text:
# Difference between revisions of "Number theory"

Main Article
Talk
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

Number theory is a branch of mathematics devoted primarily to the study of the integers. Any attempt to conduct such a study naturally leads to an examination of the properties of that which integers are made of (namely, prime numbers) as well as the properties of objects made out of integers (such as rational numbers) or defined as generalisations of the integers (such as, for example, algebraic integers).

Integers can be considered either as such or as solutions to equations (diophantine geometry). Some of the main questions are those of distribution: questions, say, on patterns or their absence (in the primes or other sequences) or, more generally, questions on size, number and growth. Such matters are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode them in some fashion (analytical number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (diophantine approximation).

The older term for number theory is arithmetic; it was superseded by "number theory" in the nineteenth century, though the adjective arithmetical is still fully current. By 1921, T. Heath had to explain: "By arithmetic Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers."[1] The general public now uses arithmetic to mean elementary calculations, whereas mathematicians use arithmetic as this article shall, viz., as an older synonym for number theory. (The use of the term arithmetic for number theory has regained some ground since Heath's time, arguably in part due to French influence.[2])

## Origins

### The dawn of arithmetic

The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 (Larsa, Mesopotamia, ca. 1800 BCE) contains a list of "Pythagorean triples", i.e., integers $\scriptstyle (a,b,c)$ such that $\scriptstyle a^2+b^2=c^2$. The triples are too many and too large to have been obtained by brute force.

The Plimpton 322 tablet.

The table's outlay suggests[3] that it was constructed by means of what amounts, in modern language, to the identity

$\left(\frac{1}{2} \left(x - \frac{1}{x}\right)\right)^2 + 1 = \left(\frac{1}{2} \left(x + \frac{1}{x}\right)\right)^2,$

which is implicit in routine Old Babylonian exercises[4]. If some other method was used, the triples were first constructed and then reordered by $c/a$, presumably for actual use as a "table", i.e., with a view to applications.

We do not know what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly flowered only later. It has been suggested instead that the table was a source of numerical examples for school problems[5][6]. Alternatively, the table could have served to demonstrate a method for solving a problem of interest to one's students or fellow scribes.

While Babylonian number theory - or what survives of Babylonian mathematics that can be called thus - consists of this single, striking fragment, Babylonian algebra (in the secondary-school sense of "algebra") was exceptionally well developed. Iamblichus states that Pythagoras learned mathematics from the Babylonians, and there is no very strong reason to believe otherwise. (Much earlier sources attest to the travels and studies of Thales and Pythagoras in Egypt.)

Pythagoras was a mystic who gave great importance to the odd and the even. Euclid IX 21--34 is very probably Pythagorean; it is very simple material ("odd times even is odd", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that $\scriptstyle \sqrt{2}$ is irrational. The discovery that $\scriptstyle \sqrt{2}$ is irrational is credited to the early Pythagoreans (pre-Theodorus). By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect. It is only here that we can start to speak of a clear, conscious division between numbers (integers and the rationals - the subjects of arithmetic) and lengths (real numbers, whether rational or not).

The Pythagorean tradition spoke also of so-called polygonal numbers and perfect numbers. The study of the latter made some look at the divisors of integers. While square numbers, cubic numbers, etc., are seen now as more natural than triangular numbers, square numbers, pentagonal numbers, etc., the study of the sums of triangular and pentagonal numbers would prove very fruitful in the early modern period (17th to early 19th century).

We know of no clearly arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both. Imperial China has left us a single result in arithmetic, namely, the basic statement known as the Chinese remainder theorem to all students of number theory. The result appears as an exercise in Sun Zi's Suan Ching (also known as Sun Tzu's Mathematical Classic; 3rd, 4th or 5th century CE). A sketch of a method of solution is given: Sun Zi finds a number given its residues much as a modern would, though we do not know whether he had a good way of taking an important intermediate step - namely, finding the inverse of a number modulo another. (The earliest full solution known to this last problem is Āryabhaṭa's; see below.)

There is also some numerical mysticism in Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere. Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation.

### Plato and Euclid

Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period. In the case of number theory, this means, large and by, Plato and Euclid, respectively.

Plato had a keen interest in mathematics, and distinguished clearly between arithmetic and calculation. (By arithmetic he meant, in part, theorising on number, rather than what arithmetic or number theory have come to mean.) It is through one of Plato's dialogues -- namely, Theaetetus -- that we know that Theodorus had proven that $\scriptstyle \sqrt{3}, \sqrt{5}, \dots, \sqrt{17}$ are irrational. Theaetetus was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kind of inconmensurables, and was thus arguably a pioneer in the study of number systems. (Book X of Euclid is described by Pappus as being largely based on Theaetetus's work.)

... Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic thereto; in particular, he gave the first known proof of the infinitude of primes ...

### Diophantus

Very little is known about Diophantus of Alexandria; he probably lived in the third century CE, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's Arithmetica survive in the original Greek; four more books survive in an Arabic translation. The Arithmetica is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form $\scriptstyle f(x,y)=z^2$ or $\scriptstyle f(x,y,z)=w^2$. Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found.

One may say that Diophantus was studying rational points -- i.e., points whose coordinates are rational -- on curves and varieties; however, unlike the Greeks of the Classical period, who did what we would now call basic algebra in geometrical terms, Diophantus did what we would now call basic algebraic geometry in purely algebraic terms. In modern terms, what Diophantus does is to find rational parametrisations of many varieties; in other words, he shows how to obtain infinitely many rational numbers satisfying a system of equations by giving a procedure that can be made into an algebraic expression (say, $\scriptstyle x=f(r,s)$, $\scriptstyle y=g(r,s)$, $\scriptstyle z=h(r,s)$, where $\scriptstyle f$, $\scriptstyle g$ and $\scriptstyle h$ are polynomials or quotients of polynomials).

Diophantus generally seems to prefer procedures that lead to invertible rational parametrisations; when his procedure leads to a non-invertible rational parametrisation (a so-called unirrational parametrisation) it is generally the case that one can show nowadays that no invertible rational parametrisation is possible. However, Diophantus never inverts his parametrisations (or procedures, as he would more likely have seen them). He would have seen the distinction between his invertible and non-invertible rational parametrisations only in so far as the procedures that give the latter make special assumptions, and (arguably) in so far as non-invertible rational parametrisations miss some rational points (very easily found ones in some of Diophantus's examples).

Diophantus also studies the equations of some non-rational curves, for which no rational parametrisation is possible. He manages to find some rational points on these curves -- elliptic curves, as it happens, in what seems to be their first known occurence -- by means of what amounts (in geometrical terms) to a tangent construction. (He also resorts to what could be called a special case of a secant construction.)

While Diophantus is concerned largely with rational solutions, he assumes some results on integer numbers; in particular, he seems to assume that every integer is the sum of four squares, though he never states as much explicitly.

### The Indian school: Āryabhaṭa, Brahmagupta, Bhāskara

... Brahmagupta (628 CE) started the systematic study of indefinite quadratic equations -- in particular, the misnamed Pell equation, in which Archimedes may have first been interested. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general method (the cakravāla) for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost) and Bhāskara (twelfth century).

### Arithmetic in the Islamic golden age

In the early ninth century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (generally presumed to be Brahmagupta's), thus giving rise to the rich tradition of Islamic mathematics. Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820-912). Part of the treatise al-Fakhri (by al-Karajī, 953 - ca. 1029) builds on it to some extent. Al-Karajī's contemporary Ibn al-Haytham knew[7] what would later be called Wilson's theorem, which, arguably, was thus the first clearly non-trivial result on congruences to prime moduli ever known.

Other than a treatise on squares in arithmetic progression by Fibonacci - who lived and studied in north Africa and Constantinople during his formative years, ca. 1175-1200 - no number theory to speak of was done in western Europe while it went through the Middle Ages. Matters started to change in Europe in the late Rennaissance, thanks to a renewed study of the works of Greek antiquity. The key catalyst was the textual emendation and translation into Latin of Diophantus's Arithmetica.

## Subfields

### Analytic number theory

Analytic number theory is generally held to denote the study of problems in number theory by analytic means, i.e., by the tools of calculus. Some would emphasize the use of complex analysis: the study of the Riemann zeta function and other L-functions can be seen as the epitome of analytic number theory. At the same time, the subfield is often held to cover studies of elementary problems by elementary means, e.g., the study of the divisors of a number without the use of analysis, or the application of sieve methods. A problem in number theory can be said to be analytic simply if it involves statements on quantity or distribution, or if the ordering of the objects studied (e.g., the primes) is crucial. Several different senses of the word analytic are thus conflated in the designation analytic number theory as it is commonly used.

The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or the Hardy-Littlewood conjectures), the Waring problem and the Riemann Hypothesis. Some of the most important tools of analytic number theory are the circle method, sieve methods and L-functions (or, rather, the study of their properties).

One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalisations of prime numbers living in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalisations of the Riemann zeta function, an all-important analytic object that controls the distribution of prime numbers.

### Algebraic number theory

Algebraic number theory studies fields of algebraic numbers, which are generalisations of the rational numbers. (Briefly, an algebraic number is any complex number that is a solution to some polynomial equation $\scriptstyle f(x)=0$ with rational coefficients; for example, every solution $x$ of $\scriptstyle x^5 + (11/2) x^3 - 7 x^2 + 9 = 0$ (say) is an algebraic number.) Fields of algebraic numbers are also called number fields.

It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and norms in quadratic fields. (A quadratic field consists of all numbers of the form $\scriptstyle a + b \sqrt{d}$, where $a$ and $b$ are rational numbers and $d$ is a fixed rational number whose square root is not rational.) For that matter, the 11th-century cakravāla method amounts - in modern terms - to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.

