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Summation where the number of terms is zero
an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the
Empty_sum
Mathematical set containing no elements
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 this is that zero is the
Empty_set
Inverse of a finite difference
{\displaystyle \sum _{\nu =x}^{y}(\lambda f(\nu )+\mu g(\nu ))=\lambda \sum _{\nu =x}^{y}f(\nu )+\mu \sum _{\nu =x}^{y}g(\nu )} . Empty Sum Condition: ∑
Indefinite_sum
Figurate number
the tetrahedral numbers. Taking T 0 = 0 {\displaystyle T_{0}=0} (see empty sum), the first few terms are 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66,
Triangular_number
Result from multiplying no factors
one), just as the empty sum — the result of adding no numbers — is by convention equal to the additive identity (zero). The term empty product is most often
Empty_product
Topics referred to by the same term
contracting tensor indices Empty sum, a sum with no terms Indefinite sum, the inverse of a finite difference Kronecker sum, an operation considered a
Sum
Problem in computer science
that the sum ∑ x = i j A [ x ] {\displaystyle \sum _{x=i}^{j}A[x]} is as large as possible. (Some formulations of the problem also allow the empty subarray
Maximum_subarray_problem
Addition of several numbers or other values
addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials
Summation
Aircraft weight without fuel or payload
Empty weight (EW) is the sum of the ‘as built’ manufacturer's empty weight (MEW), plus any standard items (SI) plus any operator items (OI), EW = MEW
Operating_empty_weight
Decomposition of an integer as a sum of positive integers
3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1 The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition
Integer_partition
Canadian rock band (1996–2025)
Sum 41 was a Canadian rock band formed in Ajax, Ontario, in 1996. The band's final lineup consisted of Deryck Whibley (lead vocals, rhythm guitar, keyboards)
Sum_41
Whole number
method, such as l'Hôpital's rule. The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates
0
Topics referred to by the same term
Joyce Carol Oates Empty sum Non Zero Sumness Zerosumfree monoid This disambiguation page lists articles associated with the title Zero sum. If an internal
Zero_sum_(disambiguation)
In linear algebra, generated subspace
empty, the only possibility is n = 0, and the previous expression for span ( S ) {\displaystyle \operatorname {span} (S)} reduces to the empty sum.
Linear_span
Chinese cuisine
Dim sum (traditional Chinese: 點心; simplified Chinese: 点心; pinyin: diǎn xīn; Jyutping: dim2 sam1) is a large range of small Chinese dishes that are traditionally
Dim_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
Conditional statement which is true because the antecedent cannot be satisfied
{\displaystyle \forall x\,P(x)\equiv \neg \exists x\,\neg P(x)} Empty sum and empty product Empty function Paradoxes of material implication, especially the
Vacuous_truth
In type theory, a type with no terms
In type theory, an empty type or absurd type, typically denoted 0 {\displaystyle \mathbb {0} } is a type with no terms. Such a type may be defined as
Empty_type
1994 film by Kevin Dowling
Australia The Sum of Us at Box Office Mojo www.imdb.com https://www.imdb.com/title/tt0111309/. Retrieved 14 May 2026. {{cite web}}: Missing or empty |title=
The_Sum_of_Us_(film)
Theorem in set theory
index set I is finite. If I is empty, then the left sum is the empty sum and therefore 0, while the right product is the empty product and therefore 1.) Kőnig's
Kőnig's_theorem_(set_theory)
(origin) belongs to all conical hulls (since the summation becomes an empty sum). The conical hull of a set S {\displaystyle S} is a convex set. In fact
Conical_combination
Sums vector sets A and B by adding each vector in A to each vector in B
of radius 0 is the empty set). The Minkowski sum of a closed ball and an open ball is an open ball. More generally, the Minkowski sum of an open subset
