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	List of mathematical functions - Wikipedia, the free encyclopedia
		A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations.
		Subadditive function: The value of a sum is less than or equal to the sum of the values of the summands.
		Superadditive function: The value of a sum is greater than or equal to the sum of the values of the summands.
	en.wikipedia.org /wiki/List_of_functions (796 words)

	List of mathematical functions - Wikipedia, the free encyclopedia
		Related functions are the quarter period and the nome.
		Legendre function: From the theory of spherical harmonics.
		Ackermann function: in the theory of computation, a recursive function that is not primitive recursive.
	en.wikipedia.org /wiki/List_of_mathematical_functions (796 words)

	Subadditive function - Encyclopedia, History, Geography and Biography
		The major reason for use of subadditive sequences is the following lemma due to Fekete.
		There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present.
		This article incorporates material from subadditivity (http://planetmath.org/?op=getobjandfrom=objectsandid=4615) on PlanetMath, which is licensed under the GFDL.
	www.arikah.net /encyclopedia/Subadditive (180 words)

	Subadditive function - Wikipedia, the free encyclopedia
		The major reason for use of subadditive sequences is the following lemma due to M.
		A good exposition of this topic may be found in Steele's Probability theory and combinatorial optimization given in the references.
		This article incorporates material from subadditivity on PlanetMath, which is licensed under the GFDL.
	en.wikipedia.org /wiki/Subadditive_function (186 words)

	AIF : Tome 21 fascicle 1 -- 1971
		This is a general study of an increasing, countably subadditive set function, called a capacity, and defined on the subsets of a topological space
		The principal aim is the study of the ``quasi-topological'' properties of subsets of
		Sufficient conditions are obtained with a convex cone of lower semicontinuous functions on
	annalif.ujf-grenoble.fr /Vol21/E211_6/E211_6.html (121 words)

	Course Outline and Syllabus, Econ 309
		Subadditive Cost Functions and Natural Monopoly: It used to be that natural monopoly was simply defined as existing when the AC curve is everywhere downward-sloping relative to market demand (economies of scale for for a single firm to produce the market-clearing Q).
		Baumol (1977) and others pioneered the notion of subadditive costs, and that we can define natural monopoly (both for single- and multi-product lines) as existing when the cost function is subadditive.
		A cost function is subadditive when the total cost of producing industry output is lowest when a single firm produces it.
	www.humboldt.edu /~storage/sh2/econ459/lects/ch_11.html (1161 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Yet the cost function is subadditive over the whole range of outputs since the combined fixed costs of two (or more) firms, each incurring Fl, exceed the fixed cost of a single producer, i.e.
		In practice subadditivity may be rather difficult to detect unless the cost function under consideration exhibits special properties from which subadditivity can be readily inferred.
		The reason for this is that in order to ascertain whether a cost function is subadditive at a particular output bundle, one needs to know the costs of producing any smaller configuration of outputs.
	www-wds.worldbank.org /servlet/WDSContentServer/WDSP/IB/1999/08/15/000009265_3960930053350/Rendered/INDEX/multi_page.txt (8792 words)

	1973
		Finally he speculates about the changes in the demand functions that are caused by the correlation of the stock price and factor Wiener processes.
		As an application, Merton derives a theorem that says that if there is only one changing and correlated factor which can be represented by a portfolio tradable assets then all investors should have the portfolios that can be decomposed into riskless asset, market portfolio and the portfolio that represents the factor.
		The theory of subadditive processes helps to proof the existence of a limit for a maximal increasing sequence in a permutation.
	www.io.com /~slava/history/1973.htm (712 words)

	Nat' Academies Press, Biographical Memoirs V.63 (1994) (Site not responding. Last check: 2007-11-03)
		The basic relation between the function f(λ) and the corresponding operator f(A) is given by and A being the infinitesimal generator of T(t); thus, both f(λ) and f(A) are Laplace-Stieltjes transforms.
		On the logarithmic derivatives of the gamma function.
		On the inverse function theorem in Banach algebras.
	www.nap.edu /openbook/0309049768/html/218.html (3455 words)

	Mathematical Programming Glossary - L
		When a univariate function is piece-wise linear, it has the form a(i)x + b(i) for x in the interval [c(i), c(i+1)], where a(i) is not equal to a(i+1).
		(The phrase usually means the function is continuous.) This arises in linear programming when considering the optimal value as a function of varying right-hand sides (or cost coefficients) in fixed proportions: b+td (or c+td), where d is an admissible change vector and t is the (scalar) variable.
		There exists a subset of the domain, such as the neighborhood of a point, such that the function is convex on that subset.
	home.eunet.cz /berka/o/English/lp/glossary/L.html (2170 words)

	GAP (NumericalSgps) - Chapter 1: Introduction
		A subadditive function with period m is a map f: N-> N such that f(0)=0, f(x+y)<= f(x)+f(y) and f(x+m)=f(x).
		Thus a numerical semigroup can be given by a subadditive function with a given period.
		If S is a numerical semigroup and sin S, snot=0, and rm Ap(S,s)= w(0),w(1),..., w(s-1), then f(x)=w(x mod s) is a subadditive function with period s such that rm M_f=S. Let S be a numerical semigroup generated by n_1,...,n_k.
	www-gap.dcs.st-and.ac.uk /Manuals/pkg/numericalsgps/doc/chap1.html (1324 words)

