Product (mathematics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Product_(mathematics) 

In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and ${\displaystyle x\cdot (2+x)}$ is the product of ${\displaystyle x}$ and ${\displaystyle (2+x)}$ (indicating that the two factors should be multiplied together). When one factor is an integer, the product is called a multiple.

The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well.

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 algebraic structures.

Product of two numbers

The product of two numbers or the multiplication between two numbers can be defined for common special cases: integers, natural numbers, fractions, real numbers, complex numbers, and quaternions.

Product of a sequence

See also: Multiplication § Product of a sequence

The product operator for the product of a sequence is denoted by the capital Greek letter pi Π (in analogy to the use of the capital Sigma Σ as summation symbol). For example, the expression ${\displaystyle \prod _{i=1}^6{i}^2}$is another way of writing ${\displaystyle 1\cdot 4\cdot 9\cdot 16\cdot 25\cdot 36}$.

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 empty product, and is equal to 1.

Commutative rings

Commutative rings have a product operation.

Residue classes of integers

Main article: residue class

Residue classes in the rings ${\displaystyle \mathbb{Z}/N\mathbb{Z}}$ can be added:

${\displaystyle (a+N\mathbb{Z})+(b+N\mathbb{Z})=a+b+N\mathbb{Z}}$

and multiplied:

${\displaystyle (a+N\mathbb{Z})\cdot (b+N\mathbb{Z})=a\cdot b+N\mathbb{Z}}$

Convolution

Main article: convolution

The convolution of the square wave with itself gives the triangular function

Two functions from the reals to itself can be multiplied in another way, called the convolution.

If

${\displaystyle \underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|f(t)|\mathrm{d}t<\mathrm{\infty }\text{and}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|g(t)|\mathrm{d}t<\mathrm{\infty },}$

then the integral

${\displaystyle (f\ast g)(t):=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(\tau )\cdot g(t-\tau )\mathrm{d}\tau }$

is well defined and is called the convolution.

Under the Fourier transform, convolution becomes point-wise function multiplication.

Polynomial rings

Main article: polynomial ring

The product of two polynomials is given by the following:

${\displaystyle \left(\sum _{i=0}^n{a}_i{X}^i\right)\cdot \left(\sum _{j=0}^m{b}_j{X}^j\right)=\sum _{k=0}^{n+m}{c}_k{X}^k}$

with

${\displaystyle c_k=\sum _{i+j=k}{a}_i\cdot b_j}$

Products in linear algebra

There are many different kinds of products in linear algebra. Some of these have confusingly similar names (outer product, exterior product) 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.

Scalar multiplication

Main article: scalar multiplication

By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map ${\displaystyle \mathbb{R}\times V\to V}$.

Scalar product

Main article: scalar product

A scalar product is a bi-linear map:

${\displaystyle \cdot :V\times V\to \mathbb{R}}$

with the following conditions, that ${\displaystyle v\cdot v>0}$ for all ${\displaystyle 0\ne v\in V}$.

From the scalar product, one can define a norm by letting ${\displaystyle \Vert v\Vert :=\sqrt{v\cdot v}}$.

The scalar product also allows one to define an angle between two vectors:

${\displaystyle \cos \mathrm{\angle }(v,w)=\frac{v\cdot w}{\Vert v\Vert \cdot \Vert w\Vert }}$

In ${\displaystyle n}$-dimensional Euclidean space, the standard scalar product (called the dot product) is given by:

${\displaystyle \left(\sum _{i=1}^n{\alpha }_i{e}_i\right)\cdot \left(\sum _{i=1}^n{\beta }_i{e}_i\right)=\sum _{i=1}^n{\alpha }_i{\beta }_i}$

Cross product in 3-dimensional space

Main article: cross product

The cross product 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.

The cross product can also be expressed as the formal determinant:

${\displaystyle \mathbf{u}\times \mathbf{v}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right|}$

Composition of linear mappings

Main article: function composition

A linear mapping can be defined as a function f between two vector spaces V and W with underlying field F, satisfying

${\displaystyle f(t_1{x}_1+t_2{x}_2)=t_{1}f(x_1)+t_{2}f(x_2),\mathrm{\forall }{x}_1,x_2\in V,\mathrm{\forall }{t}_1,t_2\in \mathbb{F}.}$

If one only considers finite dimensional vector spaces, then

${\displaystyle f(\mathbf{v})=f\left(v_i{{\mathbf{b}}_{\mathbf{V}}}^i\right)=v_{i}f\left({{\mathbf{b}}_{\mathbf{V}}}^i\right)={f^i}_j{v}_i{{\mathbf{b}}_{\mathbf{W}}}^j,}$

in which b_V and b_W denote the bases of V and W, and v_i denotes the component of v on b_Vⁱ, and Einstein summation convention is applied.

Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping f map V to W, and let the linear mapping g map W to U. Then one can get

${\displaystyle g\circ f(\mathbf{v})=g\left({f^i}_j{v}_i{{\mathbf{b}}_{\mathbf{W}}}^j\right)={g^j}_k{f^i}_j{v}_i{{\mathbf{b}}_{\mathbf{U}}}^k.}$

Or in matrix form:

${\displaystyle g\circ f(\mathbf{v})=\mathbf{G}\mathbf{F}\mathbf{v},}$

in which the i-row, j-column element of F, denoted by F_ij, is f^j_i, and G_ij=g^j_i.

The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.

Product of two matrices

Main article: matrix product

Given two matrices

${\displaystyle A=(a_{i,j}{)}_{i=1\dots s;j=1\dots r}\in {\mathbb{R}}^{s\times r}}$ and ${\displaystyle B=(b_{j,k}{)}_{j=1\dots r;k=1\dots t}\in {\mathbb{R}}^{r\times t}}$

their product is given by

${\displaystyle B\cdot A={\left(\sum _{j=1}^r{a}_{i,j}\cdot b_{j,k}\right)}_{i=1\dots s;k=1\dots t}\in {\mathbb{R}}^{s\times t}}$

Composition of linear functions as matrix product

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) dimensions of vector spaces U, V and W. Let ${\displaystyle \mathcal{U}=\{u_1,\dots ,u_r\}}$ be a basis of U, ${\displaystyle \mathcal{V}=\{v_1,\dots ,v_s\}}$ be a basis of V and ${\displaystyle \mathcal{W}=\{w_1,\dots ,w_t\}}$ be a basis of W. In terms of this basis, let ${\displaystyle A=M_{\mathcal{V}}^{\mathcal{U}}(f)\in {\mathbb{R}}^{s\times r}}$ be the matrix representing f : U → V and ${\displaystyle B=M_{\mathcal{W}}^{\mathcal{V}}(g)\in {\mathbb{R}}^{r\times t}}$ be the matrix representing g : V → W. Then

${\displaystyle B\cdot A=M_{\mathcal{W}}^{\mathcal{U}}(g\circ f)\in {\mathbb{R}}^{s\times t}}$

is the matrix representing ${\displaystyle g\circ f:U\to W}$.

In other words: the matrix product is the description in coordinates of the composition of linear functions.

Tensor product of vector spaces

Main article: Tensor product

Given two finite dimensional vector spaces V and W, the tensor product of them can be defined as a (2,0)-tensor satisfying:

${\displaystyle V\otimes W(v,m)=V(v)W(w),\mathrm{\forall }v\in V^{\ast },\mathrm{\forall }w\in W^{\ast },}$

where V^* and W^* denote the dual spaces of V and W.

For infinite-dimensional vector spaces, one also has the:

• Tensor product of Hilbert spaces
• Topological tensor product.

The tensor product, outer product and Kronecker product 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 intrinsic definition. The outer product is simply the Kronecker product, limited to vectors (instead of matrices).

The class of all objects with a tensor product

In general, whenever one has two mathematical objects that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the internal product of a monoidal category. 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 class of all things (of a given type) that have a tensor product.

Other products in linear algebra

Other kinds of products in linear algebra include:

• Hadamard product
• Kronecker product
• The product of tensors:
  • Wedge product or exterior product
  • Interior product
  • Outer product
  • Tensor product

Cartesian product

In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b)—where a ∈ A and b ∈ B.

The class of all things (of a given type) that have Cartesian products is called a Cartesian category. Many of these are Cartesian closed categories. Sets are an example of such objects.

Empty product

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 value of 0 (the identity element of addition). However, the concept of the empty product is more general, and requires special treatment in logic, set theory, computer programming and category theory.

Products over other algebraic structures

Products over other kinds of algebraic structures include:

• the Cartesian product of sets
• the direct product of groups, and also the semidirect product, knit product and wreath product
• the free product of groups
• the product of rings
• the product of ideals
• the product of topological spaces
• the Wick product of random variables
• the cap, cup, Massey and slant product in algebraic topology
• the smash product and wedge sum (sometimes called the wedge product) in homotopy

A few of the above products are examples of the general notion of an internal product in a monoidal category; the rest are describable by the general notion of a product in category theory.

