There exist a variety of formulas for either producing the th prime as a function of or taking on only prime values. However, all such formulas require either extremely accurate knowledge of some unknown constant, or effectively require knowledge of the primes ahead of time in order to use the formula (Dudley 1969; Ribenboim 1996, p. 186). There also exist simple prime-generating polynomials that generate only primes for the first (possibly large) number of integer values.There are also many beautiful formulas involving prime sums and prime products that can be done in closed form.Considering examples of formulas that produce only prime numbers (although not necessarily the complete set of prime numbers ), there exists a constant (OEIS A051021) known as Mills' constant such that(1)where is the floor function, is prime for all (Ribenboim 1996, p. 186). The first few values of are 2, 11, 1361, 2521008887, ... (OEIS A051254). It..
The prime distance of a nonnegative integer is the absolute difference between and the nearest prime. It is therefore true that for primes . The first few values for , 1, 2, ... are therefore 2, 1, 0, 0, 1, 0, 1, 0, 1, 2, ... (OEIS A051699). The values of having prime distances of 0, 1, 2, 3, ... are 2, 1, 0, 26, 93, 118, 119, 120, 531, 532, 897, ... (OEIS A077019).
If and (i.e., and are relatively prime), then has at least one primitive prime factor with the following two possible exceptions: 1. . 2. and is a power of 2. Similarly, if , then has at least one primitive prime factor with the exception .A specific case of the theorem considers the th Mersenne number , then each of , , , ... has a prime factor that does not occur as a factor of an earlier member of the sequence, except for . For example, , , , ... have the factors 3, 7, 5, 31, (1), 127, 17, 73, 11, , ... (OEIS A064078) that do not occur in earlier . These factors are sometimes called the Zsigmondy numbers .Zsigmondy's theorem is often useful, especially in group theory, where it is used to prove that various groups have distinct orders except when they are known to be the same (Montgomery 2001)...
A prime factorization algorithm also known as Pollard Monte Carlo factorization method. There are two aspects to the Pollard factorization method. The first is the idea of iterating a formula until it falls into a cycle. Let , where is the number to be factored and and are its unknown prime factors. Iterating the formula(1)or almost any polynomial formula (an exception being ) for any initial value will produce a sequence of number that eventually fall into a cycle. The expected time until the s become cyclic and the expected length of the cycle are both proportional to .However, since with and relatively prime, the Chinese remainder theorem guarantees that each value of (mod ) corresponds uniquely to the pair of values (), ). Furthermore, the sequence of s follows exactly the same formula modulo and , i.e.,(2)(3)Therefore, the sequence (mod ) will fall into a much shorter cycle of length on the order of . It can be directly verified that two values and..
A variant of the Pollard p-1 method which uses Lucas sequences to achieve rapid factorization if some factor of has a decomposition of in small prime factors.
The Lyapunov condition, sometimes known as Lyapunov's central limit theorem, states that if the th moment (with ) exists for a statistical distribution of independent random variates (which need not necessarily be from same distribution), the means and variances are finite, and(1)then if(2)where(3)the central limit theorem holds.
A plane curve given by the parametric equations(1)(2)The plots above show curves for values of from 0 to 7.The teardrop curve has area(3)
A sufficient condition on the Lindeberg-Feller central limit theorem. Given random variates , , ..., let , the variance of be finite, and variance of the distribution consisting of a sum of s(1)be(2)In the terminology of Zabell (1995), let(3)where denotes the expectation value of restricted to outcomes , then the Lindeberg condition is(4)for all (Zabell 1995).In the terminology of Feller (1971), the Lindeberg condition assumed that for each ,(5)or equivalently(6)Then the distribution(7)tends to the normal distribution with zero expectation and unit variance (Feller 1971, p. 256). The Lindeberg condition (5) guarantees that the individual variances are small compared to their sum in the sense that for given for all sufficiently large , for , ..., (Feller 1971, p. 256).
In the triangle illustrated above, side subtends angle . More generally, given a geometric object in the plane and a point , let be the angle from one edge of to the other with vertex at . Then is said to subtend an angle from .
A full angle, also called a complete angle, round angle, or perigon, is an angle equal to radians corresponding to the central angle of an entire circle.Four right angles or two straightangles equal one full angle.
An angle drawn on the coordinate plane is said to be in standard position if its initial side lies on the positive x-axis so that its vertex coincides with the origin and its rotation is in the counterclockwise direction.In the above image, the angle is in standard position due to the locations of its vertex and its initial side and because of the direction of its rotation.
A right angle is an angle equal to half the angle from one end of a line segment to the other. A right angle is radians or . A triangle containing a right angle is called a right triangle. However, a triangle cannot contain more than one right angle, since the sum of the two right angles plus the third angle would exceed the total possessed by a triangle.The patterns of cracks observed in mud that has been dried by the sun form curves that often intersect in right angles (Williams 1979, p. 45; Steinhaus 1999, p. 88; Pearce 1990, p. 12).
The exterior angle bisectors (Johnson 1929, p. 149), also called the external angle bisectors (Kimberling 1998, pp. 18-19), of a triangle are the lines bisecting the angles formed by the sides of the triangles and their extensions, as illustrated above.Note that the exterior angle bisectors therefore bisect the supplementaryangles of the interior angles, not the entire exterior angles.There are therefore three pairs of oppositely oriented exterior angle bisectors. The exterior angle bisectors intersect pairwise in the so-called excenters , , and . These are the centers of the excircles, i.e., the three circles that are externally tangent to the sides of the triangle (or their extensions).The points determined on opposite sides of a triangle by an angle bisector from each vertex lie on a straight line if either (1) all or (2) one out of the three bisectors is an external angle bisector (Johnson 1929, p. 149; Honsberger..
