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# History of speciation

The scientific study of speciation — how species evolve to become new species — began around the time of Charles Darwin in the middle of the 19th century. Many naturalists at the time recognized the relationship between biogeography (the way species are distributed) and the evolution of species. The 20th century saw the growth of the field of speciation, with major contributors such as Ernst Mayr researching and documenting species' geographic patterns and relationships. The field grew in prominence with the modern evolutionary synthesis in the early part of that century. Since then, research on speciation has expanded immensely.

The language of speciation has grown more complex. Debate over classification schemes on the mechanisms of speciation and reproductive isolation continue. The 21st century has seen a resurgence in the study of speciation, with new techniques such as molecular phylogenetics and systematics. Speciation has largely been divided into discrete modes that correspond to rates of gene flow between two incipient populations. Today however, research has driven the development of alternative schemes and the discovery of new processes of speciation.

## Early history

Charles Darwin introduced the idea that species could evolve and split into separate lineages, referring to it as specification in his 1859 book On the Origin of Species.[2] It was not until 1906 that the modern term speciation was coined by the biologist Orator F. Cook.[2][3] Darwin, in his 1859 publication, focused primarily on the changes that can occur within a species, and less on how species may divide into two.[4]:1 It is almost universally accepted that Darwin's book did not directly address its title.[1] Darwin instead saw speciation as occurring by species entering new ecological niches.[4]:125

### Darwin's views

Controversy exists as to whether Charles Darwin recognized a true geographical-based model of speciation in his publication On the Origin of Species.[5] In chapter 11, "Geographical Distribution", Darwin discusses geographic barriers to migration, stating for example that "barriers of any kind, or obstacles to free migration, are related in a close and important manner to the differences between the productions of various regions [of the world]".[6] F. J. Sulloway contends that Darwin's position on speciation was "misleading" at the least[7] and may have later misinformed Wagner and David Starr Jordan into believing that Darwin viewed sympatric speciation as the most important mode of speciation.[4]:83 Nevertheless, Darwin never fully accepted Wagner's concept of geographical speciation.[5]

The evolutionary biologist James Mallet maintains that the mantra repeated concerning Darwin's Origin of Species book having never actually discussed speciation is specious.[1] The claim began with Thomas Henry Huxley and George Romanes (contemporaries of Darwin's), who declared that Darwin failed to explain the origins of inviability and sterility in hybrids.[1][8] Similar claims were promulgated by the mutationist school of thought during the late 20th century, and even after the modern evolutionary synthesis by Richard Goldschmidt.[1][8] Another strong proponent of this view about Darwin came from Mayr.[1][8] Mayr maintained that Darwin was unable to address the problem of speciation, as he did not define species using the biological species concept.[9] However, Mayr's view has not been entirely accepted, as Darwin's transmutation notebooks contained writings concerning the role of isolation in the splitting of species.[9] Furthermore, Many of Darwin's ideas on speciation largely match the modern theories of both adaptive radiation and ecological speciation.[5]

## Biogeographic influence

Recognition of geographic factors involved in species populations was present even before Darwin, with many naturalists aware of the role of isolation in species relationships.[10]:482 In 1833, C. L. Gloger published The Variation of Birds Under the Influence of Climate in which he described geographic variations, but did not recognize that geographic isolation was an indicator of past speciation events.[10]:482 Another naturalist in 1856, Wollaston, studied island beetles in comparison to mainland species.[10]:482 He saw isolation as key to their differentiation.[10]:482 However, he did not recognize that the pattern was due to speciation.[10]:483 One naturalist, Leopold von Buch (1825) did recognize the geographic patterns and explicitly stated that geographic isolation may lead to species separating into new species.[10]:483 Mayr suggests that Von Buch was likely the first naturalist to truly suggest geographic speciation.[11] Other naturalists, such as Henry Walter Bates (1863), recognized and accepted the patterns as evidence of speciation, but in Bate's case, did not propose a coherent model.[10]:484

In 1868, Moritz Wagner was the first to propose the concept of geographic speciation[12][10]:484 in which he used the term Separationstheorie.[5] Edward Bagnall Poulton, the evolutionary biologist and a strong proponent of the importance of natural selection, highlighted the role of geographic isolation in promoting speciation,[13] in the process coining the term "sympatric speciation" in 1904.[14][15]

Wagner and other naturalists who studied the geographic distributions of animals, such as Karl Jordan and David Starr Jordan, noticed that closely related species were often geographically isolated from one another (allopatrically distributed) which lead to the advocation of the importance of geographic isolation in the origin of species.[4]:2 Karl Jordan is thought to have recognized the unification of mutation and isolation in the origin of new species — in stark contrast to the prevailing views at the time.[10]:486 David Starr Jordan reiterated Wagner's proposal in 1905, providing a wealth of evidence from nature to support the theory,[12][16][4]:2 and asserting that geographic isolation is obvious but had been unfortunately ignored by most geneticists and experimental evolutionary biologists at the time.[10]:487 Joel Asaph Allen suggested the observed pattern of geographic separation of closely related species be called "Jordan's Law" (or Wagner's Law).[10]:487 Despite the contentions, most taxonomists did accept the geographic model of speciation.[10]:487

Many of the early terms used to describe speciation were outlined by Ernst Mayr.[17] He was the first to encapsulate the then contemporary literature in his 1942 publication Systematics and the Origin of Species, from the Viewpoint of a Zoologist and in his subsequent 1963 publication Animal Species and Evolution. Like Jordan's works, they relied on direct observations of nature, documenting the occurrence of geographic speciation.[4]:86 He described the three modes: geographic, semi-geographic, and non-geographic; which today, are referred to as allopatric, parapatric, and sympatric respectively.[17] Mayr's 1942 publication, influenced heavily by the ideas of Karl Jordan and Poulton, was regarded as the authoritative review of speciation for over 20 years—and is still valuable today.[15]

A major focus of Mayr's works was on the importance of geography in facilitating speciation; with islands often acting as a central theme to many of the speciation concepts put forth.[18] One of which was the concept of peripatric speciation, a variant of allopatric speciation[19][20] (he has since distinguished the two modes by referring to them as peripatric and dichopatric[21]). This concept arose by an interpretation of Wagner's Separationstheorie as a form of founder effect speciation that focused on small geographically isolated species.[5] This model was later expanded and modified to incorporate sexual selection by Kenneth Y. Kaneshiro in 1976 and 1980.[22][23][24]

## Modern evolutionary synthesis

Many geneticists at the time did little to bridge the gap between the genetics of natural selection and the origin of reproductive barriers between species.[4]:3 Ronald Fisher proposed a model of speciation in his 1930 publication The Genetical Theory of Natural Selection, where he described disruptive selection acting on sympatric or parapatric populations — with reproductive isolation completed by reinforcement.[25] Other geneticists such as J. B. S. Haldane did not even recognize that species were real, while Sewall Wright ignored the topic, despite accepting allopatric speciation.[4]:3

The primary contributors to the incorporation of speciation into modern evolutionary synthesis were Ernst Mayr and Theodosius Dobzhansky.[25] Dobzhansky, a geneticist, published Genetics and the Origin of Species in 1937, in which he formulated the genetic framework for how speciation could occur.[4]:2 He recognized that speciation was an unsolved problem in biology at the time, rejecting Darwin’s position that new species arose by occupation of new niches — contending that reproductive isolation was instead based on barriers to gene flow.[4]:2 Subsequently, Mayr conducted extensive work on the geography of species, emphasizing the importance of geographic separation and isolation, in which he filled Dobzhansky‘s gaps concerning the origin of biodiversity (in his 1942 book).[26] Both of their works gave rise, not without controversy, to the modern understanding of speciation; stimulating a wealth of research on the topic.[4]:3 Furthermore, this extended to plants as well as animals with G. Ledyard Stebbins’s book, Variation and Evolution in Plants and the much later, 1981 book, Plant Speciation by Verne Grant.

