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# 4.5 Combining polynomials using multiplication

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be able to multiply a polynomial by a monomial, be able to simplify +(a + b) and -(a - b), be able to multiply a polynomial by a polynomial.

## Overview

• Multiplying a Polynomial by a Monomial
• Simplifying $+\left(a+b\right)$ and $-\left(a+b\right)$
• Multiplying a Polynomial by a Polynomial

## Multiplying a polynomial by a monomial

Multiplying a polynomial by a monomial is a direct application of the distributive property.

## Distributive property

The distributive property suggests the following rule.

## Multiplying a polynomial by a monomial

To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.

## Sample set a

$10a{b}^{2}c\left(125{a}^{2}\right)=1250{a}^{3}{b}^{2}c$

## Practice set a

Determine the following products.

$3\left(x+8\right)$

$3x+24$

$\left(2+a\right)4$

$4a+8$

$\left({a}^{2}-2b+6\right)2a$

$2{a}^{3}-4ab+12a$

$8{a}^{2}{b}^{3}\left(2a+7b+3\right)$

$16{a}^{3}{b}^{3}+56{a}^{2}{b}^{4}+24{a}^{2}{b}^{3}$

$4x\left(2{x}^{5}+6{x}^{4}-8{x}^{3}-{x}^{2}+9x-11\right)$

$8{x}^{6}+24{x}^{5}-32{x}^{4}-4{x}^{3}+36{x}^{2}-44x$

$\left(3{a}^{2}b\right)\left(2a{b}^{2}+4{b}^{3}\right)$

$6{a}^{3}{b}^{3}+12{a}^{2}{b}^{4}$

$5mn\left({m}^{2}{n}^{2}+m+{n}^{0}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ne 0$

$5{m}^{3}{n}^{3}+5{m}^{2}n+5mn$

$6.03\left(2.11{a}^{3}+8.00{a}^{2}b\right)$

$12.7233{a}^{3}+48.24{a}^{2}b$

## $+\left(a+b\right)$ And $-\left(a+b\right)$

Oftentimes, we will encounter multiplications of the form

$+1\left(a+b\right)$ or $-1\left(a+b\right)$

These terms will actually appear as

$+\left(a+b\right)$ and $-\left(a+b\right)$

Using the distributive property, we can remove the parentheses.

The parentheses have been removed and the sign of each term has remained the same.

The parentheses have been removed and the sign of each term has been changed to its opposite.

1. To remove a set of parentheses preceded by a " $+$ " sign, simply remove the parentheses and leave the sign of each term the same.
2. To remove a set of parentheses preceded by a “ $-$ ” sign, remove the parentheses and change the sign of each term to its opposite sign.

## Sample set b

Simplify the expressions.

$\left(6x-1\right)$ .

This set of parentheses is preceded by a “ $+$ ’’ sign (implied). We simply drop the parentheses.

$\left(6x-1\right)=6x-1$

$\left(14{a}^{2}{b}^{3}-6{a}^{3}{b}^{2}+a{b}^{4}\right)=14{a}^{2}{b}^{3}-6{a}^{3}{b}^{2}+a{b}^{4}$

$-\left(21{a}^{2}+7a-18\right)$ .

This set of parentheses is preceded by a “ $-$ ” sign. We can drop the parentheses as long as we change the sign of every term inside the parentheses to its opposite sign.

$-\left(21{a}^{2}+7a-18\right)=-21{a}^{2}-7a+18$

$-\left(7{y}^{3}-2{y}^{2}+9y+1\right)=-7{y}^{3}+2{y}^{2}-9y-1$

## Practice set b

Simplify by removing the parentheses.

$\left(2a+3b\right)$

$2a+3b$

$\left({a}^{2}-6a+10\right)$

${a}^{2}-6a+10$

$-\left(x+2y\right)$

$-x-2y$

$-\left(5m-2n\right)$

$-5m+2n$

$-\left(-3{s}^{2}-7s+9\right)$

$3{s}^{2}+7s-9$

## Multiplying a polynomial by a polynomial

Since we can consider an expression enclosed within parentheses as a single quantity, we have, by the distributive property,

For convenience we will use the commutative property of addition to write this expression so that the first two terms contain $a$ and the second two contain $b$ .

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Abstract

Throughout history, spies, soldiers, and others have relied on so-called {\em hand ciphers} to send encrypted messages. Since the creation of Pontifex (also known as Solitaire) by Bruce Schneier in 1999, a number of hand ciphers utilizing a standard deck of playing cards have emerged. Since there are $52! \approx 2^{225.58}$ possible ways to order a deck of cards, there are over 225 bits of entropy in a well-shuffled deck of cards. Theoretically, this can provide enough security to rival modern computer-based cryptosystems. In this paper, we describe and analyze one such playing card cipher, Card-Chameleon, created by Matthew McKague. Our analysis reveals new weaknesses in this cryptosystem, particularly the tendency for a letter to encrypt to itself. This bias makes it easy to recover the plaintext if it is encrypted into multiple different ciphertexts. We will describe variations of Card-Chameleon which significantly reduced these weaknesses but did not completely eliminate them.

COinS

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# Parity (physics)

In quantum mechanics, a parity transformation (also called parity inversion) is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection):

${\displaystyle \mathbf {P} :{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\-y\\-z\end{pmatrix}}.}$

It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity. The weak interaction is chiral and thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions.

A matrix representation of P (in any number of dimensions) has determinant equal to −1, and hence is distinct from a rotation, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign is not a parity transformation; it is the same as a 180°-rotation.

In quantum mechanics, wave functions which are unchanged by a parity transformation are described as even functions, while those which change sign under a parity transformation are odd functions.

## Simple symmetry relations

Under rotations, classical geometrical objects can be classified into scalars, vectors, and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group.

Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations. The word projective refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not an observable, then a projective representation reduces to an ordinary representation. All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states.

The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, which is the special orthogonal group SO(3), are ordinary representations of the special unitary group SU(2) (see Representation theory of SU(2)). Projective representations of the rotation group that are not representations are called spinors, and so quantum states may transform not only as tensors but also as spinors.

If one adds to this a classification by parity, these can be extended, for example, into notions of

• scalars (P = +1) and pseudoscalars (P = −1) which are rotationally invariant.
• vectors (P = −1) and axial vectors (also called pseudovectors) (P = +1) which both transform as vectors under rotation.

One can define reflections such as

${\displaystyle V_{x}:{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto {\begin{pmatrix}-x\\y\\z\end{pmatrix}},}$

which also have negative determinant and form a valid parity transformation. Then, combining them with rotations (or successively performing x-, y-, and z-reflections) one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an even number of dimensions, though, because it results in a positive determinant. In odd number of dimensions only the latter example of a parity transformation (or any reflection of an odd number of coordinates) can be used.

Parity forms the abelian group ${\displaystyle \mathbb {Z} _{2}}$ due to the relation ${\displaystyle {\hat {\mathcal {P}}}^{2}={\hat {1}}}$. All Abelian groups have only one-dimensional irreducible representations. For ${\displaystyle \mathbb {Z} _{2}}$, there are two irreducible representations: one is even under parity, ${\displaystyle {\hat {\mathcal {P}}}\phi =+\phi }$, the other is odd, ${\displaystyle {\hat {\mathcal {P}}}\phi =-\phi }$. These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase.

## Classical mechanics

Newton's equation of motion ${\displaystyle {\vec {F}}=m\,{\vec {a}}}$ (if the mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity.

However, angular momentum ${\displaystyle {\vec {L}}}$ is an axial vector,

${\displaystyle {\vec {L}}={\vec {r}}\times {\vec {p}}}$,
${\displaystyle {\hat {P}}\left({\vec {L}}\right)=-\,{\vec {r}}\times -\,{\vec {p}}={\vec {L}}}$.

In classical electrodynamics, the charge density ${\displaystyle \rho }$ is a scalar, the electric field, ${\displaystyle {\vec {E}}}$, and current ${\displaystyle {\vec {j}}}$ are vectors, but the magnetic field, ${\displaystyle {\vec {H}}}$ is an axial vector. However, Maxwell's equations are invariant under parity because the curl of an axial vector is a vector.

