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Set disjoint from its sumset with itself
set of all nonzero nth powers of the integers is a sum-free set. Some basic questions that have been asked about sum-free sets are: How many sum-free
Sum-free_set
Sequence of numbers avoiding sums of subsets
represented as a sum of any subset of the preceding elements of the sequence. This differs from a sum-free set, where only pairs of sums must be avoided
Sum-free_sequence
Sums vector sets A and B by adding each vector in A to each vector in B
{\textstyle (A-B)} produces a set that could be summed with B to recover A. This is defined as the complement of the Minkowski sum of the complement of A with
Minkowski_addition
Method in combinatorics
used this approach in 2002-2003 to enumerate independent sets in regular graphs, sum-free sets in abelian groups, and study a variety of other enumeration
Container_method
Theorem in combinatorics
Cameron–Erdős conjecture (now a theorem) is the statement that the number of sum-free sets contained in [ N ] = { 1 , … , N } {\displaystyle [N]=\{1,\ldots ,N\}}
Cameron–Erdős_conjecture
Theorem in arithmetic combinatorics
set A {\displaystyle A} of integers, at least one of the sets A + A {\displaystyle A+A} and A ⋅ A {\displaystyle A\cdot A} (the sets of pairwise sums
Erdős–Szemerédi_theorem
In mathematics, a module that has a basis
mathematics, a free module is a module that has a basis, that is, a generating set that is linearly independent. Every vector space is a free module, but
Free_module
+ 1 = pka pk+1b, solved by Florian Luca in 2001. The Cameron–Erdős conjecture on sum-free sets of integers, proved by Ben Green and Alexander Sapozhenko in 2003–2004
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
Type of group in abstract algebra
μk) with n = ∑ i = 1 k μ i {\textstyle n=\sum _{i=1}^{k}\mu _{i}} and μ1 ≥ μ2 ≥ ... ≥ μk, is associated the set Cμ of permutations with cycles of lengths
Symmetric_group
Mathematical term
mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a
Disjoint_union_(topology)
Algebraic curve in mathematics
repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes
Elliptic_curve
Number without repeated prime factors
encoding ∑ n = 0 ∞ a n ⋅ 2 n . {\displaystyle \sum _{n=0}^{\infty }{a_{n}}\cdot 2^{n}.} The square-free number 42 has factorization 2 × 3 × 7, or as an
Square-free_integer
Topics referred to by the same term
product Cartesian product of sets Direct product of groups Semidirect product Product of group subsets Wreath product Free product Zappa–Szép product (or
Product
Algebra of formal sums
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is
Free_abelian_group
Canadian rock band (1996–2025)
free listening on Alternative Press. The album Screaming Bloody Murder was released on March 29, 2011. On May 28, 2011, Sum 41 performed a live set for
Sum_41
Mathematical subject
progressions Schnirelmann density Shapley–Folkman lemma Sidon set Sum-free set Restricted sumset Sum-product phenomenon Green, Ben (July 2009). "Book Reviews:
Arithmetic_combinatorics
Currency of Uzbekistan
The sum (ISO code: UZS) is the official currency of the Republic of Uzbekistan. the Government replaced the Soviet Ruble with the Sum at par on 16 July
Uzbekistani_sum
In geometry, set whose intersection with every line is a single line segment
{\displaystyle \sum _{k=1}^{r}\lambda _{k}u_{k}} belongs to S. As the definition of a convex set is the case r = 2, this property characterizes convex sets. Such
Convex_set
Gluing graphs at complete subgraphs
on the notion of set sum) or possibly deleting some of the clique edges (a loosening of the definition). A k-clique-sum is a clique-sum in which both cliques
Clique-sum
Ancient Mesopotamian civilization from 3300 to 1900 BC
Sumer (/ˈsuːmər/ SOO-mər) is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging
Sumer
Number equal to the sum of its proper divisors
number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself
Perfect_number
Counting technique in combinatorics
cardinality of a set S (which may be considered as the number of elements of the set, if the set is finite). The formula expresses the fact that the sum of the
Inclusion–exclusion_principle
Index of articles associated with the same name
mathematics, a formal sum, formal series, or formal linear combination may be: In group theory, an element of a free abelian group, a sum of finitely many
Formal_sum
Mathematical theory of data types
products and sums in set theory, they are often written with the symbols Π {\displaystyle \Pi } and Σ {\displaystyle \Sigma } , respectively. Sum types are
Type_theory
British mathematician (born 1977)
progressions in sumsets, as well as a proof of the Cameron–Erdős conjecture on sum-free sets of natural numbers. He also proved an arithmetic regularity lemma for
