AI & ChatGPT searches , social queriess for DENSELY DEFINED-OPERATOR
Search references for DENSELY DEFINED-OPERATOR. Phrases containing DENSELY DEFINED-OPERATOR
See searches and references containing DENSELY DEFINED-OPERATOR!
Search references for DENSELY DEFINED-OPERATOR. Phrases containing DENSELY DEFINED-OPERATOR
See searches and references containing DENSELY DEFINED-OPERATOR!DENSELY DEFINED-OPERATOR
Linear operator on dense subset of its apparent domain
mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological
Densely_defined_operator
Conjugate transpose of an operator in infinite dimensions
been further extended to include unbounded densely defined operators, whose domain is topologically dense in, but not necessarily equal to, H . {\displaystyle
Hermitian_adjoint
Linear operator defined on a dense linear subspace
{\|x\|^{2}+\|Tx\|^{2}}}.} An operator T is said to be densely defined if its domain is dense in X. This also includes operators defined on the entire space X
Unbounded_operator
Aspect of mathematical spectrum theory
spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition
Essential_spectrum
Linear operator equal to its own adjoint
symmetric operators and A = A ∗ ∗ ⊆ A ∗ {\displaystyle A=A^{**}\subseteq A^{*}} for closed symmetric operators. The densely defined operator A {\displaystyle
Self-adjoint_operator
Typically linear operator defined in terms of differentiation of functions
functions are dense in L2, this defines the adjoint on a dense subset of L2: P* is a densely defined operator. The Sturm–Liouville operator is a well-known
Differential_operator
Operation on self-adjoint operators
{\displaystyle \operatorname {ran} (1-W(A))} is dense in A {\displaystyle A} . Conversely, given any densely defined operator U {\displaystyle U} which is isometric
Extensions of symmetric operators
Extensions_of_symmetric_operators
Set of eigenvalues of a matrix
set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, I {\displaystyle I} is the identity operator. By the closed graph theorem, λ {\displaystyle
Spectrum (functional analysis)
Spectrum_(functional_analysis)
Mathematical theorem about Banach spaces
theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in
Closed_range_theorem
Type of vector space in math
bounded operators, an unbounded operator is usually not defined on all of H. If D(T) is dense in H, then T is called a densely defined operator. The domain
Hilbert_space
Function whose actual domain of definition may be smaller than its apparent domain
Multivalued function – Generalized mathematical function Densely defined operator – Linear operator on dense subset of its apparent domain Martin Davis (1958)
Partial_function
Hilbert space H. A closed and densely defined operator A is said to be affiliated with M if A commutes with every unitary operator U in the commutant of M.
Affiliated_operator
Generalization of the exponential function
C_{0}(\mathbb {R} ):q\cdot f\in C_{0}(\mathbb {R} )\}} is a closed densely defined operator and generates the multiplication semigroup ( T q ( t ) ) t ≥ 0
C0-semigroup
Linear operator whose graph is closed
branch of mathematics, a closed linear operator or often a closed operator is a partially defined linear operator whose graph is closed (see closed graph
Closed_linear_operator
Mathematical method in functional analysis
graphs Continuous linear operator – Function between topological vector spaces Densely defined operator – Linear operator on dense subset of its apparent
Continuous_linear_extension
Hermitian operator, an operator (sometimes a symmetric operator, sometimes a symmetric densely defined operator, sometimes a self-adjoint operator) Hermitian
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Mathematical operation
addition, T has closed range. In general, if A, B are closed and densely defined operators on a Hilbert space H, and A* A = B* B, then A = UB where U is
Square_root_of_a_matrix
Theorems connecting continuity to closure of graphs
redirect targets Closed linear operator – Linear operator whose graph is closed Densely defined operator – Linear operator on dense subset of its apparent domain
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
Similar to the basis of a vector space, but not necessarily linearly independent
Benjamin; Moran, Bill; Cochran, Doug (2021). "Positive operator-valued measures and densely defined operator-valued frames". Rocky Mountain Journal of Mathematics
