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Generalized mathematical function
In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in
Multivalued_function
Function whose values are sets (mathematics)
distinguish multivalued functions from set-valued functions (which they called set-valued relations) by the fact that multivalued functions only take multiple
Set-valued_function
Arithmetic operation
to consider these functions as multivalued functions. If log z {\displaystyle \log z} denotes one of the values of the multivalued logarithm (typically
Exponentiation
Mathematical concept
hyperbolic sine function is typically written as arsinh(x). The expressions like sin−1(x) can still be useful to distinguish the multivalued inverse from
Inverse_function
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Specific values of a multivalued function
specifically complex analysis, a multivalued function has often the property that, near almost every point, the graph of the function is the disjoint union of
Principal_value
Mathematical function, denoted exp(x) or e^x
Half-exponential function, a compositional square root of an exponential function Lambert W function#Solving equations – Multivalued function in mathematics
Exponential_function
Family of solutions to related differential equations
article. The Bessel function of the first kind is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero
Bessel_function
Angle of complex number about real axis
considered, the argument is a multivalued function operating on the nonzero complex numbers. The principal value of this function is single-valued, typically
Argument_(complex_analysis)
Certain type of mistaken proof
the power i only the principal value is chosen. When treated as multivalued functions, both sides produce the same set of values, being { e 2 π n | n
Mathematical_fallacy
Number whose cube is a given number
has a unique cube root, which is zero itself. The cube root is a multivalued function. The principal cube root is its principal value, that is a unique
Cube_root
Mathematical function
exponential function, logarithm, and the trigonometric functions. Algebraic functions are usually treated more generally as multivalued functions. The example
Algebraic_function
Mathematical function
setting, am ( u | m ) {\displaystyle \operatorname {am} (u|m)} is a multivalued function (in u {\displaystyle u} ) with infinitely many logarithmic branch
Jacobi_elliptic_functions
Point of interest for complex multi-valued functions
field of complex analysis, a branch point of a multivalued function is a point such that if the function is n {\displaystyle n} -valued (has n {\displaystyle
Branch_point
Mathematical functions
}{2n\left({2n}\right)!!}}}{\frac {1}{x^{2n}}}} As functions of a complex variable, inverse hyperbolic functions are multivalued functions that are analytic except at a finite
Inverse_hyperbolic_functions
Function whose actual domain of definition may be smaller than its apparent domain
Extension of the domain of an analytic function (mathematics) Multivalued function – Generalized mathematical function Densely defined operator – Linear operator
Partial_function
Function valued in a vector space; typically a real or complex one
Vector fields are one kind of tensor field. Coordinate vector Curve Multivalued function Parametric surface Position vector Parametrization In fact, these
Vector-valued_function
Functions of an angle
trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define
Trigonometric_functions
Inverse functions of sin, cos, tan, etc.