Ben_Green_(mathematician)
2002 film by Phil Alden Robinson
The Sum of All Fears is a 2002 American spy thriller film directed by Phil Alden Robinson, based on Tom Clancy's 1991 novel of the same name. The film
The_Sum_of_All_Fears_(film)
Transformations induced by a mathematical group
decomposes as a direct sum of irreducible actions. Consider a group G acting on a set X. The orbit of an element x in X is the set of elements in X to which
Group_action
Methodic assignment of colors to elements of a graph
v} in G {\displaystyle G} , the color sum of v {\displaystyle v} , σ ( v ) {\displaystyle \sigma (v)} , is the sum of all of the adjacent vertices to v
Graph_coloring
Algebraic structure formed from a collection of algebraic structures
In mathematics, more specifically in algebra, the direct sum of a collection of abelian groups is an abelian group constructed by combining the given
Direct_sum
sum _{i}x_{i}&&=\sum _{i}\tan \theta _{i}\\[6pt]e_{2}&=\sum _{i<j}x_{i}x_{j}&&=\sum _{i<j}\tan \theta _{i}\tan \theta _{j}\\[6pt]e_{3}&=\sum
List of trigonometric identities
List_of_trigonometric_identities
Arithmetic function related to the divisors of an integer
Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. The sum of positive
Divisor_function
Number of nonzero symbols in a string
length. For the most typical case, a given set of bits, this is the number of bits set to 1, or the digit sum of the binary representation of a given number
Hamming_weight
Discrete probability distribution
from a Poisson distribution with mean λ), one would set k = ∑ i = 1 n k i , {\displaystyle k=\sum _{i=1}^{n}k_{i},} calculate an interval for μ = nλ,
Poisson_distribution
Australian mathematician
Cheryl Praeger. She was the author of several textbooks, and her work on sum-free sets became a standard reference for its subject matter. She helped found
Anne_Penfold_Street
Number in {..., –2, –1, 0, 1, 2, ...}
closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion
Integer
Mathematical group that can be generated as the set of powers of a single element
element e corresponds to 0, products correspond to sums, and powers correspond to multiples. For example, the set of complex 6th roots of unity: G = { ± 1 , ±
Cyclic_group
arithmetic progressions that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long arithmetic
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Mathematical proof expressed visually
them formulate or better understand a true proof. The statement that the sum of all positive odd numbers up to 2n − 1 is a perfect square—more specifically
Proof_without_words
the sum of reciprocals (or sum of inverses) is defined as the sum of reciprocals of some series of positive integers (counting numbers). It is a sum of
List_of_sums_of_reciprocals
conjecture to start with) Comon's conjecture Doomsday conjecture Euler's sum of powers conjecture Ganea conjecture Generalized Smith conjecture Hauptvermutung
List_of_conjectures
Theorem in number theory
) {\displaystyle \sum _{i=k+1}^{\infty }{\frac {1}{p_{i}}}<{\frac {1}{2}}\qquad (1)} For a positive integer x, let Mx denote the set of those n in {1,
Divergence of the sum of the reciprocals of the primes
Divergence_of_the_sum_of_the_reciprocals_of_the_primes
Number that is less than the sum of its proper divisors
abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the
Abundant_number
1991 thriller novel by Tom Clancy
The Sum of All Fears is a political thriller novel, written by Tom Clancy and released on August 14, 1991, as the sequel to Clear and Present Danger (1989)
The_Sum_of_All_Fears
Operation that combines groups
disjoint union plays in set theory, or that the direct sum plays in module theory. Even if the groups are commutative, their free product is not, unless
Free_product
British mathematician
5642/jhummath.202101.03, retrieved 2021-03-07 A Theorem on Maximal Sum-Free Sets in Groups (PDF), retrieved 2021-03-04 Perkins, Sarah (March 1993), "Investigating
Sarah_B._Hart
Mathematical set containing no elements
elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (the empty sum) is zero. The reason for
Empty_set
Category-theoretic construction
coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups,
Coproduct
Theorem in group theory
smallest cardinality of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained
Grushko_theorem
Algebraic structure
{\displaystyle p=\sum _{\alpha \in I}p_{\alpha }X^{\alpha },\quad q=\sum _{\beta \in J}q_{\beta }X^{\beta },} where I and J are finite sets of exponent vectors
Polynomial_ring
Integer having a non-trivial divisor
every set of consecutive composite numbers, there is an equally sized set of prime numbers, and a bijection mapping each composite in the former set to a
Composite_number
Description of the behaviour of bosons
