text:
#### Exact Reconstruction of Extended Exponential Sums using Rational Approximation of their Fourier Coefficients

##### Nadiia Derevianko, Gerlind Plonka

In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form $y(t) = \sum_{j=1}^{M} \left( \sum_{m=0}^{n_j} \, \gamma_{j,m} \, t^{m} \right) {\mathrm e}^{2\pi \lambda_j t}$, where the frequency parameters $\lambda_{j} \in {\mathbb C}$ are pairwise distinct. For the reconstruction we employ a finite set of classical Fourier coefficients of $y$ with regard to a finite interval $[0,P] \subset {\mathbb R}$ with $P>0$. Our method requires at most $2N+2$ Fourier coefficients $c_{k}(y)$ to recover all parameters of $y$, where $N:=\sum_{j=1}^{M} (1+n_{j})$ denotes the order of $y$. The recovery is based on the observation that for $\lambda_{j} \not\in \frac{{\mathrm i}}{P} {\mathbb Z}$ the terms of $y$ possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [12]. If a sufficiently large set of $L$ Fourier coefficients of $y$ is available (i.e., $L > 2N+2$), then our recovery method automatically detects the number $M$ of terms of $y$, the multiplicities $n_{j}$ for $j=1, \ldots , M$, as well as all parameters $\lambda_{j}$, $j=1, \ldots , M$ and $\gamma_{j,m}$ $j=1, \ldots , M$, $m=0, \ldots , n_{j}$, determining $y$. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method.

arrow_drop_up


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positiveRoots(RootSystem,Parabolic) -- the set of all positive roots in a parabolic sugroups

Synopsis

• Function: positiveRoots
• Usage:
positiveRoots(R,P)
• Inputs:
• Outputs:
• a set, of all positive roots of R that are written as sums of roots corresponding to reflexions in P

Description

 i1 : R=rootSystemA(4) o1 = RootSystem{...8...} o1 : RootSystem i2 : P=parabolic(R,set{1,2}) o2 = set {1, 2} o2 : Parabolic i3 : positiveRoots(R,P) o3 = set {| -1 |, | 1 |, | 2 |} | 2 | | 1 | | -1 | | -1 | | -1 | | 0 | | 0 | | 0 | | 0 | o3 : Set

format: plain


text:
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# If xy does not equal 0, what is the value of ((x^2)^3 y^4)/x

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If xy does not equal 0, what is the value of ((x^2)^3 y^4)/x [#permalink]  19 Oct 2010, 11:15
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If xy does not equal 0, what is the value of ((x^2)^3 y^4)/x ?

(1) x^3y^5 = 6y/x^2

(2) x = 2y
[Reveal] Spoiler: OA
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Re: Difficult Algebra DS Problem from Past GMAT ETS paper exam [#permalink]  19 Oct 2010, 11:23
1
KUDOS
Expert's post
tonebeeze wrote:
Hello all,

Can someone please walk me thru how they best simplified this. Thanks

If xy does not equal 0, what is the value of ((x^2)^3 y^4)/x ?

(1) x^3y^5 = 6y/x^2

(2) x = 2y

$$xy\neq{0}$$ means that neither of unknowns equals to zero.

$$\frac{(x^2)^3*y^4}{x}=\frac{x^6*y^4}{x}=x^5*y^4=?$$

(1) $$x^3y^5=\frac{6y}{x^2}$$ --> multiple by $$x^2$$ --> $$x^5y^5=6y$$ --> reduce by $$y$$ --> $$x^5y^4=6$$. Sufficient.

(2) $$x=2y$$ --> $$x^5*y^4=(2y)^5*y^4=32y^9=?$$ --> need to know the value of $$y$$ (or $$x$$). Not sufficient.

For more on exponents check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
_________________
Manager
Status: Current MBA Student
Joined: 19 Nov 2009
Posts: 126
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GMAT 1: 720 Q49 V40
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Re: Difficult Algebra DS Problem from Past GMAT ETS paper exam [#permalink]  19 Oct 2010, 13:29
Thanks Bunuel. I apologize for the bad formatting. Won't happen again!
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Re: Difficult Algebra DS Problem from Past GMAT ETS paper exam [#permalink]  15 Oct 2013, 09:06
tonebeeze wrote:
Thanks Bunuel. I apologize for the bad formatting. Won't happen again!

And just to emphasize that we can divide both sides by y ONLY BECAUSE we are explicitly told that x and y are both different from zero as Bunuel clearly explained above.
Cheers

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Re: If xy does not equal 0, what is the value of ((x^2)^3 y^4)/x [#permalink]  29 Jan 2016, 03:49
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Re: If xy does not equal 0, what is the value of ((x^2)^3 y^4)/x   [#permalink] 29 Jan 2016, 03:49
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text:
nLab Banach manifold

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

A notion of infinite-dimensional manifold. A Banach manifold is a manifold modelled on Banach spaces. By default, transition maps are taken to be smooth.

Properties

Embedding into the category of diffeological spaces

The category of smooth Banach manifolds has a full and faithful functor into the category of diffeological spaces. In terms of Chen smooth spaces this was observed in (Hain). For more see at Fréchet manifold – Relation to diffeological spaces.

References

For general references see at infinite-dimensional manifold.

The embedding into the category of diffeological spaces is discussed in

• Richard Hain, A characterization of smooth functions defined on a Banach space, Proc. Amer. Math. Soc. 77 (1979), 63-67 (web, pdf)

Last revised on March 6, 2015 at 07:40:50. See the history of this page for a list of all contributions to it.


text:
## hooverst Group Title compute the derivative y=ln((x+1)^2(3x-1)^3) 2 years ago 2 years ago

1. hooverst

$y=\ln((x+1)^2(3x-1)^3)$

2. slaaibak

${1 \over (x+1)^2 (3x-1)^3} \times [2\times(x+1)(3x-1)^3 + 3 \times 3 \times (3x-1)^2(x+1)^2]$

3. hooverst

ok I see

4. slaaibak

or easier: split it up into the following $2\ln(x+1) + 3\ln(3x-1)$ then derive: ${2 \over {x+1}} +{ 9 \over 3x-1}$

5. hooverst

so would that be the answer

6. slaaibak

yes

7. hooverst

do you know how to do $y=x \sqrt{6}$

8. slaaibak

yes

9. hooverst

can you show me how to do it? if you have time.

10. slaaibak

well, since $\sqrt6$ is a constant, the answer would be $\sqrt6$

11. hooverst

thanks @slaaibak


text:
# Mobility of Australian Flying-Foxes, Pteropus spp. (Megachiroptera): Evidence from Genetic Variation

N. J. Webb, C. R. Tidemann

## Abstract

Black (Pteropus alecto) and grey-headed (Pteropus poliocephalus) flying-foxes inhabit large ranges in coastal north and eastern Australia. P. poliocephalus is endemic to the region and is classified as vulnerable. The bats are known to migrate in response to flowering and fruiting of their food plants, but direct observation of movement patterns is difficult. Protein electrophoresis was used to investigate genetic subdivision among populations. High gene flow was inferred for both species with an estimated exchange of 15 (P. alecto) and 28 (P. poliocephalus) individuals between populations per generation. Wright's F$_{\text{ST}}$, an index of among population genetic variation, was low, 0.023 (P. alecto), 0.014 (P. poliocephalus), reflecting the homogenising action of movements across the species' ranges. An F$_{\text{ST}}$ value of 0.028 has been reported for a third Australian species, the little red flying-fox, P. scapulatus. The F$_{\text{ST}}$ values for Australian flying-foxes are closer to those found for birds than the typically higher values for mammals and we conclude that these flying-foxes are essentially panmictic and for management purposes should be treated as migratory species.


text:
# Trivialize a cocycle of a continuous Lie group-cohomology to a coboundary

Someone recently asks a question $SO(3)$ 2-cocycle trivialized to a 2-coboundary in $SU(2)$? now inspires me to revisit an earlier general question to ask an example of 3-cocycle $\omega_3^G$ of a cohomology group $H^3[G,\mathbb{R}/\mathbb{Z}]$ for the case when $G$ is a continuous group, say $G=U(1)=SO(2)$, instead of a finite group.

$$\omega_3^G \in H^3[G,\mathbb{R}/\mathbb{Z}]=\mathbb{Z}$$

Consider a $3$-cocycle $\omega_3^G \in H^3(U(1),\mathbb{R}/\mathbb{Z})=\mathbb{Z}$ in the cohomology group of a group $G$ with $U(1)=\mathbb{R}/\mathbb{Z}$ coefficient. There are $\mathbb{Z}$ classes of $\omega_3^G$.

