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Upside-down marks, simple in the era of hand typesetting, were originally recommended by the Real Academia Española (Royal Spanish Academy), in the second edition of the Ortografía de la lengua castellana (Orthography of the Castilian language) in 1754 [3] recommending it as the symbol indicating the beginning of a question in written Spanish—e.g. "¿Cuántos años tienes?"
The less-than sign is a mathematical symbol that denotes an inequality between two values. The widely adopted form of two equal-length strokes connecting in an acute angle at the left, <, has been found in documents dated as far back as the 1560s.
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
In English, the question mark typically occurs at the end of a sentence, where it replaces the full stop (period). However, the question mark may also occur at the end of a clause or phrase, where it replaces the comma (see also Question comma):
The check mark is a predominant affirmative symbol of convenience in the English-speaking world because of its instant and simple composition.
Note that the vanilla Azuma's inequality requires symmetric bounds on martingale increments, i.e. .So, if known bound is asymmetric, e.g. , to use Azuma's inequality, one need to choose = (| |, | |) which might be a waste of information on the boundedness of .
[2] Whether a space supports a Poincaré inequality has turned out to have deep connections to the geometry and analysis of the space. For example, Cheeger has shown that a doubling space satisfying a Poincaré inequality admits a notion of differentiation. [3] Such spaces include sub-Riemannian manifolds and Laakso spaces.
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 .