ConseggioLigure/seed-instruct-eng-lij huggingface dataset:
Translate this sentence to Ligurian:
Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons.
The Ligurian (Genoese) translation is:
I numeri sessageximali ean un scistema radix mesccio ch’o mantegniva l’alternansa de base 10 e base 6 inte unna sequensa de cunei cuneiformi verticali e chevrons.
Translate this sentence to Ligurian:
Conventional tallies are quite difficult to multiply and divide.
The Ligurian (Genoese) translation is:
I contezzi convençionæ son assæ diffiçili da moltiplicâ e da dividde.
Translate to Ligurian (Genoese):
Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.
The Ligurian (Genoese) translation is:
I ebrëi an comensou à deuviâ un scistema scimile (numeri ebrëi) e i exempi ciù antighi che se conosce en de monæe do 100 a.C. ca.
Translate to Ligurian (Genoese):
The Maya of Central America used a mixed base 18 and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero.
The Ligurian (Genoese) translation is:
I Maya de l’America do çentro deuviavan un sistema mesccio de base 18 e base 20, che probabilmente o l’ea stæto ereditou da-i olmechi e che comprendeivan de caratteristiche avansæ comme a nottaçion poxiçionale e un zero.
Translate from English to Ligurian (Genoese dialect):
Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.
The Ligurian (Genoese) translation is:
A conoscensa de codifiche di groppi e di coî a l’ea stæta soppressa da-i conquistadores spagnòlli do secolo XVI e a no l’é visciua ciù avanti, ben che di senci dispoxitivi de registraçion che s’assomeggian a-i quipu en deuviæ inta region andiña ancon a-a giornâ d’ancheu.
Translate to Genoese:
Zero was first used in India in the 7th century CE by Brahmagupta.
The Ligurian (Genoese) translation is:
O zero o l’é stæto deuviou pe-a primma vòtta in India do secolo VII d.C. da Brahmagupta.
Translate this sentence to Ligurian:
Arabic mathematicians extended the system to include decimal fractions, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century.
The Ligurian (Genoese) translation is:
I studioxi de matematica arabi an esteiso o scitemo fin à comprende e fraçioin deçimæ, e Muḥammad ibn Mūsā al-Ḵwārizmī o l’à scrito unn’importante travaggio à sto propòxito do secolo IX.
What’s the Genoese translation of this sentence?
The binary system (base 2), was propagated in the 17th century by Gottfried Leibniz.
The Ligurian (Genoese) translation is:
O scistema binäio (base 2) o l’é stæto diffuso do secolo XVII da Gottfred Leibniz.
What’s the Ligurian translation of this sentence?
The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation.
The Ligurian (Genoese) translation is:
E variabile pe-e quæ l’equaçion a l’à da ëse soluçionâ piggian o nomme de incògnite, e i valoî de incògnite che satisfan l’egualiansa en ciammæ soluçion de l’equaçion.
What’s the Genoese translation of this sentence?
A conditional equation is only true for particular values of the variables.
The Ligurian (Genoese) translation is:
Unn’equaçion condiçionâ a l’é vea solo che pe di valoî determinæ de variabile.
Translate from English to Ligurian:
Very often the right-hand side of an equation is assumed to be zero.
The Ligurian (Genoese) translation is:
Ben ben de spesso a parte de drita de unn’equaçion a l’é tegnua pe-o zero.
Translate this sentence to Genoese:
An equation is analogous to a scale into which weights are placed.
The Ligurian (Genoese) translation is:
Unn’equaçion a l’é analoga à unna bansa con de peise.
Translate from English to Genoese:
This is the starting idea of algebraic geometry, an important area of mathematics.
The Ligurian (Genoese) translation is:
Sta chì a l’é l’idea a-a base da geometrica algebrica, unna ramma importante da matematica.
What’s the Ligurian translation of this sentence?
To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis.
The Ligurian (Genoese) translation is:
Pe resòlve de equaçioin de tutte doe e famigge se deuvia de tecniche algoritmiche ò geometriche che ne vëgnan da l’algebra lineare ò da l’analixi matematica.
Translate from English to Ligurian (Genoese dialect):
In the illustration, x, y and z are all different quantities (in this case real numbers) represented as circular weights, and each of x, y, and z has a different weight.