Question: We ask whether there exists some group $N$ as a normal subgroup of some bigger group $J$, such that $G$ is the quotient group $$\frac{J}{N}=G$$ and such that we can always trivialize the $3$-cocycle $\omega_3^G$ of $G=U(1)$ into $3$-coboundary if we lift $G$ into a larger group $J$? Given that we know the group homomorphism $r$: $$J \overset{r}{\rightarrow} G.$$ Namely, $$\omega_3^G(g_i,\dots)=\omega_3^G(r(j_i),\dots)= \delta \beta_{2}^J(j_i,\dots).$$ with $g \in G, j \in J$. Here $\beta_{2}^J$ is a 2-cochain of $J$.

Guess: I am guessing that we may not be able to find such a $J$ to trivialize the 3-cocycle of $G=U(1)$. How can we verify or prove this?


text:
Sample Size, $n$

Power, $1-\beta$

Type I error rate, $\alpha$

X-axis

min

max

Documentation
Test 1 Mean
Compare 2 Means
Compare k Means
Test 1 Proportion
Compare 2 Proportions
Compare Paired Proportions
Compare k Proportions
Test Time-To-Event Data
Test Odds Ratio
Test Relative Incidence in Self Controlled Case Series Studies
Other

Calculate Sample Size Needed to Test 1 Proportion: 1-Sample Equivalence

This calculator is useful when we wish to test whether a proportion, $p$, is different from a gold standard reference value, $p_0$. For example, we may wish to test whether a new product is equivalent to an existing, industry standard product. Here, the 'burden of proof', so to speak, falls on the new product; that is, equivalence is actually represented by the alternative, rather than the null hypothesis.

$H_0:|p-p_0|\ge\delta$
$H_1:|p-p_0|<\delta$
.

Formulas

This calculator uses the following formulas to compute sample size and power, respectively: $$n=p(1-p)\left(\frac{z_{1-\alpha}+z_{1-\beta/2}}{|p-p_0|-\delta}\right)^2$$
$$1-\beta= 2\left[\Phi\left(z-z_{1-\alpha}\right)+\Phi\left(-z-z_{1-\alpha}\right)\right]-1 \quad ,\quad z=\frac{|p-p_0|-\delta}{\sqrt{\frac{p(1-p)}{n}}}$$
where

R Code

R code to implement these functions:

p=0.6
p0=0.6
delta=0.2
alpha=0.05
beta=0.20
(n=p*(1-p)*((qnorm(1-alpha)+qnorm(1-beta/2))/(abs(p-p0)-delta))^2)
ceiling(n) # 52
z=(abs(p-p0)-delta)/sqrt(p*(1-p)/n)
(Power=2*(pnorm(z-qnorm(1-alpha))+pnorm(-z-qnorm(1-alpha)))-1)

References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 87.


text:
# .99 = 1?

1. Jun 13, 2008

### Ed Aboud

Hi all.
I can't remember where I saw this but it really confuses me.

If

x = .999999999.....
10x = 9.9999999......
10x - x = 9x

x = 1?

So does that mean if that .9 recurring is 1 exactly?
Thanks for any help.

2. Jun 13, 2008

### Staff: Mentor

Yes, by definition and derivation. You got it.

3. Jun 13, 2008

### nicksauce

Yes it does mean that. .99999.... is an alternate representation for the number 1.

Another way of looking at it: Suppose a and b are real numbers with b>a. There there exists another number, c, such that a < c < b. Suppose we have the hypothesis that .9999.... is not equal to 1. Can you find a number between .999999.... and 1?

4. Jun 13, 2008

### Diffy

5. Jun 13, 2008

### arildno

6. Jun 13, 2008

### maze

Assuming .99999... is actually represents a number, it can't be bigger than 1. It also can't be smaller than 1 because of Nicksauce's reasoning. Therefore the only option is for .9999.. to be 1.

The question of whether .999999... is even a real number at all - the question if the sequence .9 + .09 + .009 + ... converges - is a bit more complicated and requires some basic real analysis.

7. Jun 13, 2008

### arildno

Really?

From what I know, a usual axiom appended to the real number field is that every bounded, increasing sequence converges..
Not too complicated to show that the condition holds in this case..

8. Jun 13, 2008

### Diffy

9. Jun 13, 2008

### Ed Aboud

Ok, thanks very much for the help guys.
And I am sorry if it was posted before, I'm relatively new to PF.

10. Jun 13, 2008

### GRB 080319B

11. Jun 13, 2008

### maze

Yeah, I mean it's not really too complicated, its just probably a little beyond a standard highschool level.

The possible "problem" would be if the real numbers weren't complete (they are complete, so it's not a problem). The analogy would be, the rational numbers are not complete, so the rational sequence 3, 3.1, 3.14, 3.1415, ..., a bounded increasing sequence, doesn't actually converge in the rational numbers. If the real numbers were not complete, .9, .99, .999, ... might converge to some non-real "infintessimal" sort of number, in the same way that 3.14, 3.141, ... converges to a non-rational number.

Last edited: Jun 13, 2008
12. Jun 13, 2008

### LukeD

The form of that axiom that I've always seen is that the real numbers are complete (i.e. every Cauchy sequence of real numbers has a limit in the real numbers). A bounded, increasing sequence is a Cauchy sequence though, so it has a limit.

13. Jun 14, 2008

### D H

Staff Emeritus
This is true, but be fair to the young'uns. This is far beyond high school mathematics. It is even beyond the typical Calc 1/2/3/... sequence required of all science and engineering undergraduates in most colleges. Science majors do study advanced math, but it is stuff like Green's theorem, complex analysis, numerical methods, ... That $$0.\bar{9}=1$$ is even beyond a lot of high school math teachers. I know, because long, long ago I saw the simple 9*0.111... "proof" and pointed it out to a high school math teacher who insisted I was wrong. I didn't learn that I was right until I went to college and took a real algebra course. Bottom line: We should go a bit easy on the kind of people who typically ask this question.

What we need is a standard answer to this question, and now we have the vehicle for this answer: the PF library. Unfortunately, I will be very busy for the next few weeks. Any takers on writing this up? Then when the question comes up again we can simply point to the library entry.

14. Jun 14, 2008

### cristo

Staff Emeritus
That's a good idea that perhaps someone could undertake.

There are also hundreds of threads on this topic that one can find by searching. I'm thus locking this thread.


text:
+0

# help

0
112
1

If the area of the regular hexagon is 90, what is the area of the purple region?

Jun 17, 2020

#1
+8352
+1

The area of a regular hexagon A is related to its length L by

$$A = \dfrac{3\sqrt 3}2 L^2$$

Consider the top white isosceles triangle.