Minkowski_addition
Data structure
indices corresponding to sums of k {\displaystyle k} distinct powers of 2 (with k = 0 {\displaystyle k=0} representing an empty sum 0). For example, level
Fenwick_tree
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
2021 studio album by Tantric
The Sum of All Things is the eighth full-length studio album released by the American rock band Tantric. The album was released on July 23, 2021 via Cleopatra
The_Sum_of_All_Things
Algebra of formal sums
combination of zero basis elements, according to the usual convention for an empty sum, and it must not be possible to find any other combination that represents
Free_abelian_group
Arithmetical puzzle game
Killer sudoku (also killer su doku, sumdoku, sum doku, sumoku, addoku, or samunanpure サムナンプレ sum-num(ber) pla(ce)) is a puzzle based on sudoku with elements
Killer_sudoku
Mathematical approximation of a function
infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its
Taylor_series
On the unique representation of integers as sums of non-consecutive Fibonacci numbers
representation of integers as sums of Fibonacci numbers. It states that every positive integer can be represented uniquely as the sum of one or more distinct
Zeckendorf's_theorem
Type of figurate number
− 1 ( 4 k + 1 ) {\displaystyle h_{n}=\sum _{k=0}^{n-1}{(4k+1)}} where the empty sum is taken to be 0. The sum of the reciprocal hexagonal numbers is
Hexagonal_number
Mathematical form
The empty product on numbers and most algebraic structures has the value of 1 (the identity element of multiplication), just like the empty sum has the
Product_(mathematics)
Repeated application of an operation to a sequence
e ) {\displaystyle (S,\star ,e)} is a monoid, then: Empty product = identity element Empty sum = 0 (in arithmetic monoids) In functional programming
Iterated_binary_operation
Number of partitions of an integer
convention p(0) = 1, as there is one way of representing 0 as a sum of positive integers (the empty sum). Furthermore p(n) = 0 when n is negative. The first few
Partition function (number theory)
Partition_function_(number_theory)
2026 film by Joe Carnahan
broken down as team members suspect each other of trying to steal a large sum of cash. It was released by Netflix on January 16, 2026, to generally favorable
The_Rip_(film)
Determinant of a product of rectangular matrices
{[n]}{m}}} is the empty set, and the formula says that det ( A B ) = 0 {\displaystyle \det(AB)=0} (its right hand side is an empty sum); indeed in this
Cauchy–Binet_formula
This is a list of episodes for the American television sitcom Empty Nest, a spinoff of The Golden Girls that originally aired on NBC from October 8, 1988
List_of_Empty_Nest_episodes
Special functions of several complex variables
topics such as number theory: "in how many ways can a number be written as a sum of squares?" physics: "how does heat flow on a toroidal ring?", "how do quantum
Theta_function
Node ordering for directed acyclic graphs
a set S; at least one such node must exist in a non-empty (finite) acyclic graph. Then: L ← Empty list that will contain the sorted elements S ← Set of
Topological_sorting
Infinite sum
of these sums exist via the completeness of the real numbers and whether series terms can be rearranged or not without changing their sums using absolute
Series_(mathematics)
Rational number equal to an integer plus 1/2
{\displaystyle n} is odd. This includes n = 0 {\displaystyle n=0} since the empty sum 0 is not half-integer. The negative of a half-integer is a half-integer
Half-integer
Arithmetic operation
Dedekind cut of rationals: a non-empty set of rationals that is closed downward and has no greatest element. The sum of real numbers a and b is defined
Addition
Data type defined by combining other types
data type is the singly linked list. A list type is a sum type with two variants, Nil for an empty list and Cons x xs for the combination of a new element