## Effect of spatial inversion on some variables of classical physics

### Even

Classical variables, predominantly scalar quantities, which do not change upon spatial inversion include:

${\displaystyle \ t}$, the time when an event occurs
${\displaystyle \ m}$, the mass of a particle
${\displaystyle \ E}$, the energy of the particle
${\displaystyle \ P}$, power (rate of work done)
${\displaystyle \ \rho }$, the electric charge density
${\displaystyle \ V}$, the electric potential (voltage)
${\displaystyle \ \rho }$, energy density of the electromagnetic field
${\displaystyle \mathbf {L} }$, the angular momentum of a particle (both orbital and spin) (axial vector)
${\displaystyle \mathbf {B} }$, the magnetic field (axial vector)
${\displaystyle \mathbf {H} }$, the auxiliary magnetic field
${\displaystyle \mathbf {M} }$, the magnetization
${\displaystyle \ T_{ij}}$ Maxwell stress tensor.
All masses, charges, coupling constants, and other physical constants, except those associated with the weak force

### Odd

Classical variables, predominantly vector quantities, which have their sign flipped by spatial inversion include:

${\displaystyle \ h}$, the helicity
${\displaystyle \ \Phi }$, the magnetic flux
${\displaystyle \mathbf {x} }$, the position of a particle in three-space
${\displaystyle \mathbf {v} }$, the velocity of a particle
${\displaystyle \mathbf {a} }$, the acceleration of the particle
${\displaystyle \mathbf {p} }$, the linear momentum of a particle
${\displaystyle \mathbf {F} }$, the force exerted on a particle
${\displaystyle \mathbf {J} }$, the electric current density
${\displaystyle \mathbf {E} }$, the electric field
${\displaystyle \mathbf {D} }$, the electric displacement field
${\displaystyle \mathbf {P} }$, the electric polarization
${\displaystyle \mathbf {A} }$, the electromagnetic vector potential
${\displaystyle \mathbf {S} }$, Poynting vector.

## Quantum mechanics

### Possible eigenvalues

Two dimensional representations of parity are given by a pair of quantum states which go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all irreducible representations of parity are one-dimensional.

In quantum mechanics, spacetime transformations act on quantum states. The parity transformation, ${\displaystyle {\hat {\mathcal {P}}}}$, is a unitary operator, in general acting on a state ${\displaystyle \psi }$ as follows: ${\displaystyle {\hat {\mathcal {P}}}\,\psi _{\left(r\right)}=e^{\frac {i\phi }{2}}\psi _{\left(-r\right)}}$.

One must then have ${\displaystyle {\hat {\mathcal {P}}}^{2}\,\psi _{\left(r\right)}=e^{i\phi }\psi _{\left(r\right)}}$, since an overall phase is unobservable. The operator ${\displaystyle {\hat {\mathcal {P}}}^{2}}$, which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases ${\displaystyle e^{i\phi }}$. If ${\displaystyle {\hat {\mathcal {P}}}^{2}}$ is an element ${\displaystyle e^{iQ}}$ of a continuous U(1) symmetry group of phase rotations, then ${\displaystyle e^{-iQ}}$is part of this U(1) and so is also a symmetry. In particular, we can define ${\displaystyle {\hat {\mathcal {P}}}'\equiv {\hat {\mathcal {P}}}\,e^{-{\frac {iQ}{2}}}}$, which is also a symmetry, and so we can choose to call ${\displaystyle {\hat {\mathcal {P}}}'}$ our parity operator, instead of ${\displaystyle {\hat {\mathcal {P}}}^{2}}$. Note that ${\displaystyle {{\hat {\mathcal {P}}}'}^{2}=1}$ and so ${\displaystyle {\hat {\mathcal {P}}}'}$ has eigenvalues ${\displaystyle \pm 1}$. Wave functions with eigenvalue +1 under a parity transformation are even functions, while eigenvalue -1 corresponds to odd functions.[1] However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ${\displaystyle \pm 1}$.

For electronic wavefunctions, even states are usually indicated by a subscript g for gerade (German: even) and odd states by a subscript u for ungerade (German: odd). For example, the lowest energy level of the hydrogen molecule ion (H2+) is labelled ${\displaystyle 1\sigma _{g}}$ and the next-closest (higher) energy level is labelled ${\displaystyle 1\sigma _{u}}$.[2]

The wave functions of a particle moving into an external potential, which is centrosymmetric (potential energy invariant with respect to a space inversion, symmetric to the origin), either remain invariable or change signs: these two possible states are called the even state or odd state of the wave functions.[3]

The law of conservation of parity of particle (not true for the beta decay of nuclei[4]) states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution.

The parity of the states of a particle moving in a spherically symmetric external field is determined by the angular momentum, and the particle state is defined by three quantum numbers: total energy, angular momentum and the projection of angular momentum.[3]

### Consequences of parity symmetry

When parity generates the Abelian group2, one can always take linear combinations of quantum states such that they are either even or odd under parity (see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number.

In quantum mechanics, Hamiltonians are invariant (symmetric) under a parity transformation if ${\displaystyle {\hat {\mathcal {P}}}}$ commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any potential which is scalar, i.e., ${\displaystyle V=V{\left(r\right)}}$, hence the potential is spherically symmetric. The following facts can be easily proven:

• If ${\displaystyle \left|\varphi \right\rangle }$ and ${\displaystyle \left|\psi \right\rangle }$ have the same parity, then ${\displaystyle \left\langle \varphi \right|{\hat {X}}\left|\psi \right\rangle =0}$ where ${\displaystyle {\hat {X}}}$ is the position operator.
• For a state ${\displaystyle \left|{\vec {L}},L_{z}\right\rangle }$ of orbital angular momentum ${\displaystyle {\vec {L}}}$ with z-axis projection ${\displaystyle L_{z}}$, then ${\displaystyle {\hat {\mathcal {P}}}\left|{\vec {L}},L_{z}\right\rangle =\left(-1\right)^{L}\left|{\vec {L}},L_{z}\right\rangle }$.
• If ${\displaystyle \left[{\hat {H}},{\hat {P}}\right]=0}$, then atomic dipole transitions only occur between states of opposite parity.[5]
• If ${\displaystyle \left[{\hat {H}},{\hat {P}}\right]=0}$, then a non-degenerate eigenstate of ${\displaystyle {\hat {H}}}$ is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of ${\displaystyle {\hat {H}}}$ is either invariant to ${\displaystyle {\hat {\mathcal {P}}}}$ or is changed in sign by ${\displaystyle {\hat {\mathcal {P}}}}$.

Some of the non-degenerate eigenfunctions of ${\displaystyle {\hat {H}}}$ are unaffected (invariant) by parity ${\displaystyle {\hat {\mathcal {P}}}}$ and the others will be merely reversed in sign when the Hamiltonian operator and the parity operator commute:

${\displaystyle {\hat {\mathcal {P}}}\left|\psi \right\rangle =c\left|\psi \right\rangle }$,

where ${\displaystyle c}$ is a constant, the eigenvalue of ${\displaystyle {\hat {\mathcal {P}}}}$,

${\displaystyle {\hat {\mathcal {P}}}^{2}\left|\psi \right\rangle =c\,{\hat {\mathcal {P}}}\left|\psi \right\rangle }$.

## Many-particle systems: atoms, molecules, nuclei

The overall parity of a many-particle system is the product of the parities of the one-particle states. It is -1 if an odd number of particles are in odd-parity states, and +1 otherwise. Different notations are in use to denote the parity of nuclei, atoms, and molecules.

### Atoms

Atomic orbitals have parity (-1), where the exponent ℓ is the azimuthal quantum number. The parity is odd for orbitals p, f, ... with ℓ = 1, 3, ..., and an atomic state has odd parity if an odd number of electrons occupy these orbitals. For example, the ground state of the nitrogen atom has the electron configuration 1s22s22p3, and is identified by the term symbol 4So, where the superscript o denotes odd parity. However the third excited term at about 83,300 cm−1 above the ground state has electron configuration 1s22s22p23s has even parity since there are only two 2p electrons, and its term symbol is 4P (without an o superscript).[6]

### Molecules

Only some molecules have a centre of symmetry, including all homonuclear diatomic molecules as well as certain symmetric molecules including ethylene, benzene, xenon tetrafluoride and sulphur hexafluoride. For such centrosymmetric molecules, the parity each molecular orbital is either g (gerade or even) or u (ungerade or odd). An electronic state is u if and only if it contains an odd number of electrons in u orbitals.

For molecules with no centre of symmetry, including all heteronuclear diatomics as well as the majority of polyatomics, inversion is not a symmetry operation and the orbitals and states cannot be described as even or odd.