The base is $$2L\sin 60^\circ = \sqrt 3 L$$. This is also the length of the rectangle.

The width of the rectangle is $$L$$, so the area of the rectangle is $$\sqrt 3 L^2 = \dfrac23 A$$.

The area of rectangle is two thirds of the area of the hexagon, which is 60.

Jun 17, 2020


text:
# How to disable auto-complete braces in TEXStudio?

I want to disable auto-complete for braces in TeXstudio. In its simplest form, when I type { I do not want it to automatically add a }. When it does this, many times I end up deleting the brace, which is a hassle. Also, when I hit Ctrl+I to get into italics mode I don't want it to auto-complete the ending brace either. I have gone to Options -> Configure TeXstudio... -> Completion -> Show Advanced Options and disabled everything. But it still auto-completes them.

In case it matters the version I have is TeXstudio 2.8.4.

• Sorry, I thought you are having texworks editor – Enthusiastic Engineer Nov 10 '14 at 17:58
• haha it's fine. Part of my problem is all the help out there is for different editors/versions. I have edited the post to include the version in case it comes up again. – Curtis Nov 10 '14 at 18:01

• Question on Ctrl+I: Go to Macros -> User Macros and create a new normal macro with content \textit{. Then go to Options -> Configure -> Shortcuts and assign Ctrl+I to your new macro. – Tim Hoffmann Nov 11 '14 at 22:14


text:
## Calculus

Mathematica is useful for many types of mathematical operations, but of particular use are derivatives and integrals.

### Derivatives

Derivatives can be taken using either the function D or the alias ESCpdESC.

For example, to take the first derivative of x^2 one can use either of the following:

 D[x^2,x]

 2 x

or

 Subscript[∂, x]x^2

 2 x

A derivative can be taken of a derivative:

 Subscript[∂, x]Subscript[∂, x]x^2

 2

But this is equivalent to a simple second derivative which Mathematica can also do:

 D[x^2,{x,2}]

 2

or

 Subscript[∂, x, x]x^2

 2

We can also take derivatives in multiple variables:

 Subscript[∂, x, y]1/Sqrt[x^2 + y^2]

 (3 x y)/(x^2 + y^2)^(5/2)

Here's that but formatted better:

$$\frac{3 x y}{\left(x^2+y^2\right)^{5/2}}$$

And then for fun let's plot this:

 dxy=Subscript[∂, x, y]1/Sqrt[x^2 + y^2];
Plot3D[dxy,{x,-5,5},{y,-5,5}]

### Integration

Mathematica also knows how to do more integrals than any chemist is likely to need. The function for integration is Integrate although like with derivatives there is an alias via ESCinttESC for indefinite integrals and ESCdinttESC for definite integrals

Here's the standard form for Integrate :

 Integrate[2,x]

 2 x

And here's the alias form:

 ∫2ⅆx

 2 x

And just to show that Mathematica observes the fundamental theorem of calculus like the rest of us:

 ∫∫dxyⅆxⅆy

 1/Sqrt[x^2 + y^2]

Again, here's the same but formatted better:

$$\frac{1}{\sqrt{x^2+y^2}}$$

And we'll plot this too:

 reversedxy=∫∫dxy ⅆxⅆy;
Plot3D[reversedxy,{x,-5,5},{y,-5,5}]

### Fun examples:

Mathematica is a great calculus tool and it knows many types of interesting functions so we'll have it integrate the following nasty function for us:

$$P-hash-!!!11573959464626687!!!-hash-4(\cos (\theta ))}{\partial \theta }$$

where $P_i(\cos (\theta ))$ is the $i^{\text{th}}$ Legendre polynomial in $\cos (\theta )$

 Integrate[LegendreP[3,Cos[θ]]Cot[θ]D[LegendreP[4,Cos[θ]],θ],{θ,0,π}]

 0

but Mathematica has absolutely no problem evaluating this integral. It can even tell you that switching the derivative and cot terms makes the integral impossible to evaluate.

 Integrate[LegendreP[3,Cos[θ]]D[Cot[θ]LegendreP[4, Cos[θ]],θ], {θ,0,π}]

$$\int_0^{\pi } \frac{1}{2} \left(-3 \cos (\theta )+5 \cos ^3(\theta )\right) \left(-\frac{1}{8} \left(3-30 \cos ^2(\theta )+35 \cos ^4(\theta )\right) \csc ^2(\theta )+\frac{1}{8} \cot (\theta ) \left(60 \cos (\theta ) \sin (\theta )-140 \cos ^3(\theta ) \sin (\theta )\right)\right) \, d\theta$$

And when it sees that it simply returns the integral unevaluated


text:
# help with proofs

#### michaelmike121

##### New member
show that integral Upper limit 1 and lower limit 0 of dx/(1+e^x)(1+e^-x) is equal to 1/2tanh(1/2x)

#### Subhotosh Khan

##### Super Moderator
Staff member
show that integral Upper limit 1 and lower limit 0 of dx/(1+e^x)(1+e^-x) is equal to 1/2tanh(1/2x)
Hint: express tanh(x) as a function of e[SUP]x[/SUP].


text:
Journal article Open Access

# PRESCRIBERS' ADHERENCE TO THE BASIC PRINCIPLES OF PRESCRIPTION ORDER WRITING IN A TEACHING HOSPITAL, ETHIOPIA

Teshale Ayele1, Tessema Tsehay2, Haymanot Bedada2

### Citation Style Language JSON Export

{
"publisher": "Zenodo",
"DOI": "10.5281/zenodo.1036484",
"container_title": "INDO AMERICAN JOURNAL OF PHARMACEUTICAL RESEARCH",
"title": "PRESCRIBERS' ADHERENCE TO THE BASIC PRINCIPLES OF PRESCRIPTION ORDER WRITING IN A TEACHING HOSPITAL, ETHIOPIA",
"issued": {
"date-parts": [
[
2017,
8,
30
]
]
},
"abstract": "
Background: Prescription, a therapeutic transaction between physicians and patient, must be error free, legible and contains all the information to facilitate the dispensing process. However, most of the prescriptions written and arrived at dispensing units are incomplete making the dispensing activity more complicated. Objective: To assess prescribers\u2019 adherence to the basic principles of standard prescription order writing in the different pharmacy units of Mizan Tepi University Teaching Hospital (MTUTH), Ethiopia. Methods: A descriptive cross sectional study was conducted in MTUTH from 5-24, March 2016. Data was extracted from prescription papers using structured data collection format. Three hundred ninety six prescription papers were selected using systematic random sampling method from prescription papers collected over one year period. The data was filled in well-structured checklist and analyzed using SPSS windows version 20 for frequency distribution. Results: Of 396 prescription papers analyzed, only 11(2.7%) prescription papers contained complete patient information. On the other hand, 345(87.1%), 142(35.9%), 357(90.2%), and 272(68.7%) prescription papers contained strength, dose, frequency of administration of drugs and duration of treatment, respectively. Only 17(4.3%) prescription papers contained complete dosage regimen recommendations. Complete prescribers information was found in 268(69.8%) prescription papers. Generally, none of the prescriptions papers complete; i.e., at least one variable was missing. Conclusions: The study revealed a good adherence of prescribers on the basic principles of prescription order writing for some variables and poor adherence for other variables. In order to minimize the therapeutic errors, prescribers should adhere to these principles and hence write all the necessary information required to make prescription papers valid for dispensing. Dispensers on their behalf should not dispense prescription papers with incomplete information. Large scale future studies are important on the area.
",
"author": [
{
"family": "Teshale Ayele1, Tessema Tsehay2, Haymanot Bedada2"
}
],
"page": "499-503",
"volume": "07",
"type": "article-journal",
"issue": "09",
"id": "1036484"
}
68
128
views


text:
Language:   Search:   Contact
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Zbl 1222.47123
Wang, Lin; Yildirim, Isa; Ozdemir, Murat
(Yildirim, \D{I}sa; Özdemir, Murat)
Strong convergence of an implicit iteration process for two asymptotically nonexpansive mappings in Banach spaces.
(English)
[J] An. Ştiinţ. Univ. Ovidius" Constanţa, Ser. Mat. 18, No. 2, 281-294 (2010). ISSN 1223-723X; ISSN 1224-1784/e