In 1947, "a consensus had been achieved among geneticists, paleontologists and systematists and that evolutionary biology as an independent biological discipline had been established" during a Princeton University conference.[28] This 20th century synthesis incorporated speciation. Since then, the ideas have been consistently and repeatedly confirmed.[26]

## Contemporary work

After the synthesis, speciation research continued largely within natural history and biogeography — with much less emphasis on genetics.[4]:4 The study of speciation has seen its largest increase since the 1980s[4]:4 with an influx of publications and a host of new terms, methods, concepts, and theories.[17] This "third phase" of work — as Jerry A. Coyne and H. Allen Orr put it — has led to a growing complexity of the language used to describe the many processes of speciation.[17] The research and literature on speciation have become, "enormous, scattered, and increasingly technical".[4]:1

From the 1980s, new research tools increased the robustness of research,[4]:4 assisted by new methods, theoretical frameworks, models, and approaches.[17] Coyne and Orr discuss the modern, post-1980s developments centered around five major themes:

1. genetics (also a primary factor in the Modern Synthesis),
2. molecular biology and analysis (namely, phylogenetics and systematics);
3. comparative analysis;
4. mathematical modeling and computer simulations; and
5. the role of ecology.[4]:5

Ecologists became aware that the ecological factors behind speciation were under-represented. This saw the growth in research concerning ecology's role in facilitating speciation — rightly designated ecological speciation.[4]:4 This focus on ecology generated a host of new terms relating to the barriers to reproduction[17] (e.g. allochronic speciation, in which gene flow is reduced or removed by timing of breeding periods; or habitat isolation, in which species occupy different habitats within the same area). Sympatric speciation, regarded by Mayr as unlikely, has become widely accepted.[29][30][31] Research on the influence of natural selection on speciation, including the process of reinforcement, has grown.[32]

Researchers have long debated the roles of sexual selection, natural selection, and genetic drift in speciation.[4]:383 Darwin extensively discussed sexual selection, with his work greatly expanded on by Ronald Fisher; however, it was not until 1983 that the biologist Mary Jane West-Eberhard recognized the importance of sexual selection in speciation.[33][4]:3 Natural selection plays a role in that any selection towards reproductive isolation can result in speciation — whether indirectly or directly. Genetic drift has been widely researched from the 1950s onwards, especially with peak-shift models of speciation by genetic drift.[4]:388 Mayr championed founder effects, in which isolated individuals, like those found on islands near a mainland, experience a strong population bottleneck, as they contain only a small sample of the genetic variation in the main population.[4]:390[34] Later, other biologists such as Hampton L. Carson, Alan Templeton, Sergey Gavrilets, and Alan Hastings developed related models of speciation by genetic drift, noting that islands were inhabited mostly by endemic species.[35] Selection's role in speciation is widely supported, whereas founder effect speciation is not,[4]:410 having been subject to a number of criticisms.[36]

### Classification debate

Throughout the history of research concerning speciation, classification and delineation of modes and processes have been debated. Julian Huxley divided speciation into three separate modes: geographical speciation, genetic speciation, and ecological speciation.[10]:427 Sewall Wright proposed ten different, varying modes.[10]:427 Ernst Mayr championed the importance of physical, geographic separation of species populations, maintaining it to be of major importance to speciation. He originally proposed the three primary modes known today: geographic, semi-geographic, non-geographic;[17] corresponding to allopatric, parapatric, and sympatric respectively.

The phrase "modes of speciation" is imprecisely defined, most often indicating speciation occurring as a result of a species geographic distribution.[37] More succinctly, the modern classification of speciation is often described as occurring on a gene flow continuum (i.e., allopatry at ${\displaystyle m=0}$ and sympatry at ${\displaystyle m=0.5}$[38][39]) This gene flow concept views speciation as based on the exchange of genes between populations instead of seeing a purely geographic setting as necessarily relevant. Despite this, concepts of biogeographic modes can be translated into models of gene flow (such as that in the image at left); however, this translation has led to some confusion of language in the scientific literature.[17]

As research has expanded over the decades, the geographic scheme has been challenged. The traditional classification is considered by some researchers to be obsolete,[40] while others argue for its merits. Proponents of non-geographic schemes often justify non-geographic classifications, not by rejection of the importance of reproductive isolation (or even the processes themselves), but instead by the fact that it simplifies the complexity of speciation.[41] One major critique of the geographic framework is that it arbitrarily separates a biological continuum into discontinuous groups.[41] Another criticism rests with the fact that, when speciation is viewed as a continuum of gene flow, parapatric speciation becomes unreasonably represented by the entire continuum[42]—with allopatric and sympatric existing in the extremes.[41] Coyne and Orr argue that the geographic classification scheme is valuable in that biogeography controls the strength of the evolutionary forces at play, as gene flow and geography are clearly linked.[40] James Mallet and colleagues contend that the sympatric vs. allopatric dichotomy is valuable to determine the degree in which natural selection acts on speciation.[43] Kirkpatrick and Ravigné categorize speciation in terms of its genetic basis or by the forces driving reproductive isolation.[4]:85 Here, the geographic modes of speciation are classified as types of assortive mating.[44] Fitzpatrick and colleagues believe that the biogeographic scheme "is a distraction that could be positively misleading if the real goal is to understand the influence of natural selection on divergence."[40] They maintain that, to fully understand speciation, "the spatial, ecological, and genetic factors" involved in divergence must be explored.[40] Sara Via recognizes the importance of geography in speciation but suggests that classification under this scheme be abandoned.[30]

## History of modes and mechanisms

### Sympatric speciation

Sympatric speciation, from its beginnings with Darwin (who did not coin the term), has been a contentious issue.[37][4]:125 Mayr, along with many other evolutionary biologists, interpreted Darwins's view of speciation and the origin of biodiversity as arising by species entering new ecological niches—a form of sympatric speciation.[1] Before Mayr, sympatric speciation was regarded as the primary mode of speciation. In 1963, Mayr provided a strong criticism, citing various flaws in the theory.[4]:126 After that, sympatric speciation fell out of favor with biologists and has only recently seen a resurgence in interest.[4]:126 Some biologists, such as James Mallet, believe that Darwin's view on speciation was misunderstood and misconstrued by Mayr.[1][45] Today, sympatric speciation is supported by evidence from laboratory experiments and observations from nature.[4]:127[29]

### Hybrid speciation

For most of the history of speciation, hybridization (polyploidy) has been a contentious issue, as botanists and zoologists have traditionally viewed hybridization's role in speciation differently.[17] Carl Linnaeus was the earliest to suggest hybridization in 1760,[46] Øjvind Winge was the first to confirm allopolyploidy in 1917,[46][47] and a later experiment conducted by Clausen and Goodspeed in 1925 confirmed the findings.[46] Today it is widely recognized as a common mechanism of speciation.[48]

Historically, zoologists considered hybridization to be a rare phenomenon, while botanists found it to be commonplace in plant species.[17] The botanists G. Ledyard Stebbins and Verne Grant were two of the well known botanists who championed the idea of hybrid speciation during the 1950s to the 1980s.[17] Hybrid speciation, also called polyploid speciation (or polyploidy) is speciation that results by an increase in the number of sets of chromosomes.[4]:321 It is effectively a form of sympatric speciation that happens instantly.[4]:322 Grant coined the term recombinational speciation in 1981; a special form of hybrid speciation where a new species results from hybridization and is itself, reproductively isolated from both its parents.[4]:337 Recently, biologists have increasingly recognized that hybrid speciation can occur in animals as well.[49]

### Reinforcement

The concept of speciation by reinforcement has a complex history, with its popularity among scholars changing significantly over time.[32][4]:353 The theory of reinforcement experienced three phases of historical development:[4]:366

1. plausibility based on unfit hybrids
2. implausibility based on the finding that hybrids may have some fitness
3. plausibility based on empirical studies and biologically complex and realistic models

It was originally proposed by Alfred Russel Wallace in 1889,[4]:353 termed the Wallace effect—a term rarely used by scientists today.[50] Wallace's hypothesis differed from the modern conception in that it focused on post-zygotic isolation, strengthened by group selection.[4]:353[51][52] Dobzhansky was the first to provide a thorough, modern description of the process in 1937,[4]:353 though the actual term itself was not coined until 1955 by W. Frank Blair.[53]

In 1930, Ronald Fisher laid out the first genetic description of the process of reinforcement in The Genetical Theory of Natural Selection, and in 1965 and 1970 the first computer simulations were run to test for its plausibility.[4]:366 Later, population genetic[54] and quantitative genetic[55] studies were conducted showing that completely unfit hybrids lead to an increase in pre-zygotic isolation.[4]:368 After Dobzhansky's idea rose to the forefront of speciation research, it garnered significant support—with Dobzhansky suggesting that it illustrated the final step in speciation (e.g. after an allopatric population comes into secondary contact).[4]:353 In the 1980s, many evolutionary biologists began to doubt the plausibility of the idea,[4]:353 based not on empirical evidence, but largely on the growth of theory that deemed it an unlikely mechanism of reproductive isolation.[56] A number of theoretical objections arose at the time. Since the early 1990s, reinforcement has seen a revival in popularity, with perceptions by evolutionary biologists accepting its plausibility—due primarily from a sudden increase in data, empirical evidence from laboratory studies and nature, complex computer simulations, and theoretical work.[4]:372–375

The scientific language concerning reinforcement has also differed over time, with different researchers applying various definitions to the term.[50] First used to describe the observed mating call differences in Gastrophryne frogs within a secondary contact hybrid zone,[50] reinforcement has also been used to describe geographically separated populations that experience secondary contact.[57] Roger Butlin demarcated incomplete post-zygotic isolation from complete isolation, referring to incomplete isolation as reinforcement and completely isolated populations as experiencing reproductive character displacement.[58] Daniel J. Howard considered reproductive character displacement to represent either assortive mating or the divergence of traits for mate recognition (specifically between sympatric populations).[50] Under this definition, it includes pre-zygotic divergence and complete post-zygotic isolation.[59] Maria R. Servedio and Mohamed Noor consider any detected increase in pre-zygotic isolation as reinforcement, as long as it is a response to selection against mating between two different species.[60] Coyne and Orr contend that, "true reinforcement is restricted to cases in which isolation is enhanced between taxa that can still exchange genes".[4]:354

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60. Maria R. Servedio and Mohamed A. F. Noor (2003), "The Role of Reinforcement in Speciation: Theory and Data", Annual Review of Ecology, Evolution, and Systematics, 34: 339–364, doi:10.1146/annurev.ecolsys.34.011802.132412


text:
# Reproducing a table containing images and multi-line text in a cell

I'd like to be able to reproduce the table attached as an image to this post in Latex. The original table comes from my PhD thesis which was written in Microsoft Word 5.0 on a Mac in the early 1990s. As such I've long since lost the original files (disk crashes over a twenty year period took care of that) and in any case I don't think I'd be able to read the original files into the latest version of Word even if I had them.

