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### Rough number

Finch (2001, 2003) defines a -rough (or -jagged) number to be positive integer all of whose prime factors are greater than or equal to .Greene and Knuth define "unusual numbers" as numbers whose greatest prime factor is greater than or equal to , and these number are dubbed "-rough" or "-jagged" by Finch (2001, 2003). The first few unusual numbers are 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, ... (OEIS A063538), which turn out to not be so unusual after all (Greene and Knuth 1990, Finch 2001). The first few "usual" numbers are then 8, 12, 16, 18, 24, 27, 30, ... (OEIS A063539).The probability that the greatest prime factor of a random integer is greater than is (Schroeppel 1972).

### Abundant number

An abundant number, sometimes also called an excessive number, is a positive integer for which(1)where is the divisor function and is the restricted divisor function. The quantity is sometimes called the abundance.A number which is abundant but for which all its proper divisors are deficient is called a primitive abundant number (Guy 1994, p. 46).The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (OEIS A005101).Every positive integer with is abundant. Any multiple of a perfect number or an abundant number is also abundant. Prime numbers are not abundant. Every number greater than 20161 can be expressed as a sum of two abundant numbers.There are only 21 abundant numbers less than 100, and they are all even.The first odd abundant number is(2)That 945 is abundant can be seen by computing(3)Define the density function(4)(correcting the expression in Finch 2003, p. 126) for a positive real number where gives the cardinal..

### Infinite product

A product involving an infinite number of terms. Such products can converge. In fact, for positive , the product converges to a nonzero number iff converges.Infinite products can be used to define the cosine(1)gamma function(2)sine, and sinc function.They also appear in polygon circumscribing,(3)An interesting infinite product formula due to Euler which relates and the th prime is(4)(5)(Blatner 1997). Knar's formula gives a functional equation for the gamma function in terms of the infinite product(6)A regularized product identity is given by(7)(Muñoz Garcia and Pérez-Marco 2003, 2008).Mellin's formula states(8)where is the digamma function and is the gamma function.The following class of products(9)(10)(11)(12)(13)(Borwein et al. 2004, pp. 4-6), where is the gamma function, the first of which is given in Borwein and Corless (1999), can be done analytically. In particular, for ,(14)where (Borwein et..

### Sphere tetrahedron picking

Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. random tetrahedra can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 4].Pick four points on a sphere. What is the probability that the tetrahedron having these points as polyhedron vertices contains the center of the sphere? In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the two-dimensional case, pick two points. In order for the third to form a triangle containing the center, it must lie in the quadrant bisected by a line segment passing through the center of the circle and the bisector of the two points. This happens for one quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one octant, or 1/8.Pick four points at random on the surface of a unit sphereusing(1)(2)(3)with..

### Ball triangle picking

Ball triangle picking is the selection of triples of points (corresponding to vertices of a general triangle) randomly placed inside a ball. random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].The distribution of areas of a triangle with vertices picked at random in a unit ball is illustrated above. The mean triangle area is(1)(Buchta and Müller 1984, Finch 2010). random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].The determination of the probability for obtaining an acute triangle by picking three points at random in the unit disk was generalized by Hall (1982) to the -dimensional ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals. Let be the probability that three points chosen independently and uniformly from the -ball form an acute triangle, then (2)(3)These can be combined..

### Hard square entropy constant

Let be the number of (0,1)-matrices with no adjacent 1s (in either columns or rows). For , 2, ..., is given by 2, 7, 63, 1234, ... (OEIS A006506).The hard square entropy constant is defined by(OEIS A085850). It is not known if this constanthas an exact representation.The quantity arises in statistical physics (Baxter et al. 1980, Pearce and Seaton 1988), and is known as the entropy per site of hard squares. A related constant known as the hard hexagon entropy constant can also be defined.

### Tree searching

In database structures, two quantities are generally of interest: the average number of comparisons required to 1. Find an existing random record, and 2. Insert a new random record into a data structure. Some constants which arise in the theory of digital tree searching are(1)(2)(3)(4)(5)(6)(OEIS A065442 and A065443), where is a q-polygamma function. Erdős (1948) proved that is irrational, and is sometimes known as the Erdős-Borwein constant.The expected number of comparisons for a successful search is(7)(8)and for an unsuccessful search is(9)(10)(OEIS A086309 and A086310). Here is the Euler-Mascheroni constant, , , and are small-amplitude periodic functions, and lg is the base 2 logarithm.The variance for searching is(11)(12)(OEIS A086311) and for inserting is(13)(14)(OEIS A086312).The expected number of pairs of twin vacancies in a digital search tree is(15)where(16)(17)(OEIS A086313), which can also be written(18)(Flajolet..

### Hard hexagon entropy constant

Consider an (0, 1)-matrix such as(1)for . Call two elements adjacent if they lie in positions and , and , or and for some . Call the number of such arrays with no pairs of adjacent 1s. Equivalently, is the number of configurations of nonattacking kings on an chessboard with regular hexagonal cells.The first few values of for , 2, ... are 2, 6, 43, 557, 14432, ... (OEIS A066863).The hard square hexagon constant is then given by(2)(3)(OEIS A085851).Amazingly, is algebraic and is given by(4)where(5)(6)(7)(8)(9)(10)(11)(Baxter 1980, Joyce 1988ab).The variable can be expressed in terms of the tribonacci constant(12)where is a polynomial root, as(13)(14)(15)(T. Piezas III, pers. comm., Feb. 11, 2006).Explicitly, is the unique positive root(16)where denotes the th root of the polynomial in the ordering of the Wolfram Language...

### Pólya's random walk constants

Let be the probability that a random walk on a -D lattice returns to the origin. In 1921, Pólya proved that(1)but(2)for . Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that(3)(OEIS A086230), where(4)(5)(6)(7)(8)(9)(OEIS A086231; Borwein and Bailey 2003, Ch. 2, Ex. 20) is the third of Watson's triple integrals modulo a multiplicative constant, is a complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function.Closed forms for are not known, but Montroll (1956) showed that for ,(10)where(11)(12)and is a modified Bessel function of the first kind.Numerical values of from Montroll (1956) and Flajolet (Finch 2003) are given in the following table.OEIS3A0862300.3405374A0862320.1932065A0862330.1351786A0862340.1047157A0862350.08584498A0862360.0729126..

### Cube line picking

The average distance between two points chosen at random inside a unit cube (the case of hypercube line picking), sometimes known as the Robbins constant, is(1)(2)(3)(OEIS A073012; Robbins 1978, Le Lionnais 1983).The probability function as a function of line length, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form(4)The first even raw moments for , 2, ... are 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 and A160694).Pick points on a cube, and space them as far apart as possible. The best value known for the minimum straight line distance between any two points is given in the following table. 51.118033988749861.0606601482100718190.86602540378463100.74999998333331110.70961617562351120.70710678118660130.70710678118660140.70710678118660150.625..

### Lengyel's constant

Let denote the partition lattice of the set . The maximum element of is(1)and the minimum element is(2)Let denote the number of chains of any length in containing both and . Then satisfies the recurrence relation(3)where and is a Stirling number of the second kind. The first few values of for , 2, ... are then 1, 1, 4, 32, 436, 9012, 262760, ... (OEIS A005121).Lengyel (1984) proved that the quotient(4)is bounded between two constants as , and Flajolet and Salvy (1990) improved the result of Babai and Lengyel (1992) to show that(5)(OEIS A086053).

### Jackson's theorem

Jackson's theorem is a statement about the error of the best uniform approximation to a real function on by real polynomials of degree at most . Let be of bounded variation in and let and denote the least upper bound of and the total variation of in , respectively. Given the function(1)then the coefficients(2)of its Fourier-Legendre series, where is a Legendre polynomial, satisfy the inequalities(3)Moreover, the Fourier-Legendre series of converges uniformly and absolutely to in .Bernstein (1913) strengthened Jackson's theorem to(4)A specific application of Jackson's theorem shows that if(5)then(6)

### Rényi's parking constants

Given the closed interval with , let one-dimensional "cars" of unit length be parked randomly on the interval. The mean number of cars which can fit (without overlapping!) satisfies(1)The mean density of the cars for large is(2)(3)(4)(OEIS A050996). While the inner integral canbe done analytically,(5)(6)where is the Euler-Mascheroni constant and is the incomplete gamma function, it is not known how to do the outer one(7)(8)(9)where is the exponential integral. The slowly converging series expansion for the integrand is given by(10)(OEIS A050994 and A050995).In addition,(11)for all (Rényi 1958), which was strengthened by Dvoretzky and Robbins (1964) to(12)Dvoretzky and Robbins (1964) also proved that(13)Let be the variance of the number of cars, then Dvoretzky and Robbins (1964) and Mannion (1964) showed that(14)(15)(16)(OEIS A086245), where(17)(18)and the numerical value is due to Blaisdell and Solomon..

