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Rational number equal to an integer plus 1/2
mathematics, a half-integer is a number of the form n + 1 2 , {\displaystyle n+{\tfrac {1}{2}},} where n {\displaystyle n} is an integer. For example,
Half-integer
Type of subatomic particle
subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin 3/2, etc.) and obey the Pauli exclusion principle
Fermion
Two-dimensional packing problem
a half-integer, the wasted space is at least proportional to its square root. The precise asymptotic growth rate of the wasted space, even for half-integer
Square_packing
Replacing a number with a simpler value
integer. Rounding a number x to the nearest integer requires some tie-breaking rule for those cases when x is exactly half-way between two integers –
Rounding
Tool in multivariate statistical analysis
(n-k+1)}}\left(2z\right)^{-k}.} This allows for the Matérn covariance of half-integer values of ν {\displaystyle \nu } to be expressed as C p + 1 / 2 ( d )
Matérn_covariance_function
Quantum mechanics principle
exclusion principle states that two or more identical particles with half-integer spins (i.e., fermions) cannot simultaneously occupy the same quantum
Pauli_exclusion_principle
Intrinsic quantum property of particles
exclusion principle: observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion. Spin is described mathematically
Spin_(physics)
Mathematical constants
mathematics. Its particular values can be expressed in closed form for integer, half-integer, and some other rational arguments, but no simple expressions are
Particular values of the gamma function
Particular_values_of_the_gamma_function
Class of subatomic particle
classes of subatomic particle, the other being fermions, which have half odd-integer spin (1/2, 3/2, 5/2, ...). Every observed subatomic particle is either
Boson
Theorem in quantum mechanics
for every energy eigenstate of a time-reversal symmetric system with half-integer total spin, there is another eigenstate with the same energy related
Kramers'_theorem
Hexagonal lattice made of carbon atoms
Hankinson, John; Sprinkle, Mike; Berger, Claire; de Heer, Walt A. (2009). "Half integer quantum Hall effect in high mobility single layer epitaxial graphene"
Graphene
(k+n+2)-1\right]} for positive integers m. Similar series may be obtained by exploring the Hurwitz zeta function at half-integer values. Thus, for example
Rational_zeta_series
Property of being an even or odd number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. For
Parity_(mathematics)
Functional square root of an exponential
logarithmic function iterated an integer number of times, rather than the half-integer that would be required for a half-exponential function. Crone, Lawrence
Half-exponential_function
Family of solutions to related differential equations
are when α {\displaystyle \alpha } is an integer or a half-integer. When α {\displaystyle \alpha } is an integer, the resulting Bessel functions are often
Bessel_function
British physicist (1902–1984)
a theory which describes the behaviour of fermions, particles with half-integer spin. His 1930 monograph The Principles of Quantum Mechanics is one of
Paul_Dirac
Generalization of Gaussian integers to quaternions
Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers
Hurwitz_quaternion
Symmetry between bosons and fermions
existence of a symmetry between particles with integer spin (bosons) and particles with half-integer spin (fermions). It proposes that for every known
Supersymmetry
Statistical model in quantum mechanics of magnetic materials
of the magnetic dipoles. If the spin is integer then only short-range order is present. A system of half-integer spins exhibits quasi-long range order.