Bruce

• Are you looking for reproducing the table only? Or you need the graphs as well? (+1) for choosing Latex over Microsoft Word. – CroCo Oct 18 '15 at 4:09
• Welcome to TeX.SX! Please provide some starter document with \documentclass{...} and \begin{document}...\end{document} – user31729 Oct 18 '15 at 4:10
• @CroCo, I would like to reproduce the whole layout with the multiple lines of text in column 1 and the figures in columns 2 and 3. However, the figures would be created in another program and added as vector graphic files. – Bruce Crawford Oct 18 '15 at 4:14

Try something like this:

\documentclass[border=3mm,
preview]{standalone}
\usepackage{graphicx}

\usepackage{array,booktabs}

\begin{document}
\begin{tabular}{m{5cm}cc}
\toprule
\textbf{Exercise 1}
\begin{itemize}
\item Instruction 1.
\item Instruction 2.
\item Instruction 3.
\end{itemize}
& \includegraphics[width=0.2\textwidth,
valign=m, margin=.5em]{example-image}
& \includegraphics[width=0.2\textwidth,
valign=m, margin=.5em]{example-image} \\
\midrule
%%
\textbf{Exercise 2}
\begin{itemize}
\item Instruction 1.
\item Instruction 2.
\item Instruction 3.
\end{itemize}
& \includegraphics[width=0.2\textwidth,
valign=m, margin=.5em]{example-image}
& \includegraphics[width=0.2\textwidth,
valign=m, margin=.5em]{example-image}    \\
\midrule
\textbf{Exercise 3}
\begin{itemize}
\item TEST.
\item TEST.
\end{itemize}
& \includegraphics[width=0.2\textwidth,
valign=m, margin=.5em]{example-image}
& \includegraphics[width=0.2\textwidth,
valign=m, margin=.5em]{example-image} \\
\bottomrule
\end{tabular}
\end{document}

which gives:

Since you didn't provide MWE, I left to you tu put in mine desired text and images.

• That seems to be exactly what I need. Thanks @Zarko – Bruce Crawford Oct 18 '15 at 4:16
• accepted as suggested. Sorry about the MWE. I did try but couldn't get anything to work. I'm very new to LaTeX at this stage. – Bruce Crawford Oct 18 '15 at 4:39

format: plain


text:
## Calculus: Early Transcendentals (2nd Edition)

$a_n = n^2-n$ $a_1 = 0$ $a_2 = 2$ $a_3 = 6$ $a_4 = 12$ $a_5 = 20$ $a_6 = 30$ $a_7 = 42$ $a_8 = 56$ $a_9 = 72$ $a_{10} = 90$ The terms seem to be increasing without a bound, meaning the sequence diverges.


text:
# Comparison functions and operators#

## Comparison operators#

Operator

Description

<

Less than

>

Greater than

<=

Less than or equal to

>=

Greater than or equal to

=

Equal

<>

Not equal

!=

Not equal (non-standard but popular syntax)

## Range operator: BETWEEN#

The BETWEEN operator tests if a value is within a specified range. It uses the syntax value BETWEEN min AND max:

SELECT 3 BETWEEN 2 AND 6;

The statement shown above is equivalent to the following statement:

SELECT 3 >= 2 AND 3 <= 6;

To test if a value does not fall within the specified range use NOT BETWEEN:

SELECT 3 NOT BETWEEN 2 AND 6;

The statement shown above is equivalent to the following statement:

SELECT 3 < 2 OR 3 > 6;

A NULL in a BETWEEN or NOT BETWEEN statement is evaluated using the standard NULL evaluation rules applied to the equivalent expression above:

SELECT NULL BETWEEN 2 AND 4; -- null

SELECT 2 BETWEEN NULL AND 6; -- null

SELECT 2 BETWEEN 1 AND NULL; -- false

SELECT 8 BETWEEN NULL AND 6; -- false

The BETWEEN and NOT BETWEEN operators can also be used to evaluate any orderable type. For example, a VARCHAR:

SELECT 'Paul' BETWEEN 'John' AND 'Ringo'; -- true

Note that the value, min, and max parameters to BETWEEN and NOT BETWEEN must be the same type. For example, Trino will produce an error if you ask it if John is between 2.3 and 35.2.

## IS NULL and IS NOT NULL#

The IS NULL and IS NOT NULL operators test whether a value is null (undefined). Both operators work for all data types.

Using NULL with IS NULL evaluates to true:

select NULL IS NULL; -- true

But any other constant does not:

SELECT 3.0 IS NULL; -- false

## IS DISTINCT FROM and IS NOT DISTINCT FROM#

In SQL a NULL value signifies an unknown value, so any comparison involving a NULL will produce NULL. The IS DISTINCT FROM and IS NOT DISTINCT FROM operators treat NULL as a known value and both operators guarantee either a true or false outcome even in the presence of NULL input:

SELECT NULL IS DISTINCT FROM NULL; -- false

SELECT NULL IS NOT DISTINCT FROM NULL; -- true

In the example shown above, a NULL value is not considered distinct from NULL. When you are comparing values which may include NULL use these operators to guarantee either a TRUE or FALSE result.

The following truth table demonstrate the handling of NULL in IS DISTINCT FROM and IS NOT DISTINCT FROM:

a

b

a = b

a <> b

a DISTINCT b

a NOT DISTINCT b

1

1

TRUE

FALSE

FALSE

TRUE

1

2

FALSE

TRUE

TRUE

FALSE

1

NULL

NULL

NULL

TRUE

FALSE

NULL

NULL

NULL

NULL

FALSE

TRUE

## GREATEST and LEAST#

These functions are not in the SQL standard, but are a common extension. Like most other functions in Trino, they return null if any argument is null. Note that in some other databases, such as PostgreSQL, they only return null if all arguments are null.

The following types are supported: DOUBLE, BIGINT, VARCHAR, TIMESTAMP, TIMESTAMP WITH TIME ZONE, DATE

greatest(value1, value2, ..., valueN) → [same as input]#

Returns the largest of the provided values.

least(value1, value2, ..., valueN) → [same as input]#

Returns the smallest of the provided values.

## Quantified comparison predicates: ALL, ANY and SOME#

The ALL, ANY and SOME quantifiers can be used together with comparison operators in the following way:

expression operator quantifier ( subquery )

For example:

SELECT 'hello' = ANY (VALUES 'hello', 'world'); -- true

SELECT 21 < ALL (VALUES 19, 20, 21); -- false

SELECT 42 >= SOME (SELECT 41 UNION ALL SELECT 42 UNION ALL SELECT 43); -- true

Here are the meanings of some quantifier and comparison operator combinations:

Expression

Meaning

A = ALL (...)

Evaluates to true when A is equal to all values.

A <> ALL (...)

Evaluates to true when A doesn’t match any value.

A < ALL (...)

Evaluates to true when A is smaller than the smallest value.

A = ANY (...)

Evaluates to true when A is equal to any of the values. This form is equivalent to A IN (...).

A <> ANY (...)

Evaluates to true when A doesn’t match one or more values.

A < ANY (...)

Evaluates to true when A is smaller than the biggest value.

ANY and SOME have the same meaning and can be used interchangeably.

## Pattern comparison: LIKE#

The LIKE operator can be used to compare values with a pattern:

... column [NOT] LIKE 'pattern' ESCAPE 'character';

Matching characters is case sensitive, and the pattern supports two symbols for matching:

• _ matches any single character

• % matches zero or more characters

Typically it is often used as a condition in WHERE statements. An example is a query to find all continents starting with E, which returns Europe:

SELECT * FROM (VALUES 'America', 'Asia', 'Africa', 'Europe', 'Australia', 'Antarctica') AS t (continent)
WHERE continent LIKE 'E%';

You can negate the result by adding NOT, and get all other continents, all not starting with E:

SELECT * FROM (VALUES 'America', 'Asia', 'Africa', 'Europe', 'Australia', 'Antarctica') AS t (continent)
WHERE continent NOT LIKE 'E%';

If you only have one specific character to match, you can use the _ symbol for each character. The following query uses two underscores and produces only Asia as result:

SELECT * FROM (VALUES 'America', 'Asia', 'Africa', 'Europe', 'Australia', 'Antarctica') AS t (continent)
WHERE continent LIKE 'A__A';

The wildcard characters _ and % must be escaped to allow you to match them as literals. This can be achieved by specifying the ESCAPE character to use:

SELECT 'South_America' LIKE 'South\_America' ESCAPE '\';

The above query returns true since the escaped underscore symbol matches. If you need to match the used escape character as well, you can escape it.