### Plouffe's constants

Plouffe's constants are numbers arising in summations of series related to where is a trigonometric function. Define the Iverson bracket function(1)Now define through(2)(3)then(4)(5)(6)(OEIS A086201).For(7)(8)the sum is (amazingly) given by(9)(10)(11)(OEIS A086202), where denotes the XOR of binary digits (Chowdhury 2001a; Finch 2003, p. 432). A related sum is given by(12)(13)(14)(OEIS A111953), where again denotes the XOR of binary digits (Chowdhury 2001b; Finch 2005, p. 20).Letting(15)(16)then(17)(18)(OEIS A049541).Plouffe asked if the above processes could be "inverted." He considered(19)(20)giving(21)(22)and(23)(24)giving(25)(26)and(27)(28)giving(29)(30)(31)(OEIS A086203), where the identity was conjecturedby Plouffe and proved by Borwein and Girgensohn (1995). is sometimes known as Plouffe's constant (Plouffe 1997), although this angle had arisen in the geometry of the..

### Favard constants

Let be an arbitrary trigonometric polynomial(1)with real coefficients, let be a function that is integrable over the interval , and let the th derivative of be bounded in . Then there exists a polynomial for which(2)for all , where is the smallest constant possible, known as the th Favard constant. can be given explicitly by the sum(3)which can be written in terms of the Lerch transcendentas(4)These can be expressed by(5)where is the Dirichlet lambda function and is the Dirichlet beta function. Explicitly,(6)(7)(8)(9)(10)(11)(OEIS A050970 and A050971).

### Chebyshev constants

The constantswhere and are th and th order polynomials, and is the set of all rational functions with real coefficients.

### Shapiro's cyclic sum constant

Consider the sum(1)where the s are nonnegative and the denominators are positive. Shapiro (1954) asked if(2)for all . It turns out (Mitrinovic et al. 1993) that this inequality is true for all even and odd .Define(3)and let(4)Then Rankin (1958) proved that(5) can be computed by letting be the function convex hull of the functions(6)(7)Then(8)(OEIS A086277; Drinfeljd 1971).A modified sum was considered by Elbert (1973):(9)Consider(10)where(11)and let be the convex hull of(12)(13)Then(14)(OEIS A086278).

### Shallit constant

Define with positive as(1)Then(2)as increases, where the Shallit constant is(3)(OEIS A086276; Shallit 1995). In their solution,Grosjean and De Meyer (quoted in Shallit 1995) reduced the complexity of the problem.

### Bloch constant

Let be the set of complex analytic functions defined on an open region containing the set closure of the unit disk satisfying and . For each in , let be the supremum of all numbers such that there is a subregion in on which is one-to-one and such that contains a disk of radius . In 1925, Bloch (Conway 1989) showed that .Define Bloch's constant by(1)Ahlfors and Grunsky (1937) derived(2)Bonk (1990) proved that , which was subsequently improved to (Chen and Gauthier 1996; Xiong 1998; Finch 2003, p. 456).Ahlfors and Grunsky (1937) also conjectured that the upper limit is actually the value of ,(3)(4)(5)(OEIS A085508; Le Lionnais 1983).

### Bernstein's constant

Let be the error of the best uniform approximation to a real function on the interval by real polynomials of degree at most . If(1)then Bernstein showed that(2)He conjectured that the lower limit () was . However, this was disproven by Varga and Carpenter (1987) and Varga (1990), who computed(3)For rational approximations for and of degree and , D. J. Newman (1964) proved(4)for . Gonchar (1967) and Bulanov (1975) improved the lower bound to(5)Vjacheslavo (1975) proved the existence of positive constants and such that(6)(Petrushev 1987, pp. 105-106). Varga et al. (1993) conjectured and Stahl(1993) proved that(7)

### Niven's constant

Given a positive integer , let its prime factorization be written(1)Define the functions and by , , and(2)(3)The first few terms of are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, ... (OEIS A051904), while the first few terms of are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, ... (OEIS A051903).Then the average value of tends to(4)Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 16/13, ... (OEIS A086195 and A086196).In addition, the ratio(5)where is the Riemann zeta function (Niven 1969).Niven (1969) also proved that(6)where Niven's constant is given by(7)(OEIS A033150). Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 17/13, ... (OEIS A086197 and A086198).The continued fraction of Niven's constant is 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 4, 8, 4, 1, ... (OEIS A033151). The positions at which the digits 1, 2, ... first occur in the continued fraction are 1, 3, 10, 7, 47, 41, 34,..

### Mills' constant

Mills' theorem states that there exists a real constant such that is prime for all positive integers (Mills 1947). While for each value of , there are uncountably many possible values of such that is prime for all positive integers (Caldwell and Cheng 2005), it is possible to define Mills' constant as the least such thatis prime for all positive integers , giving a value of(OEIS A051021). is therefore given by the next prime after , and the values of are known as Mills' primes (Caldwell and Cheng 2005).Caldwell and Cheng (2005) computed more than 6850 digits of assuming the truth of the Riemann hypothesis. Proof of primality of the 13 Mills prime in Jul. 2013 means that approximately digits are now known.It is not known if is irrational.

### Stephens' constant

Let and be nonzero integers such that (except when ). Also let be the set of primes for which for some nonnegative integer . Then assuming the generalized Riemann hypothesis, Stephens (1976) showed that the density of relative to the primes is a rational multiple of(OEIS A065478), where is the th prime (Finch 2003).

### Silverman constant

(1)(2)(OEIS A093827), where is the totient function and is the divisor function.

### Mertens constant

The Mertens constant , also known as the Hadamard-de la Vallee-Poussin constant, prime reciprocal constant (Bach and Shallit 1996, p. 234), or Kronecker's constant (Schroeder 1997), is a constant related to the twin primes constant and that appears in Mertens' second theorem,(1)where the sum is over primes and is a Landau symbol. This sum is the analog of(2)where is the Euler-Mascheroni constant (Gourdon and Sebah).The constant is given by the infinite sum(3)where is the Euler-Mascheroni constant and is the th prime (Rosser and Schoenfeld 1962; Hardy and Wright 1979; Le Lionnais 1983; Ellison and Ellison 1985), or by the limit(4)According to Lindqvist and Peetre (1997), this was shown independently by Meisselin 1866 and Mertens (1874). Formula (3) is equivalent to(5)which follows from (4) using the Mercator series for with . is also given by the rapidly converging series(6)where is the Riemann zeta function, and is the Möbius..

### Conway's constant

The constant(OEIS A014715) giving the asymptotic rate of growth of the number of digits in the th term of the look and say sequence, given by the unique positive real root of the polynomialillustrated in the figure above. Note that the polynomialgiven in Conway (1987, p. 188) contains a misprint.The continued fraction for is 1, 3, 3, 2, 1, 2, 1, 5, 8, 4, 14, 3, 1, ... (OEIS A014967).

### Squarefree word

A "square" word consists of two identical adjacent subwords (for example, acbacb). A squarefree word contains no square words as subwords (for example, abcacbabcb). The only squarefree binary words are , , ab, ba, aba, and bab (since aa, bb, aaa, aab, abb, baa, bba, and bbb contain square identical adjacent subwords a, b, a, a, b, a, b, and b, respectively).However, there are arbitrarily long ternary squarefree words. The number of ternary squarefree words of length , 2, ... are 1, 3, 6, 12, 18, 30, 42, 60, ... (OEIS A006156), and is bounded by(Brandenburg 1983). In addition,(Brinkhuis 1983, Noonan and Zeilberger 1999).The number of squarefree quaternary words of length , 2, ... are 4, 12, 36, 96, 264, 696, ... (OEIS A051041).

### Overlapfree word

A word is said to be overlapfree if it has no subwords of the form xyxyx. A squarefree word is overlapfree, and an overlapfree word is cubefree.The numbers of binary overlapfree words of length , 2, ... are 2, 4, 6, 10, 14, 20, ... (OEIS A007777). satisfies(1)for some constants and (Restivo and Selemi 1985, Kobayashi 1988). In addition, while(2)does not exist,(3)where(4)(5)(Cassaigne 1993).The Thue-Morse sequence is overlapfree (Alloucheand Shallit 2003, p. 15).