### Nuclei

In atomic nuclei, the state of each nucleon (proton or neutron) has even or odd parity, and nucleon configurations can be predicted using the nuclear shell model. As for electrons in atoms, the nucleon state has odd overall parity if and only if the number of nucleons in odd-parity states is odd. The parity is usually written as a + (even) or – (odd) following the nuclear spin value. For example the isotopes of oxygen include 17O(5/2+), meaning that the spin is 5/2 and the parity is even. The shell model explains this because the first 16 nucleons are paired so that each pair has spin zero and even parity, and the last nucleon is in the 1d5/2 shell which has even parity since ℓ = 2 for a d orbital.[7]

## Quantum field theory

The intrinsic parity assignments in this section are true for relativistic quantum mechanics as well as quantum field theory.

If we can show that the vacuum state is invariant under parity, ${\displaystyle {\hat {\mathcal {P}}}\left|0\right\rangle =\left|0\right\rangle }$, the Hamiltonian is parity invariant ${\displaystyle \left[{\hat {H}},{\hat {\mathcal {P}}}\right]}$ and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction.

To show that quantum electrodynamics is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. For simplicity we will assume that canonical quantization is used; the vacuum state is then invariant under parity by construction. The invariance of the action follows from the classical invariance of Maxwell's equations. The invariance of the canonical quantization procedure can be worked out, and turns out to depend on the transformation of the annihilation operator:

Pa(p, ±)P+ = −a(−p, ±)

where p denotes the momentum of a photon and ± refers to its polarization state. This is equivalent to the statement that the photon has odd intrinsic parity. Similarly all vector bosons can be shown to have odd intrinsic parity, and all axial-vectors to have even intrinsic parity.

There is a straightforward extension of these arguments to scalar field theories which shows that scalars have even parity, since

Pa(p)P+ = a(−p).

This is true even for a complex scalar field. (Details of spinors are dealt with in the article on the Dirac equation, where it is shown that fermions and antifermions have opposite intrinsic parity.)

With fermions, there is a slight complication because there is more than one spin group.

## Parity in the standard model

### Fixing the global symmetries

In the Standard Model of fundamental interactions there are precisely three global internal U(1) symmetry groups available, with charges equal to the baryon number B, the lepton number L and the electric charge Q. The product of the parity operator with any combination of these rotations is another parity operator. It is conventional to choose one specific combination of these rotations to define a standard parity operator, and other parity operators are related to the standard one by internal rotations. One way to fix a standard parity operator is to assign the parities of three particles with linearly independent charges B, L and Q. In general one assigns the parity of the most common massive particles, the proton, the neutron and the electron, to be +1.

Steven Weinberg has shown that if P2 = (−1)F, where F is the fermion number operator, then, since the fermion number is the sum of the lepton number plus the baryon number, F = B + L, for all particles in the Standard Model and since lepton number and baryon number are charges Q of continuous symmetries eiQ, it is possible to redefine the parity operator so that P2 = 1. However, if there exist Majorana neutrinos, which experimentalists today believe is possible, their fermion number is equal to one because they are neutrinos while their baryon and lepton numbers are zero because they are Majorana, and so (−1)F would not be embedded in a continuous symmetry group. Thus Majorana neutrinos would have parity ±i.

### Parity of the pion

In 1954, a paper by William Chinowsky and Jack Steinberger demonstrated that the pion has negative parity.[8] They studied the decay of an "atom" made from a deuteron (2
1
H+
) and a negatively charged pion (
π
) in a state with zero orbital angular momentum ${\displaystyle L=0}$ into two neutrons (${\displaystyle n}$).

Neutrons are fermions and so obey Fermi–Dirac statistics, which implies that the final state is antisymmetric. Using the fact that the deuteron has spin one and the pion spin zero together with the antisymmetry of the final state they concluded that the two neutrons must have orbital angular momentum ${\displaystyle L=1}$. The total parity is the product of the intrinsic parities of the particles and the extrinsic parity of the spherical harmonic function ${\displaystyle \left(-1\right)^{L}}$. Since the orbital momentum changes from zero to one in this process, if the process is to conserve the total parity then the products of the intrinsic parities of the initial and final particles must have opposite sign. A deuteron nucleus is made from a proton and a neutron, and so using the aforementioned convention that protons and neutrons have intrinsic parities equal to ${\displaystyle +1}$ they argued that the parity of the pion is equal to minus the product of the parities of the two neutrons divided by that of the proton and neutron in the deuteron, explicitly ${\displaystyle {\frac {(-1)(1)^{2}}{(1)^{2}}}=-1}$. Thus they concluded that the pion is a pseudoscalar particle.

### Parity violation

Top: P-symmetry: A clock built like its mirrored image will behave like the mirrored image of the original clock.
Bottom: P-asymmetry: A clock built like its mirrored image will not behave like the mirrored image of the original clock.

Although parity is conserved in electromagnetism, strong interactions and gravity, it turns out to be violated in weak interactions. The Standard Model incorporates parity violation by expressing the weak interaction as a chiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in weak interactions in the Standard Model. This implies that parity is not a symmetry of our universe, unless a hidden mirror sector exists in which parity is violated in the opposite way.

By the mid-20th century, it had been suggested by several scientists that parity might not be conserved (in different contexts), but without solid evidence these suggestions were not considered important. Then, in 1956, a careful review and analysis by theoretical physicists Tsung Dao Lee and Chen Ning Yang[9] went further, showing that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests. They were mostly ignored, but Lee was able to convince his Columbia colleague Chien-Shiung Wu to try it. She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards.

In 1957 Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson found a clear violation of parity conservation in the beta decay of cobalt-60.[10] As the experiment was winding down, with double-checking in progress, Wu informed Lee and Yang of their positive results, and saying the results need further examination, she asked them not to publicize the results first. However, Lee revealed the results to his Columbia colleagues on 4 January 1957 at a "Friday Lunch" gathering of the Physics Department of Columbia. Three of them, R. L. Garwin, Leon Lederman, and R. Weinrich modified an existing cyclotron experiment, and they immediately verified the parity violation.[11] They delayed publication of their results until after Wu's group was ready, and the two papers appeared back to back in the same physics journal.

After the fact, it was noted that an obscure 1928 experiment had in effect reported parity violation in weak decays, but since the appropriate concepts had not yet been developed, those results had no impact.[12] The discovery of parity violation immediately explained the outstanding τ–θ puzzle in the physics of kaons.

In 2010, it was reported that physicists working with the Relativistic Heavy Ion Collider (RHIC) had created a short-lived parity symmetry-breaking bubble in quark-gluon plasmas. An experiment conducted by several physicists including Yale's Jack Sandweiss as part of the STAR collaboration, suggested that parity may also be violated in the strong interaction.[13]

To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactions do not, one can still assign a parity to any hadron by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as rho meson decay to pions.

## References

General
• Perkins, Donald H. (2000). Introduction to High Energy Physics. ISBN 9780521621960.
• Sozzi, M. S. (2008). Discrete symmetries and CP violation. Oxford University Press. ISBN 978-0-19-929666-8.
• Bigi, I. I.; Sanda, A. I. (2000). CP Violation. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University Press. ISBN 0-521-44349-0.
• Weinberg, S. (1995). The Quantum Theory of Fields. Cambridge University Press. ISBN 0-521-67053-5.
Specific
1. ^ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice-Hall. p. 163. ISBN 0-205-12770-3.
2. ^ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice-Hall. p. 355. ISBN 0-205-12770-3.
3. ^ a b Andrew, A. V. (2006). "2. Schrödinger equation". Atomic spectroscopy. Introduction of theory to Hyperfine Structure. p. 274. ISBN 978-0-387-25573-6.
4. ^ Mladen Georgiev (November 20, 2008). ""Parity non-conservation in β-decay of nuclei: revisiting experiment and theory fifty years after. IV. Parity breaking models"". p. 26. arXiv:0811.3403.
5. ^ Bransden, B. H.; Joachain, C. J. (2003). Physics of Atoms and Molecules (2nd ed.). Prentice Hall. p. 204. ISBN 978-0-582-35692-4.
6. ^ NIST Atomic Spectrum Database To read the nitrogen atom energy levels, type "N I" in the Spectrum box and click on Retrieve data.
7. ^ Cottingham, W.N.; Greenwood, D.A. (1986). An introduction to nuclear physics. Cambridge University Press. p. 57. ISBN 0 521 31960 9.
8. ^ Chinowsky, W.; Steinberger, J. (1954). "Absorption of Negative Pions in Deuterium: Parity of the Pion". Physical Review. 95 (6): 1561–1564. Bibcode:1954PhRv...95.1561C. doi:10.1103/PhysRev.95.1561.
9. ^ Lee, T. D.; Yang, C. N. (1956). "Question of Parity Conservation in Weak Interactions". Physical Review. 104 (1): 254–258. Bibcode:1956PhRv..104..254L. doi:10.1103/PhysRev.104.254.
10. ^ Wu, C. S.; Ambler, E; Hayward, R. W.; Hoppes, D. D.; Hudson, R. P. (1957). "Experimental Test of Parity Conservation in Beta Decay". Physical Review. 105 (4): 1413–1415. Bibcode:1957PhRv..105.1413W. doi:10.1103/PhysRev.105.1413.
11. ^ Garwin, R. L.; Lederman, L. M.; Weinrich, M. (1957). "Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: The Magnetic Moment of the Free Muon". Physical Review. 105 (4): 1415–1417. Bibcode:1957PhRv..105.1415G. doi:10.1103/PhysRev.105.1415.
12. ^ Roy, A. (2005). "Discovery of parity violation". Resonance. 10 (12): 164–175. doi:10.1007/BF02835140.
13. ^ Muzzin, S. T. (19 March 2010). "For One Tiny Instant, Physicists May Have Broken a Law of Nature". PhysOrg. Retrieved 2011-08-05.
(−1)F

