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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
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
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
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
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
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
Negative of a convex function
{x+y}{2}}\right)\geq {\frac {f(x)+f(y)}{2}}} If a function f is concave, and f(0) ≥ 0, then f is subadditive on [ 0 , ∞ ) {\displaystyle [0,\infty )} . Proof:
Concave_function
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
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
In mathematics, Kingman's subadditive ergodic theorem is one of several ergodic theorems. It can be seen as a generalization of Birkhoff's ergodic theorem
Kingman's subadditive ergodic theorem
Kingman's_subadditive_ergodic_theorem
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
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
Operation in mathematical 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
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
Function made from a set
{1}{a}}|f(x)|\quad {\text{ for all }}x\in X.} The function p K {\textstyle p_{K}} has the following properties: It is subadditive: p K ( x + y ) ≤ p K ( x ) + p K (
Minkowski_functional
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
require exponentially many queries). When agents' utilities are subadditive set functions (more general than submodular), a 1 m 1 / 2 − ϵ {\displaystyle
Welfare_maximization
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)
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
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
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
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)
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
Notion in computational learning
discrepancies and subadditive processes", Annals of Probability, 6 (1): 118–227, doi:10.1214/aop/1176995615, JSTOR 2242865. Origin of "Shattered sets" terminology
Shattered_set
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
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
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
Mathematical space with a notion of distance
topologically equivalent. This can be done using a subadditive monotonically increasing bounded function which is zero at zero, e.g. d ′ ( x , y ) = d (
Metric_space
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
Function spaces generalizing finite-dimensional p norm spaces
homogeneous function for 0 < p < 1 ; {\displaystyle 0<p<1;} however, the resulting function does not define a norm, because it is not subadditive. On the
Lp_space
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
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 ) {\displaystyle |a+b|=s(a+b)}
Absolute_value
holds when the buyers' valuations are fractionally subadditive. Case 2: fractionally subadditive buyers, 2nd-price auction, incomplete information. Assuming
Price_of_anarchy_in_auctions
Convex and balanced set
is any non-negative function q : X → R {\displaystyle q:X\to \mathbb {R} } that satisfies the following conditions: Subadditivity/Triangle inequality:
Absolutely_convex_set
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)
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
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
Normed vector space that is complete
→ R {\displaystyle f:X\to \mathbb {R} } is a subadditive function (such as a norm, a sublinear function, or real linear functional), then f {\displaystyle
Banach_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
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
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
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
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
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
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
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
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
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
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
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
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
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
Highest power of p dividing a given 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
Concept in Hlibert 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
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
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
Space with topology generated by convex sets
particular, p ( 0 ) = 0 {\displaystyle p(0)=0} ; p {\displaystyle p} is subadditive. It satisfies the triangle inequality: p ( x + y ) ≤ p ( x ) + p ( y
Locally convex topological vector space
Locally_convex_topological_vector_space
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
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
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
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
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
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
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
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
Measure of information in probability and information theory
of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set. This is an example of subadditivity. This
Joint_entropy
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
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)
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)
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
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
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
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
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
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
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
{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
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
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
{\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
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
Kind of fair item allocation
O(log (n) + log (m)) of the maximum welfare. With limited supply, for subadditive valuations, a random single price achieves revenue within 2O(√(log n
Envy-free_pricing
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
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
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Female
Egyptian
, the wife of the usurper Sipthah.
Female
Egyptian
, the mother of Fai-hor-ou-oer.
Male
Hebrew
Variant spelling of Hebrew Sheth, SHET means "buttocks."
Female
Egyptian
, a sister of Sekherta.
Male
Egyptian
, the seven great spirits of the Ritual of the Dead.
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.
Boy/Male
Egyptian Hebrew Swedish
Son of Seb and Nut.
Surname or Lastname
English
English : variant spelling of See.
Male
Egyptian
, the seven great spirits of the Ritual of the Dead.
Female
Egyptian
, an uncertain goddess.
Male
English
Short form of English Stephen, STE means "crown."
Female
Egyptian
, second wife of Antef.
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’).
Female
English
Short form of English Elizabeth, BET means "God is my oath."Â
Female
Hungarian
Hungarian form of Greek Elisabet, ERZSÉBET means "God is my oath."
Female
Egyptian
, the wife of Osirtesen.
Female
Egyptian
, a wife and daughter of Antef.
Female
Egyptian
, a sister of Sekherta.
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Servant of Janaki
Girl/Female
Australian, Irish
Saint
Boy/Male
French
Named for Saint Denys.
Boy/Male
American, British, English
From the Bull Pasture; Surname
Girl/Female
American, British, Christian, English, French, Greek, Indian, Jamaican, Swedish
Pure; Form of Katherine; Virginal
Boy/Male
British, English
Bull Meadow; Meadow of the Sheep
Girl/Female
American, Christian, Gujarati, Hindu, Indian, Japanese, Kannada, Malayalam, Marathi, Sanskrit, Sindhi, Telugu
Water; Toy
Female
Scottish
Pet form of Scottish Aileen, EILIDH means "beauty, radiance."
Girl/Female
Tamil
King of the gods
Girl/Female
Indian, Punjabi, Sikh
To Stay in God's Eyes
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
SUBADDITIVE SET-FUNCTION
n.
That which is set, placed, or fixed.
v. t.
To establish as a rule; to furnish; to prescribe; to assign; as, to set an example; to set lessons to be learned.
imp. & p. p.
of Set
a.
Regular; uniform; formal; as, a set discourse; a set battle.
a.
Fixed in position; immovable; rigid; as, a set line; a set countenance.
v. i.
To fit or suit one; to sit; as, the coat sets well.
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 compose; to arrange in words, lines, etc.; as, to set type; to set a page.
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).
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 make to agree with some standard; as, to set a watch or a clock.
a.
Established; prescribed; as, set forms of prayer.
n.
Direction or course; as, the set of the wind, or of a current.
a.
Firm; unchanging; obstinate; as, set opinions or prejudices.
n.
See Set, n., 2 (e) and 3.
v. t.
To reduce from a dislocated or fractured state; to replace; as, to set a broken bone.
n.
A young plant for growth; as, a set of white thorn.
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.
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.