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Merge pull request sympy#18005 from oscarbenjamin/nonascii_rebase1.5
Nonascii rebase1.5
2 parents 9f1de8a + 3fe9b14 commit 7904e28

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‎.travis.yml

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@@ -25,6 +25,11 @@ matrix:
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sudo: true
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script: bin/test quality
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env:
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- python: 2.7
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dist: xenial
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sudo: true
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script: bin/test quality
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env:
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- stage: baseline
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python: 3.8

‎sympy/combinatorics/free_groups.py

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# -*- coding: utf-8 -*-
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from __future__ import print_function, division
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from sympy.core import S
@@ -1181,7 +1180,7 @@ def is_cyclically_reduced(self):
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r"""Returns whether the word is cyclically reduced or not.
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A word is cyclically reduced if by forming the cycle of the
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word, the word is not reduced, i.e a word w = `a_1 ... a_n`
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is called cyclically reduced if `a_1 \ne a_n^{−1}`.
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is called cyclically reduced if `a_1 \ne a_n^{-1}`.
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Examples
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========

‎sympy/combinatorics/tests/test_coset_table.py

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# -*- coding: utf-8 -*-
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from sympy.combinatorics.fp_groups import FpGroup
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from sympy.combinatorics.coset_table import (CosetTable,
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coset_enumeration_r, coset_enumeration_c)
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assert C_r.table == table1
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assert C_c.table == table1
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# E₁ from [2] Pg. 474
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# E1 from [2] Pg. 474
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F, r, s, t = free_group("r, s, t")
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E1 = FpGroup(F, [t**-1*r*t*r**-2, r**-1*s*r*s**-2, s**-1*t*s*t**-2])
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C_r = coset_enumeration_r(E1, [])
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C_r.table == table3
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C_c.table == table3
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# Group denoted by B₂,₄ from [2] Pg. 474
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# Group denoted by B2,4 from [2] Pg. 474
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F, a, b = free_group("a, b")
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B_2_4 = FpGroup(F, [a**4, b**4, (a*b)**4, (a**-1*b)**4, (a**2*b)**4, \
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(a*b**2)**4, (a**2*b**2)**4, (a**-1*b*a*b)**4, (a*b**-1*a*b)**4])

‎sympy/crypto/crypto.py

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# -*- coding: utf-8 -*-
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"""
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This file contains some classical ciphers and routines
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implementing a linear-feedback shift register (LFSR)
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######################################################################
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#################### Vigenère cipher examples ########################
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#################### Vigenere cipher examples ########################
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######################################################################
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def encipher_vigenere(msg, key, symbols=None):
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"""
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Performs the Vigenère cipher encryption on plaintext ``msg``, and
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Performs the Vigenere cipher encryption on plaintext ``msg``, and
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returns the ciphertext.
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Examples
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Notes
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=====
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The Vigenère cipher is named after Blaise de Vigenère, a sixteenth
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The Vigenere cipher is named after Blaise de Vigenere, a sixteenth
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century diplomat and cryptographer, by a historical accident.
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Vigenère actually invented a different and more complicated cipher.
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The so-called *Vigenère cipher* was actually invented
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Vigenere actually invented a different and more complicated cipher.
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The so-called *Vigenere cipher* was actually invented
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by Giovan Batista Belaso in 1553.
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This cipher was used in the 1800's, for example, during the American
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Civil War. The Confederacy used a brass cipher disk to implement the
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Vigenère cipher (now on display in the NSA Museum in Fort
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Vigenere cipher (now on display in the NSA Museum in Fort
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Meade) [1]_.
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The Vigenère cipher is a generalization of the shift cipher.
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The Vigenere cipher is a generalization of the shift cipher.
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Whereas the shift cipher shifts each letter by the same amount
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(that amount being the key of the shift cipher) the Vigenère
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(that amount being the key of the shift cipher) the Vigenere
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cipher shifts a letter by an amount determined by the key (which is
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a word or phrase known only to the sender and receiver).
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For example, if the key was a single letter, such as "C", then the
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so-called Vigenere cipher is actually a shift cipher with a
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shift of `2` (since "C" is the 2nd letter of the alphabet, if
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you start counting at `0`). If the key was a word with two
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letters, such as "CA", then the so-called Vigenère cipher will
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letters, such as "CA", then the so-called Vigenere cipher will
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shift letters in even positions by `2` and letters in odd positions
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are left alone (shifted by `0`, since "A" is the 0th letter, if
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you start counting at `0`).
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comprised of randomly selected characters -- a one-time pad -- the
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message is theoretically unbreakable.
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The cipher Vigenère actually discovered is an "auto-key" cipher
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The cipher Vigenere actually discovered is an "auto-key" cipher
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described as follows.
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ALGORITHM:
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def decipher_vigenere(msg, key, symbols=None):
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"""
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Decode using the Vigenère cipher.
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Decode using the Vigenere cipher.
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Examples
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========
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return ''.join(res)
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################ Blum–Goldwasser cryptosystem #########################
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################ Blum-Goldwasser cryptosystem #########################
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def bg_private_key(p, q):
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"""
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Check if p and q can be used as private keys for
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the Blum–Goldwasser cryptosystem.
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the Blum-Goldwasser cryptosystem.
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The three necessary checks for p and q to pass
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so that they can be used as private keys:

‎sympy/discrete/transforms.py

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# -*- coding: utf-8 -*-
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"""
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Discrete Fourier Transform, Number Theoretic Transform,
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Walsh Hadamard Transform, Mobius Transform
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#----------------------------------------------------------------------------#
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# #
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# Möbius Transform for Subset Lattice #
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# Mobius Transform for Subset Lattice #
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# #
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#----------------------------------------------------------------------------#
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def _mobius_transform(seq, sgn, subset):
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r"""Utility function for performing Möbius Transform using
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r"""Utility function for performing Mobius Transform using
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Yate's Dynamic Programming method"""
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if not iterable(seq):
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def mobius_transform(seq, subset=True):
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r"""
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Performs the Möbius Transform for subset lattice with indices of
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Performs the Mobius Transform for subset lattice with indices of
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sequence as bitmasks.
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The indices of each argument, considered as bit strings, correspond
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==========
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seq : iterable
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The sequence on which Möbius Transform is to be applied.
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The sequence on which Mobius Transform is to be applied.
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subset : bool
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Specifies if Möbius Transform is applied by enumerating subsets
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Specifies if Mobius Transform is applied by enumerating subsets
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or supersets of the given set.
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Examples

‎sympy/liealgebras/root_system.py

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# -*- coding: utf-8 -*-
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from .cartan_type import CartanType
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from sympy.core.backend import Basic
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from sympy.core.compatibility import range
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system, we first consider the Cartan subalgebra of g, which is the maximal
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abelian subalgebra, and consider the adjoint action of g on this
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subalgebra. There is a root system associated with this action. Now, a
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root system over a vector space V is a set of finite vectors Φ (called
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root system over a vector space V is a set of finite vectors Phi (called
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roots), which satisfy:
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1. The roots span V
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2. The only scalar multiples of x in Φ are x and -x
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3. For every x in Φ, the set Φ is closed under reflection
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2. The only scalar multiples of x in Phi are x and -x
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3. For every x in Phi, the set Phi is closed under reflection
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through the hyperplane perpendicular to x.
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4. If x and y are roots in Φ, then the projection of y onto
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the line through x is a half-integral multiple of x.
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4. If x and y are roots in Phi, then the projection of y onto
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the line through x is a half-integral multiple of x.
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Now, there is a subset of Φ, which we will call Δ, such that:
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1. Δ is a basis of V
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2. Each root x in Φ can be written x = Σ k_y y for y in Δ
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Now, there is a subset of Phi, which we will call Delta, such that:
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1. Delta is a basis of V
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2. Each root x in Phi can be written x = sum k_y y for y in Delta
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The elements of Δ are called the simple roots.
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The elements of Delta are called the simple roots.
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Therefore, we see that the simple roots span the root space of a given
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simple Lie algebra.
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‎sympy/liealgebras/weyl_group.py

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@@ -9,7 +9,7 @@ class WeylGroup(Basic):
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"""
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For each semisimple Lie group, we have a Weyl group. It is a subgroup of
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the isometry group of the root system. Specifically, it’s the subgroup
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the isometry group of the root system. Specifically, it's the subgroup
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that is generated by reflections through the hyperplanes orthogonal to
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the roots. Therefore, Weyl groups are reflection groups, and so a Weyl
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group is a finite Coxeter group.

‎sympy/ntheory/residue_ntheory.py

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# -*- coding: utf-8 -*-
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from __future__ import print_function, division
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from sympy.core.compatibility import as_int, range
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class mobius(Function):
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"""
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Möbius function maps natural number to {-1, 0, 1}
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Mobius function maps natural number to {-1, 0, 1}
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It is defined as follows:
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1) `1` if `n = 1`.
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It is an important multiplicative function in number theory
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and combinatorics. It has applications in mathematical series,
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algebraic number theory and also physics (Fermion operator has very
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concrete realization with Möbius Function model).
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concrete realization with Mobius Function model).
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Parameters
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==========

‎sympy/physics/mechanics/rigidbody.py

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# -*- encoding: utf-8 -*-
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from __future__ import print_function, division
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from sympy.core.backend import sympify
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The angular momentum H of a rigid body B about some point O in a frame
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N is given by:
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H = I·w + r×Mv
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H = I . w + r x Mv
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where I is the central inertia dyadic of B, w is the angular velocity
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of body B in the frame, N, r is the position vector from point O to the

‎sympy/physics/optics/polarization.py

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