In a quantum field theory with fermions, (−1)F is a unitary, Hermitian, involutive operator where F is the fermion number operator. For the example of particles in the Standard Model, it is equal to the sum of the lepton number plus the baryon number, F = B + L. The action of this operator is to multiply bosonic states by 1 and fermionic states by −1. This is always a global internal symmetry of any quantum field theory with fermions and corresponds to a rotation by 2π. This splits the Hilbert space into two superselection sectors. Bosonic operators commute with (−1)F whereas fermionic operators anticommute with it.This operator really shows its utility in supersymmetric theories. Its trace is the spectral asymmetry of the fermion spectrum, and can be understood physically as the Casimir effect.

CPT symmetry

Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T that is observed to be an exact symmetry of nature at the fundamental level. The CPT theorem says that CPT symmetry holds for all physical phenomena, or more precisely, that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by ${\displaystyle {\overrightarrow {AB}}.}$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.

Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function ${\displaystyle f(x)=x^{n}}$ is an even function if ${\displaystyle n}$ is an even integer, and it is an odd function if ${\displaystyle n}$ is an odd integer.

Glossary of civil engineering

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

This glossary of civil engineering terms pertains specifically to civil engineering and its sub-disciplines. Please see glossary of engineering for a broad overview of the major concepts of engineering.

Glossary of engineering

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

This glossary of engineering terms is a list of definitions about the major concepts of engineering. Please see the bottom of the page for glossaries of specific fields of engineering.

Glossary of structural engineering

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

This glossary of structural engineering terms pertains specifically to structural engineering and its sub-disciplines. Please see glossary of engineering for a broad overview of the major concepts of engineering.

Index of physics articles (P)

The index of physics articles is split into multiple pages due to its size.

Jet quenching

In high-energy physics, jet quenching is a phenomenon that can occur in the collision of ultra-high-energy particles. In general, the collision of high-energy particles can produce jets of elementary particles that emerge from these collisions. Collisions of ultra-relativistic heavy-ion particle beams create a hot and dense medium comparable to the conditions in the early universe, and then these jets interact strongly with the medium, leading to a marked reduction of their energy. This energy reduction is called "jet quenching".

Judd–Ofelt theory

Judd–Ofelt theory is a theory in physical chemistry describing the intensity of electron transitions within the 4f shell of rare-earth ions in solids and solutions.The theory was introduced independently in 1962 by Brian R. Judd of the University of California, Berkeley, and PhD candidate George S. Ofelt at Johns Hopkins University. Their work was published in Physical Review and the Journal of Chemical Physics, respectively.

Judd and Ofelt did not meet, however, until 2003 at a workshop in Lądek-Zdrój, Poland.Judd and Ofelt's work was cited approximately 2000 times between 1962 and 2004. Brian M. Walsh of NASA Langley places Judd and Ofelt's theory at the "forefront" of a 1960s revolution in spectroscopic research on rare-earth ions.

Kaon

In particle physics, a kaon , also called a K meson and denoted K, is any of a group of four mesons distinguished by a quantum number called strangeness. In the quark model they are understood to be bound states of a strange quark (or antiquark) and an up or down antiquark (or quark).

Kaons have proved to be a copious source of information on the nature of fundamental interactions since their discovery in cosmic rays in 1947. They were essential in establishing the foundations of the Standard Model of particle physics, such as the quark model of hadrons and the theory of quark mixing (the latter was acknowledged by a Nobel Prize in Physics in 2008). Kaons have played a distinguished role in our understanding of fundamental conservation laws: CP violation, a phenomenon generating the observed matter–antimatter asymmetry of the universe, was discovered in the kaon system in 1964 (which was acknowledged by a Nobel Prize in 1980). Moreover, direct CP violation was discovered in the kaon decays in the early 2000s by the NA48 experiment at CERN and the KTeV experiment at Fermilab.

Multiplicative quantum number

In quantum field theory, multiplicative quantum numbers are conserved quantum numbers of a special kind. A given quantum number q is said to be additive if in a particle reaction the sum of the q-values of the interacting particles is the same before and after the reaction. Most conserved quantum numbers are additive in this sense; the electric charge is one example. A multiplicative quantum number q is one for which the corresponding product, rather than the sum, is preserved.

Any conserved quantum number is a symmetry of the Hamiltonian of the system (see Noether's theorem). Symmetry groups which are examples of the abstract group called Z2 give rise to multiplicative quantum numbers. This group consists of an operation, P, whose square is the identity, P2 = 1. Thus, all symmetries which are mathematically similar to parity (physics) give rise to multiplicative quantum numbers.

In principle, multiplicative quantum numbers can be defined for any abelian group. An example would be to trade the electric charge, Q, (related to the abelian group U(1) of electromagnetism), for the new quantum number exp(2iπ Q). Then this becomes a multiplicative quantum number by virtue of the charge being an additive quantum number. However, this route is usually followed only for discrete subgroups of U(1), of which Z2 finds the widest possible use.

Parity

Parity may refer to:

Parity (computing)

Parity bit in computing, sets the parity of transmitted data for the purpose of error detection

Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the result of the last operation

Parity file in data processing, created in conjunction with data files and used to check data integrity and assist in data recovery

Parity (mathematics), indicates whether a number is even or odd

Parity of a permutation, indicates whether a permutation has an even or odd number of inversions

Parity function, a Boolean function whose value is 1 if the input vector has an odd number of ones

Parity learning, a problem in machine learning

Parity of even and odd functions

Parity (physics), a symmetry property of physical quantities or processes under spatial inversion

Parity (biology), the number of times a female has given birth; gravidity and parity represent pregnancy and viability, respectively

Parity (charity), UK equal rights organisation

Parity (law), legal principle

Mental Health Parity Act, also applies to substance use disorder

Purchasing power parity, in economics, the exchange rate required to equalise the purchasing power of different currencies

Interest rate parity, in finance, the notion that the differential in interest rates between two countries is equal to the differential between the forward exchange rate and the spot exchange rate

Put–call parity, in financial mathematics, defines a relationship between the price of a European call option and a European put option

Parity (sports), an equal playing field for all participants, regardless of their economic circumstances

Potty parity, equalization of waiting times for males and females in restroom queues

A tactic in reversi

Grid parity of renewable energy

Doctrine of parity, agricultural price controls

Military parity, equipotential readiness between foes, without gaps such as a missile gap

Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric.

Point reflection can be classified as an affine transformation. Namely, it is an isometric involutive affine transformation, which has exactly one fixed point, which is the point of inversion. It is equivalent to a homothetic transformation with scale factor equal to −1. The point of inversion is also called homothetic center.

C, P, and T symmetries

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text:
# Drect limit of sequences

Let $\mathcal{C}$ is a grothendiect category and consider all of what follows in $\mathcal{C}$.

Let $$\{\varepsilon_i: 0\to A_i \to B_i \to C_i\to 0\ ,\ \phi_i^j\}$$ be a direct system of short exact sequences.

Is it true to say that $\varepsilon: 0\to \lim A_i \to \lim B_i \to \lim C_i \to 0$ (where $lim$ denotes the direct limit) is the direct limit $\lim \varepsilon_i$.