Algebraic_data_type
The Neapolitan sums (Polish: Sumy neapolitańskie) refers to a loan made in 1557 by Bona Sforza, dowager Queen of Poland and Grand Duchess of Lithuania
Neapolitan_sums
Decision problem in computer science
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers
Subset_sum_problem
Type of vector space in math
n {\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}} consisting of vectors in R3 is absolutely convergent provided that the sum of the lengths converges
Hilbert_space
Indian film series
smuggling syndicate in the late 1990s. He starts out working for a small sum, but dreams of a better life. Pushpa franchise is the Fourth-highest-grossing
Pushpa_(film_series)
Description of flat one-vertex origami
partial alternating sum α1 − α2 + α3 − ⋯ + α2i − 1 − α2i is as small as possible. Either i = 0 and the partial sum is an empty sum that is also zero, or
Kawasaki's_theorem
Special type of continuous-time Markov process
{1}{n}}+{\frac {1}{n}}\sum _{k=1}^{K-1}{\frac {1}{\prod _{j=1}^{k}\ln _{(j)}(n)}}+{\frac {c}{n\prod _{j=1}^{K}\ln _{(j)}(n)}},} where the empty sum for K = 1 {\displaystyle
Birth–death_process
Category-theoretic construction
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces
Coproduct
Mathematical theorem
For n = 0 this is just the assumed integral inequality, because the empty sum is defined as zero. Induction step from n to n + 1: Inserting the assumed
Grönwall's_inequality
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
King of Macedon from 336 to 323 BC
prosperity during Alexander's campaign in Asia. Alexander sent back vast sums from his conquest, which stimulated the economy and increased trade across
Alexander_the_Great
Financial service company
SumZero is an online community for professional investors (collectively referred to as the "buyside"), which hosts investment research, job opportunities
SumZero
Musical comedy drama February 13, 2026 1 season, 8 episodes Pending Maa Ka Sum Drama April 3, 2026 1 season, 8 episodes Pending Matka King Historical crime
List of Amazon Prime Video original programming
List_of_Amazon_Prime_Video_original_programming
Theorem in fluid mechanics
The total time T {\displaystyle T} taken to drain all the water and hence empty the vessel is found by setting h ( T ) = 0 {\displaystyle h(T)=0} and re-arranging:
Torricelli's_law
) = 0 {\displaystyle VAL(0,\mathbf {0} )=0} - since their objective sum is empty. V A L ( 1 , n ) = f ( C ( n ) ) {\displaystyle VAL(1,\mathbf {n} )=f(C(\mathbf
Multiway_number_partitioning
1978 single by Jackson Browne
"Running on Empty" is a song by American singer-songwriter Jackson Browne. It is the title track of his 1977 live album of the same name, recorded at
Running_on_Empty_(song)
Generalizations of '"`UNIQ--math-0000002F-QINU`"' in algebraic structures
(f+g)(x)=f(x)+g(x)} The empty set under set union An empty sum or empty coproduct An initial object in a category (an empty coproduct, and so an identity
Zero_element
Hong Kong actor, martial artist, and singer (born 1975)
directed by Stephen Chow: as the goalkeeper "Empty Hand" in Shaolin Soccer (2001) and as the Axe Gang boss Brother Sum in Kung Fu Hustle (2004). In 2008, due
Danny_Chan_Kwok-kwan
In geometry, set whose intersection with every line is a single line segment
finite collection of non-empty sets: Conv ( ∑ n S n ) = ∑ n Conv ( S n ) . {\displaystyle {\text{Conv}}\left(\sum _{n}S_{n}\right)=\sum _{n}{\text{Conv}}\left(S_{n}\right)
Convex_set
Basic integral in elementary calculus
finite sums of areas of vertical rectangles. For suitable functions, including every continuous function on a closed bounded interval, these Riemann sums approach
Riemann_integral
English footballer (born 1975)
the media effect, in the press release they decided to list the potential sum that Beckham could make over the five-year period from all his revenue sources
David_Beckham
