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Linear operator equal to its own adjoint
In mathematics, a self-adjoint operator on a complex vector space V {\displaystyle V} with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot
Self-adjoint_operator
Result about when a matrix can be diagonalized
perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The
Spectral_theorem
Conjugate transpose of an operator in infinite dimensions
specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle
Hermitian_adjoint
Typically linear operator defined in terms of differentiation of functions
the adjoint on a dense subset of L2: P* is a densely defined operator. The Sturm–Liouville operator is a well-known example of a formal self-adjoint operator
Differential_operator
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose;
Hermitian_matrix
Element of *-algebra where x* equals x
In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a ∗ {\displaystyle a=a^{*}} ). Let A {\displaystyle
Self-adjoint_element
Set of eigenvalues of a matrix
\mathbb {N} } . If X is a Hilbert space and T is a self-adjoint operator (or, more generally, a normal operator), then a remarkable result known as the spectral
Spectrum (functional analysis)
Spectrum_(functional_analysis)
Linear operator defined on a dense linear subspace
essentially self-adjoint if and only if it has one and only one self-adjoint extension. A symmetric operator may have more than one self-adjoint extension
Unbounded_operator
Mathematical conjecture about the Riemann zeta function
zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of
Hilbert–Pólya_conjecture
Branch of functional analysis
Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. If T is a self-adjoint operator on a finite-dimensional
Borel_functional_calculus
Any entity that can be measured
c\in \mathbb {C} } . Observables are given by self-adjoint operators on V. Not every self-adjoint operator corresponds to a physically meaningful observable
Observable
Interaction of a quantum system with a classical observer
von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable". These observables
Measurement in quantum mechanics
Measurement_in_quantum_mechanics
Aspect of mathematical spectrum theory
{\displaystyle X} be a Hilbert space and let T {\displaystyle T} be a self-adjoint operator on X {\displaystyle X} . The essential spectrum of T {\displaystyle
Essential_spectrum
Measure used in functional analysis
for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators
Projection-valued_measure
Functional analysis concept
(finite-dimensional) self-adjoint matrices generalizes to compact self-adjoint operators on real or complex Hilbert spaces, namely such an operator can be diagonalized
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
Matrices similar to diagonal matrices
\|T-D\|_{p}\leq \epsilon } . In other words, any self-adjoint operator is an infinitesimal perturbation from a diagonal operator, where "infinitesimal" is in the sense
Diagonalizable_matrix
Linear operator scaling by a fixed function
that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L2 space. These operators are often contrasted
Multiplication_operator
In mathematics, a linear operator acting on inner product space
authors define a positive operator A {\displaystyle A} to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex
Positive_operator
Theorem relating unitary operators to one-parameter Lie groups
functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H {\displaystyle {\mathcal {H}}} and one-parameter
Stone's theorem on one-parameter unitary groups
Stone's_theorem_on_one-parameter_unitary_groups
Matrix whose conjugate transpose is its negative (additive inverse)
of as skew-adjoint (since they are like 1 × 1 {\displaystyle 1\times 1} matrices), whereas real numbers correspond to self-adjoint operators. For example
Skew-Hermitian_matrix
(on a complex Hilbert space) continuous linear operator
, self-adjoint operators): N ∗ = N {\displaystyle N^{\ast }=N} skew-Hermitian operators: N ∗ = − N {\displaystyle N^{\ast }=-N} positive operators: N
Normal_operator
Construction in functional analysis, useful to solve differential equations
the adjoint of an operator T ∈ B(H), not the transpose, and σ(T*) is not σ(T) but rather its image under complex conjugation. For a self-adjoint T ∈ B(H)
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Branch of functional analysis
context, the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include C*-algebras, von Neumann
Operator_algebra
Class of ordinary differential equations
satisfy the above regular boundary conditions. Moreover, L is a self-adjoint operator: ⟨ L f , g ⟩ = ⟨ f , L g ⟩ . {\displaystyle \langle Lf,g\rangle
Sturm–Liouville_theory
Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. It is named after the mathematician
Friedrichs_extension
Type of vector space in math
gives a precise sense in which self-adjoint operators play the role of real-valued functions: a bounded self-adjoint operator has real spectrum and can be
Hilbert_space
Mathematical study of linear operators
perspective. Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The
Operator_theory
Type of continuous linear operator
