text:
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field is isomorphic to the reals.

Every subfield of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1. Finite fields cannot be ordered.

Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and Hans Hahn. This grew eventually into the Artin–Schreier theory of ordered fields and formally real fields.

## Definitions

There are two equivalent common definitions of an ordered field. The definition of total order appeared first historically and is a first-order axiomatization of the ordering ${\displaystyle \,\leq \,}$ as a binary predicate. Artin and Schreier gave the definition in terms of positive cone in 1926, which axiomatizes the subcollection of nonnegative elements. Although the latter is higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings.

### Total order

A field ${\displaystyle (F,+,\cdot \,)}$ together with a (strict) total order ${\displaystyle \,<\,}$ on ${\displaystyle F}$ is an ordered field if the order satisfies the following properties for all ${\displaystyle a,b,c\in F:}$

• if ${\displaystyle a then ${\displaystyle a+c and
• if ${\displaystyle 0 and ${\displaystyle 0 then ${\displaystyle 0

### Positive cone

A prepositive cone or preordering of a field ${\displaystyle F}$ is a subset ${\displaystyle P\subseteq F}$ that has the following properties:[1]

• For ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle P,}$ both ${\displaystyle x+y}$ and ${\displaystyle x\cdot y}$ are in ${\displaystyle P.}$
• If ${\displaystyle x\in F,}$ then ${\displaystyle x^{2}\in P.}$ In particular, ${\displaystyle 1^{2}=1\in P.}$
• The element ${\displaystyle -1}$ is not in ${\displaystyle P.}$

A preordered field is a field equipped with a preordering ${\displaystyle P.}$ Its non-zero elements ${\displaystyle P^{*))$ form a subgroup of the multiplicative group of ${\displaystyle F.}$

If in addition, the set ${\displaystyle F}$ is the union of ${\displaystyle P}$ and ${\displaystyle -P,}$ we call ${\displaystyle P}$ a positive cone of ${\displaystyle F.}$ The non-zero elements of ${\displaystyle P}$ are called the positive elements of ${\displaystyle F.}$

An ordered field is a field ${\displaystyle F}$ together with a positive cone ${\displaystyle P.}$

The preorderings on ${\displaystyle F}$ are precisely the intersections of families of positive cones on ${\displaystyle F.}$ The positive cones are the maximal preorderings.[1]

### Equivalence of the two definitions

Let ${\displaystyle F}$ be a field. There is a bijection between the field orderings of ${\displaystyle F}$ and the positive cones of ${\displaystyle F.}$

Given a field ordering ≤ as in the first definition, the set of elements such that ${\displaystyle x\geq 0}$ forms a positive cone of ${\displaystyle F.}$ Conversely, given a positive cone ${\displaystyle P}$ of ${\displaystyle F}$ as in the second definition, one can associate a total ordering ${\displaystyle \,\leq _{P}\,}$ on ${\displaystyle F}$ by setting ${\displaystyle x\leq _{P}y}$ to mean ${\displaystyle y-x\in P.}$ This total ordering ${\displaystyle \,\leq _{P}\,}$ satisfies the properties of the first definition.

## Examples of ordered fields

Examples of ordered fields are:

• the rational numbers
• the real numbers
• any subfield of an ordered field, such as the real algebraic numbers or computable numbers
• the field of real rational functions ${\displaystyle {\frac {p(x)}{q(x)))\,}$, where ${\displaystyle p(x)}$ and ${\displaystyle q(x)}$ are polynomials with real coefficients, ${\displaystyle q(x)\neq 0\,}$, can be made into an ordered field where the polynomial ${\displaystyle p(x)=x}$ is greater than any constant polynomial, by defining that ${\displaystyle {\frac {p(x)}{q(x)))>0\,}$ whenever ${\displaystyle {\frac {p_{0)){q_{0))}>0\,}$, for ${\displaystyle p(x)=p_{0}\cdot x^{n}+\cdots }$ and ${\displaystyle q(x)=q_{0}\cdot x^{m}+\cdots \,}$. This ordered field is not Archimedean.
• The field ${\displaystyle \mathbb {R} ((x))}$ of formal Laurent series with real coefficients, where x is taken to be infinitesimal and positive
• the transseries
• real closed fields
• the superreal numbers
• the hyperreal numbers

The surreal numbers form a proper class rather than a set, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.

## Properties of ordered fields

The property ${\displaystyle a>0\land x
The property ${\displaystyle x

For every a, b, c, d in F:

• Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.
• One can "add inequalities": if ab and cd, then a + cb + d.
• One can "multiply inequalities with positive elements": if ab and 0 ≤ c, then acbc.
• Transitivity of inequality: if a < b and b < c, then a < c.
• If a < b and a, b > 0, then 1/b < 1/a.
• An ordered field has characteristic 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc. If the field had characteristic p > 0, then −1 would be the sum of p − 1 ones, but −1 is not positive.) In particular, finite fields cannot be ordered.
• Squares are non-negative: 0 ≤ a2 for all a in F.
• Every non-trivial sum of squares is nonzero. Equivalently: ${\displaystyle \textstyle \sum _{k=1}^{n}a_{k}^{2}=0\;\Longrightarrow \;\forall k\;\colon a_{k}=0.}$[2][3]

Every subfield of an ordered field is also an ordered field (inheriting the induced ordering). The smallest subfield is isomorphic to the rationals (as for any other field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves. If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean. Otherwise, such field is a non-Archimedean ordered field and contains infinitesimals. For example, the real numbers form an Archimedean field, but hyperreal numbers form a non-Archimedean field, because it extends real numbers with elements greater than any standard natural number.[4]

An ordered field F is isomorphic to the real number field R if every non-empty subset of F with an upper bound in F has a least upper bound in F. This property implies that the field is Archimedean.

### Vector spaces over an ordered field

Vector spaces (particularly, n-spaces) over an ordered field exhibit some special properties and have some specific structures, namely: orientation, convexity, and positively-definite inner product. See Real coordinate space#Geometric properties and uses for discussion of those properties of Rn, which can be generalized to vector spaces over other ordered fields.

## Which fields can be ordered?

Every ordered field is a formally real field, i.e., 0 cannot be written as a sum of nonzero squares.[2][3]

Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order need not be uniquely determined.) The proof uses Zorn's lemma.[5]

Finite fields and more generally fields of positive characteristic cannot be turned into ordered fields, because in characteristic p, the element −1 can be written as a sum of (p − 1) squares 12. The complex numbers also cannot be turned into an ordered field, as −1 is a square of the imaginary unit i. Also, the p-adic numbers cannot be ordered, since according to Hensel's lemma Q2 contains a square root of −7, thus 12+12+12+22+(−7)2=0, and Qp (p > 2) contains a square root of 1−p, thus (p−1)⋅12+(1−p)2=0.[6]

## Topology induced by the order

If F is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and × are continuous, so that F is a topological field.

## Harrison topology

The Harrison topology is a topology on the set of orderings XF of a formally real field F. Each order can be regarded as a multiplicative group homomorphism from F onto ±1. Giving ±1 the discrete topology and ±1F the product topology induces the subspace topology on XF. The Harrison sets ${\displaystyle H(a)=\{P\in X_{F}:a\in P\))$ form a subbasis for the Harrison topology. The product is a Boolean space (compact, Hausdorff and totally disconnected), and XF is a closed subset, hence again Boolean.[7][8]

## Fans and superordered fields

A fan on F is a preordering T with the property that if S is a subgroup of index 2 in F containing T − {0} and not containing −1 then S is an ordering (that is, S is closed under addition).[9] A superordered field is a totally real field in which the set of sums of squares forms a fan.[10]

## Notes

1. ^ a b Lam (2005) p. 289
2. ^ a b Lam (2005) p. 41
3. ^ a b Lam (2005) p. 232
4. ^ Bair, Jaques; Henry, Valérie. "Implicit differentiation with microscopes" (PDF). University of Liège. Retrieved 2013-05-04.
5. ^ Lam (2005) p. 236
6. ^ The squares of the square roots −7 and 1−p are in Q, but are <0, so that these roots cannot be in Q which means that their p-adic expansions are not periodic.
7. ^ Lam (2005) p. 271
8. ^ Lam (1983) pp. 1–2
9. ^ Lam (1983) p. 39
10. ^ Lam (1983) p. 45


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# H čiarový symbol latex

Learn the LaTeX commands to display the greek alphabet. A rendered preview of all letters is shown alongside all commands in a nice table. The following table shows the whole Greek alphabet along with the commands in a nice table.

Číslo šarže/čiarový kód: Neznáme Druh rizika: Riziko úrazu elektrickým prúdom Výrobok predstavuje riziko úrazu elektrickým prúdom, pretože kolíky môžu oddeliť a ponechať v stene na stene, ak je výrobok odstránený zo zásuvky na steny. Do motívu generuje inteligentné značky a čiarový kód na optimálne priradenie orezovej cesty a tlačeného PDF súboru, či upravuje výstupy z hľadiska optimálneho farebného gamutu. Web umenia je on-line katalóg výtvarných diel zo zbierok slovenských galérií. Nájdete tu základné informácie o dielach a ich autoroch, ale aj pôvodné články, videá a kolekcie. Web umenia is an on-line catalog of artworks from collections of Slovak galleries.

5. · 47768 účtový/YN účtovný/YN účtovníčka/Z účtovník/C účtovníctvo/M účtovnícky/YN účtovne/YN účtovať/WN účtenka/Z účinok/O účinný/YN účinnosť/K úč User Manual: Philips Instructions for Use . Open the PDF directly: View PDF . Page Count: 374 Feb 16, 2019 · Hyperbolic functions. The abbreviations arcsinh, arccosh, etc., are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. Symbol.

## Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTeX version is used in their description. So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article. For most symbols, the entry name is the corresponding Unicode symbol.