The grounds of the subject as we know it were set in the late nineteenth century, when ideal numbers, the theory of ideals and valuation theory were developed; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals and $\scriptstyle \sqrt{-5}$, the number $6$ can be factorised both as $\scriptstyle 6 = 2 \cdot 3$ and $\scriptstyle 6 = (1 + \sqrt{-5}) ( 1 - \sqrt{-5})$; all of $2$, $3$, $\scriptstyle 1 + \sqrt{-5}$ and $\scriptstyle 1 - \sqrt{-5}$ are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) A failure of awareness of this lack had led to an early erroneous "proof" of Fermat's Last Theorem by G. Lamé; the realisation that this proof was erroneous made others study the consequences of this lack, and ways in which it could be alleviated.

Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K. (For example, the complex numbers C are an extension of the reals R, and the reals R are an extension of the rationals Q.) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions -- that is, extensions L of K such that the Galois group[8] Gal(L/K) of L over K is an abelian group -- are relatively well understood. Their classification was the object of the programme of class field theory, which was initiated in the late 19th century (partly by Kronecker and Eisenstein) and carried out largely in 1900--1950.

The Langlands program, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.

### Diophantine geometry

Consider an equation or system of equations. Does it have rational or integer solutions, and if so, how many? This is the central question of Diophantine geometry.

We may think of this question in the following graphic way. An equation in two variables defines a curve in the plane; more generally, an equation, or system of equations, in two or more variables defines a curve, a surface or some other such object in n-dimensional space. We are asking whether there are any rational points (points all of whose coordinates are rationals) or integer points (points all of whose coordinates are integers) on the curve or surface. If there are any such points on the curve or surface, we may ask how many there are and how they are distributed. Most importantly: are there finitely or infinitely many rational points on a given curve (or surface)? What about integer points?

An example here may be helpful. Consider the equation $x^2+y^2 = 1$; we would like to study its rational solutions, i.e., its solutions $(x,y)$ such that x and y are both rational. This is the same as asking for all integer solutions to $a^2 + b^2 = c^2$; any solution to the latter equation gives us a solution $x = a/c$, $y = b/c$ to the former. It is also the same as asking for all points with rational coordinates on the curve described by $x^2 + y^2 = 1$. (This curve happens to be a circle of radius 1 around the origin.)

The rephrasing of questions on equations in terms of points on curves turns out to be felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve - that is, rational or integer solutions to an equation $f(x,y)=0$, where $f$ is a polynomial in two variables - turns out to depend crucially on the genus of the curve. The genus can be defined as follows: allow the variables in $f(x,y)=0$ to be complex numbers; then $f(x,y)=0$ defines a 2-dimensional surface in 4-dimensional surface; count[9] the number of (doughnut) holes in the surface; call this number the genus of $f(x,y)=0$. Other geometrical notions turn out to be just as crucial.

There is also the closely linked area of diophantine approximations: given a number $x$, how well can it be approximated by rationals? (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: call $a/q$ (with $gcd(a,q)=1$) a good approximation to $x$ if $\scriptstyle |x-a/q|<\frac{1}{q^c}$, where $c$ is large.) This question is of special interest if $x$ is an algebraic number. If $x$ cannot be well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) turn out to be crucial both in diophantine geometry and in the study of diophantine approximations.

Diophantine geometry should not be confused with the geometry of numbers, which is a collection of graphical methods for answering certain questions in algebraic number theory.

### Arithmetic combinatorics

Let $A$ be a set of integers. Consider the set $A+A$ consisting of all sums of two elements of $A$. Is $A+A$ much larger than A? Barely larger? If $A + A$ is barely larger than $A$, must $A$ have plenty of arithmetic structure - e.g., does it look like an arithmetic progression?

If we begin from a fairly "thick" infinite set $A$ (say, the primes), does it contain many elements in arithmetic progression: $a$, $a+b$, $a+2 b$, $a+3 b$, ... , $a+10b$, say? Should it be possible to write large integers as sums of elements of $A$?

These questions are characteristic of arithmetic combinatorics. This is a presently coalescing field; it subsumes additive number theory (which concerns itself with certain very specific sets $A$ of arithmetic significance, such as the primes or the squares) and, arguably, some of the geometry of numbers, together with some rapidly developing new material. Its focus on issues of growth and distribution make the strengthening of links with ergodic theory likely. The term additive combinatorics is also used; however, the sets $A$ being studied need not be sets of integers, but rather subsets of non-commutative groups, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of rings, in which case the growth of $A+A$ and $A$·$A$ may be compared.

### Probabilistic number theory

Take a number at random between one and a million. How likely is it to be prime? This is just another way of asking how many primes there are between one and a million. Very well; ask further: how many prime divisors will it have, on average? How many divisors will it have altogether, and with what likelihood? What is the probability that it have many more or many fewer divisors or prime divisors than the average?

Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually independent. For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite.

It is sometimes said that probabilistic combinatorics uses the fact that whatever happens with probability greater than $0$ must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one.

### Computations in number theory

While the word algorithm goes back only to certain readers of Al-Kwarismi, careful descriptions of methods of solution are older than proofs: such methods - that is, algorithms - are as old as any recognisable mathematics - ancient Egyptian, Babylonian, Vedic, Chinese - whereas proofs appeared only with the Greeks of the classical period.

There are two main questions: "can we compute this?" and "can we compute it rapidly?". Anybody can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter. We now know fast algorithms for testing primality, but, in spite of much work, no truly fast algorithm for factoring.

The difficulty of a computation can be useful: modern protocols for encrypting messages depend on functions that are known to all, but whose inverses (a) are known only to a chosen few, and (b) would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorised. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems.

On a different note - some things may not be computable at all; in fact, this can be proven. For instance, Turing showed in 1936 that there is no algorithm for deciding in finite time whether a given algorithm ends in finite time. In 1970, it was proven that there is no algorithm for solving any and all Diophantine equations. There are thus some problems in number theory that will never be solved. We even know the shape of some of them, viz., Diophantine equations in nine variables; we simply do not know, and cannot know, which coefficients give us equations for which the following two statements are both true: there are no solutions, and we shall never know that there are no solutions.

## Problems solved and unsolved

### The beginnings

What are the integers x, y, z such that $\scriptstyle x^2 + y^2 = z^2$?

A scribe from Larsa (1800 BCE) almost certainly had a full solution. The late source Proclus credits Pythagoras with the partial solution $\scriptstyle (x,(x^2-1)/2,(x^2+1)/2)$, where $x$ ranges on the odd integers. He also credits Plato with a closely related rule. A general solution makes its first fully explicit appearance in Euclid's Elements (Book X, Lemma 1).

Are there incommensurable line segments? (In our language: are there irrational numbers?)

Yes (early Pythagoreans, before Plato's day). The question belongs to the history of number systems at least as much as it belongs here. The proof in Euclid's Elements is purely arithmetical; nothing besides the "theory of the odd and the even" (likely early Pythagorean) is needed.

Are there infinitely many prime numbers?

Yes (Euclid).

Given two integers, find the largest integer that divides them both.

Euclid's algorithm does the job. It also provides the basis for the standard method for finding integer solutions to linear equations in two variables. Such equations, however, were not addressed by Euclid; the first algorithm found for solving them was Āryabhaṭa's kuṭṭaka (see below).

## References

1. Sir Thomas Heath, A History of Greek Mathematics, vol. 1, Dover, 1981, p. 13.
2. Take, e.g., Serre's A Course in Arithmetic (1970; translated into English in 1973). In 1952, Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book."
3. Eleanor Robson, "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3), p. 189.
4. Eleanor Robson, "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3), p. 184.
5. J. Friberg, Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples and the Babylonian triangle parameter equations, Historia Math. 8, p. 302.
6. Eleanor Robson, "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3), p. 201.
7. Roshdi Rashed, Ibn al-Haytham el le théorème de Wilson, Arch. Hist. Exact Sci. 22 (1980), no. 4, pp. 305-321.
8. The Galois group of an extension K/L consists of the operations (isomorphisms) that send elements of L to other elements of L while leaving all elements of K fixed. Thus, for instance, Gal(C/R) consists of two elements: the identity element (taking every element x+iy of C to itself) and complex conjugation (the map taking each element x+iy to x-iy). The Galois group of an extension tells us many of its crucial properties. The study of Galois groups started with Evariste Galois; in modern language, the main outcome of his work is that an equation f(x)=0 can be solved by radicals (that is, x can be expressed in terms of the four basic operations together with square roots, cubic roots, etc.) if and only if the extension of the rationals by the roots of the equation f(x)=0 has a Galois group that is solvable in the sense of group theory. ("Solvable", in the sense of group theory, is a simple property that can be checked easily for finite groups.)
9. It may be useful to look at an example here. Say we want to study the curve $y^2 = x^3 + 7$. We allow x and y to be complex numbers: $(a + b i)^2 = (c + d i)^3 + 7$. This is, in effect, a set of two equations on four variables, since both the real and the imaginary part on each side must match. As a result, we get a surface (two-dimensional) in four dimensional space. After we choose a convenient hyperplane on which to project the surface (meaning that, say, we choose to ignore the coordinate a), we can plot the resulting projection, which is a surface in ordinary three-dimensional space. It then becomes clear that the result is a torus, i.e., the surface of a doughnut (somewhat stretched). A doughnut has one hole; hence the genus is 1.