An exterior angle of a polygon is the angle formed externally between two adjacent sides. It is therefore equal to , where is the corresponding internal angle between two adjacent sides (Zwillinger 1995, p. 270).Consider the angles formed between a side of a polygon and the extension of an adjacent side. Since there are two directions in which a side can be extended, there are two such angles at each vertex. However, since corresponding angles are opposite, they are also equal.Confusingly, a bisector of an angle is known as an exterior angle bisector, while a bisector of an angle (which is simply a line oriented in the opposite direction as the interior angle bisector) is not given any special name.The sum of the angles in a convex polygon is equal to radians (), since this corresponds to one complete rotation of the polygon...
The (interior) bisector of an angle, also called the internal angle bisector (Kimberling 1998, pp. 11-12), is the line or line segment that divides the angle into two equal parts.The angle bisectors meet at the incenter , which has trilinear coordinates 1:1:1.The length of the bisector of angle in the above triangle is given bywhere and .The points , , and have trilinear coordinates , , and , respectively, and form the vertices of the incentral triangle.
Let be the angle between and , the angle between and , and the angle between and . Then the direction cosines are equivalent to the coordinates of a unit vector ,(1)(2)(3)From these definitions, it follows that(4)To find the Jacobian when performing integrals overdirection cosines, use(5)(6)(7)The Jacobian is(8)Using(9)(10)(11)(12)so(13)(14)(15)(16)Direction cosines can also be defined between two sets of Cartesiancoordinates,(17)(18)(19)(20)(21)(22)(23)(24)(25)Projections of the unprimed coordinates onto the primed coordinates yield(26)(27)(28)(29)(30)(31)and(32)(33)(34)(35)(36)(37)Projections of the primed coordinates onto the unprimed coordinates yield(38)(39)(40)(41)(42)(43)and(44)(45)(46)Using the orthogonality of the coordinate system, it must be true that(47)(48)giving the identities(49)for and , and(50)for . These two identities may be combined into the single identity(51)where is the..
Given two intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is called the angle between them. The term "plane angle" is sometimes used to distinguish angles in a plane from solid angles measured in space (International Standards Organization 1982, p. 5).The term "angle" can also be applied to the rotational offset between intersecting planes about their common line of intersection, in which case the angle is called the dihedral angle of the planes.Angles are usually measured in degrees (denoted ), radians (denoted rad, or without a unit), or sometimes gradians (denoted grad).The concept of an angle can be generalized from the circle to the sphere, in which case it is known as solid angle. The fraction of a sphere subtended by an object (its solid angle) is measured in steradians, with the entire sphere..
The radian is a unit of angular measure defined such that an angle of one radian subtended from the center of a unit circle produces an arc with arc length 1.A full angle is therefore radians, so there are per radians, equal to or 57./radian. Similarly, a right angle is radians and a straight angle is radians.Radians are the most useful angular measure in calculus because they allow derivative and integral identities to be written in simple terms, e.g.,for measured in radians.Unless stated otherwise, all angular quantities considered in this work are assumed to be specified in radians.
A ruled surface is called a generalized cone if it can be parameterized by , where is a fixed point which can be regarded as the vertex of the cone. A generalized cone is a regular surface wherever . The above surface is a generalized cone over a cardioid. A generalized cone is a flat surface, and is sometimes called "conical surface."
A cone with elliptical cross section. The parametric equations for an elliptic cone of height , semimajor axis , and semiminor axis are(1)(2)(3)where and .The elliptic cone is a quadratic ruledsurface, and has volume(4)The coefficients of the first fundamental form(5)(6)(7)second fundamental form coefficients(8)(9)(10)The lateral surface area can then be calculated as(11)(12)(13)where is a complete elliptic integral of the second kind and assuming .The Gaussian curvature is(14)and the mean curvature is(15)
A conical frustum is a frustum created by slicing the top off a cone (with the cut made parallel to the base). For a right circular cone, let be the slant height and and the base and top radii. Then(1)The surface area, not including the top and bottomcircles, is(2)(3)The volume of the frustum is given by(4)But(5)so(6)(7)(8)This formula can be generalized to any pyramid by letting be the base areas of the top and bottom of the frustum. Then the volume can be written as(9)The area-weighted integral of over the frustum is(10)(11)so the geometric centroid is located alongthe z-axis at a height(12)(13)(Eshbach 1975, p. 453; Beyer 1987, p. 133; Harris and Stocker 1998, p. 105). The special case of the cone is given by taking , yielding .
The mapping of a grid of regularly ruled squares onto a cone with no overlap or misalignment. Cone nets are possible for vertex angles of , , and , where the dark edges in the upper diagrams above are joined. Beautiful photographs of cone net models (lower diagrams above) are presented in Steinhaus (1999). The transformation from a point in the grid plane to a point on the cone is given by(1)(2)(3)where , 1/2, or 3/4 is the fraction of a circle forming the base, and(4)(5)(6)
Two cones placed base-to-base.The bicone with base radius and half-height has surface area and volume(1)(2)The centroid is at the origin, and the inertia tensor about the centroid is given by(3)
Nielsen's spiral, also called the sici spiral (von Seggern 1993) is the spiralwith parametric equations(1)(2)where is the cosine integral and is the sine integral.The curvature is given by(3)and the arc length measured from by(4)
Fermat's spiral, also known as the parabolic spiral, is an Archimedean spiral with having polar equation(1)This curve was discussed by Fermat in 1636 (MacTutor Archive). For any given positive value of , there are two corresponding values of of opposite signs. The left plot above shows(2)only, while the right plot shows equation (1) in red and(3)in blue. Taking both signs, the resulting spiral is symmetrical about the origin.The curvature and arc lengthof the positive branch of Fermat's spiral are(4)(5)(6)where is a hypergeometric function and is an incomplete beta function.