Products in category theory

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 product (category theory), which describes how to combine two objects of some kind to create an object, possibly of a different kind. But also, in category theory, one has:

• the fiber product or pullback,
• the product category, a category that is the product of categories.
• the ultraproduct, in model theory.
• the internal product of a monoidal category, which captures the essence of a tensor product.

Other products

• A function's product integral (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".
• Complex multiplication, a theory of elliptic curves.

Longevity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Longevity

Comparison of male and female life expectancy at birth for countries and territories as defined by WHO for 2019. The green dotted line corresponds to equal female and male life expectancy. Open the original svg-image in a separate window and hover over a bubble to see more detailed information. The square of the bubbles is proportional to countries population based on estimation of the UN.

Longevity may refer to especially long-lived members of a population, whereas life expectancy is defined statistically as the average number of years remaining at a given age. For example, a population's life expectancy at birth is the same as the average age at death for all people born in the same year (in the case of cohorts).

Longevity studies may involve putative methods to extend life. Longevity has been a topic not only for the scientific community but also for writers of travel, science fiction, and utopian novels. The legendary fountain of youth appeared in the work of the Ancient Greek historian Herodotus.

There are difficulties in authenticating the longest human life span, owing to inaccurate or incomplete birth statistics. Fiction, legend, and folklore have proposed or claimed life spans in the past or future vastly longer than those verified by modern standards, and longevity narratives and unverified longevity claims frequently speak of their existence in the present.

A life annuity is a form of longevity insurance.

Life expectancy, as of 2010

For a more comprehensive list, see List of countries by life expectancy.

LEB in OECD countries

Various factors contribute to an individual's longevity. Significant factors in life expectancy include gender, genetics, access to health care, hygiene, diet and nutrition, exercise, lifestyle, and crime rates. Below is a list of life expectancies in different types of countries:

• Developed countries: 77–90 years (e.g. Canada: 81.29 years, 2010 est.)
• Developing countries: 32–80 years (e.g. Mozambique: 41.37 years, 2010 est.)

Population longevities are increasing as life expectancies around the world grow:

• Australia: 80 years in 2002, 81.72 years in 2010
• France: 79.05 years in 2002, 81.09 years in 2010
• Germany: 77.78 years in 2002, 79.41 years in 2010
• Italy: 79.25 years in 2002, 80.33 years in 2010
• Japan: 81.56 years in 2002, 82.84 years in 2010
• Monaco: 79.12 years in 2002, 79.73 years in 2011
• Spain: 79.06 years in 2002, 81.07 years in 2010
• United Kingdom: 80 years in 2002, 81.73 years in 2010
• United States: 77.4 years in 2002, 78.24 years in 2010

Long-lived individuals

Elderly couple in Portugal

The Gerontology Research Group validates current longevity records by modern standards, and maintains a list of supercentenarians; many other unvalidated longevity claims exist. Record-holding individuals include:

• Eilif Philipsen(1682–1785, 102 years, 333 days): first person to reach the age of 100 (on July 21, 1782) and whose age could be validated.
• Geert Adriaans Boomgaard (1788–1899, 110 years, 135 days): first person to reach the age of 110 (on September 21, 1898) and whose age could be validated.
• Margaret Ann Neve, (18 May 1792 – 4 April 1903, 110 years, 346 days) the first validated female supercentenarian (on 18 May 1902).
• Jeanne Calment (1875–1997, 122 years, 164 days): the oldest person in history whose age has been verified by modern documentation. This defines the modern human life span, which is set by the oldest documented individual who ever lived.
• Sarah Knauss (1880–1999, 119 years, 97 days): the third oldest documented person in modern times and the oldest American.
• Jiroemon Kimura (1897–2013, 116 years, 54 days): the oldest man in history whose age has been verified by modern documentation.
• Kane Tanaka (1903–2022, 119 years, 107 days): the second oldest documented person in modern times and the oldest Japanese.

Major factors

Evidence-based studies indicate that longevity is based on two major factors: genetics and lifestyle.

Genetics

Further information: Genetics of aging

Twin studies have estimated that approximately 20-30% of the variation in human lifespan can be related to genetics, with the rest due to individual behaviors and environmental factors which can be modified. Although over 200 gene variants have been associated with longevity according to a US-Belgian-UK research database of human genetic variants these explain only a small fraction of the heritability.