Let $E$ be a Banach space, $K$ a nonempty closed convex subset of $E$ and $S,T:K\rightarrow K$ two asymptotically nonexpansive mappings. In order to obtain a common fixed point of $S$ and $T$ in uniformly convex Banach spaces, the authors consider an implicit iterative scheme $\{x_n\}$ of the form $$x_n=\alpha_n x_{n-1}+\beta_n T^{n-1}x_{n-1}+\gamma_n T^{n}y_{n},$$ $$y_n=\alpha_n' x_{n-1}+\beta_n' S^{n-1}x_{n-1}+\gamma_n' S^{n}x_{n},$$ for which a corresponding convergence theorem (Theorem 3.3) is proven.
[Vasile Berinde (Baia Mare)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: Banach space; asymptotically nonexpansive mapping; common fixed point; implicit iterative scheme; convergence theorem

Highlights
Master Server


text:
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text:
# What is mixpoissonreg for?

The mixpoissonreg package is useful for fitting and analysing regression models with overdispersed count responses. It provides two regression models for overdispersed count data:

• Negative-Binomial regression models;

• Poisson Inverse Gaussian regression models.

The parameters of the above models may be estimated by the Expectation-Maximization algorithm or by direct maximization of the likelihood function.

There are several functions to aid the user to check the adequacy of the fitted model, perform some inference and study influential observations.

For theoretical details we refer the reader to Barreto-Souza and Simas (2016)

# Why use the mixpoissonreg package?

Some reasons are:

• glm-like functions to fit and analyze fitted models

• Useful plots for fitted models available using R’s base graphics or ggplot2

• Estimation of parameters via EM-algorithm

• Easy to do inference on fitted models with help of lmtest package

• Regression structure for the precision parameter

• Several global and local influence measures implemented (along with corresponding plots)

• Compatibility with the tidyverse

We will now delve into the reasons above in more detail.

## glm-like functions to fit and analyze fitted models

If you know how to fit glm models then you already know how to fit mixpoissonreg models. You use a formula expression for the regression with respect to mean parameter:

library(mixpoissonreg)

fit <- mixpoissonreg(daysabs ~ gender + math + prog, data = Attendance)

If you do not provide further arguments, the mixpoissonreg package will fit a Negative-Binomial model with a log link function for the mean estimated via EM-algorithm. Just like you would do with a glm fit, you can just type fit to get a simplified output of the fit, or get a summary, by typing summary(fit):

fit
#>
#> Negative Binomial Regression - Expectation-Maximization Algorithm
#>
#> Call:
#> mixpoissonreg(formula = daysabs ~ gender + math + prog, data = Attendance)
#>
#> Coefficients modeling the mean (with log link):
#>    (Intercept)     gendermale           math   progAcademic progVocational
#>    2.707563981   -0.211066195   -0.006235448   -0.424575877   -1.252812675
#> Coefficients modeling the precision (with identity link):
#> (Intercept)
#>    1.047289

summary(fit)
#>
#> Negative Binomial Regression - Expectation-Maximization Algorithm
#>
#> Call:
#> mixpoissonreg(formula = daysabs ~ gender + math + prog, data = Attendance)
#>
#>
#> Pearson residuals:
#>      RSS      Min       1Q   Median       3Q      Max
#> 345.7857  -0.9662  -0.7356  -0.3351   0.3132   5.5512
#>
#> Coefficients modeling the mean (with  link):
#>                 Estimate Std.error z-value Pr(>|z|)
#> (Intercept)     2.707564  0.203069  13.333  < 2e-16 ***
#> gendermale     -0.211066  0.122373  -1.725   0.0846 .
#> math           -0.006235  0.002491  -2.503   0.0123 *
#> progAcademic   -0.424576  0.181772  -2.336   0.0195 *
#> progVocational -1.252813  0.201381  -6.221 4.94e-10 ***
#>
#> Coefficients modeling the precision (with  link):
#>             Estimate Std.error z-value Pr(>|z|)
#> (Intercept)   1.0473    0.1082   9.675   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Efron's pseudo R-squared:  0.1854615
#> Number of iterations of the EM algorithm = 1

If you want to fit a Poisson Inverse Gaussian regression model, just set the model argument to PIG:

fit_pig <- mixpoissonreg(daysabs ~ gender + math + prog, model = "PIG",
data = Attendance)

summary(fit_pig)
#>
#> Poisson Inverse Gaussian Regression - Expectation-Maximization Algorithm
#>
#> Call:
#> mixpoissonreg(formula = daysabs ~ gender + math + prog, data = Attendance,
#>     model = "PIG")
#>
#>
#> Pearson residuals:
#>      RSS      Min       1Q   Median       3Q      Max
#> 276.4232  -0.8225  -0.6386  -0.3219   0.3038   5.4969
#>
#> Coefficients modeling the mean (with  link):
#>                 Estimate Std.error z-value Pr(>|z|)
#> (Intercept)     3.060166  0.227621  13.444  < 2e-16 ***
#> gendermale     -0.244078  0.129011  -1.892 0.058502 .
#> math           -0.007421  0.002641  -2.810 0.004951 **
#> progAcademic   -0.719714  0.193040  -3.728 0.000193 ***
#> progVocational -1.591909  0.215375  -7.391 1.45e-13 ***
#>
#> Coefficients modeling the precision (with  link):
#>             Estimate Std.error z-value Pr(>|z|)
#> (Intercept)   0.7323    0.1121   6.533 6.46e-11 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Efron's pseudo R-squared:  0.1758126
#> Number of iterations of the EM algorithm = 20

There are also several standard methods implemented for mixpoissonreg fitted objects: plot, residuals, vcov, influence, cooks.distance, hatvalues, predict, etc.

A fitted mixpoissonreg object is compatible with the lrtest, waldtest, coeftest and coefci methods from the lmtest package, which, for instance, allows one to easily perform ANOVA-like tests for mixpoissonreg models.

We refer the reader to the Analyzing overdispersed count data with the mixpoissonreg package vignette for further details on fitting mixpoissonreg objects.

## Useful plots for fitted models available using R’s base graphics or ggplot2

Once you have a fitted mixpoissonreg object, say fit, you can easily produce useful plots by typing plot(fit):

plot(fit)

You can also have the ggplot2 version of the above plots by typing autoplot(fit):

autoplot(fit)

For further details on plots of mixpoissonreg objects, we refer the reader to the Building and customizing base-R diagnostic plots with the mixpoissonreg package and Building and customizing ggplot2-based diagnostic plots with the mixpoissonreg package vignettes.

## Estimation of parameters via EM-algorithm

The EM-algorithm also tries to find the maximum-likelihood estimate of the parameters. So, why is there advantage on using the EM-algorithm? The reason is purely numerical. Sometimes the log-likelihood function has several local maxima or almost flat, which makes the direct maximization of the likelihood function converge (and making it very difficult not to obtain early convergence) before reaching its maximum. In such situations the EM-algorithm may be able to obtain estimates closer to the maximum-likelihood estimate. However, it has a drawback that it may take much longer to converge on such situations.