### Perfect power

A perfect power is a number of the form , where is a positive integer and . If the prime factorization of is , then is a perfect power iff .Including duplications (i.e., taking all numbers up to some cutoff and taking all their powers) and taking , the first few are 4, 8, 9, 16, 16, 25, 27, 32, 36, 49, 64, 64, 64, ... (OEIS A072103). Here, 16 is duplicated since(1)As shown by Goldbach, the sum of reciprocals of perfect powers (excluding 1) withduplications converges,(2)The first few numbers that are perfect powers in more than one way are 16, 64, 81,256, 512, 625, 729, 1024, 1296, 2401, 4096, ... (OEIS A117453).The first few perfect powers without duplications are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, ... (OEIS A001597). Even more amazingly, the sum of the reciprocals of these numbers (excluding 1) is given by(3)(OEIS A072102), where is the Möbius function and is the Riemann zeta function.The numbers of perfect powers without duplications..

### Heilbronn triangle problem

The Heilbronn triangle problem is to place points in a disk (square, equilateral triangle, etc.) of unit area so as to maximize the area of the smallest of the triangles determined by the points. For points, there is only a single triangle, so Heilbronn's problem degenerates into finding the largest triangle that can be constructed from points in a square. For , there are four possible triangles for each configuration, so the problem is to find the configuration of points for which the smallest of these four triangles is the maximum possible.For a unit square, the first few maxima of minimaltriangle areas are(1)(2)(3)(4)(5)(6)(7)(8)For larger values of , proofs of optimality are open, but the best known results are(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)with the configurations leading to maximum minimal triangles illustrated above (Friedman 2006; Comellas and Yebra 2002; D. Cantrell..

### Apéry's constant

Apéry's constant is defined by(1)(OEIS A002117) where is the Riemann zeta function. Apéry (1979) proved that is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of (Hata 2000). arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.The following table summarizes progress in computing upper bounds on the irrationality measure for . Here, the exact values for is given by(2)(3)(Hata 2000).upper boundreference15.513891Rhin and Viola (2001)28.830284Hata (1990)312.74359Dvornicich and Viola (1987)413.41782Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)Beukers (1979) reproduced Apéry's rational approximation to using the triple..

### Generalized fermat equation

A generalization of the equation whose solution is desired in Fermat'slast theoremtofor , , , and positive constants, with trivial solutions having , , or being excluded. is trivial to solve by taking and . is more difficult, but can be solved by noting that solutions exist for values of which can be written as a sum of two squares, the first few of which are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, ... (OEIS A001481).

### Khinchin's constant

Let(1)be the simple continued fraction of a "generic" real number , where the numbers are the partial quotients. Khinchin (1934) considered the limit of the geometric mean(2)as . Amazingly, except for a set of measure 0, this limit is a constant independent of given by(3)(OEIS A002210), as proved in Kac (1959).The constant is known as Khinchin's constant, and is commonly also spelled "Khintchine'sconstant" (Shanks and Wrench 1959, Bailey et al. 1997).It is implemented as Khinchin, where its value is cached to 1100-digit precision. However, the numerical value of is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower.It is not known if is irrational, let alone transcendental.While it is known that almost all numbers have limits that approach , this fact has not been proven for any explicit real number , e.g., a real number cast in terms of fundamental constants..

### Continued fraction constants

A number of closed-form constants can be obtained for generalized continued fractions having particularly simple partial numerators and denominators.The Ramanujan continued fractions provide a fascinating class of continued fraction constants. The Trott constants are unexpected constants whose partial numerators and denominators correspond to their decimal digits (though to achieve this, it is necessary to allow some partial numerators to equal 0).The first in a series of other famous continued fraction constants is the infiniteregular continued fraction(1)(2)The first few convergents of the constant are 0, 1, 2/3, 7/10, 30/43, 157/225, 972/1393, 6961/9976, ... (OEIS A001053 and A001040).Both numerator and denominator satisfy the recurrence relation(3)where has the initial conditions , and has the initial conditions , . These can be solved exactly to yield(4)(5)(6)(7)where is a modified Bessel function of the first kind..

### Weakly binary tree

A weakly binary tree is a planted tree in which all nonroot graph vertices are adjacent to at most three graph vertices.Let(1)be the generating function for the number of weakly binary trees on nodes, where(2)(3)(4)(5)This gives the sequence 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, ... (OEIS A001190),sometimes also known as the Wedderburn-Etherington numbers.Otter (Otter 1948, Harary and Palmer 1973, Knuth 1997) showed that(6)where(7)(OEIS A086317; Knuth 1997, p. 583; Finch 2003, p. 297) is the unique positive root of(8)and(9)(OEIS A086318; Knuth 1997, p. 583). is also given by the rapidly converging limit(10)where is given by(11)(12)the first few terms of which are 6, 38, 1446, 2090918, 4371938082726, ... (OEIS A072191), giving(13)

### Lehmer's constant

The Lehmer cotangent expansion for whichthe convergence is slowest occurs when the inequality in the recurrence equation(1)for(2)is replaced by equality, giving and(3)for .This recurrences gives values of corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065), and defines the constant known as Lehmer's constant as(4)(5)(6)(OEIS A030125). is not an algebraic number of degree less than 4, but Lehmer's approach cannot show whether is transcendental.

### Stieltjes constants

Expanding the Riemann zeta function about gives(1)(Havil 2003, p. 118), where the constants(2)are known as Stieltjes constants.Another sum that can be used to define the constants is(3)These constants are returned by the WolframLanguage function StieltjesGamma[n].A generalization takes as the coefficient of is the Laurent series of the Hurwitz zeta function about . These generalized Stieltjes constants are implemented in the Wolfram Language as StieltjesGamma[n, a].The case gives the usual Euler-Mascheroni constant(4)A limit formula for is given by(5)where is the imaginary part and is the Riemann zeta function.An alternative definition is given by absorbing the coefficient of into the constant,(6)(e.g., Hardy 1912, Kluyver 1927).The Stieltjes constants are also given by(7)Plots of the values of the Stieltjes constants as a function of are illustrated above (Kreminski). The first few numerical values are given in the..

### Backhouse's constant

Let be defined as the power series whose th term has a coefficient equal to the th prime ,(1)(2)The function has a zero at (OEIS A088751). Now let be defined by(3)(4)(5)(OEIS A030018).Then N. Backhouse conjectured that(6)(7)(OEIS A072508). This limit was subsequently shown to exist by P. Flajolet. Note that , which follows from the radius of convergence of the reciprocal power series.The continued fraction of Backhouse's constant is [1, 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, ...] (OEIS A074269), which is also the same as the continued fraction of except for a leading 0 in the latter.

### Lebesgue constants

There are two sets of constants that are commonly known as Lebesgue constants. The first is related to approximation of function via Fourier series, which the other arises in the computation of Lagrange interpolating polynomials.Assume a function is integrable over the interval and is the th partial sum of the Fourier series of , so that(1)(2)and(3)If(4)for all , then(5)and is the smallest possible constant for which this holds for all continuous . The first few values of are(6)(7)(8)(9)(10)(11)(12)(13)Some sum formulas for include(14)(15)(Zygmund 1959) and integral formulas include(16)(17)(Hardy 1942). For large ,(18)This result can be generalized for an -differentiable function satisfying(19)for all . In this case,(20)where(21)(Kolmogorov 1935, Zygmund 1959).Watson (1930) showed that(22)where(23)(24)(25)(OEIS A086052), where is the gamma function, is the Dirichlet lambda function, and is the Euler-Mascheroni constant.Define..

### Nonaveraging sequence

A sequence of positive integers(1)is a nonaveraging sequence if it contains no three terms which are in an arithmeticprogression, i.e., terms such that(2)for distinct , , . The empty set and sets of length one are therefore trivially nonaveraging.Consider all possible subsets on the integers . There is one nonaveraging sequence on (), two on ( and ), four on , and so on. For example, 13 of the 16 subjects of are nonaveraging, with , , and excluded. The numbers of nonaveraging subsets on , , ... are 1, 2, 4, 7, 13, 23, 40, ... (OEIS A051013).Wróblewski (1984) showed that for infinite nonaveraging sequences,(3)

### Irrationality measure

Let be a real number, and let be the set of positive real numbers for which(1)has (at most) finitely many solutions for and integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and is no longer approximable by rational numbers,(2)where is the infimum. If the set is empty, then is defined to be , and is called a Liouville number. There are three possible regimes for nonempty :(3)where the transitional case can correspond to being either algebraic of degree or being transcendental. Showing that for an algebraic number is a difficult result for which Roth was awarded the Fields medal.The definition of irrationality measure is equivalent to the statement that if has irrationality measure , then is the smallest number such that the inequality(4)holds for any and all integers and with sufficiently large.The..