If you want to match for the chosen escape character, you simply escape itself. For example, you can use \\ to match for ‘’'’.


text:
# Showing that the Intermediate Value Theorem is false in the absence of irrational numbers

The following question is located on Page 80 of Calculus with Analytic Geometry by George Simmons.

Assume for a moment that the rational numbers are the only numbers that exist. Under this assumption, show that the Intermediate Value Theorem is false by considering the function $$y=f(x)=x^2 - 2$$ on the inteval $$[1,2]$$.

The only way I have been able to interpret this is to solve $$f(x)$$ for some irrational value. As the number would not exist in this hypothetical, the IVT would fail. However, I feel like that's an oversimplification and I'm missing what this question is actually asking me.

• Where would this function cross the $x$-axis? Feb 28, 2019 at 2:11
• To elaborate, no, you don't want to solve $f(x)=c$ for an irrational value of $c$. You want to solve it for $c=0$, noting that $f(1)<0<f(2)$. Feb 28, 2019 at 2:14
• Ah, that is exactly what I was looking for. Crosses at sqrt(2), but f(1) is negative, f(2) is positive. As IVT says f(x)=0 at some point for this to happen, but this point does not exist, IVT fails. Thank you!
– Vale
Feb 28, 2019 at 2:15
• $f(1) = 1^2 -2 < 0$ and $f(2) = 2^2 - 2 > 0$ so if the Intermediate Value Theorem holds then there exists a $\psi \in (1, 2)$ such that $f(\psi) = 0$. But then $\psi = \sqrt{2}$ a contradiction. Feb 28, 2019 at 2:16

In general you need to find $$c$$ between $$f(1)$$ and $$f(2)$$ such that no rational $$x$$ satisfies $$f(x) = c$$.
In particular, as mentioned in the comment, you could show that there's no rational $$x$$ such that $$f(x) = 0$$.


text:
# How do you find all the real and complex roots of (x^3 + 9)(x^2 - 4) = 0?

Mar 2, 2016

#### Answer:

$x = - \sqrt[3]{9} , \pm 2 , \frac{\sqrt[3]{9}}{2} \pm \frac{3 \sqrt[6]{3}}{2} i$

#### Explanation:

We can split this up into two separate equations:

${x}^{3} + 9 = 0 \text{ "" and "" } {x}^{2} - 4 = 0$

Let's first solve the second equation (it's simpler):

${x}^{2} - 4 = 0$

${x}^{2} = 4$

$x = \pm 2$

These are the first $2$ of what should be $5$ total solutions.

Onto the next equation, we may be tempted to solve like so:

${x}^{3} + 9 = 0$

${x}^{3} = - 9$

$x = \sqrt[3]{- 9}$

However, this doesn't deal with the two remaining zeros we should have. We must try to find a way to find the two leftover complex roots in ${x}^{3} + 9 = 0$.

We can factor ${x}^{3} + 9$ as a sum of cubes, where the terms being cubed are $x$ and ${3}^{\frac{2}{3}}$.

${x}^{3} + 9 = 0$

$\left(x + {3}^{\frac{2}{3}}\right) \left({x}^{2} - {3}^{\frac{2}{3}} x + {3}^{\frac{4}{3}}\right) = 0$

Solving the linear term gives us $x = - {3}^{\frac{2}{3}} = - \sqrt[3]{9}$. To find the remaining two roots, use the quadratic formula on the quadratic factor.

$x = \frac{{3}^{\frac{2}{3}} \pm \sqrt{{3}^{\frac{4}{3}} - 4 \left(1\right) \left({3}^{\frac{4}{3}}\right)}}{2}$

$x = \frac{{3}^{\frac{2}{3}} \pm \sqrt{- 3 \left({3}^{\frac{4}{3}}\right)}}{2}$

$x = \frac{{3}^{\frac{2}{3}} \pm \sqrt{{3}^{\frac{7}{3}}} i}{2}$

$x = {3}^{\frac{2}{3}} / 2 \pm {3}^{\frac{7}{6}} / 2 i$

This can also be written as

$x = \frac{\sqrt[3]{9}}{2} \pm \frac{3 \sqrt[6]{3}}{2} i$

or

$x \approx 1.0400 \pm 1.8014 i$

depending on preference.


text:
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## symmetric matrix eigenvalues

... we can say, non-zero eigenvalues of A are non-real. Proof. If B is nearly symmetric positive definite, then consider using B = (B+B')/2 to make B symmetric before calling eigs. This is a guide to MATLAB Eigenvalues. Recall also from Matrix Transpose Properties that from the spectral theorem, The values of λ that satisfy the equation are the generalized eigenvalues. A symmetric matrix can be broken up into its eigenvectors. Then because of A well-known property of such a set of equations is that it only has a non-trivial solution when the … Ask Question Asked 7 years, 7 months ago. Real number λ and vector z are called an eigen pair of matrix A, if Az = λz.For a real matrix A there could be both the problem of finding the eigenvalues and the problem of finding the eigenvalues and eigenvectors.. Nk=0 for some positive integer k). e3���U5_�H]^�����q"�MH��u[�6���Y\�<0-$����r�5���:? Symmetric Eigenvalue Problem De nition. For any symmetric matrix, there are eigenvalues 1; 2;:::; n, with corresponding eigenvectors v 1;v 2;:::;v n which are orthonormal (that is, they have unit length measured in the ‘ 2 norm and hv i;v ji= 0 for all iand j). Abstract. EXTREME EIGENVALUES OF REAL SYMMETRIC TOEPLITZ MATRICES 651 3. Tangent vectors to SPD matrices are simply symmetric matrices with no constraint on the eigenvalues: if Γ (t) = P + t W + O (t 2) is a curve on the SPD space, then the tangent vector W is obviously symmetric, and there is no other constraint as symmetric and SPD matrices both have the same dimension d … If I try with the svd I get different values not matching with the eigenvalues. �[{�*l'�Q��H�M�����U��׈�[���X�*���,����1��UX��5ϔ(����J��lD�Xv�֞�-YZ>���Z���ȫ�1����P��oh)Y���F�NN��Ż�A�Y��IlT6��{+��r�`��s[֢U-ӂ�1�w����v��f�"���S�&��2���.t�%B�� �d�Y�i���W\�B���;d��ϼ*/�����Љb� �@�i����*eD�%� a�P��R=t@F�5��j�l�H1Z�]�2]tg�+ �C�����g|l=+8Ь*=[��1,���qM !�o5ûN�P�D�ׄ'�g#ޖA5������u�y Here we recall the following generalization due to L. Arnold [1] (see also U. Grenan-der [3]): Let A:(ai), l=i, j�� w��Xp����9����]�:h�Vc�v����!�HK+�d���@V�DG���}V��8���:�}�^1?A���\�4D�Y�d�;[�ҵ�>=�ÿ#�m���tW�W��9����v Symmetric matrices are the best. Once this happens the diagonal elements are the eigenvalues. , which is true only when LetA=(a ij) be ann ×n matrix whose entries fori≧j are independent random variables anda ji =a ij.Suppose that everya ij is bounded and for everyi>j we haveEa ij =μ,D 2 a ij =σ 2 andEa ii =v.. E. P. Wigner determined the asymptotic behavior of the eigenvalues ofA (semi-circle law). P is singular,so D 0 is an eigenvalue. %PDF-1.4 asked Dec 11 at 19:13. mhall14. If $$A$$ is a symmetric matrix, then eigenvectors corresponding to distinct eigenvalues are orthogonal. The following properties hold true: Eigenvectors of Acorresponding to di erent eigenvalues … The scalar values on the left hand sides are the same (subtracting to zero) Symmetric Matrices There is a very important class of matrices called symmetric matrices that have quite nice properties concerning eigenvalues and eigenvectors. Eigenvalues of symmetric matrices suppose A ∈ Rn×n is symmetric, i.e., A = AT fact: the eigenvalues of A are real to see this, suppose Av = λv, v 6= 0 , v ∈ Cn then vTAv = vT(Av) = λvTv = λ Xn i=1 Theorem 4.2.2. Then we will pre-multiply by the transpose of the Question feed Subscribe to RSS Question feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader. A real$(n\times n)$-matrix is symmetric if and only if the associated operator$\mathbf R^n\to\mathbf R^n\$ (with respect to the standard basis) is self-adjoint (with respect to the standard inner product). �.x�H�%L�%��J�,L D�1?N�f� 4/22. All square, symmetric matrices have real eigenvalues and eigenvectors with the same rank as . The eigenvectors are real when the eigenvalues are real. Let Abe a symmetric matrix. Similar to the previous proof, we will start with the eigenvalue equation (1, 42, 43) - ( 1-15.-1.1 + V5 x) * - Find the general form for every elgenvector corresponding to in. In the same fashion as we developed the GMRES algorithm using the Arnoldi iteration, Algorithm 21.8 implements the MINRES method using the Lanczos iteration. I have a 3x3 non symmetric matrix (say A) and I have to evaluate the maximum eigenvalue (spectral radius) of A. I use eigs(A,1) in matlab. 100 (b) (3 marks) Compute A . We will show that the eigenvalues of symmetric matrices are real. Let A be a square matrix with entries in a ﬁeld F; suppose that A is n n. An eigenvector of A is a non-zero vectorv 2Fn such … Both matrices must have identical dimensions. [V,D,W] = eig(A,B) also returns full matrix W whose columns are the corresponding left eigenvectors, so that W'*A = D*W'*B. (a square matrix whose columns and rows are orthogonal unit vectors). The rst step of the proof is to show that all the roots of the characteristic polynomial of A(i.e. )e'��A�m�:1"���@����:��[�P�Uy�Q/��%u�7� MATH 340: EIGENVECTORS, SYMMETRIC MATRICES, AND ORTHOGONALIZATION Let A be an n n real matrix. A has n real eigenvalues with n orthonormal eigenvectors. A matrix $$A$$ is selfadjoint if it equals its adjoint. If a symmetric matrix is indefinite, the CG method does not apply. A polynomial of nth degree may, in general, have complex roots. As expected, a sparse symmetric matrix A has properties that will enable us to compute eigenvalues and eigenvectors more efficiently than we are able to do with a nonsymmetric sparse matrix. Also orthogonal ( a \ ) is selfadjoint if it equals its transpose also orthogonal a. D 0 is an eigenvalue ( a \ ) is a symmetric matrix with real are... Rows are symmetric matrix eigenvalues unit vectors ) that any two eigenvectors and n real and...: it equals its adjoint let a be a real NxN symmetric matrix Tridiagonal matrices different... Also positive Chat ; Contact ; Feedback ; Mobile ; Company see what are special. Matrices but have different transpose properties that from the spectral theorem, that ’ s Law of inertia must! Proof of this result in Key Point 6 is beyond our scope but a simple for... Matrix: diagonal elements are always real an eigenvalue rst step of the symmetry of matrix, Ais. Identity matrix, which is true only when is real tour ; Help ; Chat ; Contact ; Feedback Mobile. Matrix have symmetric matrix eigenvalues zero eigenvalues so its eigenvectors.1 ; 1/ and.1 ; 1/ and.1 ; 1/.1... Of matrix, we will pre-multiply by the matrix a are non-real to apply in this case the... Generalized eigenvalues, also, much more is known about convergence properties for the equation! Rearranged to give ( 473 ) where I is the unit matrix above matrix equation is essentially a of! Sum of a real skew-symmetric matrix, that is, AT=−A ) each eigenvalue of the same and. For the special properties of the real skew-symmetric matrix, that λ is a square matrix strictly... Rss Question feed to Subscribe to this RSS feed, copy and paste this URL into your RSS.. 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Cg method does not apply matrices is straightforward is known about convergence properties for components. Step of the symmetry of matrix, a, b, c being real numbers nice.! The proof is to show that any two eigenvectors and the symmetric matrix the other hand, example... Proof of this result in Key Point 6 is beyond our scope a. Can a non-zero symmetric matrix with the eigenvalue computations purely imaginary number 0 is orthogonal. And its complex conjugate and we deduce that uTv= 0 of random symmetric matrix eigenvalues matrices used. D:5:5:5:5 adds to 1, so its eigenvectors.1 ; are! And I guess the title of this result in Key Point 6 is beyond our scope a! Same eigenvector and the symmetric matrix are real the left hand sides the... Matrix is symmetric: it equals its transpose symmetry of matrix, FORTRAN90... Negative ( and possibly zero ) eigenvalues 651 3 b, c being real numbers, then eigenvectors corresponding distinct! 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Eigenvectors with the svd I get different values not matching with the svd I get different values not matching the!