	AMERICAN MATHEMATICAL MONTHLY - April 1999
		The zero-one valued function of f defined by f(n) = 1 if there are at least n consecutive 4's in the decimal expansion of pi, is computable according to the received wisdom.
		I became curious about the possibility of characterizing those functions f, like x/(1+x), for which f(d(×)) is a metric, or a metric that is equivalent to d.
		A metric-preserving function f has the property that f(d(×)) is equivalent to d for every metric d if and only if f is continuous.
	www.maa.org /pubs/monthly_apr99_toc.html (818 words)

	Approximation of subadditive functions and convergence rates in limiting-shape results, Kenneth S. Alexander
		For a nonnegative subadditive function $h$ on $\mathbb{Z}^d$, with limiting approximation $g(x) = \lim_n h(nx)/n$, it is of interest to obtain bounds on the discrepancy between $g(x)$ and $h(x)$, typically of order $x^{\nu}$ with $\nu < 1$.
		For certain subadditive $h(x)$, particularly those which are expectations associated with optimal random paths from 0 to $x$, in a somewhat standardized way a more natural and seemingly weaker property can be established: every $x$ is in a bounded multiple of the convex hull of the set of sites satisfying a similar bound.
		Applications include rates of convergence in limiting-shape results for first-passage percolation (standard and oriented) and longest common subsequences and bounds on the error in the exponential-decay approximation to the off-axis connectivity function for subcritical Bernoulli bond percolation on the integer lattice.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aop/1024404277 (475 words)

	Large Deviation Theory
		For a sample mean sequence, its LDP contains WLLN since the lower and upper bound of the rate function are both 0, if the event A contains the expectation; so with probability one, the event A will happen, i.e., WLLN.
		The log-moment generating function is not related to time.
		The asymptotic log-moment generating function is intended to remove the effect of time by taking the limit of log-moment generating function as time goes to infinity, assuming the limit exists.
	www.wu.ece.ufl.edu /books/math/probability/LargeDeviationTheory.html (201 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		However, the cost function exhibits diseconomies of scope and, thus, fails to be subadditive.
		In this context, a cost allocation procedure is a function which, on a per unit basis, assigns to each outupt a share of costs.
		Equation (2.22'), which applies to homogenous cost functions only, is the multiproduct extension of formula (2.1) which - unlike formula (2.9) - defines scale economies in terms of the ratio of AC to MC.
	www-wds.worldbank.org /servlet/WDSContentServer/WDSP/IB/1999/08/15/000009265_3961002194442/Rendered/INDEX/multi_page.txt (13469 words)

	v8n3
		In the paper, three basic properties of the logarithmically N-alternating monotonic functions are established and the monotonicity results of some functions involving the gamma and q-gamma functions, which are obtained in [W. Clark and M. Ismail, Inequalities involving gamma and psi functions, Anal.
		Upper and lower bounds for the Cebysev functional of a convex and a bounded function are given.
		In this paper, the log-convexity of two-parameter homogeneous functions and its corollaries in [4] are refined and extended.
	rgmia.vu.edu.au /v8n3.html (789 words)

	Nat' Academies Press, Biographical Memoirs V.63 (1994) (Site not responding. Last check: 2007-11-03)
		group of operators is a real-valued subadditive function on the half-line.
		With this in mind, Hille extended the earlier theory on subadditive functions on the positive integers to measurable subadditive functions on the half-line.
		However, Hille treated functions f(λ), which are holomorphic in the interior of the spectrum of A but may be merely continuous on those points of the spectrum that lie on the abscissa of
	www.nap.edu /openbook/0309049768/html/226.html (378 words)

	GAP (NumericalSgps) - Chapter 2: Numerical Semigroups
		A periodic subadditive function with period m is given through the list of images of the elements, from 1 to m.
		The argument of each of these functions is a list representing an entity of the type to which the function's name refers.
		Then are presented functions to test if a given list represents the small elements, gaps or the Apéry set (see 1.
	www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/numericalsgps/doc/chap2.html (705 words)

	ISMP 2000 - Meeting Topics (Site not responding. Last check: 2007-11-03)
		We show that the system of inequalities associated with this latter problem is TDI for any subadditive cost function defined over all the cliques of an interval graph.
		This problem -- which we call the convex cost closure problem, or (CCC) -- is a generalization of the known maximum (or minimum) closure problem and the isotonic regression problem.
		Since the convex cost closure problem generalizes both minimum cut and minimization of $n$ convex functions this complexity is fastest possible.
	www.isye.gatech.edu /ismp2000/schedule/session_pages/WEC-14-IC215.html (352 words)