Actually we know that there exists a morphism $\varepsilon_i\to \varepsilon$ for each $i$.

What does occur when $\varepsilon_i$ is a split short exact sequence for each $i$? In fact why in this fact the direct limit is a pure short exact sequence?

In a locally finitely presented category, an exact sequence is called pure if it is a direct limit of split short exact sequences. In a locally presentable category the notion of $\alpha$-purity arises in a similar way (In fact pure exact sequences are $\aleph$-pure exact sequences).

For example in the category of modules over a ring, an exact sequence is pure if and only if it is a direct limit of split exact sequences.

-
Your question has a large number of typos and confusing phrasing. The answer to your first question "is it true" is yes. For your second question on splittings I see no reason why splittings at each $i$ would not assemble to a splitting for $\epsilon$, but I could be wrong here. For your last question I don't understand what you're asking. You'll get better answers if you take the time to make your question clearer –  David White May 18 '13 at 10:04
@David White: There is no reason that one should be able to choose the splittings to be compatible with the direct system so as to yield a splitting in the limit. For example, the submodule $\bigoplus_{\mathbb{N}} \mathbb{Z}$ of $\prod_{\mathbb{N}} \mathbb{Z}$ is pure, but it is not a direct summand. –  Martin May 19 '13 at 4:08

I am going to concentrate on the underlying categorical issue here and leave the homological algebra to the experts in that subject.

A direct limit is also known as a colimit. Yours, I take it, is over a sequence $N$, and as such is called directed or filtered.

Being an short exact sequence $0\to A\to B\to C\to 0$ amounts three things:

• $A\to B$ is a monomorphism or $0$ is its kernel, which are kinds of limit (or projective limit in old terminology), but finitary ones;

• $B\to C$ is an epimorphism, or $C$ is its image, which are other kinds of colimit property, this time finitary ones.

• the image of $A\to B$ is the kernel of $B\to C$, which combines properties of both kinds.

Now, limits commute with limits and colimits commute with colimits.

The question is whether filtered colimits commute with finite limits.

Indeed, they do if the category described is by a finitary algebraic theory.

I presume that a Grothendieck category is like this, though Zhen Lin says otherwise.

As David White says, the interest in filtered colimits arises from the fact that they are the ones that commute with finitary stuff such as limits in $\bf Set$.

However, this is not a theorem of categories in general.

As Zhen Lin points out, ${\bf Set}^{op}$ does not have this property. Limits and colimits in $\bf Set$ behave quite differently, as I emphasise in Chapter V of my book, Practical Foundations of Mathematics.

A preorder in which this happens is called meet-continuous. See Counterexamples O 4.5 of A Compendium of Continuous Lattices for complete lattices that are not meet-continuous.

(I had difficulty with my Internet connection when I originally posted this, so some of the intended text got deleted by mistake.)

I repeat that I am not an expert on homological algebra, but I note that the Wikipedia article on Grothendieck categories says explicitly that "direct limits (a.k.a. filtered colimits) of exact sequences are exact".

-
I thought filtered colimits always commute with finite limits. Isn't that basically what filtered colimits are good for? There's an old mathoverflow question on this subject –  David White May 18 '13 at 11:15
A Grothendieck category has a lot of smallness (I think it's locally presentable for example) so that means Hom commutes with colimits, maybe with some hypothesis on the kind of colimit or its length. It seems your argument might hold more generally and not really need this smallness hypothesis –  David White May 18 '13 at 11:17
@DavidWhite Filtered colimits commute with finite limits in any finitely accessible category (hence Paul's emphasis on finitary algebraic theory). In particular it will be true in a locally finitely presentable abelian category, but Grothendieck categories need not be locally finitely presentable. Moreover it is not true in general that filtered colimits commmute with finite limits: take $\mathbf{Set}^\mathrm{op}$, for example. –  Zhen Lin May 18 '13 at 12:36
To be clear, filtered colimits do commute with finite limits in a Grothendieck category – but this is basically part of the definition! For an example of a Grothendieck category that is not locally finitely presentable, one just has to look for a topos that is not locally finitely presentable. –  Zhen Lin May 18 '13 at 13:34
I thought that might be the case, but maybe for completeness you could either say explicitly what a Grothendieck category is or provide a link to a definition. –  Paul Taylor May 18 '13 at 14:29


text:
# What is the derivatives of sec2x and tan2x?

Mar 15, 2015

$\frac{d}{\mathrm{dx}} \left(\sec 2 x\right) = 2 \sec \left(2 x\right) \tan \left(2 x\right)$
$\frac{d}{\mathrm{dx}} \left(\tan 2 x\right) = 2 {\sec}^{2} \left(2 x\right)$

Use the derivatives of trig functions and the chain rule:

$\frac{d}{\mathrm{dx}} \left(\sec \left(2 x\right)\right) = \sec \left(2 x\right) \tan \left(2 x\right) \cdot 2 = 2 \sec \left(2 x\right) \tan \left(2 x\right)$.

$\frac{d}{\mathrm{dx}} \left(\tan \left(2 x\right)\right) = {\sec}^{2} \left(2 x\right) \cdot 2 = 2 {\sec}^{2} \left(2 x\right)$.

Explanation
You'll need the derivatives of $y = \sec x$ and $y = \tan x$
$\frac{d}{\mathrm{dx}} \left(\sec x\right) = \sec x \tan x$. and $\frac{d}{\mathrm{dx}} \left(\tan x\right) = {\sec}^{2} x$

You'll also want the Chain Rule.

There are various notations for derivatives and the Chain Rule, but for this question, this is a good one:
Suppose that we know $\frac{d}{\mathrm{dx}} \left(f \left(x\right)\right) = f ' \left(x\right)$, then if we want $\frac{d}{\mathrm{dx}} \left(f \left(u\right)\right)$ , the Chain Rule tells us to find the derivative of the outside function (that's $f '$) and evaluate it, not at $x$, but at $u$. Then multiply by the derivative of the inside.

The Chain Rule:
$\frac{d}{\mathrm{dx}} \left(f \left(u\right)\right) = f ' \left(u\right) \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

Finding $\frac{d}{\mathrm{dx}} \left(\tan \left(2 x\right)\right)$

The outside function is $\tan$ and the inside function (the $u$) is $2 x$.

The derivative of the tangent function is the square of the secant function.
$\frac{d}{\mathrm{dx}} \left(f \left(u\right)\right) = \frac{d}{\mathrm{dx}} \left(\tan \left(u\right)\right) = {\sec}^{2} \left(u\right) \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

As you get more experience, you'll simply write:

$\frac{d}{\mathrm{dx}} \left(\tan \left(2 x\right)\right) = {\sec}^{2} \left(2 x\right) \cdot 2 = 2 {\sec}^{2} \left(2 x\right)$.

Finding $\frac{d}{\mathrm{dx}} \left(\sec \left(2 x\right)\right)$

The outside function is $\sec$ and the inside function (the $u$) is $2 x$.

The derivative of $f \left(x\right) = \sec x$ is the function (singular), $f ' \left(x\right) = \sec x \tan x$.

So the derivative of $f \left(2 x\right)$ is $f ' \left(2 x\right) \cdot \frac{d}{\mathrm{dx}} \left(2 x\right)$, we write:

$\frac{d}{\mathrm{dx}} \left(\sec \left(2 x\right)\right) = \sec \left(2 x\right) \tan \left(2 x\right) \cdot 2 = 2 \sec \left(2 x\right) \tan \left(2 x\right)$.


text:
Back to Mover page.

## HydrogenBondConstraintGenerator

Generates constraints to attempt to enforce hydrogen bonding between atoms in residues defined by the two residue selectors. Constraints are generated by default for all pairs of residues (r1, r2), where r1 is a residue selected by residue_selector1 and r2 is a residue selected by residue_selector2, A1 is the hydrogen bond-capable heavy atom in r1, P_A1 is the parent atom of A1, PP_A1 is the parent of P_A1, A2 is the hydrogen-bond-capable heavy atom in r2, P_A2 is the parent atom of A2, and PP_A2 is the parent of P_A2 (see below for parent definitions). The following constraints are generated for each (A1, A2) pair:

• An atom pair constraint between the A1 and A2.
• An angle constraint (angle1) to restrain the angle between P_A1, A1 and A2
• An angle constraint (angle2) to restrain the angle between A1, A2 and P_A2.