Lymphoblastoid cell lines established from blood samples of centenarians have significantly higher activity of the DNA repair protein PARP (Poly ADP ribose polymerase) than cell lines from younger (20 to 70 year old) individuals. The lymphocytic cells of centenarians have characteristics typical of cells from young people, both in their capability of priming the mechanism of repair after H₂O₂ sublethal oxidative DNA damage and in their PARP gene expression. These findings suggest that elevated PARP gene expression contributes to the longevity of centenarians, consistent with the DNA damage theory of aging.

"Healthspan, parental lifespan, and longevity are highly genetically correlated."

In July 2020 scientists, using public biological data on 1.75 m people with known lifespans overall, identify 10 genomic loci which appear to intrinsically influence healthspan, lifespan, and longevity – of which half have not been reported previously at genome-wide significance and most being associated with cardiovascular disease – and identify haem metabolism as a promising candidate for further research within the field. Their study suggests that high levels of iron in the blood likely reduce, and genes involved in metabolising iron likely increase healthy years of life in humans.

Lifestyle

Longevity is a highly plastic trait, and traits that influence its components respond to physical (static) environments and to wide-ranging life-style changes: physical exercise, dietary habits, living conditions, and pharmaceutical as well as nutritional interventions. A 2012 study found that even modest amounts of leisure time physical exercise can extend life expectancy by as much as 4.5 years.

Diet

Main article: Diet and longevity

As of 2021, there is no clinical evidence that any dietary practice contributes to human longevity.

Biological pathways

Four well-studied biological pathways that are known to regulate aging, and whose modulation has been shown to influence longevity are Insulin/IGF-1, mechanistic target of rapamycin (mTOR), AMP-activating protein kinase (AMPK), and Sirtuin pathways.

Autophagy

Autophagy plays a pivotal role in healthspan and lifespan extension.

Change over time

Post-COVID life expectancy in the US, UK, Netherlands, and Austria

In preindustrial times, deaths at young and middle age were more common than they are today. This is not due to genetics, but because of environmental factors such as disease, accidents, and malnutrition, especially since the former were not generally treatable with pre-20th-century medicine. Deaths from childbirth were common for women, and many children did not live past infancy. In addition, most people who did attain old age were likely to die quickly from the above-mentioned untreatable health problems. Despite this, there are many examples of pre-20th-century individuals attaining lifespans of 85 years or greater, including John Adams, Cato the Elder, Thomas Hobbes, Eric of Pomerania, Christopher Polhem, and Michelangelo. This was also true for poorer people like peasants or laborers. Genealogists will almost certainly find ancestors living to their 70s, 80s and even 90s several hundred years ago.

For example, an 1871 census in the UK (the first of its kind, but personal data from other censuses dates back to 1841 and numerical data back to 1801) found the average male life expectancy as being 44, but if infant mortality is subtracted, males who lived to adulthood averaged 75 years. The present life expectancy in the UK is 77 years for males and 81 for females, while the United States averages 74 for males and 80 for females.

Studies have shown that black American males have the shortest lifespans of any group of people in the US, averaging only 69 years (Asian-American females average the longest). This reflects overall poorer health and greater prevalence of heart disease, obesity, diabetes, and cancer among black American men.

Women normally outlive men. Theories for this include smaller bodies that place lesser strain on the heart (women have lower rates of cardiovascular disease) and a reduced tendency to engage in physically dangerous activities. Conversely, women are more likely to participate in health-promoting activities. The X chromosome also contains more genes related to the immune system, and women tend to mount a stronger immune response to pathogens than men. However, the idea that men have weaker immune systems due to the supposed immuno-suppressive actions of testosterone is unfounded.

There is debate as to whether the pursuit of longevity is a worthwhile health care goal. Bioethicist Ezekiel Emanuel, who is also one of the architects of ObamaCare, has argued that the pursuit of longevity via the compression of morbidity explanation is a "fantasy" and that longevity past age 75 should not be considered an end in itself. This has been challenged by neurosurgeon Miguel Faria, who states that life can be worthwhile in healthy old age, that the compression of morbidity is a real phenomenon, and that longevity should be pursued in association with quality of life. Faria has discussed how longevity in association with leading healthy lifestyles can lead to the postponement of senescence as well as happiness and wisdom in old age.