We refer the reader to the Supplementary Material of Barreto-Souza and Simas (2016) and to Barreto-Souza and Simas (2017) for numerical studies.

## Easy to do inference on fitted models with help of lmtest package

We can easily perform ANOVA-like tests using the lmtest package. Let us perform a likelihood-ratio test against the NULL model that only contains the intercept:

lmtest::lrtest(fit)
#> Likelihood ratio test
#>
#> Model 1: daysabs ~ gender + math + prog | 1
#> Model 2: daysabs ~ 1 | 1
#>   #Df  LogLik Df  Chisq Pr(>Chisq)
#> 1   6 -864.15
#> 2   2 -896.47 -4 64.638  3.068e-13 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Now let us perform the Wald test against the NULL model that only contains the intercept:

lmtest::waldtest(fit)
#> Wald test
#>
#> Model 1: daysabs ~ gender + math + prog | 1
#> Model 2: daysabs ~ 1 | 1
#>   Res.Df Df  Chisq Pr(>Chisq)
#> 1    308
#> 2    312 -4 74.257  2.861e-15 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Let us now consider a reduced model:

fit_red <- mixpoissonreg(daysabs ~ math, data = Attendance)

We can perform a likelihood ratio test of full fit model against the NULL of the reduced model as:

lmtest::lrtest(fit, fit_red)
#> Likelihood ratio test
#>
#> Model 1: daysabs ~ gender + math + prog | 1
#> Model 2: daysabs ~ math | 1
#>   #Df  LogLik Df  Chisq Pr(>Chisq)
#> 1   6 -864.15
#> 2   3 -888.15 -3 47.999  2.131e-10 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

In the same fashion, we can perform a Wald test of full fit model against the NULL of the reduced model as:

lmtest::waldtest(fit, fit_red)
#> Wald test
#>
#> Model 1: daysabs ~ gender + math + prog | 1
#> Model 2: daysabs ~ math | 1
#>   Res.Df Df  Chisq Pr(>Chisq)
#> 1    308
#> 2    311 -3 52.537  2.301e-11 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We can obtain confidence intervals for the estimated parameters by using the coefci method:

lmtest::coefci(fit)
#>                             2.5 %       97.5 %
#> (Intercept)            2.30955534  3.105572621
#> gendermale            -0.45091355  0.028781163
#> math                  -0.01111851 -0.001352381
#> progVocational        -1.64751140 -0.858113947
#> (Intercept).precision  0.83512850  1.259450101

## Regression structure for the precision parameter

The mixpoissonreg package allows the regression models to also have a regression structure on the precision parameter. It is very simple to fit such models, one simply write the formula as formula_1 | formula_2, where formula_1 corresponds to the regression structure with respect to the mean parameter and contains the response variable and formula_2 corresponds to the regression structure with respect to the precision parameter and does not contain the response variable.

Consider the following example:

fit_prec <- mixpoissonreg(daysabs ~ gender + math + prog | gender + math + prog,
data = Attendance)

summary(fit_prec)
#>
#> Negative Binomial Regression - Expectation-Maximization Algorithm
#>
#> Call:
#> mixpoissonreg(formula = daysabs ~ gender + math + prog | gender +
#>     math + prog, data = Attendance)
#>
#>
#> Pearson residuals:
#>      RSS      Min       1Q   Median       3Q      Max
#> 322.0142  -1.1751  -0.6992  -0.3600   0.3014   4.7178
#>
#> Coefficients modeling the mean (with  link):
#>                 Estimate Std.error z-value Pr(>|z|)
#> (Intercept)     2.746123  0.147412  18.629  < 2e-16 ***
#> gendermale     -0.245113  0.117967  -2.078  0.03773 *
#> math           -0.006617  0.002317  -2.856  0.00429 **
#> progAcademic   -0.425983  0.132144  -3.224  0.00127 **
#> progVocational -1.269755  0.174444  -7.279 3.37e-13 ***
#>
#> Coefficients modeling the precision (with  link):
#>                 Estimate Std.error z-value Pr(>|z|)
#> (Intercept)     1.414227  0.343243   4.120 3.79e-05 ***
#> gendermale     -0.208397  0.203692  -1.023 0.306262
#> math           -0.005123  0.004181  -1.225 0.220458
#> progAcademic   -1.084418  0.305479  -3.550 0.000385 ***
#> progVocational -1.422051  0.343812  -4.136 3.53e-05 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Efron's pseudo R-squared:  0.1860886
#> Number of iterations of the EM algorithm = 11

In the abscence of link.precision argument, a log link will be used as link function for the precision parameter.

## Several global and local influence measures implemented (along with corresponding plots)

There are several global influence measures implemented. See influence, hatvalues, cooks.distance and their arguments.

The easiest way to perform global influence analysis on a mixpoissonreg fitted object is to obtain the plots related to global influence:

plot(fit_prec, which = c(3,4,5))

One can also get the ggplot2 version of the above plots:

autoplot(fit_prec, which = c(3,4,5))

In a similar fashion, there are several local influence measures implemented in the local_influence method. In the same spirit as above, the easiest way to perform local influence analysis on a mixpoissonreg fitted object is to obtain plots by using local_influence_plot method:

local_influence_plot(fit_prec)

One can also obtain ggplot2 versions of the above plots by using the local_influence_autoplot method:

local_influence_autoplot(fit_prec)

For further details on global and local influence analyses of mixpoissonreg objects, we refer the reader to the Global and local influence analysis with the mixpoissonreg package vignette.

## Compatibility with the tidyverse

We provide the standard methods from the broom package, namely augment, glance and tidy, for mixpoissonreg objects:

augment(fit_prec)
#> # A tibble: 314 x 12
#>    daysabs gender  math prog       .fitted .fittedlwrconf .fitteduprconf .resid
#>      <dbl> <fct>  <dbl> <fct>        <dbl>          <dbl>          <dbl>  <dbl>
#>  1       4 male      63 Academic      5.25           4.14           6.66 -0.200
#>  2       4 male      27 Academic      6.66           5.40           8.22 -0.370
#>  3       2 female    20 Academic      8.92           7.21          11.0  -0.814
#>  4       3 female    16 Academic      9.15           7.34          11.4  -0.713
#>  5       3 female     2 Academic     10.0            7.76          13.0  -0.772
#>  6      13 female    71 Academic      6.36           5.02           8.07  0.956
#>  7      11 female    63 Academic      6.71           5.39           8.34  0.599
#>  8       7 male       3 Academic      7.81           6.04          10.1  -0.102
#>  9      10 male      51 Academic      5.68           4.58           7.05  0.660
#> 10       9 male      49 Vocational    2.48           1.86           3.30  1.86
#> # … with 304 more rows, and 4 more variables: .resfit <dbl>, .hat <dbl>,
#> #   .cooksd <dbl>, .gencooksd <dbl>

glance(fit_prec)
#> # A tibble: 1 x 9
#>   efron.pseudo.r2 df.null logLik   AIC   BIC df.residual  nobs model.type
#>             <dbl>   <dbl>  <dbl> <dbl> <dbl>       <int> <int> <chr>
#> 1           0.186     312  -853. 1726. 1764.         304   314 NB
#> # … with 1 more variable: est.method <chr>

tidy(fit_prec)
#> # A tibble: 10 x 6
#>    component term           estimate std.error statistic  p.value
#>    <chr>     <chr>             <dbl>     <dbl>     <dbl>    <dbl>
#>  1 mean      (Intercept)     2.75      0.147       18.6  1.87e-77
#>  2 mean      gendermale     -0.245     0.118       -2.08 3.77e- 2
#>  3 mean      math           -0.00662   0.00232     -2.86 4.29e- 3
#>  4 mean      progAcademic   -0.426     0.132       -3.22 1.27e- 3
#>  5 mean      progVocational -1.27      0.174       -7.28 3.37e-13
#>  6 precision (Intercept)     1.41      0.343        4.12 3.79e- 5
#>  7 precision gendermale     -0.208     0.204       -1.02 3.06e- 1
#>  8 precision math           -0.00512   0.00418     -1.23 2.20e- 1
#>  9 precision progAcademic   -1.08      0.305       -3.55 3.85e- 4
#> 10 precision progVocational -1.42      0.344       -4.14 3.53e- 5

