186:
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301:
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181:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
2141:
1677:
344:
534:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
1942:
1697:
229:
2790:
4529:
623:
546:
4765:
all convey the same general idea. The differences between these are that the
Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its
2240:) with very different meanings, while others have very different names (outer product, tensor product, Kronecker product) and yet convey essentially the same idea. A brief overview of these is given in the following sections.
3936:
3833:
4442:
4370:
4713:
1660:
856:
4790:. That is, the monoidal category captures precisely the meaning of a tensor product; it captures exactly the notion of why it is that tensor products behave the way they do. More precisely, a monoidal category is the
3670:
3179:
2592:
770:
2557:
1571:
5154:(as a continuous equivalent to the product of a sequence or as the multiplicative version of the normal/standard/additive integral. The product integral is also known as "continuous product" or "multiplical".
3437:
2965:
3947:
2353:
1416:
4298:
4236:
4170:
1974:
411:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
2222:
2292:
1364:
326:
881:
5215:
Here, "formal" means that this notation has the form of a determinant, but does not strictly adhere to the definition; it is a mnemonic used to remember the expansion of the cross product.
442:
714:
1485:
795:
2469:
211:
4570:
1081:
There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different
1842:
1826:{\displaystyle \int \limits _{-\infty }^{\infty }|f(t)|\,\mathrm {d} t<\infty \qquad {\mbox{and}}\qquad \int \limits _{-\infty }^{\infty }|g(t)|\,\mathrm {d} t<\infty ,}
296:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
4106:
There is a relationship between the composition of linear functions and the product of two matrices. To see this, let r = dim(U), s = dim(V) and t = dim(W) be the (finite)
2420:
2388:
985:
2922:{\displaystyle \mathbf {u\times v} ={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\\end{vmatrix}}}
1197:
4450:
1037:
908:
44:
684:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}
599:{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}
1267:
1220:
2580:
1244:
1005:
5170:
2774:
of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.
5095:
All of the previous examples are special cases or examples of the general notion of a product. For the general treatment of the concept of a product, see
3838:
3735:
4375:
4303:
4603:
3342:{\displaystyle f(\mathbf {v} )=f\left(v_{i}\mathbf {b_{V}} ^{i}\right)=v_{i}f\left(\mathbf {b_{V}} ^{i}\right)={f^{i}}_{j}v_{i}\mathbf {b_{W}} ^{j},}
1582:
4949:
has the value of 0 (the identity element of addition). However, the concept of the empty product is more general, and requires special treatment in
2749:{\displaystyle \left(\sum _{i=1}^{n}\alpha _{i}e_{i}\right)\cdot \left(\sum _{i=1}^{n}\beta _{i}e_{i}\right)=\sum _{i=1}^{n}\alpha _{i}\,\beta _{i}}
813:
3617:
735:
5384:
2480:
1493:
3601:{\displaystyle g\circ f(\mathbf {v} )=g\left({f^{i}}_{j}v_{i}\mathbf {b_{W}} ^{j}\right)={g^{j}}_{k}{f^{i}}_{j}v_{i}\mathbf {b_{U}} ^{k}.}
901:
37:
5297:
1423:
The product of a sequence consisting of only one number is just that number itself; the product of no factors at all is known as the
1116:
4782:
that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the
3163:{\displaystyle f(t_{1}x_{1}+t_{2}x_{2})=t_{1}f(x_{1})+t_{2}f(x_{2}),\forall x_{1},x_{2}\in V,\forall t_{1},t_{2}\in \mathbb {F} .}
4745:
4091:{\displaystyle B\cdot A=\left(\sum _{j=1}^{r}a_{i,j}\cdot b_{j,k}\right)_{i=1\ldots s;k=1\ldots t}\;\in \mathbb {R} ^{s\times t}}
2318:
2136:{\displaystyle \left(\sum _{i=0}^{n}a_{i}X^{i}\right)\cdot \left(\sum _{j=0}^{m}b_{j}X^{j}\right)=\sum _{k=0}^{n+m}c_{k}X^{k}}
5355:
5330:
1371:
894:
30:
4241:
4179:
4113:
4767:
2152:
3407:
Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping
1142:
at the end of the 15th century, it became common to consider the multiplication of numbers that are either unspecified (
2263:
4922:
4810:
1319:
307:
862:
5264:
4750:
3715:
The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.
423:
5189:
5096:
5084:
4926:
695:
1455:
776:
5100:
4989:
4779:
2432:
2260:
By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map
192:
4575:
In other words: the matrix product is the description in coordinates of the composition of linear functions.
4537:
5411:
4107:
2232:
There are many different kinds of products in linear algebra. Some of these have confusingly similar names (
934:
5157:
4864:
4173:
3367:
1139:
1045:
942:
449:
1937:{\displaystyle (f*g)(t)\;:=\int \limits _{-\infty }^{\infty }f(\tau )\cdot g(t-\tau )\,\mathrm {d} \tau }
5103:
of some kind to create an object, possibly of a different kind. But also, in category theory, one has:
2249:
1151:
1135:
1075:
23:
4958:
2938:
1067:
4974:
4942:
2393:
2361:
1082:
1071:
938:
1078:, for example, is non-commutative, and so is multiplication in other algebras in general as well.
952:
5179:
4993:
4791:
2778:
2426:
2255:
5322:
4770:. The outer product is simply the Kronecker product, limited to vectors (instead of matrices).
5390:
5380:
5351:
5326:
5293:
5137:
5080:
5020:
4982:
4868:
4860:
4815:
4787:
4762:
4524:{\displaystyle B\cdot A=M_{\mathcal {W}}^{\mathcal {U}}(g\circ f)\in \mathbb {R} ^{s\times t}}
3401:
1951:
1164:
5184:
5151:
5133:
5115:
5076:
5026:
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1202:
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5001:
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3724:
2565:
2309:
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1676:
1298:
1292:
1229:
1099:
Originally, a product was and is still the result of the multiplication of two or more
1094:
1057:
1039:(indicating that the two factors should be multiplied together). When one factor is an
990:
930:
611:
332:
5405:
5108:
5061:
5054:
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4837:
4758:
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2765:
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1447:
1424:
1062:
1684:
Two functions from the reals to itself can be multiplied in another way, called the
5126:
5122:
5032:
5007:
4997:
4894:
1124:
3931:{\displaystyle B=(b_{j,k})_{j=1\ldots r;k=1\ldots t}\in \mathbb {R} ^{r\times t}}
3828:{\displaystyle A=(a_{i,j})_{i=1\ldots s;j=1\ldots r}\in \mathbb {R} ^{s\times r}}
5046:
5042:
4918:
4795:
4437:{\displaystyle B=M_{\mathcal {W}}^{\mathcal {V}}(g)\in \mathbb {R} ^{r\times t}}
4365:{\displaystyle A=M_{\mathcal {V}}^{\mathcal {U}}(f)\in \mathbb {R} ^{s\times r}}
2781:
2583:
1685:
1671:
1302:
1280:
1275:
have been introduced, which do not involve numbers at all, and have been called
1143:
1053:
949:. For example, 21 is the product of 3 and 7 (the result of multiplication), and
922:
217:
4708:{\displaystyle V\otimes W(v,m)=V(v)W(w),\forall v\in V^{*},\forall w\in W^{*},}
1655:{\displaystyle (a+N\mathbb {Z} )\cdot (b+N\mathbb {Z} )=a\cdot b+N\mathbb {Z} }
4954:
4856:
4727:
851:{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
5240:
5065:
4946:
3665:{\displaystyle g\circ f(\mathbf {v} )=\mathbf {G} \mathbf {F} \mathbf {v} ,}
1680:
The convolution of the square wave with itself gives the triangular function
1313:
1147:
801:
5394:
4945:
has the value of 1 (the identity element of multiplication), just like the
2474:
The scalar product also allows one to define an angle between two vectors:
1060:
numbers are multiplied has no bearing on the product; this is known as the
4597:, the tensor product of them can be defined as a (2,0)-tensor satisfying:
5069:
1154:). These multiplications that cannot be effectively performed are called
1074:
are multiplied, the product usually depends on the order of the factors.