To resolve this error, it is best to relax this demand to ht i.e. place the float either here or at the top of the page. The demand can even be relaxed further to htbp or !htbp if necessary. The Comprehensive LATEX Symbol List Scott Pakin ∗ 25 June 2020 Abstract This document lists 14599 symbols and the corresponding LATEX commands that produce them.

### TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. It only takes a minute to sign up. Sign up to join this community

We've documented and categorized hundreds of macros! TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. It only takes a minute to sign up. Sign up to join this community Symbol Script Comment ∅ \emptyset \varnothing: empty set: ℕ \mathbb{N} set of natural numbers: ℤ \mathbb{Z} set of integers: ℚ \mathbb{Q} set of rational numbers: ℝ \mathbb{R} set of real numbers: ℂ \mathbb{C} set of complex numbers: 𝔸 \mathbb{A} set of algebraic numbers: ℍ \mathbb{H} set of quaternions: 𝕆 \mathbb{O} set of Jul 11, 2020 · Latex plus or minus symbol; Latex symbol for all x; Latex symbol exists; Latex symbol not exists; Latex horizontal space: qquad,hspace, thinspace,enspace; Latex square root symbol; Latex degree symbol; LateX Derivatives, Limits, Sums, Products and Integrals; Latex copyright, trademark, registered symbols; Latex euro symbol Learn the LaTeX commands to display the greek alphabet.

Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience.

See the “Comprehensive LaTeX Symbol List” package at https://ctan.org/pkg/comprehensive. Here is a list of commonly-used symbols. It is by no means exhaustive. The Comprehensive LATEX Symbol List Scott Pakin 9 November 2009 Abstract This document lists 5913 symbols and the corresponding LATEX commands that produce them. Some of these symbols are guaranteed to be available in every LATEX2"system; others require fonts and packages These LaTeX's symbols are grouped together more or less according to function. Some of these symbols are primarily for use in text; most of them are mathematical symbols and can only be used in LaTeX's math mode. Most tables are excerpted from the LaTeX Command Summary (Botway & Biemesderfer 1989, Providence, RI: TeX Users Group) and reproduced Dec 05, 2020 · Latex plus or minus symbol; Latex symbol for all x; Latex symbol exists; Latex symbol not exists; Latex horizontal space: qquad,hspace, thinspace,enspace; Latex square root symbol; Latex degree symbol; LateX Derivatives, Limits, Sums, Products and Integrals; Latex copyright, trademark, registered symbols; Latex euro symbol The Comprehensive LATEX Symbol List Scott Pakin ∗ 8 October 2002 Abstract This document lists 2590 symbols and the corresponding LATEX commands that produce them.

\rotr q \varg. TABlE 16: t4phonet Phonetic Symbols. of these symbols are guaranteed to be available in every LATEX 2ε system; h \ oleft. Table 33: Variable-sized Math Operators. T \. \bigcap.

Sign up to join this community Symbol Script Comment ∅ \emptyset \varnothing: empty set: ℕ \mathbb{N} set of natural numbers: ℤ \mathbb{Z} set of integers: ℚ \mathbb{Q} set of rational numbers: ℝ \mathbb{R} set of real numbers: ℂ \mathbb{C} set of complex numbers: 𝔸 \mathbb{A} set of algebraic numbers: ℍ \mathbb{H} set of quaternions: 𝕆 \mathbb{O} set of Jul 11, 2020 · Latex plus or minus symbol; Latex symbol for all x; Latex symbol exists; Latex symbol not exists; Latex horizontal space: qquad,hspace, thinspace,enspace; Latex square root symbol; Latex degree symbol; LateX Derivatives, Limits, Sums, Products and Integrals; Latex copyright, trademark, registered symbols; Latex euro symbol Learn the LaTeX commands to display the greek alphabet. A rendered preview of all letters is shown alongside all commands in a nice table. The following table shows the whole Greek alphabet along with the commands in a nice table. The float specifier h by itself is usually too strict of a command for LaTeX. To resolve this error, it is best to relax this demand to ht i.e. place the float either here or at the top of the page.

9. 5. · 47768 účtový/YN účtovný/YN účtovníčka/Z účtovník/C účtovníctvo/M účtovnícky/YN účtovne/YN účtovať/WN účtenka/Z účinok/O účinný/YN účinnosť/K úč User Manual: Philips Instructions for Use . Open the PDF directly: View PDF . Page Count: 374 Feb 16, 2019 · Hyperbolic functions. The abbreviations arcsinh, arccosh, etc., are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. Symbol.

bank of america transakcie m&a
60 000 usd v aud
prázdny znak png

format: plain


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In this paper we apply recent developments in transpermanent magnetics to the problem of ultra-low-power valve control. Whereas the traditional approach to ultra-low-power valve control is based on latching mechanisms that turn off valves during inactive periods, in this paper we describe an approach that eliminates the need for latching mechanisms. Instead of latching mechanisms, the principles of transpermanent magnetics are employed to switch the states of permanent magnets; the use of permanent magnets instead of electromagnets eliminates power loads during inactive periods, thereby reducing power consumption to ultralow levels. The permanent magnets in a transpermanent magnet valve are configured in a stack. The relationships between the strength and number of permanent magnets in the stack and the stroke and resolution of the valve are developed. In this paper we show that the alternating uniform linear stack is well suited for digital process valves having a small number of states. Then in the paper we report on the design and testing of a laboratory prototype valve that uses an alternating uniform linear stack. The prototype valve had five states yielding a range of flow rates between 0 and $1.58m∕s$ with a resolution of $0.3m∕s$. In this paper we find that transpermanent valves represent a promising valve technology for digital process valves.

1.
Sethson
,
K. M. R.
, 1993, “
Identifying the Dynamics of Fast Response Solenoid Valves
,”
The 3rd International Conference on Fluid Power
271
287
.
2.
Becker
,
U.
, and
Schnieder
,
E.
, 1993 “
Different Mode of Operation of Pulse-Modulation with the Drive of 2∕2 Fast Switching Valves
,”
The 3rd Scandinavian International Conference of Fluid Power
, May 1993, Sweden, pp.
237
254
.
3.
Muramatsu
,
N.
, and
Tsutsumi
,
K.
, 2001, “
Measure to Reduce the Bounce of a Movable Part in an Electromagnetic Contactor
,”
ASME J. Mech. Des.
1050-0472
123
, pp
289
293
.
4.
Silverberg
,
L.
, and
Farmer
,
D.
, 2003, “
On Trans-Permanent Magnetic Actuation for Spacecraft Pointing, Shape Control, and Deployment
,”
J. Spacecr. Rockets
0022-4650, to appear.
5.
Farmer
,
D.
, 2003, “
Trans-Permanent Actuators and Motors
,” Ph.D. dissertation, Mechanical Engineering, North Carolina State University, Raleigh, NC, 27695-7910.
6.
Silverberg
,
L.
, and
Duval
,
L.
, 2003, “
Analysis of Trans-Permanent Magnetic Systems
,”
J. Dyn. Syst., Meas., Control
0022-0434,
125
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143
146
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Zycki
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Z.
, and
Pawluk
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, 1998, “
Design of RLC Circuit for Pulse Magnetizer of Permanent Magnets
,”
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,
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, pp.
412
417
.
8.
BasicX-24 Microcontroller, NetMedia Inc., (www. basicx. comwww. basicx. com).
9.
L298 Dual Full-Bridge Driver Datasheet, STMicroelectronics (www.st.comwww.st.com).
10.
Duval
,
L.
, 2003, “
Low Power Valve Actuation Using Trans-Permanent Magnetics
,” Ph.D. dissertation, Mechanical Engineering, North Carolina State University, Raleigh, NC, 27695-7910.
11.
Spring Catalog 2002, Century Spring Corp. (www.centuryspring.comwww.centuryspring.com).
This content is only available via PDF.


text:
# Confusion about “Seemingly not ～”

So there are several ways to express something is "seemingly not ～":

• ～なさそう
• ～そうにない
• ～そうもない
• ～そうにもない (is this one even real?)

I was always taught ～なさそう in my Japanese classes, and it was not until I heard one of the other ones that I ever become confused about them.

After doing a little research on this, it seems that adjectives can only take ～なさそう. ex.

○ 料理がおいしくなさそう
？／×　料理がおいしそうにない

but that verbs can take any of the forms

○ 雨が　降らなさそう／降りそう（に・も・にも？）ない

So, my questions are a) is the above reasoning correct, and b) what are the differences/nuances of the latter three forms? Someone please set me straight on this; it's been bugging me for a couple years.

-
This is not the key point of the question, but I am not sure about the correctness of 雨が降らなさそうだ. Although I may say 雨が降らなさそうだ myself, I believe that the correct form is 雨が降らなそうだ and that in general, when the そうだ meaning speculation follows the auxiliary verb ない (instead of the i-adjective ない), it takes the form なそうだ. (more) –  Tsuyoshi Ito Jul 3 '11 at 23:26
(cont’d) Both Daijirin and Daijisen (sense 1) agree on this. –  Tsuyoshi Ito Jul 3 '11 at 23:27
イトウさん、フィードバックありがとう。「問題な日本語」っていう辞典はお持ちなら、「～なさそう　vs.　～なそう」をトピックとした面白い項目が載ってる。「知らなさ‌そうだ」っていうものだ。 –  istrasci Jul 3 '11 at 23:39
Thank you for the pointer. I do not have that book, but (in retrospect) I can imagine that it contains some story about the forms ～なさそう and ～なそう. –  Tsuyoshi Ito Jul 3 '11 at 23:45
@istrasci You can see for yourself that 〜そうにもない is real if you search BCCWJ. –  snailboat Jul 13 '14 at 18:46

Looks like you have done your research well. You are correct with verbs only taking そうに/もない。Although そうにもない　is not grammatical though.

雨が降らなさそう　you make an assumption based on current state of the sky.

There is another common way to express the same idea, ～みたいです。only used in the affirmative. (ie, 「～みたいじゃない」ってだめです）

あの男の人は学生じゃないみたいです　on campus you plainly state that guy over there doesn't seem like a student.