The inverse curve of the epispiralwith inversion center at the origin and inversion radius is the rose
The epispiral is a plane curve with polar equationThere are sections if is odd and if is even.A slightly more symmetric version considers instead
The Theodorus spiral is a discrete spiral formed by connecting the ends of radial spokes corresponding to the hypotenuses of a sequence of adjoining right triangles. The initial spoke is of length , the next spoke is of length , etc., and each segment of the spiral (corresponding to the outer leg of a triangle) has unit length. It is also known as the square root spiral, Einstein spiral, Pythagorean spiral, or--to contrast it with certain continuous analogs--the discrete spiral of Theodorus.The slope of a continuous analog of the discrete Theodorus spiral due to Davis (1993) at the point is sometimes known as Theodorus's constant.
The pedal curve of a logarithmicspiral with parametric equation(1)(2)for a pedal point at the pole is an identical logarithmicspiral(3)(4)so(5)
The inverse curve of the logarithmicspiralwith inversion center at the origin and inversion radius is the logarithmic spiral
For a logarithmic spiral given parametricallyas(1)(2)evolute is given by(3)(4)As first shown by Johann Bernoulli, the evolute of a logarithmic spiral is therefore another logarithmic spiral, having and ,In some cases, the evolute is identical to the original,as can be demonstrated by making the substitution to the new variable(5)Then the above equations become(6)(7)(8)(9)which are equivalent to the form of the original equation if(10)(11)(12)where only solutions with the minus sign in exist. Solving gives the values summarized in the following table.10.2744106319...20.1642700512...30.1218322508...40.0984064967...50.0832810611...60.0725974881...70.0645958183...80.0583494073...90.0533203211...100.0491732529...
A spiral that gives the solution to the central orbitproblem under a radial force law(1)where is a positive constant. There are three solution regimes,(2)where and are constants,(3)(4)and is the specific angular momentum (Whittaker 1944, p. 83). The case gives an epispiral, while leads to a hyperbolic spiral.
The catacaustic of a logarithmic spiral, where the origin is taken as the radiant point, is another logarithmic spiral. For an original spiral with parametric equations(1)(2)the catacaustic with radiant point at the originis(3)(4)
The logarithmic spiral is a spiral whose polarequation is given by(1)where is the distance from the origin, is the angle from the x-axis, and and are arbitrary constants. The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira mirabilis. It can be expressed parametrically as(2)(3)This spiral is related to Fibonacci numbers, the golden ratio, and the golden rectangle, and is sometimes called the golden spiral.The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3).The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw..
The pedal curve of a sinusoidalspiralwith pedal point at the center is another sinusoidalspiral with polar equationA few examples are illustrated above.
The inverse curve of a sinusoidalspiralwith inversion center at the origin and inversion radius is another sinusoidal spiral
The most common statement known as Steiner's theorem (Casey 1893, p. 329) states that the Pascal lines of the hexagons 123456, 143652, and 163254 formed by interchanging the vertices at positions 2, 4, and 6 are concurrent (where the numbers denote the order in which the vertices of the hexagon are taken). The 20 points of concurrence so generated are known as Steiner points.Another theorem due to Steiner lets lines and join a variable point on a conic section to two fixed points on the same conic section. Then and are projectively related.A third "Steiner's theorem" states that if two opposite edges of a tetrahedron move on two fixed skew lines in any way whatsoever but remain fixed in length, then the volume of the tetrahedron remains constant (Altshiller-Court 1979, p. 87)...
The Lemoine hexagon is a cyclic hexagon with vertices given by the six concyclic intersections of the parallels of a reference triangle through its symmedian point . The circumcircle of the Lemoine hexagon is therefore the first Lemoine circle. There are two definitions of the hexagon that differ based on the order in which the vertices are connected.The first definition is the closed self-intersecting hexagon in which alternate sides , , and pass through the symmedian point (left figure). The second definition (Casey 1888, p. 180) is the hexagon formed by the convex hull of the first definition, i.e., the hexagon (right figure).The sides of this hexagon have the property that, in addition to , , and , the remaining sides , , and are antiparallel to , , and , respectively.For the self-intersecting Lemoine hexagon, the perimeter and area are(1)(2)and for the simple hexagon, they are given by(3)(4)(Casey 1888, p. 188), where is the..
Let the opposite sides of a convex cyclic hexagon be , , , , , and , and let the polygon diagonals , , and be so chosen that , , and have no common polygon vertex (and likewise for , , and ), thenThis is an extension of Ptolemy's theorem tothe hexagon.
The dual of Brianchon's theorem (Casey 1888, p. 146), discovered by B. Pascal in 1640 when he was just 16 years old (Leibniz 1640; Wells 1986, p. 69). It states that, given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line.In 1847, Möbius (1885) published the following generalization of Pascal's theorem: if all intersection points (except possibly one) of the lines prolonging two opposite sides of a -gon inscribed in a conic section are collinear, then the same is true for the remaining point.
