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mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most
Subadditive_set_function
Property of some mathematical functions
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain
Subadditivity
subadditive set function. Let Ω {\displaystyle \Omega } be a set and f : 2 Ω → R {\displaystyle f\colon 2^{\Omega }\rightarrow \mathbb {R} } be a set
Superadditive_set_function
Mapping function
measure in mathematics Submodular set function – Set-to-real map with diminishing returns Subadditive set function τ-additivity – Property of certain
Sigma-additive_set_function
A set function is called fractionally subadditive, or XOS (not to be confused with OXS), if it is the maximum of several non-negative additive set functions
Fractionally subadditive valuation
Fractionally_subadditive_valuation
Set-to-real map with diminishing returns
x_{2}\})-f(X\cup \{x_{2}\})} . A nonnegative submodular function is also a subadditive function, but a subadditive function need not be submodular. If Ω {\displaystyle
Submodular_set_function
Function from sets to numbers
mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values
Set_function
Concept in game theory
not contain i {\displaystyle i} . If v {\displaystyle v} is a subadditive set function, i.e., if v ( S ∪ T ) ≤ v ( S ) + v ( T ) {\displaystyle v(S\cup
Shapley_value
{\displaystyle u} is a subadditive set function. Assuming u ( ∅ ) {\displaystyle u(\emptyset )} is non-negative, every submodular function is subadditive. However,
Utility functions on indivisible goods
Utility_functions_on_indivisible_goods
Function made from a set
then pK is a seminorm on X. It may be shown that a real-valued subadditive function f:X→ℝ on an arbitrary topological vector space X is continuous at
Minkowski_functional
Negative of a convex function
f(x+y2)≥f(x)+f(y)2 If a function f is concave, and f(0) ≥ 0, then f is subadditive on [0,∞). Proof: Since f is concave and 1 ≥ t ≥ 0, letting y = 0 we have
Concave_function
space (Ω,Σ,μ), and let {gn}n∈ℕ be a sequence of L1 functions such that gn+m(x)≤gn(x)+gm(Tnx) (subadditivity relation). Then limn→∞gn(x)n=:g(x)≥−∞ for μ-a.e
Kingman's subadditive ergodic theorem
Kingman's_subadditive_ergodic_theorem
Fractal curve resembling a blancmange pudding
combinations and point-wise limits of subadditive functions are subadditive, the Takagi function is subadditive for any value of the parameter w {\displaystyle
Blancmange_curve
Relationship of various quantum subsystems
In quantum information theory, strong subadditivity of quantum entropy (SSA) is the relation among the von Neumann entropies of various quantum subsystems
Strong subadditivity of quantum entropy
Strong_subadditivity_of_quantum_entropy
operation and an order: Subadditive function: for which the value of f (x + y) is less than or equal to f (x) + f (y). Superadditive function: for which the value
List_of_types_of_functions
Distance from origin of tangent hyperplanes
In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of
Support_function
Lemma concerning the limit of subadditive sequences
calculus, Fekete's lemma (also called Fekete's subadditive lemma) is a lemma concerning the limit of subadditive sequences. The lemma provides an estimate
Fekete's_lemma
Average uncertainty in variable's states
properties of entropy as a function of random variables (subadditivity and additivity), rather than the properties of entropy as a function of the probability
Entropy_(information_theory)
require exponentially many queries). When agents' utilities are subadditive set functions (more general than submodular), a 1 m 1 / 2 − ϵ {\displaystyle
Welfare_maximization
Measurable set whose measure is zero
null sets is itself a null set (by countable subadditivity of μ {\displaystyle \mu } ). Any (measurable) subset of a null set is itself a null set (by
Null_set
Type of function in linear algebra
symmetric function if p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} for all x ∈ X . {\displaystyle x\in X.} Every subadditive symmetric function is necessarily
Sublinear_function
Concept within complex analysis
a norm, as it is not subadditive. The p {\displaystyle p} -th power ‖ f ‖ H p p {\displaystyle \|f\|_{H^{p}}^{p}} is subadditive for p < 1 {\displaystyle
Hardy_space
Set disjoint from its sumset with itself
such that any set of n nonzero integers has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, lim n f
Sum-free_set
Property of a function
superadditive integral Inner measure Subadditivity – Property of some mathematical functions Sublinear function – Type of function in linear algebra M. Marcus
Superadditivity
Operation in calculus
processes such as the fractional Brownian motion. The Choquet integral, a subadditive or superadditive integral created by Gustave Choquet in 1953. The Bochner
Integral
Topological vector space whose topology can be defined by a metric
sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then
Metrizable topological vector space
Metrizable_topological_vector_space
Bounds of a sequence
in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively.
