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Process to make sure wave functions can induce probability distributions
In quantum field theory, wave function renormalization is a rescaling (or renormalization) of quantum fields to take into account the effects of interactions
Wave_function_renormalization
Concept in theoretical physics
in the value of the charge is determined by the renormalization group equation. The renormalization group was initially developed for particle physics
Renormalization_group
Identity in abelian theories due to gauge invariance
and Yasushi Takahashi to relate the wave function renormalization of the electron to its vertex renormalization factor, guaranteeing the cancellation
Ward–Takahashi_identity
Method in physics used to deal with infinities
skepticism, it was Paul Dirac who pioneered renormalization. Today, on the basis of the breakthrough renormalization group insights of Nikolay Bogolyubov and
Renormalization
Elementary particle or quantum of light
photons have zero rest mass, no wave function defined for a photon can have all the properties familiar from wave functions in non-relativistic quantum mechanics
Photon
Symmetry breaking through the vacuum state
"Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order". Phys. Rev. B. 82 (15) 155138. arXiv:1004
Spontaneous_symmetry_breaking
Theoretical framework in physics
Costello's monograph Renormalization and Effective Field Theory provides a rigorous formulation of perturbative renormalization that combines both the
Quantum_field_theory
Type of order at absolute zero
"Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order". Phys. Rev. B. 82 (15) 155138. arXiv:1004
Topological_order
Mathematical transform that expresses a function of time as a function of frequency
(or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well
Fourier_transform
function reduces, yielding a renormalization group flow that flows away from the topological phase transitions. The name "curvature renormalization group"
Curvature renormalization group method
Curvature_renormalization_group_method
Theories, models and concepts that go back to the quantum hypothesis of Max Planck
"Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order". Phys. Rev. B. 82 (15) 155138. arXiv:1004
Applications of quantum mechanics
Applications_of_quantum_mechanics
named renormalization. This "divergence problem" was solved in the case of quantum electrodynamics through the procedure known as renormalization in 1947–49
History of quantum field theory
History_of_quantum_field_theory
Mathematical wave functions
variational renormalization group methods for quantum spin systems. In 2006, Vidal developed the multi-scale entanglement renormalization ansatz (MERA)
Tensor_network
Divergences arising from high energy physics
Infrared divergence Cutoff (physics) Renormalization group UV fixed point Causal perturbation theory Zeta function regularization J.D. Bjorken, S. Drell
Ultraviolet_divergence
Quantum field theory of electromagnetism
though renormalization works well in practice, Feynman was never entirely comfortable with its mathematical validity, referring to renormalization as a
Quantum_electrodynamics
Numerical variational technique
renormalization group method, because they all happened to fail with this simple problem. The DMRG overcame the problems of previous renormalization group
Density matrix renormalization group
Density_matrix_renormalization_group
Italian theoretical physicist
Rattazzi (1998). "Extracting supersymmetry breaking effects from wave function renormalization". Nucl. Phys. B. 511 (1–2): 25–44. arXiv:hep-ph/9706540. Bibcode:1998NuPhB
Gian_Francesco_Giudice
Phenomenon in many-body quantum systems
"Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order". Physical Review B. 82 (15) 155138.
Topological_degeneracy
Hypothetical elementary particle that mediates gravity
gravitational wave energy. There is no complete quantum field theory of gravitons due to the unsolved mathematical problem of renormalization in general
Graviton
Relativistic wave equation describing massless fermions
particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles which have an inherent
Weyl_equation
of a quantum field theory may be modified by renormalization in the full quantum theory. Renormalization theorems are common in theories with a sufficient
Supersymmetry nonrenormalization theorems
Supersymmetry_nonrenormalization_theorems
Chinese-American physicist
"Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order". Physical Review B. 82 (15) 155138.
