ConseggioLigure/seed-instruct-eng-lij huggingface dataset:
Translate this sentence to Genoese:
Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions.
The Ligurian (Genoese) translation is:
L’utilitæ do conçetto de fonçioin con ciù valoî a l’é ciù ciæa se se conscidera de fonçioin complesse, che son de fonçioin tipicamente analitiche.
Translate to Ligurian:
Such a function is called the principal value of the function.
The Ligurian (Genoese) translation is:
Sta fonçion a l’é ciammâ valô prinçipâ da fonçion.
Translate this sentence to Ligurian:
Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions.
The Ligurian (Genoese) translation is:
A programmaçion fonçionale a l’é o paradigma de programmaçion ch’o consciste inta costruçion de programmi co-o deuviâ solo de subroutine che se compòrtan comme fonçioin matematiche.
What’s the Genoese translation of this sentence?
Except for computer-language terminology, "function" has the usual mathematical meaning in computer science.
The Ligurian (Genoese) translation is:
Feua che pe-a terminologia do lenguaggio informatico, anche pe-e sciense che deuvian o computer «fonçion» a l’à o mæximo scignificato che inta matematica.
What’s the Genoese translation of this sentence?
Terms are manipulated through some rules, (the -equivalence, the -reduction, and the -conversion), which are the axioms of the theory and may be interpreted as rules of computation.
The Ligurian (Genoese) translation is:
I termini vëgnan manipolæ atraverso çerte regole (l’equivalensa, a reduçion e a converscion), che son i assiömi da teoria e peuan vegnî interpretæ comme regole de carcolo.
Translate to Ligurian:
Nicolas Chuquet used a form of exponential notation in the 15th century, which was later used by Henricus Grammateus and Michael Stifel in the 16th century.
The Ligurian (Genoese) translation is:
Nicolas Chuquet o l’à deuviou unna forma de nottaçion esponençiale do secolo chinze, che dapeu a l’é stæta deuviâ da Henricus Grammateus e Michael Stifel do secolo sezze.
Translate this sentence to Ligurian:
Thus they would write polynomials, for example, as .
The Ligurian (Genoese) translation is:
Coscì scriveivan i polinòmi, pe fâ un exempio, comme .
Translate this sentence to Ligurian:
The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents.
The Ligurian (Genoese) translation is:
O resultato o l’é sempre un numeo poxitivo reale e e identitæ e propietæ illustræ chì de d’ato pe-i esponenti intreghi arrestan vee con ste definiçioin pe-i esponenti reali.
What’s the Ligurian translation of this sentence?
This function equals the usual th root for positive real radicands.
The Ligurian (Genoese) translation is:
Sta fonçion a l’é pægia a-a reixe de tersa pe-i radicandi reali poxitivi.
Translate this sentence to Genoese:
This is the starting point of the mathematical theory of semigroups.
The Ligurian (Genoese) translation is:
Sto chì o l’é o gruppo de partensa da teoria matematica di semigruppi.
Translate from English to Ligurian:
We can again replace the set N with a cardinal number n to get Vn, although without choosing a specific standard set with cardinality n, this is defined only up to isomorphism.
The Ligurian (Genoese) translation is:
Poemmo sostituî torna l’insemme N con un numeo cardinale n pe ottegnî Vn; sciben che se no çerne un insemme speçifico standard da-a cardinalitæ n, questo o l’é definio solo fin à l’isomorfismo.
Translate this sentence to Ligurian:
Nicolas Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.
The Ligurian (Genoese) translation is:
Nicolas Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.
Translate this sentence to Ligurian:
Iterating tetration leads to another operation, and so on, a concept named hyperoperation.
The Ligurian (Genoese) translation is:
D’iterâ a tetraçion s’arriva à unn’atra operaçion e coscì avanti; sto conçetto o l’é ciammou iperoperaçion.
Translate to Ligurian:
In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable.
The Ligurian (Genoese) translation is:
Inte di ambienti applicæ, e fonçioin esponençiale modellan unna relaçion donde un cangio costante inta variabile independente o dà o mæximo cangio proporçionale (saieiva à dî l’aumento ò a diminuçion perçentuale) inta variabile independente.