The Ligurian (Genoese) translation is:
Inta figua, x, y e z en tutte quantitæ despæge (inte sto caxo numeri reali) representæ comme peise rionde, e ciascheduña de x, y e z a l’à un peiso despægio.
Translate this sentence to Ligurian:
Hence, the equation with R unspecified is the general equation for the circle.
The Ligurian (Genoese) translation is:
Donca, l’equaçion con R no speçificâ a l’é l’equaçion generale pe-o çercio.
Translate this sentence to Ligurian:
The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called solving the equation.
The Ligurian (Genoese) translation is:
O processo pe mezo do quæ che çerca d’attrovâ a soluçion ò, into caxo di parametri, d’esprimme e incògnite pe mezo di parametri, a l’é a resoluçion de l’equaçion.
Translate this sentence to Ligurian:
Multiplying or dividing both sides of an equation by a non-zero quantity.
The Ligurian (Genoese) translation is:
Co-o moltiplicâ ò dividde tutti doî i loei de unn’equaçion pe unna quantitæ ch’a no segge pægia a-o zero.
Translate from English to Genoese:
An algebraic equation is univariate if it involves only one variable.
The Ligurian (Genoese) translation is:
Unn’equaçion algebrica a l’é univariabile s’a comprende solo che unna variabile.
Translate to Ligurian (Genoese):
This formalism allows one to determine the positions and the properties of the focuses of a conic.
The Ligurian (Genoese) translation is:
Sto formalismo o permette a determinaçion de poxiçioin e de propietæ di feughi de unna cònica.
Translate from English to Ligurian (Genoese dialect):
This point of view, outlined by Descartes, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.
The Ligurian (Genoese) translation is:
Sto ponto de vista, deliniou da-o Cartexio, o l’inricchisce e o modifica o tipo de geometria stæta conçepia da-i antighi matematichi greghi.
Translate to Ligurian:
An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.
The Ligurian (Genoese) translation is:
Unn’equaçion diofantiña esponençiale a l’é unn’equaçion pe-a quæ i esponenti di termini de l’equaçion peuan ëse de incògnite.
What’s the Ligurian translation of this sentence?
Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.
The Ligurian (Genoese) translation is:
A geometria algebrica moderna a se basa in sce de tecniche ciù astræte de l’algebra astræta, mascime l’algebra commutativa, co-o lenguaggio e i problemi da geometria.
Translate from English to Ligurian:
A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation.
The Ligurian (Genoese) translation is:
Un ponto de un cian o l’appartëgne à unna curva algebrica se e seu coordinæ satisfan unn’equaçion polinomiale determinâ.
Translate to Ligurian (Genoese):
In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation.
The Ligurian (Genoese) translation is:
Inta matematica pua, e equaçioin differensiæ vëgnan studiæ da tante prospettive, interessæ pe-o ciù a-e seu soluçioin (l’insemme de fonçioin che satisfan l’equaçion).
Translate to Ligurian (Genoese):
Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained.
The Ligurian (Genoese) translation is:
E equaçioin differençiale lineare, che an de soluçioin che peuan vegnî sommæ e moltiplicæ pe-i coeffiçenti, en ben definie e compreise, e s’ottëgne de soluçioin esatte de forma serrâ.
What’s the Genoese translation of this sentence?
PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics.
The Ligurian (Genoese) translation is:
E PDE peuan vegnî deuviæ pe descrive unna larga varietæ de fenòmeni comme o son, o cado, l’elettrostatica, l’elettrodinamica, o flusso de fluidi, l’elastiçitæ e a mecanica quantistica.
Translate to Ligurian:
A solution is an assignment of values to the unknown variables that makes the equality in the equation true.
The Ligurian (Genoese) translation is:
Unna soluçion a l’é l’assegnaçion de valoî a-e variabile desconosciue ch’a l’assegua a veitæ de l’egualiansa inte l’equaçion.
Translate to Ligurian (Genoese):
The set of all solutions of an equation is its solution set.
The Ligurian (Genoese) translation is:
L’insemme de tutte e soluçioin de unn’equaçion o l’é dito insemme de soluçioin.
Translate to Ligurian:
The variety in types of equations is large, and so are the corresponding methods.
The Ligurian (Genoese) translation is:
A varietæ inta qualitæ d’equaçioin a l’é grande, coscì comme i metodi correspondenti.