many nonzero eigenvalues. Thus compact self-adjoint operators behave much like finite-dimensional self-adjoint matrices, except that the eigenvalues may
Compact_operator
Operation on self-adjoint operators
symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions
Extensions of symmetric operators
Extensions_of_symmetric_operators
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 a self-adjoint operator in ( A − ) 1 {\displaystyle (A^{-})_{1}} , then h {\displaystyle h} is in the strong-operator closure of the set of self-adjoint
Kaplansky_density_theorem
typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint operator, endowed with supplemental structures. It was conceived
Spectral_triple
Mathematical tool from spectral theory and functional analysis
functions of self-adjoint operators. Named after Bernard Helffer and Johannes Sjöstrand, this formula provides a way to calculate functions of operators without
Helffer–Sjöstrand_formula
Operator in quantum mechanics
square-integrable functions on the real line. The position operator is defined as the self-adjoint operator Q : D Q → L 2 ( R , C ) : ψ ↦ q ψ , {\displaystyle
Position_operator
Conjecture on zeros of the zeta function
the eigenvalues of some self-adjoint operator, which would imply the Riemann hypothesis. All attempts to find such an operator have failed. There are several
Riemann_hypothesis
{\displaystyle V^{*}} is the adjoint of V. If T is a self-adjoint operator, then the compression T W {\displaystyle T_{W}} is also self-adjoint. When V is replaced
Compression (functional analysis)
Compression_(functional_analysis)
Mathematical structures that allow quantum mechanics to be explained
(positive semi-definite) self-adjoint operator ρ {\displaystyle \rho } normalized to be of trace 1. In turn, any density operator of a mixed state can be
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Differential operator in mathematics
conditions, then the corresponding realization of the Laplacian is a self-adjoint operator with compact resolvent. Consequently its spectrum is discrete: there
Laplace_operator
Relates 2 second-order elliptic operators on a manifold with the same principal symbol
elliptic operators on a manifold with the same principal symbol. Usually Weitzenböck formulae are implemented for G-invariant self-adjoint operators between
Weitzenböck_identity
Self-adjoint operator that arises in physical transition problems
}u](n)=u(n+1)+u(n-1)+2\lambda \cos(2\pi (\omega +n\alpha ))u(n),\,} acting as a self-adjoint operator on the Hilbert space ℓ 2 ( Z ) {\displaystyle \ell ^{2}(\mathbb
Almost_Mathieu_operator
Mathematical tool in quantum physics
_{ij})} be a positive semi-definite operator, see below. A density operator is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert
Density_matrix
Mathematical function
x=0} for every x ∈ R 2 {\displaystyle x\in \mathbb {R} ^{2}} . A self-adjoint operator A : H → H , {\displaystyle A:H\to H,} where H {\displaystyle H}
Coercive_function
Theorem in functional analysis
associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below. Let A be a n × n Hermitian matrix. As with
Min-max_theorem
Collection of mathematical theories
spectra of transformations in a Hilbert space. In particular, for self-adjoint operators, the spectrum lies on the real line and (in general) is a spectral
Spectral_theory
Description of a quantum-mechanical system
momentum, energy, spin – are represented by observables, which are self-adjoint operators acting on the Hilbert space. A wave function can be an eigenvector
Schrödinger_equation
Operator in quantum mechanics
quantum state then the operator is self-adjoint. In physics the term Hermitian often refers to both symmetric and self-adjoint operators. (In certain artificial
Momentum_operator
Matrix equal to its transpose
negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space
Symmetric_matrix
Concept in quantum mechanics
The term "observable" has gained a technical meaning, denoting a self-adjoint operator that represents the possible results of a random variable. The theoretical
Observer_(quantum_physics)
Generalized function whose value is zero everywhere except at zero
position operator are called the eigenkets and are denoted by φy = |y⟩. Similar considerations apply to any other (unbounded) self-adjoint operator with continuous
Dirac_delta_function
expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations
Hilbert–Schmidt_theorem
Square roots of the eigenvalues of the self-adjoint operator
non-negative) eigenvalues of the self-adjoint operator T ∗ T {\displaystyle T^{*}T} (where T ∗ {\displaystyle T^{*}} denotes the adjoint of T {\displaystyle T}
Singular_value
Summability method in physics
particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but
Zeta_function_regularization
Scheme for obtaining the position operator
discovered when attempting to define a self adjoint operator in the relativistic setting that resembled the position operator in basic quantum mechanics in the
Newton–Wigner_localization
Expected value of a quantum measurement
A\rangle _{\sigma }} . Mathematically, A {\displaystyle A} is a self-adjoint operator on a separable complex Hilbert space. In the most commonly used
Expectation value (quantum mechanics)
Expectation_value_(quantum_mechanics)
Matrix decomposition
