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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
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
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
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
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
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
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
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
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
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
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
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)
American rapper (born 2006)
"Tell U Sum" and "Nosferatu". On June 19, Che released another experimental EP, which is likely a continuation of "Fully Loaded", titled "Empty Clip",
Che_(rapper)
Canadian rock band
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
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
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
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
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
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 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
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
(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
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
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)
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
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)
Country in South America
(PDF) from the original on 7 February 2025. "URUGUAY A HAVEN FOR REFUGEE SUMS; Gold Flows to 'Switzerland of Americas' Since Korean War – Foreign Trade
Uruguay
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
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)
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
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)
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
Problem in computational complexity theory
n} real numbers contains three elements that sum to zero. A generalized version, k {\displaystyle k} -SUM, asks the same question on k {\displaystyle k}
3SUM
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
Sequence of equally spaced numbers
arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series. According
Arithmetic_progression
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
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
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
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
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
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)
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
Film by Matt Bettinelli-Olpin and Tyler Gillett
set ablaze, and Grace is attacked by Charity, who she knocks out with her empty gun. Alex escapes his restraints to find Grace killed his mother Becky with
Ready_or_Not_(2019_film)
Country in West Asia
Southeast Asia. To go to work in Saudi Arabia, they must often pay large sums to recruitment agencies in their home countries. The agencies then handle
Saudi_Arabia
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
American singer-songwriter (born 1941)
so that no definitive or ideal version exists. Dylan's legacy will be the sum of thousands of performances, over many decades ... Every night, whether
Bob_Dylan
Finger-counting system
hour using 4 fingers (1-12) and the minutes using 6 fingers (0-59). 0 = empty sum 1 = 1 2 = 2 4 = 4 6 = 4 + 2 7 = 4 + 2 + 1 14 = 8 + 4 + 2 16 = 16 19 =
Finger_binary
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
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
Australian comedy-drama television series
to pay him back for sparing him. Alex convinces them that he has a large sum of money hidden away and negotiates his life for it, and they drive off to
Mr_Inbetween
South African footballer and coach (born 1977)
Hotspur ahead of the 2003–04 season, Porto finally acquired McCarthy for a sum of €7.856 million, (later re-sold part of the rights to GestiFute and First
Benni_McCarthy
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
Word processor
"averages" each sentence by adding the scores of its words and dividing the sum by the number of words in the sentence – the higher the average, the higher
Microsoft_Word
Prime Minister of the United Kingdom in 2022
it". In 2021 Truss stated that the Conservatives should "reject the zero-sum game of identity politics, [reject] the illiberalism of cancel culture, and
Liz_Truss
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)
2016 film by Martin Scorsese
(1988) and Kundun (1997), and it feels like a self-conscious masterpiece, a summing-up from a filmmaker who, at 74, may be thinking of his legacy." Joshua
Silence_(2016_film)
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
Generalizations of '"`UNIQ--math-00000000-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
American rap rock band
pop-punk, their aesthetic and subject matter is straight horrorcore. I tried to sum up their sound to a friend by calling it 'Good Charlotte fronted by Insane
Hollywood_Undead
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
Sequence of words formed by specific rules
the empty word, which is often denoted by e, ε, λ or even Λ. By concatenation one can combine two words to form a new word, whose length is the sum of
Formal_language
Mathematical concept