Translate this sentence to Ligurian:
This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort.
The Ligurian (Genoese) translation is:
Sto fæto o peu ëse dovuo a-a mancansa de conoscense matematiche; çerti problemi an attrovou unna soluçion dòppo di sfòrsci duæ pe de çentanæa d’anni.
What’s the Ligurian translation of this sentence?
Polynomials appear in many areas of mathematics and science.
The Ligurian (Genoese) translation is:
I polinòmmi s’attreuvan inte tante aree da matematica e de sciense.
What’s the Genoese translation of this sentence?
Many authors use these two words interchangeably.
The Ligurian (Genoese) translation is:
Tanti autoî an deuviou ste doe paròlle sensa distinçion.
Translate this sentence to Genoese:
Formally, the name of the polynomial is P, not P(x), but the use of the functional notation P(x) dates from a time when the distinction between a polynomial and the associated function was unclear.
The Ligurian (Genoese) translation is:
In sciô cian formale, o nomme do polinòmio o l’é P, e no P(x), e l’utilizzaçion da nottaçion fonçionale P(x) a remonta à unn’epoca donde a distinçion tra un polinòmio e a fonçion assoçiâ a no l’ea ancon ciæa.
What’s the Genoese translation of this sentence?
However, one may use it over any domain where addition and multiplication are defined (that is, any ring).
The Ligurian (Genoese) translation is:
À tutti i mòddi, o se peu deuviâ inte tutti i dominni donde l’addiçion e a moltiplicaçion en definie (saieiva à dî, un anello chesesegge).
What’s the Ligurian translation of this sentence?
Polynomials of small degree have been given specific names.
The Ligurian (Genoese) translation is:
A-i polinòmmi d’entitæ picciña l’é stæto dæto di nommi speçifichi.
What’s the Ligurian translation of this sentence?
The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial.
The Ligurian (Genoese) translation is:
O polinòmio 0, ch’o se peu considerâ sensa termini, o piggia o nomme de polinòmio zero.
Translate from English to Ligurian:
Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.
The Ligurian (Genoese) translation is:
Da-o momento che o graddo de un polinòmio ch’o no l’à valô de zero o l’é o graddo ciù grande de un atro termine chesesegge, sto polinòmio o l’é de graddo doî.
Translate from English to Ligurian (Genoese dialect):
When it is used to define a function, the domain is not so restricted.
The Ligurian (Genoese) translation is:
Quande se deuvia pe definî unna fonçion, o dominio o no l’é coscì restreito.
Translate to Genoese:
A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial.
The Ligurian (Genoese) translation is:
Un polinòmio inte un indeterminou o piggia o nomme de polinòmio univariou; un polinòmio in de ciù de un indeterminou o piggia o nomme de polinòmio moltivariou.
What’s the Ligurian translation of this sentence?
In the case of the field of complex numbers, the irreducible factors are linear.
The Ligurian (Genoese) translation is:
Into caxo do campo di numeri complesci, i fattoî irreduçibili en lineari.
Translate from English to Ligurian (Genoese dialect):
If the degree is higher than one, the graph does not have any asymptote.
The Ligurian (Genoese) translation is:
Se o graddo o l’é ciù grande de un, o grafico o no l’à d’ascindoti.
Translate to Ligurian (Genoese):
In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable.
The Ligurian (Genoese) translation is:
Inte l’algebra elementare se mostra di metodi comme a formola quaddratica pe resòlve tutte e equaçion polinomiale de primmo e segondo graddo inte unna variabile.
Translate to Genoese:
However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.
The Ligurian (Genoese) translation is:
À tutti i mòddi, i algoritmi de root-finding peuan vegnî deuviæ pe trovâ de approscimaçioin numeriche de reixe de unn’esprescion polinomiale de un graddo chesesegge.
Can you translate this text to Ligurian?
Since the 16th century, similar formulas (using cube roots in addition to square roots), but much more complicated are known for equations of degree three and four (see cubic equation and quartic equation).
The Ligurian (Genoese) translation is:
Da-o secolo XVI se conosce de formole pæge (che deuvian de reixe cubbiche in azzonta à quelle quaddræ), ma ben ben ciù complicæ pe de equaçioin de graddo trei e quattro (s’ammie equaçion cubbica e equaçioin quartica).