inverse functions are proper (i.e. strict) subsets of the domains of the original functions. For example, using function in the sense of multivalued functions
Inverse trigonometric functions
Inverse_trigonometric_functions
Type of mathematical function
be multivalued. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function e
Elementary_function
Arctangent function with two arguments
correct branch of the multivalued function Arctan ( y / x ) {\displaystyle \operatorname {Arctan} (y/x)} . The atan2 function is useful in many applications
Atan2
One-dimensional complex manifold
glued together. Examples of Riemann surfaces include graphs of multivalued functions such as z {\displaystyle {\sqrt {z}}} or log ( z ) {\displaystyle
Riemann_surface
Function with a smaller domain
is no need to restrict the domain if we allow the inverse to be a multivalued function. In relational algebra, a selection (sometimes called a restriction
Restriction_(mathematics)
Dirac delta function: useful to describe physical phenomena such as point charges. Multivalued function: one-to-many relation. Random function: Random element
List_of_types_of_functions
About mathematical functions
are ordinarily considered functions in contemporary mathematics. For example, Hardy's definition includes multivalued functions and what in computability
History of the function concept
History_of_the_function_concept
Number with a real and an imaginary part
(real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is often
Complex_number
Operation on mathematical functions
two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘
Function_composition
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
One-to-one correspondence
surjection Bijective numeration Bijective proof Category theory Multivalued function Hall 1959, p. 3 There are names associated to properties (1) and
Bijection
logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined
List of logarithmic identities
List_of_logarithmic_identities
Representation of a mathematical function
In mathematics, the graph of a function f {\displaystyle f} is the set of ordered pairs ( x , y ) {\displaystyle (x,y)} , where f ( x ) = y . {\displaystyle
Graph_of_a_function
Logarithm of a complex number
of this is that the "inverse" of the complex exponential function is a multivalued function taking each nonzero complex number z to the set of all logarithms
Complex_logarithm
Mathematical function with multiple real-number arguments
In mathematics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being
Function of several real variables
Function_of_several_real_variables
Mathematical function of two positive real arguments
the square root are allowed to be taken inconsistently, it is a multivalued function. To find the arithmetic–geometric mean of a0 = 24 and g0 = 6, iterate
Arithmetic–geometric_mean
Generalization of elliptic integrals
also on the path along which the integral is taken; it is thus a multivalued function of z {\displaystyle z} . Abelian integrals are natural generalizations
Abelian_integral
Arithmetic operation, inverse of nth power
complex number x can thus be taken to be a multivalued function. By convention the principal value of this function, called the principal root and denoted
Nth_root
Locally defined function in general relativity
domains larger than convex normal neighborhoods generally leads to a multivalued function since there may be several geodesic segments joining a pair of points
Synge's_world_function
Sigmoid shape special function
{\displaystyle \operatorname {erf} (w)=z} , so a true inverse function would be multivalued. However, for − 1 < x < 1 {\displaystyle -1<x<1} , there is
Error_function
Type of uncertainty of meaning where several interpretations are possible
notation. The ambiguity in the style of writing a function should not be confused with a multivalued function, which can (and should) be defined in a deterministic
Ambiguity
Design pattern in functional programming to build generic types
with no need to rewrite functions in the pipeline. A second situation where List shines is composing multivalued functions. For instance, the nth complex
Monad (functional programming)
Monad_(functional_programming)
Concept in probability theory
kernel as a generalised function with a (in general) random rather than certain value. That is, it is a multivalued function where the values are not
Markov_kernel
Representation of the relationship between taxation and government revenue
considering a single marginal tax rate. In addition, revenue may well be a multivalued function of tax rate; for instance, an increase in tax rate to a certain percentage
Laffer_curve
Mathematical function
mathematics, a function of a real variable is a function whose domain is a subset of R {\displaystyle \mathbb {R} } . Many real functions that are often
Function_of_a_real_variable
Cohomology with real coefficients computed using differential forms
going once around the circle in the positive direction implies a multivalued function θ {\displaystyle \theta } . Removing one point of the circle obviates
De_Rham_cohomology
needed] mathematics See mathematics. multivalued A "multivalued function” from a set A to a set B is a function from A to the subsets of B. It has typically
Glossary of mathematical jargon
Glossary_of_mathematical_jargon
Mathematical function relating circular and hyperbolic functions