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[You are correct with verbs only taking そうに/もない]. Did you mean "adjectives"? Because that's what I put in the OP. –  istrasci Jul 3 '11 at 19:25
i thought that そうもない should a higher degree, like 100% confidence that it wasn't going to rain? –  Mark Hosang Jul 4 '11 at 0:58

## Different types of ～そう

I think it's best to consider adjective + そう and verb + そう separately, because they have slightly different meanings. In particular, with verbs, そう has the sense of something close to happening or about to happen in the near future, but with adjectives... not really.

雨が降りそうだ。
It looks like it's about to rain. (near future)

She waved happily. (not the future)

あの料理は（まだ食べていないけど、）おいしそうだ。
(I haven't eaten that food yet, but) it looks tasty. (kind of maybe the future?)

## Negative form of ～そう with verbs

Other answers have already mentioned that the さ in 降りなさそう is not the officially correct form, and I found an authoritative-looking answer on Oshiete! goo which agrees.

I believe that

• 雨が降りそうにない
• 雨が降りそうもない
• 雨が降りそうにもない

all mean the same thing, and I found a link for that one too.

## Difference between ～そうにない and ～なそう (verbs only)

「雨が降らなそう」is the normal negative, which just means "it doesn't look like it's going to rain. I think ～そうにない (and そうもない and そうにもない) not only add emphasis, but also imply "I wish it would happen, but it looks hopeless." Your earlier example 「僕にもわかりそうにない」, "It looks like there's no way I'm going to be able to understand it either" also fits this pattern. However, it's more difficult to back this idea up. I did find a link where someone wrote that 「授業に遅れそうにない」 seemed unnatural, but they couldn't explain why.

-

I just realized I didn't know the difference so I asked a native Japanese.

Here is the explation

normal straigh meaning: "it doesn't seem to rain"

this sounds like: "I though it would rain, but now it doesn't seem so"

this is like the previous but も makes it more strong, like put an "absolutely" in front.

this is correct (same meaning that 降りそうもない) but a bit too complex for casual conversation. It's かきことば.

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Hehe, so it's your native speaker vs my native speaker. –  Gerard Sexton Jul 3 '11 at 23:25
:) well only the にもない thingy it's still unclear. My native said it's bit unusual and formal but not wrong. –  Uberto Jul 4 '11 at 7:49


text:
Home / Help

Author Goerss, Paul G. author Simplicial Homotopy Theory [electronic resource] / by Paul G. Goerss, John F. Jardine Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1999 http://dx.doi.org/10.1007/978-3-0348-8707-6 XV, 510 p. online resource

CONTENT

I Simplicial sets -- 1. Basic definitions -- 2. Realization -- 3. Kan complexes -- 4. Anodyne extensions -- 5. Function complexes -- 6. Simplicial homotopy -- 7. Simplicial homotopy groups -- 8. Fundamental groupoid -- 9. Categories of fibrant objects -- 10. Minimal fibrations -- 11. The closed model structure -- II Model Categories -- 1. Homotopical algebra -- 2. Simplicial categories -- 3. Simplicial model categories -- 4. The existence of simplicial model category structures -- 5. Examples of simplicial model categories -- 6. A generalization of Theorem 4.1 -- 7. Quillenโ{128}{153}s total derived functor theorem -- 8. Homotopy cartesian diagrams -- III Classical results and constructions -- 1. The fundamental groupoid, revisited -- 2. Simplicial abelian groups -- 3. The Hurewicz map -- 4. The Ex? functor -- 5. The Kan suspension -- IV Bisimplicial sets -- 1. Bisimplicial sets: first properties -- 2. Bisimplicial abelian groups -- 3. Closed model structures for bisimplicial sets -- 4. The Bousfield-Friedlander theorem -- 5. Theorem B and group completion -- V Simplicial groups -- 1. Skeleta -- 2. Principal fibrations I: simplicial G-spaces -- 3. Principal fibrations II: classifications -- 4. Universal cocycles and $$\bar W$$G -- 5. The loop group construction -- 6. Reduced simplicial sets, Milnorโ{128}{153}s FK-construction -- 7. Simplicial groupoids -- VI The homotopy theory of towers -- 1. A model category structure for towers of spaces -- 2. The spectral sequence of a tower of fibrations -- 3. Postnikov towers -- 4. Local coefficients and equivariant cohomology -- 5. On k-invariants -- 6. Nilpotent spaces -- VII Reedy model categories -- 1. Decomposition of simplicial objects -- 2. Reedy model category structures -- 3. Geometric realization -- 4. Cosimplicial spaces -- VIII Cosimplicial spaces: applications -- 1. The homotopy spectral sequence of a cosimplicial space -- 2. Homotopy inverse limits -- 3. Completions -- 4. Obstruction theory -- IX Simplicial functors and homotopy coherence -- 1. Simplicial functors -- 2. The Dwyer-Kan theorem -- 3. Homotopy coherence -- 4. Realization theorems -- X Localization -- 1. Localization with respect to a map -- 2. The closed model category structure -- 3. Bousfield localization -- 4. A model for the stable homotopy category -- References

Mathematics Topology Algebraic topology Mathematics Algebraic Topology Topology

Location

Office of Academic Resources, Chulalongkorn University, Phayathai Rd. Pathumwan Bangkok 10330 Thailand


text:
spin geometry

string geometry

bundles

# Contents

## Definition

A spinor bundle on a smooth manifold with spin structure is a $\rho$-associated bundle associated to the spin group-principal bundle lifting the tangent bundle, for $\rho : \mathbf{B} Spin \to$ Vect a representation of the spin group.

A section of a spinor bundle is called a spinor.

A Dirac operator acts on sections of a spinor bundle.

In physics, sections of spinor bundles model matter particles: fermion. See spinors in Yang-Mills theory.

standard model of particle physics and cosmology

| | gravity | electroweak and strong nuclear force | fermionic matter | scalar field | | field content: | vielbein field $e$ | principal connection $\nabla$ | spinor $\psi$ | scalar field $H$ | | Lagrangian: | scalar curvature density | field strength squared | Dirac operator component density | field strength squared + potential density | | $L =$ | $R(e) vol(e) +$ | $\langle F_\nabla \wedge \star_e F_\nabla\rangle +$ | $(\psi , D_{(e,\nabla)} \psi) vol(e) +$ | $\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$ |

## References

Revised on October 15, 2013 19:48:26 by Urs Schreiber (80.237.234.132)


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# Critical apparatus

Page 1 of the works of Gaius Musonius Rufus, with critical apparatus
Stemma codium constructed by Cesare Questa, representing relationships of textual witnesses (with their sigla; the lost source text is indicated by Ω) to the works of Roman playwright Plautus, with estimated dates from the IVth through Xth centuries.

A critical apparatus (Latin: apparatus criticus) in textual criticism of primary source material, is an organized system of notations to represent, in a single text, the complex history of that text in a concise form useful to diligent readers and scholars. The apparatus typically includes footnotes, standardized abbreviations for the source manuscripts, and symbols for denoting recurring problems (one symbol for each type of scribal error).

The fullness of the critical apparatus may excite surprise. The object of a critical apparatus, as I conceive it, is to enable the earnest student to form his own text. It is therefore an editor's duty to give all the material variants of all the independent manuscripts. It is particularly his duty in a case like the present where there are many passages of which he can give no satisfactory explanation.[1]

## Shakespearean studies

Many editions employ a standard format for a critical apparatus, as illustrated by a line from Hamlet, which the Oxford Complete Works (1988) prints as follows:[2]

LAERTES. Alas, then is she drowned.

The apparatus for the line might be rendered as:[3]

4.7.156 Alas, then is she drowned.] HIBBARD; Alas then, is she drown'd? F;
Alas then is she drownd. Q3; Alas, then, she is drownd. Q2;
So, she is drownde: Q1.

The format of the apparatus has several parts:

• The location of the variant in the text (act, scene, line number)
• The lemma, which is the portion of the text to which the note applies
• A right bracket (])
• The source from which the edition took its reading
• A list of variants, in each case followed by the source in which the variant is found, and set off with a semicolon.

To save space, frequently cited sources are usually assigned an abbreviation called a siglum. In Shakespeare editions, F always signifies the First Folio; the second through fourth folios are referred to as F2 through F4 respectively. Similarly, Q1 is the first quarto, Q2 the second quarto, and Q3 the third.

In the example given, the first folio (F) and the three early quartos (Q1 to Q3) each have a different reading of the line in question. The editors have concluded that all four early sources are corrupt, and instead have adopted a reading suggested by G. R. Hibbard. Other editors of the play may choose a different reading of the line. The apparatus summarizes all of the textual evidence, allowing readers to assess for themselves whether the editor has made the best choice. Sometimes the editor will add a commentary, defending the choice made, explaining why other readings were rejected, or discussing how other editors have treated the passage.

This format has been used for critical apparatuses of Shakespeare and many other authors. In variorum editions, the apparatus is often placed at the bottom of the page. Sometimes a three-part format is employed, with the main text at the top of the page, textual variants in the middle, and the editor's commentary at the bottom. This remains the most common format for Shakespeare editions, although the Oxford Complete Works breaks with tradition by putting its critical apparatus in a separately published volume.

## New Testament studies

The first printed edition of the New Testament with critical apparatus, noting variant readings among the manuscripts, was produced by the printer Robert Estienne of Paris in 1550. The Greek text of this edition and of those of Erasmus became known as the Textus Receptus (Latin for "received text"), a name given to it in the Elzevier edition of 1633, which termed it as the text nunc ab omnibus receptum ("now received by all").

