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Some mathematicians, especially in set theory, do not consider the sets and to be part of the relation, and therefore define a binary relation as being a subset of x, that is, just the graph.

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## Some Related Sentences

Some and mathematicians

__Some__

**of**

**the**

__mathematicians__who were involved

**in**

**the**selection

**and**presentation

**of**

**the**seven problems were Atiyah

**,**Bombieri

**,**Connes

**,**Deligne

**,**Fefferman

**,**Milnor

**,**Mumford

**,**Wiles

**,**

**and**Witten

**.**

__Some__notable

__mathematicians__include Archimedes

**of**Syracuse

**,**Leonhard Euler

**,**Carl Gauss

**,**Johann Bernoulli

**,**Jacob Bernoulli

**,**Aryabhata

**,**Brahmagupta

**,**Bhaskara II

**,**Nilakantha Somayaji

**,**Omar Khayyám

**,**Muhammad ibn Mūsā al-Khwārizmī

**,**Bernhard Riemann

**,**Gottfried Leibniz

**,**Andrey Kolmogorov

**,**Euclid

**of**Alexandria

**,**Jules Henri Poincaré

**,**Srinivasa Ramanujan

**,**Alexander Grothendieck

**,**David Hilbert

**,**Alan Turing

**,**von Neumann

**,**Kurt Gödel

**,**Joseph-Louis Lagrange

**,**Georg Cantor

**,**William Rowan Hamilton

**,**Carl Jacobi

**,**Évariste Galois

**,**Nikolay Lobachevsky

**,**Rene Descartes

**,**Joseph Fourier

**,**Pierre-Simon Laplace

**,**Alonzo Church

**,**Nikolay Bogolyubov

**and**Pierre de Fermat

**.**

__Some__

**of**these

__mathematicians__were influential

**in**arranging

**the**Elements

**,**

**that**Euclid later published

**.**

__Some__

__mathematicians__

**,**such

**as**Carl Boyer

**,**hold

**that**Zeno's paradoxes are simply mathematical problems

**,**for which modern calculus provides

**a**mathematical solution

**.**

__Some__

__mathematicians__

**,**

**therefore**

**,**

**do**

**not**recognize blackboard bold

**as**

**a**separate style from bold: Jean-Pierre Serre

**,**for example

**,**has publicly inveighed against

**the**use

**of**" blackboard bold " anywhere other than on

**a**blackboard

**,**

**and**uses double-struck letters when writing bold on

**the**blackboard

**,**whereas his published works consistently use ordinary bold for

**the**same symbols

**.**

__Some__

__mathematicians__independently made

**the**related hypothesis ( sometimes incorrectly called

**the**Chinese Hypothesis )

**that**p

**is**

**a**prime if

**and**only if

**.**

__Some__

__mathematicians__

**as**late

**as**

**the**sixteenth century continued

**to**view halving

**as**

**a**separate operation

**,**

**and**it often continues

**to**

**be**treated separately

**in**modern computer programming

**.**

__Some__theories tend

**to**focus on mathematical practice

**,**

**and**aim

**to**describe

**and**analyze

**the**actual working

**of**

__mathematicians__

**as**

**a**social group

**.**

Formally

**,**Minkowski space**is****a**four-dimensional real vector space equipped with**a**nondegenerate**,**symmetric bilinear form with signature < tt >(−,+,+,+)</ tt > (__Some__may also prefer**the**alternative signature < tt >(+,−,−,−)</ tt >;**in**general**,**__mathematicians__**and**general relativists prefer**the**former while particle physicists tend**to**use**the**latter**.**__Some__working

__mathematicians__have espoused mathematical realism

**as**they see themselves experiencing naturally-occurring objects

**.**

__Some__

__mathematicians__

**and**scientists are critical

**of**using closed source mathematical software such

**as**Mathematica for mathematical proofs

**,**because these

**do**

**not**provide transparency

**,**

**and**thus are

**not**verifiable

**.**

On 21 November 1930

**the**declaration**of****the**“ initiative group ”**of****the**Moscow Mathematical Society which consisted**of**former Luzin's students Lazar Lyusternik**and**Lev Shnirelman along with Alexander Gelfond**and**Lev Pontryagin claimed**that**“ there appeared active counter-revolutionaries among__mathematicians__.”__Some__**of**these__mathematicians__were pointed out**,**including**the**advisor**of**Luzin**,**Dmitri Egorov**.**__Some__