Frame_(linear_algebra)
Surjective bounded operator on a Hilbert space preserving the inner product
notion serves to define the concept of isomorphism between Hilbert spaces. Definition 1. A unitary operator is a bounded linear operator U : H → H on a
Unitary_operator
Measure of the "size" of linear operators
Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ‖ T ‖ {\displaystyle
Operator_norm
} That is, in studying operators that are not everywhere-defined, one may restrict one's attention to densely defined operators without loss of generality
Discontinuous_linear_map
Set of isolated points in the spectrum of an operator with finite-rank Riesz projectors
specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank
Discrete spectrum (mathematics)
Discrete_spectrum_(mathematics)
Mathematical method in functional analysis
=S^{*}S=FS} is a positive (hence, self-adjoint) and densely defined operator called the modular operator. The main result of Tomita–Takesaki theory states
Tomita–Takesaki_theory
Any real function on R admits a continuous restriction on a dense subset of R
Theorems connecting continuity to closure of graphs Densely defined operator – Linear operator on dense subset of its apparent domain Hahn–Banach theorem –
Blumberg_theorem
Topic in mathematics
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A : H → H {\displaystyle A\colon H\to
Hilbert–Schmidt_operator
Kind of linear transformation
Mathematical study of linear operators Seminorm – Mathematical function Unbounded operator – Linear operator defined on a dense linear subspace Proof: Assume
Bounded_operator
extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician Kurt Friedrichs. This
Friedrichs_extension
Compact operator for which a finite trace can be defined
specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent
Trace_class
a positive operator, whereas Δ is a dissipative operator. Using spectral theory, one can define a square root (1 − Δ)1/2 for the operator (1 − Δ). This
Ornstein–Uhlenbeck_operator
analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex
Singular integral operators on closed curves
Singular_integral_operators_on_closed_curves
Subset whose closure is the whole space
linear operator between topological vector spaces X {\displaystyle X} and Y {\displaystyle Y} is said to be densely defined if its domain is a dense subset
Dense_set
Part of Fredholm theories in integral equations
unilateral shift operator S on H is defined by S ( e n ) = e n + 1 , n ≥ 0. {\displaystyle S(e_{n})=e_{n+1},\quad n\geq 0.\,} This operator S is injective
Fredholm_operator
Branch of functional analysis
operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if T is an operator
Borel_functional_calculus
graph is closed. 3. The closed range theorem says that a densely defined closed operator has closed image (range) if and only if the transpose of it
Glossary of functional analysis
Glossary_of_functional_analysis
Linear operator in algebra and operator theory
of a bounded linear operator L is an open set. More generally, the resolvent set of a densely defined closed unbounded operator is an open set. Reed
Resolvent_set
Theorem on boundedness of symmetric operators
everywhere-defined operators are necessarily self-adjoint, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded
Hellinger–Toeplitz_theorem
is also densely defined, and it is self-adjoint. That is, ( T ∗ T ) ∗ = T ∗ T {\displaystyle \left(T^{*}T\right)^{*}=T^{*}T} and the operators on the right-
Von_Neumann's_theorem
Mathematical function, in linear algebra
Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space
Linear_map
Bounded operators with sub-unit norm
following basic objects associated with T can be defined. The defect operators of T are the operators DT = (1 − T*T)1⁄2 and DT* = (1 − TT*)1⁄2. The square
Contraction_(operator_theory)
Operator in quantum mechanics
quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation
Momentum_operator
Theorem relating unitary operators to one-parameter Lie groups
this derivative exists—i.e., that A {\displaystyle A} is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional
Stone's theorem on one-parameter unitary groups
Stone's_theorem_on_one-parameter_unitary_groups
= F|D| of D into a self adjoint unitary operator F (the 'phase' of D) and a densely defined positive operator |D| (the 'metric' part). If ( A , H , D
Spectral_triple