as the sum of a sequence of positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while
Composition_(combinatorics)
U.S. state
000 km2) for sixty million Francs (approximately $15 million). Part of this sum, $3.5 million, was used to forgive debts owed by France to the United States
Louisiana
Numeric quantity representing the center of a collection of numbers
purpose. The arithmetic mean, also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set
Mean
Political party in the United States
the rest of the western world. ... A government seemingly driven by zero-sum ideology and a commitment to reducing international co-operation is one whose
Republican Party (United States)
Republican_Party_(United_States)
Bet sizing formula for long-term growth
{\sum _{k\notin S}p_{k}}{\left(D-\sum _{k\in S}\beta _{k}\right)}}.} One may prove that R ( S o ) = 1 − ∑ i ∈ S o f i o {\displaystyle R(S^{o})=1-\sum _{i\in
Kelly_criterion
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
Jewish nationalist movement
the priority of a Jewish claim to the land of Palestine-Eretz Yisrael. In sum, Shimoni argues that most Zionists have adopted the utilitarian moral principle
Zionism
American known for bulldozer rampage (1951–2004)
scrap paper. These ranged from the Docheffs' unwillingness to pay the large sums he requested, the sanitation district's fining of him, and that the town
Marvin_Heemeyer
1978 film directed by Michael Cimino
to go home. Steven shows Mike that he has been regularly receiving large sums of cash from Saigon. Realizing that Nick is the source of these payments
The_Deer_Hunter
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
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
American multinational retail corporation operating department stores
acquired Tel Aviv-based product review start-up Aspectiva for an undisclosed sum. In May 2019, Walmart announced the launch of free one-day shipping on more
Walmart
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
British anthology television series
Arcade Fire song of the same name, with Brooker using it to refer to the empty screen of an electronic device once it is turned off. Brooker recognised
Black_Mirror
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
2025 Indian film by Milap Zaveri
him about his impending death and flinging photographs of him around an empty funeral pyre to show how much she welcomes the prospect of his demise. Meanwhile
Ek_Deewane_Ki_Deewaniyat
Vast empty spaces between filaments with few or no galaxies
~15% of the average matter density of the universe, the voids look almost empty to an observer. Voids typically have a diameter of 10 to 100 megaparsecs
Void_(astronomy)
English polymath (1642–1727)
were almost always empty. Humphrey Newton, his sizar (assistant), noted that Newton would arrive on time and, if the room was empty, he would reduce his
Isaac_Newton
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
American broadcast sports network
million over a period of several years in the 2010s, even while paying large sums for the broadcast rights to properties like the NFL, NBA and College Football
ESPN
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)
French historian (1886–1944)
example of how technology impacts society was the watermill. This can be summed up as illustrating how it was known of but little used in the classical
Marc_Bloch
A. Confidential (1997), The Green Mile (1999), Space Cowboys (2000), The Sum of All Fears (2002), I, Robot (2004), and The Artist (2011), Jurassic World:
James_Cromwell_filmography
Unidentified serial killer in London in 1888
Wilson, Colin; Odell, Robin; Gaute, J. H. H. (1988). Jack the Ripper: Summing up and Verdict. London: Corgi Books. ISBN 978-0-552-12858-2 Woods, Paul;
Jack_the_Ripper
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
American heavy metal band
the band with artists performing its songs was held. Performances included Sum 41 and a medley of "For Whom the Bell Tolls", "Enter Sandman", and "Master
Metallica
(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
French polymath (1596–1650)
matters before." His best-known philosophical statement is "cogito, ergo sum" ("I think, therefore I am", French: "Je pense, donc je suis"). Descartes
René_Descartes
Casino gambling machine
to the sides of early mechanical machines, and to the games' ability to empty players' pockets and wallets as thieves would. In the 1890s a number of
Slot_machine
2019 single by the Weeknd
the most time totaled in the Hot 100's top three, 21 weeks, equaling the sums of The Chainsmokers' "Closer" and Mark Ronson's "Uptown Funk!". For a short
Blinding_Lights
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
American rock band
lacked online presence such as a website or a Facebook page. As Thayil summed up, "We kind of had neglected our merchandise over the last decade". Eventually
Soundgarden
South Korean mass murderer (1984–2007)
bullying by his peers, which he seeks to escape from after winning a large sum of money from a casino. In his PDF mailed to NBC, Cho states: "Children of
Seung-Hui_Cho
President of Iraq from 1979 to 2003
exchange reserves, and state-controlled enterprises and diverted significant sums of Iraqi state funds through opaque financial networks under his authority
Saddam_Hussein
Major river in northeast Africa
The Nile is a major north-flowing river in northeast Africa which empties into the Mediterranean Sea. At 7,088 kilometers (4,404 mi) long, it is the longest
Nile
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
EMPTY SUM
EMPTY SUM
Boy/Male
Arabic
Empty.
