Bernhard RiemannGeorg Friedrich Bernhard Riemann (ˈɡeːɔʁk ˈfʁiːdʁɪç ˈbɛʁnhaʁt ˈʁiːman; 17 September 1826 – 20 July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis.
Absolute convergenceIn mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely".
Square-integrable functionIn mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line is defined as follows. One may also speak of quadratic integrability over bounded intervals such as for . An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable.
Frobenius theorem (differential topology)In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields.
Arabic diacriticsArabic script has numerous diacritics, which include consonant pointing known as ALA (إِعْجَام), and supplementary diacritics known as ALA (تَشْكِيل). The latter include the vowel marks termed ALA (حَرَكَات; singular: حَرَكَة, ALA). The Arabic script is a modified abjad, where short consonants and long vowels are represented by letters but short vowels and consonant length are not generally indicated in writing. ALA is optional to represent missing vowels and consonant length.
Arabic alphabetThe Arabic alphabet (الْأَبْجَدِيَّة الْعَرَبِيَّة, ALA ʔælʔæbʒædijːæ-lʕɑrɑbijːæ or الْحُرُوف الْعَرَبِيَّة, ALA), or Arabic abjad, is the Arabic script as it is codified for writing Arabic. It is written from right to left in a cursive style and includes 28 letters. Most letters have contextual letterforms. The Arabic alphabet is considered an abjad, meaning it only uses consonants, but it is now considered an "impure abjad". As with other impure abjads, such as the Hebrew alphabet, scribes later devised means of indicating vowel sounds by separate vowel diacritics.
Persian alphabetThe Persian alphabet (Alefbâye Fârsi), also known as the Perso-Arabic script, is the right-to-left alphabet used for the Persian language. It is a variation of the Arabic alphabet with four additional letters added: پ چ ژ گ. It was the basis of many Arabic-based scripts used in Central and South Asia. It is used for the Iranian and Dari standard varieties of Persian; and is one of two official writing systems for the Persian language, alongside the Cyrillic-based Tajik alphabet.
Jawi scriptJawi (; Jawoë; Kelantan-Pattani: Yawi; d͡ʒä.wi) is a writing system used for writing Malay and several languages of Southeast Asia, such as Acehnese, Banjarese, Kerinci, Maguindanaon, Minangkabau, Tausūg, and Ternate. Jawi is based on the Arabic script, consisting of all of the original 31 Arabic letters, and six additional letters constructed to fit the phonemes native to Malay, and an additional phoneme used in foreign loanwords, but not found in Classical Arabic, which are ca /t͡ʃ/, nga /ŋ/, pa /p/, ga /ɡ/, va /v/, and nya /ɲ/.