As seen above, the autoplot method from ggplot2 package is also implemented for mixpoissonreg objects.

autoplot(fit_prec)

Finally, we provide additional methods related to local influence analysis: tidy_local_influence, local_influence_benchmarks and local_influence_autoplot:

tidy_local_influence(fit_prec)
#> # A tibble: 314 x 5
#>    case_weights hidden_variable mean_explanatory precision_explanatory
#>           <dbl>           <dbl>            <dbl>                 <dbl>
#>  1     0.00146         0.000353         0.000533              0.00217
#>  2     0.000970        0.000268         0.000782              0.00129
#>  3     0.000712        0.000369         0.000535              0.00132
#>  4     0.000350        0.000328         0.000875              0.000130
#>  5     0.000680        0.000562         0.00101               0.000701
#>  6     0.00118         0.00387          0.00494               0.000740
#>  7     0.00125         0.00230          0.00396               0.00139
#>  8     0.00243         0.00127          0.00230               0.00185
#>  9     0.00107         0.00180          0.00266               0.00160
#> 10     0.00249         0.00325          0.000340              0.000788
#> # … with 304 more rows, and 1 more variable: simultaneous_explanatory <dbl>

local_influence_benchmarks(fit_prec)
#> # A tibble: 1 x 5
#>   case_weights hidden_variable mean_explanatory precision_explanatory
#>          <dbl>           <dbl>            <dbl>                 <dbl>
#> 1       0.0140          0.0206           0.0182                0.0195
#> # … with 1 more variable: simultaneous_explanatory <dbl>

local_influence_autoplot(fit_prec)

For further details on tidyverse compatibility of mixpoissonreg objects and how to use it to produce nice diagnostics and plots, we refer the reader to the mixpoissonreg in the tidyverse vignette.


text:
## Electronic Theses and Dissertations

1994

Dissertation

Ph.D.

#### Department

Chemistry and Biochemistry

#### Keywords

Chemistry, Biochemistry.

CC BY-NC-ND 4.0

#### Abstract

The purpose of this dissertation was to investigate the genetics and cellular expression of human apolipoproteins. The genetics aspect was an investigation of the Hind III apolipoprotein B gene DNA polymorphism and its association with lipids and lipoprotein parameters in a selected Canadian Caucasian population with documented coronary artery disease, and a healthy control population. To carry out this project, a simple and rapid method for the purification of human DNA from whole blood was developed. Typical 260 nm/280 nm absorbance ratio and yield for DNA so purified were 1.84 and 24.5 $\mu$g/mL, respectively. Patients had significantly (p $<$ 0.05) higher levels of cholesterol, triglycerides, LDL-cholesterol, apolipoprotein B, and lower level of apoAI compared to the controls. Restriction fragment length polymorphism analysis detected 9 hybridizable fragments denoted as H1 to H9. The H1, H2, H3, and H7 alleles were polymorphic. The (H4-H9) genotype was detected in 69% of the control group while the (H1-H9) genotype was observed in 68% of the patients. In the second phase of the studies the effects of insulin and thyroid hormone on apolipoprotein B, E, and AI mRNA expression were investigated using hepatoma cell-line. Thyroid hormone increased apolipoprotein B mRNA levels by about 25-36% as determined by slot- and Northern-blot analysis of total cellular RNA. No change in the amount of apolipoprotein B mRNA, measured by either slot- or Northern-blot analysis of total RNA was found in insulin treated cells. Both insulin and thyroid hormone were found to have no significant effect on apoE mRNA levels. The levels of apolipoprotein AI mRNA were found to be decreased by thyroid hormone and increased by insulin by about 19% and 30% respectively. A competitive chemiluminescent immunoassay was also developed for the sensitive measurement of apolipoprotein B. The lower limit of detection was 2.17 $\mu$g/L (4.0 pmol/L). Comparison of the assay with an immunoturbidimetric assay gave a slope of 0.84, y-intercept at $-$0.94 and a correlation coefficient of 0.86. In summary, the observed mutations in apolipoprotein B gene may not affect the rate of apolipoprotein B synthesis, clearance or the affinity of low-density lipoprotein for structural elements on the arterial wall. Furthermore, thyroid hormone appears to regulate apolipoprotein B and AI transcription while insulin modulates only apoAI mRNA levels.Dept. of Chemistry and Biochemistry. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1993 .O42. Source: Dissertation Abstracts International, Volume: 56-01, Section: B, page: 0223. Adviser: K. Adeli. Thesis (Ph.D.)--University of Windsor (Canada), 1994.

COinS


text:
# Thread: Which of two the following three sets is a basis of R3?

1. ## Which of two the following three sets is a basis of R3?

5. Which of two the following three sets is a basis of R3?

B1 = {(1, 0, 1), (6, 4, 5), (4, 4, 7)}
B2 ={(2, 1, 3), (3, 1, 3), (1, 1, 9)}
B3 ={(3, 1, 2), (5, 1, 1), (1, 1, 1)}

2. ## Re: Which of two the following three sets is a basis of R3?

Any basis, for a vector space of dimension n, has three properties:
1) The vectors span the space.
2) The vectors are independent.
3) There are n vectors in the set.

Further, any two of those is sufficient to prove the third.