__mathematicians__

**and**historians have wrongfully claimed

**that**Lobachevsky

**in**his studies

**in**non-Euclidean geometry was influenced by Gauss

**,**which

**is**untrue-Gauss himself appreciated Lobachevsky's published works very highly

**,**but they never had personal correspondence between them prior

**to**

**the**publication

**.**

__Some__jokes are based on stereotypes

**of**

__mathematicians__tending

**to**think

**in**complicated

**,**abstract terms

**,**causing them

**to**lose touch with

**the**" real world ".

Some and especially

__Some__alphabets disregard tone entirely

**,**

__especially__when it does

**not**carry

**a**heavy functional load

**,**

**as**

**in**Somali

**and**many other languages

**of**Africa

**and**

**the**Americas

**.**

__Some__are willing

**to**travel

**to**

**the**other designated areas

**,**

__especially__during slow periods

**of**

**the**day

**.**

__Some__birds

**,**

__especially__corvids

**and**parrots

**,**are among

**the**most intelligent animal species ;

**a**number

**of**bird species have been observed manufacturing

**and**using tools

**,**

**and**many social species exhibit cultural transmission

**of**knowledge across generations

**.**

__Some__carbohydrates (

__especially__after condensation

**to**oligo-and polysaccharides ) contain less carbon relative

**to**H

**and**O

**,**which still are present

**in**2: 1 ( H: O ) ratio

**.**

__Some__biological agents (

__especially__smallpox

**,**plague

**,**

**and**tularemia ) have

**the**capability

**of**person-to-person transmission via aerosolized respiratory droplets

**.**

__Some__kinds

**of**raw beans

**,**

__especially__red

**and**kidney beans

**,**contain

**a**harmful toxin ( lectin phytohaemagglutinin )

**that**must

**be**removed by cooking

**.**

__Some__constitutions

**,**

__especially__written constitutions

**,**also act

**as**limiters

**of**state power

**,**by establishing lines which

**a**state's rulers cannot cross

**,**such

**as**fundamental rights

**.**

__Some__are styled " national " banks

**,**such

**as**

**the**National Bank

**of**Ukraine ; but

**the**term " national bank "

**is**more often used by privately owned commercial banks

**,**

__especially__

**in**

**the**United States

**.**

__Some__calves are ear tagged soon after birth

**,**

__especially__those

**that**are stud cattle

**in**order

**to**correctly identify their dams ( mothers ), or

**in**areas ( such

**as**

**the**EU ) where tagging

**is**

**a**legal requirement for cattle

**.**

__Some__canid species have also been trapped

**and**hunted for their fur

**and**

**,**

__especially__

**the**gray wolf

**and**

**the**red fox

**,**for sport

**.**

__Some__

**of**

**the**most elaborate columns

**in**

**the**ancient world were those

**of**

**the**Persians

**,**

__especially__

**the**massive stone columns erected

**in**Persepolis

**.**

__Some__displays

**,**

__especially__newer LCD monitors

**,**replace

**the**traditional anti-glare matte finish with

**a**glossy one

**.**

__Some__players perform with

**the**sides

**of**one

**,**two

**,**or three fingers

**,**

__especially__for walking basslines

**and**slow tempo ballads

**,**because this

**is**purported

**to**create

**a**stronger

**and**more solid tone

**.**

__Some__

**of**

**the**main topics

**of**

**the**field are moral responsibility

**,**moral development

**,**moral character (

__especially__

**as**related

**to**virtue ethics ), altruism

**,**psychological egoism

**,**moral luck

**,**

**and**moral disagreement

**.**

__Some__roads

**,**

__especially__

**in**

**the**Sinai

**and**southeastern

**part**

**of**

**the**country

**,**are off-limits

**to**foreigners

**.**

__Some__aficionados who prefer class-A amplifiers also prefer

**the**use

**of**thermionic valve ( or " tube ") designs instead

**of**transistors

**,**

__especially__

**in**Single-ended triode output configurations for several claimed reasons:

__Some__arthropods however

**,**

__especially__large insects with tracheal respiration

**,**expand their new exoskeleton by swallowing or otherwise taking

**in**air

**.**

__Some__Directors

**,**

__especially__more established ones

**,**take on many

**of**

**the**roles

**of**

**a**Producer

**,**

**and**

**the**distinction between

**the**two roles

**is**sometimes blurred

**.**

__Some__authors use

**,**or Cpq

**,**instead

**of**

**,**

**and**

**,**or Epq

**,**instead

**of**

**,**

__especially__

**in**contexts where

**is**used for other purposes

**.**

__Some__people use oral sex

**as**

**a**means

**of**preserving virginity

**,**

__especially__

**in**

**the**case

**of**adolescent girls who fellate their boyfriends

**to**create

**and**maintain intimacy while preserving their virginity

**,**avoiding pregnancy

**,**or both

**.**

__Some__people interpret

**the**Bible

**,**

__especially__

**the**New Testament

**,**

**as**teaching belief

**in**

**,**

**and**practice

**of**

**,**faith healing

**.**

__Some__electrics

**,**

__especially__those with odd-shaped bodies

**,**have one or both strap buttons on

**the**back

**of**

**the**body

**.**

__Some__predictions

**of**general relativity differ significantly from those

**of**classical physics

**,**

__especially__concerning

**the**passage

**of**time

**,**

**the**geometry

**of**space

**,**

**the**motion

**of**bodies

**in**free fall

**,**

**and**

**the**propagation

**of**light

**.**

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