Translate to Ligurian (Genoese):
In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it.
The Ligurian (Genoese) translation is:
Do 1830 Évariste Galois o l’à demostrou che a ciù parte de equaçioin de graddo ciù erto che o quattro no se peuan resòlve co-i radicali e o l’à demostrou che pe ògni equaçion se peu deçidde s’a se peu resòlve co-i radicali e, s’a l’é coscì, resòlvila.
Translate from English to Genoese:
Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).
The Ligurian (Genoese) translation is:
Segge comm’a segge, l’é stæto pubricou de fòrmole pe de equaçioin che se peuan resòlve de graddo 5 ò 6 (s’ammie fonçion quintica e equaçion sessica).
Translate to Ligurian (Genoese):
The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1,000 (see Root-finding algorithm).
The Ligurian (Genoese) translation is:
I algoritmi ciù effiçenti permettan de risòlve de lengê (in sce un computer) de equaçioin polinomiali de graddo ciù erto che 1000 (vedde algoritmo de reçerca de reixe).
Translate to Ligurian:
For a set of polynomial equations in several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number is finite, for computing the solutions.
The Ligurian (Genoese) translation is:
Pe un insemme d’equaçioin polinomiale inte varie incògnite l’existe di algoritmi pe stabilî s’aggian un numeo de soluçioin complesse e, se sto numeo o l’é finio, pe carcolâ e soluçioin.
What’s the Genoese translation of this sentence?
A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation.
The Ligurian (Genoese) translation is:
Un’equaçion polinomiale pe-a quæ s’é interessæ solo che a-e soluçioin intreghe a piggia o nomme d’equaçion diafantiña.
Translate from English to Ligurian (Genoese dialect):
The coefficients may be taken as real numbers, for real-valued functions.
The Ligurian (Genoese) translation is:
I coeffiçenti peuan ëse piggiæ comme numeri reali, pe fonçioin con un valô reale.
What’s the Ligurian translation of this sentence?
This equivalence explains why linear combinations are called polynomials.
The Ligurian (Genoese) translation is:
St’equivalensa a spiega perché e combinaçioin lineare se ciamma polinòmmi.
Can you translate this text to Ligurian?
In the case of coefficients in a ring, "non-constant" must be replaced by "non-constant or non-unit" (both definitions agree in the case of coefficients in a field).
The Ligurian (Genoese) translation is:
Into caxo di coeffiçenti inte un anello, «no constante» o dev’ëse sostituito da «no constante ò no-unitæ» (tutte doe e definiçioin concòrdan into caxo de coeffiçenti inte un campo).
Translate to Genoese:
When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials).
The Ligurian (Genoese) translation is:
Quande i coeffiçenti appartëgnan à di intreghi, à di numeri raçionali ò à un campo finio, l’existe di algoritmi pe verificâ l’irreduçibilitæ e pe carcolâ a fattorizzaçion inte di polinòmmi irreduçibili (s’ammie Fattorizzaçion de polinòmmi).
Translate to Genoese:
The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues.
The Ligurian (Genoese) translation is:
O polinòmio caratteristico de unna matrixe ò operatô linieare o contëgne de informaçioin in scî autovaloî de l’operatô.
Translate to Genoese:
However, the elegant and practical notation we use today only developed beginning in the 15th century.
The Ligurian (Genoese) translation is:
À tutti i mòddi, a nottaçion elegante e prattica che deuviemmo ancheu a s’é sviluppâ solo che a-o comenso do secolo XV.
Translate this sentence to Genoese:
This "completes the square", converting the left side into a perfect square.
The Ligurian (Genoese) translation is:
Sto fæto o «completa o quaddrou», pe mezo da converscion do lou da manciña inte un quaddrou perfetto.
Translate to Genoese:
Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.
The Ligurian (Genoese) translation is:
O teorema do Cartexo o l’enonçia che pe ògni quattro çerci combaxanti (saieiva à dî tangenti l’un con l’atro), i seu raggi satisfan unna particolâ equaçion quaddratica.
Translate this sentence to Ligurian:
Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots.
The Ligurian (Genoese) translation is:
I studioxi de matematica babiloneixi do 400 a.C. ca. e quelli cineixi do 200 a.C. ca. deuviavan di metodi geometrichi de disseçion pe resòlve de equaçioin quaddratiche con de reixe poxitive.