The following dihedral constraints will be generated only if a planar chemical group is involved:

• A dihedral constraint (dihedral1) to restrain the dihedral PP_A1, P_A1, A1 and A2 (only if PP_A1-P_A1-A1 are coplanar, e.g. if all are in a His ring, Asp/Glu carboxylate, etc)
• A dihedral constraint (dihedral2) to restrain the dihedral A1, A2, P_A2, PP_A2 (only if A2-P_A2-PP_A2 are coplanar, e.g. if all are in a His ring, Asp/Glue carboxylate, etc)

The functions to be used for these constraints are user-configurable. By default, harmonic functions are used for atom pair constraints and circular harmonic functions are used for angle/dihedral constraints. See the constraint documentation for more information about functions.

### Selection of multiple residues

It is possible for each residue selector to select multiple residues. If this occurs, Ambiguous constraints will be generated for each (r1, r2) pair, where r1 is selected by residue_selector1 and r2 is selected by residue_selector2. The result is that only the constraints from the best-scoring residue pair will be used.

If you want to constrain multiple hydrogen bonds simulataneously, you should use a seperate HydrogenBondConstraintGenerator for each.

### Selection of multiple atoms

It is also possible to select multiple atoms in each residue. If this occurs, Ambiguous constraints are generated for each (A1, A2) pair in each (r1, r2) pair, where r1 is a residue selected by residue_selector1, r2 is a residue selected by residue_selector2, A1 is a hydrogen-bond-capable atom in r1, and A2 is a hydrogen-bond-capable atom in r2. The resulting score will only include the best-scoring hydrogen bonding atom pair. If you want to constrain multiple hydrogen bonds simulataneously, you should use a seperate HydrogenBondConstraintGenerator for each.

### Parent atom selection

The atoms involved in the angle and dihedral constraints are given below:

 Residue Name A1 P_A1 PP_A1 Distance (1/2) Angle Dihedrals ARG NE CZ NH1 1.4 120.0 0.0,180.0 ARG NH1 CZ NE 1.4 120.0 0.0,180.0 ARG NH2 CZ NE 1.4 120.0 0.0,180.0 ASP OD1 CG OD2 1.4 120.0 0.0,180.0 ASP OD2 CG OD1 1.4 120.0 0.0,180.0 CYS SG CB CA 2.0 109.5 Any GLU OE1 CD OE2 1.4 120.0 0.0,180.0 GLU OE2 CD OE1 1.4 120.0 0.0,180.0 HIS ND1 CE1 NE2 1.4 120.0 180.0 HIS NE2 CE1 ND1 1.4 120.0 180.0 LYS NZ CE CD 1.4 109.5 Any SER OG CB CA 1.4 109.5 Any THR OG1 CB CA 1.4 109.5 Any TRP NE1 CD1 CG 1.4 120.0 180.0 TYR OH CZ CE1 1.4 109.5 Any

### Caveats

To have effect, the atom_pair_constraint, angle_constraint, and dihedral_constraint scoreterms must be on in the scorefunction.

This ConstraintGenerator currently only works with canonical amino acids.

## Definition

<HydrogenBondConstraintGenerator name="(&string)"
residue_selector1="(&string TrueSelector)"
residue_selector2="(&string TrueSelector)"
atoms1="(&string '')"
atoms2="(&string '')"
atom_pair_func="(&string 'HARMONIC 2.8 0.5')"
angle1_func="(&string 'CIRCULARHARMONIC 2.0 0.5')"
angle2_func="(&string 'CIRCULARHARMONIC 2.0 0.5')"
atom_definitions="(&string '' )" />
• residue_selector1 - If given, the residues r1 (see above) will come from the subset of residues selected by residue_selector1. This should probably always be given to avoid huge numbers of constraints being generated.
• residue_selector2 - If given, the residues r2 (see above) will come from the subset of residues selected by residue_selector2. This should probably always be given to avoid huge numbers of constraints being generated.
• atoms1 - Comma-seperated list of atom names to consider constraining for each residue r1 selected by residue_selector1. Atom names that are not present in the pose will not be constrained -- therefore, if one specifies "OD1,OE1" as atoms1, it could be used for both Asp and Glu residues -- OD1 would be constrained in Asp residues, and OE1 would be constrained in Glu residues.
• atoms2 - Comma-seperated list of atom names to consider constraining for each residue r2 selected by residue_selector2. Atom names not present in the pose will not be constrained (see "atoms1" above).
• atom_pair_func -- Function to be used for atom pair constraints. By default, "HARMONIC x 0.5" is used, where x is the ideal A1-A2 distance.
• angle1_func - Function used for angle constraints. By default, "CIRCULARHARMONIC x 0.5" is used, where x is the ideal P_A1-A1-A2 angle.
• angle2_func - Function used for angle2 constraints. By default, "CIRCULARHARMONIC x 0.5" is used, where x is the ideal A1-A2-P_A2 angle.
• atom_definitions - Definitions of atoms to be used in hydrogen bonding. Only required for non-canonical amino acids. The format is: NAME3,ATOM1,ATOM2,ATOM3,DIST,ANGLE[,DIHEDRALS]*. For hydrogen bonding to OD1 of ASP, this line might read:
  "ASP,OD1,CG,OD2,1.4,120.0,0.0,180.0"
Note the 'ideal' dihedrals of 0.0 and 180.0 which indicates that hydrogen bonds to ASP should be in-plane. For hydrogen bonding to OG of SER, this line might read:
  "SER,OG,CB,CA,1.4,109.5"

### Example

This example creates constraints to enforce a hydrogen bond between atom NE2 on residue 6 and atom OE1 on residue 50. The atom pair constraint between these atoms uses a custom function -- a flat-bottom function with width 0.4 and sd 0.5.

<RESIDUE_SELECTORS>
<Index name="his6" resnums="6" />
<Index name="glu50" resnums="50" />
</RESIDUE_SELECTORS>
<MOVERS>
<HydrogenBondConstraintGenerator name="gen_my_csts"
residue_selector1="his6"
residue_selector2="glu50"
atoms1="NE2"
atoms2="OE1"
atom_pair_func="FLAT_HARMONIC 2.8 0.5 0.4" />
</PROTOCOLS>