\lambda } in its spectrum is an eigenvalue. Furthermore, a compact self-adjoint operator can be diagonalized by its eigenvectors. If M {\displaystyle \mathbf
Singular_value_decomposition
Analog of the continuous Laplace operator
Laplacian on an infinite grid is of key interest; since it is a self-adjoint operator, it has a real spectrum. For the convention Δ = I − M {\displaystyle
Discrete_Laplace_operator
(1909)) or Hilbert–Schmidt operator (von Neumann (1935)) of arbitrarily small norm, a bounded self-adjoint operator or unitary operator on a Hilbert space is
Weyl–von_Neumann_theorem
concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras
Jordan_operator_algebra
Operators useful in quantum mechanics
by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions
Creation and annihilation operators
Creation_and_annihilation_operators
Foundational principle in quantum physics
mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits
Uncertainty_principle
First-order differential linear operator on spinor bundle, whose square is the Laplacian
of smooth, square-integrable functions. It can be extended to a self-adjoint operator on that domain. The square, in this case, is not the Laplacian,
Dirac_operator
Differential operator
In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues
Eta_invariant
Theory of logic to account for observations from quantum theory
observable is represented by some (possibly unbounded) densely defined self-adjoint operator A on a Hilbert space H. A has a spectral decomposition, which is
Quantum_logic
Theorem about the dual of a Hilbert space
} Self-adjoint operators A continuous linear operator A : H → H {\displaystyle A:H\to H} is called self-adjoint if it is equal to its own adjoint; that
Riesz_representation_theorem
Compact operator for which a finite trace can be defined
always self-adjoint (i.e. A = A ∗ = | A | {\displaystyle A=A^{*}=|A|} ) though the converse is not necessarily true. Given a bounded linear operator T :
Trace_class
Numerical method
The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. It
Adjoint_state_method
Integral transform and linear operator
{\displaystyle L^{p}(\mathbb {R} )} . The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between L p ( R ) {\displaystyle
Hilbert_transform
Theory of quantum gravity merging quantum mechanics and general relativity
Hamiltonian operator is not self-adjoint, in fact it is not even a normal operator (i.e. the operator does not commute with its adjoint) and so the spectral
Loop_quantum_gravity
Generalized notion of measure in mathematics
(1959), Linear Operators. Part I: General Theory. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. Part III: Spectral Operators., Pure and
Signed_measure
Mathematical inequality relating inner products and norms
spectral theorem for self-adjoint operators in the finite-dimensional case. Let A {\displaystyle A} be a self-adjoint operator on a finite-dimensional
Cauchy–Schwarz_inequality
Linear operator
a tridiagonal matrix. The most important case is the one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over
Jacobi_operator
Fact that observing a situation changes it
The term "observable" has gained a technical meaning, denoting a self-adjoint operator that represents the possible results of a random variable. Observer
Observer_effect_(physics)
Theorem of convex functions
n‑tuple of bounded self‐adjoint operators x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} with spectra in I and an n‑tuple of operators a 1 , … , a n {\displaystyle
Jensen's_inequality
zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. Lindelöf hypothesis that for all ε > 0 {\displaystyle \varepsilon
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Function over linear operators
trace TrW(T) is defined to be this operator. The partial trace of a self-adjoint operator is defined if and only if the partial traces of the positive and
Partial_trace
Bounded operators with sub-unit norm
\displaystyle {-\Re (A\xi ,\xi )\geq 0}} on its domain. When A is a self-adjoint operator T ( t ) = e A t , {\displaystyle \displaystyle {T(t)=e^{At},}} in
Contraction_(operator_theory)
Branch of mathematics
Hilbert space H {\displaystyle H} , together with a usually unbounded self-adjoint operator D {\displaystyle D} , such that ( 1 + D 2 ) − 1 / 2 {\displaystyle
Noncommutative_geometry
Systematic procedure of turning a classical theory into a quantum one
attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase
Quantization_(physics)
Statistical mechanics of quantum-mechanical systems
freedom. A density operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting
Quantum_statistical_mechanics
Topic in mathematics
trace tr {\displaystyle \operatorname {tr} } of the nonnegative self-adjoint operator T ∗ T {\displaystyle T^{*}T} is finite, in which case ‖ T ‖ HS 2
Hilbert–Schmidt_operator
Calculation rule in quantum mechanics
{\displaystyle |\psi \rangle } (see Bra–ket notation), corresponds to a self-adjoint operator A {\displaystyle A} whose spectrum is discrete if: the measured
Born_rule
*-algebra of bounded operators on a Hilbert space
projections; this is a consequence of the spectral theorem for self-adjoint operators. The projections of a finite factor form a continuous geometry.
