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...hematics]], a '''Pythagorean triple''' is a set of three [[Positive number|positive]] [[integer|integers]] which satisfy the [[equation]] (make the equation wo • all even numbers, or ...

...mat number''' is a special [[positive number|positive]] [[number]]. Fermat numbers are named after [[Pierre de Fermat]]. The formula that generates them is where ''n'' is a nonnegative integer. The first nine Fermat numbers are {{OEIS|id=A000215}}: ...

...say "number" they usually mean "real number". The official symbol for real numbers is a bold '''R''' or a blackboard bold <math>\mathbb{R}</math>. ...ven minus signs (–) so that they are labeled differently from the positive numbers. ...

A '''positive-definite matrix''' is a [[matrix (mathematics)|matrix]] with special proper ...se]] and be greater than zero. The vector chosen must be filled with real numbers. ...

...vely prime means that they do not have common divisors, other than 1. Such numbers are named after [[Robert Daniel Carmichael|Robert Carmichael]]. ...many Carmichael numbers. We can say that Carmichael numbers are composite numbers that behave a little bit like they would be a prime number. ...

...l number instead of a positive real number. This is not possible with real numbers, as there is no real number that will multiply by itself to get a negative ...''' are to '''negative numbers''' what negative numbers are to '''positive numbers'''. If I say "go east by -1 mile" it is the same as if I had said "go west ...

...but in the [[integers]] subtraction does have closure. Subtraction of two numbers can produce a negative number, which is not a natural number, but it is an ...]] are a closure of the real numbers by including square roots of negative numbers. ...

.... A number bigger than zero has two square roots: one is [[positive number|positive]] (bigger than zero) and the other is [[negative number|negative]] (smaller ...umber]]s – they are [[imaginary numbers]]. Imaginary numbers are basically numbers that cannot be square rooted and get a real result. Every [[complex number] ...

...numbers have a [[:wikt:minus|minus]] (−) in front of the number. Positive numbers have no sign or a [[:wikt:plus|plus]] (+) sign in front. Zero usually has n Zero is also an integer but it is not positive nor negative. "Integer" is another word for "whole". An integer is a [[rati ...

...]] [0;1] is a proper subset of the set of [[real number]]s or the set of [[positive number]]s. ...

...at any polynomial function <math>y=f(x)</math> on the real line that takes positive and negative values for <math>y</math> has to cross <math>x</math> axis. ...

...]. Rational numbers are all [[real numbers]], and can be [[Positive_number|positive]] or [[negative]]. A number that is not rational is called [[Irrational_num Most of the numbers that people use in everyday life are rational. These include fractions and ...

...le_function|differentiable]] at this point. These are all of your critical numbers. Plug these numbers into your original function to find the exact [[:en:Coordinate_system|coord ...

...atics]], the '''norm''' of a [[vector]] is its [[length]]. For the [[real numbers]] the only norm is the [[absolute value]]. For [[vector space|spaces]] wit # [[Positive homogeneity|Scales]] for real numbers <math>a</math>, that is <math>p(ax) = |a|p(x)</math> ...

|<center>'''Natural numbers example''' Numbers less than [[0]] (such as [[−1]]) are not natural numbers. ...

<math>\sum d^2</math> means that we take the total of all the numbers that were in the column <math>d^2</math>. This is because <math>\sum</math> == What the numbers mean == ...

...e number of a number, is the [[negative number|negative]] version of the [[positive]] number. This is what the opposite of 7 is or the opposite of -7. [[Category:Numbers]] ...

:''For the book in the Bible, see [[Numbers (Bible)]]''. ...] or [[measurement|measure]]. Depending on the field of mathematics, where numbers are used, there are different definitions: ...

In [[number theory]], the '''totient''' of a [[positive number|positive]] [[integer]] is the number of integers smaller than ''n'' which are [[copr For example, <math>\phi(8) = 4</math>, because the four numbers: 1, 3, 5 and 7 don't share any factors with 8. ...

...icab number is the smallest number that can be expressed as the sum of two positive cubes in ''n'' distinct ways. It has nothing to do with [[taxi]]s, but the *27+8=35, so 35 is the “sum of two cubes” (“[[sum]]” in this sense means “numbers that are added together”). ...

...t in a value of [[infinity]], which is itself undefined. Usually when two numbers are equal to the same thing, they are equal to each other. That is not true ...] in computer programming. Dividing [[Floating point unit|floating point]] numbers (decimals) by zero will usually result in either [[infinity]] or a special ...