Limit inferior and limit superior
Limit_inferior_and_limit_superior
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
Generalization of mass, length, area and volume
(cf. Dirac delta function) is given by δa(S) = χS(a), where χS is the indicator function of S . {\displaystyle S.} The measure of a set is 1 if it contains
Measure_(mathematics)
Type of entropy in quantum theory
is a continuous, nonincreasing function of the parameter α {\displaystyle \alpha } . A weak version of subadditivity can be proven: S α ( ρ A ) − S 0
Von_Neumann_entropy
Distance from zero to a number
multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that |a+b|=s(a+b) where s=±1, with its sign chosen
Absolute_value
Mathematical space with a notion of distance
space is a set together with a notion of distance between its points. The distance is measured by a function called a metric or distance function. Metric
Metric_space
Function that counts distinct factors of a string
p_{u}(n)}{n\log k}}\ .} The limit exists as the logarithm of the complexity function is subadditive. Every real number between 0 and 1 occurs as the topological entropy
Complexity_function
Mathematical function
measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying
Outer_measure
Subadditive or superadditive integral
A Choquet integral is a subadditive or superadditive integral created by the French mathematician Gustave Choquet in 1953. It was initially used in statistical
Choquet_integral
Notion in computational learning
discrepancies and subadditive processes", Annals of Probability, 6 (1): 118–227, doi:10.1214/aop/1176995615, JSTOR 2242865. Look up shattered set in Wiktionary
Shattered_set
Type of measure on Euclidean spaces
outer measure (only countably subadditive), becomes a full measure (countably additive) if restricted to the Borel sets. Evans, Lawrence C.; Gariepy,
Borel_regular_measure
Convex and balanced set
non-negative function q:X→ℝ that satisfies the following conditions: Subadditivity/Triangle inequality: q(x+y)≤q(x)+q(y) for all x,y∈X. Absolute homogeneity
Absolutely_convex_set
Function spaces generalizing finite-dimensional p norm spaces
absolutely homogeneous function for 0<p<1; however, the resulting function does not define a norm, because it is not subadditive. On the other hand, the
Lp_space
Left-invariant (or right-invariant) measure on locally compact topological group
problem is that the function given by the lim sup formula is not countably subadditive in general and in particular is infinite on any set without compact
Haar_measure
Concept in financial economics
Darkiewicz, G.; Goovaerts, M.J. (2008). "Can a Coherent Risk Measure be too Subadditive?". Journal of Risk and Insurance. 75 (2): 365–386. doi:10.1111/j.1539-6975
Coherent_risk_measure
Vector space with a notion of nearness
sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then
Topological_vector_space
Mathematical theorem
0<p<1} requires some modifications, because the p-norm is no longer subadditive. One starts with the stronger assumption that ∑ ‖ u n ‖ p p < ∞ {\displaystyle
Riesz–Fischer_theorem
Mathematical notion
topological equivalence: there exists a strictly increasing, continuous, and subadditive f : R → R + {\displaystyle f:\mathbb {R} \to \mathbb {R} _{+}} such that
Equivalence_of_metrics
Mathematical function
real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } is called a seminorm if it satisfies the following two conditions: Subadditivity/Triangle
Seminorm
Length in a vector space
denotes the usual absolute value of a scalar s {\displaystyle s} : Subadditivity / Triangle inequality: p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle
Norm_(mathematics)
Measure of algorithmic complexity
writing out the string itself. Theorem. (extra information bounds, subadditivity) K ( x | y ) ≤ K ( x ) ≤ K ( x , y ) ≤ max ( K ( x | y ) + K ( y )
Kolmogorov_complexity
mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. Universal algebra a field studying the
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Normed vector space that is complete