765:{\displaystyle \scriptstyle {\sqrt{\scriptstyle {\text{radicand}}}}\,=\,}
720:
543:
60:
1040:
2582:-dimensional Euclidean space, the standard scalar product (called the
2552:{\displaystyle \cos \angle (v,w)={\frac {v\cdot w}{\|v\|\cdot \|w\|}}}
5075:
A few of the above products are examples of the general notion of an
4821:
3386:
1566:{\displaystyle (a+N\mathbb {Z} )+(b+N\mathbb {Z} )=a+b+N\mathbb {Z} }
1100:
5317:
An introduction to differentiable manifolds and
Riemannian geometry
4950:
1675:
1283:. Most of this article is devoted to such non-numerical products.
1128:
5290:
Functional analysis, calculus of variations and optimal control
470:
3173:
If one only considers finite dimensional vector spaces, then
4479:
4472:
4398:
4391:
4326:
4319:
4247:
4185:
4119:
629:
592:
467:
350:
235:
78:
5192: β Repeated application of an operation to a sequence
4741:
For infinite-dimensional vector spaces, one also has the:
1968:
The product of two polynomials is given by the following:
1954:, convolution becomes point-wise function multiplication.
2348:{\displaystyle \cdot :V\times V\rightarrow \mathbb {R} }
5175:
Pages displaying short descriptions of redirect targets
1411:{\displaystyle 1\cdot 4\cdot 9\cdot 16\cdot 25\cdot 36}
5083:; the rest are describable by the general notion of a
4293:{\displaystyle {\mathcal {W}}=\{w_{1},\ldots ,w_{t}\}}
4231:{\displaystyle {\mathcal {V}}=\{v_{1},\ldots ,v_{s}\}}
4165:{\displaystyle {\mathcal {U}}=\{u_{1},\ldots ,u_{r}\}}
2813:
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75:
1271:
Later and essentially from the 19th century on, new
4806:Other kinds of products in linear algebra include:
2217:{\displaystyle c_{k}=\sum _{i+j=k}a_{i}\cdot b_{j}}
808:
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730:
719:
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295:
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5321:(2nd ed.). Orlando: Academic Press. p.
5140:, which captures the essence of a tensor product.
4102:Composition of linear functions as matrix product
2287:{\displaystyle \mathbb {R} \times V\rightarrow V}
5350:(2nd ed.). New York: Springer. p. 13.
4444:be the matrix representing g : V β W. Then
1359:{\displaystyle \textstyle \prod _{i=1}^{6}i^{2}}
321:{\displaystyle \scriptstyle {\text{difference}}}
5118:, a category that is the product of categories.
4372:be the matrix representing f : U β V and
2777:The cross product can also be expressed as the
1947:is well defined and is called the convolution.
876:{\displaystyle \scriptstyle {\text{logarithm}}}
4778:In general, whenever one has two mathematical
4774:The class of all objects with a tensor product
2943:A linear mapping can be defined as a function
4300:be a basis of W. In terms of this basis, let
902:
437:{\displaystyle \scriptstyle {\text{product}}}
38:
8:
5171:Deligne tensor product of abelian categories
4287:
4255:
4225:
4193:
4159:
4127:
2543:
2537:
2531:
2525:
2442:
2436:
1308:(in analogy to the use of the capital Sigma
4921:) that have Cartesian products is called a
4589:Given two finite dimensional vector spaces
1293:Multiplication Β§ Product of a sequence
709:{\displaystyle \scriptstyle {\text{power}}}
4066:
2425:From the scalar product, one can define a
1870:
1480:{\displaystyle \mathbb {Z} /N\mathbb {Z} }
909:
895:
790:{\displaystyle \scriptstyle {\text{root}}}
56:
45:
31:
18:
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2511:
2482:
2464:{\displaystyle \|v\|:={\sqrt {v\cdot v}}}
2448:
2434:
2395:
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2018:
2008:
1998:
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206:{\displaystyle \scriptstyle {\text{sum}}}
197:
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137:
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116:
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95:
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74:
5068:(sometimes called the wedge product) in
4969:Products over other algebraic structures
4875:) from multiple sets. That is, for sets
5228:
5208:
4929:. Sets are an example of such objects.
4565:{\displaystyle g\circ f:U\rightarrow W}
1301:is denoted by the capital Greek letter
5292:. Dordrecht: Springer. pp. 9β10.
5099:, which describes how to combine two
1316:symbol). For example, the expression
7:
5234:
5232:
4917:The class of all things (of a given
2760:Cross product in 3-dimensional space
2358:with the following conditions, that
4683:
4661:
4110:of vector spaces U, V and W. Let
3123:
3088:
2490:
1927:
1887:
1882:
1817:
1807:
1778:
1773:
1753:
1743:
1714:
1709:
14:
4828:Wedge product or exterior product
1117:fundamental theorem of arithmetic
5265:"Summation and Product Notation"
4802:Other products in linear algebra
4746:Tensor product of Hilbert spaces
3655:
3650:
3645:
3634:
3583:
3579:
3508:
3504:
3454:
3324:
3320:
3269:
3265:
3225:
3221:
3190:
2831:
2824:
2817:
2801:
2795:
4579:Tensor product of vector spaces
1764:
1756:
5198: β Arithmetical operation
5160:, a theory of elliptic curves.
4798:) that have a tensor product.
4655:
4649:
4643:
4637:
4628:
4616:
4556:
4497:
4485:
4410:
4404:
4338:
4332:
3868:
3848:
3765:
3745:
3638:
3630:
3458:
3450:
3194:
3186:
3082:
3069:
3050:
3037:
3018:
2972:
2933:Composition of linear mappings
2505:
2493:
2337:
2278:
1922:
1910:
1901:
1895:
1867:
1861:
1858:
1846:
1801:
1797:
1791:
1784:
1737:
1733:
1727:
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1609:
1603:
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1537:
1520:
1514:
1497:
1026:
1014:
974:
962:
839:
831:
1:
5346:Moschovakis, Yiannis (2006).
5173: β Belgian mathematician
5027:product of topological spaces
4973:Products over other kinds of
3419:, and let the linear mapping
3402:Einstein summation convention
2415:{\displaystyle 0\not =v\in V}
2383:{\displaystyle v\cdot v>0}
1452:Residue classes in the rings
1297:The product operator for the
5313:Boothby, William M. (1986).
1070:or members of various other
980:{\displaystyle x\cdot (2+x)}
5379:. Stuttgart: B.G. Teubner.
5091:Products in category theory
4927:Cartesian closed categories
4534:is the matrix representing
1442:Residue classes of integers
945:) to be multiplied, called
5428:
5085:product in category theory
4794:of all things (of a given
4751:Topological tensor product
4582:
3941:their product is given by
3722:
2947:between two vector spaces
2936:
2763:
2301:
2253:
2247:
2228:Products in linear algebra
1961:
1669:
1445:
1438:have a product operation.