### Artin's constant

Let be a positive nonsquare integer. Then Artin conjectured that the set of all primes for which is a primitive root is infinite. Under the assumption of the generalized Riemann hypothesis, Artin's conjecture was solved by Hooley (1967; Finch 2003, p. 105).Let be not an th power for any such the squarefree part of satisfies (mod 4). Let be the set of all primes for which such an is a primitive root. Then Artin also conjectured that the density of relative to the primes is given independently of the choice of by , where(1)(OEIS A005596), and is the th prime.The significance of Artin's constant is more easily seen by describing it as the fraction of primes for which has a maximal period repeating decimal, i.e., is a full reptend prime (Conway and Guy 1996) corresponding to a cyclic number. is connected with the prime zeta function by(2)where is a Lucas number (Ribenboim 1998, Gourdon and Sebah). Wrench (1961) gave 45 digits of , and Gourdon and Sebah..

### Moving sofa problem

What is the sofa of greatest area which can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that(1)(OEIS A086118). Gerver (1992) found a sofa with larger area and provided arguments indicating that it is either optimal or close to it. The boundary of Gerver's sofa is a complicated shape composed of 18 arcs. Its area can be given by defining the constants , , , and by solving(2)(3)(4)(5)This gives(6)(7)(8)(9)Now define(10)where(11)(12)(13)Finally, define the functions(14)(15)(16)The area of the optimal sofa is then given by(17)(18)(Finch 2003).

### Hermite constants

The Hermite constant is defined for dimension as the value(1)(Le Lionnais 1983). In other words, they are given by(2)where is the maximum lattice packing density for hypersphere packing and is the content of the -hypersphere. The first few values of are 1, 4/3, 2, 4, 8, 64/3, 64, 256, ... (OEIS A007361 and A007362; Gruber and Lekkerkerker 1987, p. 518). Values for larger are not known.For sufficiently large ,(3)

### Disk covering problem

Given a unit disk, find the smallest radius required for equal disks to completely cover the unit disk. The first few such values are(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)Here, values for , 8, 9, 10 are approximate values obtained using computer experimentation by Zahn (1962).For a symmetrical arrangement with (known as the five disks problem), , where is the golden ratio. However, rather surprisingly, the radius can be slightly reduced in the general disk covering problem where symmetry is not required; this configuration is illustrated above (Friedman). Neville (1915) showed that the value is equal to , where and are solutions to(11)(12)(13)(14)These solutions can be found exactly as(15)(16)where (17)(18)are the smallest positive roots of the given polynomials, with denoting the th root of the polynomial in the ordering of the Wolfram Language. This gives (OEIS A133077) exactly as(19)where the root is the smallest positive one of the..

### Reciprocal fibonacci constant

Closed forms are known for the sums of reciprocals of even-indexed Fibonaccinumbers(1)(2)(3)(4)(5)(6)(7)(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is the golden ratio, is a q-polygamma function, and is a Lambert series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci numbers(8)(9)(10)(11)(12)(13)(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi elliptic function. Together, these give a closed form for the reciprocal Fibonacci constant of(14)(15)(16)(17)(18)(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of was formally raised by Paul Erdős and this sum was proved to be irrational by André-Jeannin (1989).Borwein..

### Equianharmonic case

The case of the Weierstrass elliptic function with invariants and . The corresponding real half-period is given by(1)(2)(OEIS A064582), known as the omega2-constant, where is the gamma function. The other half-period is then given by(3)(4)(5)(OEIS A094961 and A094962).

### Power tower

The power tower of order is defined as(1)where is Knuth up-arrow notation (Knuth 1976), which in turn is defined by(2)together with(3)(4)Rucker (1995, p. 74) uses the notation(5)and refers to this operation as "tetration."A power tower can be implemented in the WolframLanguage as PowerTower[a_, k_Integer] := Nest[Power[a, #]&, 1, k]or PowerTower[a_, k_Integer] := Power @@ Table[a, {k}]The following table gives values of for , 2, ... for small .OEIS1A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...2A0003121, 4, 27, 256, 3125, 46656, ...3A0024881, 16, 7625597484987, ...41, 65536, ...The following table gives for , 2, ... for small .OEIS1A0000121, 1, 1, 1, 1, 1, ...2A0142212, 4, 16, 65536, , ...3A0142223, 27, 7625597484987, ...44, 256, , ...Consider and let be defined as(6)(Galidakis 2004). Then for , is entire with series expansion:(7)Similarly, for , is analytic for in the domain of the principal branch of , with series expansion:(8)For..

### Euclidean algorithm

The Euclidean algorithm, also called Euclid's algorithm, is an algorithm for finding the greatest common divisor of two numbers and . The algorithm can also be defined for more general rings than just the integers . There are even principal rings which are not Euclidean but where the equivalent of the Euclidean algorithm can be defined. The algorithm for rational numbers was given in Book VII of Euclid's Elements. The algorithm for reals appeared in Book X, making it the earliest example of an integer relation algorithm (Ferguson et al. 1999).The Euclidean algorithm is an example of a P-problem whose time complexity is bounded by a quadratic function of the length of the input values (Bach and Shallit 1996).Let , then find a number which divides both and (so that and ), then also divides since(1)Similarly, find a number which divides and (so that and ), then divides since(2)Therefore, every common divisor of and is a common divisor of and , so the procedure..

### Dottie number

The Dottie number is the name given by Kaplan (2007) to the unique real root of (namely, the unique real fixed point of the cosine function), which is 0.739085... (OEIS A003957). The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator uses before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian..

### Porter's constant

Porter's constant is the constant appearing in formulasfor the efficiency of the Euclidean algorithm,(1)(2)(3)(OEIS A086237), where is the Euler-Mascheroni constant, is the Riemann zeta function, and is the Glaisher-Kinkelin constant (Knuth 1998, p. 357). The notation is generally used for this constant (Knuth 1998, p. 357, Finch 2003, pp. 156-157), though other authors use (Ustinov 2010) or (Dimitrov et al. 2000).The related constant originally considered by Porter (1975) and Knuth (1976) was denoted and , respectively, and defined by(4)(5)Knuth (1976) suggested be called the Lochs-Porter constant due to the work of Lochs (1961).

### Extreme value distribution

There are essentially three types of Fisher-Tippett extreme value distributions. The most common is the type I distribution, which are sometimes referred to as Gumbel types or just Gumbel distributions. These are distributions of an extreme order statistic for a distribution of elements .The Fisher-Tippett distribution corresponding to a maximum extreme value distribution (i.e., the distribution of the maximum ), sometimes known as the log-Weibull distribution, with location parameter and scale parameter is implemented in the Wolfram Language as ExtremeValueDistribution[alpha, beta].It has probability density functionand distribution function(1)(2)The moments can be computed directly by defining(3)(4)(5)Then the raw moments are(6)(7)(8)(9)(10)(11)where are Euler-Mascheroni integrals. Plugging in the Euler-Mascheroni integrals gives(12)(13)(14)(15)(16)where is the Euler-Mascheroni constant and is Apéry's..

### Abelian group

An Abelian group is a group for which the elements commute (i.e., for all elements and ). Abelian groups therefore correspond to groups with symmetric multiplication tables.All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.In the Wolfram Language, the function AbelianGroup[n1, n2, ...] represents the direct product of the cyclic groups of degrees , , ....No general formula is known for giving the number of nonisomorphic finite groups of a given group order. However, the number of nonisomorphic Abelian finite groups of any given group order is given by writing as(1)where the are distinct prime factors, then(2)where is the partition function, which is implemented in the Wolfram Language as FiniteAbelianGroupCount[n]...

### Magic geometric constants

Let be a compact connected subset of -dimensional Euclidean space. Gross (1964) and Stadje (1981) proved that there is a unique real number such that for all , , ..., , there exists with(1)The magic constant of is defined by(2)where(3)These numbers are also called dispersion numbers and rendezvous values. For any , Gross (1964) and Stadje (1981) proved that(4)If is a subinterval of the line and is a circular disk in the plane, then(5)If is a circle, then(6)(OEIS A060294). An expression for the magic constant of an ellipse in terms of its semimajor and semiminor axes lengths is not known. Nikolas and Yost (1988) showed that for a Reuleaux triangle (7)Denote the maximum value of in -dimensional space by . Thenwhere is the gamma function (Nikolas and Yost 1988).An unrelated quantity characteristic of a given magicsquare is also known as a magic constant...

### Landau constant

Let be the set of complex analytic functions defined on an open region containing the closure of the unit disk satisfying and . For each in , let be the supremum of all numbers such that contains a disk of radius . ThenThis constant is called the Landau constant, or the Bloch-Landau constant. Robinson (1938, unpublished) and Rademacher (1943) derived the bounds(OEIS A081760), where is the gamma function, and conjectured that the second inequality is actually an equality.