In the arbelos, consider the semicircles and with centers and passing through . The Apollonius circle of , and the large semicircle of the arbelos is an Archimedean circle . This circle has radius(as it must), and centerThe line perpendicular to and passing through the center of is called the Schoch line.Now let and be two semicircles through with radii proportional to and respectively. The circle tangent to and with its center on the Schoch line is an Archimedean circle. These circles are called Woo circles.Let be the radical axis of the great semicircle of the arbelos and . From a point on consider the tangents to the circle on diameter . The circle with center on the Schoch line and tangent to these tangents is a Woo circle (Okumura and Watanabe 2004).An applet for investigating Woo circles and Schoch lines has been prepared by Schoch (2005)...
The circle through the cusp of the arbelos and the tangent points of the first Pappus circle, which is congruent to the two Archimedes' circles. If and , then the radius of the Bankoff circle is
Draw the perpendicular line from the intersection of the two small semicircles in the arbelos. The two circles and tangent to this line, the large semicircle, and each of the two semicircles are then congruent and known as Archimedes' circles.For an arbelos with outer semicircle of unit radius and parameter , Archimedes' circles have radii(1)and centers(2)(3)Circles that are constructed in a natural way using an arbelos and are congruent to Archimedes' circles are known as Archimedean circles.
An Archimedean circle is a circle defined in the arbelos in a natural way and congruent to Archimedes' circles, i.e., having radiusfor an arbelos with outer semicircle of unit radius and parameter .
The term "arbelos" means shoemaker's knife in Greek, and this term is applied to the shaded area in the above figure which resembles the blade of a knife used by ancient cobblers (Gardner 1979). Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. The position of the central notch is arbitrary and can be located anywhere along the diameter.The arbelos satisfies a number of unexpected identities (Gardner 1979, Schoch). 1. Call the diameters of the left and right semicircles and , respectively, so the diameter of the enclosing semicircle is 1. Then the arc length along the bottom of the arbelos is(1)so the arc length along the enclosing semicircle is the same as the arc length along the two smaller semicircles. 2. Draw the perpendicular from the tangent of the two semicircles to the edge of the large circle. Then the area of the arbelos is the same as the area of the circle with..
A proof by contradiction establishes the truth of a given proposition by the supposition that it is false and the subsequent drawing of a conclusion that is contradictory to something that is proven to be true. That is, the supposition that is false followed necessarily by the conclusion from not-, where is false, which implies that is true.For example, the second of Euclid's theorems starts with the assumption that there is a finite number of primes. Cusik gives some other nice examples.
A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model). This is often expressed by saying that the object is uniquely determined by a certain set of data. The word unique is sometimes replaced by essentially unique, whenever one wants to stress that the uniqueness is only referred to the underlying structure, whereas the form may vary in all ways that do not affect the mathematical content.The object of many uniqueness theorems is the solution to a problem or an equation; in such cases, a uniqueness theorem is normally combined with an existence theorem.
A theorem stating the existence of an object, such as the solution to a problem or equation. Strictly speaking, it need not tell how many such objects there are, nor give hints on how to find them. Some existence theorems give explicit formulas for solutions (e.g., Cramer's rule), others describe in their proofs iteration processes for approaching them (e.g., Bolzano-Weierstrass theorem), while others are settled by nonconstructive proofs which simply deduce the necessity of solutions without indicating any method for determining them (e.g., the Brouwer fixed point theorem, which is proved by reductio ad absurdum, showing that the nonexistence would lead to a contradiction).
Transfinite induction, like regular induction, is used to show a property holds for all numbers . The essential difference is that regular induction is restricted to the natural numbers , which are precisely the finite ordinal numbers. The normal inductive step of deriving from can fail due to limit ordinals.Let be a well ordered set and let be a proposition with domain . A proof by transfinite induction uses the following steps (Gleason 1991, Hajnal 1999): 1. Demonstrate is true. 2. Assume is true for all . 3. Prove , using the assumption in (2). 4. Then is true for all . To prove various results in point-set topology, Cantor developed the first transfinite induction methods in the 1880s. Zermelo (1904) extended Cantor's method with a "proof that every set can be well-ordered," which became the axiom of choice or Zorn's Lemma (Johnstone 1987). Transfinite induction and Zorn's lemma are often used interchangeably (Reid 1995), or are strongly..
The truth of an infinite sequence of propositions for , ..., is established if (1) is true, and (2) implies for all . This principle is sometimes also known as the method of induction.
According to G. Pólya, the method of finding geometric objects by intersection. 1. For example, the centers of all circles tangent to a straight line at a given point lie on a line that passes through and is perpendicular to . 2. In addition, the circle centered at with radius is the locus of the centers of all circles of radius passing through . The intersection of and consists of two points and which are the centers of two circles of radius tangent to at .Many constructions with straightedge and compass are based on this method, as, for example, the construction of the center of a given circle by means of the perpendicular bisector theorem.
Modus tollens is a valid argument form in propositional calculus in which and are propositions. If implies , and is false, then is false. Also known as an indirect proof or a proof by contrapositive.For example, if being the king implies having a crown, not having a crown implies not being the king.
A description of an object by properties that are different from those mentioned in its definition, but are equivalent to them. The following list gives a number of examples.1. A rational number is defined as the quotient of two integers, but it can be characterized as a number admitting a finite or repeating decimal expansion. 2. An equilateral triangle is defined as a triangle having three equal sides, but it can be characterized as a triangle having two angles of . 3. A real square matrix is nonsingular, by definition, if it admits a matrix inverse, but it can be characterized by the condition that its determinant be nonzero. Of course, a characterization should not merely be a rephrasing of the definition, but should give an entirely new description, which is useful because it contains a simpler formulation, can be verified more easily, is interesting because it places the object in another context, or unveils unexpected links between different..
Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these triangles is the same independent of the polygon vertex chosen (Johnson 1929, p. 193).If a triangle is inscribed in a circle, another circle inside the triangle, a square inside the circle, another circle inside the square, and so on. Then the equation relating the inradius and circumradius of a regular polygon,(1)gives the ratio of the radii of the final to initial circles as(2)Numerically,(3)(OEIS A085365), where is the corresponding constant for polygon circumscribing. This constant is termed the Kepler-Bouwkamp constant by Finch (2003). Kasner and Newman's (1989) assertion that is incorrect, as is the value of 0.8700... given by Prudnikov et al. (1986, p. 757)...
Circumscribe a triangle about a circle, another circle around the triangle, a square outside the circle, another circle outside the square, and so on. The circumradius and inradius for an -gon are then related by(1)so an infinitely nested set of circumscribed polygons and circles has(2)(3)(4)Kasner and Newman (1989) and Haber (1964) state that , but this is incorrect, and the actual answer is(5)(OEIS A051762).By writing(6)it is possible to expand the series about infinity, change the order of summation, do the sum symbolically, and obtain the quickly converging series(7)where is the Riemann zeta function.Bouwkamp (1965) produced the following infinite productformulas for the constant,(8)(9)(10)where is the sinc function (cf. Prudnikov et al. 1986, p. 757), is the Riemann zeta function, and is the Dirichlet lambda function. Bouwkamp (1965) also produced the formula with accelerated convergence(11)where(12)(cited in Pickover..
A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior.The study of triangles is sometimes known as triangle geometry, and is a rich area of geometry filled with beautiful results and unexpected connections. In 1816, while studying the Brocard points of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?" (Wells 1991, p. 21).It is..
A Heronian triangle is a triangle having rational side lengths and rational area. The triangles are so named because such triangles are related to Heron's formula(1)giving a triangle area in terms of its side lengths , , and semiperimeter . Finding a Heronian triangle is therefore equivalent to solving the Diophantine equation(2)The complete set of solutions for integer Heronian triangles (the three side lengths and area can be multiplied by their least common multiple to make them all integers) were found by Euler (Buchholz 1992; Dickson 2005, p. 193), and parametric versions were given by Brahmagupta and Carmichael (1952) as(3)(4)(5)(6)(7)This produces one member of each similarity class of Heronian triangles for any integers , , and such that , , and (Buchholz 1992).The first few integer Heronian triangles sorted by increasing maximal side lengths, are ((3, 4, 5), (5, 5, 6), (5, 5, 8), (6, 8, 10), (10, 10, 12), (5, 12, 13), (10, 13,..
Bertelsen's number is an erroneous name erroneously given to the erroneous value of , where is the prime counting function. This value is 56 lower than the correct value of . Ore (1988, p. 69) states that the erroneous value 478 originated in Bertelsen's application of Meissel's method in 1893 (MathPages; Prime Curios!). However, the incorrect value actually first appears in Meissel (1885) rather than Bertelsen in 1893, as correctly noted by Lagarias et al. 1985. (Note that MathPages incorrectly states that Lagarias et al. attribute the result to Bertelsen.)Unfortunately, the incorrect value has continued to be propagated in modern works such as Hardy and Wright (1979, p. 9), Davis and Hersch (1981, p. 175; but actually given correctly in the table on p. 213), Sondheimer (1981), Kramer (1983), Ore (1988, p. 77), and Cormen et al. (1990)...
(1)The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (OEISA001223). Rankin has shown that(2)for infinitely many and for some constant (Guy 1994). At a March 2003 meeting on elementary and analytic number in Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof that(3)(Montgomery 2003). Unfortunately, this proof turned out to be flawed.An integer is called a jumping champion if is the most frequently occurring difference between consecutive primes for some (Odlyzko et al.).
The set containing no elements, commonly denoted or , the former of which is used in this work. These correspond to Wolfram Language and TeX characters summarized in the table below.symbolTeXWolfram Language\varnothing\[Diameter]\emptyset\[EmptySet]Unfortunately, some authors use the notation 0 instead of for the empty set (Mendelson 1997). The empty set is generally designated using (i.e., the empty list) in the Wolfram Language.A set that is not the empty set is called a nonemptyset. The empty set is sometimes also known as the null set (Mendelson 1997).The complement of the empty set is the universal set.Strangely, the empty set is both open and closed for any set and topology.A groupoid, semigroup, quasigroup, ringoid, and semiring can be empty. Monoids, groups, and rings must have at least one element, while division algebras and fields must have at least two elements...
If is a member of a set , then is said to be an element of , written . If is not an element of , this is written .The term element also refers to a particular member of a group, or entry in a matrix or unevaluated determinant .
Two sets and are disjoint if their intersection , where is the empty set. sets , , ..., are disjoint if for . For example, and are disjoint, but and are not. Disjoint sets are also said to be mutually exclusive or independent.
A set containing all elements of a smaller set. If is a subset of , then is a superset of , written . If is a proper superset of , this is written .
A set together with a relation which is both transitive and reflexive such that for any two elements , there exists another element with and . In this case, the relation is said to "direct" the set.
A subset is a portion of a set. is a subset of (written ) iff every member of is a member of . If is a proper subset of (i.e., a subset other than the set itself), this is written . If is not a subset of , this is written . (The notation is generally not used, since automatically means that and cannot be the same.)The subsets (i.e., power set) of a given set can befound using Subsets[list].An efficient algorithm for obtaining the next higher number having the same number of 1 bits as a given number (which corresponds to computing the next subset) is given by Gosper (1972) in PDP-10 assembler.The set of subsets of a set is called the power set of , and a set of elements has subsets (including both the set itself and the empty set). This follows from the fact that the total number of distinct k-subsets on a set of elements is given by the binomial sumFor sets of , 2, ... elements, the numbers of subsets are therefore 2, 4, 8, 16, 32, 64, ... (OEIS A000079). For example, the set..