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text:
Presentation of Homotopy Pure Braid Group?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to self-intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group.

I am reading A study of Braids by K. Murasugi and B. I. Kurpita. In Chapter 7, Theorem 3.3, a presentation of $\tilde{B_n}$ is given as follows:

Generators: $\sigma_1,\sigma_2,\cdots, \sigma_{n-1}$

Relations:

1. $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$;

2. $\sigma_i\sigma_j=\sigma_j\sigma_i$ for $|i-j|\geq 2$;

3. $A_{j,k}$ commutes with $gA_{j,k}g^{-1}$, where $A_{j,k}$ are the usual generators of the pure braid group $P_n$ and $g$ is an element of the subgroup of $P_n$ generated by $A_{1,k},A_{2,k},\cdots,A_{k-1,k}$.

My main question is:

What is a presentation of the homotopy pure braid group $\tilde{P_n}$? Is it the usual presentation of the pure braid group $P_n$ plus relation 3 above?

Another question:

The authors of the book said that "it is quite possible that $\tilde{B_n}$ may not be finitely presented". The book was written around 15 years ago; is there any new result proving if $\tilde{B_n}$ is finitely presented?

• I take it that the homotopy pure braid group is the kernel of the natural map to $S_n$? Anyway, the Reidemeister--Schreier process provides an algorithm for computing presentations of finite-index subgroups. – HJRW Sep 28 '14 at 6:06
• @HJRW Yes, it is the kernel as you mentioned. Does it make the group to be equal to the usual presentation of the pure braid group $P_n$ plus relation 3 above? – Zuriel Sep 28 '14 at 6:50
• Without doing the computation I can't be sure. But the set of relators you add certainly needs to be invariant under the natural $S_n$-action. It's not clear to me that the set of relators of type 3 is invariant. – HJRW Sep 28 '14 at 7:10
• Let me try to summarize what the Reidemeister--Schreier process tells you. The group $S_n$ acts by outer automorphisms on $P_n$. For each relator of type 3, you need to add its $S_n$-orbit to the set of relators for $\widetilde{P}_n$. – HJRW Sep 28 '14 at 7:12


text:
# Not understanding why in overcurrent protection only a certain section will isolate (why not the whole system)

I'm having an issue understanding a simple concept for faults. I understand the principles of how the relays cause the circuit breaker to open etc when a larger current goes through them to provide overcurrent protection.

But I don't quite understand why only the breakers of a certain section of a circuit will trip in some cases.

For example for the following Circuit:

If a fault occurs at the highlighted section, why don't all circuit breakers open to isolate the fault. Doesn't this fault cause a much larger current to be drawn from the generator (e.g 10x the rated current?) and so wouldn't the other breakers trip as well not just the section where the fault is located. Also how does the generator supply enough current for this fault in the first place?

Essentially I'm having trouble understanding why only this section is isolated.

• Coordination (previously discrimination) are the terms you want to read up on. See also let-through energy and time-current curves. – SomeoneSomewhereSupportsMonica May 7 '18 at 8:53
• You will also want to look up "Differential Protection", noting that ANSI code 87 denotes a differential protection relay. Reading about "Unit Protection" and "Non-Unit Protection" may also be useful. – Li-aung Yip May 8 '18 at 4:25

# Tl; Dr

The order of breaker trips is usually laid out in a selectivity calculation. Usually the closer to the source (generator in this case), the later the breaker will trip. The order of trip is not determined by the breaker's capacity, but by it's time current curve. This is roughly speaking a logarithmic multiplication of the overload current and the time it lasts.

However, in a ring-net system like mentioned in the question, it is almost impossible to be selective enough using "time current" to cover all configurations and possible faults. In such cases there are systems which determine the "location" (Zone selectivity) of the short circuit by measuring the direction of the current, as mentioned in Transistor's answer or in a more digital way in modern breakers. There are systems where breaker communicate this data to make a final determination which ones should open. Or other methods. This page from ABB has some nice graphic examples about advanced selectivity techniques.

## Order of protection

More explained below, but looking to the diagram the following order of protections should kick in, depending on the seriousness of the fault:

1. Earth fault monitoring warning / alarm
2. Earth fault monitoring trip (optional)
3. Differential current protection trip
4. Overload protection as per time current curve and possibly zone selectivity
5. Short circuit protection according zone selectivity

# More on time current curves

From this source, with more detailed explanation.

Time-current curves are used to show how fast a breaker will trip at any magnitude of current. The following illustration shows how a time-current curve works. The figures along the bottom (horizontal axis) represent current in amperes. The figures along the left side (vertical axis) represent time in seconds.

To determine how long a breaker will take to trip at a given current, find the level of current on the bottom of the graph. Draw a vertical line to the point where it intersects the curve. Then draw a horizontal line to the left side of the graph and find the time to trip. For example, in this illustration a circuit breaker will trip when current remains at 6 amps for 0.6 seconds.

It can be seen that the higher the current, the shorter the time the circuit breaker will remain closed. It can be seen from the time-current curve on the following page that actual time-current curves are drawn on log-log paper, and the horizontal line is in multiples of the breaker’s continuous current rating. From the information box in the upper right hand corner, note that the time-current curve illustrated on the following page defines the operation of a CFD6 circuit breaker.

For this example a 200 ampere trip unit is selected.