scarlet red. In Poland, school grades range from 1 (worst) to 6 (best). Sum of individual polling results of the following parties: Coalition Civic
Opinion polling for the next Polish parliamentary election
Opinion_polling_for_the_next_Polish_parliamentary_election
Chinese martial artist (1893–1972)
they had several children: sons Ip Chun and Ip Ching, and daughters Ip Nga-sum (葉雅心) and Ip Nga-wun (葉雅媛).[additional citation(s) needed] During the Second
Ip_Man
(Mathematical) ring with a unique maximal ideal
then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0). If these properties hold, then
Local_ring
U.S. state
Latino origin are not distinguished between total and partial ancestry. Sum of all 1,813,885 respondents "of foreign stock" (either foreign-born or with
New_York_(state)
In computer vision and image processing
{\begin{aligned}\mu _{0}(t)&=\sum _{i=0}^{t-1}iP(i|C_{0})=\sum _{i=0}^{t-1}{\frac {iP(i)}{\omega _{0}(t)}}={\frac {\sum _{i=0}^{t-1}iP(i)}{\omega _{0}(t)}}
Otsu's_method
Criterion for the convergence of a series
{1}{n}}\sum _{i=1}^{K-1}{\frac {1}{\prod _{k=1}^{i}\ln _{(k)}(n)}}+{\frac {\rho _{n}}{n\prod _{k=1}^{K}\ln _{(k)}(n)}},\quad K\geq 1.} (The empty sum is assumed
Ratio_test
Theorem of convex functions
log ( x i ) n {\displaystyle \log \!\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)\geq {\frac {\sum _{i=1}^{n}\log \!\left(x_{i}\right)}{n}}} exp ( log
Jensen's_inequality
Far-right authoritarian political ideology
part of the larger collectivity, which should not be viewed as a numerical sum of atomized individuals. They condemned the rationalistic, liberal individualism
Fascism
American actress
'PERFECT PARTY,' NEW GURNEY COMEDY". New York Times. Retrieved April 1, 2024. "Sum of Us, The". Williamstown Theatre Festival. Retrieved April 1, 2024. Debra
Debra_Mooney
Musical artist
Music Formats: CD, digital download Track listing 家家歌 (feat. Ilid Kaolo and Suming) 看透 還是想念 漂流 HEY 相愛無事 愛人的自我修養 帶我回家 I Love You Bon Bon 無關了 家家酒 曾經美麗過
Jia_Jia_(singer)
1976 torture and murder of a woman in Miami-Dade County, Florida
money from her family. After she gave them $25 — far less than the promised sum of $200 to $300 — they tortured her and killed her. The murder was witnessed
Murder_of_Sally_Ivester
Brazilian footballer (born 1992)
5 million), with Neymar's parents confirmed to have received a €40 million sum. In the aftermath, Barcelona and Bartomeu were charged with tax fraud. Neymar
Neymar
Qarmatian attack on the Abbasids
escort ensured the safety of the pilgrims, but the authorities paid a hefty sum to the Qarmatians to be allowed through. During the following Hajj, the caravan
Sack_of_Mecca_(930)
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
American Founding Father and polymath (1706–1790)
two, three, four or five full centuries, with the resulting astronomical sums to be spent on impossibly elaborate utopian projects. Franklin, who was 79
Benjamin_Franklin
Criterion for the convergence of an infinite series
{1}{n}}+{\frac {1}{n}}\sum _{i=1}^{K-1}{\frac {1}{\prod _{k=1}^{i}\ln _{(k)}(n)}}+{\frac {\rho _{n}}{n\prod _{k=1}^{K}\ln _{(k)}(n)}}.} (The empty sum is assumed
Root_test
Function related to statistics and probability theory
log-likelihood, given no event, is 0, which corresponds to the value of the empty sum: without any data, there is no support for any models. The graph of the
Likelihood_function
Theorem in arithmetic combinatorics
{\displaystyle A+A} and A ⋅ A {\displaystyle A\cdot A} (the sets of pairwise sums and pairwise products, respectively) form a significantly larger set. More
Erdős–Szemerédi_theorem
2007 film by Ethan and Joel Coen
Moss (Brolin), a Vietnam War veteran and welder who stumbles upon a large sum of money in the desert, Anton Chigurh (Bardem), a hitman who is sent to recover
No_Country_for_Old_Men
Mathematical theory of data types
intersection. The sum type is written as either + {\displaystyle +} or ⊔ {\displaystyle \sqcup } . In programming languages, sum types may be referred
Type_theory