Von_Neumann_algebra
Mathematical operator
is a closure operator. "Closure operators are lower adjoints of embeddings." Note however that not every embedding has a lower adjoint. Any partially
Closure_operator
Concept in Hlibert spaces mathematics
Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar
Trace_inequality
Mathematical theorem
correspondence between self-adjoint operators and (strongly continuous) one-parameter unitary groups. Let Q and P be two self-adjoint operators satisfying the
Stone–von_Neumann_theorem
Notation for quantum states
Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by self-adjoint operators, such
Bra–ket_notation
Property of a mass in motion
properties of the media. In quantum mechanics, momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines
Momentum
Part of spectral theory
von Neumann established a general spectral theorem for unbounded self-adjoint operators, which Kunihiko Kodaira used to streamline Weyl's method. Kodaira
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
Diacritical mark (◌̂)
observable; the observable A ^ {\displaystyle {\hat {A}}} is the self-adjoint operator implying a measurement whose n t h {\displaystyle n^{th}} moment
Circumflex
Operator generalizing the Laplacian in differential geometry
consequence, the Laplace–Beltrami operator is negative and formally self-adjoint, meaning that for compactly supported functions f {\displaystyle f} and
Laplace–Beltrami_operator
Concept in quantum mechanics
self-adjoint operator A {\displaystyle A} corresponding to the set of real-valued eigenvalues { a n } {\displaystyle \{a_{n}\}} . If the self-adjoint
Complete set of commuting observables
Complete_set_of_commuting_observables
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
n×n self-adjoint real matrices, as above. The Jordan algebra of n×n self-adjoint complex matrices, as above. The Jordan algebra of n×n self-adjoint quaternionic
Jordan_algebra
(I) General Theory; (II) Spectral Theory, Self Adjoint Operators in Hilbert Space; and (III) Spectral Operators. The first volume was published in 1958
Linear_Operators_(book)
Concepts from linear algebra
eigenstate of H, and E represents the eigenvalue. H is an observable self-adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the
Eigenvalues_and_eigenvectors
Generalized measurement in quantum mechanics
defined on M {\displaystyle M} whose values are positive bounded self-adjoint operators on H {\displaystyle {\mathcal {H}}} such that for every ψ ∈ H {\displaystyle
POVM
Technique for solving differential equations
problems for the operators for T {\displaystyle T} and S {\displaystyle S} . If T {\displaystyle T} is a compact, self-adjoint operator on the space L 2
Separation_of_variables
Type of entropy in quantum theory
freedom. A density operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting
Von_Neumann_entropy
Operator in probability theory
representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is self-adjoint. Even more generally
Covariance_operator
Probability problem
μ is the spectral measure of a self-adjoint operator. (More precisely stated, μ is the spectral measure for an operator T ¯ {\displaystyle {\overline {T}}}
Hamburger_moment_problem
Formulation of quantum mechanics on a Hilbert Space
of a quantum system are defined to be the (possibly unbounded) self-adjoint operators A {\displaystyle A} on H {\displaystyle \mathbb {H} } . A state
Dirac–von_Neumann_axioms
SELF ADJOINT-OPERATOR
SELF ADJOINT-OPERATOR
Girl/Female
African, Australian, British, Chinese, Christian, English, French, Greek, Hawaiian, Hebrew
Saviour; Ewe of West Africa; Goddess of the Moon; Cliff; Rock
Female
Egyptian
, a form of Isis.