...th>k</math>. Let <math>S</math> be a set with <math>|S|=n</math>. Find the numbers of <math>k</math>-tuples <math>(T_1,T_2,\dots,T_k)</math> of subsets <math> Let <math>p</math> be a prime integer and <math>a</math> be a positive integer. Show '''combinatorially''' that <math>a^p-a</math> is divisible by ...

...|300px|upright=1.2|The coloring of the complex function-values used above: positive real values are presented in red.]] ...ecause of its relation to the [[prime number theorem|distribution of prime numbers]]. It also has applications in other areas such as [[physics]], [[probabili ...

...an then be represented by a stream of Gödel numbers (also called effective numbers). [[Rogers' equivalence theorem]] states criteria for which those numberi ...s of strings as well. Given a sequence <math>x_1 x_2 x_3 ... x_n</math> of positive integers, the Gödel encoding is the product of the first n primes raised to ...

...jCNGuZ93z06kDmxXutGU8S_ADA6FgZw&cad=rja On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson], Ann. of. Math. '''39'''(1938), 350 * {{cite book | title=Prime Numbers: A Computational Perspective | author=Richard E. Crandall | coauthors=Carl ...

...something is true for all the [[natural numbers]] (all the positive whole numbers). The idea is that Prove that for all natural numbers ''n'': ...

is an algebraic equation over the rational numbers. ...s in the field of complex numbers. One can also look for solutions in real numbers. ...

...Much like hours on a [[clock]], which repeat every twelve hours, once the numbers reach a certain value, called the ''modulus'', they go back to zero. ...same [[remainder]] when both are [[division (mathematics)|divided]] by the positive integer ''n''. Congruence can be written this way: ...

...er above and to the right (if any) to find the new value. For example, the numbers 1 and 3 in the fourth row are added to make 4 in the fifth row. In general, when a binomial is raised to a positive integer power we have: ...

.../n</math>, the probability that <math>R</math> is good is larger than some positive constant. ...>p</math>, the probability that <math>R</math> is good is larger than some positive constant. <STRIKE>(Hint: Use the second moment.)</STRIKE> ...

...s is negative, and the sum of every <math>m</math> consecutive elements is positive. ...)\geq m+n-2</math>, for any <math>n,m</math> are relatively prime(the only positive integer factor that divides both of them is 1). ...

.../n</math>, the probability that <math>R</math> is good is larger than some positive constant. ...>p</math>, the probability that <math>R</math> is good is larger than some positive constant. ...

A [[number]] is called a '''perfect number''' if by adding all the positive [[divisor]]s of the number (except itself), the result is the number itself ...he number itself: 6) are 1, 2, and 3 and 1 + 2 + 3 equals 6. Other perfect numbers include 28, 496 and 8128. ...

...wo of the simplest infinite sets, that of [[natural number]]s, and that of positive [[fraction (mathematics)|fractions]]. The idea is to show that both of thes Next, the numbers are counted, as shown. Fractions which can be simplified are left out: ...

...point out the dot that is sometimes used to separate the positions of the numbers in this [[system]]. Almost everyone uses this nowadays and prefers the conv ...icates the start of a fractional part, and with one of the sign symbols + (positive) or − (negative) in front of the numerals to indicate sign. ...

...utions but you need to find the real solutions. Inequality is solving real numbers. The proper way to read inequality is from left to right, just like the oth ...A number line is one way to show the location relative to all other real numbers.<ref>{{Cite web|url=http://go.galegroup.com/ps/retrieve.do?sort=RELEVANCE&i ...

...ion is shifted down by one. This means that if n is a [[Sign_(mathematics)|positive]] [[integer]] The gamma function is defined for all [[complex numbers]]. But it is not defined for negative integers and zero. For a complex numb ...

A '''negative exponent''' is the reciprocal of a number with a positive exponent which can be mathematically represented as <math>x^{-1}=\frac{1}{x ...ent from -1. In this case the negative exponent can be separated from the positive exponent, so <math>x^{-2}=(x^{-1})^2=\left(\frac{1}{x}\right)^{2}=\frac{1} ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

...ed a [[countable set]]. But if a set has the same cardinality as the real numbers, it is called an [[uncountable set]]. * <math>\mathbb{Z}</math>, denoting the set of all [[integer]]s (whether positive, negative or zero). So <math>\mathbb{Z}</math> = {..., -2, -1, 0, 1, 2, ... ...