## Bibliography

B. Datta and A. N. Singh, History of Hindu mathematics: a source book, Parts I and II, Asia publishing house, Bombay, 1962.

T. Heath, A history of Greek mathematics, Vol. I: From Thales to Euclid, Dover, New York, 1981.

T. Heath, A history of Greek mathematics, Vol. II: From Aristarchus to Diophantus, Dover, New York, 1981.

T. Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra, 2nd. ed, Cambridge University Press, 1910.

Y. Mikami, The development of mathematics in China and Japan, Chelsea, New York, 1974.

R. Rashed, Diophante: Les Arithmetiques, Tome III: livre iv, Paris, Les Belles Lettres, 1984.

R. Rashed, Diophante: Les Arithmetiques, Tome IV: livre v, vi and vii, Paris, Les Belles Lettres, 1984.

E. Robson, "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322", Historia Math. 28 (3), pp. 167-206, 2001.

B. L. van der Waerden, Science awakening, Oxford University Press, New York, 1961.

A. Weil, Number theory. An approach through history. From Hammurapi to Legendre. Birkhäuser, Boston, MA, 1984.


text:
## Juan S. Auli ; Sergi Elizalde - Consecutive Patterns in Inversion Sequences

dmtcs:5350 - Discrete Mathematics & Theoretical Computer Science, November 4, 2019, Vol. 21 no. 2, Permutation Patters 2018 - https://doi.org/10.23638/DMTCS-21-2-6
Consecutive Patterns in Inversion Sequences

Authors: Juan S. Auli ; Sergi Elizalde

An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of patterns in inversion sequences, focusing on the enumeration of those that avoid classical patterns of length 3. We initiate an analogous systematic study of consecutive patterns in inversion sequences, namely patterns whose entries are required to occur in adjacent positions. We enumerate inversion sequences that avoid consecutive patterns of length 3, and generalize some results to patterns of arbitrary length. Additionally, we study the notion of Wilf equivalence of consecutive patterns in inversion sequences, as well as generalizations of this notion analogous to those studied for permutation patterns. We classify patterns of length up to 4 according to the corresponding Wilf equivalence relations.

Volume: Vol. 21 no. 2, Permutation Patters 2018
Published on: November 4, 2019
Accepted on: November 4, 2019
Submitted on: April 5, 2019
Keywords: Mathematics - Combinatorics,05A05 (Primary) 05A15, 05A19 (Secondary)

format: plain


text:
# set of infinite cardinals admits an injective regressive function

Let $A$ be a set of infinite cardinals. Assume that for every regular $\lambda$, the subset $A \cap \lambda$ of $\lambda$ is not stationary. Then I want to prove that there is an injective function $g$ on $A$ which satisfies $g(\alpha) < \alpha$ for all $\alpha \in A$.

This is an exercise in Kunen's set theory. I dont' understand if $g$ should be a map from $A \to A$. But then the claim would be false if $A = \{\kappa\}$. But if $g$ may have arbitrary ordinal values, then the claim is somehow trivial, also without the assumptions. So I don't know what this exercise about.

-
What does "stationary" mean? Why is a singleton not stationary? –  Arturo Magidin Sep 24 '10 at 16:35
Thanks. Unfortunately, it does not help me answer your question. –  Arturo Magidin Sep 24 '10 at 17:27
@Arturo: if you look at the measure which gives all sets containing a closed and unbounded set a measure of 1, and all sets which are complements of such a measure of 0, then stationary sets are those of positive measure. –  Asaf Karagila Sep 24 '10 at 22:47

The exercise is asking for a map from $A$ to the ordinals, not to itself. Why do you think the exercise is trivial? If $\lambda$ is regular and $A\cap\lambda$ is non-stationary (in $\lambda$) then there is a club subset of $\lambda$ disjoint from $A$. You can then pick $g(\lambda)$ in this club. You need to ensure, however, that this element is not $g(\tau)$ for some prior $\tau$, and this requires some argument: If, as suggested in another answer, you are to pick something larger than $\sup(g(\alpha)\mid\alpha\in A\cap\lambda)$, you need to argue that this supremum is strictly below $\lambda$. But this is not automatic (or true in general), as $A$ may contain all cardinals up to and including some inaccessible. Also, $A$ may contain singular cardinals, and you need to define $g$ on them as well.
Ok, so how do we define $g$ in general? –  Martin Brandenburg Sep 25 '10 at 9:59
Hm no I don't know this lemma. And why is your $\lambda$ regular? –  Martin Brandenburg Sep 27 '10 at 10:29
@AndresCaicedo You wrote "Do you know how to prove that a subset A of a cardinal lambda is non-stationary iff there is an injective regressive function on A?" But this cannot be true for silly reasons like $A = \omega_1 \subseteq \lambda = \omega_2$. I think the best one can do is the following: For any subset $A$ of a regular cardinal $\lambda$, $A$ is non stationary iff there is a regressive map $h:A \to \lambda$ which is not constant on an unbounded subset of $\lambda$. –  hot_queen Jul 7 at 13:58


text:
Our universe expands, and this expansion is accelerating. Current consensus is to attribute this acceleration to a mysterious form of energy: dark energy. This dark energy density is very tiny and therefore only notable at cosmic length scales. When expressed in natural units, the cosmic dark energy density has a value of 10-123. This tiny value presents a big mystery. Straightforward estimates for the dark energy density based on quantum field theoretical considerations result in values (again in natural units) close to unity. So, the measured dark energy density is smaller than this by a gigantic factor of 123 orders of magnitude. This discrepancy has been termed "the biggest embarrassment in the history of theoretical physics".

Recently, Erik Verlinde proposed a revolutionary new perspective to gravity extensively discussed in this blog. He notes that gravity has all the characteristics of an entropic effect, and presents clear arguments that gravitational attraction between massive bodies can be considered as resulting from nothing more than a growth in number of bits required to describe the system of massive bodies. Key is that this description of the physical system is holographic in nature, with the bits describing reality residing at holographic screens.

I previously posed the question “can dark energy, just like gravity, be understood as an entropic effect?”.

To my astonishment, it appears that a quick 10 minute exercise in determining the entropic force exerted on the entire observable universe indeed yields an effect with the right order of magnitude to explain the cosmic dark energy (or cosmic acceleration). It seems that a dark energy effect emerges from Verlinde's holographic description. All 123 orders of magnitude of the dark energy mismatch evaporate when considering the cosmic acceleration as a result of a holographic entropic force.

The line of reasoning is as follows (for ease of notation I will work in natural units and leave out factors c, G and h-bar):

1. Consider the cosmic horizon (the edge of the observable universe: a sphere with radius R approximately equal to 2.7 1061 in natural units)

2. According to the holographic principle the observable universe is encoded in N = $\pi R^2$bits located at the cosmic horizon,

3. A finite temperature T is associated with this horizon. This temperature is determined by an equipartition of the energy Mc2 contained within the horizon over the bits (degrees of freedom) associated with the horizon. Here M = 1.4 1060 is the observable mass of the universe. Using the equipartition expression ½kT = M/piR2, it follows that kT = 3 10-64.

4. An entropic force F = kT grad(N) = kT grad(pi R2) =2 pi kT R is associated with the horizon. This results in a cosmic expansion (1/R) d2 R/dt2 = F/MR = 2 pi kT / M = 1.3 10-123. Presto!

--- Postscript ---

Two critical notes to the above speculative derivation need to be mentioned here:

1. As I set out to explain more than a hundred orders of magnitude mismatch, I have not bothered myself with numerical factors of order unity. As a result,the end result could be off by a factor of two or so.

2. More importantly, a full evaluation of the above simple expansion model yields an expansion (1/R)d2R/dt2 = (2c/R)2. Whilst this expression yields the right order of magnitude for the expansion, it is not constant and therefore not in line with the full Lambda-CDM model cosmological model. Is it a coincidence that the current value of the dark energy density comes out right? Or does the Lambda-CDM model need a modification?