Xiao-Gang_Wen
Technique in computational quantum field theory
computed from the right and left LFCC eigenstates. Renormalization concepts, especially the renormalization group methods in quantum theories and statistical
Light-front computational methods
Light-front_computational_methods
Force resulting from the quantisation of a field
of the physics. This argument is the underpinning of the theory of renormalization. Dealing with infinite quantities in this way was a cause of widespread
Casimir_effect
Pictorial representation of the behavior of subatomic particles
procedure, to include particle self-interactions. The technique of renormalization, suggested by Ernst Stueckelberg and Hans Bethe and implemented by
Feynman_diagram
Formulation of the quantum many-body problem
known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to
Second_quantization
Lowest possible energy of a quantum system or field
theory led to the idea of incorporating renormalization into QED to deal with zero-point infinities. Renormalization was originally developed by Hans Kramers
Zero-point_energy
Problem in physics and quantum mechanics
system can be extracted from its wave function. Solving Schrödinger's equation allows you to find the wave function associated with a system, and therefore
Many-body_problem
Quantum state of multiple particles represented as complex matrices
&0&0\end{bmatrix}}.} Density matrix renormalization group Variational method (quantum mechanics) Renormalization Markov chain Tensor network Perez-Garcia
Matrix_product_state
Toy model in quantum field theory
they solve the renormalization group equations for the propagator of the bifermion field, using the fact that the only renormalization of the coupling
Gross–Neveu_model
Technique in computational quantum field theory
both cases the success of the renormalization program requires that the theory has a fixed point of the renormalization group; however, the details of
Light_front_quantization
Quantum state with the lowest possible energy
not a simple empty space, but instead contains fleeting electromagnetic waves and particles that pop into and out of the quantum field. The QED vacuum
Quantum_vacuum_state
Japanese physicist (1906-1979)
term, the theory gave finite results; thus Tomonaga discovered the renormalization method independently of Julian Schwinger and calculated physical quantities
Shin'ichirō_Tomonaga
Dimensionality of space at which the character of the phase transition changes
In the renormalization group analysis of phase transitions in physics, a critical dimension is the dimensionality of space at which the character of the
Critical_dimension
Formulation of quantum mechanics
Changing the scale of the regulator leads to the renormalization group. In fact, renormalization is the major obstruction to making path integrals well-defined
Path_integral_formulation
Attempt to find a consistent theory of quantum gravity
observation that nontrivial renormalization group fixed points can be used to generalize the procedure of perturbative renormalization. In an asymptotically
Asymptotic_safety
Physical theory with fields invariant under the action of local "gauge" Lie groups
to the computations of certain correlation functions in the vacuum state. This involves a renormalization of the theory. When the running coupling of
Gauge_theory
American physicist
physics and quantum field theory, with major contributions to the renormalization group theory of Fermi liquids, the fractional quantum Hall effect,
Ramamurti_Shankar
Hypothetical composite Higgs model
(1990), who connected the theory to the renormalization group, and improved its predictions. The renormalization group reveals that top quark condensation
Top_quark_condensate
Function in quantum field theory showing probability amplitudes of moving particles
propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is given by ψ ( x , t ) =
Propagator
Quantum field theory with four-point interactions
Jacobi elliptic function with p {\displaystyle p} satisfying the proper dispersion relation. Scalar field theory Landau pole Renormalization Higgs mechanism
Quartic_interaction
Non-mathematical introduction
later, renormalization largely solved this problem. Initially viewed as a provisional, suspect procedure by some of its originators, renormalization eventually
Introduction to quantum mechanics
Introduction_to_quantum_mechanics
Type of approximation to an underlying physical theory
Presently, effective field theories are discussed in the context of the renormalization group (RG) where the process of integrating out short distance degrees
Effective_field_theory
Field theory of scalar fields
quantum scale invariance, because of the renormalization group involved – see the discussion of the beta function below. A transformation x → x ~ ( x ) {\displaystyle
Scalar_field_theory
Dutch physicist (1894–1952)
method. He is also credited with introducing in 1948 the concept of renormalization into quantum field theory, although his approach was nonrelativistic
Hans_Kramers
Wave equations respecting special and general relativity