<p>Let <math>n\ge 1</math> be a positive integer and <math>A_1,A_2,\ldots,A_n</math> be <math>n</math> events.</p> <li>[<strong>Surjection</strong>] For positive integers <math>m\ge n</math>, prove that the probability of a uniform rando ...

...al-valued argument ''x''. (This means both the input and output are [[real numbers]].) :Proof: Let ''x''_o and ''x''₁ be real numbers. Suppose the line maps these two ''x''-values to the same ''y''-value. Thi ...

...variables, and see negative association in action by considering occupancy numbers in the balls and bins model. ...et <math>I_1,\cdots,I_k\subseteq[n]</math> be disjoint index sets for some positive integer <math>k</math>. For <math>j\in[k]</math>, let <math>f_j:\mathbb{R}^ ...

Adding the numbers on the left, subtracting <math>216.75</math> from both sides, and dividing ...ed to use conservation of momentum to figure out whether <math>v</math> is positive or negative. ...

...uniformly at random <strong>without replacement</strong> and add up their numbers. Find the mean and variance of the sum. Let <math>N</math> be an integer-valued, positive random variable and let <math>\{X_i\}_{i=1}^{\infty}</math> be indepedently ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

...variables, and see negative association in action by considering occupancy numbers in the balls and bins model. ...et <math>I_1,\cdots,I_k\subseteq[n]</math> be disjoint index sets for some positive integer <math>k</math>. For <math>j\in[k]</math>, let <math>f_j:\mathbb{R}^ ...

...of a real-valued argument ''x''. (This means both the input and output are numbers.) ...''x''&sup2; is <nowiki>[0,+∞) </nowiki>, that is, the set of non-negative numbers. (Also, this function is not an injection.) ...

...n|exponential functions]], and are useful in multiplying or dividing large numbers. Logarithms can make multiplication and division of large numbers easier because adding logarithms is the same as multiplying, and subtractin ...

...input a set <math>S</math> of <math>n</math> numbers, we want to sort the numbers in <math>S</math> in increasing order. One of the most famous algorithm for ...re all numbers in <math>S_1</math> are smaller than <math>x</math> and all numbers in <math>S_2</math> are larger than <math>x</math>; ...

...almost unavoidable in many [[computation]]s, especially when dividing two numbers in [[integer]] or fixed-point arithmetic; when computing mathematical funct ...t occurs when [[physical quantity|physical quantities]] must be encoded by numbers or [[digital signal]]s. ...

...uniformly at random <strong>without replacement</strong> and add up their numbers. Find the mean and variance of the sum. Let <math>N</math> be an integer-valued, positive random variable and let <math>\{X_i\}_{i=1}^{\infty}</math> be indepedently ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> ...

...nd method, we consider the number of choices in one step, and multiply the numbers of choices in all steps. This is done as follows. Multiplying the numbers of choices in all steps, the number of sequences of adding directed edges t ...

...") and <math>F</math> a function mapping subsets of <math>N</math> to real numbers. ...of size <math>n=|U|</math>, and each subset <math>S_i</math> is assigned a positive weight <math>w_i>0</math>, the goal is to find a <math>C\subseteq\{1,2,\ldo ...

...eild (you may think of it as the filed <math>\mathbb{Q}</math> of rational numbers, or the finite field <math>\mathbb{Z}_p</math> of integers modulo prime <ma ...ath>r \in\{0, 1\}^n</math>, thus the algorithm will return a "yes" for any positive instance (<math>AB=C</math>). ...

...h>f\not\equiv 0</math>, the algorithm may wrongly return "yes" (a '''false positive'''). But this happens only when the random <math>r</math> is a root of <mat ...an arbitrary subset of size <math>|S|=2d</math>, this probability of false positive is at most <math>1/2</math>. We can reduce it to an arbitrarily small const ...

...h>f\not\equiv 0</math>, the algorithm may wrongly return "yes" (a '''false positive'''). But this happens only when the random <math>r</math> is a root of <mat ...an arbitrary subset of size <math>|S|=2d</math>, this probability of false positive is at most <math>1/2</math>. We can reduce it to an arbitrarily small const ...