1366:is another way of writing
1290:
1131:the order of the factors.
1092:
1043:, the product is called a
5190:Iterated binary operation
5097:product (category theory)
5288:Clarke, Francis (2013).
4990:direct product of groups
4883:, the Cartesian product
1066:of multiplication. When
3719:Product of two matrices
1192:{\displaystyle ax+b=0,}
1134:With the introduction
5375:Jarchow, Hans (1981).
5269:math.illinoisstate.edu
5158:Complex multiplication
4865:mathematical operation
4709:
4566:
4525:
4438:
4366:
4294:
4232:
4166:
4092:
3989:
3932:
3829:
3666:
3602:
3343:
3164:
2955:with underlying field
2923:
2750:
2724:
2675:
2621:
2576:
2553:
2465:
2416:
2384:
2349:
2288:
2218:
2137:
2112:
2057:
2003:
1938:
1891:
1827:
1782:
1718:
1681:
1656:
1567:
1481:
1412:
1360:
1344:
1263:
1240:
1216:
1193:
1158:. For example, in the
1089:Product of two numbers
1033:
1001:
981:
877:
852:
791:
766:
710:
685:
600:
535:
438:
412:
322:
297:
207:
182:
5377:Locally convex spaces
5245:mathworld.wolfram.com
5057:in algebraic topology
4710:
4567:
4526:
4439:
4367:
4295:
4238:be a basis of V and
4233:
4167:
4093:
3969:
3933:
3830:
3667:
3603:
3344:
3165:
2924:
2751:
2704:
2655:
2601:
2577:
2554:
2466:
2417:
2385:
2350:
2289:
2254:Further information:
2250:Scalar multiplication
2244:Scalar multiplication
2219:
2138:
2086:
2037:
1983:
1939:
1874:
1828:
1765:
1701:
1679:
1657:
1568:
1482:
1427:, and is equal to 1.
1413:
1361:
1324:
1299:product of a sequence
1287:Product of a sequence
1264:
1241:
1217:
1194:
1136:mathematical notation
1076:Matrix multiplication
1034:
1032:{\displaystyle (2+x)}
1002:
982:
878:
853:
792:
767:
711:
686:
601:
536:
439:
413:
323:
298:
208:
183:
24:Arithmetic operations
4975:algebraic structures
4959:computer programming
4943:algebraic structures
4941:on numbers and most
4925:. Many of these are
4768:intrinsic definition
4757:The tensor product,
4604:
4538:
4451:
4376:
4304:
4242:
4180:
4114:
3948:
3839:
3736:
3618:
3438:
3180:
2966:
2939:Function composition
2791:
2593:
2566:
2481:
2433:
2394:
2362:
2319:
2312:is a bi-linear map:
2264:
2153:
1975:
1843:
1698:
1583:
1494:
1456:
1372:
1320:
1279:; for example, the
1250:
1230:
1203:
1165:
1083:algebraic structures
1072:associative algebras
1011:
991:
953:
863:
814:
777:
736:
696:
624:
547:
462:
424:
345:
308:
230:
193:
73:
5348:Notes on set theory
5239:Weisstein, Eric W.
4484:
4403:
4331:
3729:Given two matrices
3683:-column element of
3611:Or in matrix form:
3431:. Then one can get
1226:of the coefficient
1150:), or to be found (
1052:The order in which
5180:Indefinite product
4994:semidirect product
4923:Cartesian category
4893:is the set of all
4705:
4562:
4521:
4466:
4434:
4385:
4362:
4313:
4290:
4228:
4162:
4088:
3928:
3825:
3662:
3598:
3339:
3160:
2919:
2913:
2746:
2572:
2549:
2461:
2412:
2380:
2345:
2284:
2256:Scaling (geometry)
2214:
2190:
2133:
1934:
1836:then the integral
1823:
1762:
1682:
1652:
1563:
1477:
1408:
1356:
1355:
1262:{\displaystyle x.}
1259:
1236:
1215:{\displaystyle ax}
1212:
1189:
1119:states that every
1107:is the product of
1029:
997:
987:is the product of
977:
873:
872:
848:
847:
787:
786:
762:
761:
748:
706:
705:
681:
680:
669:
666:
648:
596:
595:
590:
587:
576:
565:
531:
530:
519:
516:
513:
506:
492:
489:
482:
434:
433:
408:
407:
396:
393:
372:
318:
317:
293:
292:
281:
278:
257:
203:
202:
178:
177:
166:
163:
142:
121:
100:
5386:978-3-519-02224-4
5138:monoidal category
5081:monoidal category
5021:product of ideals
4983:Cartesian product
4861:Cartesian product
4851:Cartesian product
4816:Kronecker product
4788:monoidal category
4763:Kronecker product
2575:{\displaystyle n}
2547:
2459:
2169:
1952:Fourier transform
1761:
1436:Commutative rings
1431:Commutative rings
1273:binary operations
1239:{\displaystyle a}
1127:, that is unique
1000:{\displaystyle x}
929:is the result of
919:
918:
886:
885:
870:
837:
825:
784:
754:
752:
746:
703:
663:
658:
645:
640:
585:
574:
563:
514:
511:
504:
490:
487:
480:
431:
391:
381:
370:
360:
315:
276:
266:
255:
245:
200:
161:
151:
140:
130:
119:
109:
98:
88:
16:Mathematical form
5419:
5398:
5362:
5361:
5343:
5337:
5336:
5320:
5310:
5304:
5303:
5285:
5279:
5278:
5276:
5275:
5261:
5255:
5254:
5252:
5251:
5236:
5216:
5213:
5185:Infinite product
5176:
5152:product integral
5134:internal product
5116:product category
5077:internal product
5037:random variables
5015:product of rings
4913:
4906:
4899:
4892:
4867:which returns a
4833:Interior product
4811:Hadamard product
4784:internal product
4714:
4712:
4711:
4706:
4701:
4700:
4679:
4678:
4571:
4569:
4568:
4563:
4530:
4528:
4527:
4522:
4520:
4519:
4508:
4483:
4482:
4476:
4475:
4443:
4441:
4440:
4435:
4433:
4432:
4421:
4402:
4401:
4395:
4394:
4371:
4369:
4368:
4363:
4361:
4360:
4349:
4330:
4329:
4323:
4322:
4299:
4297:
4296:
4291:
4286:
4285:
4267:
4266:
4251:
4250:
4237:
4235:
4234:
4229:
4224:
4223:
4205:
4204:
4189:
4188:
4171:
4169:
4168:
4163:
4158:
4157:
4139:
4138:
4123:
4122:
4097:
4095:
4094:
4089:
4087:
4086:
4075:
4065:
4064:
4029:
4025:
4024:
4023:
4005:
4004:
3988:
3983:
3937:
3935:
3934:
3929:
3927:
3926:
3915:
3906:
3905:
3866:
3865:
3834:
3832:
3831:
3826:
3824:
3823:
3812:
3803:
3802:
3763:
3762:
3671:
3669:
3668:
3663:
3658:
3653:
3648:
3637:
3607:
3605:
3604:
3599:
3594:
3593:
3588:
3587:
3586:
3575:
3574:
3565:
3564:
3559:
3558:
3557:
3546:
3545:
3540:
3539:
3538:
3524:
3520:
3519:
3518:
3513:
3512:
3511:
3500:
3499:
3490:
3489:
3484:
3483:
3482:
3457:
3348:
3346:
3345:
3340:
3335:
3334:
3329:
3328:
3327:
3316:
3315:
3306:
3305:
3300:
3299:
3298:
3284:
3280:
3279:
3274:
3273:
3272:
3254:
3253:
3241:
3237:
3236:
3235:
3230:
3229:
3228:
3217:
3216:
3193:
3169:
3167:
3166:
3161:
3156:
3148:
3147:
3135:
3134:
3113:
3112:
3100:
3099:
3081:
3080:
3065:
3064:
3049:
3048:
3033:
3032:
3017:
3016:
3007:
3006:
2994:
2993:
2984:
2983:
2928:
2926:
2925:
2920:
2918:
2917:
2910:
2909:
2898:
2897:
2886:
2885:
2872:
2871:
2860:
2859:
2848:
2847:
2834:
2827:
2820:
2804:
2755:
2753:
2752:
2747:
2745:
2744:
2734:
2733:
2723:
2718:
2700:
2696:
2695:
2694:
2685:
2684:
2674:
2669:
2646:
2642:
2641:
2640:
2631:
2630:
2620:
2615:
2581:
2579:
2578:
2573:
2558:
2556:
2555:
2550:
2548:
2546:
2523:
2512:
2470:
2468:
2467:
2462:
2460:
2449:
2421:
2419:
2418:
2413:
2389:
2387:
2386:
2381:
2354:
2352:
2351:
2346:
2344:
2293:
2291:
2290:
2285:
2271:
2238:exterior product
2223:
2221:
2220:
2215:
2213:
2212:
2200:
2199:
2189:
2165:
2164:
2142:
2140:
2139:
2134:
2132:
2131:
2122:
2121:
2111:
2100:
2082:
2078:
2077:
2076:
2067:
2066:
2056:
2051:
2028:
2024:
2023:
2022:
2013:
2012:
2002:
1997:
1958:Polynomial rings
1943:
1941:
1940:
1935:
1930:
1890:
1885:
1832:
1830:
1829:
1824:
1810:
1804:
1787:
1781:
1776:
1763:
1759:
1746:
1740:
1723:
1717:
1712:
1661:
1659:
1658:
1653:
1651:
1625:
1602:
1576:and multiplied:
1572:
1570:
1569:
1564:
1562:
1536:
1513:
1486:
1484:
1483:
1478:
1476:
1468:
1463:
1419:
1417:
1415:
1414:
1409:
1365:
1363:
1362:
1357:
1354:
1353:
1343:
1338:
1311:
1307:
1268:
1266:
1265:
1260:
1246:and the unknown
1245:
1243:
1242:
1237:
1221:
1219:
1218:
1213:
1198:
1196:
1195:
1190:
1123:is a product of
1121:composite number
1114:
1110:
1106:
1038:
1036:
1035:
1030:
1006:
1004:
1003:
998:
986:
984:
983:
978:
937:that identifies
911:
904:
897:
882:
880:
879:
874:
871:
868:
857:
855:
854:
849:
838:
835:
827:
826:
823:
796:
794:
793:
788:
785:
782:
771:
769:
768:
763:
755:
753:
750:
747:
744:
741:
715:
713:
712:
707:
704:
701:
690:
688:
687:
682:
674:
670:
665:
664:
661:
659:
656:
647:
646:
643:
641:
638:
605:
603:
602:
597:
594:
591:
586:
583:
575:
572:
564:
561:
540:
538:
537:
532:
524:
520:
515:
512:
509:
505:
502:
499:
491:
488:
485:
481:
478:
475:
443:
441:
440:
435:
432:
429:
417:
415:
414:
409:
401:
397:
392:
389:
382:
379:
371:
368:
361:
358:
327:
325:
324:
319:
316:
313:
302:
300:
299:
294:
286:
282:
277:
274:
267:
264:
256:
253:
246:
243:
212:
210:
209:
204:
201:
198:
187:
185:
184:
179:
171:
167:
162:
159:
152:
149:
141:
138:
131:
128:
120:
117:
110:
107:
99:
96:
89:
86:
57:
47:
40:
33:
26:
19:
5427:
5426:
5422:
5421:
5420:
5418:
5417:
5416:
5402:
5401:
5387:
5374:
5371:
5366:
5365:
5358:
5345:
5344:
5340:
5333:
5312:
5311:
5307:
5300:
5287:
5286:
5282:
5273:
5271:
5263:
5262:
5258:
5249:
5247:
5238:
5237:
5230:
5225:
5220:
5219:
5214:
5210:
5205:
5174:
5167:
5147:
5093:
4992:, and also the
4971:
4963:category theory
4935:
4908:
4901:
4897:
4884:
4853:
4820:The product of
4804:
4776:
4692:
4670:
4602:
4601:
4587:
4581:
4536:
4535:
4503:
4449:
4448:
4416:
4374:
4373:
4344:
4302:
4301:
4277:
4258:
4240:
4239:
4215:
4196:
4178:
4177:
4149:
4130:
4112:
4111:
4104:
4070:
4009:
3990:
3968:
3964:
3963:
3946:
3945:
3910:
3867:
3851:
3837:
3836:
3807:
3764:
3748:
3734:
3733:
3727:
3721:
3710:
3706:
3699:
3692:
3616:
3615:
3578:
3576:
3566:
3549:
3547:
3530:
3528:
3503:
3501:
3491:
3474:
3472:
3471:
3467:
3436:
3435:
3398:
3383:
3364:
3357:
3319:
3317:
3307:
3290:
3288:
3264:
3262:
3258:
3245:
3220:
3218:
3208:
3207:
3203:
3178:
3177:
3139:
3126:
3104:
3091:
3072:
3056:
3040:
3024:
3008:
2998:
2985:
2975:
2964:
2963:
2941:
2935:
2912:
2911:
2901:
2899:
2889:
2887:
2877:
2874:
2873:
2863:
2861:
2851:
2849:
2839:
2836:
2835:
2828:
2821:
2809:
2789:
2788:
2768:
2762:
2736:
2725:
2686:
2676:
2654:
2650:
2632:
2622:
2600:
2596:
2591:
2590:
2586:) is given by:
2564:
2563:
2524:
2513:
2479:
2478:
2431:
2430:
2392:
2391:
2360:
2359:
2317:
2316:
2306:
2300:
2262:
2261:
2258:
2252:
2246:
2230:
2204:
2191:
2156:
2151:
2150:
2123:
2113:
2068:
2058:
2036:
2032:
2014:
2004:
1982:
1978:
1973:
1972:
1966:
1964:Polynomial ring
1960:
1841:
1840:
1696:
1695:
1674:
1668:
1581:
1580:
1492:
1491:
1454:
1453:
1450:
1444:
1433:
1370:
1369:
1367:
1345:
1318:
1317:
1309:
1305:
1295:
1289:
1248:
1247:
1228:
1227:
1201:
1200:
1163:
1162:
1160:linear equation
1112:
1108:
1104:
1103:. For example,
1097:
1091:
1063:commutative law
1009:
1008:
989:
988:
951:
950:
915:
861:
860:
818:
812:
811:
775:
774:
734:
733:
694:
693:
668:
667:
654:
650:
649:
636:
628:
622:
621:
589:
588:
578:
577:
567:
566:
551:
545:
544:
518:
517:
494:
493:
466:
460:
459:
422:
421:
395:
394:
374:
373:
349:
343:
342:
306:
305:
280:
279:
259:
258:
234:
228:
227:
191:
190:
165:
164:
144:
143:
123:
122:
102:
101:
77:
71:
70:
51:
22:
17:
12:
11:
5:
5425:
5423:
5415:
