Möbius planeIn mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any generalized circle, and thus a natural setting for planar inversive geometry. An inversion of the Möbius plane with respect to any circle is an involution which fixes the points on the circle and exchanges the points in the interior and exterior, the center of the circle exchanged with the point at infinity.
Polar ice capA polar ice cap or polar cap is a high-latitude region of a planet, dwarf planet, or natural satellite that is covered in ice. There are no requirements with respect to size or composition for a body of ice to be termed a polar ice cap, nor any geological requirement for it to be over land, but only that it must be a body of solid phase matter in the polar region. This causes the term "polar ice cap" to be something of a misnomer, as the term ice cap itself is applied more narrowly to bodies that are over land, and cover less than 50,000 km2: larger bodies are referred to as ice sheets.
Plane curveIn mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form for some specific function f.
Geophysical definition of planetThe International Union of Geological Sciences (IUGS) is the internationally recognized body charged with fostering agreement on nomenclature and classification across geoscientific disciplines. However, they have yet to create a formal definition of the term planet. As a result, there are various geophysical definitions in use among professional geophysicists, planetary scientists, and other professionals in the geosciences. Many professionals opt to use one of several of these geophysical definitions instead of the definition voted on by the International Astronomical Union.
Kalam cosmological argumentThe Kalam cosmological argument is a modern formulation of the cosmological argument for the existence of God. It is named after the Kalam (medieval Islamic scholasticism) from which its key ideas originated. William Lane Craig was principally responsible for giving new life to the argument, due to his The Kalām Cosmological Argument (1979), among other writings. The argument's key underpinning idea is the metaphysical impossibility of actual infinities and of a temporally past-infinite universe, traced by Craig to 11th-century Persian Muslim scholastic philosopher Al-Ghazali.
Euler's identityIn mathematics, Euler's identity (also known as Euler's equation) is the equality where e is Euler's number, the base of natural logarithms, i is the imaginary unit, which by definition satisfies i2 = −1, and π is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula when evaluated for x = π. Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics.
Polar regions of EarthThe polar regions, also called the frigid zones or polar zones, of Earth are Earth's polar ice caps, the regions of the planet that surround its geographical poles (the North and South Poles), lying within the polar circles. These high latitudes are dominated by floating sea ice covering much of the Arctic Ocean in the north, and by the Antarctic ice sheet on the continent of Antarctica and the Southern Ocean in the south.
Partially ordered groupIn abstract algebra, a partially ordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a + g ≤ b + g and g + a ≤ g + b. An element x of G is called positive if 0 ≤ x. The set of elements 0 ≤ x is often denoted with G+, and is called the positive cone of G. By translation invariance, we have a ≤ b if and only if 0 ≤ -a + b.
SinIn a religious context, 'sin' is a transgression against divine law or a law of God. Each culture has its own interpretation of what it means to commit a sin. While sins are generally considered actions, any thought, word, or act considered immoral, selfish, shameful, harmful, or alienating might be termed "sinful".
Euler's constantEuler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log: Here, ⌊ ⌋ represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43).
Well-defined expressionIn mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if takes real numbers as input, and if does not equal then is not well defined (and thus not a function).
Euler numbersIn mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion where is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely: The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.