\operatorname {gd} ^{-1}} functions to remain invertible with these extended domains, we might consider each to be a multivalued function (perhaps Gd {\textstyle
Gudermannian_function
Set of functions between two fixed sets
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is
Function_space
Mathematical function
{\displaystyle X} and Y {\displaystyle Y} , let F {\displaystyle F} be a multivalued map from X {\displaystyle X} to Y {\displaystyle Y} (equivalently, F
Choice_function
tile by translation. Goodman's conjecture on the coefficients of multivalued functions Invariant subspace problem – does every bounded operator on a complex
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Mathematical function that outputs real values
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each
Real-valued_function
Goodman's conjecture on the coefficients of multivalued functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician
Goodman's_conjecture
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Formula for inverting a Taylor series
′ ( a ) = 0 , {\displaystyle f'(a)=0,} where the inverse g is a multivalued function. The theorem was proved by Lagrange and generalized by Hans Heinrich
Lagrange_inversion_theorem
Mathematical series
<1 to a function satisfying an inhomogeneous linear equation with singularities at z = 0 and z=1, so can be continued to a multivalued function with these
Bilateral hypergeometric series
Bilateral_hypergeometric_series
Mathematical model of the time dependence of a point in space
allow multiple future states (i.e. the maps are generalized into multivalued functions and not uniquely defined everywhere) and the system can be subject
Dynamical_system
Overview of and topical guide to discrete mathematics
apparent domain Multivalued function – Generalized mathematical function Binary function – Function that takes two inputs Floor function – Nearest integers
Outline of discrete mathematics
Outline_of_discrete_mathematics
Mathematical approximation of a function
complex arguments x. For multivalued complex functions, such as logarithms, fractional powers, and inverse trigonometric functions, a principal branch is
Taylor_series
Property that an increasing voltage results in a decreasing current
the current is a single valued, continuous function of the voltage, but the voltage is a multivalued function of the current. In the most common type there
Negative_resistance
Topics referred to by the same term
refer to: Fat link, a one-to-many link in hypertext Multivalued function, a one-to-many function in mathematics One-to-many (data model), a type of relationship
One-to-many
Mathematical principles
by a multivalued function M ( t , h ) {\displaystyle M(\mathbf {t} ,h)} ; a particular M {\displaystyle M} -solution is a single-valued function f ( t
Mathematics_of_apportionment
Normal form used in database normalization
of dependency known as a multivalued dependency. A table is in 4NF if and only if, for every one of its non-trivial multivalued dependencies X ↠ {\displaystyle
Fourth_normal_form
Type of mathematical function
mathematics, a constant function is a function whose (output) value is the same for every input value. As a real-valued function of a real-valued argument
Constant_function
Relationship between elements of two sets
authors)[citation needed] in Properties. Such a binary relation is called a multivalued function. For example, the red and green binary relations in the diagram are
Binary_relation
Type of 2D conformal field theory
doi:10.1007/BF01215276. S2CID 122018499. Novikov, S. P. (1981). "Multivalued functions and functionals. An analogue of the Morse theory". Sov. Math., Dokl
Wess–Zumino–Witten_model
Prussian mathematician (1832–1925)
abelian integrals. This book popularized Bernhard Riemann’s work on multivalued functions among mathematicians. Neumann was elected a member of the Göttingen
Carl_Neumann
Function between metric spaces that does not increase any distance
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the morphisms
Metric_map
On the existence of a continuous selection of a multivalued map from a paracompact space
Repovš, Dušan; Semenov, Pavel V. (2014). "Continuous Selections of Multivalued Mappings". In Hart, K. P.; van Mill, J.; Simon, P. (eds.). Recent Progress
Michael_selection_theorem
Vector used in astronomy
{\mathcal {A}}} rarely has a simple definition and is generally a multivalued function of the angle θ between r and A {\displaystyle {\mathcal {A}}} . The
Laplace–Runge–Lenz_vector
Provides conditions for a parametric optimization problem to have continuous solutions
Theorem 3.5 in Shouchuan Hu; Nikolas S. Papageorgiou (1997). Handbook of Multivalued Analysis. Vol. 1: Theory. Springer-Science + Business Media, B. V. p
Maximum_theorem
Theory in differential topology
Lecture Notes:From Morse Homology to Floer Homology Novikov, Sergei. Multivalued functions and functionals. An analogue of the Morse theory, Soviet Math. Dokl
Morse_homology
British scientist (born 1958)
Potential Theory Institutions Université Laval Thesis Analytic Multivalued Functions (1984) Doctoral advisor Graham Allan Website www.mat.ulaval
Thomas_Ransford
and Q is a right hand side. Interval solution can be defined as a multivalued function u = ◊ { u : K ( p ) u = Q ( p ) , p ∈ p } {\displaystyle {\mathbf
Interval_finite_element
American mathematician