5414:
5412:Multiplication
5404:
5403:
5400:
5399:
5385:
5370:
5367:
5364:
5363:
5356:
5338:
5331:
5305:
5299:978-1447148203
5298:
5280:
5256:
5227:
5226:
5224:
5221:
5218:
5217:
5207:
5206:
5204:
5201:
5200:
5199:
5196:Multiplication
5193:
5187:
5182:
5177:
5166:
5163:
5162:
5161:
5155:
5146:
5145:Other products
5143:
5142:
5141:
5130:
5119:
5112:
5092:
5089:
5073:
5072:
5058:
5039:
5029:
5023:
5017:
5011:
5004:
5002:wreath product
4986:
4970:
4967:
4934:
4931:
4852:
4849:
4848:
4847:
4846:
4845:
4843:Tensor product
4840:
4835:
4830:
4818:
4813:
4803:
4800:
4775:
4772:
4755:
4754:
4748:
4716:
4715:
4704:
4699:
4695:
4691:
4688:
4685:
4682:
4677:
4673:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4585:Tensor product
4583:Main article:
4580:
4577:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4532:
4531:
4518:
4515:
4512:
4507:
4502:
4499:
4496:
4493:
4490:
4487:
4481:
4474:
4469:
4465:
4462:
4459:
4456:
4431:
4428:
4425:
4420:
4415:
4412:
4409:
4406:
4400:
4393:
4388:
4384:
4381:
4359:
4356:
4353:
4348:
4343:
4340:
4337:
4334:
4328:
4321:
4316:
4312:
4309:
4289:
4284:
4280:
4276:
4273:
4270:
4265:
4261:
4257:
4254:
4249:
4227:
4222:
4218:
4214:
4211:
4208:
4203:
4199:
4195:
4192:
4187:
4161:
4156:
4152:
4148:
4145:
4142:
4137:
4133:
4129:
4126:
4121:
4103:
4100:
4099:
4098:
4085:
4082:
4079:
4074:
4069:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4028:
4022:
4019:
4016:
4012:
4008:
4003:
4000:
3997:
3993:
3987:
3982:
3979:
3976:
3972:
3967:
3962:
3959:
3956:
3953:
3939:
3938:
3925:
3922:
3919:
3914:
3909:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3870:
3864:
3861:
3858:
3854:
3850:
3847:
3844:
3822:
3819:
3816:
3811:
3806:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3767:
3761:
3758:
3755:
3751:
3747:
3744:
3741:
3725:Matrix product
3723:Main article:
3720:
3717:
3708:
3704:
3697:
3690:
3673:
3672:
3661:
3657:
3652:
3647:
3643:
3640:
3636:
3632:
3629:
3626:
3623:
3609:
3608:
3597:
3592:
3585:
3581:
3573:
3569:
3563:
3556:
3552:
3544:
3537:
3533:
3527:
3523:
3517:
3510:
3506:
3498:
3494:
3488:
3481:
3477:
3470:
3466:
3463:
3460:
3456:
3452:
3449:
3446:
3443:
3396:
3381:
3362:
3355:
3350:
3349:
3338:
3333:
3326:
3322:
3314:
3310:
3304:
3297:
3293:
3287:
3283:
3278:
3271:
3267:
3261:
3257:
3252:
3248:
3244:
3240:
3234:
3227:
3223:
3215:
3211:
3206:
3202:
3199:
3196:
3192:
3188:
3185:
3171:
3170:
3159:
3155:
3151:
3146:
3142:
3138:
3133:
3129:
3125:
3122:
3119:
3116:
3111:
3107:
3103:
3098:
3094:
3090:
3087:
3084:
3079:
3075:
3071:
3068:
3063:
3059:
3055:
3052:
3047:
3043:
3039:
3036:
3031:
3027:
3023:
3020:
3015:
3011:
3005:
3001:
2997:
2992:
2988:
2982:
2978:
2974:
2971:
2937:Main article:
2934:
2931:
2930:
2929:
2916:
2908:
2904:
2900:
2896:
2892:
2888:
2884:
2880:
2876:
2875:
2870:
2866:
2862:
2858:
2854:
2850:
2846:
2842:
2838:
2837:
2833:
2829:
2826:
2822:
2819:
2815:
2814:
2812:
2807:
2803:
2800:
2797:
2764:Main article:
2761:
2758:
2757:
2756:
2743:
2739:
2732:
2728:
2722:
2717:
2714:
2711:
2707:
2703:
2699:
2693:
2689:
2683:
2679:
2673:
2668:
2665:
2662:
2658:
2653:
2649:
2645:
2639:
2635:
2629:
2625:
2619:
2614:
2611:
2608:
2604:
2599:
2571:
2560:
2559:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2522:
2519:
2516:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2458:
2455:
2452:
2447:
2444:
2441:
2438:
2411:
2408:
2405:
2402:
2399:
2379:
2376:
2373:
2370:
2367:
2356:
2355:
2343:
2339:
2336:
2333:
2330:
2327:
2324:
2310:scalar product
2304:Scalar product
2302:Main article:
2299:
2298:Scalar product
2296:
2283:
2280:
2277:
2274:
2270:
2248:Main article:
2245:
2242:
2229:
2226:
2225:
2224:
2211:
2207:
2203:
2198:
2194:
2188:
2185:
2182:
2179:
2176:
2172:
2168:
2163:
2159:
2144:
2143:
2130:
2126:
2120:
2116:
2110:
2107:
2104:
2099:
2096:
2093:
2089:
2085:
2081:
2075:
2071:
2065:
2061:
2055:
2050:
2047:
2044:
2040:
2035:
2031:
2027:
2021:
2017:
2011:
2007:
2001:
1996:
1993:
1990:
1986:
1981:
1962:Main article:
1959:
1956:
1945:
1944:
1933:
1929:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1889:
1884:
1881:
1877:
1873:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1834:
1833:
1822:
1819:
1816:
1813:
1809:
1803:
1799:
1796:
1793:
1790:
1786:
1780:
1775:
1772:
1768:
1755:
1752:
1749:
1745:
1739:
1735:
1732:
1729:
1726:
1722:
1716:
1711:
1708:
1704:
1670:Main article:
1667:
1664:
1663:
1662:
1650:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1624:
1620:
1617:
1614:
1611:
1608:
1605:
1601:
1597:
1594:
1591:
1588:
1574:
1573:
1561:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1535:
1531:
1528:
1525:
1522:
1519:
1516:
1512:
1508:
1505:
1502:
1499:
1487:can be added:
1475:
1471:
1467:
1462:
1446:Main article:
1443:
1440:
1432:
1429:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1352:
1348:
1342:
1337:
1334:
1331:
1327:
1288:
1285:
1258:
1255:
1235:
1211:
1208:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1095:Multiplication
1093:Main article:
1090:
1087:
1028:
1025:
1022:
1019:
1016:
996:
976:
973:
970:
967:
964:
961:
958:
931:multiplication
917:
916:
914:
913:
906:
899:
891:
888:
887:
884:
883:
858:
845:
841:
836:anti-logarithm
833:
830:
821:
809:
806:
805:
798:
797:
772:
759:
731:
728:
727:
717:
716:
691:
678:
673:
652:
651:
634:
633:
630:
619:
616:
615:
612:Exponentiation
608:
607:
593:
580:
579:
569:
568:
558:
557:
554:
541:
528:
523:
496:
495:
472:
471:
468:
457:
454:
453:
446:
445:
418:
405:
400:
386:
376:
375:
365:
355:
354:
351:
340:
337:
336:
333:Multiplication
329:
328:
303:
290:
285:
271:
261:
260:
250:
240:
239:
236:
225:
222:
221:
214:
213:
188:
175:
170:
156:
146:
145:
135:
125:
124:
114:
104:
103:
93:
83:
82:
79:
68:
65:
64:
53:
52:
50:
49:
42:
35:
27:
15:
13:
10:
9:
6:
4:
3:
2:
5424:
5413:
5410:
5409:
5407:
5396:
5392:
5388:
5382:
5378:
5373:
5372:
5368:
5359:
5353:
5349:
5342:
5339:
5334:
5328:
5324:
5319:
5318:
5309:
5306:
5301:
5295:
5291:
5284:
5281:
5270:
5266:
5260:
5257:
5246:
5242:
5235:
5233:
5229:
5222:
5212:
5209:
5202:
5197:
5194:
5191:
5188:
5186:
5183:
5181:
5178:
5172:
5169:
5168:
5164:
5159:
5156:
5153:
5150:A function's
5149:
5148:
5144:
5139:
5135:
5131:
5128:
5124:
5120:
5117:
5113:
5110:
5109:fiber product
5106:
5105:
5104:
5102:
5098:
5090:
5088:
5086:
5082:
5078:
5071:
5067:
5063:
5062:smash product
5059:
5056:
5055:slant product
5052:
5048:
5044:
5040:
5038:
5034:
5030:
5028:
5024:
5022:
5018:
5016:
5012:
5009:
5005:
5003:
4999:
4995:
4991:
4987:
4984:
4980:
4979:
4978:
4976:
4968:
4966:
4964:
4960:
4956:
4952:
4948:
4944:
4940:
4939:empty product
4933:Empty product
4932:
4930:
4928:
4924:
4920:
4915:
4912:
4905:
4896:
4895:ordered pairs
4891:
4887:
4882:
4878:
4874:
4870:
4866:
4862:
4858:
4850:
4844:
4841:
4839:
4838:Outer product
4836:
4834:
4831:
4829:
4826:
4825:
4823:
4819:
4817:
4814:
4812:
4809:
4808:
4807:
4801:
4799:
4797:
4793:
4789:
4785:
4781:
4773:
4771:
4769:
4764:
4760:
4759:outer product
4752:
4749:
4747:
4744:
4743:
4742:
4739:
4737:
4733:
4729:
4725:
4721:
4702:
4697:
4693:
4689:
4686:
4680:
4675:
4671:
4667:
4664:
4658:
4652:
4646:
4640:
4634:
4631:
4625:
4622:
4619:
4613:
4610:
4607:
4600:
4599:
4598:
4596:
4592:
4586:
4578:
4576:
4573:
4559:
4553:
4550:
4547:
4544:
4541:
4516:
4513:
4510:
4500:
4494:
4491:
4488:
4467:
4463:
4460:
4457:
4454:
4447:
4446:
4445:
4429:
4426:
4423:
4413:
4407:
4386:
4382:
4379:
4357:
4354:
4351:
4341:
4335:
4314:
4310:
4307:
4282:
4278:
4274:
4271:
4268:
4263:
4259:
4252:
4220:
4216:
4212:
4209:
4206:
4201:
4197:
4190:
4175:
4154:
4150:
4146:
4143:
4140:
4135:
4131:
4124:
4109:
4101:
4083:
4080:
4077:
4067:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4026:
4020:
4017:
4014:
4010:
4006:
4001:
3998:
3995:
3991:
3985:
3980:
3977:
3974:
3970:
3965:
3960:
3957:
3954:
3951:
3944:
3943:
3942:
3923:
3920:
3917:
3907:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3862:
3859:
3856:
3852:
3845:
3842:
3820:
3817:
3814:
3804:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3759:
3756:
3753:
3749:
3742:
3739:
3732:
3731:
3730:
3726:
3718:
3716:
3713:
3711:
3700:
3693:
3687:, denoted by
3686:
3682:
3678:
3675:in which the
3659:
3641:
3627:
3624:
3621:
3614:
3613:
3612:
3595:
3590:
3571:
3567:
3561:
3554:
3550:
3542:
3535:
3531:
3525:
3521:
3515:
3496:
3492:
3486:
3479:
3475:
3468:
3464:
3461:
3447:
3444:
3441:
3434:
3433:
3432:
3430:
3426:
3422:
3418:
3414:
3410:
3405:
3403:
3399:
3392:
3388:
3384:
3377:
3373:
3369:
3365:
3358:
3336:
3331:
3312:
3308:
3302:
3295:
3291:
3285:
3281:
3276:
3259:
3255:
3250:
3246:
3242:
3238:
3232:
3213:
3209:
3204:
3200:
3197:
3183:
3176:
3175:
3174:
3157:
3149:
3144:
3140:
3136:
3131:
3127:
3120:
3117:
3114:
3109:
3105:
3101:
3096:
3092:
3085:
3077:
3073:
3066:
3061:
3057:
3053:
3045:
3041:
3034:
3029:
3025:
3021:
3013:
3009:
3003:
2999:
2995:
2990:
2986:
2980:
2976:
2969:
2962:
2961:
2960:
2959:, satisfying
2958:
2954:
2950:
2946:
2940:
2932:
2914:
2906:
2902:
2894:
2890:
2882:
2878:
2868:
2864:
2856:
2852:
2844:
2840:
2810:
2805:
2798:
2787:
2786:
2785:
2783:
2780:
2775:
2773:
2772:cross product
2767:
2766:Cross product
2759:
2741:
2737:
2730:
2726:
2720:
2715:
2712:
2709:
2705:
2701:
2697:
2691:
2687:
2681:
2677:
2671:
2666:
2663:
2660:
2656:
2651:
2647:
2643:
2637:
2633:
2627:
2623:
2617:
2612:
2609:
2606:
2602:
2597:
2589:
2588:
2587:
2585:
2569:
2540:
2534:
2528:
2520:
2517:
2514:
2508:
2502:
2499:
2496:
2487:
2484:
2477:
2476:
2475:
2472:
2456:
2453:
2450:
2445:
2439:
2428:
2423:
2409:
2406:
2403:
2400:
2397:
2377:
2374:
2371:
2368:
2365:
2334:
2331:
2328:
2325:
2322:
2315:
2314:
2313:
2311:
2305:
2297:
2295:
2281:
2275:
2272:
2257:
2251:
2243:
2241:
2239:
2235:
2234:outer product
2227:
2209:
2205:
2201:
2196:
2192:
2186:
2183:
2180:
2177:
2174:
2170:
2166:
2161:
2157:
2149:
2148:
2147:
2128:
2124:
2118:
2114:
2108:
2105:
2102:
2097:
2094:
2091:
2087:
2083:
2079:
2073:
2069:
2063:
2059:
2053:
2048:
2045:
2042:
2038:
2033:
2029:
2025:
2019:
2015:
2009:
2005:
1999:
1994:
1991:
1988:
1984:
1979:
1971:
1970:
1969:
1965:
1957:
1955:
1953:
1948:
1931:
1919:
1916:
1913:
1907:
1904:
1898:
1892:
1879:
1875:
1871:
1864:
1855:
1852:
1849:
1839:
1838:
1837:
1820:
1814:
1811:
1794:
1788:
1770:
1766:
1750:
1747:
1730:
1724:
1706:
1702:
1694:
1693:
1692:
1689:
1687:
1678:
1673:
1665:
1644:
1641:
1638:
1635:
1632:
1629:
1618:
1615:
1612:
1606:
1595:
1592:
1589:
1579:
1578:
1577:
1555:
1552:
1549:
1546:
1543:
1540:
1529:
1526:
1523:
1517:
1506:
1503:
1500:
1490:
1489:
1488:
1469:
1465:
1449:
1448:Residue class
1441:
1439:
1437:
1430:
1428:
1426:
1425:empty product
1421:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1350:
1346:
1340:
1335:
1332:
1329:
1325:
1315:
1304:
1300:
1294:
1286:
1284:
1282:
1278:
1274:
1269:
1256:
1253:
1233:
1225:
1209:
1206:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1161:
1157:
1153:
1149:
1145:
1141:
1137:
1132:
1130:
1126:
1125:prime numbers
1122:
1118:
1102:
1096:
1088:
1086:
1084:
1079:
1077:
1073:
1069:
1065:
1064:
1059:
1055:
1050:
1048:
1047:
1042:
1023:
1020:
1017:
994:
971:
968:
965:
959:
956:
948:
944:
940:
936:
932:
928:
924:
912:
907:
905:
900:
898:
893:
892:
890:
889:
859:
843:
828:
819:
810:
807:
803:
799:
773:
757:
732:
729:
725:
723:
718:
692:
676:
671:
620:
617:
613:
609:
606:
552:
542:
526:
521:
458:
455:
451:
447:
444:
419:
403:
398:
384:
363:
341:
338:
334:
330:
304:
288:
283:
269:
248:
226:
223:
219:
215:
189:
173:
168:
154:
133:
112:
91:
69:
66:
62:
58:
55:
54:
48:
43:
41:
36:
34:
29:
28:
25:
21:
20:
5376:
5369:Bibliography
5347:
5341:
5316:
5308:
5289:
5283:
5272:. Retrieved
5268:
5259:
5248:. Retrieved
5244:
5211:
5127:model theory
5123:ultraproduct
5111:or pullback,
5094:
5074:
5033:Wick product
5008:free product
4998:knit product
4972:
4936:
4916:
4910:
4903:
4889:
4885:
4880:
4876:
4872:
4854:
4805:
4777:
4756:
4740:
4735:
4731:
4723:
4719:
4717:
4594:
4590:
4588:
4574:
4533:
4105:
3940:
3728:
3714:
3702:
3695:
3688:
3684:
3680:
3676:
3674:
3610:
3428:
3424:
3420:
3416:
3412:
3408:
3406:
3404:is applied.
3394:
3390:
3385:denotes the
3379:
3375:
3371:
3360:
3353:
3351:
3172:
2956:
2952:
2948:
2944:
2942:
2776:
2769:
2561:
2473:
2424:
2357:
2307:
2259:
2231:
2145:
1967:
1949:
1946:
1835:
1690:
1683:
1575:
1451:
1434:
1422:
1296:
1276:
1270:
1223:
1222:denotes the
1155:
1144:coefficients
1133:
1098:
1080:
1061:
1051:
1044:
946:
941:(numbers or
926:
920:
721:
420:
390:multiplicand
4873:product set
4728:dual spaces
4726:denote the
3366:denote the
2782:determinant
2584:dot product
2429:by letting
1686:convolution
1672:Convolution
1666:Convolution
1281:dot product
923:mathematics
510:denominator
218:Subtraction
5357:0387316094
5332:0080874398
5274:2020-08-16
5250:2020-08-16
5223:References
4955:set theory
4857:set theory
4108:dimensions
1950:Under the
1291:See also:
1148:parameters
935:expression
380:multiplier
314:difference
275:subtrahend
5241:"Product"
5066:wedge sum
5010:of groups
4977:include:
4947:empty sum
4698:∗
4690:∈
4684:∀
4676:∗
4668:∈
4662:∀
4611:⊗
4557:→
4545:∘
4514:×
4501:∈
4492:∘
4458:⋅
4427:×
4414:∈
4355:×
4342:∈
4272:…
4210:…
4144:…
4081:×
4068:∈
4059:…
4041:…
4007:⋅
3971:∑
3955:⋅
3921:×
3908:∈
3900:…
3882:…
3818:×
3805:∈
3797:…
3779:…
3625:∘
3445:∘
3387:component
3352:in which
3150:∈
3124:∀
3115:∈
3089:∀
2799:×
2738:β
2727:α
2706:∑
2678:β
2657:∑
2648:⋅
2624:α
2603:∑
2544:‖
2538:‖
2535:⋅
2532:‖
2526:‖
2518:⋅
2491:∠
2488:
2454:⋅
2443:‖
2437:‖
2407:∈
2369:⋅
2338:→
2332:×
2323:⋅
2279:→
2273:×
2202:⋅
2171:∑
2088:∑
2039:∑
2030:⋅
1985:∑
1932:τ
1920:τ
1917:−
1905:⋅
1899:τ
1888:∞
1883:∞
1880:−
1876:∫
1853:∗
1818:∞
1779:∞
1774:∞
1771:−
1767:∫
1754:∞
1715:∞
1710:∞
1707:−
1703:∫
1636:⋅
1607:⋅
1403:⋅
1397:⋅
1391:⋅
1385:⋅
1379:⋅
1326:∏
1314:summation
1199:the term
1140:variables
960:⋅
943:variables
869:logarithm
829:
802:Logarithm
503:numerator
385:×
364:×
270:−
249:−
5406:Category
5165:See also
5070:homotopy
2401:≠
2390:for all
1277:products
1156:products
1152:unknowns
1068:matrices
1046:multiple
933:, or an
745:radicand
644:exponent
573:quotient
562:fraction
479:dividend
450:Division
61:Addition
5395:8210342
5101:objects
4985:of sets
4900:βwhere
4822:tensors
4780:objects
4176:of U,
1418:
1368:
1224:product
1101:numbers
1058:complex
1041:integer
947:factors
939:objects
927:product
724:th root
486:divisor
430:product
265:minuend
118:summand
108:summand
5393:
5383:
5354:
5329:
5296:
5051:Massey
4898:(a, b)
4718:where
3701:, and
3679:-row,
3400:, and
3378:, and
2779:formal
1115:. The
751:degree
369:factor
359:factor
160:addend
150:augend
139:addend
129:addend
5203:Notes
5136:of a
5125:, in
5079:in a
4951:logic
4863:is a
4792:class
4786:of a
4174:basis
4172:be a
3694:, is
3368:bases
2146:with
1129:up to
804:(log)
702:power
662:power
584:ratio
5391:OCLC
5381:ISBN
5352:ISBN
5327:ISBN
5294:ISBN
5132:the
5121:the
5114:the
5107:the
5064:and
5060:the
5053:and
5041:the
5031:the
5025:the
5019:the
5013:the
5006:the
5000:and
4988:the
4981:the
4961:and
4937:The
4919:type
4909:b β
4907:and
4902:a β
4879:and
4871:(or
4859:, a
4796:type
4761:and
4734:and
4722:and
4593:and
3835:and
3423:map
3411:map
3374:and
3359:and
2951:and
2770:The
2427:norm
2375:>
1815:<
1751:<
1146:and
1138:and
1111:and
1054:real
1007:and
925:, a
824:base
783:root
657:base
639:base
254:term
244:term
97:term
87:term
5323:200
5047:cup
5043:cap
5035:of
4869:set
4855:In
4730:of
3427:to
3415:to
3393:on
3389:of
3370:of
2562:In
2485:cos
1760:and
1691:If
1312:as
1056:or
921:In
820:log
726:(β)
614:(^)
452:(Γ·)
335:(Γ)
220:(β)
199:sum
63:(+)
5408::
5389:.