Naturally limited longevity

Most biological organisms have a naturally limited longevity due to aging, unlike a rare few that are considered biologically immortal.

Given that different species of animals and plants have different potentials for longevity, the disrepair accumulation theory of aging tries to explain how the potential for longevity of an organism is sometimes positively correlated to its structural complexity. It suggests that while biological complexity increases individual lifespan, it is counteracted in nature since the survivability of the overall species may be hindered when it results in a prolonged development process, which is an evolutionarily vulnerable state.

According to the antagonistic pleiotropy hypothesis, one of the reasons biological immortality is so rare is that certain categories of gene expression that are beneficial in youth become deleterious at an older age.

Myths and claims

Main articles: Longevity myths and Longevity claims

Longevity myths are traditions about long-lived people (generally supercentenarians), either as individuals or groups of people, and practices that have been believed to confer longevity, but for which scientific evidence does not support the ages claimed or the reasons for the claims. A comparison and contrast of "longevity in antiquity" (such as the Sumerian King List, the genealogies of Genesis, and the Persian Shahnameh) with "longevity in historical times" (common-era cases through twentieth-century news reports) is elaborated in detail in Lucian Boia's 2004 book Forever Young: A Cultural History of Longevity from Antiquity to the Present and other sources.

After the death of Juan Ponce de León, Gonzalo Fernández de Oviedo y Valdés wrote in Historia General y Natural de las Indias (1535) that Ponce de León was looking for the waters of Bimini to cure his aging. Traditions that have been believed to confer greater human longevity also include alchemy, such as that attributed to Nicolas Flamel. In the modern era, the Okinawa diet has some reputation of linkage to exceptionally high ages.

Longevity claims may be subcategorized into four groups: "In late life, very old people often tend to advance their ages at the rate of about 17 years per decade .... Several celebrated super-centenarians (over 110 years) are believed to have been double lives (father and son, relations with the same names or successive bearers of a title) .... A number of instances have been commercially sponsored, while a fourth category of recent claims are those made for political ends ...." The estimate of 17 years per decade was corroborated by the 1901 and 1911 British censuses. Time magazine considered that, by the Soviet Union, longevity had been elevated to a state-supported "Methuselah cult". Robert Ripley regularly reported supercentenarian claims in Ripley's Believe It or Not!, usually citing his own reputation as a fact-checker to claim reliability.

Non-human biological longevity

Main article: List of long-living organisms

Longevity in animals can shed light on the determinants of life expectancy in humans, especially when found in related mammals. However, important contributions to longevity research have been made by research in other species, ranging from yeast to flies to worms. In fact, some closely related species of vertebrates can have dramatically different life expectancies, demonstrating that relatively small genetic changes can have a dramatic impact on aging. For instance, Pacific Ocean rockfishes have widely varying lifespans. The species Sebastes minor lives a mere 11 years while its cousin Sebastes aleutianus can live for more than 2 centuries. Similarly, a chameleon, Furcifer labordi, is the current record holder for shortest lifespan among tetrapods, with only 4–5 months to live. By contrast, some of its relatives, such as Furcifer pardalis, have been found to live up to 6 years.

There are studies about aging-related characteristics of and aging in long-lived animals like various turtles and plants like Ginkgo biloba trees. They have identified potentially causal protective traits and suggest many of the species have "slow or [times of] negligible senescence" (or aging). The jellyfish T. dohrnii is biologically immortal and has been studied by comparative genomics.

Examples of long lived plants and animals

Currently living

• A 5,073-year-old member of the species Pinus longaeva: Oldest known currently living non-clonal tree.
• Methuselah: 4,800-year-old bristlecone pine in the White Mountains of California, the second oldest currently living non-clonal tree.

Dead

• The quahog clam (Arctica islandica) is exceptionally long-lived, with a maximum recorded age of 507 years, the longest of any animal. Other clams of the species have been recorded as living up to 374 years.
• Lamellibrachia luymesi, a deep-sea cold-seep tubeworm, is estimated to reach ages of over 250 years based on a model of its growth rates.
• A bowhead whale killed in a hunt was found to be approximately 211 years old (possibly up to 245 years old), the longest-lived mammal known.
• Possibly 250-million year-old bacteria, Bacillus permians, were revived from stasis after being found in sodium chloride crystals in a cavern in New Mexico.

Artificial animal longevity extension

Gene editing via CRISPR-Cas9 and other methods has significantly altered lifespans in animals.