Given a Jacobi amplitude and a elliptic modulus in an elliptic integral,
An Abelian integral, are also called a hyperelliptic integral, is an integral of the formwhere is a polynomial of degree .
Newton's iteration is an algorithm for computing the square root of a number via the recurrence equation(1)where . This recurrence converges quadratically as .Newton's iteration is simply an application of Newton'smethod for solving the equation(2)For example, when applied numerically, the first few iterations to Pythagoras's constant are 1, 1.5, 1.41667, 1.41422, 1.41421, ....The first few approximants , , ... to are given by(3)These can be given by the analytic formula(4)(5)These can be derived by noting that the recurrence can be written as(6)which has the clever closed-form solution(7)Solving for then gives the solution derived above.The following table summarizes the first few convergents for small positive integer OEIS, , ...11, 1, 1, 1, 1, 1, 1, 1, ...2A001601/A0510091, 3/2, 17/12, 577/408, 665857/470832, ...3A002812/A0715791, 2, 7/4, 97/56, 18817/10864, 708158977/408855776, .....
Min Max Min Max Re Im Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as(1)where and are the invariants of the Weierstrass elliptic function with modular discriminant(2)(Klein 1877). If , where is the upper half-plane, then(3)is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).Klein's absolute invariant is implemented in the WolframLanguage as KleinInvariantJ[tau].The function is the same as the j-function, modulo a constant multiplicative factor.Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).Klein's invariant can be given explicitly by(4)(5)(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function(6) is a Jacobi theta function, the are..
The second-order ordinarydifferential equation(1)The solutions are the modified Bessel functions of the first and second kinds, and can be written(2)(3)where is a Bessel function of the first kind, is a Bessel function of the second kind, is a modified Bessel function of the first kind, and is modified Bessel function of the second kind.If , the modified Bessel differential equation becomes(4)which can also be written(5)
(1)(Abramowitz and Stegun 1972; Zwillinger 1997, p. 125), having solution(2)where and are Mathieu functions. The equation arises in separation of variables of the Helmholtz differential equation in elliptic cylindrical coordinates. Whittaker and Watson (1990) use a slightly different form to define the Mathieu functions.The modified Mathieu differential equation(3)(Iyanaga and Kawada 1980, p. 847; Zwillinger 1997, p. 125) arises in separation of variables of the Helmholtz differential equation in elliptic cylindrical coordinates, and has solutions(4)The associated Mathieu differential equation is given by(5)(Ince 1956, p. 403; Zwillinger 1997, p. 125).
Underdamped simple harmonic motion is a special case of dampedsimple harmonic motion(1)in which(2)Since we have(3)it follows that the quantity(4)(5)is positive. Plugging in the trial solution to the differential equation then gives solutions that satisfy(6)i.e., the solutions are of the form(7)Using the Euler formula(8)this can be rewritten(9)We are interested in the real solutions. Since we are dealing here with a linear homogeneous ODE, linear sums of linearly independent solutions are also solutions. Since we have a sum of such solutions in (9), it follows that the imaginary and real parts separately satisfy the ODE and are therefore the solutions we seek. The constant in front of the sine term is arbitrary, so we can identify the solutions as(10)(11)so the general solution is(12)The initial values are(13)(14)so and can be expressed in terms of the initial conditions by(15)(16)The above plot shows an underdamped simple harmonic..
The ordinary differential equation(1)It has solution(2)where(3)and is a modified Bessel function of the first kind.
A method for solving ordinary differential equations using the formulawhich advances a solution from to . Note that the method increments a solution through an interval while using derivative information from only the beginning of the interval. As a result, the step's error is . This method is called simply "the Euler method" by Press et al. (1992), although it is actually the forward version of the analogous Euler backward method.While Press et al. (1992) describe the method as neither very accurate nor very stable when compared to other methods using the same step size, the accuracy is actually not too bad and the stability turns out to be reasonable as long as the so-called Courant-Friedrichs-Lewy condition is fulfilled. This condition states that, given a space discretization, a time step bigger than some computable quantity should not be taken. In situations where this limitation is acceptable, Euler's forward method becomes..
The general nonhomogeneous differential equation is given by(1)and the homogeneous equation is(2)(3)Now attempt to convert the equation from(4)to one with constant coefficients(5)by using the standard transformation for linear second-order ordinary differential equations. Comparing (3) and (5), the functions and are(6)(7)Let and define(8)(9)(10)(11)Then is given by(12)(13)(14)which is a constant. Therefore, the equation becomes a second-orderordinary differential equation with constant coefficients(15)Define(16)(17)(18)(19)and(20)(21)The solutions are(22)In terms of the original variable ,(23)Zwillinger (1997, p. 120) gives two other types of equations known as Euler differential equations,(24)(Valiron 1950, p. 201) and(25)(Valiron 1950, p. 212), the latter of which can be solved in terms of Bessel functions...