classical field theory for background). In the Schrödinger picture, the wave function or field is the solution to the Schrödinger equation, i ℏ ∂ ∂ t ψ =
Relativistic_wave_equations
Probabilistic algorithms to simulate quantum many-body systems
treatment and description of complex many-body effects encoded in the wave function, going beyond mean-field theory. In particular, there exist numerically
Quantum_Monte_Carlo
Formulation of classical mechanics using momenta
{p}}'_{i}\end{aligned}}} In quantum mechanics, the wave function will also undergo a local U(1) group transformation during the Gauge
Hamiltonian_mechanics
Process in quantum mechanical theories
{\displaystyle (\mathbf {r} _{j},\mathbf {r} _{k})} of the state function. The usual wave function is obtained using the Slater determinant and the identical
Canonical_quantization
Indian academic
to the area of few-body scattering in nuclear and atomic physics, renormalization in nonrelativistic quantum mechanics, and the physics of cold atoms
Sadhan_Kumar_Adhikari
Quantum field theory
physics community after Gerard 't Hooft, in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor
Yang–Mills_theory
British theoretical physicist and mathematician (1923–2020)
theory and developed rules for the diagrams that completely solved the renormalization problem. Dyson's paper and his lectures presented Feynman's theories
Freeman_Dyson
Concept in quantum mechanics
and symmetries is very powerful, and is also behind the concept of renormalization. The concept of the scattering length can also be extended to potentials
Scattering_length
Action of a massive abelian gauge field
f} is an arbitrary function. Electromagnetic field Photon Quantum electrodynamics Quantum gravity Vector boson Relativistic wave equations Klein–Gordon
Proca_action
Physical effect in few-body systems
emergent discrete scaling symmetry. This phenomenon, exhibiting a renormalization group limit cycle, is closely related to the scale invariance of the
Efimov_state
Attempts to develop a quantum mechanical theory of cosmology
causal set theory. In quantum cosmology, the universe is treated as a wave function instead of classical spacetime. String cosmology Brane cosmology Loop
Quantum_cosmology
Taiwanese-American condensed matter physicist
on the surface of a bulk topological insulator. Using a numerical renormalization group technique, they showed that, despite sharing the p +ip pairing
Chin-Sen_Ting
Partial differential equations with random force terms and coefficients
core problem of such a theory. This leads to the need of some form of renormalization. An early attempt to circumvent such problems for some specific equations
Stochastic partial differential equation
Stochastic_partial_differential_equation
Effect in quantum electrodynamics
microwave spectrum and this measurement provided the stimulus for renormalization theory to handle the divergences. The calculation of the Lamb shift
Lamb_shift
Spanish theoretical physicist, author, and academic
context of conformal field theories, two-dimensional physics, and renormalization groups. He demonstrated that the representation theory of the q-deformation
Germán_Sierra
Procedure of coping with redundant degrees of freedom in physical field theories
application to quantum field theory is fraught with complications related to renormalization, especially when the computation is continued to higher orders. Historically
Gauge_fixing
Energy difference between ground state and lightest excited state(s)
assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle. Since the energies of
Mass_gap
Field equation from quantum gravity
{\displaystyle |\psi \rangle } is no longer a spatial wave function in the traditional sense of a complex-valued function that is defined on a 3-dimensional space-like
Wheeler–DeWitt_equation
Features that do not change if length or energy scales are multiplied by a common factor
This energy dependence is described by the renormalization group, and is encoded in the beta-functions of the theory. For a QFT to be scale-invariant
Scale_invariance
equation Wave field synthesis Wave flume Wave function Wave function collapse Wave function renormalization Wave height Wave impedance Wave loading Wave packet
Index_of_physics_articles_(W)
Approximating method in quantum mechanics
to the ground state energy. The Hartree–Fock method, density matrix renormalization group, and Ritz method apply the variational method. Suppose we are
Variational method (quantum mechanics)
Variational_method_(quantum_mechanics)
Japanese-American nobel-winning physicist
formulation extended classical dynamics by introducing multiple Hamiltonian functions and a higher-order structure called the Nambu bracket. Unlike traditional
Yoichiro_Nambu
Unsolved physics problem
individual sources of angular momentum. These values depend on the renormalization scale, because their operators are not separately conserved. Physicists
Proton_spin_crisis
Physiological nature of sleep
1997). "Regional cerebral blood flow changes as a function of delta and spindle activity during slow wave sleep in humans". The Journal of Neuroscience.
