Initial conditionIn mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time.
Tangent half-angle formulaIn trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following: From these one can derive identities expressing the sine, cosine, and tangent as functions of tangents of half-angles: Using double-angle formulae and the Pythagorean identity gives Taking the quotient of the formulae for sine and cosine yields Combining the Pythagorean identity with the double-angle formula for the cosine, rearranging, and taking the square roots yields and which, upon division gives Alternatively, It turns out that the absolute value signs in these last two formulas may be dropped, regardless of which quadrant α is in.
Bulgarian grammarBulgarian grammar is the grammar of the Bulgarian language. Bulgarian is a South Slavic language that evolved from Old Church Slavonic—the written norm for the Slavic languages in the Middle Ages which derived from Proto-Slavic. Bulgarian is also a part of the Balkan sprachbund, which also includes Greek, Macedonian, Romanian, Albanian and the Torlakian dialect of Serbian. It shares with them several grammatical innovations that set it apart from most other Slavic languages, even other South Slavic languages.
Initial value problemIn multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem.
Emperor Gaozu of HanEmperor Gaozu of Han (Liu Bang; 256 – 1 June 195 BC), courtesy name Ji (季), was the founder and first emperor of the Han dynasty, reigning 202–195 BC. His temple name was "Taizu" while his posthumous name was Emperor Gao, or Gaodi; "Gaozu of Han", derived from the Records of the Grand Historian, is the common way of referring to this sovereign even though he was not accorded the temple name "Gaozu", which literally means "High Founder". Liu Bang was one of the few dynasty founders in Chinese history who was born into a peasant family.
Macedonian languageMacedonian (ˌmæsᵻˈdoʊniən ; македонски јазик, , maˈkɛdɔnski ˈjazik) is an Eastern South Slavic language. It is part of the Indo-European language family, and is one of the Slavic languages, which are part of a larger Balto-Slavic branch. Spoken as a first language by around two million people, it serves as the official language of North Macedonia. Most speakers can be found in the country and its diaspora, with a smaller number of speakers throughout the transnational region of Macedonia.
Neutral particle oscillationIn particle physics, neutral particle oscillation is the transmutation of a particle with zero electric charge into another neutral particle due to a change of a non-zero internal quantum number, via an interaction that does not conserve that quantum number. Neutral particle oscillations were first investigated in 1954 by Murray Gell-mann and Abraham Pais. For example, a neutron cannot transmute into an antineutron as that would violate the conservation of baryon number.
Bulgarian languageBulgarian (bʌlˈɡɛəriən, bʊlˈ- ; bǎlgarski, ˈbɤɫɡɐrski) is an Eastern South Slavic language spoken in Southeast Europe, primarily in Bulgaria. It is the language of the Bulgarians. Along with the closely related Macedonian language (collectively forming the East South Slavic languages), it is a member of the Balkan sprachbund and South Slavic dialect continuum of the Indo-European language family.
Balkan sprachbundThe Balkan sprachbund or Balkan language area is an ensemble of areal features—similarities in grammar, syntax, vocabulary and phonology—among the languages of the Balkans. Several features are found across these languages though not all apply to every single language. The Balkan sprachbund is a prominent example of the sprachbund concept. The languages of the Balkan sprachbund share their similarities despite belonging to various separate language family (genetic) branches.