The Novum Testamentum Graece (1st edition 1898, 28th edition 2014) uses a system of sigla created by Caspar René Gregory in 1908, and extended by Kurt Aland, known as Gregory–Aland numbering; these sigla are widely used in academic writing about the Greek text. The most important uncials are given Hebrew, Roman, or Greek letter names: א ( Codex Sinaiticus), A ( Codex Alexandrinus), Dp (Codex Claromontanus), or Ξ (Codex Zacynthius). The papyri are assigned the Blackletter character ${\displaystyle {\mathfrak {P}}}$ followed by a superscript number. Here, Papyrus Oxyrhynchus 208 + 1781 (parts of the Gospel of John) is 𝔓5 and the Chester Beatty Papyri, which contains the Gospels and Acts, is 𝔓45. The superscript numbers follow the order of registration, and do not reflect the age of the manuscript or order of importance. The minuscules are given plain numbers, and the lectionaries are notated by script (), ranging from 1 up to 2463.

## From footnotes to endnotes

In the United States, bibliographer Fredson Bowers (1905–1991) established a tradition of putting the critical apparatus at the back of the book, leaving the edited text clear of apparatus. This has the advantage of leaving the main text uncluttered with editorial details that may not be of interest to the general reader. However, this format is a disadvantage to scholarly readers, who are not able to see all of the textual evidence in one place.

## Electronic representation

The de facto standard for the representation of critical apparatus in digital scholarly editions is to follow the recommendations of the Text Encoding Initiative.[4] While other formats are also used in digital literary studies this has become the most accepted storage format.[5]

## References

1. ^ Ashburner, Walter, ed. (1909). Rhodian Sea-Law. Oxford: Clarendon Press. p. iii.
2. ^ William Shakespeare, The Complete Works (Oxford: Oxford University Press, 1991), p. 682
3. ^ Suggested by Stanley Wells and Gary Taylor, William Shakespeare: A Textual Companion (New York: W. W. Norton & Company, 1997), p. 408.
4. ^ See "12 Critical Apparatus", Guidelines for Electronic Text Encoding and Interchange, Text Encoding Initiative, version 4.3.0 (updated 31 August 2021), archived 2 September 2021 at archive.today.
5. ^ See also James Cummings, "The Text Encoding Initiative and the Study of Literature" in 'A Companion to Digital Literary Studies', ed. Susan Schreibman and Ray Siemens. Oxford: Blackwell, 2008. http://www.digitalhumanities.org/companionDLS/


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# Tag Info

As both AdamO and Sheep mention, the events would have to be independent. If the events were independent, one cannot simply take the sum/product of odds ratios to find the odds ratio of both events combined. However, one can take the product of probabilities of independent events to calculate the probability of the combined event. P(A \bigcap B) = P(A) * ...


text:
# StoreOnDisk¶

class StoreOnDisk(filename=None, mode=None, working_directory=None)

Class indicating that the self energies should be cached to disk.

The given file will be used for both reading and writing self energies if mode is Append. If no filename is given, or mode is ReadOnly, newly calculated self energies are stored in a temporary cache file for the duration of the calculation (note that this file will be deleted at the end of the calculation).

Parameters: filename (str) – The filename to read/write self energies to. mode (Append | ReadOnly) – Whether newly calculated self energies are added to the given file. With Append, new self energies are appended to the file. With ReadOnly, they are stored in a temporary file that is discarded at the end of the calculation. Default: Append working_directory (str) – The directory to store temporary cache files in. Default: The system’s temporary directory.

## Usage Examples¶

Basic examples on how to use StoreOnDisk() and other storage strategies can be found in the Usage Examples of RecursionSelfEnergy.

### Temporarily save self energies on disk¶

Define that the self energies are saved on disk in a temporary file after the first iteration of the SCF loop and reused in the following iterations.

self_energy_calculator = RecursionSelfEnergy(
storage_strategy=StoreOnDisk()
)

By default, a temporary cache file is created in the system’s temporary directory. The location of the temporary cache file can also be specified.

self_energy_calculator = RecursionSelfEnergy(
storage_strategy=StoreOnDisk(working_directoy='/home/user/')
)

### Reuse self energies between calculations¶

Specify the path of the file that stores the calculated self energies after the first iteration of the SCF loop.

self_energy_calculator = RecursionSelfEnergy(
storage_strategy=StoreOnDisk(filename='/home/user/selfenergycache.hdf5')
)

Note

Self energies are stored on disk using the HDF5 format. The filename provided to StoreOnDisk() can have any extension.

In subsequent calculations, the previously calculated self energies can be reused by providing the same path again.

Important

QuantumATK does not verify the validity of the self energies in the provided file. Providing a cache file containing self energies calculated for a different electrode will give incorrect results.

By default, self energies not present in the cache file are calculated and added to the cache file. This can be avoided by marking the cache file read-only.

self_energy_calculator = RecursionSelfEnergy(
storage_strategy=StoreOnDisk(
filename='/home/user/selfenergycache.hdf5'),

The StoreOnDisk() storage strategy specifies that each self energy is calculated only once and stored on disk. The self energies are reused in subsequent iterations. This strategy can significantly improve performance, but requires sufficient disk space.
In calculations involving a large number of parallel processes, storing the self energies to disk is equally fast as the SaveInMemory() storage strategy inside a single node. However, when a custom filename is used, the performance can slightly slow down when multiple nodes are involved in the calculation. If a temporary file is used instead, there is no slowdown across nodes. After the self energies have been stored, the StoreOnDisk() and SaveInMemory() storage strategies provide a similar performance improvement regardless of the parameters used.
Note that the StoreOnDisk() storage strategy requires less memory than the SaveInMemory() strategy, and is therefore the recommended choice for memory-intensive calculations when there is plenty of disk space available.


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# $\omega^2+\omega+1$divides a polynomial

The question is

Show that $f(n)=n^5+n^4+1$ is not prime for $n>4$.

The solution is given as

Let $\omega$ be the third root of unity. Then $\omega^2+\omega+1=0$. Since $\omega^5+\omega^4+1=\omega^2+\omega+1$, we see that $\omega^2+\omega+1$ is a *factor of the polynomial. So *$n^2+n+1|n^5+n^4+1$.

Which polynomial are we referring to in the bold typeface above? And how is $n^2+n+1|n^5+n^4+1$ true due to $\omega^5+\omega^4+1=\omega^2+\omega+1$?

-
What is $g(x)$ here and what are the common factors? – user117913 Dec 27 '13 at 18:15

This is rather poor phrasing (in the highlighted solution). What it is saying is that that $\omega,$ which is a root of the irreducible polynomial $x^2+x+1$ is also a root of $x^5+x^4+1,$ and so the gcd of the two polynomials is not $1,$ and since the first polynomial is irreducible, it must divide the second. Now, that said, you can verify this without any fanciness by long division.

-

As the 3rd root of identity $\omega=\mathrm{e}^{2\pi i/3}$ satisfies $x^5+x^4+1=0$, then so does its conjugate $\bar\omega=\mathrm{e}^{-2\pi i/3}$, and hence $(x-\omega)(x-\bar\omega)=x^2+x+1$ divides $x^5+x^4+1$.

Note that $$x^5+x^4+1=(x^2+x+1)(x^3-x+1).$$

-
I know other solutions. Please help me understand the given one. – user117913 Dec 27 '13 at 18:16
The given solution tells you how to get this factorization. – Igor Rivin Dec 27 '13 at 18:20
See edited version. – Yiorgos S. Smyrlis Dec 27 '13 at 18:23
This is the solution that I was looking for. – user117913 Dec 27 '13 at 18:28

See that $\omega$ is a root of both $f(x)=x^5+x^4+1$ and $g(x)=x^2+x+1$. Now $g(x)$ has all coefficients real, and it is a polynomial of degree 2. Hence it has two roots and the second root must be the complex conjugate of $\omega$, that is $\omega^2$. Hence $g(x)=(x-\omega)(x-\omega^2)$. Also note that $f(\omega^2)=0$. Hence $f(x)=(x-\omega)(x-\omega^2)h(x)=g(x)h(x)$ for some real polynomial $h(x)$. Now you can easily show by equating coefficients that $h(x)$ has integer coefficients. Now put $x=n$ on both sides. Then since $h(n)$ turns out to be an integer $n^2+n+1|n^5+n^4+1$.

-

Let $$g(n)=n^2+n+1$$ and $$h(n)=n^3-n+1$$.

Clearly $$f(n)=g(n) \cdot h(n)$$

Now $h(1)=1$ and $h'(n) = 3 n^2-1 > 0$ for $n\ge 1$. So, for $n>1$ both $g(n)$ and $h(n)$ are greater than $1$. So

$f(n)$ **is composite for all $n>1$

-


text:
# Prove that

Question:

$15 x^{2}-28=x$

Solution:

Given:

$15 x^{2}-28=x$

$\Rightarrow 15 x^{2}-x-28=0$

$\Rightarrow 15 x^{2}-(21 x-20 x)-28=0$

$\Rightarrow 15 x^{2}-21 x+20 x-28=0$

$\Rightarrow 3 x(5 x-7)+4(5 x-7)=0$

$\Rightarrow(3 x+4)(5 x-7)=0$

$\Rightarrow 3 x+4=0$ or $5 x-7=0$

$\Rightarrow x=\frac{-4}{3}$ or $x=\frac{7}{5}$

Hence, the roots of the equation are $\frac{-4}{3}$ and $\frac{7}{5}$.


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# The bare minimum guide to Matplotlib

If you want to work with arrays in Python, you use NumPy. If you want to work with tabular data, you use Pandas. The quintessential Python library for data visualization is Matplotlib. It’s easy to use, flexible, and a lot of other visualization libraries build on the shoulders of Matplotlib. This means that learning Matplotlib will make it easier to understand and work with some of the more fancy visualization libraries.

### Getting started

You’ll need to install the Matplotlib library. Assuming you have some terminal at your disposal and you have pip installed, you can install Matplotlib with the following commaned: pip install matplotlib. You can read more about the installation in Matplotlib’s installation guide.

### Two approaches

We’ll begin by making a simple scatter plot in two different ways: the ‘naive’ way and the object-oriented way. Both approaches have their pros and cons. Generally, we can say that the object-oriented approach is best when you need multiple plots next to each other.I almost always use the object-oriented approach though, even when I don’t need to make multiple plots.