5325:.
5267:.
5243:.
5231:^
5087:.
5049:,
5045:,
4996:,
4965:.
4957:,
4953:,
4914:.
4888:Γ
4824::
4738:.
4572:.
3712:.
3707:=g
3705:ij
3691:ij
2784::
2471:.
2446::=
2422:.
2308:A
2294:.
2236:,
1872::=
1688:.
1420:.
1406:36
1400:25
1394:16
1303:pi
1105:15
1085:.
1049:.
5397:.
5360:.
5335:.
5302:.
5277:.
5253:.
5129:.
4911:B
4904:A
4890:B
4886:A
4881:B
4877:A
4753:.
4736:W
4732:V
4724:W
4720:V
4703:,
4694:W
4687:w
4681:,
4672:V
4665:v
4659:,
4656:)
4653:w
4650:(
4647:W
4644:)
4641:v
4638:(
4635:V
4632:=
4629:)
4626:m
4623:,
4620:v
4617:(
4614:W
4608:V
4595:W
4591:V
4560:W
4554:U
4551::
4548:f
4542:g
4517:t
4511:s
4506:R
4498:)
4495:f
4489:g
4486:(
4480:U
4473:W
4468:M
4464:=
4461:A
4455:B
4430:t
4424:r
4419:R
4411:)
4408:g
4405:(
4399:V
4392:W
4387:M
4383:=
4380:B
4358:r
4352:s
4347:R
4339:)
4336:f
4333:(
4327:U
4320:V
4315:M
4311:=
4308:A
4288:}
4283:t
4279:w
4275:,
4269:,
4264:1
4260:w
4256:{
4253:=
4248:W
4226:}
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4207:,
4202:1
4198:v
4194:{
4191:=
4186:V
4160:}
4155:r
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4147:,
4141:,
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4132:u
4128:{
4125:=
4120:U
4084:t
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4047:;
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4038:1
4035:=
4032:i
4027:)
4021:k
4018:,
4015:j
4011:b
4002:j
3999:,
3996:i
3992:a
3986:r
3981:1
3978:=
3975:j
3966:(
3961:=
3958:A
3952:B
3924:t
3918:r
3913:R
3903:t
3897:1
3894:=
3891:k
3888:;
3885:r
3879:1
3876:=
3873:j
3869:)
3863:k
3860:,
3857:j
3853:b
3849:(
3846:=
3843:B
3821:r
3815:s
3810:R
3800:r
3794:1
3791:=
3788:j
3785:;
3782:s
3776:1
3773:=
3770:i
3766:)
3760:j
3757:,
3754:i
3750:a
3746:(
3743:=
3740:A
3709:i
3703:G
3698:i
3696:f
3689:F
3685:F
3681:j
3677:i
3660:,
3656:v
3651:F
3646:G
3642:=
3639:)
3635:v
3631:(
3628:f
3622:g
3596:.
3591:k
3584:U
3580:b
3572:i
3568:v
3562:j
3555:i
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3543:k
3536:j
3532:g
3526:=
3522:)
3516:j
3509:W
3505:b
3497:i
3493:v
3487:j
3480:i
3476:f
3469:(
3465:g
3462:=
3459:)
3455:v
3451:(
3448:f
3442:g
3429:U
3425:W
3421:g
3417:W
3413:V
3409:f
3397:V
3395:b
3391:v
3382:i
3380:v
3376:W
3372:V
3363:W
3361:b
3356:V
3354:b
3337:,
3332:j
3325:W
3321:b
3313:i
3309:v
3303:j
3296:i
3292:f
3286:=
3282:)
3277:i
3270:V
3266:b
3260:(
3256:f
3251:i
3247:v
3243:=
3239:)
3233:i
3226:V
3222:b
3214:i
3210:v
3205:(
3201:f
3198:=
3195:)
3191:v
3187:(
3184:f
3158:.
3154:F
3145:2
3141:t
3137:,
3132:1
3128:t
3121:,
3118:V
3110:2
3106:x
3102:,
3097:1
3093:x
3086:,
3083:)
3078:2
3074:x
3070:(
3067:f
3062:2
3058:t
3054:+
3051:)
3046:1
3042:x
3038:(
3035:f
3030:1
3026:t
3022:=
3019:)
3014:2
3010:x
3004:2
3000:t
2996:+
2991:1
2987:x
2981:1
2977:t
2973:(
2970:f
2957:F
2953:W
2949:V
2945:f
2915:|
2907:3
2903:v
2895:2
2891:v
2883:1
2879:v
2869:3
2865:u
2857:2
2853:u
2845:1
2841:u
2832:k
2825:j
2818:i
2811:|
2806:=
2802:v
2796:u
2742:i
2731:i
2721:n
2716:1
2713:=
2710:i
2702:=
2698:)
2692:i
2688:e
2682:i
2672:n
2667:1
2664:=
2661:i
2652:(
2644:)
2638:i
2634:e
2628:i
2618:n
2613:1
2610:=
2607:i
2598:(
2570:n
2541:w
2529:v
2521:w
2515:v
2509:=
2506:)
2503:w
2500:,
2497:v
2494:(
2457:v
2451:v
2440:v
2410:V
2404:v
2398:0
2378:0
2372:v
2366:v
2342:R
2335:V
2329:V
2326::
2282:V
2276:V
2269:R
2210:j
2206:b
2197:i
2193:a
2187:k
2184:=
2181:j
2178:+
2175:i
2167:=
2162:k
2158:c
2129:k
2125:X
2119:k
2115:c
2109:m
2106:+
2103:n
2098:0
2095:=
2092:k
2084:=
2080:)
2074:j
2070:X
2064:j
2060:b
2054:m
2049:0
2046:=
2043:j
2034:(
2026:)
2020:i
2016:X
2010:i
2006:a
2000:n
1995:0
1992:=
1989:i
1980:(
1928:d
1923:)
1914:t
1911:(
1908:g
1902:)
1896:(
1893:f
1868:)
1865:t
1862:(
1859:)
1856:g
1850:f
1847:(
1821:,
1812:t
1808:d
1802:|
1798:)
1795:t
1792:(
1789:g
1785:|
1748:t
1744:d
1738:|
1734:)
1731:t
1728:(
1725:f
1721:|
1649:Z
1645:N
1642:+
1639:b
1633:a
1630:=
1627:)
1623:Z
1619:N
1616:+
1613:b
1610:(
1604:)
1600:Z
1596:N
1593:+
1590:a
1587:(
1560:Z
1556:N
1553:+
1550:b
1547:+
1544:a
1541:=
1538:)
1534:Z
1530:N
1527:+
1524:b
1521:(
1518:+
1515:)
1511:Z
1507:N
1504:+
1501:a
1498:(
1474:Z
1470:N
1466:/
1461:Z
1388:9
1382:4
1376:1
1351:2
1347:i
1341:6
1336:1
1333:=
1330:i
1310:Ξ£
1306:Ξ
1257:.
1254:x
1234:a
1210:x
1207:a
1187:,
1184:0
1181:=
1178:b
1175:+
1172:x
1169:a
1113:5
1109:3
1027:)
1024:x
1021:+
1018:2
1015:(
995:x
975:)
972:x
969:+
966:2
963:(
957:x
910:e
903:t
896:v
844:=
840:)
832:(
758:=
722:n
677:=
672:}
553:{
527:=
522:}
404:=
399:}
289:=
284:}
174:=
169:}
155:+
134:+
113:+
92:+
46:e
39:t
32:v
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