Quantum_Heisenberg_model
Hadron (subatomic particle) that is composed of three quarks
particles. Baryons are also classified as fermions because they have half-integer spin. The name "baryon", introduced by Abraham Pais, comes from the Greek
Baryon
Theorem in quantum mechanics
elementary particles with integer spin (bosons) is symmetric under the exchange of any two particles, whereas for particles with half-integer spin (fermions),
Spin–statistics_theorem
Statistical description for the behavior of fermions
statistics applies to identical and indistinguishable particles with half-integer spin (1/2, 3/2, etc.), called fermions, in thermodynamic equilibrium
Fermi–Dirac_statistics
Every natural number can be represented as the sum of four integer squares
conjecture, states that every nonnegative integer can be represented as a sum of four non-negative integer squares. That is, the squares form an additive
Lagrange's four-square theorem
Lagrange's_four-square_theorem
List of particles in matter including fermions and bosons
are classified according to their spin. Fermions have half-integer spin while bosons have integer spin. All the elementary particles of the Standard Model
List_of_particles
Geometric arrangements of points, foundational to Lie theory
\beta } onto the line through α {\displaystyle \alpha } is an integer or half-integer multiple of α {\displaystyle \alpha } . Equivalent ways of writing
Root_system
Extension of the factorial function
z} except non-positive integers, and satisfies Γ ( n ) = ( n − 1 ) ! {\displaystyle \Gamma (n)=(n-1)!} for every positive integer n {\displaystyle n}
Gamma_function
Subatomic particle having no substructure
is differentiated via the spin–statistics theorem: it is half-integer for fermions, and integer for bosons. v t e Notes: [†] An anti-electron (e+ ) is conventionally
Elementary_particle
First case of a Lie group that is both compact and non-abelian
{\displaystyle l=m/2} , where l {\displaystyle l} is then either an integer or a half-integer, and the dimension is 2 l + 1 {\displaystyle 2l+1} . The representations
Representation theory of SU(2)
Representation_theory_of_SU(2)
Electromagnetic effect in physics
The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems
Quantum_Hall_effect
Notation for conserved quantities in physics and chemistry
atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and proton are half-integer multiples. The combination
Quantum_number
Quantum number parameterizing spin and angular momentum
such as s = 1/2 for all electrons. It is an integer for all bosons, such as photons, and a half-odd-integer for all fermions, such as electrons and protons
Spin_quantum_number
Elementary subatomic particle with negative electric charge
neutrinos, the tau is a lepton, and like all elementary particles with half-integer spin, the tau has a corresponding antiparticle of opposite charge but
Tau_(particle)
Description of the behaviour of bosons
which have integer values of spin. In contrast, particles that follow Fermi-Dirac statistics are called fermions and have half-integer spins. At low
Bose–Einstein_statistics
Physical model of non-interacting fermions
like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statistics determine the energy distribution of fermions
Fermi_gas
Class of elementary particles
symbols. In particle physics, a lepton is an elementary particle of half-integer spin (spin 1/2) that does not undergo strong interactions. Two main
Lepton
State of matter
corresponding excitation energies, the half-integer total spin of the electronic shell (one outer electron) and half-integer total spin of the nucleus are coupled
Bose–Einstein_condensate
Lattice in 8-dimensional space with special properties
that all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and the sum of
E8_lattice
Music description and organization project
render useful results. Each gene is assigned a number between 0 and 5, in half-integer increments. The Music Genome Project's database is built using a methodology
Music_Genome_Project
248-dimensional exceptional simple Lie group
coordinates are either all integers or all half-integers and the sum of the coordinates is even. Explicitly, there are 112 roots with integer entries obtained from
E8_(mathematics)
Triangle with integer side lengths
An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational
Integer_triangle
Type of light sensor
meaning a pure-moire model of ASP function is insufficient. However, at half-integer multiples of the Talbot depth, the periodicity of the diffraction grating
Angle-sensitive_pixel
Greatest integer less than or equal to square root
number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal
Integer_square_root
Mathematical entity to describe the probability of each possible measurement on a system
in units of the reduced Planck constant ħ, is either an integer (0, 1, 2, ...) or a half-integer (1/2, 3/2, 5/2, ...). For a massive particle with spin
Quantum_state
Nuclear physics classification method
nuclear spin is zero for even-Z, even-N nuclei, integer for all even-A nuclei, and odd half-integer for all odd-A nuclei. The neutron–proton ratio is
Even_and_odd_atomic_nuclei
Synchronisation signal in colour video
frequency and the audio carrier frequency a half-integer multiple of the line rate. The sum of these two half-integers implies that the distance between the
Color_burst
Quantum mechanical spectroscopic effect
energies of these are determined by the ZFS interaction strengths. For half-integer spin systems, ZFS splits the states into pairs of Kramers doublets (