text: For example: lim x→∞ 5 = 5. hope that helped. That is, f (x,mx) = 3x2 x2 +2m2x2 = 3 1+2m2. (Substitute $$\frac{1}{2}\sin θ$$ for $$\sin\left(\frac{θ}{2}\right)\cos\left(\frac{θ}{2}\right)$$ in your expression. Since $$f(x)=(x−3)^2$$for all $$x$$ in $$(2,+∞)$$, replace $$f(x)$$ in the limit with $$(x−3)^2$$ and apply the limit laws: $\lim_{x→2^+}f(x)=\lim_{x→2^−}(x−3)^2=1. Keep in mind there are $$2π$$ radians in a circle. So, let's look once more at the general expression for a limit on a given function f(x) as x approaches some constant c.. However, not all limits can be evaluated by direct substitution. The derivative of a constant function is zero. Since is constantly equal to 5, its value does not change as nears 1 and the limit is equal to 5. Proving a limit of a constant function. Evaluate each of the following limits, if possible. The first of these limits is $$\displaystyle \lim_{θ→0}\sin θ$$. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The limit has the form $$\displaystyle \lim_{x→a}f(x)g(x)$$, where $$\displaystyle\lim_{x→a}f(x)=0$$ and $$\displaystyle\lim_{x→a}g(x)=0$$. Example $$\PageIndex{8A}$$ illustrates this point. About "Limit of a Function Examples With Answers" Limit of a Function Examples With Answers : Here we are going to see some example questions on evaluating limits. For example, if, , dominates. In fact, if we substitute 3 into the function we get $$0/0$$, which is undefined. Think of the regular polygon as being made up of $$n$$ triangles. Figure $$\PageIndex{4}$$ illustrates this idea. If the exponent is negative, then the limit of the function … $$\displaystyle \lim_{x→3^+}\sqrt{x−3}$$. In this case, we find the limit by performing addition and then applying one of our previous strategies. Since $$\displaystyle \lim_{θ→0^+}1=1=\lim_{θ→0^+}\cos θ$$, we conclude that $$\displaystyle \lim_{θ→0^+}\dfrac{\sin θ}{θ}=1$$. Last, we evaluate using the limit laws: \[\lim_{x→1}\dfrac{−1}{2(x+1)}=−\dfrac{1}{4}.\nonumber$. The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time. In mathematics, a constant function is a function whose (output) value is the same for every input value. 3) The limit as x approaches 3 is 1. And we have proved that exists, and is equal to 4. Examples of polynomial functions of varying degrees include constant functions, linear functions, and quadratic functions. We then multiply out the numerator. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions. \nonumber\]. You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction. In the first step, we multiply by the conjugate so that we can use a trigonometric identity to convert the cosine in the numerator to a sine: \begin{align*} \lim_{θ→0}\dfrac{1−\cos θ}{θ} &=\displaystyle \lim_{θ→0}\dfrac{1−\cos θ}{θ}⋅\dfrac{1+\cos θ}{1+\cos θ}\\[4pt] Since is constantly equal to 5, its value does not change as nears 1 and the limit is equal to 5. Quadratic Function A polynomial function of the second degree. Let’s now revisit one-sided limits. Let $$f(x)$$ and $$g(x)$$ be defined for all $$x≠a$$ over some open interval containing $$a$$. These two results, together with the limit laws, serve as a foundation for calculating many limits. }\\[4pt] To do this, we may need to try one or more of the following steps: If $$f(x)$$ and $$g(x)$$ are polynomials, we should factor each function and cancel out any common factors. Step 1. In Example $$\PageIndex{8B}$$ we look at one-sided limits of a piecewise-defined function and use these limits to draw a conclusion about a two-sided limit of the same function. &= \frac{2(4)−3(2)+1}{(2)^3+4}=\frac{1}{4}. The function need not even be defined at the point. + a n x n, with a n ̸ = 0, then the highest order term, namely a n x n, dominates. That is, $$f(x)/g(x)$$ has the form $$K/0,K≠0$$ at a. 2. Consequently, $$0<−\sin θ<−θ$$. The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions. } Product Rule. When taking limits with exponents, you can take the limit of the function first, and then apply the exponent. Let $$a$$ be a real number. Therefore the limit as x approaches c can be similarly found by plugging c into the function. \end{align*}\], Example $$\PageIndex{2B}$$: Using Limit Laws Repeatedly, Use the limit laws to evaluate $\lim_{x→2}\frac{2x^2−3x+1}{x^3+4}. Evaluate the limit of a function by factoring. Thus, we see that for $$0<θ<\dfrac{π}{2}$$, $$\sin θ<θ<\tanθ$$. b. How to evaluate limits of Piecewise-Defined Functions explained with examples and practice problems explained step by step. Thus. Notice that this figure adds one additional triangle to Figure $$\PageIndex{7}$$. 풙→풄? The limit of a composition is the composition of the limits, provided the outside function is continuous at the limit of the inside function. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Prove, using delta and epsilon, that \lim\limits_{x\to 4} (5x-7)=13. for all $$L$$ if $$n$$ is odd and for $$L≥0$$ if $$n$$ is even. \nonumber$. Consequently, the magnitude of $$\dfrac{x−3}{x(x−2)}$$ becomes infinite. The limit of a constant is the constant. But you have to be careful! We see that the length of the side opposite angle $$θ$$ in this new triangle is $$\tan θ$$. Examples 1 The limit of a constant function is the same constant 2 Limit of the. Since $$\displaystyle \lim_{x→0}(−x)=0=\lim_{x→0}x$$, from the squeeze theorem, we obtain $$\displaystyle \lim_{x→0}x \cos x=0$$. For example: ""_(xtooo)^lim 5=5 hope that helped Example 1 Evaluate each of the following limits. Instead, we need to do some preliminary algebra. Formal definitions, first devised in the early 19th century, are given below. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Use the fact that $$−x^2≤x^2\sin (1/x) ≤ x^2$$ to help you find two functions such that $$x^2\sin (1/x)$$ is squeezed between them. We begin by restating two useful limit results from the previous section. The graphs of these two functions are shown in Figure $$\PageIndex{1}$$. 287−212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. But if your function is continuous at that x value, you will get a value, and you’re done; you’ve found your limit!   Terms. If the degree of the numerator is equal to the degree of the denominator ( n = m ) , then the limit of the rational function is the ratio a n /b m of the leading coefficients. Example: Solution: We can’t find the limit by substituting x = 1 because is undefined. The derivative of a constant function is zero. Use the methods from Example $$\PageIndex{9}$$. Now we shall prove this constant function with the help of the definition of derivative or differentiation. The graphs of $$f(x)=−x,\;g(x)=x\cos x$$, and $$h(x)=x$$ are shown in Figure $$\PageIndex{5}$$. Course Hero, Inc. \begin{align*} \lim_{x→2}\frac{2x^2−3x+1}{x^3+4}&=\frac{\displaystyle \lim_{x→2}(2x^2−3x+1)}{\displaystyle \lim_{x→2}(x^3+4)} & & \text{Apply the quotient law, make sure that }(2)^3+4≠0.\\[4pt] This is not always true, but it does hold for all polynomials for any choice of $$a$$ and for all rational functions at all values of $$a$$ for which the rational function is defined. Note: We don’t need to know all parts of our equation explicitly in order to use the product and quotient rules. The definition is analogous to the one for sequences. &= \lim_{θ→0}\dfrac{1−\cos^2θ}{θ(1+\cos θ)}\\[4pt] The concept of a limit is the fundamental concept of calculus and analysis. We can also stretch or shrink the limit. Example: Suppose that we consider . To find the formulas please visit "Formulas in evaluating limits". Legal. University of Missouri, St. Louis • MATH 1030, Copyright © 2021. The limit of a function at a point a a a in its domain (if it exists) is the value that the function approaches as its argument approaches a. a. a. Limit of a Constant Function. Example 13 Find the limit Solution to Example 13: Multiply numerator and denominator by 3t. A few are somewhat challenging. To see this, carry out the following steps: 1.Express the height $$h$$ and the base $$b$$ of the isosceles triangle in Figure $$\PageIndex{6}$$ in terms of $$θ$$ and $$r$$. When its arguments are constexpr values, a constexpr function produces a compile-time constant. : A limit o n the left (a left-hand limit) and a limit o n the right (a right-hand limit): The limit of a function where the variable x approaches the point a from the left or, where x is restricted to values less than a, is written ), 3. Let $$p(x)$$ and $$q(x)$$ be polynomial functions. The following Problem-Solving Strategy provides a general outline for evaluating limits of this type. To give an example, consider the limit (of a rational function) L:= lim x … So what's the limit as x approaches negative one from the right? For example, take the line f(x) = x and see what happens if we multiply it by 3: As the function gets stretched, so does the limit. Alright, now let's do this together. We can also stretch or shrink the limit. If, for all $$x≠a$$ in an open interval containing $$a$$ and, where $$L$$ is a real number, then $$\displaystyle \lim_{x→a}g(x)=L.