Functional analysis concept
operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Type of matrix representation
The operator A being closed and densely defined ensures that the operator A*A is self-adjoint (with dense domain) and therefore allows one to define (A*A)1/2
Polar_decomposition
Boundary condition for generalized functions
H^{1}(\Omega )} -regularity of u {\textstyle u} is sufficient for the well-definedness of this integral equation. It is not apparent, however, in which sense
Trace_operator
Unification of discrete and continuous theories of calculus
a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers
Time-scale_calculus
it follows from the other two conditions. Let A be a linear operator defined on a dense linear subspace D(A) of the reflexive Banach space X. Then A
Lumer–Phillips_theorem
Operator on a Hilbert space that shifts basis vectors
In operator theory, the unilateral shift is a one-sided shift operator, that is, a shift operator acting on one-sided sequences or shift spaces. The term
Unilateral_shift_operator
Vector space with generalized dot product
{\overline {H}},} and H {\displaystyle H} is dense in H ¯ {\displaystyle {\overline {H}}} for the topology defined by the norm. In this article, F denotes
Inner_product_space
Construction in functional analysis, useful to solve differential equations
Equivalently, the inverse linear operator (T − λ)−1, which is defined on the dense subset R, is not a bounded operator, and therefore cannot be extended
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Quasiparticle of mechanical vibrations
{\displaystyle \Pi _{k}} defined in the quantum treatment section above, we can define creation and annihilation operators: b k = m ω k 2 ℏ ( Q k + i
Phonon
Linear operator in functional analysis
mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional. Finite-rank operators are matrices (of
Finite-rank_operator
especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples
Subnormal_operator
Theorem
operator defined on a dense linear subspace of X. The Hille–Yosida theorem provides a necessary and sufficient condition for a closed linear operator
Hille–Yosida_theorem
In mathematics, a dissipative operator is a linear operator A defined on a linear subspace D(A) of Banach space X, taking values in X such that for all
Dissipative_operator
Vector space in functional analysis
a dense subset of X . {\displaystyle X.} This condition arises frequently in many theorems of functional analysis. Unbounded self-adjoint operators on
Total_subset
*-algebra of bounded operators on a Hilbert space
*-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type
Von_Neumann_algebra
Theorem on operator interpolation
simple functions, it is dense in both L1(Rd) and L2(Rd). Densely defined continuous operators admit unique extensions, and so we are justified in considering
Riesz–Thorin_theorem
Von Neumann
of bounded linear operators in H, an element Ω of H is said to be cyclic for A if the linear space AΩ = {aΩ: a ∈ A} is norm-dense in H. The element Ω
Cyclic_and_separating_vector
Probability problem
identify en with xn. In the model, the operator T is multiplication by x and a densely defined symmetric operator. It can be shown that T always has self-adjoint
Hamburger_moment_problem
Order-preserving mathematical function
precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators and and or (in particular
Monotonic_function
Linear operator related to topological vector spaces
nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately
Nuclear_operator
Result about when a matrix can be diagonalized
functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix
Spectral_theorem
Mapping between functions in the quantum phase space
space, then Φ[f] is trace-class. More generally, Φ[f] is a densely defined unbounded operator. The map Φ[f] is one-to-one on the Schwartz space (as a subspace
Wigner–Weyl_transform
(on a complex Hilbert space) continuous linear operator
functional analysis, a normal operator on a complex Hilbert space H {\displaystyle H} is a continuous linear operator N : H → H {\displaystyle N\colon
Normal_operator
Identifies the commutant of a specific von Neumann algebra
\Gamma } is defined to be the von Neumann algebra on H1 generated by the algebra A ⊗ I {\displaystyle A\otimes I} and the normalising operators U g ⊗ λ (
Commutation theorem for traces
Commutation_theorem_for_traces