#### ‘Naive’

To start with we have to import matplotlib though. The plt framework is what we’ll use for Python plotting.

We also import numpy, so we can easily generate points to plot! Let’s pick some points on the sine function. We choose some x-values and then calculate the y-values with np.sin.

Now that we’ve generated our points, we can make our scatter plot! We use the scatter function from the plt framework to make the plot, and we use show to visualize our plot.

By running these $6$ lines, a window with the following plot should appear.

If we don’t want a scatter plot but a line plot, we can switch out scatter for plot.

This gives us the following plot.

However, this line is very jagged. We can make it more smooth by generating more points.

#### Object-oriented

Now that we know how to make and visualize a plot, let’s look at the object-oriented way of producing the same plot. However, why would we want to know this? Simply because the object-oriented way is more powerful and allows for more complicated plots, as will be evident when we want to make multiple plots.

If we want to replicate the previous plot, we start by making a Figure object and an Axes object.We assume, we have generated our data.

We can think of the Figure object as the frame, we want to put plots into, and the Axes object is an actual plot in our frame. We then add the line plot to the Axes object and use show again to visualize the plot.

This generates the same plot as before.

### Line plots

Here are examples of colours that we can use. We can specify colours in many different ways; hex code, RGB, plain old names.

There are also many predefined linestyles that we can use. Note that without defining colours, Matplotlib will automatically choose some distinct default colors for our lines.

We can also adjust the width of our lines!

### Scatter plots

For scatter plots, we can change the markers and their size. Here’s an example

We can also combine line and scatter plots using the ax.plot function by changing the fmt parameter. The fmt parameter consists of a part for marker, line, and color: fmt = [marker][line][color]. If fmt = "s--m", then we have square markers, a dashed line, and they’ll be coloured magenta.

### Histograms

We can make histograms easily using the ax.hist function.

We can change a lot of things in the histogram to make it nicer - we can even add multiple!

### Legends

Naturally, we’ll want to add a legend to our plot. This is simply done with the ax.legend function.

Matplotlib will automatically try and find the best position for the legend on your plot, but we can change it by providing an argument for the loc parameter. Also, a common preference is to not have a frame around the legend, and we can disable it by setting the frameon parameter to False. Additionally, Matplotlib lists the elements of the legend in one column, but we can provide the number of columns to use in the ncol parameter.

### Final tips

There are so many quirks and different things you can do with Matplotlib, and unfortunately I cannot provide them all here. However, a few guidelines to get you started:

1. You save figures with the plt.savefig() function.
2. There are a bunch of libraries that build on the shoulders of Matplotlib that could be beneficial to the specific plot you’re trying to create, e.g. Seaborn, Bokeh, Plotly, and many more.
3. Look at the gallery. Please, please, look at the gallery! Don’t waste 3 hours working on a plot, if someone has already made it.


text:
# Homework Help: Physical asymptote

1. Feb 1, 2015

### LagrangeEuler

1. The problem statement, all variables and given/known data
For example particle performs a motion in x-y plane. In y there are walls from both side so particle can go in y direction from zero to $$h$$. I need to plot trajectory. If I got trajectory $$y=x^4-x^2$$ then
$$\lim_{x\to \infty}y(x)=\infty 2. Relevant equations 3. The attempt at a solution If I got trajectory [tex]y=x^4-x^2$$ then
$$\lim_{x\to \infty}y(x)=\infty$$, but because of the condition I may say that $$\lim_{x\to \infty}y(x)=h$$. Maybe then $$y=h$$ is some natural horisontal asymptote?

2. Feb 1, 2015

### Goddar

Hi. It's hard to understand what your problem exactly is. Could you clarify it maybe by giving its original statement?
If you are asked to plot a two-dimensional trajectory parametrized by:
y = x4–x2,
Under the constraint: ymax= h,
Then it gives you a natural constraint on x as well, as a function of h; if you need to plot this you'll have to assign an arbitrary value to h so that you can plot something.
Now depending on this value, your plot will not always look the same but that's all you can do with the given information...


text:
# Gain of multipath fading components

For the Rayleigh fading model, I want use four multipath components, each component has its delay and gain.

• Can the gain in $\textrm{dB}$ of any of the individual four components be positive value?

I can understand that it can be zero or maybe negative indicating reduction in its amplitude due to delay and effects of channel, but I can not understand how it can be positive.

• The overall signal resulting from the four components at the receiver may be with positive gain. This is reasonable, due to constructive addition of maybe inphase components, but how can a single component be with positive gain?
• In addition to Maximilian's answer: in a frequency-flat channel, what is simulated as a single channel tap is actually the sum of many scattered signals, whose delays are very similar and much smaller than the symbol period. These may also add constructively, resulting in a larger-than-one gain. – MBaz Nov 10 '16 at 13:47


text:
Opuscula Math. 33, no. 4 (2013), 641-646
http://dx.doi.org/10.7494/OpMath.2013.33.4.641

Opuscula Mathematica

# A note on k-Roman graphs

Ahmed Bouchou
Mostafa Blidia
Mustapha Chellali

Abstract. Let $$G=\left(V,E\right)$$ be a graph and let $$k$$ be a positive integer. A subset $$D$$ of $$V\left( G\right)$$ is a $$k$$-dominating set of $$G$$ if every vertex in $$V\left( G\right) \backslash D$$ has at least $$k$$ neighbours in $$D$$. The $$k$$-domination number $$\gamma_{k}(G)$$ is the minimum cardinality of a $$k$$-dominating set of $$G.$$ A Roman $$k$$-dominating function on $$G$$ is a function $$f\colon V(G)\longrightarrow\{0,1,2\}$$ such that every vertex $$u$$ for which $$f(u)=0$$ is adjacent to at least $$k$$ vertices $$v_{1},v_{2},\ldots ,v_{k}$$ with $$f(v_{i})=2$$ for $$i=1,2,\ldots ,k.$$ The weight of a Roman $$k$$-dominating function is the value $$f(V(G))=\sum_{u\in V(G)}f(u)$$ and the minimum weight of a Roman $$k$$-dominating function on $$G$$ is called the Roman $$k$$-domination number $$\gamma_{kR}\left( G\right)$$ of $$G$$. A graph $$G$$ is said to be a $$k$$-Roman graph if $$\gamma_{kR}(G)=2\gamma_{k}(G).$$ In this note we study $$k$$-Roman graphs.

Keywords: Roman $$k$$-domination, $$k$$-Roman graph.

Mathematics Subject Classification: 05C69.

Full text (pdf)

• Ahmed Bouchou
• University Dr Yahia Fares, Médéa, Algeria
• Mostafa Blidia
• University of Blida, LAMDA-RO, Department of Mathematics, B.P. 270, Blida, Algeria
• Mustapha Chellali
• LAMDA-RO, Department of Mathematics, B.P. 270, Blida, Algeria
• Revised: 2013-02-25.
• Accepted: 2013-04-04.


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You are currently browsing the monthly archive for January 2008.

We now begin our study of measure-preserving systems $(X, {\mathcal X}, \mu, T)$, i.e. a probability space $(X, {\mathcal X}, \mu)$ together with a probability space isomorphism $T: (X, {\mathcal X}, \mu) \to (X, {\mathcal X}, \mu)$ (thus $T: X \to X$ is invertible, with T and $T^{-1}$ both being measurable, and $\mu(T^n E) = \mu(E)$ for all $E \in {\mathcal X}$ and all n). For various technical reasons it is convenient to restrict to the case when the $\sigma$-algebra ${\mathcal X}$ is separable, i.e. countably generated. One reason for this is as follows:

Exercise 1. Let $(X, {\mathcal X}, \mu)$ be a probability space with ${\mathcal X}$ separable. Then the Banach spaces $L^p(X, {\mathcal X}, \mu)$ are separable (i.e. have a countable dense subset) for every $1 \leq p < \infty$; in particular, the Hilbert space $L^2(X, {\mathcal X}, \mu)$ is separable. Show that the claim can fail for $p = \infty$. (We allow the $L^p$ spaces to be either real or complex valued, unless otherwise specified.) $\diamond$

Remark 1. In practice, the requirement that ${\mathcal X}$ be separable is not particularly onerous. For instance, if one is studying the recurrence properties of a function $f: X \to {\Bbb R}$ on a non-separable measure-preserving system $(X, {\mathcal X}, \mu, T)$, one can restrict ${\mathcal X}$ to the separable sub-$\sigma$-algebra ${\mathcal X}'$ generated by the level sets $\{ x \in X: T^n f(x) > q \}$ for integer n and rational q, thus passing to a separable measure-preserving system $(X, {\mathcal X}', \mu, T)$ on which f is still measurable. Thus we see that in many cases of interest, we can immediately reduce to the separable case. (In particular, for many of the theorems in this course, the hypothesis of separability can be dropped, though we won’t bother to specify for which ones this is the case.) $\diamond$

We are interested in the recurrence properties of sets $E \in {\mathcal X}$ or functions $f \in L^p(X, {\mathcal X}, \mu)$. The simplest such recurrence theorem is

Theorem 1. (Poincaré recurrence theorem) Let $(X,{\mathcal X},\mu,T)$ be a measure-preserving system, and let $E \in {\mathcal X}$ be a set of positive measure. Then $\limsup_{n \to +\infty} \mu( E \cap T^n E ) \geq \mu(E)^2$. In particular, $E \cap T^n E$ has positive measure (and is thus non-empty) for infinitely many n.

(Compare with Theorem 1 of Lecture 3.)