### Grossman's constant

Define the sequence , , and(1)for . The first few values are(2)(3)(4)(5)Janssen and Tjaden (1987) showed that this sequence converges for exactly one value , where (OEIS A085835), confirming Grossman's conjecture. However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. The plot above shows the first few iterations of for to 30, with odd shown in red and even shown in blue, for ranging from 0 to 1. As can be seen, the solutions alternate by parity. For each fixed , the red values go to 0, while the blue values go to some positive number.Nyerges (2000) has generalized the recurrence to the functional equation(6)

A quadratic recurrence is a recurrence equation on a sequence of numbers expressing as a second-degree polynomial in with . For example,is a quadratic recurrence equation.A quadratic recurrence equation of the formin which no cross terms are present is known as a quadraticmap.

### First category

A subset of a topological space is said to be of first category in if can be written as the countable union of subsets which are nowhere dense in , i.e., if is expressible as a unionwhere each subset is nowhere dense in . Informally, one thinks of a first category subset as a "small" subset of the host space and indeed, sets of first category are sometimes referred to as meager. Sets which are not of first category are of second category.An important distinction should be made between the above-used notion of "category" and category theory. Indeed, the notions of first and second category sets are independent of category theory.The rational numbers are of first category and the irrational numbers are of second category in with the usual topology. In general, the host space and its topology play a fundamental role in determining category. For example, the set of integers with the subset topology inherited from is (vacuously) of..

### Meager set

A subset of a topological space is said to be meager if is of first category in , i.e., if can be written as the countable union of subsets which are nowhere dense in . The terms "thin set," "meager set," and "first category" are equivalent.

### Second category

A subset of a topological space is said to be of second category in if cannot be written as the countable union of subsets which are nowhere dense in , i.e., if writing as a unionimplies that at least one subset fails to be nowhere dense in . Said differently, any set which fails to be of first category is necessarily second category and unlike sets of first category, one thinks of a second category subset as a "non-small" subset of its host space. Sets of second category are sometimes referred to as nonmeager.An important distinction should be made between the above-used notion of "category" and category theory. Indeed, the notions of first and second category sets are independent of category theory.The irrational numbers are of second category and the rational numbers are of first category in with the usual topology. In general, the host space and its topology play a fundamental role in determining category. For example, the..

### De bruijn constant

The de Bruijn constant, also called the Copson-de Bruijn constant, is the minimal constant(OEIS A113276) such that the inequalityalways holds.

### Hilbert's constants

Extend Hilbert's inequality by letting and(1)so that(2)Levin (1937) and Stečkin (1949) showed that(3)and(4)Mitrinovic et al. (1991) indicate that this constant is the best possible.

### Traveling salesman constants

Let be the smallest tour length for points in a -D hypercube. Then there exists a smallest constant such that for all optimal tours in the hypercube,(1)and a constant such that for almost all optimal tours in the hypercube,(2)These constants satisfy the inequalities(3)(4)(5)(6)(7)(8)(9)(Fejes Tóth 1940, Verblunsky 1951, Few 1955, Beardwood et al. 1959),where(10) is the gamma function, is an expression involving Struve functions and Bessel functions of the second kind,(11)(OEIS A086306; Karloff 1989), and(12)(OEIS A086307; Goddyn 1990).In the limit ,(13)(14)(15)and(16)where(17)and is the best sphere packing density in -D space (Goddyn 1990, Moran 1984, Kabatyanskii and Levenshtein 1978). Steele and Snyder (1989) proved that the limit exists.Now consider the constant(18)so(19)Nonrigorous numerical estimates give (Johnson et al. 1996) and (Percus and Martin 1996).A certain self-avoiding space-filling function..

### Fundamental unit

Let be a number field with real embeddings and imaginary embeddings and let . Then the multiplicative group of units of has the form(1)where is a primitive th root of unity, for the maximal such that there is a primitive th root of unity of . Whenever is quadratic, (unless , in which case , or , in which case ). Thus, is isomorphic to the group . The generators for are called the fundamental units of . Real quadratic number fields and imaginary cubic number fields have just one fundamental unit and imaginary quadratic number fields have no fundamental units. Observe that is the order of the torsion subgroup of and that the are determined up to a change of -basis and up to a multiplication by a root of unity.The fundamental unit of a number field is intimatelyconnected with the regulator.The fundamental units of a field generated by the algebraic number can be computed in the Wolfram Language using NumberFieldFundamentalUnits[a].In a real quadratic field,..

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# Point process (PP) neurons¶

class pp_pop_psc_delta : public Node
#include <pp_pop_psc_delta.h>

Name: pp_pop_psc_delta - Population of point process neurons with leaky integration of delta-shaped PSCs.

Description:

pp_pop_psc_delta is an effective model of a population of neurons. The N component neurons are assumed to be spike response models with escape noise, also known as generalized linear models. We follow closely the nomenclature of [1]. The component neurons are a special case of pp_psc_delta (with purely exponential rate function, no reset and no random dead_time). All neurons in the population share the inputs that it receives, and the output is the pooled spike train.

The instantaneous firing rate of the N component neurons is defined as

$rate(t) = \rho_0 * \exp( (h(t) - \eta(t))/\delta_u ),$

where h(t) is the input potential (synaptic delta currents convolved with an exponential kernel with time constant tau_m), eta(t) models the effect of refractoriness and adaptation (the neuron’s own spike train convolved with a sum of exponential kernels with time constants tau_eta), and delta_u sets the scale of the voltages.

To represent a (homogeneous) population of N inhomogeneous renewal process neurons, we can keep track of the numbers of neurons that fired a certain number of time steps in the past. These neurons will have the same value of the hazard function (instantaneous rate), and we draw a binomial random number for each of these groups. This algorithm is thus very similar to ppd_sup_generator and gamma_sup_generator, see also [2].

However, the adapting threshold eta(t) of the neurons generally makes the neurons non-renewal processes. We employ the quasi-renewal approximation [1], to be able to use the above algorithm. For the extension of [1] to coupled populations see [3].

In effect, in each simulation time step, a binomial random number for each of the groups of neurons has to be drawn, independent of the number of represented neurons. For large N, it should be much more efficient than simulating N individual pp_psc_delta models.

pp_pop_psc_delta emits spike events like other neuron models, but no more than one per time step. If several component neurons spike in the time step, the multiplicity of the spike event is set accordingly. Thus, to monitor its output, the multiplicity of the spike events has to be taken into account. Alternatively, the internal variable n_events gives the number of spikes emitted in a time step, and can be monitored using a multimeter.

EDIT Nov 2016: pp_pop_psc_delta is now deprecated, because a new and presumably much faster population model implementation is now available, see gif_pop_psc_exp.

Parameters:

The following parameters can be set in the status dictionary.

 N integer Number of represented neurons tau_m ms Membrane time constant C_m pF Capacitance of the membrane rho_0 1/s Base firing rate delta_u mV Voltage scale parameter I_e pA Constant input current tau_eta list of ms Time constants of post-spike kernel val_eta list of mV Amplitudes of exponentials in post-spike-kernel len_kernel real Post-spike kernel eta is truncated after max(tau_eta) * len_kernel

The parameters correspond to the ones of pp_psc_delta as follows.

 c_1 0.0 c_2 rho_0 c_3 1/delta_u q_sfa val_eta tau_sfa tau_eta I_e I_e dead_time simulation resolution dead_time_random False with_reset False t_ref_remaining 0.0

References:

1

Naud R, Gerstner W (2012). Coding and decoding with adapting neurons: a population approach to the peri-stimulus time histogram. PLoS Compututational Biology 8: e1002711. DOI: https://doi.org/10.1371/journal.pcbi.1002711

2

Deger M, Helias M, Boucsein C, Rotter S (2012). Statistical properties of superimposed stationary spike trains. Journal of Computational Neuroscience 32:3, 443-463. DOI: https://doi.org/10.1007/s10827-011-0362-8

3

Deger M, Schwalger T, Naud R, Gerstner W (2014). Fluctuations and information filtering in coupled populations of spiking neurons with adaptation. Physical Review E 90:6, 062704. DOI: https://doi.org/10.1103/PhysRevE.90.062704

Sends: SpikeEvent

Author: May 2014, Setareh, Deger

SeeAlso: gif_pop_psc_exp, pp_psc_delta, ppd_sup_generator, gamma_sup_generator

class pp_psc_delta : public Archiving_Node
#include <pp_psc_delta.h>

Name: pp_psc_delta - Point process neuron with leaky integration of delta-shaped PSCs.

Description:

pp_psc_delta is an implementation of a leaky integrator, where the potential jumps on each spike arrival. It produces spike stochastically, and supports spike-frequency adaptation, and other optional features.

Spikes are generated randomly according to the current value of the transfer function which operates on the membrane potential. Spike generation is followed by an optional dead time. Setting with_reset to true will reset the membrane potential after each spike.