text:
# Write A Verbal Expression For Each Algebraic Expression

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given Reference Tables. Then, write a mathematical expression for each verbal expression. I can write verbal expressions for algebraic expression Power an expression containing a base and an exponent such as x Exponent the small number that tells how many times the base is used as a factor. two more than The product of 7 and n 2. 9, 17, 20, 24,. How do you write a verbal expression for the algebraic expression #6-4y#? Algebra Expressions, Equations, and Functions Variable Expressions. 4 evaluate. OATE PERIOD Student Edition pages 1. How it went: This is a fun activity to get them to think more about what the operations mean and to get them to write about them. In the last section, we simplified expressions using the order of operations. Verbal and Algebraic expressions. 9 less than g to the fourth power Evaluate each expression. Our online algebraic expression trivia quizzes can be adapted to suit your requirements for taking some of the top algebraic expression quizzes. Words that indicate multiplication: 1. This question is asking us to evaluate the expression for five different values. 2 cos 2 θ + cos θ − 1. the quotient of g and 15 3. In the next few examples, we will be working solely with algebraic expressions. Write and simplify an. 15 more than half a number Example Write an expression with the same meaning as “add a number times 2 to 5 ” Find operation words to help you write the expression Add a number times 2 to 5 This expression will be an addition of two terms First term. Translate verbal expressions into algebraic expressions and vice versa. OBJECTIVE: SWBAT write an algebraic expression from a written expression. Identify the units of your expression. The next step is to compare two expressions. F) I can translate verbal expressions to algebraic expressions. In the last section, we simplified expressions using the order of operations. 20 hours of work divided equally among w workers d – 6 s + 9 12b 20 w 2-1. We present ideas using verbal, numerical, and graphical tools. As we could see this problem could be turned into the expression $$\left ( 4+3 \right )\cdot 2=14$$ When we want to translate a verbal phrase like the example above into a mathematical expression we can look for words that indicate a mathematical operation. If x is 6, then the expression has a value of 15. Properties of equality. 10 added to twice a number n -. ☐ Understand the following terms: Member (or element) of a set, subset, Universal set, Null (or empty) set, intersection of sets (no more than three sets), union of sets (no more than three sets), the difference between two sets, the complement of a set. To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given Reference Tables. Conversion Calculator. linear algebraic expression D. If x is 2, then the expression 9 + x has a value of 11. the sum of a number and 10 10. We have a great hope these Translating Verbal Expressions Worksheets images gallery can be a guide for you, bring you more inspiration and of course help you get bright day. 11 – 2 x y Page 182 letter B Mental Math Boys 11, 13, 15, 17, 19 Girls 12, 14, 16, 18, 20,. Distributive property. 3 Setting Up Algebraic Expressions and Equations from Verbal Statements So far you have named terms and created expressions. Translate equations into English sentences and translate English sentences into equations. Forthe equation 3(2 + 6) =24, 3 and (2 + 6) are the factors, and 24 is the product. Examples: A. Evaluate Algebraic Expressions. · 3 ± 3 $16:(5 13$16:(5 4. Six less than the product of 12 and y. Conversion Calculator. Key word : quotient. Algebra Questions and Problems for Grade 7. The expression 9 + 8 represents a single number (17). Let us see the next example on "Write algebraic expression for verbal expression". PERFECT for math workshop and centers. Write algebraic expressions from verbal descriptions. (B) simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents. Evaluate the expression for x 20 and x 60. In the next few examples, we will be working solely with algebraic expressions. Always look for the placement of commas in the verbal statements. what I hope to do in this video is give ourselves some practice interpreting statements and writing them as mathematical expressions possibly using parentheses so let's get started and for any of these statements if you get so inspired and encourage you to get so inspired pause the video and see if you can write them as mathematical expressions so this first one says seven hundred minus 19. Brian McLogan. For each algebraic expression, identify the number of terms. Write a mathematical expression for each verbal expression. Now we’ll reverse the process and translate word phrases into algebraic expressions. Write a verbal expression or equation for each algebraic expression. Write Algebraic Expressions Represented Verbally. Full color and black and. In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Warmup (Go) Vocabulary Review Forthe equation 3 • 8 = 24, 3 and 8 are factors, and 24 is the product. Enduring Understandings 1. An algebraic expression is an expression that may contain numbers, operations, and one or more symbols. Algebraic expression: This is the assortment of terms mingled together by the involvement of addition or deduction and forms GMAT algebra formulas like (x + 2), (x – 3c), and (2x –3y). Translate verbal expression into. 1 20 Classroom Modifications for Students with Autism by Karen Wright, M. 3-3 Algebraic Expressions. For example, write this equation: Today a boy is twice as old as he was 4 years ago. Example 2: Write each phrase as an algebraic expression. Each minute on the phone costs 12 cents. 10 added to twice a number n -. A major goal of algebra is for students to acquire fluency with symbols, expressions, and equations, and to be able to represent various situations using algebraic expressions and equations. How do you write a verbal expression for the algebraic expression #5x+8#? Algebra Expressions, Equations, and Functions Variable Expressions. (For review, see Lesson 2-2. Writing Expressions from a Menu Use the menu from this restaurant to write an algebraic expression for each question. 21 decreased by a number x. Students deepen their understanding of expressions by matching verbal expressions with their algebraic equivalent. Write and simplify an. (c + d) + (x – y) 1. Write Algebraic Expression 8th Grade - Displaying top 8 worksheets found for this concept. Solvers Solvers. Write Algebraic Expressions Represented Verbally. The expression 9 + x represents a value that can change. Below is the algebraic expression that can represent the math phrase above. 4 less than a number. How do you write the algebraic expression 14 less than the quantity K times 6? Asked by Wiki User. a Translate verbal quantitative situations into algebraic expressions and vice versa. MONEY Lorenzo bought several pounds of chocolate-covered peanuts and gave the cashier a $20 bill. 3 Setting Up Algebraic Expressions and Equations from Verbal Statements So far you have named terms and created expressions. Then use the order of operations to find the value of the numerical expression. The quotient of S and 33 e. Concept Cheat Sheet Adding Integers 1 Subtracting Integers 2 Multiplying Integers 3 Dividing Integers 4 Absolute Value 5 Combining Like Terms 6 Distributive Property I 7 Adding Expressions 8 Subtracting Expressions 9 Writing Expressions 10. Write a verbal expression for each algebraic expression. By (date), (name) will correctly write algebraic expressions from verbal descriptions, including those that require viewing multiple parts of the expression as a single entity (e. 04 Write expressions in equivalent forms to solve problems. the sum of squared and 3 Example 2: Write Algebraic Expressions A. 16 more than 3 times a number b. Example #7. Full color and black and. (a - b) 2 = a 2 - 2ab + b 2. Translate each verbal phrase into an algebraic expression 1. 2m 62/87,21 Because the 2 and the m are written next to each other, they are being multiplied. A comprehensive database of more than 20 algebraic expression quizzes online, test your knowledge with algebraic expression quiz questions. -interpret expressions within the context of a problem. 15 less than k 11. Variable expression (A. − 3y 8 the quotient of 3 times y and 8 A model can be used to aid in translating a verbal expression into an algebraic expression. 11(B) simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents. Letters are used in algebra in place of an unknown number, giving us algebraic terms like 2x. Let x stand for the number of dollars Xavier has. Write the verbal expression for each algebraic expression. So 9 + x is an algebraic expression. The expression, equation, or inequality is a mathematical model. Half of the seventh graders and one third of the eighth graders were divided into ten teams. 12 divided by the sum of h and 2 7. (Name) will correctly identify parts of the algebraic expression using mathematical terms (e. An algebraic expression is an expression that may contain numbers, operations, and one or more symbols. See Answer. 2) The length of a rectangle is 3 feet less than 4 times its width. M splits into 12 equal parts f. Wiki User Answered 2012-09-04 07:38:17. The phrase is$5 more than Xavier. Start Here; Our Story; Podcast; Hire a Tutor; Upgrade to Math Mastery. and Algebra Goal Il Key Activities Students write and evaluate algebraic expressions for situations described in words. Students had 3 minutes to find their expression twin. Example 2: Write as an algebraic expression. notebook September 07, 2016 Ex 2: Translate each. When algebraic terms are combined with mathematical operations, eg + or - we get an algebraic expression. How do you write the algebraic expression 14 less than the quantity K times 6? Asked by Wiki User. Write an algebraic expression for each verbal expression. Evaluate Algebraic Expressions. 2m 62/87,21 Because the 2 and the m are written next to each other, they are being multiplied. Evaluate each of the expressions for the given value(s) of the variable(s). [Patterns, Functions, and Algebra Goal Il Use a variable to represent unknown quantities. a factor that is asum of two terms in parentheses). Write algebraic expressions for verbal expressions. Concept Cheat Sheet Adding Integers 1 Subtracting Integers 2 Multiplying Integers 3 Dividing Integers 4 Absolute Value 5 Combining Like Terms 6 Distributive Property I 7 Adding Expressions 8 Subtracting Expressions 9 Writing Expressions 10. 9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (5 points) Part B: Write a verbal expression for 2(n + 9). e Example 5. Multiply 4 by a number and then subtract 5 b. The difference is that an algebraic expression has numbers, symbols and variables, but a verbal expression has only words. ten less than the product of 7 and f two fifths of a square of a number 3x4. For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. the difference of a number squared and 8 The expression difference ofimplies subtraction. These Algebraic Expressions Worksheets will create algebraic statements for the student to simplify. · 3 ± 3 $16:(5 13$16:(5 4. Students had 3 minutes to find their expression twin. In this module, students extend their study of functions to include function notation and the concepts of domain and range. ) five less than a number c. For example, write this equation: Today a boy is twice as old as he was 4 years ago. Add 4 to n then multiply by 5. What do you call an expression in fraction form in which the numerator and the denominator are polynomials? A. Write an algebraic expression to. Then use the order of operations to find the value of the numerical expression. Phone Plans - Students will compare two linear functions in the context of evaluating phone plans. what is the numerical expression for the verbal phrase the cost of 5 books at 4. 6 more than twice m 5. 11 Simplify expressions using order of operations. This expression can be shown. Understand the meaning of the variable in the context of the situation. the sum of j and 13 5. Write a verbal expression for each algebraic expression. Algebraic Expressions Worksheets and Quizzes Combining Like Terms Algebraic Expression: Parts of an Expression Writing Expressions Algebraic Expressions Worksheets: Combining Like Terms Variables And Expressions Worksheets Simplify Expressions Worksheets Evaluating Expression Worksheets Pre Algebra Word Problem Worksheets Distributive Property. Write an algebraic expression for each of the following. 04 Write expressions in equivalent forms to solve problems. $5 more than. Given the expression: (𝑥+ 3) ∙(𝑦+ 1) ∙(𝑥+ 2) a. So, the verbal expression six more than a number can be written as the algebraic expression n + 6. English expressions can be more complicated than a simple relation between two items. The product of and 6 decreased by 100. Write Sentences as Equations Write each sentence as an algebraic equation. a x 2 + b x + c. a number increased by 9 62/87,21 Let x represent a number. A symbol that represents one or more numbers is called a variable.$16:(5 x + 9 two thirds of a number d to the third power 62/87,21 The words two -thirds of suggest multiplication. Translate each word phrase into an algebraic expression: 1. Day 3 notes algebraic expressions. quadruple f 4f Write your answer to the word problems in the form of an algebraic expression. , Algebra 2 (Holt, Rinehart, and Winston, Orlando: 2007) Part I: Algebraic Expressions and Equations Unit 1: Number Lines, Intervals, and Sets (approx. Solution Notice that the pattern displayed has the same form as a standard quadratic expression, a x 2 + b x + c. MONEY Lorenzo bought several pounds of chocolate-covered peanuts and gave the cashier a $20 bill. _2 3 r 4 3. 