The generalized minimal residual (GMRES) method (Saad and Schultz 1986) is an extension of the minimal residual method (MINRES), which is only applicable to symmetric systems, to unsymmetric systems. Like MINRES, it generates a sequence of orthogonal vectors, but in the absence of symmetry this can no longer be done with short recurrences; instead, all previously computed vectors in the orthogonal sequence have to be retained. For this reason, "restarted" versions of the method are used.In the conjugate gradient method, theresiduals form an orthogonal basis for the spaceIn GMRES, this basis is formed explicitly:The reader may recognize this as a modified Gram-Schmidt orthonormalization. Applied to the Krylov sequence this orthogonalization is called the "Arnoldi method" (Arnoldi 1951). The inner product coefficients and are stored in an upper Hessenberg matrix.The GMRES iterates are constructed aswhere..
The biconjugate gradient stabilized (BCGSTAB) method was developed to solve nonsymmetric linear systems while avoiding the often irregular convergence patterns of the conjugate gradient squared method (van der Vorst 1992). Instead of computing the conjugate gradient squared method sequence , BCGSTAB computes where is an th degree polynomial describing a steepest descent update.BCGSTAB often converges about as fast as the conjugate gradient squared method (CGS), sometimes faster and sometimes not. CGS can be viewed as a method in which the biconjugate gradient method (BCG) "contraction" operator is applied twice. BCGSTAB can be interpreted as the product of BCG and repeated application of the generalized minimal residual method. At least locally, a residual vector is minimized, which leads to a considerably smoother convergence behavior. On the other hand, if the local generalized minimal residual method step stagnates,..
The conjugate gradient method is not suitable for nonsymmetric systems because the residual vectors cannot be made orthogonal with short recurrences, as proved in Voevodin (1983) and Faber and Manteuffel (1984). The generalized minimal residual method retains orthogonality of the residuals by using long recurrences, at the cost of a larger storage demand. The biconjugate gradient method (BCG) takes another approach, replacing the orthogonal sequence of residuals by two mutually orthogonal sequences, at the price of no longer providing a minimization.The update relations for residuals in the conjugate gradient method are augmented in the biconjugate gradient method by relations that are similar but based on instead of . Thus we update two sequences of residuals(1)(2)and two sequences of search directions(3)(4)The choices(5)(6)ensure the orthogonality relations(7)if .Few theoretical results are known about the convergence..
In the biconjugate gradient method, the residual vector can be regarded as the product of and an th degree polynomial in , i.e.,(1)This same polynomial satisfies(2)so that(3)(4)(5)This suggests that if reduces to a smaller vector , then it might be advantageous to apply this "contraction" operator twice, and compute . The iteration coefficients can still be recovered from these vectors (as shown above), and it turns out to be easy to find the corresponding approximations for . This approach is the conjugate gradient squared (CGS) method (Sonneveld 1989).Often one observes a speed of convergence for CGS that is about twice as fast as for the biconjugate gradient method, which is in agreement with the observation that the same "contraction" operator is applied twice. However, there is no reason that the contraction operator, even if it really reduces the initial residual , should also reduce the once reduced vector . This..
The conjugate gradient method can be applied on the normal equations. The CGNE and CGNR methods are variants of this approach that are the simplest methods for nonsymmetric or indefinite systems. Since other methods for such systems are in general rather more complicated than the conjugate gradient method, transforming the system to a symmetric definite one and then applying the conjugate gradient method is attractive for its coding simplicity.CGNE solves the system(1)for and then computes the solution(2)CGNR solves(3)for the solution vector , where(4)If a system of linear equations has a nonsymmetric, possibly indefinite (but nonsingular) coefficient matrix, one obvious attempt at a solution is to apply the conjugate gradient method to a related symmetric positive definite system . While this approach is easy to understand and code, the convergence speed of the conjugate gradient method now depends on the square of the condition..
Let , , and be square matrices with small, and define(1)where is the identity matrix. Then the inverse of is approximately(2)This can be seen by multiplying(3)(4)(5)(6)Note that if we instead let , and look for an inverse of the form , we obtain(7)(8)(9)(10)In order to eliminate the term, we require . However, then , so so there can be no inverse of this form.The exact inverse of can be found as follows.(11)so(12)Using a general matrix inverse identity then gives(13)
The power of a matrix for a nonnegative integer is defined as the matrix product of copies of ,A matrix to the zeroth power is defined to be the identity matrix of the same dimensions, . The matrix inverse is commonly denoted , which should not be interpreted to mean .
The usual number of scalar operations (i.e., the total number of additions and multiplications) required to perform matrix multiplication is(1)(i.e., multiplications and additions). However, Strassen (1969) discovered how to multiply two matrices in(2)scalar operations, where is the logarithm to base 2, which is less than for . For a power of two (), the two parts of (2) can be written(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)so (◇) becomes(13)Two matrices can therefore be multiplied(14)(15)with only(16)scalar operations (as it turns out, seven of them are multiplications and 18 are additions). Define the seven products (involving a total of 10 additions) as(17)(18)(19)(20)(21)(22)(23)Then the matrix product is given using the remaining eight additions as(24)(25)(26)(27)(Strassen 1969, Press et al. 1989).Matrix inversion of a matrix to yield can also be done in fewer operations than expected using the formulas(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(Strassen..
The product of two matrices and is defined as(1)where is summed over for all possible values of and and the notation above uses the Einstein summation convention. The implied summation over repeated indices without the presence of an explicit sum sign is called Einstein summation, and is commonly used in both matrix and tensor analysis. Therefore, in order for matrix multiplication to be defined, the dimensions of the matrices must satisfy(2)where denotes a matrix with rows and columns. Writing out the product explicitly,(3)where(4)(5)(6)(7)(8)(9)(10)(11)(12)Matrix multiplication is associative, as can be seenby taking(13)where Einstein summation is again used. Now, since , , and are scalars, use the associativity of scalar multiplication to write(14)Since this is true for all and , it must be true that(15)That is, matrix multiplication is associative. Equation(13) can therefore be written(16)without ambiguity. Due to associativity,..