dealing primarily with Banach spaces. If f:X→ℝ is a subadditive function (such as a norm, a sublinear function, or real linear functional), then f is continuous
Banach_space
Criterion for fair division
Alice's opinion, Bob's share is worth 2/3. When the valuations are only subadditive, EF still implies PR, but PR no longer implies EF even with two partners:
Envy-freeness
Measure of divisibility by a prime number
also | − r | p = | r | p . {\displaystyle |{-r}|_{p}=|r|_{p}.} The subadditivity | r + s | p ≤ | r | p + | s | p {\displaystyle |r+s|_{p}\leq |r|_{p}+|s|_{p}}
P-adic_valuation
Branch of mathematics that studies dynamical systems
particle over time. A generalization of Birkhoff's theorem is Kingman's subadditive ergodic theorem. Birkhoff–Khinchin theorem. Let ƒ be measurable, E(|ƒ|)
Ergodic_theory
Alice's opinion, Bob's share is worth 2/3. When the valuations are only subadditive, EF still implies PR, but PR no longer implies EF even with two partners:
Proportional_division
Estimated potential loss for an investment under a given set of conditions
that for anchoring reasons VaR leads to higher risk taking. VaR is not subadditive: VaR of a combined portfolio can be larger than the sum of the VaRs of
Value_at_risk
Theory of generalized measures in mathematics
∪ F ) ≥ g ( E ) + g ( F ) {\displaystyle g(E\cup F)\geq g(E)+g(F)} ; subadditive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that
Fuzzy_measure_theory
Concept in Hilbert spaces mathematics
Ruskai, Mary Beth (2007). "Another short and elementary proof of strong subadditivity of quantum entropy". Reports on Mathematical Physics. 60 (1). Elsevier
Trace_inequality
Paradox in decision theory
expected utility: Created by French mathematician Gustave Choquet was a subadditive integral used as a way of measuring expected utility in situations with
Ellsberg_paradox
Distance between two statistical objects
definiteness) d(x, y) = d(y, x) (symmetry) d(x, z) ≤ d(x, y) + d(y, z) (subadditivity / triangle inequality). Many statistical distances are not metrics,
Statistical_distance
Measure of information in probability and information theory
sum of the individual entropies of the variables in the set. This is an example of subadditivity. This inequality is an equality if and only if X and Y
Joint_entropy
Mathematical folklore
open cover of X {\displaystyle X} by null sets because by choosing a countable subcover, the σ-subadditivity of μ {\displaystyle \mu } will imply that
Infinite-dimensional Lebesgue measure
Infinite-dimensional_Lebesgue_measure
Concept in the field of computational social choice
to S\T. If an agent's preference relation is represented by a subadditive utility function u, then for any agreeable subset T, u(T) ≥ u(S)/2. As an example
Agreeable_subset
holds when the buyers' valuations are fractionally subadditive. Case 2: fractionally subadditive buyers, 2nd-price auction, incomplete information: Assuming
Price_of_anarchy_in_auctions
Space with topology generated by convex sets
scalable: p(sx)=|s|p(x) for every scalar s. So, in particular, p(0)=0; p is subadditive. It satisfies the triangle inequality: p(x+y)≤p(x)+p(y). If p satisfies
Locally convex topological vector space
Locally_convex_topological_vector_space
Axioms for defining a topology
[K4] as an inclusion, giving the weaker axiom [K4''] (subadditivity): [K4''] It is subadditive: for all A , B ⊆ X {\displaystyle A,B\subseteq X} , c (
Kuratowski_closure_axioms
Generalization of metric spaces in mathematics
} Symmetry: d ( x , y ) = d ( y , x ) {\displaystyle d(x,y)=d(y,x)} Subadditivity/Triangle inequality: d ( x , z ) ≤ d ( x , y ) + d ( y , z ) {\displaystyle
Pseudometric_space
Difference in valuation of a payoff when receiving it earlier versus later
2024-11-18 Read, Daniel (2001-07-01). "Is Time-Discounting Hyperbolic or Subadditive?". Journal of Risk and Uncertainty. 23 (1): 5–32. doi:10.1023/A:1011198414683
Time_preference
Property of geometry, also used to generalize the notion of "distance" in metric spaces
norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.