text: ax² + bx + c â¡ 0 (mod n) a . This function will seed the global default random number generator, and any call to a function in numpy.random will use and alter its state. Code: lcg.c Park-Miller random number generator Description: Also known as the Lehmer random number generator. Linear congruential generators (LCGs) are a class of pseudorandom number generator (PRNG) algorithms used for generating sequences of random-like numbers. linear congruential generator Search and download linear congruential generator open source project / source codes from CodeForge.com Upgrade to Math Mastery. mod p a is a quadratic â¦ We use analytics cookies to understand how you use our websites so we can make them better, e.g. A random number generator (RNG) is a computational or physical device designed to generate a sequence of numbers or symbols that lack any pattern, i.e. â¢ Provides code examples updated and written in Python and C# Essential Algorithms has been updated and revised and offers professionals and students a hands-on guide to analyzing algorithms as well as the techniques and applications. they're used to gather information about the pages you visit and how many clicks you need to accomplish a task. IIR Increases the resolution of an image by interpolation B= IIR(inputfile,f) returns the image stored in file 'inputfile' with resolution increased by factor f in both dimensions. I have generated a small sample of numbers with this algorithm in python: x=123492981749283749834750984327 for j in range(20): x=(1664525*x+1013904223)%(2E32) print x/2E32 ... a residue is a primitive root if and only if it is not a quadratic residue. Shub Generator - - Concept Quadratic residues Let p be an odd prime and a be an integer a is a quadratic residue modulo p if a is not congruent to 0 modif a is not congruent to 0 mod p and there exists an integer x such that a â¡ x. Budap. linear congruential generator ç¨åºæºä»£ç åä¸è½½é¾æ¥ã The linear congruential method is discussed in detail in Donald Knuth's "The art of computer programming", Volume 2 "Seminumerical Algorithms" 2nd ed. Generate a Random Number. A. maths, radix2_fft. These functions are embedded within the random module of Python. The book also includes a collection of questions that may appear in a â¦ Seed: a: b: n: Join our newsletter for the latest updates. Quadratic modular equation solver. The convergence of Monte Carlo integration is $$\mathcal{0}(n^{1/2})$$ and independent of the dimensionality. Convert Celsius To Fahrenheit. Python Program to Generate a Random Number In this example, you will learn to generate a random number in Python. Linear Congruential Method Codes and Scripts Downloads Free. Comput. Rolando EÅtvÅs, Sect. Linear congruential generator Description: Another simple pseudo-random number generator used for the rand() function in glibc. â¦d lists () * Create merge_two_lists.py that implements merging of two sorted linked lists * Update merge_two_lists.py Fixed formatting errors * Fixed trailing whitespace * Change name of function to def __str__() * updating DIRECTORY.md * Imported classes from singly_linked_list.py * Update merge_two_lists.py * Update merge_two_lists.py Co-authored-by: github-actions <${GITHUBâ¦ Solve Quadratic Equation.$\begingroup\$ well written code. Alpertron. Sci. other, nested_brackets. Analytics cookies. ... other, linear_congruential_generator. Simply seed the random number generator with a fixed value, e.g. Digits per group . other, magicdiamondpattern. Quadratic Congruential Generator I (QCG-I) Quadratic Congruential Generator II (QCG-II) Cubic Congruential Generator II (CCG) ï¼åçå¸®å©ææ¡£ 116 é¡µ. That tighten the understanding of the root in an extension field. Quadratic modular equation solver. The generation of random numbers plays a large role in many applications ranging from cryptography to Monte Carlo methods. ... maths, quadratic_equations_complex_numbers. The convergence of Monte Carlo integration is $$\mathcal{0}(n^{1/2})$$ and independent of the dimensionality. GPA = 2 Since your GPA = 2, then your total letter grade on a 4.0 scale is C Programs. Solves a common algebraic Riccati equation using Schur decomposition. This is one of the 100+ free recipes of the IPython Cookbook, Second Edition, by Cyrille Rossant, a guide to numerical computing and data science in the Jupyter Notebook.The ebook and printed book are available for purchase at Packt Publishing. c . q, then x (mod n) corresponds uniquely to the pair of integers x (mod p) and x (mod q). Linear Congruential Generator Calculator. Actually there are several more efficient ways to find factors of big numbers (for smaller ones trial division works reasonably well). 32, 151-176 (2010). n . Linear congruential generator A linear congruential generator (LCG) was rst proposed in 1949 by D. H. Lehmer [ 14 ]. Convert Kilometers to Miles. numpy.random.seed(42) This way, you'll always get the same random number sequence. Hence Monte Carlo integration gnereally beats numerical intergration for moderate- and high-dimensional integration since numerical integration (quadrature) converges as $$\mathcal{0}(n^{d})$$.Even for low dimensional problems, Monte Carlo integration may have an â¦ So there don't need to exist a complex number which can solve this but it behaves like this (+-*conj(x))). maths, radians. Solving equations and inequalities. Rosetta Code is a programming chrestomathy site. Online Pseudo Random Number Generator This online tool generates pseudo random numbers based on the selected algorithm. Join. maths, relu. Exclusive OR Generator (XORG) Modular Exponentiation Generator (MODEXPG) Secure Hash Generator (G-SHA1) ï¼åçå¸®å©ææ¡£ 117 é¡µ. random.shuffle (x [, random]) ¶ Shuffle the sequence x in place.. Why my linear congruential generator generate low quality random numbers? Swap Two Variables. A pseudorandom number generator (PRNG), also known as a deterministic random bit generator â¦ One method which is very fast if the input number has two factors very close to its square root is known as Fermat factorisation .It makes use of the identity N = (a + b)(a - b) = a^2 - b^2 and is easy to understand and implement. b . Python 2.0 was released on 16 October 2000 with many major new features, including a cycle-detecting garbage collector and support for Unicode. Linear congruential generators are one of the oldest and most well-known methods for generating random â¦ 22 mod p . Random Number Generator in Python are built-in functions that help you generate numbers as and when required. Roots of a quadratic function; Roots of unity; Rosetta Code/Rank languages by popularity; Rot-13; RPG attributes generator; RSA code; Rule30; Run as a daemon or service; Run-length encoding; Runge-Kutta method The optional argument random is a 0-argument function returning a random float in [0.0, 1.0); by default, this is the function random().. To shuffle an immutable sequence and return a new shuffled list, use sample(x, k=len(x)) instead. 1980. Take a look at the following table that consists of some important random number generator functions along with their description present in the random module: Univ. Hence Monte Carlo integration gnereally beats numerical intergration for moderate- and high-dimensional integration since numerical integration (quadrature) converges as $$\mathcal{0}(n^{d})$$.Even for low dimensional problems, Monte Carlo integration may have an â¦ The algorithm II.1 is given by formula: x n +1 = ( ax n + c) mod m; n > 0; where m is the mask or the modulus m > 1, a is the multiplier (0 6 a < m ), c is the increment (0 6 c < m ), x 0 is the seed or initial value. On inversive congruential generator with a variable shift for pseudorandom numbers with prime power modulus, Ann. appear random. This Web application can solve equations of the form aâ¢x² + bx + c â¡ 0 (mod n) where the integer unknown x is in the range 0 â¤ x < n. A hardware (true) random number generator is a piece of electronics that plugs into a computer and produces genuine random numbers as opposed â¦ Text on GitHub with a CC-BY-NC-ND license Code on GitHub with a MIT license The idea is to present solutions to the same task in as many different languages as possible, to demonstrate how languages are similar and different, and to aid a person with a grounding in one approach to a problem in learning another. A tool for solving linear congruences of the form ax â¡ b (mod m). Major new features, including a cycle-detecting garbage collector and support for.... Using Schur decomposition generator Description: Also known as the Lehmer random number generator This online tool generates Pseudo number! Letter grade on a 4.0 scale is c Solve Quadratic equation October 2000 with many major new features including! 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text:
# Math Help - word ratio question

1. ## word ratio question

this is my queston:
374 divided in ratio of 1:2. The second person shared his section between two people in ratio of 2:3. How much did each person get?

for the first bit i have to do, 374 divided 3= 124.7 (i think) 124.7+124.7= 249.3.
after that part im not sure what to do, could someone please explain! thanks

2. does anyone know how to do this?

3. Originally Posted by andyboy179
this is my queston:
374 divided in ratio of 1:2. The second person shared his section between two people in ratio of 2:3. How much did each person get?

for the first bit i have to do, 374 divided 3= 124.7 (i think) 124.7+124.7= 249.3.
after that part im not sure what to do, could someone please explain! thanks
HI

the second person in this question refers to who ? The one who got 1 part of 374 or 2 parts of 374 ?

4. Originally Posted by mathaddict
HI

the second person in this question refers to who ? The one who got 1 part of 374 or 2 parts of 374 ?
i don't know, what i wrot down is what is in my exercise book, its prob the person who got 2 parts

5. Originally Posted by andyboy179
this is my queston:
374 divided in ratio of 1:2. The second person shared his section between two people in ratio of 2:3. How much did each person get?

for the first bit i have to do, 374 divided 3= 124.7 (i think) 124.7+124.7= 249.3. correct
after that part im not sure what to do, could someone please explain! thanks
well , let me make a guess here .

The person who got the 2 parts (249.3) shared with another person in the ratio of 2:3

Use the same method you used to solve the first part of the question ,

249.3 divide by 5 , then times 2 and 3 respectively to get the amount each person gets .

6. Originally Posted by mathaddict
well , let me make a guess here .

The person who got the 2 parts (249.3) shared with another person in the ratio of 2:3

Use the same method you used to solve the first part of the question ,

249.3 divide by 5 , then times 2 and 3 respectively to get the amount each person gets .

so would the answer be: 124.7:249.3, 99.72:149.58?