Proof. For any integer $N > 1$, observe that $\int_X \sum_{n=1}^N 1_{T^n E}\ d\mu = N \mu(E)$, and thus by Cauchy-Schwarz

$\int_X (\sum_{n=1}^N 1_{T^n E})^2\ d\mu \geq N^2 \mu(E)^2.$ (1)

The left-hand side of (1) can be rearranged as

$\sum_{n=1}^N \sum_{m=1}^N \mu( T^n E \cap T^m E ).$ (2)

On the other hand, $\mu( T^n E \cap T^m E) = \mu( E \cap T^{m-n} E )$. From this one easily obtains the asymptotic

$(2)\leq (\limsup_{n \to \infty} \mu( E \cap T^n E ) + o(1)) N^2,$ (3)

where o(1) denotes an expression which goes to zero as N goes to infinity. Combining (1), (2), (3) and taking limits as $N \to +\infty$ we obtain

$\limsup_{n \to \infty} \mu( E \cap T^n E ) \geq \mu(E)^2$ (4)

as desired. $\Box$

Remark 2. In classical physics, the evolution of a physical system in a compact phase space is given by a (continuous-time) measure-preserving system (this is Hamilton’s equations of motion combined with Liouville’s theorem). The Poincaré recurrence theorem then has the following unintuitive consequence: every collection E of states of positive measure, no matter how small, must eventually return to overlap itself given sufficient time. For instance, if one were to burn a piece of paper in a closed system, then there exist arbitrarily small perturbations of the initial conditions such that, if one waits long enough, the piece of paper will eventually reassemble (modulo arbitrarily small error)! This seems to contradict the second law of thermodynamics, but the reason for the discrepancy is because the time required for the recurrence theorem to take effect is inversely proportional to the measure of the set E, which in physical situations is exponentially small in the number of degrees of freedom (which is already typically quite large, e.g. of the order of the Avogadro constant). This gives more than enough opportunity for Maxwell’s demon to come into play to reverse the increase of entropy. (This can be viewed as a manifestation of the curse of dimensionality.) The more sophisticated recurrence theorems we will see later have much poorer quantitative bounds still, so much so that they basically have no direct significance for any physical dynamical system with many relevant degrees of freedom. $\diamond$

Exercise 2. Prove the following generalisation of the Poincaré recurrence theorem: if $(X, {\mathcal X}, \mu, T)$ is a measure-preserving system and $f \in L^1(X, {\mathcal X},\mu)$ is non-negative, then $\limsup_{n \to +\infty} \int_X f T^n f \geq (\int_X f\ d\mu)^2$. $\diamond$

Exercise 3. Give examples to show that the quantity $\mu(E)^2$ in the conclusion of Theorem 1 cannot be replaced by any larger quantity in general, regardless of the actual value of $\mu(E)$. (Hint: use a Bernoulli system example.) $\diamond$

Exercise 4. Using the pigeonhole principle instead of the Cauchy-Schwarz inequality (and in particular, the statement that if $\mu(E_1) + \ldots + \mu(E_n) > 1$, then the sets $E_1,\ldots,E_n$ cannot all be disjoint), prove the weaker statement that for any set E of positive measure in a measure-preserving system, the set $E \cap T^n E$ is non-empty for infinitely many n. (This exercise illustrates the general point that the Cauchy-Schwarz inequality can be viewed as a quantitative strengthening of the pigeonhole principle.) $\diamond$

For this lecture and the next we shall study several variants of the Poincaré recurrence theorem. We begin by looking at the mean ergodic theorem, which studies the limiting behaviour of the ergodic averages $\frac{1}{N} \sum_{n=1}^N T^n f$ in various $L^p$ spaces, and in particular in $L^2$.

In our final lecture on topological dynamics, we discuss a remarkable theorem of Furstenberg that classifies a major type of topological dynamical system – distal systems – in terms of highly structured (from an algebraic point of view) systems, namely towers of isometric extensions. This theorem is also a model for an important analogous result in ergodic theory, the Furstenberg-Zimmer structure theorem, which we will turn to in a few lectures. We will not be able to prove Furstenberg’s structure theorem for distal systems here in full, but we hope to illustrate some of the key points and ideas.

It’s now been almost a year since I moved my “What’s new?” page from my home page to this blog. Since then, I’ve been quite happy with the directions this blog has been headed in (most of which I had not anticipated when I started), and also with the level of feedback, some of which has been extremely informative to me (and, I hope, to other readers as well).

Anyway, after discussing things with some of my friends and colleagues, I have decided to convert some of the posts here from 2007 into a book format, in order to place some of the mathematical content here in a more formal and traditional context (with accurate citations and references, etc.). After some thought, I decided not to transcribe all of my posts from last year (there are 93 of them!), but instead to restrict attention to those articles which (a) have significant mathematical content, (b) are not announcements of material that will be published elsewhere, and (c) are not primarily based on a talk given by someone else. As it turns out, this still leaves about 33 articles from 2007, leading to a decent-sized book of a couple hundred pages in length. For various reasons (including legal reasons), I have decided not to incorporate the comments to each post directly into the book format, although corrections, mention of relevant references, etc. will be added with acknowledgments in the endnotes to each article.

I’ve converted a couple articles into a book format, and also created a table of contents, to see what it would look like. (The format comes from the American Mathematical Society, with whom I am planning to publish the book.) It looks like it will be relatively straightforward to convert the rest (at least compared to writing books from scratch, which I know from experience to be quite time-consuming!). I’ll of course post updates here on this blog when the book is closer to completion.

At present, the structure and content of the book is still rather flexible; like all things related to this blog, it is an experiment. As such, I am open to suggestions on these matters. (For instance, I do not have any particularly imaginative title for the book, other than “What’s new – 2007”. )

In this lecture, we move away from recurrence, and instead focus on the structure of topological dynamical systems. One remarkable feature of this subject is that starting from fairly “soft” notions of structure, such as topological structure, one can extract much more “hard” or “rigid” notions of structure, such as geometric or algebraic structure. The key concept needed to capture this structure is that of an isometric system, or more generally an isometric extension, which we shall discuss in this lecture. As an application of this theory we characterise the distribution of polynomial sequences in torii (a baby case of a variant of Ratner’s theorem due to (Leon) Green, which we will cover later in this course).

In this lecture, we use topological dynamics methods to prove some other Ramsey-type theorems, and more specifically the polynomial van der Waerden theorem, the hypergraph Ramsey theorem, Hindman’s theorem, and the Hales-Jewett theorem. In proving these statements, I have decided to focus on the ultrafilter-based proofs, rather than the combinatorial or topological proofs, though of course these styles of proof are also available for each of the above theorems.

I’ve just uploaded to the arXiv my joint paper with Tim Austin, “On the testability and repair of hereditary hypergraph properties“, which has been submitted to Random Structures and Algorithms. In this paper we prove some positive and negative results for the testability (and the local repairability) of various properties of directed or undirected graphs and hypergraphs, which can be either monochromatic or multicoloured.

The negative results have already been discussed in a previous posting of mine, so today I will focus on the positive results. The property testing results here are finitary results, but it turns out to be rather convenient to use a certain correspondence principle (the hypergraph version of the Furstenberg correspondence principle) to convert the question into one about exchangeable probability measures on spaces of hypergraphs (i.e. on random hypergraphs whose probability distribution is invariant under exchange of vertices). Such objects are also closely related to the”graphons” and “hypergraphons” that emerge as graph limits, as studied by Lovasz-Szegedy, Elek-Szegedy, and others. Somewhat amusingly, once one does so, it then becomes convenient to keep track of objects indexed by vertex sets and how they are exchanged via the language of category theory, and in particular using the concept of a natural transformation to describe such objects as exchangeable measures, graph colourings, and local modification rules. I will try to sketch out some of these connections, after describing the main positive results.

In the previous lecture, we established single recurrence properties for both open sets and for sequences inside a topological dynamical system $(X, {\mathcal F}, T)$. In this lecture, we generalise these results to multiple recurrence. More precisely, we shall show

Theorem 1. (Multiple recurrence in open covers) Let $(X,{\mathcal F},T)$ be a topological dynamical system, and let $(U_\alpha)_{\alpha \in A}$ be an open cover of X. Then there exists $U_\alpha$ such that for every $k \geq 1$, we have $U_\alpha \cap T^{-r} U_\alpha \cap \ldots \cap T^{-(k-1)r} U_\alpha \neq \emptyset$ for infinitely many r.

Note that this theorem includes Theorem 1 from the previous lecture as the special case $k=2$. This theorem is also equivalent to the following well-known combinatorial result:

Theorem 2. (van der Waerden’s theorem) Suppose the integers ${\Bbb Z}$ are finitely coloured. Then one of the colour classes contains arbitrarily long arithmetic progressions.

Exercise 1. Show that Theorem 1 and Theorem 2 are equivalent. $\diamond$

Exercise 2. Show that Theorem 2 fails if “arbitrarily long” is replaced by “infinitely long”. Deduce that a similar strengthening of Theorem 1 also fails. $\diamond$

Exercise 3. Use Theorem 2 to deduce a finitary version: given any positive integers m and k, there exists an integer N such that whenever $\{1,\ldots,N\}$ is coloured into m colour classes, one of the colour classes contains an arithmetic progression of length k. (Hint: use a “compactness and contradiction” argument, as in my article on hard and soft analysis.) $\diamond$

We also have a stronger version of Theorem 1:

Theorem 3. (Multiple Birkhoff recurrence theorem) Let $(X,{\mathcal F},T)$ be a topological dynamical system. Then for any $k \geq 1$ there exists a point $x \in X$ and a sequence $r_j \to \infty$ of integers such that $T^{i r_j} x \to x$ as $j \to \infty$ for all $0 \leq i \leq k-1$.

These results already have some application to equidistribution of explicit sequences. Here is a simple example (which is also a consequence of Weyl’s equidistribution theorem):

Corollary 1. Let $\alpha$ be a real number. Then there exists a sequence $r_j \to \infty$ of integers such that $\hbox{dist}(r_j^2 \alpha,{\Bbb Z}) \to 0$ as $j \to \infty$.