Triangle_inequality
Probability theory for low quality data
gap. A lower probability function is superadditive but not necessarily additive, whereas an upper probability is subadditive. To get a general understanding
Imprecise_probability
Mathematical theorem in real analysis
the ball of radius R {\displaystyle R} centered at 0. By countable subadditivity, there exists at least one R 0 {\displaystyle R_{0}} so that m ( A ∩
Steinhaus_theorem
Systematic pattern of deviation from norm or rationality in judgment
superiority (better-than-average effect) and worse-than-average effect, subadditivity effect, exaggerated expectation, overconfidence, and the hard–easy effect
Cognitive_bias
well-known for giving rise to other tools of mathematics, including the Subadditive Ergodic Theorem, a fundamental result in ergodic theory. Outside mathematics
First_passage_percolation
results to more complex valuation classes:Subadditive functions: Functions where the value of the union of two sets is no more than the sum of their individual
Budget-feasible_mechanism
actually were. Based on the evidence, memories are not extreme enough. Subadditivity effect: The tendency to estimate that the likelihood of a remembered
List_of_cognitive_biases
Generalization of Riemannian manifolds
F(v + w) ≤ F(v) + F(w) for every two vectors v,w tangent to M at x (subadditivity). F(λv) = λF(v) for all λ ≥ 0 (but not necessarily for λ < 0) (positive
Finsler_manifold
{E} [X]+\mathbb {E} [Y]\leq \mathbb {E} [X+Y]} Choquet expectation: a subadditive or superadditive integral that is used in image processing and behavioral
Nonlinear_expectation
Dimension of the column space of a matrix
(BC)\leq \operatorname {rank} (B)+\operatorname {rank} (ABC).} Rank subadditivity: if A and B are m × n matrices, then | rank ( A ) − rank ( B ) |
Rank_(linear_algebra)
Generalization of the concept of a norm
{\displaystyle X} is a function p : X → [ 0 , + ∞ ) {\displaystyle p:X\to [0,+\infty )} that has the following properties: Subadditivity, or the triangle inequality:
Asymmetric_norm
Measure of connection disorder in a network
not satisfy the property of sub-additivity (see Von Neumann entropy's subadditivity), expected to hold theoretically. A more grounded definition, satisfying
Network_entropy
Fair division problem for discrete items
bundles are weakly worse than his current share. For every agent with subadditive utility, the mFS is worth at least 1 / n {\displaystyle 1/n} . Hence
Fair_item_allocation
Mathematical function often applied to matrices
the operator norm is submultiplicative, and the logarithmic norm is subadditive. These properties do not hold for the spectral radius and abscissa, which
Logarithmic_norm
Theorem in probability theory
_{N\to \infty }\Pr \left(\bigcup _{n=N}^{\infty }E_{n}\right).} By subadditivity, Pr ( ⋃ n = N ∞ E n ) ≤ ∑ n = N ∞ Pr ( E n ) . {\displaystyle \Pr \left(\bigcup
Borel–Cantelli_lemma
Criterion of fair item allocation
existence proof for 1/5-fraction, and an upper bound of 1/2-fraction. For subadditive valuations: an existence proof for log(m)/10-fraction MMS-fairness, and
Maximin_share
{\displaystyle G_{2}} on the same vertex set, then H ( G 1 ) ≤ H ( G 2 ) {\displaystyle H(G_{1})\leq H(G_{2})} . Subadditivity. Given two graphs G 1 = ( V , E
Graph_entropy
Mathematical monograph
theorem on the structure of sequences whose lower asymptotic density is subadditive. It studies essential components, sequences that when added to another
Sequences_(book)
Vanishing theorem for multiplier ideals
Demailly, Jean-Pierre; Ein, Lawrence; Lazarsfeld, Robert (2000). "A subadditivity property of multiplier ideals". Michigan Mathematical Journal. 48. arXiv:math/0002035
Nadel_vanishing_theorem
mathematician, translator and biographer Mary Beth Ruskai (1944–2023), proved subadditivity of quantum entropy, bounded the electrons in an atom, advocate for women
List_of_women_in_mathematics
History of the development of microeconomics as a study
monopoly" (p. 810): mathematically this equivalent to subadditivity of the cost function. He then sets out to prove 12 propositions related to strict economies
History_of_microeconomics
Kind of fair item allocation
a set of buyers who are interested in these items. The buyers have different valuations to the items, and they have a quasilinear utility function; this
Envy-free_pricing
equilibrium does not always exist, even if the bidders have subadditive utility functions. Once we know that a subgame perfect equilibrium exists, the
Sequential_auction
of being together), which normally arises when their cooperation is subadditive. Bogomolnaia, Anna; Jackson, Matthew O. (Feb 2002). "The Stability of
Hedonic_game
Ecological mechanism enabling species to coexist
suffers more from competition in bad years than in good years. If γ < 0 (subadditivity, or buffered population growth), it means that the harm done by competition
Storage_effect
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SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
Female
Egyptian
, a sister of Sekherta.