Neuroscience_of_sleep
Background energy existing in space
for centuries. This argument is the underpinning of the theory of renormalization. In all practical calculations, this is how the infinity is handled
Vacuum_energy
Equations of motion for viscous fluids
23A, doi:10.1017/jfm.2020.126, S2CID 216463266 McComb, W. D. (2008), Renormalization methods: A guide for beginners, Oxford University Press, pp. 121–128
Navier–Stokes_equations
Chinese-American physicist (1926–2024)
LSZ reduction formula Partition function Path Integral Formulation Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman
Tsung-Dao_Lee
Limit of sequence of smooth functions
Cheon, T.; Shigehara, T. (1998), "Realizing discontinuous wave functions with renormalized short-range potentials", Physics Letters A, 243 (3): 111–116
Laplacian_of_the_indicator
Dimensionless number that quantifies the strength of the electromagnetic interaction
quantum field theory underlying the electromagnetic coupling, the renormalization group dictates how the strength of the electromagnetic interaction
Fine-structure_constant
Artificial helium-like atom with a harmonic instead of Coulomb potential
of variables technique to be applied to further a solution for the wave function in the form Ψ ( r 1 , r 2 ) = χ ( R ) Φ ( u ) {\displaystyle \Psi (\mathbf
Hooke's_atom
Nonlinear partial differential equation
a function φ {\displaystyle \varphi } dependent on two variables typically denoted x {\displaystyle x} and t {\displaystyle t} , involving the wave operator
Sine-Gordon_equation
Damping of electric fields
functional theory (DFT). Bjerrum length Debye length McComb, W.D. (2007). Renormalization methods: a guide for beginners (Reprinted with corrections, Reprinted ed
Electric-field_screening
Interpretation of electrodynamics
process of "addition and subtraction of infinities" associated with renormalization. This model leads to the same type of Bethe logarithm (an essential
Wheeler–Feynman absorber theory
Wheeler–Feynman_absorber_theory
Textbook by Paul Dirac
learning physics developed after 1940 due to Dirac's refusal to accept renormalization—which he described as something that merely "swept infinities under
The Principles of Quantum Mechanics
The_Principles_of_Quantum_Mechanics
Quantum field theory equations
if one views the two equations as two compatible constraints on the wave function. (See the discussion below on constraint dynamics.) If the two operators
Two-body_Dirac_equations
State of matter
1067S. doi:10.1103/PhysRevA.3.1067. E. Kolomeisky; J. Straley (1992). "Renormalization-group analysis of the ground-state properties of dilute Bose systems
Bose–Einstein_condensate
Methods of mathematical approximation
equations, differential equations (e.g., the equations of motion and commonly wave equations), thermodynamic free energy in statistical mechanics, radiative
Perturbation_theory
Experimental technique to determine the distribution of electrons in solids
{k} ,E)+i\Sigma ''(\mathbf {k} ,E)} . This function contains the full information about the renormalization of the electronic dispersion due to interactions
Angle-resolved photoemission spectroscopy
Angle-resolved_photoemission_spectroscopy
Wave equation for arbitrary spin particles
after them in a paper on a group theoretical discussion of relativistic wave equations. For a free particle of spin j without electric charge, the BW
Bargmann–Wigner_equations
Introductory article
the gauge function θ(x). We say that if the function θ oscillates, it represents a new type of quantum-mechanical wave, and this new wave has its own
Introduction_to_gauge_theory
Theoretical model in physics
with a transfer of particles between different momentum states. The renormalization of the mass of a fluid of interacting fermions can be calculated from
Fermi_liquid_theory
Difference between logarithm and harmonic series
Riemann zeta function and Dirichlet beta function. In connection to the Laplace and Mellin transform. In the regularization/renormalization of the harmonic
Euler's_constant
Singularities in the parameter space
The definition of fidelity is the inner product of the ground state wave functions of two adjacent points in parameter space, F = | ⟨ ψ 0 ( λ ) | ψ 0 (
Exceptional_point
Connection between correlation functions and the S-matrix
elements (the scattering amplitudes) from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the