Translate this sentence to Ligurian:
This function property leads to exponential growth or exponential decay.
The Ligurian (Genoese) translation is:
Sta propietæ da fonçion a pòrta a-o crescio esponençiale ò a-o descrescio esponençiale.
Translate to Ligurian (Genoese):
Similarly, the composition of onto (surjective) functions is always onto.
The Ligurian (Genoese) translation is:
Da mæxima, a compoxiçion de fonçioin onto (surgettive) a l’é sempre onto.
Translate this sentence to Ligurian:
Then one can form chains of transformations composed together, such as .
The Ligurian (Genoese) translation is:
Dapeu se peu formâ de cadeñe de trasformaçioin compòste insemme, comme .
Translate to Genoese:
The category of sets with functions as morphisms is the prototypical category.
The Ligurian (Genoese) translation is:
A categoria di insemmi con de fonçioin comme morfismi a l’é a categoria prototipica.
Translate to Ligurian:
For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example).
The Ligurian (Genoese) translation is:
Pe fâ un exempio, o decibel (dB) a l’é unn’unitæ deuviâ pe esprimme o rappòrto pe logaritmi, mascime pe-a potensa e l’ampiessa do segnâ (do quæ a prescion do son o l’é un exempio commun).
Can you translate this text to Ligurian?
They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.
The Ligurian (Genoese) translation is:
Aggiuttan à descrive i rappòrti de frequensa di intervalli de muxica, s’attreuvan inte fòrmole che contan i nummeri primmi ò appròsciman i fattoriali, informan çerti modelli in psicofixica e peuan ëse d’aggiutto inta contabilitæ forense.
Translate from English to Ligurian:
The next integer is 4, which is the number of digits of 1430.
The Ligurian (Genoese) translation is:
O pròscimo integro o l’é 4, ch’o l’é o numeo de giffre de 1430.
Translate to Ligurian (Genoese):
Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600.
The Ligurian (Genoese) translation is:
Primma de l’invençion de Napier gh’ea stæto de atre tecniche de unna portâ scimile, comme a prostaferexi ò l’utilizzaçion de töe de progrescioin, stæte sviluppæ into detaggio da Jost Bürgi verso o 1600.
Translate to Ligurian (Genoese):
Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the “order of a number”.
The Ligurian (Genoese) translation is:
Parlâ de un numeo comme s’o domandasse tante giffre a l’é unn’alluxon approscimativa a-o logaritmo commun, e o l’ea indicou da Archimede comme «ordine de un numero».
Translate to Ligurian (Genoese):
Such methods are called prosthaphaeresis.
The Ligurian (Genoese) translation is:
Sti metodi en ciammæ prosthaphaeresis.
Translate from English to Genoese:
For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor.
The Ligurian (Genoese) translation is:
Pe fâ un exempio, tutte e parte da sguscia de un nautilo son unna còpia approscimativa de quella seguente, scallâ da un fattô costante.
Can you translate this text to Ligurian?
Logarithms are also linked to self-similarity.
The Ligurian (Genoese) translation is:
I logaritmi en ligæ ascì à l’autosimilaritæ.
What’s the Ligurian translation of this sentence?
It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics, and the absorbance of light in the fields of spectrometry and optics.
The Ligurian (Genoese) translation is:
Se deuvian pe quantificâ a perdia di livelli de tenscion inta trasmiscion di segnæ elettrichi, pe descrive i livelli de potensa di soin inte l’acustica e a sciorbensa da luxe inte l’ambito da spettrometria e de l’òttica.
Translate to Ligurian:
This "law", however, is less realistic than more recent models, such as Stevens's power law.)
The Ligurian (Genoese) translation is:
Sta «lezze», à tutti i mòddi, a l’é meno realistica de modelli ciù reçenti, comme a lezze da potensa de Stevens.
Translate from English to Ligurian (Genoese dialect):
When the logarithm of a random variable has a normal distribution, the variable is said to have a log-normal distribution.