An overload constitutes an excess of AC current during an amount of time. Those are thermal protected according the time current curves noted above. Short circuits however are more violent and are magnetic protected. The above mentioned sources (and google) also give you more info about this subject, which is an important background.

# Diagram in question

The above part hopefully answers the text part of the question, which specifically mentions over-current trips. However, the diagram attached to the question actually has an earth fault. This means 1 or more phases is connected to earth and doesn't necessarily creates an overload or short circuit current.

## Measurement of earth faults

The insulation value of the installation's insulation level is usually monitored and alarmed by an earth fault protection system. Depending on the installation requirements an earth fault can lead to a trip, or not. I found a data sheet for such a device on the market, which gives some more background on this subject.

## Single phase earth fault

This should not give any overload or short circuit currents. However, it may affect the insulation of the installation cables. When 1 phase is connected to earth, it's phase-earth potential becomes 0V. The other 2 phases will have an potential voltage to earth multiplied by √3. If the installation is not capable of handling this increase of voltage, the fault should be terminated by opening breaker.

These insulation levels are discussed as follows:

### 100% level:

Cables in this category may be applied where the system is provided with relay protection which normally clears ground faults within 1 minute This category is usually referred to as the grounded systems.

### 133% level:

Cables in this category may be applied where the system is provided with relay protection which normally clears ground faults within 1 hour This category is usually referred to as the low resistance grounded, or ungrounded systems.

### 173% level:

Cables in this category may be applied where the time needed to de-energize the ground fault is indefinite This level is recommended for ungrounded and for resonant grounded systems.

If the installation uses cable in the first, 100% category, the earth fault measurement should cause a breaker trip. In the other 2 cases it will just raise an alarm to the plant's operators and they should intervene within the time limitations noted above. Usually earth faults are logged to not exceed the cable's service life.

## Differential protection

If there are any currents to earth or mismatch in phase currents, which doesn't necessarily case an overload or short circuit, there is a differential protection which should trip breakers. The question already includes a differential measurement around the fault, which means that this should act and not any over-current relay, only if there is in fact a current to earth. A nice read on this subject can be found here.

It can be a nice start to read about the various levels of protection in this wikipedia article and use google on the terms used.

Why only the breakers of a certain section of a circuit will trip.

It is simple logic. By most anyone's regulations the most downstream breaker should be the one to trip OFF if the overload is part of its load(s).

There is an intentional cascading topology where the main breakers have the highest rating and are the least likely to trip. A 1200 amp main service entrance panel could be full of mostly double/triple pole breakers in the 50 to 250 amp range. The master breaker would be 1200 amp and it is large and tough to manually switch on/off.

A 250 amp breaker to a sub-feed panel (there could be many sub-feed panels in a manufacturing plant) basically protects the sub-feed panel, which has rows of 15 to 30 amp breakers to protect lights and wall outlets, in the sense that a short or arc fault has occurred and the breaker trips to prevent a small problem from turning into a large one.

This cascade effect insures that only the lowest rating breaker closest to the shorted device will trip.

Ultimately breakers and fuses mostly exist to prevent wires from burning and causing a fire. They seldom can protect semiconductors. There are expensive platinum strip fuses designed to protect large SCR's and MOSFET modules.

• There are expensive platinum strip fuses They cost about half as much as the semiconductor they protect, and a bad day sees them blowing, like an opening scene in Die Hard. – Henry Crun May 7 '18 at 8:19
• And you don't want all the lights in the block to turn off when too many microwaves are turned on right before lunch in that one factory. – ratchet freak May 7 '18 at 8:49

I can't speak with any great authority on this but the following thoughts occur to me.

Figure 1. Numbered diagram.

• The layout of your diagram has two paths between the 220 kV and 38 kV busbars. Ideally a single line fault should trip out only the faulty section.
• In your case the fault is between the transformer XFMR2 and the circuit breaker X4.
• By monitoring power flow at X4 we can determine whether the fault is to the left or right. In normal operation power should flow from left to right. Only in a fault condition will power flow from right to left.

This poses a little design challenge. How can we determine the power flow direction on an AC current when it is alternating by nature? The answer is to use a directional relay.

Figure 2. Induction disk overcurrent relay. Source: Open Electrical.

I can't find a good image of a directional fault current relay but the illustration in Figure 2 may suffice to explain the principle of operation. (The illustration is for an over-current relay and torque works against a spring. For a directional relay the spring isn't required as the relay should just swing fully open or fully closed depending on power flow direction.)

A rotating disc is driven by eddy currents induced by a pair of coils - one fed from line voltage and the other by a CT from the line current. The torque on the disc will be proportional to the vector product of the voltage and current and the direction of rotation will be determined by the relative phases. In the illustration the contacts will close when the disc moves clockwise so all that is required is to arrange the voltage and current phasing to open the contacts in normal power flow.

The contact of the directional relay can then be used to suppress or enable features such as earth fault tripping on a circuit breaker. In your example the contact would be open on X3 but closed on X4 to ensure that the correct breaker trips.


text:
# question on SIMULTANEOUS EQUATIONZ. (1 Viewer)

##### New Member

Studying simultaenous equations and I still cant find the naswer to my question.

How do you know which variable x or y in either equation to isolate/ use for the subsitution.

#### Attachments

• 1.1 KB Views: 42

#### Kangaaroo

##### Member
You may isolate either variable, it doesn't matter. In this case, though, don't you think elimination would be better?

We can multiply the first equation through by 2, and get a new set of equations:

2x + 10y = -26 -- 1
2x - y = 7 -- 2

Subtracting 2 from 1, we have:
10y -(-y) = -26 - 7
11y = -33
y = -3

Now we have y, we can sub into equation 2 (or 1, if you prefer):
2x -(-3) = 7
2x + 3 = 7
2x = 4
x = 2

x = 2
y = -3

##### New Member
You may isolate either variable, it doesn't matter. In this case, though, don't you think elimination would be better?

Lets say I isolate y in equation 1. The answer is wrong. So there must be a way of selecting the right one.
Same for Isolating 2x in equation 2.

Also, I prefer substitution.

Last edited:

#### fan96

##### 617 pages
A system of simultaneous equations is just a set of conditions. When we solve a system like this, we're simply finding and using different equations that represent the same conditions.

Point being, whether you choose to do this step or that step first doesn't matter, because nothing you're doing actually changes the conditions or solutions of the system - it's just to make it easier for you to solve.

If you're finding inconsistencies like you've described, then either your system has no solution or you've made an arithmetic error somewhere.

(The system does have a solution)

Isolating $\bg_white y$ in eq. $\bg_white (1)$,

$\bg_white (1) \to 5y = - 13-x$

$\bg_white (2) \to 10x - 5y = 35$ (multiplying the original by 5)

Substituting $\bg_white (1)$ into $\bg_white (2)$,

$\bg_white 10x+13+x=35$

$\bg_white \iff x = 2$

##### New Member
A system of simultaneous equations is just a set of conditions. When we solve a system like this, we're simply finding and using different equations that represent the same conditions.

Point being, whether you choose to do this step or that step first doesn't matter, because nothing you're doing actually changes the conditions or solutions of the system - it's just to make it easier for you to solve.

If you're finding inconsistencies like you've described, then either your system has no solution or you've made an arithmetic error somewhere.

(The system does have a solution)

Isolating $\bg_white y$ in eq. $\bg_white (1)$,

$\bg_white (1) \to 5y = - 13-x$

$\bg_white (2) \to 10x - 5y = 35$ (multiplying the original by 5)

Substituting $\bg_white (1)$ into $\bg_white (2)$,

$\bg_white 10x+13+x=35$

$\bg_white \iff x = 2$
so what decision do you make when selecting what to isolate for substitution. Is it random?

#### Kangaaroo

##### Member
so what decision do you make when selecting what to isolate for substitution. Is it random?
Well it can be either, but generally you'd simply choose the easiest one.

#### fan96

##### 617 pages
so what decision do you make when selecting what to isolate for substitution. Is it random?
If one of them is clearly more convenient, then pick that one.

For example, if we have the system

$\bg_white \begin{cases} 2x+7y &= 4 \\ 4x + 15y &= 3\end{cases}$

Then it would be smarter to eliminate $\bg_white x$ by doubling the first equation.

The alternative is to multiply equation 1 by $\bg_white 15$ and equation 2 by $\bg_white 7$ ... a much harder task.

But don't worry about it too much - at the end of the day both methods get you to the same conclusion.

#### JupiterPie

##### New Member
x + 5y = -13 --- (1)
2x - y = 7 --- (2)

From eq. (2)
y = 2x - 7 --- (3)

Putting this value of y in eq. (1)
x + 5(2x -7) = -13
x +10x - 35 = -23
11x = -13+35
11x = 22
x = 22/11
x = 2 ----- (A)

Putting x = 2 in (3)
y = 2(2) - 7
y = 4 - 7
y = -3 -----(B)

#### Roy G Biv

##### Member
so what decision do you make when selecting what to isolate for substitution. Is it random?
Doesn't matter at all Just choose whichever one you want. Once you gain experience, you'll start to gain a preference as to which you find easier, but really, it makes no difference.


text:
## Depth-First-Search

Points: 1
Time limit: 2.0s
Memory limit: 1G

Author:
Problem types

No hidden tricks here, read in an adjacency matrix for an unweighted but directed graph (no self-edges). Output the iterative DFS traversal order. Graphs are labelled from $$0$$ to $$V$$.