6th - 7th grade Control the pace so everyone advances through each question together. thirty-one increased by fourteen 3. A number x increased by 3. Verbal Phrases and Algebraic Expressions In translating a verbal phrase to an algebraic expression, assign a variable to an unknown quantity. Three times x more than three times 33 b. Oct 1, 2013 - Learn to write algebraic expressions with this worksheet! Review key math phrases, then practice translating verbal expressions into algebraic expressions. A math expression is different from a math equation. The symbols and variables we’ve talked about will help us do that. Algebraic expressions involve unknowns. Write an algebraic expression for each verbal expression. Write a verbal expression for each algebraic expression. Upon completion of this lesson, students will: have been introduced to functions; have learned the terminology used with functions; have practiced describing functions with one operation in English sentences, data tables, and with simple algebraic expressions. After you enter the expression, Algebra Calculator will evaluate 2x for x=3: 2(3) = 6. Although they are closely related, a Great Dane weighs about 40 times as much as a Chihuahua. In the table below, the letter “x” is used to represent the unknown. Coefficient : It is a number or symbol that caters as a measure of a property or characteristics. This simplifying algebraic expressions calculator will give you the result automatically but for manual calculation, follow the steps given below. (For review, see Lesson 2-2. For example, if we have the algebraic expression 3𝑥𝑥+ 5, we can create several verbal expressions that are slightly different. What is the property used in verbal expression three times the sum of x squared and y squared?. Algebraic Expressions Match Center Common Core Standard: 6. These expressions consist of numbers, variables, and different algebraic operations. I can write verbal expressions for algebraic expression Power an expression containing a base and an exponent such as x Exponent the small number that tells how many times the base is used as a factor. Write an algebraic expression to show the number of dollars Abdul has. teacher, the student has. Assuming that r riders take slide 1 and that there are 100 riders in all, Simplify any Algebraic Expression - powered by WebMath. Algebraic Expressions And Terms 814657 PPT. Verbal Expressions to Algebraic Expressions. A symbol that represents one or more numbers is called a variable. EXAMPLES 3 Write a numerical expression to represent the amount it eats in 5 months. Our company with inventive copy writers get fantastic skills throughout verbal along with composed communicating, which in turn turn for you to any type of content you simply will not uncover wherever else. Next, write an expression for the time in minutes it will take r riders to go down slide 1. This expression can be shown. Write each as an algebraic expression. What was that variable? It's a good idea to write down what your variables mean. Example 1 Evaluate. the primary difference between the two is an equals sign. The next step is to compare two expressions. For example, 3x 2 − 2xy + c is an algebraic expression. An equation is a string of mathematical symbols stating the equality of two expressions. 1) Kim has 7 more than 5 times the amount Sylvia has. Write an algebraic expression for the verbal description. Terms are the parts of the expression that are separated by addition signs. Sum of x and twice y = x + 2y. 04 Write expressions in equivalent forms to solve problems. Write an algebraic expression for each verbal expression. In the table below, the letter “x” is used to represent the unknown. ! Evaluate algebraic expressions by substituting a value into the expression. This algebraic expressions can be also written in verbal phrase: 1. Writing reinforces maths learnt. 45 less than y c. How do you write the algebraic expression 14 less than the quantity K times 6? Asked by Wiki User. Write verbal expressions for algebraic expressions. Example 2: Write each phrase as an algebraic expression. Given an area model, identify the appropriate algebraic expression: 10: Relate a polynomial expression with three or fewer terms to a verbal expression. Write 2 verbal descriptions of x92 3. Writing algebraic expressions, equations, or inequalities that represent real-life situations is called modeling. 1: Expressions 3 Name: _____ www. Example 5: Write an algebraic expression for the math phrase " the quotient of 1, and 1 decreased by a number". In the last section, we simplified expressions using the order of operations. Physical Constants. 1 Rational Equations and Functions (6 days) Cranston!Public!Schools,!with!process!support!from!the!Charles!A. An algebraic expression has terms that are separated by plus and minus signs. Evaluate Algebraic Expressions. After you finish this lesson, view all of our Pre-Algebra and Algebra 1 lessons and practice problems. In a rational algebraic expression written in the form of. The algebraic expression is 10x. 10 more than y. For example words like "sum", "increased by" and "plus" indicates that we are to use. Math Vids offers free math help, free math videos, and free math help online for homework with topics ranging from algebra and geometry to calculus and college math. Define verbal expression. It takes the Earth about 365 days to orbit the Sun. 1) Lesson 3 After completing Lesson 3: Introduction to Graphing, the student will be able to do the following: Define origin, x-axis, y-axis, and quadrants. 1 Order of Operations 1. First type the expression 2x. There are x students trying out for a solo in a chorus concert. So, the verbal expression a number increased by 9 can be written as the algebraic expression x + 9. 1 Rational Equations and Functions (6 days) Cranston!Public!Schools,!with!process!support!from!the!Charles!A. o The variable(s) must have the same exponents on them. Translating between math expressions and word statements. It is often necessary to translate verbal expressions into algebraic expressions. F) I can translate verbal expressions to algebraic expressions. txt) or read online for free. Write each verbal phrase as an algebraic expression. five less than t 4. More practice Evaluate algebraic expressions for a given replacement set to include rational numbers. The equations below were created by students who were asked to write equivalent. Equations Solve linear equations in one variable. Divide the first number n by the second number 5. This math worksheet was created on 2019-02-08 and has been viewed 1,323 times this week and 5,609 times this month. 1: Expressions 3 Name: _____ www. The expression 9 + x represents a value that can change. 4 B Algebraic Expressions February 07, 2011 CONSTRUCTING EXPRESSIONS See box on page 41 of text for suggestions ③ EXAMPLES: Write an algebraic expression for each of these. Conversion Calculator. Modeling with Expressions Reteach A numerical expression contains operations and numbers.$16:(5 The sum of five times a number and 7 equals 18. The algebraic expression is 4 ! d or 4d. Translate verbal phrases into expressions Translate the phrase into an expression. Simplify any Algebraic Expression If you have some tough algebraic expression to simplify, this page will try everything this web site knows to simplify it. We will practice translating word sentences into algebraic equations. Write an algebraic expression for each verbal expression. Examples: A. The measure of an angle is (5 )x q. Example 2: Write each phrase as an algebraic expression. Write an algebraic expression for this: two points fewer than 3 times the number of points scored before. g divided by 4 5. Some of the worksheets for this concept are Lesson plan vats grade 8 write algebraic expressions, Variable and verbal expressions, Unit 1 grade 8 integers and algebraic expressions, Evaluating algebraic expressions, Translating phrases, Numbers and expressions, Cranford public schools summer math. To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given Reference Tables. Write each algebraic expression in words: 18. An algebraic expression contains operations, numbers, and at least one variable. If the terms have exactly the same variables to like powers, you can combine them to simplify the expressions. 4m3 (look for an operation first) the product of 4 and m to the third power OR the product of 4 and m cubed Example 3 Write a verbal expression for each algebraic expression. The table below lists some key words and phrases that are used to describe common mathematical operations. the difference of a number squared and 8 The expression difference ofimplies subtraction. Lessons Lessons. Evaluate each of the expressions for the given value(s) of the variable(s). If x is 6, then the expression has a value of 15. Rewrite each phrase as an algebraic expression. An algebraic expression is said to be in Standard Form if its terms are arranged from the term with the highest degree, up to the term with the lowest degree. When you simplify an algebraic expression, you need to keep a few basic rules in mind. a) 12 subtracted from a number *What is the importance of the word "FROM"? b) 12 subtracted from 5 times a number c) Twice a number subtracted from 7 d) The sum of 6 and a number subtracted from 3 times the number. Write an algebraic expression for each verbal expression. where f(y) is called the value of f at y and function value can be obtained by putting the value of y in above expression. In early grades, children are expected to write math expressions using numbers and. b) Two consecutive even integers. In this case, we want to divide the number 1 by the quantity 1 decreased by a number. Write the verbal expression for each algebraic expression. The sum of 10 times a number and fifteen is added to eleven times the same number. Write each algebraic expression. 12 divided by the sum of h and 2 7. the product of 5 and b 5b 4. a verbal expressions is a math questions. The boy is 8 years old today. Example 2: Expressions with Parentheses. Assuming that r riders take slide 1 and that there are 100 riders in all, Simplify any Algebraic Expression - powered by WebMath. Write an expression for the amount of change he will receive if p represents the cost of the peanuts. − 3y 8 the quotient of 3 times y and 8 A model can be used to aid in translating a verbal expression into an algebraic expression. reflects them. Write an algebraic expression to represent each verbal expression. a) 12 subtracted from a number *What is the importance of the word "FROM"? b) 12 subtracted from 5 times a number c) Twice a number subtracted from 7 d) The sum of 6 and a number subtracted from 3 times the number. Jan 8, 2012 - As an introduction to writing variable expressions, I had each student in my class choose a card out of a basket. COMPUTERS A company offers each of its 80 workers either a desktop computer that costs $900 or a laptop that costs$1550. Write an expression and draw a bar model to represent Tina’s score and Juan’s score. On this page, you will find Algebra worksheets mostly for middle school students on algebra topics such as algebraic expressions, equations and graphing functions. Write an algebraic expression for each of the following. Write, read, and evaluate expressions in which letters stand for numbers. the number of seconds in hhours Holt Algebra 2 1-4 Simplifying Algebraic Expressions To evaluate an algebraic expression, substitute a number for each variable and simplify by using the order of operations. Then use the order of operations to find the value of the numerical expression. Be able to write an algebraic expression for a word phrase or write a word phrase for an expression. the product of 5 and b 5b 4. Write an algebraic expression for this: two points fewer than 3 times the number of points scored before. Writing Algebraic Expressions Write the verbal phrase or word that mean the same thing as the algebraic symbol at the top of the column. nMultiply by 5 then add 4. An algebraic expression is an expression that may contain numbers, operations, and one or more symbols. Full text: Part A: Write an algebraic expression for 4 more than 3 times a number. • Demonstrates deep knowledge of the structure of algebraic expressions, how algebraic manipulations are governed by properties of operations and exponents, the nature of solutions to equations, and reasoning processes for manipulating expressions and solving equations. four more than a number n The algebraic expression is an + 4 Exercises b. What is the property used in verbal expression three times the sum of x squared and y squared?. Lessons Lessons. 1) the quotient of 40 and 5 2) the difference of 20 and x 3) the product of 6 and 11 4) y squared 5) the product of 2 and x 6) the quotient of 40 and n 7) the difference of 21 and 16 8) the sum of b and 12 9) v more than 5 10) 5 cubed Write each as an algebraic equation. Students will represent and analyze algebraically a wide variety of problem solving situations. Define verbal expression. Write an algebraic expression for each verbal expression. Write each as an algebraic expression. Since we need to know the values of both variables to answer this, both statements are necessary and sufficient. 1 Verified answer. with more related ideas like writing algebraic expressions word problems, writing algebraic expressions word worksheet and algebraic expressions worksheets. Write a verbal expression for each algebraic expression. Let us see the next example on "Write algebraic expression for verbal expression". 1) Translate verbal phrases and real-world scenarios into algebraic expressions 2) Solve simple (one-variable) linear equations, 3) Translate real-world scenarios into algebraic equations 4) Analyze word problems to generate algebraic equations, and 5) Synthesize knowledge of solving and writing algebraic equations. the difference of 12 and x c. There are x students trying out for a solo in a chorus concert. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. If you feel difficulty in solving some tough algebraic expression, this page will help you to solve the equation in a second. OATE PERIOD Student Edition pages 1. Follow the steps below to successfully translate the words into an expression. Name each algebraic expression for each verbal expression. • Name the parts of the expression. Create an expression for “four less than three times a number” b. the sum of two consecutive natural numbers, the first of which is n - 8884674. Write an algebraic expression for the verbal expression: five times the sum of three and f. ) y + 3z for y = 5 and z = 6 46 50 86 8 y 23 Although they are closely related, a Great. 