The Ligurian (Genoese) translation is:
Quande un logaritmo con unna variabile random o l’à unna distribuçion normale, se dixe che a variabile a l’à unna distrubiçion log-normale.
Translate this sentence to Genoese:
For such a model, the likelihood function depends on at least one parameter that must be estimated.
The Ligurian (Genoese) translation is:
Pe un modello comme questo, a fonçion se someggiansa a depende a-o manco da un parametro ch’o l’à da vegnî stimmou.
Translate to Genoese:
Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results.
The Ligurian (Genoese) translation is:
Do mæximo, l’algoritmo merge sort o crea unna lista no ordenâ co-o dividde a lista à meitæ e ordinando ste chì primma d’azzonze i resultati.
What’s the Ligurian translation of this sentence?
Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system.
The Ligurian (Genoese) translation is:
I esponenti de Lyapunov deuvian i logaritmi pe mesuâ o graddo de caotiçitæ de un scistema dinamico.
Translate from English to Genoese:
The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length.
The Ligurian (Genoese) translation is:
O triangolo de Sierpinski (inta figua) o peu vegnî coverto da træ còpie de lê mæximo, ciascheduña co-i loei longhi da meitæ da longhixe de quelli originæ.
Translate this sentence to Genoese:
Another example is the p-adic logarithm, the inverse function of the p-adic exponential.
The Ligurian (Genoese) translation is:
Un atro exempio o l’é o logaritmo p-adico, a fonçion inversa de l’esponençiale p-adico.
Translate this sentence to Ligurian:
Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups.
The Ligurian (Genoese) translation is:
L’execuçion de l’esponençiaçion a peu ëse fæta de unna mainea efficaçe, ma o logaritmo discreto, pe çerti gruppi, o l’é tegnuo pe assæ de grammo carcolâ.
Translate this sentence to Ligurian:
Square roots of negative numbers can be discussed within the framework of complex numbers.
The Ligurian (Genoese) translation is:
E reixe quaddræ di numei negativi peuan vegnî discusse inte l’ambito di numei complesci.
Translate from English to Ligurian (Genoese dialect):
In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier).
The Ligurian (Genoese) translation is:
Inte l’India antiga, a conoscensa di aspetti de teoria e applicaçion do quaddrou e da reixe quaddrâ a l’ea vegia a-o manco quante i Sulba Sutra, che remontan a-o 800-500 a.C. pöcassæ (ma fòscia ancon assæ primma).
Can you translate this text to Ligurian?
The letter jīm resembles the present square root shape.
The Ligurian (Genoese) translation is:
A lettia jīm a s’assomeggia a-a forma da reixe quaddrâ che se deuvia ancheu.
What’s the Ligurian translation of this sentence?
It defines an important concept of standard deviation used in probability theory and statistics.
The Ligurian (Genoese) translation is:
O definisce un importante conçetto de deviaçion standard deuviou inta teoria da probabilitæ e inta statistica.
What’s the Genoese translation of this sentence?
Most pocket calculators have a square root key.
The Ligurian (Genoese) translation is:
A ciù parte de carcolatoe da stacca an un tasto pe-a reixe quaddrâ.
Can you translate this text to Ligurian?
The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.
The Ligurian (Genoese) translation is:
A complescitæ de tempo pe-o carcolo de unna reixe quaddrâ con n giffre de preçixon a l’é pægia de quella da moltiplicaçion de n giffre.
Translate to Ligurian (Genoese):
Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.
The Ligurian (Genoese) translation is:
I problemi de Hilbert en vintitrei problemi d’aritmetica pubricæ da-o matematico todesco David Hilbert do 1900.
Translate this sentence to Genoese:
For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist.
The Ligurian (Genoese) translation is:
Pe di atri problemi, comme o chinto, pe tradiçion i esperti s’en missi d’accòrdio in sce unn’interpretaçion univoca e l’é stæto dæto unna soluçion à l’interpretaçion stæta açettâ; à tutti i mòddi exite di problemi, ligæ streitamente à questi, che n’an ancon de soluçion.
Translate to Ligurian:
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards.