Proof. Consider the skew shift system $X = ({\Bbb R}/{\Bbb Z})^2$ with $T(x,y) := (x+\alpha,y+x)$. By Theorem 3, there exists $(x,y) \in X$ and a sequence $n_j \to \infty$ such that $T^{r_j}(x,y)$ and $T^{2r_j}(x,y)$ both convege to $(x,y)$. If we then use the easily verified identity

$(x,y) - 2T^{r_j}(x,y) + T^{2r_j}(x,y) = (0, r_j^2 \alpha)$ (1)

we obtain the claim. $\Box$

Exercise 4. Use Theorem 1 or Theorem 2 in place of Theorem 3 to give an alternate derivation of Corollary 1. $\diamond$

As in the previous lecture, we will give both a traditional topological proof and an ultrafilter-based proof of Theorem 1 and Theorem 3; the reader is invited to see how the various proofs are ultimately equivalent to each other.

This weekend I was (once again) in San Diego, this time for the Southern California Analysis and PDE (SCAPDE) meeting. I gave a talk on “The asymptotic behaviour of large data solutions to NLS”, which is based on two of my previous papers on what solutions to focusing nonlinear Schrödinger equations behave like as time goes to infinity. (Note that this is a specialist conference, and this talk will be a bit more technical than some of the general-audience talks that I have blogged about previously.)

Avi Wigderson‘s final talk in his Distinguished Lecture Series on “Computational complexity” was entitled “Arithmetic computation“; the complexity theory of arithmetic circuits rather than boolean circuits.

We now begin the study of recurrence in topological dynamical systems $(X, {\mathcal F}, T)$ – how often a non-empty open set U in X returns to intersect itself, or how often a point x in X returns to be close to itself. Not every set or point needs to return to itself; consider for instance what happens to the shift $x \mapsto x+1$ on the compactified integers $\{-\infty\} \cup {\Bbb Z} \cup \{+\infty\}$. Nevertheless, we can always show that at least one set (from any open cover) returns to itself:

Theorem 1. (Simple recurrence in open covers) Let $(X,{\mathcal F},T)$ be a topological dynamical system, and let $(U_\alpha)_{\alpha \in A}$ be an open cover of X. Then there exists an open set $U_\alpha$ in this cover such that $U_\alpha \cap T^n U_\alpha \neq \emptyset$ for infinitely many n.

Proof. By compactness of X, we can refine the open cover to a finite subcover. Now consider an orbit $T^{\Bbb Z} x = \{ T^n x: n \in {\Bbb Z} \}$ of some arbitrarily chosen point $x \in X$. By the infinite pigeonhole principle, one of the sets $U_\alpha$ must contain an infinite number of the points $T^n x$ counting multiplicity; in other words, the recurrence set $S := \{ n: T^n x \in U_\alpha \}$ is infinite. Letting $n_0$ be an arbitrary element of S, we thus conclude that $U_\alpha \cap T^{n_0-n} U_\alpha$ contains $T^{n_0} x$ for every $n \in S$, and the claim follows. $\Box$

Exercise 1. Conversely, use Theorem 1 to deduce the infinite pigeonhole principle (i.e. that whenever ${\Bbb Z}$ is coloured into finitely many colours, one of the colour classes is infinite). Hint: look at the orbit closure of c inside $A^{\Bbb Z}$, where A is the set of colours and $c: {\Bbb Z} \to A$ is the colouring function.) $\diamond$

Now we turn from recurrence of sets to recurrence of individual points, which is a somewhat more difficult, and highlights the role of minimal dynamical systems (as introduced in the previous lecture) in the theory. We will approach the subject from two (largely equivalent) approaches, the first one being the more traditional “epsilon and delta” approach, and the second using the Stone-Čech compactification $\beta {\Bbb Z}$ of the integers (i.e. ultrafilters).


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# technical writing domains

The domain is the set of the first coordinates of the ordered pairs. Perhaps you have basic knowledge of technical writing and are looking to build a career as a technical writer, or perhaps you have been working as a technical … Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Yes. Deliver and test. Find the domain of the function $f\left(x\right)=\dfrac{x+1}{2-x}$. This is the most permissive style. $\left(-\infty ,\infty \right)$. Exclude from the domain any input values that have nonreal (or undefined) number outputs. Watch the following video to see more examples of how to find the domain of a rational function (one with a fraction). There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result. Look at the function graph and table values to confirm the actual function behavior. Find the domain of the function $f\left(x\right)=\sqrt{5+2x}$. A technical writer who becomes a subject … Oftentimes, finding the domain of such functions involves remembering three different forms. You don’t have to do it all the time: a balance … Many jobs that involve written communication require you to explain complex things in a way that is easy to understand. The courses are industry developed, giving … It was a refresher to spend a friday evening listening to Leah Guren sharing her insights acquired from a 30 year old career in Technical communication. Work Experience in a Related Occupation. Let’s turn our attention to finding the domain of a function whose equation is provided. The graph below represents the function $f$. The technical writer is the interface between the computer professional who created the technology and its user. Once your documentation is put together and live, it’s time to get some real-world … We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. Technical communication—or technical writing, as the course is often called—is not writing about a specific technical topic such as computers, but about any technical topic. To choose a domain, open any document in the Grammarly Editor and click. For example, if a person has 100 to spend, he or she would need to express the interval that is more than 0 and less than or equal to 100 and write $\left(0,\text{ }100\right]$. Casual is designed for informal types of writing and ignores most style issues. Technical communication is a means to convey scientific, engineering, or other technical information. Some individuals are designated as technical communicators or technical writers.These individuals use a set of methods to research, document, and present technical … I'd Rather Be Writing. $\left\{\left(-5,4\right),\left(0,0\right),\left(5,-4\right),\left(10,-8\right),\left(15,-12\right)\right\}$, $\left\{-5,0,5,10,15\right\}$. If possible, write the answer in interval form. Creative doesn’t flag sentence fragments (missing subjects or verbs), wordy sentences, colloquialisms, informal pronouns, passive voice, incomplete comparisons, or run-on sentences. Ex1: Evaluate a Function and Solve for a Function Value Given a Graph. Domain: $\left(-\infty ,2\right]$ Range: $\left(-\infty ,0\right]$, $\left[-\frac{5}{2},\infty \right)$. Technical Writing . We can divide this journey from ideas to dots into three domains: the subject domain, the document domain, and the media domain. Use the valid input values to determine the range of the output values. \begin{align}2-x&=0 \\ -x&=-2 \\ x&=2 \end{align}. It catches grammar, punctuation, and spelling mistakes but allows some leeway for those who want to intentionally bend grammar rules to achieve certain effects. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Write domain and range using standard notations. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. For example, the function $f\left(x\right)=-\frac{1}{\sqrt{x}}$ has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. The passive voice (‘the trials were conducted …’) may be the default in most technical writing, but the active voice is more direct (‘we conducted the trials …’). Below is a list that details the variations among the styles, from most to least formal: Academic is the strictest style of writing. The Project Team approach is what makes our technical writing consultants at InfoPros so successful. The various technical writing courses and certificates are taught by instructors with more than 20 years of professional, technical, and career training experience. When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand. Technical writing widely covers various forms of application. For you to have a broader understanding of what technical writing … https://www.youtube.com/watch?v=lHTLjfPpFyQ&feature=youtu.be. Ex: Evaluate a Function and Solve for a Function Value Given a Table. Since our private and public sectors are functioning in a vast domain, the technical writing requirement also changes as per domain … To choose a domain, open any document in the Grammarly Editor and click Goals: All the styles apply most of Grammarly’s spelling, grammar, and punctuation checks. However, unlike the Academic domain, it allows the use of some informal expressions, informal pronouns, and unclear antecedents. These are the values that cannot be inputs in the function. It doesn’t simply mean writing technical content, as most people outside the niche assume. Explain technical … We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. includes (but is not limited to) computer skills and abilities associated with computing We use the word 'domains' to refer to these areas of … Website Design & Content Writing Projects for50 - \$100. Third, if there is an even root, consider excluding values that would make the radicand negative. So, we will set the denominator equal to 0 and solve for $x$. Identify any restrictions on the input and exclude those values from the domain. $x+4\ge 0\text{ when }x\ge -4$. Now, we will exclude 2 from the domain. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. On top of catching grammar and punctuation issues, it detects passive voice, contractions, informal pronouns (I, you), and locates unclear antecedents (e.g., sentences starting with “This is…”). In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. Find the domain and range of $f\left(x\right)=\dfrac{2}{x+1}$. The next video gives more examples of how to define the domain of a function that contains an even root. Some technical writers begin their careers as specialists or research assistants in a technical field. It includes reports, executive summary statements, briefs. Can there be functions in which the domain and range do not intersect at all? Technical Writing 101: A Real-World Guide to Planning and Writing Technical … The largest term in the interval is written second, following a comma. We cannot take the square root of a negative number, so the value inside the radical must be nonnegative. The solution(s) are the domain of the function. They start out with the idea of a soup with chicken and noodles… I have attached … Here is a significant class of technical writers which has some collective subcategories listed below: Industry oriented: Based on the type of industry, the categories of specialized writing types vary. Identify any restrictions on the input. Technical writing jobs may require candidates to have both a degree and knowledge of a technical field, such as engineering, computer science, or medicine. Individuals in a variety of contexts and with varied professional credentials engage in technical communication. There are no restrictions, as the ordered pairs are simply listed. Because the function is never zero, we exclude 0 from the range. It does not flag contractions, passive voice, informal pronouns, who-vs-whom usage, split infinitives, or run-on sentences. looking name or word. The answers are all real numbers where $x<2$ or $x>2$. Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive. like sleep night any word. San Jose, California, United States About Blog Technical writing blog focusing … Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0. We know that $f\left(-4\right)=0$, and the function value increases as $x$ increases without any upper limit. The smallest term from the interval is written first. First identify the input values. Unfortunately, the preponderance of technically trained persons in the … Career One Stop, an information center sponsored by the U.S. Department of Labor, identifies three levels of competencies that are needed before a worker can begin to build technical competencies. In this section we will practice determining domains and ranges for specific functions. The technical writer still … Technical Writing Standards 1. Find the domain of the function $f\left(x\right)={x}^{2}-1$. … Technical skills are a specific type of ability and practical knowledge of … Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. https://www.youtube.com/watch?v=_fO9gx1ncyg. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common. Writing skills – For a technical writer, writing skills can never be overlooked. August 19, 2020 Technical writing is a very specific term. The domain is the set of real numbers. When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. Write the domain in interval form, making sure to exclude any restricted values from the domain. Grammarly Premium includes six document domains: General is a good one to start with if you are unsure which style to pick. Interview Subject Matter Experts on regular basis, to learn more about the product. The domain of $f\left(x\right)$ is $\left[-4,\infty \right)$. Did you have an idea for improving this content? The answers are all real numbers less than or equal to $7$, or $\left(-\infty ,7\right]$. You might have to send messages to clients or … See amazon.com. The input value is the first coordinate in an ordered pair. The business style setting checks the text against formal writing criteria. However, unlike the Academic domain, it allows the use of some informal expressions, informal pronouns, and unclear antecedents. We conclude that the range of $f$ is $\left[0,\infty \right)$. Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive. Our technical content writers and illustrators are led by project managers to maintain the project schedule and standards, allowing you the flexibility to be more productive without having to worry about your company’s technical writing … Categories of Technical Writers. 4 cm, … First, determine the domain restrictions for the following functions, then graph each one to check whether your domain agrees with the graph. This style is suitable for personal communication. Technical Writing. Technical writing has always been an integral part of the product … We will discuss interval notation in greater detail later. $\left(-\infty ,\frac{1}{2}\right)\cup \left(\frac{1}{2},\infty \right)$. Find the domain of the function: $f\left(x\right)=5-x+{x}^{3}$. Let’s suppose that an author is writing a recipe for chicken noodle soup. Ideas for Technical Writing. Find the domain of the function: $f\left(x\right)=\dfrac{1+4x}{2x - 1}$. Set the radicand greater than or equal to zero and solve for $x$. Technical writing is one of the fastest growing professions and the demand for technical writers shows no signs of slowing down. We then find the range. Next, use an online graphing tool to evaluate your function at the domain restriction you found. Find the domain and range of $f\left(x\right)=-\sqrt{2-x}$. The business style setting checks the text against formal writing criteria. In Functions and Function Notation, we were introduced to the concepts of domain and range. The domain is $\left(-\infty ,-1\right)\cup \left(-1,\infty \right)$. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for $x$ . Your Technical Documentation Project Team. In interval notation, we write the solution: $\left(\mathrm{-\infty },2\right)\cup \left(2,\infty \right)$. Find Function Inputs for a Given Quadratic Function Output. What Are Technical Skills? There are many books on technical writing, writing for the Web, managing documentation projects, and writing on-line help. Any time technical information is conveyed in writing at work, it is, by definition, technical writing. What function value does Desmos give you? Find the domain of the function $f\left(x\right)=\sqrt{7-x}$. The range is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$. This is the default style and uses a medium level of strictness. Note: This feature comes with Grammarly Premium. In interval form, the domain of $f$ is $\left(-\infty ,2\right)\cup \left(2,\infty \right)$. Find the domain and range of $f\left(x\right)=2\sqrt{x+4}$. We can use a symbol known as the union, $\cup$, to combine the two sets. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. This form of composition is utilized from engineering, culinary, education, and even to sciences and medical purposes. When you are defining the domain of a function, it can help to graph it, especially when you have a rational or a function with an even root. Write the domain in interval form, if possible. We cannot evaluate the function at $-1$ because division by zero is undefined. Oftentimes, finding the domain of such functions involves remembering three different forms. It’s a more specific term, which refers to the technical communication in fields like computer software and hardware, aeronautics, chemistry, robotics, engineering, and other technical fields. We are mattress and sleep accessories company Looking domain Name. Nowadays it’s not necessary to head to university to get domain knowledge because there are many online courses on the Internet. Documenting all the essentials. Apart from writing a technical writer performs the following roles and responsibilities: Learn about the product and analyze data to get knowledge about the product. We’d love your input. Find the domain of a function defined by an equation. If that is not what you meant then just add it in the Comments and I will … Before we begin, let us review the conventions of interval notation: Find the domain of the following function: $\left\{\left(2,\text{ }10\right),\left(3,\text{ }10\right),\left(4,\text{ }20\right),\left(5,\text{ }30\right),\left(6,\text{ }40\right)\right\}$ . \begin{align}7-x&\ge 0 \\ -x&\ge -7 \\ x&\le 7 \end{align}. The term “technical” refers to Set the radicand greater than or equal to zero and solve for $x$. Technical competencies aren't the same as foundational competencies – basic skills that are required in any career field. is a good one to start with if you are unsure which style to pick. Find the domain and range of $f\left(x\right)=2{x}^{3}-x$. We also need to consider what is mathematically permitted. In interval form the domain of $f$ is $\left(-\infty ,\infty \right)$. In addition to catching grammar, spelling, and punctuation mistakes, this style also points out the use of overly direct language that may sound harsh to a reader. You can select the style of the document that best corresponds to the type of writing you are checking in order to get the most accurate and relevant suggestions. Write out all numbers below 10. five automobiles • The exceptions to this rule are numbers used with units of measure, dates, times, percentages, page numbers, money. Technical writing isn’t an exception. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The focus was on what process is followed, what are the right questions to be raised, how well the information is analyzed rather than any particular tools or domain. The email genre is similar to the General domain and helps ensure that your email communication is engaging. Writing for a pilot or an aircraft maintenance person is not exactly the same as writing for a lawyer, or for a senior trying to understand a medicine leaflet. Exclude from the domain any input values that result in division by zero. Technical writers may also gain expertise in a particular technical domain and branch into related forms, such as software quality analysis or business analysis. New Definition of Technical Writing Today technical writing encompasses all documentation of complex technical processes. She talked about the fundamentals that lay at the core of Technical Writing. Now, we will exclude any number greater than 7 from the domain. Second, if there is a denominator in the function’s equation, exclude values in the domain … The input value, shown by the variable $x$ in the equation, is squared and then the result is lowered by one. 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# AMAT 415 — Matlab activity 1: My first fractal