1 Verified answer. Sieling’s Signature: (A) Level 4 1. In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). Evaluate the following expressions when a = 6 and b = 4. matching verbal expressions with their corresponding algebraic expressions. Elementary Algebra: Equations and Graphs was written with two goals in mind: to present the skills of algebra in the context of modeling and problem solving, and to engage students in an activity-based course. a factor that is asum of two terms in parentheses). Define verbal expression. Some of the worksheets for this concept are Variable and verbal expressions, Write a verbal expression for each algebraic, Translating verbal phrases to algebraic expressions, Numbers and expressions, Translate to an algebraic expression, Translating key words and phrases into algebraic expressions. Write algebraic. Example 2: Write each phrase as an algebraic expression. As we could see this problem could be turned into the expression $$\left ( 4+3 \right )\cdot 2=14$$ When we want to translate a verbal phrase like the example above into a mathematical expression we can look for words that indicate a mathematical operation. At times, an expression is the result of applying operations to simpler expressions. 10 x x - 10 x + 10 x / 10 Chris had $200. a number increased by 9 62/87,21 Let x represent a number. 15 + r 62/87,21 The expression shows the sum of two terms. Three times the quantity eight plus x. Then, simplify the expression. How do you write a verbal expression for the algebraic expression #5x+8#? Algebra Expressions, Equations, and Functions Variable Expressions. Algebra I Recipe: Translating Verbal Phrases into Algebraic Expressions OR Translating English into Math: A. Wiki User Answered 2012-09-04 07:38:17. To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given Reference Tables. Add 4 to n then multiply by 5. Three times x more than three times 33 b. The standard form of 2x 2 -5x 5 -2x 3 + 3x -10 is -5x 5 -2x 3 + 2x. Examples Write a verbal expression for each algebraic expression. n 5 20 Three times Jack’s age equals 12. The fact that, if you work correctly, you get the same answer for questions 1 (a) and 1 (b) is an example of a certain property of addition and multiplication called the distributive property. Whether you are learning algebra in school or examining a certain test, you will notice that almost all mathematical problems are represented in words. each student is given a sheet with a table with 3 columns: algebraic expressions, verbal expressions, and names. One way to remember the. a number n divided into 9 5. 1) the difference of 10 and 5 2) the quotient of 14 and 7 3) u decreased by 17 4) half of 14 5) x increased by 6 6) the product of x and 7 7) the sum of q and 8 8) 6 squared 9) twice q 10) the product of 8 and 12 11) the quotient of 18 and n 12) n cubed Write each as a verbal expression. In the last section, we simplified expressions using the order of operations. Physical Constants. Sample answers are given. com Tel: 800-234-2933;. 5 + 2(p  3). Algebraic expressions involve unknowns. Solvers Solvers. 8 increased by three times a number 6. ] Correct Answer is : (x7- 2) ft. Algebraic Methods 11 A. Expressions and Variables - Verbal Expressions - ALGEBRA I. more than x _x +8 6. Top Answer. Write an algebraic expression to. It also has three factors: 5, a, and (b + c). Coefficient : It is a number or symbol that caters as a measure of a property or characteristics. the sum of squared and 3 Example 2: Write Algebraic Expressions A. Always look for the placement of commas in the verbal statements. 4x + 10 10 more than four times x 2. All worksheets are created by experienced and qualified teachers. 11 Simplify expressions using order of operations. How do you write the algebraic expression 14 less than the quantity K times 6? Asked by Wiki User. Lindsay gets paid a base salary of$400 per week plus a 0. Write a verbal expression for each algebraic expression. 2:Write, read, and evaluate expressions in which letters stand for numbers. A number increased by 6 Let x represent the number; hence x 6 3 times a, subtracted from b 5 more than y x times the sum of y and z a decreased by b A number decreased by 12. A python eats 4 pounds of meat each month. half of p p 2 11. Coefficient : It is a number or symbol that caters as a measure of a property or characteristics. more than x _x +8 6. Evaluating and Writing Algebraic Expressions COURSE 2 LESSON 2-1 Write an algebraic expression for each word phrase. Given an area model, identify the appropriate algebraic expression: 10: Relate a polynomial expression with three or fewer terms to a verbal expression. Of course, answer keys are provided with each free algebra worksheet. Free and printable writing worksheets for 5th grade the 5th grade writing curriculum is demanding and requires kids to be well versed in a lot of concepts and theories. Write down the place value of the digit 8 in the following numbers. Translate each verbal phrase into an algebraic expression. where a, b and c can be any numbers. Step 1 : Write an algebraic expression to represent the points scored by Tina. 2m 62/87,21 Because the 2 and the m are written next to each other, they are being multiplied. EXAMPLE 1: WRITING AN ALGEBRAIC EXPRESSION Write an algebraic expression that models each word phrase. Mathematics » 6th grade » Operations & Algebraic Thinking Apply and extend previous understandings of arithmetic to algebraic expressions. Examples: A. the sum of and 3 squared d. the quotient of g and 15 3. Variable and Verbal Expressions - Kuta Software Worksheet by Kuta Software LLC. Example 5: Write an algebraic expression for the math phrase ” the quotient of 1, and 1 decreased by a number”. These sheets teach the concepts as they are presented in the Algebra Class Software. A number increased by 6 Let x represent the number; hence x 6 3 times a, subtracted from b 5 more than y x times the sum of y and z a decreased by b A number decreased by 12. Then use the order of operations to find the value of the numerical expression. Regents Exam Questions A. 2(4x + y) + 6(2x – 3y) b) Simplify the expression. These Algebraic Expressions Worksheets will create algebraic statements for the student to simplify. the product of 5 and b 5b 4. Instead of "8 plus 9" (with two given numbers), you would see. Conversion Calculator. Look at the following expression: x = 1x. Write each as an algebraic expression. For an algebraic expression calculator to work, it is typically required to type the above type of math expression in, by using common symbols. Three times x more than three times 33 b. We'll need to be clear about what the expression will represent. 2) Combine like terms by adding coefficients. Words that indicate multiplication: 1. Algebraic expression: 2n + 5. 7m23, 7m60, 7m61, 7m62, 7m65, 7m66, 7m67, 7m68, CGE 4b, 4c 2 Models of Linear Relationships ! Given concrete models of linear growing patterns, create verbal, numerical, graphical, and algebraic models. Sentence Equation Five more than a number is 20. Write algebraic expressions and equations to represent real-world situations. Solution: Before writing an algebraic expression for the weekly income, it is helpful to construct an informal verbal model. The difference is we will have an equal sign between the two expressions. Algebra I Module 4: Polynomial and Quadratic Expressions, Equations, and Functions. In a rational algebraic expression written in the form of. So, the verbal expression a number increased by 9 can be written as the algebraic expression x + 9. Write Algebraic Expressions from Statements: Form ax+b and a (x+b) Later in this course, we'll apply our skills in algebra to solving equations. Making a Table Make a table for the income whenn 5 5, 10, and 15. Write Algebraic Expressions Represented Verbally. each student is given a sheet with a table with 3 columns: algebraic expressions, verbal expressions, and names. Equations and Word Problems ( Two Step Equations ) Worksheets Equations and Word Problems (Two Step Equations) Worksheet 1 – This 10 problem worksheet will help you practice writing and solving two step equations that match real world situations. Definition of an expression. Examples: Three more than a number = x + 3. Martinez orders 250 key chains printed with his athletic team’s logo and 500 pencils printed with their Web Address. w-1 81+2x 12d 3x+4. You can translate word phrases into variable expressions. We have a great hope these Translating Verbal Expressions Worksheets images gallery can be a guide for you, bring you more inspiration and of course help you get bright day. More Examples Here are more examples of how to evaluate expressions in Algebra Calculator. For example, = 25 + 7 = 64 ÷ 2 = 20 × 5 = 102. a) Fifteen decreased by 3 times a number, n. You have a \$10 phone card. An expression is a string of mathematical symbols representing a quantity. Simplify the expression 9j j. Translating an algebraic expression to a verbal expression. Write each subtraction expression as an addition expression. Note: Expressions may include absolute value and/or integral exponents greater than 0. a) 12 subtracted from a number *What is the importance of the word "FROM"? b) 12 subtracted from 5 times a number c) Twice a number subtracted from 7 d) The sum of 6 and a number subtracted from 3 times the number. Example #7. An online algebra calculator simplifies expression for the input you given in the input box. sum, term, product, factor, quotient, coefficient) for (4 out of 5) problems. Martinez orders 250 key chains printed with his athletic team’s logo and 500 pencils printed with their Web Address. In order to write a numerical expression for the given verbal phrase, observe that the cost of four boxes of cereal is 12 dollars. 12) Student/Teacher Actions: What should students be doing? What should teachers be doing? 1. 13 increased by y 6. The 5 is called the coefficient of the term and the x is a variable. the product of seven and four times a number multiplied by three 41. Verbal Expressions to Algebraic Expressions. Each element and operator in a mathematical equation has its own name and there are standardized phrases that are used to describe the relationships between them. Such an expression is known as an algebraic expression. 15 less than k 11. Solvers Solvers. 2:Write, read, and evaluate expressions in which letters stand for numbers. In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). Write Phrases as Algebraic Expressions To write verbal phrases as algebraic expressions, follow the steps below. Write an algebraic expression for each verbal expression. Sentence Equation Five more than a number is 20. x + 4 = 20 or 5x = 30 LT #10 Translate and Write Equivalent Expressions E) I can identify verbal words for each operation. This expression can be shown. If x is 2, then the expression 9 + x has a value of 11. Example 3: Write each phrase as an algebraic expression using the variable n. Six less than p __p -6___. 13 a) write verbal expressions as algebraic expressions and sentences as equations and vice versa; b) evaluate algebraic expressions 8. Evaluate each of the expressions for the given value(s) of the variable(s). An algebraic expression is an expression that may contain numbers, operations, and one or more symbols. Formulas can be written and equations solved to solve a range of problems in science and engineering. 49 increased by twice a number. Let the keywords and logic help you find the proper corresponding math expressions. If x is 6, then the expression has a value of 15. 6 more than twice m 13. 6 times the sum of a number and 3. Below is the algebraic expression that can represent the math phrase above. Write yes or no. It takes the Earth about 365 days to orbit the Sun. a Translate between algebraic expressions and verbal phrases that include variables. The main difficulty is in finding the least common denominator. algebraic expression, algebraic equation, numerical expression, variable expression, verbal expression, verbal sentence (7. Writing Algebraic Expressions Objective: Be able to write an algebraic expression for a word phrase or write a word phrase for an expression. We present ideas using verbal, numerical, and graphical tools. Write a verbal expression for each algebraic expression. See Answer. Principle in order to simplify algebraic expressions. Double = multiply by 2. Translate this phrase into an algebraic expression. The launch introduces the importance of clearly defining a variable. Verbal phrases can be written as algebraic expressions. 3n2 - x Write an algebraic expression for each verbal expression. Distributive property. How to interpret parts of an expression. Write an algebraic expression for each verbal expression. Evaluate each expression. Write an algebraic expression for the following verbal expression. They will glue or tape their matches to a sheet of colored paper. Price of meal (dollars) Price of meal (dollars) 15%. Writing reinforces maths learnt. Therefore, the need to translate […]. Write an algebraic expression for this: two points fewer than 3 times the number of points scored before. Create an expression for “four less than three times a number” b. Simplify the expression 9j j. 05p is the sum of the simpler expressions p and 0. ) five less than a number c. Write each as a verbal expression. 3n2 - x Write an algebraic expression for each verbal expression. Identify the units of your expression. Student will be able to identify a variable, term, constant, coefficient, and an algebraic expression or equation. Nine-fourteenths of 6x reduced by 10. 6 less a number t 6. a) Fifteen decreased by 3 times a number, n. Set the expressions equal to each other. You can translate word phrases into variable expressions. After you enter the expression, Algebra Calculator will evaluate 2x for x=3: 2(3) = 6. (a + b) 2 = a 2 + 2ab + b 2. Martinez orders 250 key chains printed with his athletic team’s logo and 500 pencils printed with their Web Address. ℝ, their sum is a. a2 + (a + 5) 21. Write an expression that shows how much money is left on the card after x minutes on the phone. More practice Evaluate algebraic expressions for a given replacement set to include rational numbers. ExampleExample11 Write an algebraic expression for each verbal expression. ) five less than a number c. 9 less than e e - 9 9. Translating an algebraic expression to a verbal expression. Write Phrases as Algebraic Expressions To write verbal phrases as algebraic expressions, follow the steps below. 3) Describing algebraic expressions in words. (a - b) 2 = a 2 - 2ab + b 2. Closing the Activity: Show illustrations of Algebra Tile pieces and have students write a simplified expression. Presentation Summary : An algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication,. Translate each word phrase into an algebraic expression: 1. Physical Constants. 2 cos 2 θ + cos θ − 1. For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger.