Algebra of possibly unbounded operators
mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were described
O*-algebra
Generalized function whose value is zero everywhere except at zero
of the function at every point. The Dirac delta distribution is a densely defined unbounded linear functional on the Hilbert space L2 of square-integrable
Dirac_delta_function
Mathematical compact operator
product. The operator K defines a formally self-adjoint operator on the dense subspace H of HS. As Krein (1947) and Reid (1951) noted, the operator has the
Symmetrizable compact operator
Symmetrizable_compact_operator
Function between topological vector spaces
functional Topologies on spaces of linear maps Unbounded operator – Linear operator defined on a dense linear subspace Narici & Beckenstein 2011, pp. 126–128
Continuous_linear_operator
Topological complex vector space
(equivalently continuous) linear operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x : H → H. In fact, every
C*-algebra
Mathematical concept
d\zeta ,} where δ = |1 – eiε|. Since it is defined as convolution with a bounded function, it is a bounded operator on L2(T). Now H ε 1 = i π ∫ ε π 2 ℜ ( 1
Singular integral operators of convolution type
Singular_integral_operators_of_convolution_type
Partial differential equation
relies on the Lp theory of the Beurling transform, a singular integral operator defined on Lp(C) for all 1 < p < ∞. The same method applies equally well on
Beltrami_equation
Approach used in computer vision systems
Gaussian derivative operators (Gauss-SIFT and Gauss-SURF) instead of original SIFT as defined from an image pyramid or original SURF as defined from Haar wavelets
Corner_detection
Concept in set theory
countable ordinal obtained by ordinal exponentiation). The Baire space is defined to be the Cartesian product of countably infinitely many copies of the
Baire_space_(set_theory)
Particular task in computer vision
detection. Some basic properties of blobs defined from scale-space maxima of the normalized Laplacian operator are that the responses are covariant with
Blob_detection
Mathematical transform that expresses a function of time as a function of frequency
each σ ∈ Σ. For f ∈ L1(G), the Fourier transform of f at σ is the operator on Hσ defined by f ^ ( σ ) = ∫ G f ( g ) U g − 1 ( σ ) d λ ( g ) . {\displaystyle
Fourier_transform
is defined by {a,b,c} = (a ∘ b) ∘ c + (c ∘ b) ∘ a − (a ∘ c) ∘ b. A JW algebra is a Jordan subalgebra of the Jordan algebra of self-adjoint operators on
Jordan_operator_algebra
Objects that generalize functions
An alternative way to define the convolution of a function f and a distribution T is to use the translation operator τx defined on test functions by τ
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Type of strongly continuous semigroup
operator topology. The infinitesimal generators of analytic semigroups have the following characterization: A closed, densely defined linear operator
Analytic_semigroup
Axiomatization of quantum field theory
set of operators A 1 ( f ) , … , A n ( f ) {\displaystyle A_{1}(f),\ldots ,A_{n}(f)} which, together with their adjoints, are defined on a dense subset
Wightman_axioms
Theory of logic to account for observations from quantum theory
unbounded) densely defined self-adjoint operator A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure E defined on the
Quantum_logic
D(A)\cap D(B)} is dense in X {\displaystyle X} , the operator U ( t ) {\displaystyle U(t)} can be extended to a bounded linear operator defined on the entire
Abstract differential equation
Abstract_differential_equation
sometimes an operator is called locally finite when the sum of the { V i | i ∈ I } {\displaystyle \{V_{i}\vert i\in I\}} is only dense in V {\displaystyle
Locally_finite_operator
Mathematics of smooth surfaces
tangent space is an inner product space, the shape operator Sx can be defined as a linear operator on this space by the formula ( S x v , w ) = ( d n
Differential geometry of surfaces
Differential_geometry_of_surfaces
Mathematical term
linear functional on Y defined by y ↦ b(x, y). Similarly, for all y ∈ Y, let b(•, y) : X → K {\displaystyle \mathbb {K} } be defined by x ↦ b(x, y). Definition
Weak_topology
All points and limit points in a subset of a topological space
closed subset C ⊆ X . {\displaystyle C\subseteq X.} One may define the closure operator in terms of universal arrows, as follows. The powerset of a set