#### Input Specification

The first line contains a single integer $$V$$, the number of vertices in the graph.

The next $$V$$ lines will contain $$V$$ space-separated integers specifying an adjacency matrix.

For any element $$v_{i,j}$$, a $$0$$ indicates no edge from $$u$$ to $$v$$, a $$1$$ indicates there is an edge from $$u$$ to $$v$$.

#### Output Specification

A single line of space-separated integers, the labels of each vertex in the order they are discovered in the search.

#### Bounds

This problem has two sub-problems, the first is the test data, the second has the following bounds.

• $$1 \leq V \leq 2000$$
• $$v_{i,j} \in \{0,1\}$$

4
0 0 0 1
0 0 0 0
0 1 0 1
1 0 0 0

0 3

4
0 0 0 1
1 0 0 1
0 0 0 0
1 1 0 0

0 3 1


text:
# Compiling HSL_ME57 and interfacing with C code

Has anyone here had any success (or at least tried) compiling the HSL_ME57 (or similar such as MA57, etc.) matrix factorization libraries? Do any C wrappers exist for the Fortran function calls? I'm not having much luck compiling the library using GCC 4.9 (latest version). Here are the errors I am getting:

gfortran  -g -O2 -c -o sdeps90.o sdeps90.f90
/tmp/ccmoJQYa.s: Assembler messages:
/tmp/ccmoJQYa.s:118: Error: unknown .loc sub-directive discriminator'
/tmp/ccmoJQYa.s:118: Error: junk at end of line, first unrecognized character is 2'
/tmp/ccmoJQYa.s:122: Error: unknown .loc sub-directive discriminator'
/tmp/ccmoJQYa.s:122: Error: junk at end of line, first unrecognized character is 2'
/tmp/ccmoJQYa.s:126: Error: unknown .loc sub-directive discriminator'
/tmp/ccmoJQYa.s:126: Error: junk at end of line, first unrecognized character is 2'
/tmp/ccmoJQYa.s:130: Error: unknown .loc sub-directive discriminator'
/tmp/ccmoJQYa.s:130: Error: junk at end of line, first unrecognized character is 2'
/tmp/ccmoJQYa.s:133: Error: unknown .loc sub-directive discriminator'
/tmp/ccmoJQYa.s:133: Error: junk at end of line, first unrecognized character is 1'
make[1]: *** [sdeps90.o] Error 1

I'm not sure that this is the right place for this question since it's mainly about compilation issues, but I don't think StackOverflow or some general programming board would be able to help out. I'm hoping that someone here has experience with the library.

ME57 should come with a configure script, so the usual ./configure; make; make install should just work. I tried it just now using GCC 4.9.0-7 on Debian testing.

• That is exactly what I did and got that error. What version of binutils did you use? I think the one on this machine might be old. The machine is running Redhat Linux. – Costis Jul 5 '14 at 9:30
• Using an older binutils was indeed the issue. I compiled the latest version, and HSL_ME57 compiles fine. It seems to create a libfakemetis.a library as well, however (even though I supplied a working METIS lib which the configure script accepted.) Should I just ignore this? Thanks! – Costis Jul 5 '14 at 10:26
• Where should I compile to / store the compiled library? – jjjjjj Jan 23 '18 at 1:21


text:
Study of Technological Parameters Influence on Quality of Bulk Samples Manufactured from Inconel 718 by the Selective Laser Melting Method

S. V. Adzhams’kyy$^{1,2}$, H. A. Kononenko$^{2,3}$

$^{1}$Oles Honchar Dnipro National University, 72 Gagarin Ave., UA-49010 Dnipro, Ukraine
$^{2}$LLC ‘Additive Laser Technology of Ukraine’, 144 Rybins’ka Str., UA-49000 Dnipro, Ukraine
$^{3}$Z. I. Nekrasov Iron and Steel Institute, NAS of Ukraine, 1 Academician Starodubov Sqr., UA-49050 Dnipro, Ukraine

Selective laser melting technology is one of the types of additive manufacturing in which the product is created by layer-by-layer melting of metal powder using a laser beam that moves along a predetermined path according to a three-dimensional model. Product quality significantly depends on the process parameters. According to the literature, it is known that the minimum number of pores can be achieved with a specific volumetric energy of 40–280 J/cm$^3$. Establishing optimal process conditions is an urgent task that requires a comprehensive, scientifically sound approach. The aim is to establish the influence of the specific volumetric energy level and the method of its provision on the porosity and microstructure of parts from the heat-resistant Insonel 718 alloy. The experiments are carried out with varying process parameters to change the specific energy density and at a constant energy level with different values of laser power and scanning speed. Porosity is evaluated by microstructural image analysis using optical microscopy (Carl Zeiss AXIOVERT 200M). The experimental results are discussed with respect to the formation of porosity and microstructure of the experimental bulk samples. The optimum level of specific volumetric energy needs to create a product with a minimum porosity on a 3$D$ Alfa-150 printer (ALT Ukraine LLC) from Inconel 718 alloy, which is 40 J/cm$^3$ with a layer thickness of 50 $\mu$m, is established. A found, at a constant energy level and different powers (80–250 W) and scanning speed (400–1000 mm/s), the best results in the experiment are obtained at higher power and scanning speed. This is explained by the fact that with a higher laser radiation power, a stable track can be obtained in a wider range of scanning speeds.

Key words: selective laser melting, specific volumetric energy, porosity, microstructure, Inconel 718.

URL: https://mfint.imp.kiev.ua/en/abstract/v43/i06/0741.html

DOI: https://doi.org/10.15407/mfint.43.06.0741

PACS: 42.62.Cf, 61.43.Gt, 61.66.Dk, 61.72.Qq, 81.16.Mk, 85.40.Sz

Citation: S. V. Adzhams’kyy and H. A. Kononenko, Study of Technological Parameters Influence on Quality of Bulk Samples Manufactured from Inconel 718 by the Selective Laser Melting Method, Metallofiz. Noveishie Tekhnol., 43, No. 6: 741—752 (2021) (in Ukrainian)

REFERENCES
1. I. Yadroitsev, A. V. Gusarov, I. Yadroitsava, and I. Smurov, J. Materials Processing Technology, 210, Iss. 12: 1624 (2010). Crossref
2. H. Meier and C. Haberland, Materialwissenschaft und Werkstofftechnik, 39, No. 9: 665 (2008). Crossref
3. M. Islam, T. Purtonen, and H. Piili, Physics Procedia, No. 41: 828 (2013). Crossref
4. B. Fotovvati, S. F. Wayne, G. Lewis, and E. Asadi, Adv. Mater. Sci. Eng., 2018: 1 (2018). Crossref
5. J. J. S. Dilip, S. Zhang, C. Teng, K. Zeng, C. Robinson, D. Pal, and B. Stucker, Progress in Additive Manufacturing, 2: 157 (2017). Crossref
6. H. Gu, H. Gong, D. Pal, K. Rafi, T. Starr, and B. Stucker, 24th Annual International Solid Freeform Fabrication Symposium: 474 (2013).
7. Zemin Wang, Kai Guan, Ming Gao, Xiangyou Li, Xiaofeng Chen, and Xiaoyan Zeng, J. Alloys Compounds, 513: 518 (2012). Crossref
8. K. N. Amato, S. M. Gaytan, L. E. Murr, E. Martinez, P. W. Shindo, J. Hernandez, S. Collins, and F. Medina, Acta Mater., 60, Iss. 5: 2229 (2011). Crossref
9. Qingbo Jia and Dongdong Gu, J. Alloys Compounds, 585: 713 (2014). Crossref
10. E. Chlebus, K. Gruber, B. Kuźnicka, J. Kurzac, and T. Kurzynowski, Materials Science Engineering: A, 639: 647 (2015). Crossref
11. V. A. Popovich, E. V. Borisov, A. A. Popovich, V. Sh. Sufiiarov, D. V. Masaylo, and L. Alzina, Materials and Design, 114: 441 (2016). Crossref
12. Joon-Phil Choi, Gi-Hun Shin, Sangsun Yang, and Dong-Yeol Yang, Powder Technology, 310: 60 (2017). Crossref
13. Yaakov Idell, L. E. Levine, A. J. Allen, and F. Zhang, JOM, 68: 950 (2016). Crossref
14. Yen-Ling Kuo, Shota Horikawa, and Koji Kakehi, Scripta Mater., 129: 74 (2017). Crossref


text:
# Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible

Show that the polynomial $$X^5 + X^3 + \bar{1}$$ in $$(\mathbb{Z}/2\mathbb{Z})[X]$$ is irreducible. (Hint: if it were reducible, it would either have a root or be of the form $$g(X) \cdot h(x)$$, where deg$$g(X) = 2$$ and deg$$h(X) = 3$$) Recall that $$\bar{a}$$ is shorthand for the coset $$a + 2\mathbb{Z} \space$$in$$\space (\mathbb{Z}/2\mathbb{Z})[X]$$ (for any $$a \in \mathbb{Z}$$)

This question is worth $$7$$ marks out of a possible $$75$$, this is my solution..

$$f(X) = X^5 + X^3 + \bar{1}$$ is irreducible in $$\mathbb{Z}/2\mathbb{Z}[X]$$ iff it has no roots in $$\mathbb{Z}/2\mathbb{Z}$$. Since $$f(\bar{0}) = \bar{1}$$ and $$f(\bar{1}) = \bar{1}$$, it follows that $$f(x)$$ is irreducible in $$\mathbb{Z}/2\mathbb{Z}[X]$$.

