the positive
Utility
Procedure to estimate standard deviation from a sample
a nonlinear function, and only linear functions commute with taking the expectation. Since the square root is a strictly concave function, it follows
Unbiased estimation of standard deviation
Unbiased_estimation_of_standard_deviation
Zero of the derivative of a function
concavity changes from concave downwards to concave upwards and the sign of f′(x) does not change; it stays positive. For the function f(x) = x3 we have f′(0)
Stationary_point
Congenital deformity of the chest
widely known as the Nuss procedure, involves slipping in one or more concave steel bars into the chest, underneath the sternum. The bar is flipped to
Pectus_excavatum
the global maximum. A convex program is to maximize a concave function or minimize a convex function on a convex set. A set S is convex if ∀ u , v ∈ S {\displaystyle
Quadratic_knapsack_problem
Concept in psychology
Bernoulli proposed that subjective value, or utility, is a concave function of money. In such a function, the difference between the utilities of $200 and $100
Risk_aversion_(psychology)
Top border of the hip
crest is convex superiorly but is sinuously curved, being concave inward in front, concave outward behind. It is sigmoid in shape, such that viewed from
Iliac_crest
is nonempty and it never attains minus infinity. Concave function - the negative of a convex function. Convex geometry - the branch of geometry studying
List_of_convexity_topics
Property of functions which is weaker than continuity
is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively
Semi-continuity
Mathematical concept
of the entire Pareto front, convex or concave. Definition For a minimization problem with objective functions f 1 , … , f k {\displaystyle f_{1},\dots
Multi-objective_optimization
an important research topic. By traditional convention, a TUF is a concave function, including linear ones. See the depiction of some example TUFs. TUF/UA
Time-utility_function
Measure of distinguishability between two quantum states
{p_{j}}{q_{j}}}=\sum _{j}(-\log {\frac {q_{j}}{p_{j}}})(p_{j}).} Notice that log is a concave function. Therefore -log is convex. Applying Jensen's inequality, we obtain
Quantum_relative_entropy
Number of subsets of a given size
previous generating function after the substitution x → x y {\displaystyle x\to xy} . A symmetric exponential bivariate generating function of the binomial
Binomial_coefficient
Four-sided polygon
(self-intersecting, or crossed). Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ABCD add up
Quadrilateral
CONCAVE FUNCTION
CONCAVE FUNCTION
Boy/Male
Shakespearean German
Much Ado About Nothing' Follower of Don John.
Male
Egyptian
, a high Egyptian functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, the son of the functionary Heknofre.
Biblical
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Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, a great functionary.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, Functionary of the Interior.
Boy/Male
German, Shakespearean
Brave Adviser
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
CONCAVE FUNCTION
CONCAVE FUNCTION
Boy/Male
Arabic
Variant used for Mohammad - founder of Islamic religion. praiseworthy; glorified.
Boy/Male
Tamil
Highest Dharma
Girl/Female
Tamil
(Wife of Manu)
Biblical
foxes; fists; path
Girl/Female
Tamil
Pearl, Precious stone or gem
Boy/Male
Tamil
Pirnav | பீரà¯à®¨à®¾à®µÂ
Start of something new
Surname or Lastname
English
English : variant spelling of Shackelford.
Boy/Male
Muslim
Prophet, Jesus
Boy/Male
Tamil
Sampurna Nand | ஸஂபூரà¯à®£Â நஂதÂ
Boy/Male
Muslim Arabic
Faithful. Trustworthy.
CONCAVE FUNCTION
CONCAVE FUNCTION
CONCAVE FUNCTION
CONCAVE FUNCTION
CONCAVE FUNCTION
a.
Slightly concave.
a.
Concave on both sides; as, biconcave vertebrae.
n.
A concave molding.
n.
Also used adjectively; as, the conacre system or principle.
v. t.
To form in the mind; to plan; to devise; to generate; to originate; as, to conceive a purpose, plan, hope.
p. pr. & vb. n.
of Concave
a.
Plane or flat on one side, and concave on the other; as, a plano-concave lens. See Lens.
v. t.
To make hollow or concave.
v. t.
To yield or suffer; to surrender; to grant; as, to concede the point in question.
a.
Concave or hollow on both sides; double concave.
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 body of cardinals shut up in the conclave for the election of a pope; hence, the body of cardinals.
a.
Concave on one side and convex on the other, as an eggshell or a crescent.
a.
Arched; concave.
a.
Concave.
a.
Hollow and curved or rounded; vaulted; -- said of the interior of a curved surface or line, as of the curve of the of the inner surface of an eggshell, in opposition to convex; as, a concave mirror; the concave arch of the sky.
a.
Congenitally united; growing from one base, or united at their bases; united into one body; as, connate leaves or athers. See Illust. of Connate-perfoliate.
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.
imp. & p. p.
of Concave
a.
Having both ends concave; biconcave; -- said of vertebrae.