Zero-field_splitting
Method to solve optimization problems
case, integer programming problems are in many practical situations (those with bounded variables) NP-hard. 0–1 integer programming or binary integer programming
Linear_programming
Integer side lengths of a right triangle
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known
Pythagorean_triple
Deformation of the group algebra of a Coxeter group
(T_{s}-q_{s}^{1/2})(T_{s}+q_{s}^{-1/2})=0.} After extending the scalars to include the half-integer powers q±1/2 s the resulting Hecke algebra is isomorphic to the previously
Iwahori–Hecke_algebra
Israeli electrical engineer and condensed matter physicist (born 1947)
fractional charge of the fractional quantum Hall effect and measuring the of half-integer thermal conductance quantum at filling fraction 5/2. For these discoveries
Mordehai_Heiblum
Type of field appearing in the Lagrangian
integer spin". Physical Review D. 18 (10): 3624–3629. doi:10.1103/PhysRevD.18.3624. Fang, J.; Fronsdal, C. (1978-11-15). "Massless fields with half-integral
Source_field
Number with a half-integer abundancy index
hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the sum-of-divisors
Hemiperfect_number
Formula for area of a grid polygon
formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result
Pick's_theorem
Chemical element with atomic number 2 (He)
(lines at 455.1, 541.1, and 1012.3 nm) to a new form of hydrogen with half-integer transition levels. In 1912, Alfred Fowler managed to produce similar
Helium
Complex number whose real and imaginary parts are both integers
number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and
Gaussian_integer
Wave equation for arbitrary spin particles
particles with non-zero mass and arbitrary spin j, an integer for bosons (j = 1, 2, 3 ...) or half-integer for fermions (j = 1⁄2, 3⁄2, 5⁄2 ...). The solutions
Bargmann–Wigner_equations
Natural number
1867. In computing, this number is the largest value that a signed 32-bit integer field can hold. At the time of its discovery, 2,147,483,647 was the largest
2,147,483,647
Natural number
square triangular number 1,673,196,525 : Least common multiple of the odd integers from 1 to 25 1,677,922,740 : number of series-reduced planted trees with
1,000,000,000
mesons and baryons is that mesons have integer spin (thus are bosons) while baryons are fermions (half-integer spin). Because mesons are bosons, the Pauli
List_of_mesons
Least-weight tree connecting graph vertices
ellipsoid method. However, if we add a requirement that f(e) must be half-integer (that is, f(e) must be in {0, 1/2, 1}), then the problem becomes NP-hard
Minimum_spanning_tree
Unsolved problem in mathematics
). Place a centered copy of C {\displaystyle C} at every point with half-integer coordinates. A ray from the origin may either miss all of the copies
Lonely_runner_conjecture
Natural number
= the number of possible arrangements of the integers 1, 2, ..., 10 with the restriction that the integer j must not be in the j-th postition for each
1,000,000
Subatomic particle with atypical quark content
baryons are a type of hadron (bound states of quarks and gluons) with half-integer spin, but with a quark content different from the three quarks (qqq)
Exotic_baryon
Coefficients in angular momentum eigenstates of quantum systems
=\delta _{j,j'}\delta _{m,m'}.} Here the italicized j and m denote integer or half-integer angular momentum quantum numbers of a particle or of a system.
Clebsch–Gordan_coefficients
Open problem on 3x+1 and x/2 functions
numbers, multiply by 3 and add 1. With enough repetition, do all positive integers converge to 1? More unsolved problems in mathematics The Collatz conjecture
Collatz_conjecture
Quantum mechanical operator related to rotational symmetry
is a half-integer (1/2, 3/2, etc.), R ( n ^ , 360 ∘ ) = − 1 {\displaystyle R\left({\hat {n}},360^{\circ }\right)=-1} , and when it is an integer, R (
Angular_momentum_operator
Pickering attributed the observation to a new form of hydrogen with half-integer transition levels. Fowler managed to produce similar lines from a hydrogen–helium
Pickering_series
Shape in mathematics of domino tiling
fixed integer, and the square lattice consists of unit squares with the origin as a vertex of 4 of them, so that both x and y are half-integers. The Aztec
Aztec_diamond
Function in statistics
denote the pair of half-integer rounding operators that map a real x {\displaystyle x} to its nearest left and right half-odd integer, respectively, according
Marcum_Q-function
Nearest integers from a number
returns the greatest integer less than or equal to x, written ⌊x⌋ or floor(x). Similarly, the ceiling function returns the least integer greater than or equal
Floor_and_ceiling_functions
Product of numbers from 1 to n
factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers less than or equal to
Factorial
Particle smaller than an atom
of quantum mechanics, can be either a boson (with integer spin) or a fermion (with odd half-integer spin). In the Standard Model, all the elementary fermions
Subatomic_particle
Mathematical problem
+2^{n-1}\\&=2^{n}-1\end{aligned}}} for n {\displaystyle n} being any positive integer. The exercise of working through this problem may be used to explain and
Wheat_and_chessboard_problem
Elementary particle with extremely low mass
leaving a theorized diffuse supernova neutrino background. Neutrinos have half-integer spin (1/2ħ); therefore they are fermions. Neutrinos are leptons; therefore