same time. The central role of set-valued mappings (also called multivalued functions) was also recognized in Rockafellar's dissertation and, in fact
R._Tyrrell_Rockafellar
Axiomatic set theories based on the principles of mathematical constructivism
{\mathcal {R}}_{ab}} represents the space of "non-unique-valued functions" or "multivalued functions" from a {\displaystyle a} to b {\displaystyle b} , but as
Constructive_set_theory
Unit-distance-preserving maps are isometries
Formally, the result is as follows. Let f {\displaystyle f} be a function or multivalued function from a d {\displaystyle d} -dimensional Euclidean space to
Beckman–Quarles_theorem
Design of experiments in statistics
and statistics literature has focused on estimation and inference for multivalued treatments and ignorability conditions for identifying the treatment
Multivalued_treatment
Numeric method in quantum chemistry
molecular ion – Two hydrogen nuclei sharing one electron Lambert W function – Multivalued function in mathematics Quantum tunnelling – Quantum mechanical phenomenon
Holstein–Herring_method
Type of NoSQL database
the language, to ensure that DataBASIC functioned to his satisfaction. Three of the implementations of MultiValue - PICK version R77, Microdata Reality
MultiValue_database
curve, note that the curve can be represented as the graph of the multivalued function y = x ( x − 1 ) ( x − λ ) {\displaystyle y={\sqrt {x(x-1)(x-\lambda
Period_mapping
Soviet ternary computer
Binary and Multivalued ICs According to VLSI Criteria written by Daniel Etiemble & Michel Israël, the authors compared binary and multivalued integrated
Setun
In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member
Integer-valued_function
Function that outputs either true or false
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B
Boolean-valued_function
Mathematical framework to model epistemic uncertainty
Dempster, A. P. (1967). "Upper and lower probabilities induced by a multivalued mapping". The Annals of Mathematical Statistics. 38 (2): 325–339. doi:10
Dempster–Shafer_theory
Quantized unit of magnetic flux
be as well but increments of flux can be quantized. The wave function can be multivalued as it happens in the Aharonov–Bohm effect or quantized as in
Magnetic_flux_quantum
Type of function
both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but
Pseudoconvex_function
Generalization of gamma distribution to multiple dimensions
formula requires a little care, because noninteger complex powers are multivalued; when n is noninteger, the correct branch must be determined via analytic
Wishart_distribution
Variable that represents an argument to a function
error status – see Semipredicate problem: Multivalued return. For example, to return two variables from a function in C, one may write: int width; int height;
Parameter (computer programming)
Parameter_(computer_programming)
) ) , {\displaystyle {\frac {dx}{dt}}(t)\in F(t,x(t)),} where F is a multivalued map, i.e. F(t, x) is a set rather than a single point in R d {\displaystyle
Differential_inclusion
Theoretical framework in physics
the original on 2012-07-22. Retrieved 2009-07-02. Kleinert, H. (2008). Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation (PDF). World
Quantum_field_theory
Topics referred to by the same term
structure equipped with at least one multivalued operation Hyperbolic functions, analogues of trigonometric functions defined using the hyperbola rather
Hyper
Connected input and output streams for computer programs
distinguished, and is analogous to a function returning a pair of values – see Semipredicate problem § Multivalued return. The usual destination is the
Standard_streams
Edge case where the signalling of failure is ambiguous
pointers – can allow for multivalued return by designating some parameters as output parameters. In this case, the function could just return the error
Semipredicate_problem
Algorithms in numerical analysis
from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at
Predictor–corrector_method
Mathematical abstraction of level sets
as a tool in Morse theory, Reeb graphs are the natural tool to study multivalued functional relationships between 2D scalar fields ψ {\displaystyle \psi
Reeb_graph
Method of differentiating single-term polynomials
derivative and the binomial theorem. However, due to the multivalued nature of complex power functions for non-integer exponents, one must be careful to specify
Power_rule
Type of data model
Inverted index Flat file Other models include: Multidimensional model Multivalue model Semantic model XML database Named graph Triplestore A given database
Database_model
Pakistani academician (born 1951)
Mathematical Association 1992. "Fixed points of asymptotically regular multivalued mappings", J. Austral. Math. Soc., (Series-A) 53(3), 313–326. 1995. "Random
Ismat_Beg
Semicontinuity for set-valued functions
Analysis. Basel: Birkhäuser. ISBN 3-7643-3478-9. Deimling, Klaus (1992). Multivalued Differential Equations. Walter de Gruyter. ISBN 3-11-013212-5. Mas-Colell
Hemicontinuity
Operating system released in 1965
as the general name of all operating environments which employ this multivalued database and have some implementation of Pick/BASIC and ENGLISH/Access
Pick_operating_system
MULTIVALUED FUNCTION
MULTIVALUED FUNCTION
Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, an Egyptian 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.