form a single value of more significant meaning or measurement, such as a sum, count, or max. agile software development An approach to software development
Glossary_of_computer_science
Medical device used to count cells
on the size (see figure on the right). The number of cells counted is the sum of all cells counted across squares in one chamber. The proportion of the
Hemocytometer
Vector calculus formulas relating the bulk with the boundary of a region
{\displaystyle \sum _{m}\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right)=\nabla \cdot \left(\sum _{m}p_{m}\nabla q_{m}-\sum _{m}q_{m}\nabla
Green's_identities
operations over zones, such as testing emptiness, inclusion, equality, and computing the intersection and the sum of two zones. It is, for example, used
Difference_bound_matrix
2019 American disappearance, filicide and multiple murders case
as being under Chad Daybell's psychological control. Prosecutor Rob Wood summed up the evidence that showed Lori had participated in the conspiracy to commit
Vallow-Daybell_murders
Function with variable number of arguments
an empty interface as a catch-all type. package main import "fmt" // This variadic function takes an arbitrary number of ints as arguments. func sum(nums
Variadic_function
American TV series
featuring home buyers who have won lotteries or suddenly inherited a large sum of money. It is hosted by David Bromstad. It premiered on March 7, 2015.
My_Lottery_Dream_Home
Natural number between 89 and 91
dividing a set of six objects into three non-empty subsets. 90 is also the sixteenth Perrin number from a sum of 39 and 51, whose difference is 12. The members
90_(number)
Indian Army soldier
Every year a seat is left empty for the journey to his hometown and three soldiers accompany the Baba to his home. A small sum of money is contributed by
Baba_Harbhajan_Singh
Cast of neo-Western crime media franchise
Martinez). After the break-up, she confronts Jesse at his home about a large sum of money he had left for her at her home. He tells her the money is for her
List of characters in the Breaking Bad franchise
List_of_characters_in_the_Breaking_Bad_franchise
Sheikh
seventy rupees and one bag of rice on haulers. This was quarter of the normal sum advanced to the pearlers. The boats started out for the banks late and the
Saeed bin Maktoum bin Hasher Al Maktoum
Saeed_bin_Maktoum_bin_Hasher_Al_Maktoum
Jewish noble banking family
the Rothschild family had at their disposal, this profit was an enormous sum. Nathan Mayer Rothschild started his business in Manchester in 1806 and gradually
Rothschild_family
English football player and manager (1935–2004)
time by Dave Mackay, who in 2010 received an apology and an undisclosed sum from the producers. The Clough family declined to co-operate with the film
Brian_Clough
Christian tradition about the tomb of Jesus
The empty tomb is the Christian tradition that the tomb of Jesus was found empty after his crucifixion. The canonical gospels each describe the visit of
Empty_tomb
Theorem in combinatorics
the other hand, either the "0" hole, the "n − 1" hole, or both must be empty, for it is impossible (if n > 1) for some person to shake hands with everybody
Pigeonhole_principle
US Army strategy concept (1982 – late 1990s)
were well received, especially its newly offensive orientation, which it summed up neatly with this statement: ... once political authorities commit military
AirLand_Battle
2010 film by Srđan Spasojević
shoots all but Raša, whom he kills by shoving his erect penis into his empty eye socket. A dying Vukmir praises Miloš's actions as truly worthy of a
A_Serbian_Film
EF5 tornado in North Dakota, US
collected some tens of thousands of USD. Johnson cited that one man donated a sum of $10,000 USD at the cafe. With the abundance of money, Johnson contacted
2025_Enderlin_tornado
travel, tourism, insurance
EMPTY SUM
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EMPTY SUM
travel, tourism, insurance