The Ligurian (Genoese) translation is:
Gh’é doî problemi che no solo no an ancon avuo de soluçion, ma che porrieivan fiña ëse sensa soluçion pe-i standard moderni.
Can you translate this text to Ligurian?
The other twenty-one problems have all received significant attention, and late into the twentieth century work on these problems was still considered to be of the greatest importance.
The Ligurian (Genoese) translation is:
I atri vintun problemi an reçevuo tutti unn’attençion scignificativa, e a-a fin do secolo vintun o travaggio in sce sti problemi o l’ea ancon consciderou da mascima importansa.
Translate from English to Ligurian (Genoese dialect):
Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.
The Ligurian (Genoese) translation is:
Hilbert o l’à visciuo pe dozz’anni dòppo che Kurt Gödel o l’aiva pubricou o seu teorema, ma pâ ch’o no l’agge scrito nisciuña respòsta formale a-o travaggio de Gödel.
What’s the Genoese translation of this sentence?
In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.
The Ligurian (Genoese) translation is:
Into discutte a seu opinion che tutti i problemi de matematica dovieivan avei unna soluçion, Hilbert o l’ammette a poscibilitæ che a soluçion a pòsse ëse unna preuva che o problema originâ o l’é imposcibile.
Translate from English to Genoese:
The first of these was proved by Bernard Dwork; a completely different proof of the first two, via l-adic cohomology, was given by Alexander Grothendieck.
The Ligurian (Genoese) translation is:
A primma de queste a l’é stæta demostrâ da Bernard Dwork; unna preuva do tutto despægia da-e primme doe, pe mezo da coomologia l-adica, a l’é stæta fornia da Alexander Grothendieck.
Translate from English to Ligurian:
However, the Weil conjectures were, in their scope, more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics.
The Ligurian (Genoese) translation is:
À tutti i mòddi, e ipòtexi de Weil, inta seu portâ, s’assomeggiavan de ciù à un problema sencio de Hilbert, e Weil o no ê à mai inteise comme un programma ch’o l’anesse ben pe tutta a matematica.
What’s the Ligurian translation of this sentence?
Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.
The Ligurian (Genoese) translation is:
Erdős de spesso o l’offriva de recompense monetäie; l’entitæ da recompensa a dependeiva da-a difficoltæ perçepia do problema.
Translate this sentence to Ligurian:
At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute.
The Ligurian (Genoese) translation is:
A-o manco inti media mainstream, l’analogo de facto di problemi de Hilbert do secolo vintun a l’é a lista di sette problemi do Premio do Millennio stæti çernui do 2000 da-o Clay Mathematics Institute.
Translate to Ligurian:
The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise.
The Ligurian (Genoese) translation is:
L’ipòtexi de Riemann a meita de vegnî nottâ pe-o fæto de figuâ inta lista di problemi de Hilbert, inta lista de Smale, inta lista di problemi do Premio do Millennio e inte ipòtexi de Weil ascì, inta seu forma geometrica.
Translate this sentence to Ligurian:
1931, 1936 3rd Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second?
The Ligurian (Genoese) translation is:
1931, 1936 3° Dæto doî poliedri chesegesse con un mæximo volumme, l’é delongo poscibile taggiâ o primmo inte di tòcchi poliedrichi finii che peuan vegnî missi torna insemme pe ottegnî o segondo?
What’s the Genoese translation of this sentence?
1959 15th Rigorous foundation of Schubert's enumerative calculus.
The Ligurian (Genoese) translation is:
1959 15° Fondaçion rigorosa do carcolo enumerativo de Schubert.
Translate this sentence to Genoese:
1927 18th (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions?(b) What is the densest sphere packing?
The Ligurian (Genoese) translation is:
1927 18 (a) Existe un poliedro ch’o l’ammette solo unna covertua anisoedrica à træ dimenscioin? (b) Quæ o l’é l’imballaggio à sfera ciù denso?
Translate to Genoese:
A number is a mathematical object used to count, measure, and label.
The Ligurian (Genoese) translation is:
Un numeo o l’é un oggetto matematico deuviou pe contâ, mesuâ e dâ de etichette.