...h>f\not\equiv 0</math>, the algorithm may wrongly return "yes" (a '''false positive'''). But this happens only when the random <math>r</math> is a root of <mat ...an arbitrary subset of size <math>|S|=2d</math>, this probability of false positive is at most <math>1/2</math>. We can reduce it to an arbitrarily small const ...

...h>f\not\equiv 0</math>, the algorithm may wrongly return "yes" (a '''false positive'''). But this happens only when the random <math>r</math> is a root of <mat ...an arbitrary subset of size <math>|S|=2d</math>, this probability of false positive is at most <math>1/2</math>. We can reduce it to an arbitrarily small const ...

...h>f\not\equiv 0</math>, the algorithm may wrongly return "yes" (a '''false positive'''). But this happens only when the random <math>r</math> is a root of <mat ...an arbitrary subset of size <math>|S|=2d</math>, this probability of false positive is at most <math>1/2</math>. We can reduce it to an arbitrarily small const ...

...h>f\not\equiv 0</math>, the algorithm may wrongly return "yes" (a '''false positive'''). But this happens only when the random <math>r</math> is a root of <mat ...an arbitrary subset of size <math>|S|=2d</math>, this probability of false positive is at most <math>1/2</math>. We can reduce it to an arbitrarily small const ...

...h>f\not\equiv 0</math>, the algorithm may wrongly return "yes" (a '''false positive'''). But this happens only when the random <math>r</math> is a root of <mat ...an arbitrary subset of size <math>|S|=2d</math>, this probability of false positive is at most <math>1/2</math>. We can reduce it to an arbitrarily small const ...

...d. This is because there is no value for the natural logarithm of negative numbers. ...

<p>Let <math>n\ge 1</math> be a positive integer and <math>A_1,A_2,\ldots,A_n</math> be <math>n</math> events.</p> <li>[<strong>Surjection</strong>] For positive integers <math>m\ge n</math>, prove that the probability of a uniform rando ...

...}^n\overline{A_i}</math>. How can we guarantee this rare event occurs with positive probability? The ''Lovász Local Lemma'' provides an answer to this fundamen ..._1,A_2,\ldots,A_n</math> be a sequence of events. Suppose there exist real numbers <math>x_1,x_2,\ldots, x_n</math> such that <math>0\le x_i<1</math> and for ...

...that makes <math>A(x,r)=1</math> a '''witness''' for <math>x</math>. For a positive <math>x</math>, at least half of <math>[p]</math> are witnesses. The random Sampling <math>t</math> mutually independent random numbers from <math>[p]</math> can be quite expensive since it requires <math>\Omega ...

...") and <math>F</math> a function mapping subsets of <math>N</math> to real numbers. ...of size <math>n=|U|</math>, and each subset <math>S_i</math> is assigned a positive weight <math>w_i>0</math>, the goal is to find a <math>C\subseteq\{1,2,\ldo ...

:'''False positive''': If the true answer is "yes" then the algorithm returns "yes" with proba ...ver half running instances return "yes", and output "no" if otherwise. The numbers of "yes"s and "no"s in the <math>t</math> trials follow the Binomial distri ...

:'''False positive''': If the true answer is "yes" then the algorithm returns "yes" with proba ...ver half running instances return "yes", and output "no" if otherwise. The numbers of "yes"s and "no"s in the <math>t</math> trials follow the Binomial distri ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

...h> many <math>k</math>-compositions (the ''ordered'' sum of <math>k</math> positive integers). There are <math>{n-1\choose k-1}</math> many <math>k</math>-comp ...math>n</math>. This fact is very useful in proving theorems for partitions numbers. ...

...the independence of random variables, such as the Chernoff bounds. On the positive side, there are tools that require less independence. ...that makes <math>A(x,r)=1</math> a '''witness''' for <math>x</math>. For a positive <math>x</math>, at least half of <math>[p]</math> are witnesses. The random ...

...G(V, E)</math>, where every edge <math>e \in E</math> is associated with a positive real weight <math>w_e</math>; (b) Use Jensen's inequality to prove log-sum inequality: For nonnegative numbers, ...

...that makes <math>A(x,r)=1</math> a '''witness''' for <math>x</math>. For a positive <math>x</math>, at least half of <math>[p]</math> are witnesses. The random Sampling <math>t</math> mutually independent random numbers from <math>[p]</math> can be quite expensive since it requires <math>\Omega ...