Boy/Male
Welsh
peace'.
Boy/Male
Biblical
A rock.
Boy/Male
African, Arabic, Hindu, Indian, Muslim, Sindhi, Swahili
Sword; Brave; Sword of Religion
Surname or Lastname
English (East Anglia)
English (East Anglia) : from the Middle English personal name Saulf, Old English Sǣwulf, composed of the elements sǣ ‘sea’ + wulf ‘wolf’.
Boy/Male
Indian
Sword
Boy/Male
Muslim/Islamic
Sword
Girl/Female
British, English
Soft
Girl/Female
Egyptian
Surname or Lastname
English
English : from Middle English selle, a rough hut of the type normally occupied by animals, hence a topographic name for someone who lived in a hut like this. In many cases the name may have been in effect a metonymic occupational name for a herdsman.Americanized spelling of Hungarian and Hungarian Jewish Széll, a topographic name for someone who lived in a spot exposed to the wind, from Hungarian szél ‘wind’.German : variant of Selle.
Boy/Male
British, English, Nigerian, Norwegian
Rock
Girl/Female
Hebrew Biblical
Rock.
Boy/Male
British, English, Hebrew
A Tree
Male
English
(סֶלַע) Anglicized form of Hebrew Cela, SELA means "a rock." In the Old Testament bible, this is the name of the capital city of Edom, possibly an early name for Petra. In use as a unisex name.
Male
Yiddish
(סֶעף) Variant spelling of Yiddish Zeff, SEFF means "wolf."
Biblical
a rock
Male
Welsh
Welsh form of Greek SolomÅn, SELYF means "peaceable."Â
Female
Irish
Variant spelling of Irish Éadan, ÉADAOIN means "face" or perhaps "against" or "opposite."
Boy/Male
Muslim
Sword
SELF ADJOINT-OPERATOR
SELF ADJOINT-OPERATOR
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
The Earth; Broad; Spacious; Lover; A King from the Epics
Girl/Female
Australian, Czechoslovakian, German, Slavic
Golden
Female
English
Elaborated form of English Andrea, ANDRINA means "man; warrior."
Girl/Female
Hindu
Decorated with flowers, One that has flowered
Surname or Lastname
English
English : variant of Whitlock.
Boy/Male
Arabic, Muslim
God Fearing; Devout; Pious
Boy/Male
Hindu, Indian
Protector
Boy/Male
American, Anglo, Australian, British, Chinese, Christian, English, French, German, Irish
From Irvine; Scotland; Sea Lover; Boar Friend
Boy/Male
Tamil
Pleasing to eyes
Girl/Female
Hindu, Indian
Prosperous
SELF ADJOINT-OPERATOR
SELF ADJOINT-OPERATOR
SELF ADJOINT-OPERATOR
SELF ADJOINT-OPERATOR
SELF ADJOINT-OPERATOR
n.
Self.
a.
Disposed to self-assertion; self-asserting.
n.
Control of one's self; restraint exercised over one's self; self-command.
a.
Refusing to gratify one's self; self-sacrificing.
n.
Self-love.
a.
Dependent on one's self; self-depending; self-reliant.
n.
Self-deceit.
n.
Restraint over one's self; self-control; self-command.
n.
Imposture practiced on one's self; self-deceit.
n.
The act of estimating one's self; self-esteem.
n.
Enjoyment of one's self; self-satisfaction.
n.
Self-denial; self-renunciation; self-sacrifice.
v. i.
To join one's self.
imp. & p. p.
of Adjoin
n.
Faith in one's self; self-reliance.
n.
The act of governing one's self, or the state of being governed by one's self; self-control; self-command.
n.
The idolizing of one's self; immoderate self-conceit.
n.
Self-devotion.
a.
Self-repelling.
n.
Self-communion.