What’s the Genoese translation of this sentence?
More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five.
The Ligurian (Genoese) translation is:
Ciù universalmente, i numei individuali peuan vegnî representæ da di scimboli, diti numerali: pe fâ un exempio, «5» o l’é un numerale ch’o representa o numeo çinque.
Translate this sentence to Genoese:
Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation.
The Ligurian (Genoese) translation is:
I carcoli co-i numei se fan con de operaçioin aritmetiche; e ciù commun en l’addiçion, a sottraçion, a moltiplicaçion, a divixon e l’esponençiaçion.
Translate from English to Genoese:
Gilsdorf, Thomas E. Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas, John Wiley & Sons, Feb 24, 2012.Restivo, S. Mathematics in Society and History, Springer Science & Business Media, Nov 30, 1992.
The Ligurian (Genoese) translation is:
Gilsdorf, Thomas E. Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas, John Wiley & Sons, 24 frevâ 2012.Restivo, S. Mathematics in Society and History, Springer Science & Business Media, 30 novembre 1992.
What’s the Ligurian translation of this sentence?
During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept.
The Ligurian (Genoese) translation is:
Durante o secolo dixineuve, i studioxi de matematica an comensou à sviluppâ ben ben d’astraçioin despæge che condividdan çerte propietæ di numei e che peua vegnî viste comme unn’estenscion do conçetto.
Translate this sentence to Genoese:
A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers.
The Ligurian (Genoese) translation is:
Un scistema de contezzo o no l’à o conçetto de valô de leugo (comme inta nottaçion deximâ moderna), e pe questo a seu representaçion de numei grendi a l’é limitâ.
Can you translate this text to Ligurian?
Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero.
The Ligurian (Genoese) translation is:
O Brāhmasphuṭasiddhānta de Brahmagupta o l’é o primmo libbro ch’o mensuñe o zero in qualitæ de numeo; pe questo Brahmagupta o l’é consciderou, pe-o sòlito, comme a primma persoña ch’a l’agge formolou o conçetto do zero.
Translate to Ligurian:
In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi, an early example of an algebraic grammar for the Sanskrit language (also see Pingala).
The Ligurian (Genoese) translation is:
À un mòddo pægio, Pāṇini (secolo V a.C.) o l’à deuviou l’operatô zero inte l’Ashtadhyayi, un primmo exempio de grammatica algebrica pe-a lengua sanscrita (s’ammie ascì Pingala).
Translate to Ligurian (Genoese):
By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals.
The Ligurian (Genoese) translation is:
Do 130 d.C. Tolomeo, influensou da Ipparco e da-i babiloneixi, o deuviava un scimbolo pe-o 0 (un çercio piccinetto con unna longa bara in çimma) drento de un scistema numerico sessageximale che pe-o resto o deuviava di numei greghi alfabetichi.
Translate this sentence to Genoese:
Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today.
The Ligurian (Genoese) translation is:
A referensa à Diofanto stæta mensunâ a l’é stæta discussa à un mòddo ciù espliçito da-o matematico indian Brahmagupta, inte Brāhmasphuṭasiddhānta do 628, ch’o l’à deuviou i numei negativi pe produe a formola quaddratica de forma generale ch’a se deuvia ancon a-a giornâ d’ancheu.
Translate this sentence to Genoese:
At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral.
The Ligurian (Genoese) translation is:
A-o mæximo tempo i cineixi indicavan i numei negativi con un tratto diagonale atraverso a giffra ch’a no segge zero e ch’a s’attreuva ciù a-a drita do numeo poxitivo correspondente.
Translate to Ligurian (Genoese):
Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory.
The Ligurian (Genoese) translation is:
I studioxi de matematica greghi e indien aivan fæto di studdi in sciâ teoria di numei raçionæ, comme parte do studio generale da teoria di numei.
Translate from English to Ligurian (Genoese dialect):
The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem.
The Ligurian (Genoese) translation is:
O conçetto de fraçioin deximæ o l’é ligou streitamente a-a nottaçion deximâ do valô de pòsto; pâ che i doî se seggian sviluppæ insemme.