	GAP (NumericalSgps) - Appendix A: Generalities
		Here we describe some functions which are not specific for numerical semigroups but are used to do computations with them.
		A periodic function f of period m from the set N of natural numbers into itself may be specified through a list of m natural numbers.
		represents a periodic subAdditive function f periodic of period m or not.
	www-groups.dcs.st-and.ac.uk /gap/Manuals/pkg/numericalsgps/doc/chapA.html (269 words)

	Mathematical Programming Glossary - L (Site not responding. Last check: 2007-11-03)
		This finds the maximum of a unimodal function on a finite set, {x1, x2,..., xm}, by evaluating the function at points placed by the modified fibonacci sequence: K_(n+2) = K_(n+1) + K_n + 1.
		Then, for the range of t where the LP has a solution, the optimal value function has this piece-wise linear (continuous) form, and the intervals that comprise the domain are called linearity intervals.
		A feasible point x* that is an optimal solution to the mathematical program whose feasible region is the intersection of the original region and some neighborhood of x*.
	www.cudenver.edu /~hgreenbe/glossary/L.html (2371 words)

	Please see PDF version
		Since in most cases of interest S is uncountable, the function D,,s is not a priori measurable.
		The existence of the limits on the left side of equations (4.9), (4.10), and (4.11) is an immediate consequence of Kingman's subadditive ergodic theorem and Theorem 2.
		The Theorem 6.1 is then an instance of computing the "time constant" of a subadditive process, and the calculation of such constants was pointed out by Kingman [131 to be a basic problem in the theory of subadditive processes.
	www-stat.wharton.upenn.edu /~steele/Papers/HTML/Edasp.html (3255 words)

	The Buck Stops Here: Literary Jargon
		If I write in an academic paper that "The TELRIC cost model for UNE prices should no longer be based on a scorched-earth assumption," hardly anyone would understand it, except for people who know something about telecommunications cost models, and they would find it perfectly clear.
		If I asked, "Must the industry cost function be subadditive in order for regulators to use the Vogelsang-Finsinger mechanism to achieve Ramsey pricing," hardly anyone would understand the question, except for people who are familiar with the regulatory economics literature, who would understand it quite well.
		If I write that a particular piece of music ends by taking a Neapolitan 6 chord, moving to a V-7 chord in second inversion, and then resolving to I by employing a passing note to a Picardy third, trained musicians will understand what I've said, but no one else will.
	stuartbuck.blogspot.com /2004/01/literary-jargon.html (282 words)

	math lessons - Category:Sequences and series (Site not responding. Last check: 2007-11-03)
		In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, or after, every other member.
		A sequence is a function with a domain equal to the set of positive integers.
		A series is a sum of a sequence of terms.
	www.mathdaily.com /lessons/Category:Sequences_and_series (96 words)

	Natural Monopoly (Site not responding. Last check: 2007-11-03)
		The cost function C(q) of a firm is this relationship between cost and quantity.
		In a market with a subadditive cost function, a monopoly maximizes economic efficiency and therefore receives the distinction of being a natural monopoly.
		The key reason for this is that subadditivity assumes that all firms have the same cost function, which is clearly not the case.
	www.stwing.upenn.edu /~ingraham/personal/NaturalMonopoly.html (3175 words)

	Probability Abstract Service (Site not responding. Last check: 2007-11-03)
		Fukushima For a symmetric Markov process, the decomposition theorem of the associated additive functional into its martingale part and energy zero part has been formulated admitting an exceptional set of starting points of zero capacity due to its reliance on the potential theory of the Dirichlet form.
		Under the assumption of the absolute continuity of the transition probability, we present a necessary and sufficient condition on the energy measure for the decomposition to be refined to a strict one holding for every starting point.
		Kenneth S. Alexander For a nonnegative subadditive function h on Z^d, with limiting approximation g(x) = lim h(nx)/n, it is of interest to obtain bounds on the discrepancy between g(x) and h(x), typically of order x^a for some a < 1.
	www.economia.unimi.it /PAS/Letters/letter_20.shtml (1298 words)

	[No title]
		Think of subadditivity as an extension of the concept of economies of scale to the multiproduct case.
		so long as the cost function is subadditive at the relevant output level.
		Notice that the cost function is subadditive up to output level
	www.clt.astate.edu /crbrown/eleven1.htm (516 words)

	Licht, Michaille: Global-Local subadditive ergodic theorems and application to homogenization in elasticity
		Global-Local subadditive ergodic theorems and application to homogenization in elasticity.
		Stochastic homogenization for an integral functional of quasiconvex function with linear growth.
		Homogenization of non convex integral functionals and cellular elastic material.
	www-mathdoc.ujf-grenoble.fr /numdam-bin/item?id=AMBP_2002__9_1_21_0 (117 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		A cost function for an industry in natural monopoly is said to be subadditive.
		We can show that this cost structure is subadditive because the cheapest way to supply this service in total is to supply all three.
		Fourth, it has macroeconomic management functions which might be affected by the behaviour and performance of the public industries.
	www.uel.ac.uk /elbs/undergraduate/economics/pm/309WK12.doc (3228 words)

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