w(n,g)=\sum _{k_{1}=0}^{n}\sum _{k_{2}=0}^{n-k_{1}}w(n-k_{1}-k_{2},g-2)=\sum _{k_{1}=0}^{n}\sum _{k_{2}=0}^{n-k_{1}}\cdots \sum _{k_{g}=0}^{n-\sum
Bose–Einstein_statistics
Means of constructing a group from two subgroups
a unique finite set S and a unique set {hi ∈ Hi : i ∈ S} such that g = Π {hi : i in S}. Direct sum Coproduct Free product Direct sum of topological groups
Direct_sum_of_groups
Problem in computer science
when each set contains at most k elements (the k-set packing problem), the intersection graph is (k+1)-claw-free. This is because, if a set intersects
Set_packing
English mathematician
University of Waterloo where he was awarded a PhD (1988) for his thesis "Sum-Free Sets and Measure Spaces" written under the supervision of Ian Peter Goulden
Neil_J._Calkin
Statement that is taken to be true
Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative
Axiom
Pair of integers related by their divisors
that the sum of the proper divisors of each is equal to the other number. That is, s(a)=b and s(b)=a, where s(n)=σ(n) − n is equal to the sum of positive
Amicable_numbers
Product of two prime numbers
) = ∑ k = 1 π ( n ) [ π ( n p k ) − k + 1 ] {\displaystyle \pi _{2}(n)=\sum _{k=1}^{\pi \left({\sqrt {n}}\right)}\left[\pi \left({\frac {n}{p_{k}}}\right)-k+1\right]}
Semiprime
Process of finding a spatial transformation that aligns two point clouds
set χ {\displaystyle {\mathcal {\chi }}} is defined as the sum of the kernel correlations of every point in the set to every other point in the set:
Point-set_registration
Periodic set of points
summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n {\displaystyle n}
Lattice_(group)
Number that cannot be written as an aliquot sum
expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their
Untouchable_number
Integer divisible by sum of its digits
Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also
Harshad_number
Numbers parameterizing ways to partition a set
\atop k}\right\}} counts set partitions of an n-element set into k parts, the sum B n = ∑ k = 0 n { n k } {\displaystyle B_{n}=\sum _{k=0}^{n}\left\{{n \atop
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Numbers whose sum of divisors is twice the number plus 1
a quasiperfect number is a natural number n for which the sum of all its divisors (the sum-of-divisors function σ ( n ) {\displaystyle \sigma (n)} ) is
Quasiperfect_number
Statistical measure of how far values spread from their average
n ∑ i = 1 n x i . {\displaystyle \mu ={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.} The variance of a set of n {\displaystyle n} equally likely values can be equivalently
Variance
Certain type of divisor of an integer
(s-k)}{\zeta (2s-k)}}=\sum _{n\geq 1}{\frac {\sigma _{k}^{*}(n)}{n^{s}}}.} Every divisor of n is unitary if and only if n is square-free. The set of all unitary
Unitary_divisor
British drag performer
Sum Ting Wong is the stage name of Bo Zeng, a drag queen from Birmingham, England. She is best known for her appearances on the first series of RuPaul's
Sum_Ting_Wong_(drag_queen)
Decision rule used for minimizing the possible loss for a worst-case scenario
maximize the minimum gain. Originally formulated for several-player zero-sum game theory, covering both the cases where players take alternate moves and
Minimax
Number of arguments required by a function
into register AX the contents of a calculated memory location that is the sum (parenthesis) of the registers BX and CX. The arithmetic mean of n real numbers
Arity
Type of computational problem
rainbow set is a conflict-free set in the special case in which GO is made of disjoint cliques, where each clique represents a color. Conflict-free set cover
Covering_problems
Type of thermodynamic potential
T+\sum _{i=1}^{k}\mu _{i}\,\mathrm {d} N_{i}-\sum _{i=1}^{n}X_{i}\,\mathrm {d} a_{i}+\cdots \\\mathrm {d} G&=V\,\mathrm {d} p-S\,\mathrm {d} T+\sum _{i=1}^{k}\mu
Gibbs_free_energy
Components of a mathematical or logical formula
&n); printf("%d\n", sum(1, n, square)); // applies sum operator to sum up squares return 0; } Given a set V of variable symbols, the set of lambda terms is
Term_(logic)
Positive integer that is an integer power of another positive integer
According to Euler, Goldbach showed (in a now-lost letter) that the sum of 1/p − 1 over the set of perfect powers p, excluding 1 and excluding duplicates, is
Perfect_power
Number whose sums of distinct divisors represent all smaller numbers
of which the sum of all divisors (including 1 and itself) is less than twice the number unless the deficiency is one. If the ordered set of all divisors
Practical_number
Number that is more than the sum of its proper divisors
for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n
Deficient_number
Operations on ordinals that extend classical arithmetic