LSZ_reduction_formula
Mathematical device used in statistical mechanics
with the wave vector k {\displaystyle \mathbf {k} } identified with n {\displaystyle n} . The Zwanzig projection operator relies on these functions but does
Zwanzig_projection_operator
Method for approximating many-body systems
reference wave function, which is typically a Slater determinant constructed from Hartree–Fock molecular orbitals, though other wave functions such as configuration
Coupled_cluster
Branch of mathematics
which deals with functions. Harmonic analysis deals with the representation of functions or signals as the superposition of basic waves. Geometric analysis
Mathematical_analysis
Energy driving the accelerated expansion of the universe
problem of constructing models of cosmological inflation) occurs: renormalization theory predicts that scalar fields should acquire large masses. The
Dark_energy
American theoretical physicist
radiation, demonstrating the renormalization of multipole moments and the cancellation of infrared divergences in gravitational wave observables at 3PN order
Walter_D._Goldberger
Theory of fundamental physics
quantum electrodynamics, these problems can be "solved" using the renormalization technique, namely, replacing the diverging physical values by their
Superfluid_vacuum_theory
Type of unphysical field in quantum field theory which provides mathematical consistency
LSZ reduction formula Partition function Path Integral Formulation Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman
Faddeev–Popov_ghost
Equation for arbitrary spin particles
mechanics and quantum field theory, the Joos–Weinberg equation is a relativistic wave equation applicable to free particles of arbitrary spin j, an integer for
Joos–Weinberg_equation
Lattice model of statistical mechanics
methods of quantum field theory, such as the renormalization group and the conformal bootstrap. Renormalization group methods are applicable because the critical
Classical_XY_model
Formulation of general relativity
no need for renormalization and the elimination of infinities. However, in other work, Thomas Thiemann admitted the need for renormalization as a way to
Canonical_quantum_gravity
Mechanism that explains the generation of mass for gauge bosons
16 June 2012. 't Hooft, G.; Veltman, M. (1972). "Regularization and renormalization of gauge fields". Nuclear Physics B. 44 (1): 189–219. Bibcode:1972NuPhB
Higgs_mechanism
WAVE FUNCTION-RENORMALIZATION
WAVE FUNCTION-RENORMALIZATION
Boy/Male
Anglo, British, English
Alert; Watchman
Girl/Female
Irish
Joy.
Boy/Male
Indian
Friction
Male
English
English short form of Hebrew David, DAVE means "beloved."
Boy/Male
Anglo Saxon American English Scandinavian
Moving.
Surname or Lastname
English
English : variant spelling of Way.
Male
English
 English topographical surname transferred to forename use, WADE means "lives near the river crossing." Middle English form of Anglo-Saxon Wada (the name of a sea giant), meaning "to go," in the sense of going forward, proceeding.
Surname or Lastname
English
English : topographic name for someone who lived by a dam or weir on a river (Old English wær, wer), or a habitational name from a place named with this word, such as Ware in Hertfordshire.English : nickname for a cautious person, from Middle English war(e) ‘wary’, ‘prudent’ (Old English (ge)wær).English : Robert Ware came to Dedham, MA, from England in or before 1642. Henry Ware (1764–1845), born in Sherborn, MA, was a Unitarian clergyman and theologian and father of the physician John Ware (b. 1795) and two clergymen, Henry (b. 1794) and William (b. 1797).
Girl/Female
Slavic
Stranger. Pet name formed from Varvara; the Russian form of Barbara.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Boy/Male
Hebrew American Scottish Welsh
Cherished; Beloved.
Boy/Male
Hindu
Variant of David beloved
Surname or Lastname
English
English : from a Germanic personal name Walo, either a byname meaning ‘foreigner’ (see Wallace), or else a short form of the various compound names with this first element.English : nickname for a well-liked person, from Middle English wale ‘good’, ‘excellent’ (originally meaning ‘choice’).English : topographic name for someone who lived near an embankment, Middle English wale (Old English walu).
Boy/Male
Anglo Saxon English
Wise.