Girl/Female
Biblical
Empty, temple of the head.
Biblical
den; making empty; watching
Biblical
who is empty, exhausted;free, empty, exhausted;
Boy/Male
Tamil
One who is empty, Hollow, Vain
Surname or Lastname
English
English : occupational name for a carrier, from Middle English sum(p)ter ‘(driver of a) pack animal’.
Boy/Male
Arabic, Australian, German, Greek, Kurdish
Empty; Void
Boy/Male
American, Australian, Danish, French, Jamaican, Latin
Vain; Empty; Poor; Robbed; Hollow
Girl/Female
Biblical
Den, cave, making empty.
Girl/Female
Muslim
Fruit, Summer fruit
Boy/Male
Hindu
One who is empty, Hollow, Vain
Biblical
empty; temple of the head
Surname or Lastname
English
English : nickname for a foolish or eccentric person, from a diminutive of Foll, from Old French fol ‘mad’, ‘stupid’ (Late Latin follis, originally a noun denoting any of various objects filled with air, but later transferred to vain and empty-headed notions).
Boy/Male
Biblical
Who is empty or exhausted.
Biblical
den; cave; making empty
Girl/Female
Biblical
Den, making empty, watching.
Boy/Male
Hindu, Indian
Empty
Girl/Female
Biblical
Void, empty.
Surname or Lastname
English
English : variant of Sumpter.Fort Sumter, SC, was named in honor of Thomas Sumter, known as the ‘Gamecock of the Revolution’ for the fear he inspired in the British and Tory forces and the pivotal role he played in key American victories. Born in 1734 near Charlottesville, VA, he was of Welsh heritage; his ancestors probably emigrated to America in the late 17th century.
Boy/Male
British, English, Spanish
Strong Leader; Empty
EMPTY SUM
EMPTY SUM
Female
English
Possibly a contracted form of Old High German Betlinde, BELINDA means "bright serpent" or "bright linden tree."
Boy/Male
Hindu
Boy/Male
Indian, Sanskrit
United
Boy/Male
Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lord Vishnu; Lucky
Girl/Female
English
Combination of Deana (divine) and Dina (from the valley; avenged).
Boy/Male
Tamil
As bright as the Sun
Girl/Female
Arabic, Muslim
Healthy Girl; Fertile Land without Rock and Stone
Boy/Male
Hindu, Indian, Tamil, Telugu
Learned; Shiva
Girl/Female
Gujarati, Hindu, Indian, Kannada, Mythological, Telugu
An Ancient City
Girl/Female
Hindu
EMPTY SUM
EMPTY SUM
EMPTY SUM
EMPTY SUM
EMPTY SUM
a.
Empty; frivolous.
v. i.
To become empty.
a.
Empty.
superl.
Destitute of, or lacking, sense, knowledge, or courtesy; as, empty brains; an empty coxcomb.
a.
Empty.
superl.
Destitute of reality, or real existence; unsubstantial; as, empty dreams.
v. t.
To empty.
v. t.
To deprive of the contents; to exhaust; to make void or destitute; to make vacant; to pour out; to discharge; as, to empty a vessel; to empty a well or a cistern.
v. t.
To empty.
n.
An empty box, crate, cask, etc.; -- used in commerce, esp. in transportation of freight; as, "special rates for empties."
superl.
Destitute of effect, sincerity, or sense; -- said of language; as, empty words, or threats.
n.
Love of empty of empty talk or noise.
pl.
of Empty
v. t.
To empty.
a.
To empty.
p. pr. & vb. n.
of Empty
superl.
Containing nothing; not holding or having anything within; void of contents or appropriate contents; not filled; -- said of an inclosure, as a box, room, house, etc.; as, an empty chest, room, purse, or pitcher; an empty stomach; empty shackles.
superl.
Producing nothing; unfruitful; -- said of a plant or tree; as, an empty vine.
compar.
of Empty.
imp. & p. p.
of Empty