operations on the infinite ordinals are more complicated. The sum of two well-ordered sets S and T is the ordinal representing the variant of lexicographical
Ordinal_arithmetic
Number equal to the sum of all or some of its divisors
is a natural number n equal to the sum of all or some of its proper divisors. A semiperfect number equal to the sum of all its proper divisors is a perfect
Semiperfect_number
Number that is abundant but not semiperfect
the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to
Weird_number
Mathematical set with repetitions allowed
(similar to the one for sets), stating that a finite union of finite multisets is the difference of two sums of multisets: in the first sum we consider all possible
Multiset
Number, product of consecutive integers
+ 1 ) ( n + 2 ) 3 = 2 T n {\displaystyle \sum _{k=1}^{n}k(k+1)={\frac {n(n+1)(n+2)}{3}}=2T_{n}} . The sum of the reciprocals of the positive pronic numbers
Pronic_number
Number taken as representative of a list of numbers
average is the arithmetic mean, also known as "arithmetic average" i.e. the sum divided by the count, so the "average" of the list of numbers [2, 3, 4, 7
Average
Numbers obtained by adding the two previous ones
mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci
Fibonacci_sequence
Commutative group (mathematics)
abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the
Abelian_group
American mathematician (born 1959)
including patterns and periodicity in 1-additive sequences and related sum-free sets. He became interested in constants, and studied those arising in models
Steven_R._Finch
Extremal graph theory bound on clique-free graph edges
∑ i | S i | 2 ) , {\displaystyle \sum _{i\neq j}\left|S_{i}\right|\left|S_{j}\right|={\frac {1}{2}}\left(n^{2}-\sum _{i}\left|S_{i}\right|^{2}\right)
Turán's_theorem
Any one of the distinct objects that make up a set in set theory
extension, the axiom of separation, and the union axiom (Suppes calls it the sum axiom) are needed for a more thorough understanding of "set element".
Element_of_a_set
Mathematical models of strategic interactions
| σ i = 1 {\displaystyle \sum _{i=1}^{|A|}\sigma _{i}=1} . Given an action set A {\displaystyle {\mathcal {A}}} , the set of valid strategies (also known
Game_theory
Integer having only small prime factors
although that name has other more widely used meanings, most notably for the sum of the reciprocals of the natural numbers. 5-smooth numbers are also called
Smooth_number
Blue chip stock market index
1 K t {\displaystyle I_{t}=1000\times {\frac {\sum _{i=1}^{N}Q_{i,t}\,F_{i,t}\,f_{i,t}\,C_{i,t}\,}{\sum _{i=1}^{N}Q_{i,0}\,C_{i,0}\,}}\times {\frac {1}{K_{t}}}}
CAC_40
Numbers whose aliquot sums form a cyclic sequence
the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known
Sociable_number
Study of parts and the wholes they form
the beginnings of set theory, there has been a dispute between conceiving of sets "mereologically", where a set is the mereological sum of its elements
Mereology
Class of natural numbers with many divisors
meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither set, however, is a subset of the other. All
Superior highly composite number
Superior_highly_composite_number
Numbers whose prime factors all divide the number more than once
uniquely defined by this property. The sum of the reciprocals of the powerful numbers converges. The value of this sum may be written in several other ways
Powerful_number
Free object in the category of associative algebras
denotes the free monoid on X (i.e. words on the letters Xi), ⊕ {\displaystyle \oplus } denotes the external direct sum, and Rw denotes the free R-module
Free_algebra
Type of composite integer
the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the
Smith_number
Set of elements common to all of some sets
ISBN 0-13-181629-2. Rosen, Kenneth (2007). "Basic Structures: Sets, Functions, Sequences, and Sums". Discrete Mathematics and Its Applications (Sixth ed.).
Intersection_(set_theory)
Algorithm for statistical inference on graphical models
Belief propagation, also known as sum–product message passing, is a message-passing algorithm for performing inference on graphical models, such as Bayesian
Belief_propagation
Old Babylonian poem
his demand and begs his favor to be returned. Gilgamesh, before Utu, sets Aga free to return to Kish. The poem is divided into two segments. The first
Gilgamesh_and_Aga
Set of elements in any of some sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations
Union_(set_theory)
Mathematical set of all subsets of a set
mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed
Power_set
Number divisible only by 1 and itself
than squares of natural numbers, although both sets are infinite. Brun's theorem states that the sum of the reciprocals of twin primes, ( 1 3 + 1 5 )
Prime_number
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