The transfer function can be chosen to be linear, exponential or a sum of both by adjusting three parameters:

$rate = Rect[ c_1 * V' + c_2 * \exp(c_3 * V') ],$

where the effective potential $$V' = V_m - E_{sfa}$$ and $$E_{sfa}$$ is called the adaptive threshold. Here Rect means rectifier: $$Rect(x) = {x \text{ if } x>=0, 0 \text{ else}}$$ (this is necessary because negative rates are not possible).

By setting c_3 = 0, c_2 can be used as an offset spike rate for an otherwise linear rate model.

The dead time enables to include refractoriness. If dead time is 0, the number of spikes in one time step might exceed one and is drawn from the Poisson distribution accordingly. Otherwise, the probability for a spike is given by $$1 - \exp(-rate*h)$$, where h is the simulation time step. If dead_time is smaller than the simulation resolution (time step), it is internally set to the resolution.

Note that, even if non-refractory neurons are to be modeled, a small value of dead_time, like dead_time=1e-8, might be the value of choice since it uses faster uniform random numbers than dead_time=0, which draws Poisson numbers. Only for very large spike rates (> 1 spike/time_step) this will cause errors.

The model can optionally include an adaptive firing threshold. If the neuron spikes, the threshold increases and the membrane potential will take longer to reach it. Here this is implemented by subtracting the value of the adaptive threshold E_sfa from the membrane potential V_m before passing the potential to the transfer function, see also above. E_sfa jumps by q_sfa when the neuron fires a spike, and decays exponentially with the time constant tau_sfa after (see [2] or [3]). Thus, the E_sfa corresponds to the convolution of the neuron’s spike train with an exponential kernel. This adaptation kernel may also be chosen as the sum of n exponential kernels. To use this feature, q_sfa and tau_sfa have to be given as a list of n values each.

The firing of pp_psc_delta is usually not a renewal process. For example, its firing may depend on its past spikes if it has non-zero adaptation terms (q_sfa). But if so, it will depend on all its previous spikes, not just the last one so it is not a renewal process model. However, if “with_reset” is True, and all adaptation terms (q_sfa) are 0, then it will reset (“forget”) its membrane potential each time a spike is emitted, which makes it a renewal process model (where “rate” above is its hazard function, also known as conditional intensity).

pp_psc_delta may also be called a spike-response model with escape-noise [6] (for vanishing, non-random dead_time). If c_1>0 and c_2==0, the rate is a convolution of the inputs with exponential filters which is a model known as a Hawkes point process (see [4]). If instead c_1==0, then pp_psc_delta is a point process generalized linear model (with the canonical link function, and exponential input filters) (see [5,6]).

This model has been adapted from iaf_psc_delta. The default parameters are set to the mean values given in [2], which have been matched to spike-train recordings. Due to the many features of pp_psc_delta and its versatility, parameters should be set carefully and conciously.

Parameters:

The following parameters can be set in the status dictionary.

References:

1

Cardanobile S, Rotter S (2010). Multiplicatively interacting point processes and applications to neural modeling. Journal of Computational Neuroscience 28(2):267-284 DOI: https://doi.org/10.1007/s10827-009-0204-0

2

Jolivet R, Rauch A, Luescher H-R, Gerstner W. (2006). Predicting spike timing of neocortical pyramidal neurons by simple threshold models. Journal of Computational Neuroscience 21:35-49. DOI: https://doi.org/10.1007/s10827-006-7074-5

3

Pozzorini C, Naud R, Mensi S, Gerstner W (2013). Temporal whitening by power-law adaptation in neocortical neurons. Nature Neuroscience 16:942-948. (Uses a similar model of multi-timescale adaptation) DOI: https://doi.org/10.1038/nn.3431

4

Grytskyy D, Tetzlaff T, Diesmann M, Helias M (2013). A unified view on weakly correlated recurrent networks. Frontiers in Computational Neuroscience, 7:131. DOI: https://doi.org/10.3389/fncom.2013.00131

5

Deger M, Schwalger T, Naud R, Gerstner W (2014). Fluctuations and information filtering in coupled populations of spiking neurons with adaptation. Physical Review E 90:6, 062704. DOI: https://doi.org/10.1103/PhysRevE.90.062704

6

Gerstner W, Kistler WM, Naud R, Paninski L (2014). Neuronal Dynamics: From single neurons to networks and models of cognition. Cambridge University Press

Sends: SpikeEvent

format: plain


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# Infinitely many planets on a line, with Newtonian gravity

(I apologize if this question is too theoretical for this site.)

This is related to the answer here, although I came up with it independently of that. $\:$ Suppose we
have a unit mass planet at each integer point in 1-d space. $\:$ As described in that answer, the sum
of the forces acting on any particular planet is absolutely convergent. $\;\;$ Suppose we move planet_0
to point $\epsilon$, where $\: 0< \epsilon< \frac12 \:$. $\;\;$ For similar reasons, those sums will still be absolutely convergent.
Now we let Newtonian gravity apply. $\:$ What will happen?

If it's unclear what an answer might look like, you could consider the following more specific questions:

planet_0 will start out moving right, and all of the other planets will start out moving to the left.
Will there be a positive amount of time before any of them turn around?
(As opposed to, for example, each planet_n for $\: n\neq 0 \:$ turning around at time 1/|n|.)

Will there be a positive amount of time before any collisions occur?

"Obviously" (at least, I hope I'm right), planet_0 will collide with planet_1. $\:$ Will that be the first collision?

How long will it be before there are any collisions? $\:\:$ (perhaps just an approximation for small $\:\epsilon\:$)

-

The acceleration of planet number $n$ except for the planet $0$ will go like $-1/n^3$ because the shift of planet $0$ from zero to $\epsilon$ is equivalent to adding a "dipole" (a pair of positive and negative mass, relatively shifted) at the location $0$ relatively to the balanced (but unstable) uniform chain and this dipole acts with inverse cube, instead of the inverse law.

We see that indeed the planets $+1$ and $-1$ are most affected and fastest to get some acceleration. However, planet $-1$ will move to the left, away from a potential collision. Nevertheless, planet $-2$ is trying to escape from planet $-1$, although by a smaller speed, but that will be enough to guarantee that the $0-1$ collision will be the first one. Other collisions will follow. You may numerically simulate it – the problem isn't integrable even for small $\epsilon$, I think, simply because you're interested in the moments when the distance $\epsilon$ grew to a large number $O(1)$, anyway.

-

Lumo's answer is good. I would accept it. Only because I'm bored and it's easy here is a numerical simulation of 31 planets (a poor man's infinity), placed initially at rest at the integers, except the centre body is displaced upward by 0.1:

Note that, of course, the outer bodies fall toward the middle since there are only finitely many planets here. Planets 0 and 1 fall towards each other much faster than any of the other bodies (apart from the outer two), and the distance between 0 and -1 is increasing. This agrees with Lumo's prediction. The other bodies are nearly stationary, which gives confidence that the boundary conditions aren't numerically important for understanding what is happening in the middle. Shortly after this plot ends bodies start colliding and this simple simulation breaks down. So in units where everything is 1 it takes about 1 unit of time for bad things to happen. :)

-
Good job. But you could easily treat the planets at the end of your truncated sequence much more accurately, so that the collisions near the end of the world - which are artifacts of your truncation, I hope you agree - wouldn't be there etc. – Luboš Motl Mar 15 '13 at 7:04
@LubošMotl Thanks. :) Yes, of course. I gave half a thought to doing periodic boundary conditions or artificially pinning the end ones in place (please don't egg me), but I had already uploaded the picture. :) – Michael Brown Mar 15 '13 at 7:08


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# Homework Help: Finding a Limit in two variables

1. Jun 25, 2012

### hsetennis

1. The problem statement, all variables and given/known data
f(1/u,0)=1 and f(0,1/u)=-1 for all positive (integer) values of u. Prove whether or not the limit as (x,y) ->(0,0) exists.

2. Relevant equations

none.

3. The attempt at a solution

I argue that 0,0 is not in the domain of the function, but this neglects the behavior of f(x,y). So I feel like I'm missing a clue. It seems similar to the binary function: f(x)=-1 when x>0 and f(x)=-1 when x<0. But I'm unsure how to solve and prove the limit.

Edit:[ok, I'm saying that if (1/u) were to equal 0, then n would end up being infinity (not a positive integer), but again, this doesn't account for the -1/1 values of f(x,y)]

Last edited: Jun 25, 2012
2. Jun 25, 2012

### algebrat

By definition of limit, x and y would never be equal to zero.