Biblical
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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
, the son of the functionary Heknofre.
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.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, a great functionary.
Male
Celtic
, great justiciary, or 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.
MULTIVALUED FUNCTION
MULTIVALUED FUNCTION
Girl/Female
Australian
Clear
Girl/Female
English
A , meaning gift of god. Famous bearer: British writer Dodie Smith, author of the children's...
Boy/Male
Hindu
Controller of destiny
Girl/Female
Hindu, Indian
One of the Name of Goddess Durga
Surname or Lastname
English
English : habitational name from a minor place in Somerset, an area of land in the marshes near Markham. This is first recorded in the form Rodenye; it derives from the genitive case of the Old English personal name Hroda (a short form of the various compound names with the first element hrÅð ‘renown’) + Old English Ä“g ‘island’, ‘dry land (in a fen)’.
Male
Greek
(ΘεÏιστής) Greek name THERISTÃS means "mowing month," referring to the month of June.
Surname or Lastname
English
English : habitational name from a place in Suffolk named Cavendish, from an Old English byname CÄfna (meaning ‘bold’, ‘daring’) + Old English edisc ‘enclosed pasture’.
Female
English
Pet form of English Jane, JANELLA means "God is gracious."
Boy/Male
Hindu
Creator of the universe
Boy/Male
Arabic, Muslim
Grateful; Very Thankful
MULTIVALUED FUNCTION
MULTIVALUED FUNCTION
MULTIVALUED FUNCTION
MULTIVALUED FUNCTION
MULTIVALUED FUNCTION
v. i.
Alt. of Functionate
prep.
Acting as a substitute; -- said of abnormal action which replaces a suppressed normal function; as, vicarious hemorrhage replacing menstruation.
a.
Pertaining to, or connected with, a function or duty; official.
n.
One of the pieces or divisions of bivalve or multivalve shells.
n.
Any mollusk which has a shell composed of more than two pieces.
a.
Pertaining to the function of an organ or part, or to the functions in general.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
v. t.
To assign to some function or office.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
a.
Alt. of Multivalvular
n.
The doctrine that all the functions of a living organism are due to an unknown vital principle distinct from all chemical and physical forces.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
a.
Destitute of function, or of an appropriate organ. Darwin.
pl.
of Functionary
n. pl.
One of the grand divisions of the animal kingdom, including the classes Cephalopoda, Gastropoda, PteropodaScaphopoda, and Lamellibranchiata, or Conchifera. These animals have an unsegmented bilateral body, with most of the organs and parts paired, but not repeated longitudinally. Most of them develop a mantle, which incloses either a branchial or a pulmonary cavity. They are generally more or less covered and protected by a calcareous shell, which may be univalve, bivalve, or multivalve.
n.
One deputed or authorized to perform the functions of another; a substitute in office; a deputy.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
adv.
In a functional manner; as regards normal or appropriate activity.