Surname or Lastname
English
English : occupational name for a servant, from Middle English knave ‘boy’, ‘youth’, ‘servant’.English : possibly a metonymic occupational name for a maker of wheel-hubs, Middle English nave (from Old English nafa, nafu).German (also Näve) : variant of Neff (see Neve).Dutch (de Nave) : variant of Naef 1.In some cases possibly Portuguese : topographic name from nave ‘plain’ (a variant of nava), or a habitational name from a place named with this word. Compare Nava.
Female
Irish
Variant spelling of Irish Maeve, MAVE means "intoxicating."Â
Boy/Male
English
Alert.
Surname or Lastname
English (of Norman origin) and northern French
English (of Norman origin) and northern French : nickname for a bald man, from Anglo-Norman French cauf ‘bald’. Compare Chaffee.English : habitational name from a place in East Yorkshire called Cave, apparently from a river name derived from Old English cÄf ‘swift’.French : metonymic occupational name for someone employed in or in charge of the wine cellars of a great house, from Old French cave ‘cave’, ‘cellar’ (Latin cavea, a derivative of cavus ‘hollow’).French, possibly also English : topographic name for someone who lived in or near a cave, from the same word as in 3 in an older sense.
Boy/Male
Anglo, British, English, Jamaican
Wise; Watchful; Aware; Watchman; Careful
Surname or Lastname
English
English : from the Middle English personal name Wade, Old English Wada, from wadan ‘to go’. (Wada was the name of a legendary sea-giant.)English : topographic name for someone who lived near a ford, Old English (ge)wæd (of cognate origin to 1), or a habitational name from a place named with this word, as for example Wade in Suffolk.Dutch and North German : occupational name or nickname from Middle Dutch, Middle Low German wade ‘garment’, ‘large net’.Jonathan Wade emigrated from Norfolk, England, to Medford, MA, in 1632. Benjamin Franklin Wade (1800–1878), born near Springfield, MA, was a prominent U.S. senator from OH during the Civil War.
WAVE FUNCTION-RENORMALIZATION
WAVE FUNCTION-RENORMALIZATION
Boy/Male
British, English
From the Large Fields or Granta's Fields
Boy/Male
Hindu, Indian, Marathi
Lovely Scene
Girl/Female
Hindu, Indian
Brilliant
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lord Shiva
Girl/Female
Indian, Tamil
Bright Face
Girl/Female
Indian, Sanskrit
Sight
Girl/Female
Tamil
Goddess Lakshmi, The Sun
Boy/Male
Australian, French, German, Italian, Latin
Savior
Boy/Male
Muslim/Islamic
Reward Compensation
Male
Polish
Pet form of Polish Czcibor, CIBOR means "battle of honor."
WAVE FUNCTION-RENORMALIZATION
WAVE FUNCTION-RENORMALIZATION
WAVE FUNCTION-RENORMALIZATION
WAVE FUNCTION-RENORMALIZATION
WAVE FUNCTION-RENORMALIZATION
v. t.
To move like a wave, or by floating; to waft.
n.
The place or point of union, meeting, or junction; specifically, the place where two or more lines of railway meet or cross.
n.
The act of joining, or the state of being joined; union; combination; coalition; as, the junction of two armies or detachments; the junction of paths.
v. t.
See Waive.
n.
The things sold by auction or put up to auction.
a.
Pertaining to the function of an organ or part, or to the functions in general.
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.
n.
The act of anointing, smearing, or rubbing with an unguent, oil, or ointment, especially for medical purposes, or as a symbol of consecration; as, mercurial unction.
v. t.
The act of uniting, or the state of being united; junction.
a.
Pertaining to, or connected with, a function or duty; official.
imp. & p. p.
of Wave
v. i.
To play loosely; to move like a wave, one way and the other; to float; to flutter; to undulate.
n.
A wave.
v. t.
To give sanction to; to ratify; to confirm; to approve.
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.
Exhibiting a wavelike form or outline; undulating; intended; wavy; as, waved edge.
v. t.
To sell by auction.
v. i.
To fluctuate; to waver; to be in an unsettled state; to vacillate.
n.
A wave.