If you haven’t used Matlab before, I recommend you spend half-an-hour going through the first few sections of the Matlab Primer to familiarize yourself with the basis syntax.  In particular, you should learn how to enter and manipulate matrices (with real or complex entries) and generate basic plots.

To test your understanding, can you — without using any loops — generate a 10-by-10 matrix whose $(i,j)$-th entry is $i$? How about one whose $(i,j)$-th entry is $i^2+j$? And, finally, one whose $(i,j)$-th entry is $0$ if $i^2+j<20.5$ and $1$ if $i^2+j>20.5$? (Hint: Consider $\begin{pmatrix}1\\2\\3\end{pmatrix}\begin{pmatrix}1&1&1\end{pmatrix}$.)  Now plot a 10-by-10 grid so that the box at position $(i,j)$ is one color if $i^2+j>20.5$ and another color if $i^2+j < 20.5$.  You’ll want to use the “image” command or something similar.  If you want to choose the colors, look up the “colormap” command.

As our first nontrivial Matlab activity, we’ll generate pictures of Mandelbrot sets.  The classical Mandelbrot set is defined as follows.  For complex numbers $z$ and $c$, define $f_c(z)=z^2+c$. For $n\geq 1$, we define $f_c^{n}(z)=f_c(f_c^{n-1}(z))$. We define the $f_c$-orbit of $0$ to be the set $\{0,f_c(0),f_c(f_c(0)),f_c(f_c(f_c(0))),\ldots\}$. The Mandelbrot set $M$ is defined to be the set of complex numbers $c$ for which the $c$-orbit of $0$ is bounded. In other words, $M$ is the set of $c$ such that $|f_c^{n}(0)|$ does not tend to infinity as $n$ grows.

Let $h$ be a small number, say $h=0.01$ to start with. To generate a picture of the Mandelbrot set (with resolution $h$) we want to build a matrix $A=(a_{pq})$ such that $a_{pq}=1$ if $h((p-1)+(q-1)i)\in M$ and $a_{pq}=0$ if $h((p-1)+(q-1)i)\notin M$.  For simplicity, you can assume that $c=h((p-1)+(q-1)i)\in M$ if $|f_c^{20}(0)|>5$.  I suggest you start by building a 101-by-101 matrix $C$ whose $(p,q)$-th entry is $h((p-1) + (q-1)i)$, and a zero matrix $Z$ of the same size. With this setup, the recursion is simply Z=Z.^2+C. (In Matlab, Z.^2 squares each entry of the matrix Z.)  After you can generate a picture, try playing with the resolution parameter $h$ and the range of values in $C$ to get a “nicer” picture.  In particular, try making the entries of $C$ symmetric about $0$ instead of lying in the first quadrant. (Nicer is subjective, of course.)

Now, generate yourself a personal Mandelbrot set by replacing $f_c$ by $f_c(z)=z^2+0.(\text{your student number})z+c$ or some other interesting function of your choice involving a parameter $c$.  Make a nice picture of your personal Mandelbrot set.  Submit printout of the picture, together with the Matlab code you used to generated it, as part of HW 1. Here’s mine, drawn with $h=0.001$ symmetrically around $0$:

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Search: All articles in the CJM digital archive with keyword Weyl fractional integral

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1. CJM 1997 (vol 49 pp. 1010)

Lorente, Maria
 A characterization of two weight norm inequalities for one-sided operators of fractional type In this paper we give a characterization of the pairs of weights $(\w,v)$ such that $T$ maps $L^p(v)$ into $L^q(\w)$, where $T$ is a general one-sided operator that includes as a particular case the Weyl fractional integral. As an application we solve the following problem: given a weight $v$, when is there a nontrivial weight $\w$ such that $T$ maps $L^p(v)$ into $L^q(\w )$? Keywords:Weyl fractional integral, weightsCategories:26A33, 42B25
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