Closure_(topology)
Construction for adding objects to a Hilbert space
eigenstates of the position and momentum operators are not in the Hilbert space, but are in a suitably defined rigged Hilbert space. Informally, the term
Rigged_Hilbert_space
Topological vector spaces
distributions and spaces of distributions are often defined by means of the transpose of a linear operator. This is because the transpose allows for a unified
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Mathematical theory by discovered by Józef Marcinkiewicz
operators, but also applies to non-linear operators. Let f be a measurable function with real or complex values, defined on a measure space (X, F, ω). The distribution
Marcinkiewicz interpolation theorem
Marcinkiewicz_interpolation_theorem
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
so does define a bounded operator between Sobolev spaces on ∂Ω, decreasing the order by 1. It allows a 2 × 2 matrix of operators to be defined by C = (
Neumann–Poincaré_operator
Geometric theory based on regions
by the infix operator "=", is part of the background logic, the binary relation Proper Part, denoted by the infix operator "<", is defined as: x < y ↔
Whitehead's point-free geometry
Whitehead's_point-free_geometry
Correspondence in functional analysis
on operators π {\displaystyle \pi } is nondegenerate, that is the space of vectors π ( x ) {\displaystyle \pi (x)} ξ {\displaystyle \xi } is dense as
Gelfand–Naimark–Segal construction
Gelfand–Naimark–Segal_construction
Representation theory of the symplectic group
space onto the Hn(x). The heat operator for the harmonic oscillator is the operator on L2(R) defined as the diagonal operator e − D t H n = e − ( 2 n + 1
Oscillator_representation
Part of spectral theory
{\displaystyle U:L^{2}(a,b)\mapsto L^{2}(\psi (a),\psi (b))} is the unitary operator defined by ( U f ) ( ψ ( x ) ) = f ( x ) × ( ψ ′ ( x ) ) − 1 / 2 , ∀ x
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
a dense subset in general topology. No topology is needed to define a dense submodule, and a dense submodule may or may not be topologically dense in
Dense_submodule
Mathematical objects that generalise the notion of Hilbert spaces
F)} , normed by the operator norm. The adjointable and compact adjointable operators are subspaces of this Banach space defined using the inner product
Hilbert_C*-module
Induced map between the dual spaces of the two vector spaces
transpose or algebraic adjoint of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two
Transpose_of_a_linear_map
Concept in mathematics
Analytic vectors are dense by a classical argument of Edward Nelson, amplified by Roe Goodman, since vectors in the image of a heat operator e–tD, corresponding
Unitary_representation
DENSELY DEFINED-OPERATOR
DENSELY DEFINED-OPERATOR
Girl/Female
German
Noble; Kind
Surname or Lastname
English
English : habitational name from Hensall in North Yorkshire, originally named with the unattested Old English personal name Heþīn or Old Scandinavian Heþinn + Old English halh ‘nook’.English : Huguenot surname, of unexplained origin, which was taken to England by a Protestant refugee who fled France after the Massacre of St. Bartholomew’s Day (24 August 1572) and settled in Newcastle-upon-Tyne.
Girl/Female
Irish
Dark-haired.
Boy/Male
English American
A place-name in Cornwall.
Surname or Lastname
Irish
Irish : reduced Anglicized form of either of two Gaelic names, Ó DuibhÃn ‘descendant of DuibhÃn’, a byname meaning ‘little black one’, or Ó DaimhÃn ‘descendant of DaimhÃn’, a byname meaning ‘fawn’, ‘little stag’. These are attenuated versions of Ó Dubháin and Ó Damháin, and are the phonetic origin of Anglicizations with an internal v (as opposed to w, as in Dewan, or monosyllabic forms with an o or u) (see Doane).English and French : nickname, of literal or ironic application, from Middle English, Old French devin, divin ‘excellent’, ‘perfect’ (Latin divinus ‘divine’).
Girl/Female
French, German, Greek
Dolphin
Surname or Lastname
English
English : habitational name from a lost or unidentified place, possibly, in view of the present-day concentration of the name in Norwich, in East Anglia.
Surname or Lastname
English
English : apparently a habitational name from an unidentified place, probably so named from Old English denu ‘valley’ + lēah ‘woodland clearing’. It may well be an altered form of Delly End in Oxfordshire.