3. Jun 25, 2012

### hsetennis

Why is that? Can you not take: {Lim as u→∞[f(1/u,0)]}=f(0,0)

I was thinking that if I evaluate: $Lim_{u\rightarrow\infty}$f(1/u,0)$\neq$$Lim_{u\rightarrow\infty}$f(0,1/u)
Because 1≠-1

Any help is appreciated

4. Jun 25, 2012

### LCKurtz

You could only claim that if f(0,0) was defined, which it isn't, and if it were and it was actually equal to that limit.

You are getting close. Remember that for a two variable function, if$$\lim_{(x,y)\rightarrow (a,b)} f(x,y)= L$$ that requires that you get $L$ as $(x,y)\rightarrow (a,b)$ along any path. What can you conclude from that?

5. Jun 25, 2012

### hsetennis

I conclude that because a limit for a point must be equivalent from any real path, $\lim_{(x,y)\rightarrow (a,b)} f(x,y)= 1$ via path $f(x,y)=(1/u,0)$ but $\lim_{(x,y)\rightarrow (a,b)} f(x,y)= -1\neq 1$ via path $f(x,y)=(0,1/u)$.

Is such a conclusion sound ?

6. Jun 25, 2012

### LCKurtz

Your original question was whether or not the limit exists. What do you conclude about that?

7. Jun 25, 2012

### hsetennis

I conclude that because a limit for a point must be equivalent from any real path and $\lim_{(x,y)\rightarrow (a,b)} f(x,y)= 1$ via path $f(x,y)=(1/u,0)$ but $\lim_{(x,y)\rightarrow (a,b)} f(x,y)= -1\neq 1$ via path $f(x,y)=(0,1/u)$, the limit at (x,y)=(0,0) does not exist.

8. Jun 25, 2012

### algebrat

I should have said x OR y are non zero.
That is, by definition of limit, (x,y)≠(0,0)

9. Jun 25, 2012

### LCKurtz

That's right.

10. Jun 25, 2012

### hsetennis

Thanks for the help !


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text:
# CAT Practice : Speed Time, Races

## Increasing Speed

Q.21: Distance between the office and the home of Alok is 100 Km. One day, he was late by an hour than the normal time to leave for the office, so he increased his speed by 5 Km/hr and reached office at the normal time. What is the changed speed of Alok?
1. 25 Km/hr
2. 20 Km/hr
3. 16 Km/hr
4. 50 Km/hr

Choice A. 25 Km/hr

## Detailed Solution

Here, again the distance is constant. However if we write the equation as we did in previous question, we get:
s x t = (s + 5) x (t - 1) = 100
In this case, we can identify other constant as the time which Alok took less to cover the distance i.e. 1 hour. So, we can write:
$\frac{100}{s} - \frac{100}{s+5} = 1$ (Difference of time take in both the cases is 1 hour)
=) 100s + 500 - 100s = s(s+5)
=) s(s+5) = 500 (One should practice to solve these kind of equations directly by factorizing 500 into 20 X 25, instead of solving the entire quadratic. However, if you are more comfortable with quadratic, keep using it. More important thing is to avoid silly mistakes once you have got the concept right)
Solving we get, s = 20 Km/hr
Therefore, Increased speed = s + 5 = 25 km/hr
We could have alternatively identified increase in speed by 5 Km/hr as a constant. Therefore, we could write the equations as:
$\frac{100}{t - 1} - \frac{100}{t} = 5$
Solving, we get t = 5 hrs, t-1 = 4hrs. Therefore, increased speed = $\frac{100}{t - 1}$ = 25 Km/hr.

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# Properties

 Modulus $1872$ Structure $$C_{12}\times C_{12}\times C_{2}\times C_{2}$$ Order $576$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(1872)

pari: g = idealstar(,1872,2)

## Character group

 sage: G.order()  pari: g.no Order = 576 sage: H.invariants()  pari: g.cyc Structure = $$C_{12}\times C_{12}\times C_{2}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1872}(703,\cdot)$, $\chi_{1872}(469,\cdot)$, $\chi_{1872}(209,\cdot)$, $\chi_{1872}(145,\cdot)$

## First 32 of 576 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character Orbit Order Primitive $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$
$$\chi_{1872}(1,\cdot)$$ 1872.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1872}(5,\cdot)$$ 1872.hh 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$i$$
$$\chi_{1872}(7,\cdot)$$ 1872.gh 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1872}(11,\cdot)$$ 1872.eo 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{1872}(17,\cdot)$$ 1872.dx 6 no $$-1$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1872}(19,\cdot)$$ 1872.ed 12 no $$1$$ $$1$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{1872}(23,\cdot)$$ 1872.dq 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1872}(25,\cdot)$$ 1872.cn 6 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$
$$\chi_{1872}(29,\cdot)$$ 1872.gt 12 yes $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{1872}(31,\cdot)$$ 1872.fg 12 no $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$
$$\chi_{1872}(35,\cdot)$$ 1872.fc 12 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{5}{12}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{1872}(37,\cdot)$$ 1872.eb 12 no $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{1872}(41,\cdot)$$ 1872.fu 12 no $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{1872}(43,\cdot)$$ 1872.er 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{1872}(47,\cdot)$$ 1872.gl 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$
$$\chi_{1872}(49,\cdot)$$ 1872.cx 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{1872}(53,\cdot)$$ 1872.z 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-i$$ $$-1$$ $$-i$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$i$$
$$\chi_{1872}(55,\cdot)$$ 1872.bw 6 no $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{1872}(59,\cdot)$$ 1872.eo 12 yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{1872}(61,\cdot)$$ 1872.eq 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{1872}(67,\cdot)$$ 1872.hk 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$
$$\chi_{1872}(71,\cdot)$$ 1872.fq 12 no $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{1872}(73,\cdot)$$ 1872.bc 4 no $$-1$$ $$1$$ $$-i$$ $$-i$$ $$i$$ $$-1$$ $$-i$$ $$-1$$ $$-1$$ $$-1$$ $$i$$ $$-1$$
$$\chi_{1872}(77,\cdot)$$ 1872.eu 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$
$$\chi_{1872}(79,\cdot)$$ 1872.cp 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$
$$\chi_{1872}(83,\cdot)$$ 1872.ej 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$
$$\chi_{1872}(85,\cdot)$$ 1872.hn 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
$$\chi_{1872}(89,\cdot)$$ 1872.fm 12 no $$1$$ $$1$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{1872}(95,\cdot)$$ 1872.cf 6 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1872}(97,\cdot)$$ 1872.fw 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1872}(101,\cdot)$$ 1872.fb 12 yes $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{1872}(103,\cdot)$$ 1872.ci 6 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$


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#### FPF-DieHard

• DDO Junkie
• Captain
• Posts: 9459
##### Beyond sucked
« on: July 23, 2016, 05:25:54 pm »
It's no wonder Paramount is scared of Axanar, this steaming pile of poo was just awful.  I saw it for free and felt ripped off.

With any luck the IP will be devalued enough that nobody will care anymore.
Who'd thunk that Star-castling was the root of all evil . . .

#### Javora

• America for Americans first.
• Commander
• Posts: 2986
• Gender:
##### Re: Beyond sucked
« Reply #1 on: July 23, 2016, 05:52:55 pm »
I view JJ Trek as a parody.  But given that, I think it was the best one JJ has put out so far... as parodies go.

#### Tumulorum Fossor

• Posts: 147
##### Re: Beyond sucked
« Reply #2 on: July 23, 2016, 06:22:55 pm »
I'm interpreting Paramount's unusually recent announcement of a fourth film in the franchise indicating they realize this isn't a strong film.

I'm interpreting the fact that they're even announcing the storyline of the fourth film at this point, Kirk meeting his father presumably through time travel, as their way of reassuring a rabid fan base that they are going to reset the timeline, so to speak, with the fourth film, and probably retire this cast (which is a shame, because I think the cast and music were the strongest things Abramstrek had going for it).  This will also keep the possibilities open for the new series/future films, etc.

I'm interpreting Abrams' comments about not recasting Chekov [with this generation of actors in the Star Trek Universe] as reinforcing my theory that the fourth film will be the final film with this group of actors/this interpretation of the Star Trek Universe, and that Kirk will probably reset everything, saving Vulcan, Chekov, his father, and his Shatnerish timeline (remember, he knows about this via Nimoyspock) by the end of the fourth film.  Heard it hear first: the fourth film initiates the reboot of the reboot, and ends Abrams' foray in the Trek universe.

#### bobfreedom

• Ensign
• Posts: 1
##### Re: Beyond sucked
« Reply #3 on: July 23, 2016, 07:00:19 pm »
Why do you guys hate beyond? I watched last night and thought it was awesome. I didn't really like the new movies or the new cast until this one, I'm totally turned around, I thought it was great.