text:
# Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem (Memoirs of the American Mathematical Society)

Paperback

Up to 2 WeeksUsually despatched within 2 weeks

£44.50

### Description

In this volume, the authors demonstrate under some assumptions on $f^+$, $f^-$ that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure $\mu{^+}=f^+dx$ onto $\mu^-=f^-dy$ can be constructed by studying the $p$-Laplacian equation $- \mathrm {div}(\vert DU_p\vert^{p-2}Du_p)=f^+-f^-$ in the limit as $p\rightarrow\infty$. The idea is to show $u_p\rightarrow u$, where $u$ satisfies $\vert Du\vert\leq 1,-\mathrm {div}(aDu)=f^+-f^-$ for some density $a\geq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f^+$ and $f^-$.

### Contents

Introduction Uniform estimates on the $p$-Laplacian, limits as $p\to\infty$ The transport set and transport rays Differentiability and smoothness properties of the potential Generic properties of transport rays Behavior of the transport density along rays Vanishing of the transport density at the ends of rays Approximate mass transfer plans Passage to limits a.e. Optimality Appendix: Approximating semiconcave and semiconvex functions by $C^2$ functions Bibliography.

### Product Details

• ISBN13: 9780821809389
• Format: Paperback
• Number Of Pages: 66
• ID: 9780821809389
• ISBN10: 0821809385

### Delivery Information

• Saver Delivery: Yes
• 1st Class Delivery: Yes
• Courier Delivery: Yes
• Store Delivery: Yes

### Calculus and AnalysisView More

Prices are for internet purchases only. Prices and availability in WHSmith Stores may vary significantly

Close


text:
# Microscopic Processes in Global Relativistic Jets Containing Helical Magnetic Fields

DOI:
10.3390/galaxies4040038
Publication date:
01/09/2016
Main author:
Nishikawa, Ken-Ichi
IAA authors:
Gómez, Jose L.
Authors:
Nishikawa, Ken-Ichi;Mizuno, Yosuke;Niemiec, Jacek;Kobzar, Oleh;Pohl, Martin;Gómez, Jose;Duţan, Ioana;Pe'er, Asaf;Frederiksen, Jacob;Nordlund, Åke;Meli, Athina;Sol, Helene;Hardee, Philip;Hartmann, Dieter
Journal:
Galaxies
Refereed:
Yes
Publication type:
Article
Volume:
4
Pages:
38
Issue:
4
Abstract:

In the study of relativistic jets one of the key open questions is their interaction with the environment on the microscopic level. Here, we study the initial evolution of both electron$-$proton ($e^{-}-p^{+}$) and electron$-$positron ($e^{\pm}$) relativistic jets containing helical magnetic fields, focusing on their interaction with an ambient plasma. We have performed simulations of 'global' jets containing helical magnetic fields in order to examine how helical magnetic fields affect kinetic instabilities such as the Weibel instability, the kinetic Kelvin-Helmholtz instability (kKHI) and the Mushroom instability (MI). In our initial simulation study these kinetic instabilities are suppressed and new types of instabilities can grow. In the $e^{-}-p^{+}$ jet simulation a recollimation-like instability occurs and jet electrons are strongly perturbed. In the $e^{\pm}$ jet simulation a recollimation-like instability occurs at early times followed by a kinetic instability and the general structure is similar to a simulation without helical magnetic field. Simulations using much larger systems are required in order to thoroughly follow the evolution of global jets containing helical magnetic fields.