Boy/Male
African, American, Australian, British, English
From Denzell; A Place in Cornwall
Boy/Male
English
a place in Cornwall.
Surname or Lastname
English
English : probably a habitational name from either of two places in Devon: Hensley in East Worlington, which is named with the Old English personal name Hēahmund + Old English lēah ‘(woodland) clearing’, or Hensleigh in Tiverton, which is named from Old English hengest ‘stallion’ (or the Old English personal name Hengest) + lēah.English : possibly also a variant of Hemsley.
Boy/Male
Gaelic, Hindu, Indian
Oxen; Bard
Surname or Lastname
English and Scottish
English and Scottish : variant spelling of Denny.
Boy/Male
African, American, Australian, British, Chinese, Christian, English, Jamaican
From Denzell; A Place-name in Cornwall; Fort; Fertile Land
Surname or Lastname
English
English : habitational name from Wensleydale in North Yorkshire.
Female
French
Short form of French Adeline, DELINE means "noble."Â
Surname or Lastname
English (Somerset)
English (Somerset) : apparently a habitational name from an unidentified place. It is probably a variant of Denslow or possibly Denley, neither of which are of identified origin.
Girl/Female
Greek
From Delphi.
Girl/Female
Welsh
From 'cilun' meaning idol.
Boy/Male
American, British, English, Greek, Scandinavian
Follower of Dionysius; Greek God of Wine
DENSELY DEFINED-OPERATOR
DENSELY DEFINED-OPERATOR
Boy/Male
Indian, Telugu
Interest
Girl/Female
German
Noble; Kind
Boy/Male
Hindu, Indian
Lord Vinayagar
Girl/Female
Muslim/Islamic
A virgin maiden of Paradise for its dwellers
Boy/Male
Arabic, Muslim
Tall; Towering; Lofty
Boy/Male
Christian, French, German, Greek, Gujarati, Hindu, Indian, Kannada, Sikh, Swedish
Famous Egyptian King; Ruler over Heroes
Boy/Male
Muslim
The guide to the right path
Boy/Male
Muslim/Islamic
Of noble birth
Boy/Male
Indian
Surname or Lastname
English
English : variant spelling of Emery.
DENSELY DEFINED-OPERATOR
DENSELY DEFINED-OPERATOR
DENSELY DEFINED-OPERATOR
DENSELY DEFINED-OPERATOR
DENSELY DEFINED-OPERATOR
n.
One who defiles; one who corrupts or violates; that which pollutes.
n.
The quality of being dense; density.
imp. & p. p.
of Refine
a.
Having the constituent parts massed or crowded together; close; compact; thick; containing much matter in a small space; heavy; opaque; as, a dense crowd; a dense forest; a dense fog.
a.
Full of veins; streaked; variegated; as, veined marble.
n.
One who dares and defies; a contemner; as, a defier of the laws.
a.
Freed from impurities or alloy; purifed; polished; cultured; delicate; as; refined gold; refined language; refined sentiments.
imp. & p. p.
of Defend
a.
Free from ambiguity; unequivocal; unmistakable; unquestionable; clear; evident; as, a decided advantage.
v. t.
To repel danger or harm from; to protect; to secure against; attack; to maintain against force or argument; to uphold; to guard; as, to defend a town; to defend a cause; to defend character; to defend the absent; -- sometimes followed by from or against; as, to defend one's self from, or against, one's enemies.
imp. & p. p.
of Defile
n.
One who defines or explains.
v. t.
To determine the precise signification of; to fix the meaning of; to describe accurately; to explain; to expound or interpret; as, to define a word, a phrase, or a scientific term.
n.
The quality of being dense, close, or thick; compactness; -- opposed to rarity.
imp. & p. p.
of Define
adv.
In a compact manner; with close union of parts; densely; tersely.
n.
One who, or that which, refines.
a.
Stupid; gross; crass; as, dense ignorance.
adv.
In a dense, compact manner.
v. t.
To determine or clearly exhibit the boundaries of; to mark the limits of; as, to define the extent of a kingdom or country.