7. Originally Posted by andyboy179
so would the answer be: 124.7:249.3, 99.72:149.58?
yes .


text:
# Weighted Uniform Convergence of Entire Grünwald Operators on the Real Line

Weighted Uniform Convergence of Entire Grünwald Operators on the Real Line We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type τ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau >0$$\end{document} at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# Weighted Uniform Convergence of Entire Grünwald Operators on the Real Line

, Volume 22 (4): 17 – Dec 1, 2022
17 pages

/lp/springer-journals/weighted-uniform-convergence-of-entire-gr-nwald-operators-on-the-real-mq35TfxQTX
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-021-00408-2
Publisher site
See Article on Publisher Site

### Abstract

We consider weighted uniform convergence of entire analogues of the Grünwald operator on the real line. The main result deals with convergence of entire interpolations of exponential type τ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau >0$$\end{document} at zeros of Bessel functions in spaces with homogeneous weights. We discuss extensions to Grünwald operators from de Branges spaces.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Dec 1, 2022

Keywords: Grünwald operator; Hermite–Fejér interpolation; Weighted uniform approximation; de Branges space; Exponential type; 41A05; 41A17; 30E05

### References

Access the full text.


text:
Hot carrier and hot phonon coupling during ultrafast relaxation of photoexcited electrons in graphene

@article{Iglesias2016HotCA,
title={Hot carrier and hot phonon coupling during ultrafast relaxation of photoexcited electrons in graphene},
author={Jos{\'e} M. Iglesias and Mar'ia J. Mart'in and Elena Pascual and Ra{\'u}l Rengel},
journal={arXiv: Mesoscale and Nanoscale Physics},
year={2016}
}
• J. M. Iglesias, +1 author R. Rengel
• Published 27 January 2016
• Physics, Materials Science
• arXiv: Mesoscale and Nanoscale Physics
We study, by means of a Monte Carlo simulator, the hot phonon effect on the relaxation dynamics in photoexcited graphene and its quantitative impact as compared to considering an equilibrium phonon distribution. Our multi-particle approach indicates that neglecting the hot phonon effect significantly underestimates the relaxation times in photoexcited graphene. The hot phonon effect is more important for a higher energy of the excitation pulse and photocarrier densities between $1$ and \$3\times…
22 Citations

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Ultrafast dynamics of hot electrons and phonons in chemical vapor deposited graphene
• Materials Science
• 2013
The relaxation dynamics of photoexcited carriers in a chemical vapor deposited graphene transferred on quartz substrate has been investigated by using ultrafast optical-pump terahertz (THz)-probe
Hot carriers relaxation in highly excited polar semiconductors: Hot phonons versus phonon-plasmon coupling
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We present a study of the photo-excited charge carriers relaxation dynamics in polar semiconductors comparing calculations to pump probe experiments. Hot carrier densities in the 1018cm-3 range can
Ultrafast relaxation of hot optical phonons in monolayer and multilayer graphene on different substrates
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Abstract Hot carrier cooling in few-layer and multilayer epitaxial graphene on SiC, and chemical vapor deposition (CVD) grown graphene transferred onto a glass substrate was investigated by transient
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The dominant role of carrier\char21{}polar-optical-phonon interaction is shown in determining the nature of the carrier dynamics in the low-density limit, and the miniband width turns out to be a relevant quantity in predicting the existence of Bloch oscillations.
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text:
9.4.3 DistanceLeft, DistanceRight

Distances between the rolls on both sides (in the direction of the material flow). The MonitoredParameterType is defined in OPC 40083.

Unit: mm or inch


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# Tag Info

0

Assuming you want to allow headings part way down a page - just not at the very bottom, one solution could be the needspace package. Needspace allows you to specify how much space must be on the page before placing the next item. However, this may not catch all instances of your issue. Add \usepackage{needspace} and then do \needspace{X} before your ...

1

This version resembles the version of egreg, but it regards the (total) section number, such, that there is no pagebreak just before the first section occurs (unless manually forced) \documentclass[12pt]{article} \usepackage{etoolbox}% \usepackage{blindtext}% \usepackage{assoccnt}% \let\LaTeXStandardSection\section \newcounter{totalsection}% ...

3

2

I take over the relay from Christian Hupfer and give a tcolorbox based solution. I tried to mimicry the original macros to get an equivalent appearance. The main difference is that lemma and proof are now breakable boxes. I left out the \qed since I don't know what is used for that one. The macros take an optional parameter (any tcolorbox option). E.g., use ...

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I don't think an automated solution would be wise, because you would run into problems with functions that span multiple pages, but here is an ad-hoc solution. You can invoke (within an "escape to LaTeX" in your listings) the two macros defined below, \forbidpgbr and \allowpgbr, to locally forbid/allow pagebreaks. Note that the change takes effect only on ...

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Here's a way to get better breaks using the development version of minted. I will add this to the development version with a new option, but that may take a little while since I expect that some refactoring and additional refining will be necessary. Add the code below in your preamble, after you have set the highlighting style that you want. If you aren't ...

1

A Rather Prelimary Answer In order to make it work, I downloaded the latest version of minted.sty. There are some ways to make it work, but I feel that there need to be some improvements, so everybody please be my guest! Short code using environment/inputminted seems to be working fine! \documentclass{book} \usepackage[]{minted} \usepackage{xcolor} ...

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This is a response to a comment on Jaun's answer. It's a modified version of his suggestion that works with description environments. % ========================================================== % Adapted from TeX.StackExchange.com user Juan A. Navarro's % solution for itemize as seen at % http://tex.stackexchange.com/a/4493/327 % ...

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You can use one of the hooks provided by titling and set up some commands to define things conveniently. I use \foreignauthor{} and \foreigntitle{} which you use to defined the English versions. Then \maketitle will do what you want, I think. \documentclass[twocolumn]{article} \usepackage{titling} \pretitle{\begin{center}\LARGE} ...

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The dual title in twocolumn mode causes the page breaks. Perhaps you'd just be interested in setting the title manually. Below I've taken the \maketitle constructions from the titling package code: \documentclass[twocolumn]{article} \usepackage{lipsum}% Just for this example \begin{document} \twocolumn[{ \begin{center} \LARGE First title \end{center} ...

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LaTeX is trying to typeset what amounts to a poorly formed document. Normally headings are "tied" to the text which follows. This is to avoid the unsightly situation where a heading appears alone at the bottom of a page, with its following text on the next page. Your document has no content, just headings, therefore all break options are equal and no ...

0

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It can be done by switching off the boolean @twoside for the front matter. However, it is not to be recommended. \documentclass[11pt,a4paper,twoside]{book} \usepackage[pdftex]{graphicx} \usepackage[T1]{fontenc} \usepackage{fouriernc} \usepackage{mathtools} \usepackage{amsfonts,amsmath,amssymb,amsthm} \usepackage[a4paper, hmargin={3.5cm,3cm}, ...

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You should use a \mark command e.g. with the extramarks package: \documentclass[10pt,a4paper]{article} \usepackage{extramarks} \usepackage{amsmath} \usepackage{amssymb} \usepackage{fancyhdr} \rfoot{\large{$\dfrac{}{\lastxmark}$}} \usepackage{perpage} \newcounter{PageTestCounter} \MakePerPage{PageTestCounter} \newcommand{\addTest}[1]{% ...

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\nopagebreak makes it infinitely bad to break at that point. So TeX will not break at that node However it is possible that it breaks at that point visually for example blah blah \nopagebreak \hrule height 0pt \pagebreak[2] blah blah Then TeX will not break above the invisible rule, but may break after the invisible rule. So it may seem that ...

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This is not a full solution, rather a starting point, due to lack of time and lack of knowledge on TikZ ;-) I tried to use the tcolorbox breakable and overlay features and its theorem style to make the boxes. The logos from bclogo packages can be inserted, but I fail at the positioning them correctly. This leads to a right shift of the box content as well. ...

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The package endfloat will move all floats to the end; setting totalnumber to 1 will ensure only one float per page. \documentclass{article} \usepackage[nomarkers,nolists]{endfloat} \usepackage{lipsum} \setcounter{totalnumber}{1} \begin{document} \lipsum[1] \begin{figure} \centering \fbox{\rule{0pt}{1cm}\rule{4cm}{0pt}} \caption{A caption} \end{figure} ...

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This is what the package endfloat is for.

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\clearpage will flush each float on to a new page

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For quotes like that I usually include the attribution into the quote environment and make it "flush right" with an \hfill \begin{quote} Now I am become Death, the destroyer of worlds. Also, I like bananas! I like bananas so much that my quotes about them run to \emph{multiple paragraphs}!\hfill Ronald MacDonald \end{quote} A MWE would help. ...

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You should not break the two parts into two environments: \documentclass{article} \usepackage{xparse} \NewDocumentEnvironment{mquote}{m} {\begin{quote}} {{\interlinepenalty 10000 \brokenpenalty 0 \par}% <--- change to \par, if you like \nopagebreak\makebox[\linewidth][r]{#1}\end{quote}} \setlength\textheight{2cm} \begin{document} \noindent ...

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Wow, that's fantastic. Thanks so much for your help - I hadn't realised the enumitem package was so powerful. I tried it some time ago and it kept screwing up the numbering because I was trying to prepend the section number rather than using its internal counters (doh!) I've taken the liberty of posting my final version here in case anyone else finds this ...

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Don't you think that semantically, your subsections look like items in a list? I propose this solution, with enumitem, in which enumerate mimicks more or less (it's up to you to tell) your subsections. It allows easy page breaks and, adding labels, you still can cross-reference items. \documentclass{article} \usepackage{fontspec} ...

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