Translate this sentence to Ligurian:
Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC.
The Ligurian (Genoese) translation is:
Euclide, o matematico grego, o l’à enonçiou un metodo geometrico ciù astræto verso o 300 a.C.
Translate this sentence to Genoese:
Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process.
The Ligurian (Genoese) translation is:
Al-Khwarizmi o va ciù in là into fornî unna soluçion completa de l’equaçion quaddratica generale, açettando uña ò doe respòste numeriche pe ciascheduña equaçion quaddratica e fornindo a-o mæximo tempo de preuve geometriche.
Translate to Genoese:
Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.
The Ligurian (Genoese) translation is:
Ciù into detaggio, Abū Kāmil Shujā ibn Aslam (Egitto, secolo X) o l’é stæto o primmo ch’o l’à açettou di numeri irraçionali (de spesso sotta a forma de reixe quaddrâ, cubbica ò reixe quata) comme soluçion d’equaçioin quaddratiche ò comme coeffiçenti inte unn’equaçion.
Translate from English to Genoese:
He used a prime meridian through the Canary Islands, so that all longitude values would be positive.
The Ligurian (Genoese) translation is:
O l’à deuviou un meridian primmo pe-e isoe Canäie, de mòddo che tutti i valoî da longitudine fïsan poxitivi.
Translate from English to Ligurian:
Hindu and Muslim astronomers continued to develop these ideas, adding many new locations and often improving on Ptolemy's data.
The Ligurian (Genoese) translation is:
I astrònomi indien en mosulmoen an continuo à sviluppâ ste idee, azzonzendo de neuve localitæ e, de spesso, megioando i dæti de Tolomeo.
Translate from English to Genoese:
In the later Middle Ages, interest in geography revived in the west, as travel increased, and Arab scholarship began to be known through contact with Spain and North Africa.
The Ligurian (Genoese) translation is:
Inta tarda Etæ de Mezo l’interesse pe-a geografia in occidente o l’é torna montou, con l’aumentâ di viægi, e a scheua araba a l’à comensou à ëse conosciua pe mezo do contatto co-a Spagna e l’Africa do Nòrd.
Translate to Genoese:
Christopher Columbus made two attempts to use lunar eclipses to discover his longitude, the first in Saona Island, on 14 September 1494 (second voyage), and the second in Jamaica on 29 February 1504 (fourth voyage).
The Ligurian (Genoese) translation is:
O Cristòffa Combo o l’à provou doe vòtte à deuviâ a luña in negrisso pe descrovî a seu longitidune; a primma vòtta in sce l’isoa de Saona, a-i 14 de settembre do 1494 (durante o segondo viægio), e a segonda in Giamaica, a-i 29 de frevâ do 1504 (quarto viægio).
Translate from English to Genoese:
Initially an observation device, developments over the next half century transformed it into an accurate measurement tool.
The Ligurian (Genoese) translation is:
I sviluppi che gh’é stæto inti çinquant’anni apreuvo l’an trasformou da un appægio d’osservaçion à un instrumento esatto de mesuaçion.
Translate from English to Ligurian (Genoese dialect):
On land, the period from the development of telescopes and pendulum clocks until the mid 18th-Century saw a steady increase in the number of places whose longitude had been determined with reasonable accuracy, often with errors of less than a degree, and nearly always within 2-3°.
The Ligurian (Genoese) translation is:
In sciâ tæra, o periodo de sviluppo di telescòppi e di releui à pendola fin a-a meitæ do secolo XVIII o l’à visto un cresce continuo de localitæ con unna longitudine stæta edeterminâ con unna boña accuratessa, de spesso con di erroî ciù piccin de un graddo, e delongo de dötrei graddi pöcassæ.
Translate to Genoese:
Making accurate observations in an ocean swell is much harder than on land, and pendulum clocks do not work well in these conditions.
The Ligurian (Genoese) translation is:
Fâ de osservaçioin accuræ into mâ remesciou l’é ben ben ciù diffiçile che in sciâ tæra, e i releui à pendola no fonçioñan ben inte quelle condiçioin.
Translate from English to Genoese:
It offered two levels of rewards, for solutions within 1° and 0.5°.
The Ligurian (Genoese) translation is:
O l’òffre doî livelli de compenso, pe soluçioin tra 1° e 0,5°.