...>(a_1,a_2,\ldots,a_n)</math> be a sequence of <math>n</math> distinct real numbers.A '''subsequence''' of <math>(a_1,a_2,\ldots,a_n)</math> is an <math>(a_{i_ :A sequence of more than <math>mn</math> different real numbers must contain either an increasing subsequence of length <math>m+1</math>, o ...

...>(a_1,a_2,\ldots,a_n)</math> be a sequence of <math>n</math> distinct real numbers.A '''subsequence''' of <math>(a_1,a_2,\ldots,a_n)</math> is an <math>(a_{i_ :A sequence of more than <math>mn</math> different real numbers must contain either an increasing subsequence of length <math>m+1</math>, o ...

...>(a_1,a_2,\ldots,a_n)</math> be a sequence of <math>n</math> distinct real numbers.A '''subsequence''' of <math>(a_1,a_2,\ldots,a_n)</math> is an <math>(a_{i_ :A sequence of more than <math>mn</math> different real numbers must contain either an increasing subsequence of length <math>m+1</math>, o ...

...>(a_1,a_2,\ldots,a_n)</math> be a sequence of <math>n</math> distinct real numbers.A '''subsequence''' of <math>(a_1,a_2,\ldots,a_n)</math> is an <math>(a_{i_ :A sequence of more than <math>mn</math> different real numbers must contain either an increasing subsequence of length <math>m+1</math>, o ...

...>(a_1,a_2,\ldots,a_n)</math> be a sequence of <math>n</math> distinct real numbers.A '''subsequence''' of <math>(a_1,a_2,\ldots,a_n)</math> is an <math>(a_{i_ :A sequence of more than <math>mn</math> different real numbers must contain either an increasing subsequence of length <math>m+1</math>, o ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...ational problem <math>f:\{0,1\}^*\to \mathbb{Z}^+</math> whose outputs are positive integers. ...ational problem <math>f:\{0,1\}^*\to \mathbb{Z}^+</math> whose outputs are positive integers. ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

Give an efficient randomized algorithm with bounded one-sided error (false positive), for testing isomorphism between rooted trees with <math>n</math> vertices ...h>x_1,x_2,\ldots,x_n</math> is an unsorted list of <math>n</math> distinct numbers. We sample (with replacement) <math>t</math> items uniformly at random from ...

...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. :Given positive integers <math>m</math> and <math>k</math>, there exists a unique represent ...

...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. :Given positive integers <math>m</math> and <math>k</math>, there exists a unique represent ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

...the independence of random variables, such as the Chernoff bounds. On the positive side, there are tools that require less independence. ...that makes <math>A(x,r)=1</math> a '''witness''' for <math>x</math>. For a positive <math>x</math>, at least half of <math>[p]</math> are witnesses. The random ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

...other other for +1. Each <math>c_i</math> gives the difference between the numbers of subjects with feature <math>i</math> in the two groups. By minimizing <m ...less than <math>\frac{k}{2}-\sqrt{2m\ln n}=\left(1-\delta\right)\mu</math> positive <math>b_i</math>'s, where <math>\delta=\frac{2\sqrt{2m\ln n}}{k}</math>. Ap ...

...eild (you may think of it as the filed <math>\mathbb{Q}</math> of rational numbers, or the finite field <math>\mathbb{Z}_p</math> of integers modulo prime <ma ...ath>r \in\{0, 1\}^n</math>, thus the algorithm will return a "yes" for any positive instance (<math>AB=C</math>). ...

...ximation ratio of 2 to being 2-universal. The proof uses the fact that odd numbers are relative prime to a power of 2. We exploit that C-multiplication (*) of unsigned u-bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing th ...

and hence this probability is positive if <math>\Pr[A_i]<1</math> for all <math>i</math>. which is positive if <math>\sum_{i=1}^m\Pr\left[A_i\right]<1</math>. This is a very loose bou ...

...that makes <math>A(x,r)=1</math> a '''witness''' for <math>x</math>. For a positive <math>x</math>, at least half of <math>[p]</math> are witnesses. The random Sampling <math>t</math> mutually independent random numbers from <math>[p]</math> can be quite expensive since it requires <math>\Omega ...

...ject, we define a probability space of objects in which the probability is positive that a randomly selected object has the required property. *If an object chosen randomly from a universe satisfies a property with positive probability, then there must be an object in the universe that satisfies th ...

...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. :Given positive integers <math>m</math> and <math>k</math>, there exists a unique represent ...

:n = [[integer|positive integer]]. [[Category:Numbers]] ...