Female
Hungarian
Hungarian form of Greek Elisabet, ERZSÉBET means "God is my oath."
Surname or Lastname
English and German
English and German : topographic name for someone who lived by the sea-shore or beside a lake, from Middle English see ‘sea’, ‘lake’ (Old English sǣ), Middle High German sē. Alternatively, the English name may denote someone who lived by a watercourse, from an Old English sēoh ‘watercourse’, ‘drain’.
Surname or Lastname
English
English : perhaps a variant of Sait, from the Old English personal name Sǣgēat (‘sea Geat’).
Male
Hebrew
Variant spelling of Hebrew Sheth, SHET means "buttocks."
Male
Egyptian
, the seven great spirits of the Ritual of the Dead.
Female
Egyptian
, second wife of Antef.
Boy/Male
Egyptian Hebrew Swedish
Son of Seb and Nut.
Male
English
Anglicized form of Hebrew Sheth, SETH means "buttocks." In the bible, this is the name of the third son of Adam and Eve. Compare with other forms of Seth.
Female
Egyptian
, a sister of Sekherta.
Female
Egyptian
, a wife and daughter of Antef.
Female
Egyptian
, the wife of the usurper Sipthah.
Female
Egyptian
, the wife of Osirtesen.
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Female
English
Short form of English Elizabeth, BET means "God is my oath."Â
Male
Egyptian
, the seven great spirits of the Ritual of the Dead.
Female
Egyptian
, the mother of Fai-hor-ou-oer.
Female
Egyptian
, an uncertain goddess.
Male
English
Short form of English Stephen, STE means "crown."
Surname or Lastname
English
English : variant spelling of See.
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
v. t.
To make to agree with some standard; as, to set a watch or a clock.
v. t.
To establish as a rule; to furnish; to prescribe; to assign; as, to set an example; to set lessons to be learned.
a.
Established; prescribed; as, set forms of prayer.
a.
Firm; unchanging; obstinate; as, set opinions or prejudices.
a.
Regular; uniform; formal; as, a set discourse; a set battle.
v. t.
To cause to sit; to make to assume a specified position or attitude; to give site or place to; to place; to put; to fix; as, to set a house on a stone foundation; to set a book on a shelf; to set a dish on a table; to set a chest or trunk on its bottom or on end.
n.
See Set, n., 2 (e) and 3.
v. t.
To put in order in a particular manner; to prepare; as, to set (that is, to hone) a razor; to set a saw.
v. t.
To compose; to arrange in words, lines, etc.; as, to set type; to set a page.
v. t.
To determine; to appoint; to assign; to fix; as, to set a time for a meeting; to set a price on a horse.
v. t.
To extend and bring into position; to spread; as, to set the sails of a ship.
v. i.
To be fixed for growth; to strike root; to begin to germinate or form; as, cuttings set well; the fruit has set well (i. e., not blasted in the blossom).
imp. & p. p.
of Set
n.
A series of as many games as may be necessary to enable one side to win six. If at the end of the tenth game the score is a tie, the set is usually called a deuce set, and decided by an application of the rules for playing off deuce in a game. See Deuce.
n.
That which is set, placed, or fixed.
n.
Direction or course; as, the set of the wind, or of a current.
v. i.
To fit or suit one; to sit; as, the coat sets well.
n.
A young plant for growth; as, a set of white thorn.
a.
Fixed in position; immovable; rigid; as, a set line; a set countenance.
v. t.
To reduce from a dislocated or fractured state; to replace; as, to set a broken bone.
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