Neutrino
Interaction between subatomic particles
when two particles (typically, but not necessarily, half-integer spin fermions) exchange integer-spin, force-carrying bosons. The fermions involved in
Weak_interaction
Analytic function in mathematics
essential singularity. For sums involving the zeta function at integer and half-integer values, see rational zeta series. The reciprocal of the zeta function
Riemann_zeta_function
Mathematical function
\qquad \psi _{1}(3)={\frac {\pi ^{2}}{6}}-{\frac {5}{4}}.} At positive half integer values we have that ψ 1 ( n + 1 2 ) = π 2 2 − 4 ∑ k = 1 n 1 ( 2 k − 1
Trigamma_function
Constants of the mathematical zeta function
{\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)} We have simple relations for half-integer arguments ζ ( 3 / 2 ) ζ ( − 1 / 2 ) = − 4 π ζ ( 5 / 2 ) ζ ( − 3 / 2 )
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
Calculation and prediction of eclipses
in which a certain number of synodic months closely equals an integer or half-integer number of draconic months: one such period after an eclipse, a
Eclipse_cycle
Group of rotations in 3 dimensions
eigenvalues of this Casimir operator are 𝑱2=−j(j+1)𝑰2j+1, where j is integer or half-integer, and referred to as the spin or angular momentum. So, the 3 × 3
3D_rotation_group
Superconducting qubit implementation
(screen or enhance) a non-integer external flux bias. When the applied flux through the loop area is close to a half integer number of flux quanta, the
Flux_qubit
Coupling in quantum physics
has half-integer values, such as 1/2 for an electron, then the total (orbital plus spin) angular momentum will also be restricted to half-integer values
Angular_momentum_coupling
Ratio of two numbers
are, for example, one-half, eight-fifths, three-quarters. A simple fraction (examples: 1/2 and 17/3) consists of an integer numerator, displayed above
Fraction
Family of subatomic particles
and they are required to have an intrinsic spin of 3 /2 or higher (half-integer units). Ordinary nucleons (symbol N, meaning either a proton or neutron)
Delta_baryon
Polynomial with all terms of degree two
matrix with integer coefficients twos out a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients
Quadratic_form
Representation of the symmetry group of spacetime in special relativity
2 ) {\displaystyle {\mathfrak {su}}(2)} are labeled by nonnegative half-integers. Consequently, the irreducible representations of the Lorentz group's
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Composite subatomic particle
an odd number of quarks, they are also all fermions, i.e., they have half-integer spin. As quarks possess baryon number B = 1/3, baryons have baryon
Hadron
Noncentral generalization of the chi-squared distribution
where n is non-negative integer, Q is the Gaussian Q-function, and I is the modified Bessel function of first kind with half-integer order. The modified Bessel
Noncentral chi-squared distribution
Noncentral_chi-squared_distribution
Book by John Gribbin
bosons, or particles with whole-integer spin) and Fermi-Dirac statistics (describing fermions, or particles with half-integer spin). 6. Matrices & Waves discusses
In Search of Schrödinger's Cat
In_Search_of_Schrödinger's_Cat
Important atomic emission spectra
series was originally attributed to an unknown form of hydrogen with half-integer transition levels by both Pickering and Fowler, but Bohr correctly recognised
Hydrogen_spectral_series
Shape with four equal sides and angles
with half-integer vertex coordinates is NP-complete. Squaring the square involves subdividing a given square into smaller squares, all having integer side
Square
Austrian physicist (1900–1958)
field theory that states that particles with half-integer spin are fermions, while particles with integer spin are bosons. In 1949, he published a paper
Wolfgang_Pauli
Binary representation for signed numbers
most common method of representing signed (positive, negative, and zero) integers on computers, and more generally, fixed point binary values. As with the
Two's_complement
Physical phenomenon
spin tunneling splitting is only possible for integer spins, and is certainly absent for half-integer spins, as ensured by Kramers degeneracy theorem
Quantum_spin_tunneling
Curved triangle with constant width
integer lattice, the one with the largest width is a Reuleaux triangle. It has one of its axes of symmetry parallel to the coordinate axes on a half-integer
Reuleaux_triangle
Physical and science fiction concept
disks of brass, a material made up largely of elements with a total half-integer nuclear spin. He claimed that by rapidly rotating a disk of such material
Anti-gravity
with integer values. However, if we examine these formulae for toroidal harmonics, i.e. where the degree is half-integer, the order is integer, and the
Whipple_formulae
Fraction with denominator a power of two
Every integer, and every half-integer, is a dyadic rational. They both meet the definition of being an integer divided by a power of two: every integer is
Dyadic_rational
Use of mathematical groups in magnetochemistry
The irreducible representations of SU(2) are indexed by non-negative half integers j. If V is the two-dimensional vector representation, then Vj = S2j
Finite_subgroups_of_SU(2)
a majority of its element. Unpaired protons or neutrons contribute a half-integer nuclear spin, which in this case is a spin 1/2 orbital, giving the nucleus
Isotopes_of_nitrogen
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