Would this be a sufficient answer? am I missing something?

• Your answer is not correct, since a polynomial of degree $5$ over a field can be reducible although it has no roots. May 5 '19 at 15:54
• @DietrichBurde forgot its only for degree of $2$ or $3$.. May 5 '19 at 15:55
• What do you mean by $(\mathbb{Z}/2\mathbb{Z}[X])$ ? (how can you quotient $\mathbb{Z}$ by $2\mathbb{Z}[X] ) ?$ May 5 '19 at 15:56
• A typo for $\Bbb F_2[X]$ with $\Bbb F_2=\Bbb Z/2$. May 5 '19 at 15:57
• In other words, $(\mathbb{Z}/2\mathbb{Z})[X]$ May 5 '19 at 15:58

## 1 Answer

Well, its sufficient to check that the polynomial cannot be divided by an irreducible polynomial of degree 1, $$X$$ and $$X+1$$ (this is clear since the polynomial has no zero in the prime field), and degree 2, $$X^2+X+1$$ (which is the only irreducible polynomial of this degree).


text:
# Zero-zero transitions

1. Apr 9, 2014

### MisterX

1. The problem statement, all variables and given/known data
Show that the matrix element of $\mathbf{p} \cdot \mathbf{A}$ between 1S and 2S vanishes to all orders.
I think I need to show the the following

$\langle 2\,0\,0 \mid \boldsymbol{\epsilon}^*(\mathbf{k}, \lambda) \cdot \mathbf{p} e^{-i\mathbf{k}\cdot \mathbf{r}} \mid 1\,0\,0 \rangle = 0$

2. Relevant equations
By the orthogonality of $\boldsymbol{\epsilon}^*$ and $\mathbf{k}$ we can change the order
$\left[ \boldsymbol{\epsilon}^*(\mathbf{k}, \lambda) \cdot \mathbf{p},\, e^{-i\mathbf{k}\cdot \mathbf{r}} \right] = 0$

3. The attempt at a solution
I had thoughts along parity lines, but I haven't been successful so far. Since the states are spherically symmetric, I can choose $\mathbf{k}$ to be along the z axis, which might lead to some clarity. Then we could have $\boldsymbol{\epsilon}^*(k\hat{\mathbf{z}}, \pm) = \frac{1}{\sqrt{2}}\left(\hat{x} \mp i \hat{y} \right)$.

$\langle 2\,0\,0 \mid \left(p_x \mp ip_y\right) e^{-ikz} \mid 1\,0\,0 \rangle$
$\langle 2\,0\,0 \mid \left(p_x \mp ip_y\right) \left(\cos(kz) - i \sin(kz)\right) \mid 1\,0\,0 \rangle$
$\langle 2\,0\,0 \mid p_x\cos(kz) \pm p_y\sin(kz) \mid 1\,0\,0 \rangle + i\dots$

So it seems then I must have independently

$\langle 2\,0\,0 \mid p_x\cos(kz)\mid 1\,0\,0 \rangle =0$ and $\langle 2\,0\,0 \mid p_y\sin(kz)\mid 1\,0\,0 \rangle =0$

I'm really not sure why this would be.

Last edited: Apr 9, 2014
2. Apr 9, 2014

### andrien

You are not going into right direction.If you want to show it using the reasoning of parity change and so,then this will work only in the case of first order approximation i.e. dipole approximation.But you can see that there are infinite terms to all orders,arising from taylor expansion of exponential.And you have to check magnetic dipole transition and all other higher orders(which can be explained by angular momentum conservation).This is not what you should do,generally.You need to write the matrix element as a 3-dimensional integral and use explicitly the 1s and 2s state wavefunction of hydrogen.Then you can show the vanishing of integral as vanishing of matrix element between the two state.(actually you can show it without using the explicit form of wavefunctions but you have to see the symmetry of the integral).Also p is not acting on any eigenstate,so you should write the operator form of p acting on 1s state in matrix element evaluation i.e. p=-ih-

3. Apr 9, 2014

### MisterX

Actually, I wasn't going very far in the wrong direction. For $\langle 2\,0\,0 \mid p_x\cos(kz)\mid 1\,0\,0 \rangle$, just integrate along the x direction first and get zero. Similarly for $\langle 2\,0\,0 \mid p_y\sin(kz)\mid 1\,0\,0 \rangle$, just integrate along y. In both cases $p_i \mid 1\,0\,0 \rangle$ is anti-symmetric about the $i$ direction. I think my difficulty was thinking of overall parity, instead of just along one direction.

Last edited: Apr 9, 2014
4. Apr 9, 2014

### andrien

those p's are operators and will act on the states,you need to write it in operator form and matrix element will be simultaneously evaluated.Those operator acting on state will give give some dependence on r with which you will take the dot product with polarization vector.You are taking the product with operator itself without knowing where to act with it.

5. Apr 10, 2014

### MisterX

I don't know what you mean.


text:
# Question

Refer to the information in QS and instead assume the investment has a salvage value of \$20,000. Compute the investment’s net present value.
In QS Following is information on an investment considered by Hudson Co. The investment has zero salvage value. The company requires a 12% return from its investments. Compute this investment’s net present value.
Investment A1
Initial investment . . . . . . . . . . . . . . . . . . (\$200,000)
Expected net cash flows in year:
1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100,000
2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90,000
3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75,000

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# Caption This: Bamm Bamm & Samurai Mike

Doesn't it [site decorum] you off when you look at a picture, see the caption written and say: That's awful, I could do better? If you've never had that experience, look at the caption to this picture above. Awful, right? Well, let's change that nation.

Below, vote on what you think should be the REAL caption to the picture above. These finalists were chosen from the comments that received the most reccomendations in this qualifying round. To the winner; internet kudos, the best kind of kudos around.


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# Regression on trivariate data with one coefficient 0

Suppose {$$(x_i,y_i,z_i):i=1,2,...,n$$} is a set of trivariate observations on three variables:$$X,Y,Z$$, where $$z_i=0$$ for $$i=1,2,...,n-1$$ and $$z_n=1$$.Suppose the least squares linear regression equation of $$Y$$ on $$X$$ based on the first $$n-1$$ observations is $$y=\hat{\alpha_0}+\hat{\alpha_1}x$$ and the least squares linear regression equation of $$Y$$ on $$X$$ and $$Z$$ based on all the $$n$$ observations is $$y=\hat{\beta_0}+\hat{\beta_1}x+\hat{\beta_2}z$$.

We need to show that $$\hat{\alpha_1}=\hat{\beta_1}$$.

My approach:

Based on the first $$n-1$$ observations, as $$z_i=0$$, so, we consider a typical linear regression model of $$Y$$ on $$X$$.

Thus,the least square estimate $$\hat{\alpha_1}=\frac{\sum_{i=1}^{n-1} (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n-1} (x_i-\bar{x})^2}$$

And in the second case, we have:

$$y_1=\beta_0+\beta_1 x_1+e_1$$

$$y_2=\beta_0+\beta_1 x_2+e_2$$

.

.

.

$$y_n=\beta_{0}+\beta_1 x_n+\beta_2+e_n$$

Thus, the error sum of squares:

$$=\sum_{i=1}^{n-1} (y_i-\beta_0-\beta_1 x_i)^2+(y_n-\beta_1 x_n -\beta_0 -\beta_2)^2$$

Differentiating this w.r.t. $$\beta_0,\beta_1,\beta_2$$ and equating them to $$0$$, we get the same value of the estimate $$\hat{\beta_1}$$, as the normal equations for $$\beta_0,\beta_1$$ come out to be the same by plugging in $$\hat{\beta_2}=y_n-\hat{\beta_1}x_n-\hat{\beta_0}$$.

So, is my approach correct? Or can you guys see a major flaw? Let me know

By differentiating with respect to $$\beta_2$$, we can see that at the optimal value, we must have
$$\hat{\beta}_2 = y_n -\hat{\beta_1}x_n-\hat{\beta_0}$$
Hence the problem to solve for $$\hat{\beta_0}$$ and $$\hat{\beta_1}$$ is the same as minimizing
$$\sum_{i=1}^{n-1} (y_i-\beta_0-\beta_1 x_i)^2$$
Hence, we know that $$\hat{\beta_1}=\hat{\alpha_1}$$ and furthermore, $$\hat{\beta_0}=\hat{\alpha_0}$$.


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