Every complex matrix can be broken into a Hermitianpart(i.e., is a Hermitian matrix) and an antihermitian part(i.e., is an antihermitian matrix). Here, denotes the adjoint.
The square root method is an algorithm which solves the matrixequation(1)for , with a symmetric matrix and a given vector. Convert to a triangular matrix such that(2)where is the transpose. Then(3)(4)so(5)giving the equations(6)(7)(8)(9)(10)These give(11)(12)(13)(14)(15)giving from . Now solve for in terms of the s and ,(16)(17)(18)which gives(19)(20)(21)Finally, find from the s and ,(22)(23)(24)giving the desired solution,(25)(26)(27)
The inverse of a square matrix , sometimes called a reciprocal matrix, is a matrix such that(1)where is the identity matrix. Courant and Hilbert (1989, p. 10) use the notation to denote the inverse matrix.A square matrix has an inverse iff the determinant (Lipschutz 1991, p. 45). The so-called invertible matrix theorem is major result in linear algebra which associates the existence of a matrix inverse with a number of other equivalent properties. A matrix possessing an inverse is called nonsingular, or invertible.The matrix inverse of a square matrix may be taken in the Wolfram Language using the function Inverse[m].For a matrix(2)the matrix inverse is(3)(4)For a matrix(5)the matrix inverse is(6)A general matrix can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination, or LU decomposition.The inverse of a product of matrices and can be expressed in terms of and . Let(7)Then(8)and(9)Therefore,(10)so(11)where..
The power series that defines the exponential map also defines a map between matrices. In particular,(1)(2)(3)converges for any square matrix , where is the identity matrix. The matrix exponential is implemented in the Wolfram Language as MatrixExp[m].The Kronecker sum satisfies the nice property(4)(Horn and Johnson 1994, p. 208).Matrix exponentials are important in the solution of systems of ordinary differential equations (e.g., Bellman 1970).In some cases, it is a simple matter to express the matrix exponential. For example, when is a diagonal matrix, exponentiation can be performed simply by exponentiating each of the diagonal elements. For example, given a diagonal matrix(5)The matrix exponential is given by(6)Since most matrices are diagonalizable,it is easiest to diagonalize the matrix before exponentiating it.When is a nilpotent matrix, the exponential is given by a matrix polynomial because some power of vanishes...
Two matrices and are said to be equal iff(1)for all . Therefore,(2)while(3)
The matrix direct sum of matrices constructs a block diagonal matrix from a set of square matrices, i.e.,(1)(2)
Denote the sum of two matrices and (of the same dimensions) by . The sum is defined by adding entries with the same indicesover all and . For example,Matrix addition is therefore both commutative andassociative.
The Kronecker sum is the matrix sum defined by(1)where and are square matrices of order and , respectively, is the identity matrix of order , and denotes the Kronecker product.For example, the Kronecker sum of two matrices and is given by(2)The Kronecker sum satisfies the nice property(3)where denotes a matrix exponential.
Every complex matrix can be broken into a Hermitian part(i.e., is a Hermitian matrix) and an antihermitian part(i.e., is an antihermitian matrix). Here, denotes the conjugate transpose.
Let be a vector norm of a vector such thatThen is a matrix norm which is said to be the natural norm induced (or subordinate) to the vector norm . For any natural norm,where is the identity matrix. The natural matrix norms induced by the L1-norm, L2-norm, and L-infty-norm are called the maximum absolute column sum norm, spectral norm, and maximum absolute row sum norm, respectively.
The natural norm induced by the L-infty-normis called the maximum absolute row sum norm and is defined byfor a matrix . This matrix norm is implemented as Norm[m, Infinity].
The natural norm induced by the L1-normis called the maximum absolute column sum norm and is defined byfor a matrix . This matrix norm is implemented as MatrixNorm[m, 1] in the Wolfram Language package MatrixManipulation` .
Given a square complex or real matrix , a matrix norm is a nonnegative number associated with having the properties 1. when and iff , 2. for any scalar , 3. , 4. . Let , ..., be the eigenvalues of , then(1)The matrix -norm is defined for a real number and a matrix by(2)where is a vector norm. The task of computing a matrix -norm is difficult for since it is a nonlinear optimization problem with constraints.Matrix norms are implemented as Norm[m, p], where may be 1, 2, Infinity, or "Frobenius".The maximum absolute column sum norm is defined as(3)The spectral norm , which is the square root of the maximum eigenvalue of (where is the conjugate transpose),(4)is often referred to as "the" matrix norm.The maximum absolute row sum norm isdefined by(5), , and satisfy the inequality(6)..
The Frobenius norm, sometimes also called the Euclidean norm (a term unfortunately also used for the vector -norm), is matrix norm of an matrix defined as the square root of the sum of the absolute squares of its elements,(Golub and van Loan 1996, p. 55).The Frobenius norm can also be considered as a vectornorm.It is also equal to the square root of the matrix trace of , where is the conjugate transpose, i.e.,The Frobenius norm of a matrix is implemented as Norm[m, "Frobenius"] and of a vector as Norm[v, "Frobenius"].
Let be the matrix norm associated with the matrix and be the vector norm associated with a vector . Let the product be defined, then and are said to be compatible if
If is an square matrix and is an eigenvalue of , then the union of the zero vector and the set of all eigenvectors corresponding to eigenvalues is a subspace of known as the eigenspace of .