#### Tumulorum Fossor

• Posts: 147
##### Re: Beyond sucked
« Reply #4 on: July 23, 2016, 08:35:42 pm »
Why do you guys hate beyond? I watched last night and thought it was awesome. I didn't really like the new movies or the new cast until this one, I'm totally turned around, I thought it was great.

I have NOT seen it yet. Sorry if I gave that impression. Was underwhelmed at first two, but hope springs eternal for the third. Better than no Trek at all, I think...

Anyhow, generally I don't like the wide open holes in story lines that Abrams' Trek creates.  The latest was Khan Superblood as a cure-all. But being able to transport across the galaxy also creates some troublesome story problems.  I didn't like the red matter thang.  Would also think Nero might have used his ability to go back in time to warn Romulus of the supernova. Is Abrams' Spock Vulcan or merely Asperger's?  Engineering filmed in a brewery ruined a bit of the suspension of disbelief for me.  I don't know if the Klingons could have been more unimaginatively realized, especially their ships.

That's just me. And just off the top of my head.

But the third Abramstrek film, I intend to see and has been getting good mainstream reviews. I just like the extrapolation of REAL science that Star Trek represents, and don't really feel that way about these Treks.  Cotton Candy instead of a meal.

That said, I do LOVE the casting, music, and bridge aesthetics.

#### FPF-DieHard

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• Posts: 9459
##### Re: Beyond sucked
« Reply #5 on: July 23, 2016, 09:54:38 pm »
Why do you guys hate beyond? I watched last night and thought it was awesome. I didn't really like the new movies or the new cast until this one, I'm totally turned around, I thought it was great.

Were you intoxicated?   Then maybe this sh*tburger could have been tolerable.
Who'd thunk that Star-castling was the root of all evil . . .

#### FPF-DieHard

• DDO Junkie
• Captain
• Posts: 9459
##### Re: Beyond sucked
« Reply #6 on: July 23, 2016, 09:55:44 pm »

That said, I do LOVE the casting, music, and bridge aesthetics.

Cast is great, when I see them I think "Kirk, Spock, Bones, etc . . ."

LOVED the 2009 movie, other two were abortions.
Who'd thunk that Star-castling was the root of all evil . . .

#### TAnimaL

• Lt.
• Posts: 766
• Gender:
##### Re: Beyond sucked
« Reply #7 on: July 25, 2016, 11:00:18 am »
I do believe it's possible to disagree about things and not resort to insulting people, no matter how strongly you feel on the topic.

I saw Beyond last night, and having gone in with very (very) lowered expectations found myself pleasantly surprised how much I liked it. I disliked a lot of the choices made in ST 2009 and hated "Into Darkness" as Trek and as a movie, but Beyond seemed to work better. It still feels a bit like Star Trek "Lite" and more action than sci-fi but that's hardly surprising with franchise movies these days. The cast is pretty good in their roles; altho I've never liked Quinto's Spock he was better this time and Pegg's Scotty was more than the jokey version he did previously, and the music and FX were good. I could pick at it as a filmmaker, and really nitpick it as a Trekkie, but hell, I did that with TWOK too. WIthout giving away any spoilers I saw a few plot holes  but the dialogue was pretty tight. All in all, if I had never seen the other two JJ movies I could accept this as "big screen" Trek, the same way I accepted "The Voyage Home" (which I didn't care for) or "Final Frontier" (which I personally love but have to admit it's not a good movie at all) - maybe JJ was the real problem.

IMHO, recasting TOS and starting over was inevitable. This series of movies will probably only last a few more as the cast gets expensive and box office goes down. ST and movies have always been a mixed bag, the sooner it returns to "TV"/episodic media, the better.

• should know better by now
• Posts: 52
• Gender:
##### Re: Beyond sucked
« Reply #8 on: August 12, 2016, 10:34:56 am »
I saw it (and am planning to see it again before my vacation is over).  I didn't hate it.  I don't know if I liked it, but this is the first Trek film since Nemesis that feels like a Trek film.  My gripes are ones that have been unique to the Abramsverse films (ignorance of science, bland soundtrack, etc.).

maybe JJ was the real problem.

He freely admitted during the media blitz leading up to the release of ST 09 that he wasn't a fan of the franchise at all, and had never seen any of it save for TMP (as a kid).  He didn't/doesn't understand what he was working with.  Keep him out of the directors chair and away from the script room and things might be salvageable.
« Last Edit: August 16, 2016, 10:38:17 am by Captain Spadaro »
"Don't try to be a great man; just be a man, and let history make its own judgments." - William Riker, quoting Zefram Cochrane, to Zefram Cochrane, Star Trek: First Contact

#### FPF-DieHard

• DDO Junkie
• Captain
• Posts: 9459
##### Re: Beyond sucked
« Reply #9 on: August 16, 2016, 10:29:01 am »
Who'd thunk that Star-castling was the root of all evil . . .

• should know better by now
• Posts: 52
• Gender:
##### Re: Beyond sucked
« Reply #10 on: August 16, 2016, 10:40:36 am »
http://www.inquisitr.com/3409505/its-official-star-trek-beyond-is-a-box-office-flop/http://www.inquisitr.com/3409505/its-official-star-trek-beyond-is-a-box-office-flop/

#### Don Karnage

• Lt. Commander
• Posts: 2327
• Gender:
##### Re: Beyond sucked
« Reply #28 on: September 11, 2016, 08:18:16 am »
How did they get 450 peoples in a ship that (beside the crews quarters) can carry like 50 + poeples?

How did that ship crash on that planet?

I find it weird that they where able to blowup the other ship with VHF and something like "rap" music.

#### TAnimaL

• Lt.
• Posts: 766
• Gender:
##### Re: Beyond sucked
« Reply #29 on: September 11, 2016, 10:45:05 am »
let's not quibble over how the music was used, but the Beastie Boys are hip-hop. They deserve that much respect.

I do wonder how many Enterprise crew survived the attack and escape. They beam out several groups of 20-30 and you see them crowded into the "POW" camp, but, what, maybe 100-200? More? Did they ever give a number? IIRC, there was some passing comment on how Franklin crashed; but I just wonder that if it was fixable, why Elba didn't just use that for his sneak attack on Yorktown Starbase a la Reliant in TWOK? One of the many plotholes for sure.

#### FPF-DieHard

• DDO Junkie
• Captain
• Posts: 9459
##### Re: Beyond sucked
« Reply #30 on: September 12, 2016, 02:10:23 pm »
This dumb piece of sh*t isn't worth that much thought.
Who'd thunk that Star-castling was the root of all evil . . .


text:
# Cost Planning; High-End Copiers Companies that have a high demand for making copies, both color and

1 answer below »

1. Cost Planning; High-End Copiers Companies that have a high demand for making copies, both color and black and white, often choose to lease a high-end copier that provides fast and reliable service at a reasonable cost. The lease is usually for three to five years and the cost to the user is $0.01 per page for black and white copies and typically$0.10 per page for color copies. These are the terms of your current three-year lease contract with Ricoh Company, which is up for renewal this month; the lease terms are expected to be the same for the next three years, if renewed.

Hewlett-Packard Company (H-P) has recently developed an innovative copier that can reduce the costs of color copies. The copier measures exactly how much color is used in a color copy, so that the price of the copy can be determined by the amount of color used rather than a fixed price per page, so the cost could be as low as $.070 per page for a color copy. H-P calls this a “flexible-pricing” approach. Assume for this example that the cost of the leased copier (three-year lease) is only the per page charge—the initial lease cost is negligible, and the service costs would not differ between the H-P copier and the copier you are using now. Your company is an advertising agency, Tanner and Jones, LLC, and the quality of the color copies is critical to your business success. The ability to rely on the copier at any time is also very important, as some customer requests require urgent attention. You believe that the Ricoh and H-P printers are of the same reliability, but you have not had experience with the H-P copier to be sure of the copy qual- ity. The demonstration of the H-P copier has shown as good or better copy quality, but you have not had three years’ experience with it to know what it would be like day to day. Required 1. Assume that your company is considering the lease of one of these H-P copiers, and you expect that the average price for a color copy for your company would be$0.075 because you would carefully prioritize color copy jobs and reduce the number of copies requiring a large amount of color. You expect that train- ing your copy center staff to properly use the new copier would cost about $12,400 for materials and lost work time. What is the breakeven number of color copies per year that would make you indifferent between the new H-P copier and your current copier? 2. As in part 1 above, assume you expect that your per-copy cost for color copies with the H-P copier will be$0.075, the training costs are \$12,400, and you expect to make 200,000 copies per year for the next three years. In your negotiations with Ricoh concerning the new lease and the cost of color copies, what price would you bargain for?

3. Consider your choice between the copiers within the context of the competitive and business environ- ment of Tanner and Jones. What are the issues you should consider in addition to cost of copies?

## 1 Approved Answer

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