Database:


text:
# How to compute the charge as a function of time in a simple $CL$ circuit?

I am trying to compute $q(t)$ in a series circuit consisting of a capacitor of plate charges $q_0$ and $-q_0$ and an inductor. The switch is closed at time $t=0$.

By Kirchoff's Law, $$\frac{q}{C}+L\frac{dI}{dt}=0.$$ There is a conservation of charge in this circuit, so $I=\dot{q}$, which gives us the differential equation $$\ddot{q}+\frac{1}{LC}q=0.$$ I recognise that this is the equation of a harmonic oscillator with angular frequency $\frac{1}{\sqrt{LC}}$, so $$\ddot{q}+\omega^2 q=0.$$ To solve this I construct a characteristic equation $$\lambda^2+\omega^2=0 \implies \lambda=\pm i\omega$$ and taking only the positive root gives the solution $q(t)$ as $$q(t)=A\cos{\omega t}+Bi\sin{\omega t}.$$ The total initial charge in the circuit is from the capacitor and is given by $q(0)=CV$, so we get $A\cos{\omega t}=CV \implies A=CV.$ Then $$q(t)=CV\cos(\omega t)+Bi\sin(\omega t).$$ I know that the answer should not have the imaginary term in it, but how to I eliminate it or show that B=0?

• This is a second order differential equation, so it's only well defined if you have two boundary conditions. You listed one: $q(0)=CV$, but there must be another. – DanielSank Dec 18 '17 at 21:20
• Hint: What is the current at time $t=0$?, or, more explicitly, $I(0)=\dot{q}(0)$. – jim Jun 28 at 8:53

Recognize the initial condition: at t=0, no current is flowing. That means that the charge is a maximum at t=0, and that means that B=0. For a different initial condition, the equation just means there is a phase shift - that is, you could write $\cos\omega t + \phi$ instead of introducing a complex amplitude.
When we use complex numbers for solving harmonic equations, it is really just because it usually makes the math easier (more so if you use $Ae^{i\omega t}$ notation with $A$ a complex number): the actual amplitude at a given moment in time is the real part of the expression.


text:
# Error implementing Harmonic Product Spectrum algorithm

I'm trying to find the fundamental frequencies present in a piano recording using MATLAB. These are the steps I've followed;

1. Find the envelop of the signal

2. Find the note onsets

3. perform FFT between each onset

4. Harmonic product spectrum.

It's when I try to implement the HPS algorithm that I face a "dimensions don't agree" error. This is the whole code that implements the HPS algorithm .

**% In harmonic prodcut spectrum, you downsample the fft data several times and multiply all those with the original fft data to get the maximum peak.
%HPS
seg_fft = seg_fft(1 : size(seg_fft,1)/2 );
seg_fft = abs(seg_fft);

%HPS: downsampling
for i = 1:length(seg_fft)
seg_fft2(i,1) = 1;
seg_fft3(i,1) = 1;
seg_fft4(i,1) = 1;
%   seg_fft5(i,1) = 1;
end
for i = 1:floor((length(seg_fft)-1)/2)
seg_fft2(i,1) = (seg_fft(2*i,1) + seg_fft((2*i)+1,1))/2;
end
for i = 1:floor((length(seg_fft)-2)/3)
seg_fft3(i,1) = (seg_fft(3*i,1) + seg_fft((3*i)+1,1) + seg_fft((3*i)+2,1))/3;
end

for i = 1:floor((length(seg_fft)-3)/4)
seg_fft4(i,1) = (seg_fft(4*i,1) + seg_fft((4*i)+1,1) + seg_fft((4*i)+2,1) + seg_fft((4*i)+3,1))/4;
end

%HPS, PartII: calculate product
p1 = (seg_fft3)  .* (seg_fft4);
p2 = p1.* (seg_fft2);
p3 = p2.* (seg_fft);

HPS, PartIII: find max
[f_y1,I] = max(p3)

for c = 1 : length(p3)
if(p3(c,1) == f_y1)
index = c;
end
end

% Convert that to a frequency
f_y(h) = (index / NFFT) * FS;

h = 1;
h=h+1;
f_y = abs(f_y)';

end**

Before implementing p3 = p2.* (seg_fft); the sizes of seg_fft, seg_fft2, seg_fft3, seg_fft4 all have the same dimensions of 16384 1. Then when I DO implement p3 = p2.* (seg_fft); the size of seg_fft changes to 8192 1 while the sizes of rest remain at 16384 1 thus causing an error in multiplication as the dimensions aren't the same.

I'm really confused as to why this keeps happening and nothing I try seems to work out. Would really REALLY appreciate some help here... If someone could fix this code it'd be a GREAT help... Thanx in advance.. I'm real desperate here......


text:
1.5

# A few preliminaries

Just before we start, it would be a good idea that we all ‘speak’ the same math notation language and refresh our minds with some basic math.

For addition, we will be using the short and long addition formats with the usual ‘+’ sign.

For subtraction, we will be using long subtraction formats with the usual ‘-‘ sign.

For multiplication, we will be using short and long multiplication formats with the usual ‘X’ sign.

For division, we will be using short and long division formats. Division can be denoted either using two dots ‘:’, or two dots with a horizontal line between them (this is the usual standard) ‘÷’.

It will also be useful to remember that every whole number has factors - numbers that divide the given number without leaving a remainder. For example: The factors of the number 10 are 1, 2, 5 and 10. All these numbers divide 10 without a remainder. The factors of the number 5 are 1 and 5 because only 1 and 5 divide the number 5 without a remainder. A number that has only two factors is a Prime number - it divides only itself and the number 1. Here is a fun explanation about factors and an online ‘factor calculator’.


text:
# Lesson 1: Basic Concepts¶

Note

To follow this tutorial it is recommended to first to watch the introductory video found in this linkTutorial Overview”.

To create an AnyScript model from scratch, go to “File menu -> New from template…” this will bring up a new window in which you choose “Basic Main” and provide a “Target Name” (e.g., NewModel) and click OK. This is similar to this step from the previous tutorial.

The new file that opens up in the text editor contains a basic skeleton for your model, based on a built-in template.

## What are Classes?¶

Let’s have a look at what the system has generated for you. If you ignore most of the text and comments (green lines beginning with //), the overall structure of the model looks like this:

Main = {

AnyFolder MyModel = {
}; // MyModel

AnyBodyStudy MyStudy = {
};

}; // Main

What you see is a hierarchy of braces - the outermost pair of braces is named “Main”. Everything else in the model goes between these braces.

When you load the model, the name assigned to a pair of braces, and all the contents between the braces will show up as folders and sub-folders in the Model Tree.

The code creates two objects - “MyModel” & “MyStudy” - which perform very specific functions, which depend on the pre-preprogrammed object templates (AnyFolder & AnyBodyStudy) used to create these objects. These inbuilt object templates are also known as CLASSES.

“MyModel” (of type AnyFolder) is simply an organizational folder for containing the entire model you are going to build. Let us change the folder name “MyModel” to “ArmModel”.

The object named “MyStudy” (of type AnyBodyStudy) is a collection of simulation tasks that you want to perform with your model. The Study of Studies tutorial contains much more information on simulation studies.

Warning

Rename “MyStudy” to “ArmModelStudy”, and replace all occurences of “MyModel” with “ArmModel”. All changes to your model will be highlighted in red hereon.

## What does this file contain so far?¶

 // The actual body model goes in this folder
AnyFolder ArmModel = {

// Global Reference Frame
AnyFixedRefFrame GlobalRef = {

// Todo: Add points for grounding
// of the model here

}; // Global reference frame

// Todo. Add the model elements such as
// segments, joints, and muscles here.

}; // §ArmModel§

Within “ArmModel” is an object named “GlobalRef”, created with the AnyFixedRefFrame class. This object is the global reference frame for your model.

You will notice “to do” comments inside the braces, to which we will return later.

Note

The model objects that you create henceforth must be placed within the “ArmModel” folder and should go between its pair of braces.

You should be ready to load the model now. If cannot recollect how this is done, refer to this section from a previous tutorial.

You may get message similar to the one below, in the Output Window.

Loading Main : "C:\\...\\NewModel.main.any"
Scanning...
Parsing...
Constructing model tree...
Evaluating constants...
Configuring model...
Evaluating model...
Elapsed Time : 0.063000

## Basic troubleshooting¶

If you mistype something, you will get an error message. A common mistake is to forget a semicolon somewhere. Try removing a semicolon and re-load the model, which may give you an error message like this:

ERROR(SCR.PRS11) : C:\\...\\NewModel.main.any(26) : 'EOF' unexpected Model loading skipped

The message contains clickable links for both error code and the location of the error in your file. Upon clicking the file link, the text cursor jumps to the exact problematic line in the file. Remember that an error can sometimes be caused by something you mistyped earlier in the file.

Clicking the error code, e.g: ERROR(SCR.PRS11) opens a pop-up window with a complete explanation:

We now assume that you have removed the errors and have loaded the model successfully.


text:
## Chemistry: A Molecular Approach (3rd Edition)

$^{40}_{19}K$
Atomic number of potassium (A) = 19 Number of neutrons = 21 So, mass number of potassium isotope (Z) = 21 + 19 = 40 Hence, isotopic symbol = $^{40}_{19}K$