$$, Example $$\PageIndex{10}$$: Applying the Squeeze Theorem. Evaluate $$\displaystyle\lim_{x→3}\dfrac{x^2−3x}{2x^2−5x−3}$$. & & \text{Apply the basic limit results and simplify.} To find the formulas please visit "Formulas in evaluating limits". The left limit also follows the same argument (but with, We end this section by looking also at limits of functions as. Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. To find that delta, we begin with the final statement and work backwards. Since 3 is in the domain of the rational function $$f(x)=\displaystyle \frac{2x^2−3x+1}{5x+4}$$, we can calculate the limit by substituting 3 for $$x$$ into the function. We then need to find a function that is equal to $$h(x)=f(x)/g(x)$$ for all $$x≠a$$ over some interval containing a. (2) Limit of the identity function lim x → a x = a. Step 3. }\\[4pt] Course Hero is not sponsored or endorsed by any college or university. In the figure, we see that $$\sin θ$$ is the $$y$$-coordinate on the unit circle and it corresponds to the line segment shown in blue. C tutorial for beginners with examples - Learn C programming language covering basic C, literals, data types,C Constants with examples, functions etc. Ask Question Asked 5 years, 6 months ago. The highest power that the variable x is raised to is the second power. Notes. Use the same technique as Example $$\PageIndex{7}$$. We now calculate the first limit by letting T = 3t and noting that when t approaches 0 so does T. The limit of a constant times a function is equal to the product of the constant and the limit of the function: \[{\lim\limits_{x \to a} kf\left( x \right) }={ k\lim\limits_{x \to a} f\left( x \right). The proofs that these laws hold are omitted here. Because $$−1≤\cos x≤1$$ for all $$x$$, we have $$−x≤x \cos x≤x$$ for $$x≥0$$ and $$−x≥x \cos x ≥ x$$ for $$x≤0$$ (if $$x$$ is negative the direction of the inequalities changes when we multiply). Have questions or comments? Limit Laws. The limit of a constant is that constant: $$\displaystyle \lim_{x→2}5=5$$. By dividing by $$\sin θ$$ in all parts of the inequality, we obtain, \[1<\dfrac{θ}{\sin θ}<\dfrac{1}{\cos θ}.\nonumber. Evaluate the limit of a function by using the squeeze theorem. Problem-Solving Strategy: Calculating a Limit When $$f(x)/g(x)$$ has the Indeterminate Form $$0/0$$. A limit is used to examine the behavior of a function near a point but not at the point. Simple modifications in the limit laws allow us to apply them to one-sided limits. and the function $$g(x)=x+1$$ are identical for all values of $$x≠1$$. Thus, $\lim_{x→3}\frac{2x^2−3x+1}{5x+4}=\frac{10}{19}. 2. Deriving the Formula for the Area of a Circle. Use the limit laws to evaluate the limit of a polynomial or rational function. Recall from the Limits of Functions of Two Variables page that \lim_{(x,y) \to (a,b)} f(x,y) = L if: \forall \epsilon > 0 \exists \delta > 0 such that if (x, y) \in D(f) and 0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta then \mid f(x,y) - L \mid < epsilon. Thus, since $$\displaystyle \lim_{θ→0^+}\sin θ=0$$ and $$\displaystyle \lim_{θ→0^−}\sin θ=0$$, Next, using the identity $$\cos θ=\sqrt{1−\sin^2θ}$$ for $$−\dfrac{π}{2}<θ<\dfrac{π}{2}$$, we see that, \[\lim_{θ→0}\cos θ=\lim_{θ→0}\sqrt{1−\sin^2θ}=1.\nonumber$. 5. To see that $$\displaystyle \lim_{θ→0^−}\sin θ=0$$ as well, observe that for $$−\dfrac{π}{2}<θ<0,0<−θ<\dfrac{π}{2}$$ and hence, $$0<\sin(−θ)<−θ$$. The technique of estimating areas of regions by using polygons is revisited in Introduction to Integration.   Privacy Example 5 lim x → 3(8x) The limit of a function at a point a a a in its domain (if it exists) is the value that the function approaches as its argument approaches a. a. a. (b) Typically, people tend to use a circular argument involving L’Hˆopital’s. Step 4. Then . Pages 11. Example $$\PageIndex{2A}$$: Evaluating a Limit Using Limit Laws, Use the limit laws to evaluate $\lim_{x→−3}(4x+2). For all $$x≠3,\dfrac{x^2−3x}{2x^2−5x−3}=\dfrac{x}{2x+1}$$. Use the method in Example $$\PageIndex{8B}$$ to evaluate the limit. 2) The limit of a product is equal to the product of the limits. Consider the unit circle shown in Figure $$\PageIndex{6}$$. The Constant Rule can be understood by noting that the graph of a constant function is a horizontal line, i.e., has slope 0. plot( 2.3, x=-3..3, title="Constant functions have slope 0" ); The defintion of the derivative of a constant function is simple to apply. You may press the plot button to view a graph of your function. Don’t forget to factor $$x^2−2x−3$$ before getting a common denominator. Most problems are average. If the numerator or denominator contains a difference involving a square root, we should try multiplying the numerator and denominator by the conjugate of the expression involving the square root. (Use radians, not degrees.). Step 1. Here is a set of practice problems to accompany the Computing Limits section of the Limits chapter of the notes for Paul Dawkins Calculus I course at Lamar University. For $$f(x)=\begin{cases}4x−3, & \mathrm{if} \; x<2 \\ (x−3)^2, & \mathrm{if} \; x≥2\end{cases}$$, evaluate each of the following limits: Figure illustrates the function $$f(x)$$ and aids in our understanding of these limits. Limit of a Product. So the right limit exists and equals 1. Active 5 years, 6 months ago. Because $$\displaystyle \lim_{θ→0^+}0=0$$ and $$\displaystyle \lim_{x→0^+}θ=0$$, by using the squeeze theorem we conclude that. Factoring And Canceling. + = + The limit of a sum is equal to the sum of the limits. Follow the steps in the Problem-Solving Strategy and. Now we factor out −1 from the numerator: \[=\lim_{x→1}\dfrac{−(x−1)}{2(x−1)(x+1)}.\nonumber$. We also noted that $\lim_{(x,y) \to (a,b)} f(x,y)$ does not exist if either: That is, as $$x$$ approaches $$2$$ from the left, the numerator approaches $$−1$$; and the denominator approaches $$0$$. We factor the numerator as a difference of squares … If $$f(x)/g(x)$$ is a complex fraction, we begin by simplifying it. The following are some other techniques that can be used. We now use the squeeze theorem to tackle several very important limits. Functions with Direct Substitution Property are called continuous at a. Since this function is not defined to the left of 3, we cannot apply the limit laws to compute $$\displaystyle\lim_{x→3^−}\sqrt{x−3}$$. In this case the function that we’ve got is simply “nice enough” so that what is happening around the point is exactly the same as what is happening at the point. The limit of $$x$$ as $$x$$ approaches $$a$$ is a: $$\displaystyle \lim_{x→2}x=2$$. Example 2 Math 114 – Rimmer 14.2 – Multivariable Limits LIMIT OF A FUNCTION • Let’s now approach (0, 0) along another line, say y= x. rule. You may NOT use a constant function. 풙→풄 풌 ∙ ? If we originally had . Example 13 Find the limit Solution to Example 13: Multiply numerator and denominator by 3t. To limit the complexity of compile-time constant computations, ... or to provide a non-type template argument. Evaluate $$\displaystyle \lim_{x→3}\left(\dfrac{1}{x−3}−\dfrac{4}{x^2−2x−3}\right)$$. Missed the LibreFest? Declaration. $f(x)=\dfrac{x^2−1}{x−1}=\dfrac{(x−1)(x+1)}{x−1}\nonumber$. The first two limit laws were stated previosuly and we repeat them here. Multiply numerator and denominator by $$1+\cos θ$$. Examples (1) The limit of a constant function is the same constant. For example, take the line f(x) = x and see what happens if we multiply it by 3: As the function gets stretched, so does the limit. Follows from the corresponding statement for sequences. Before we start differentiating trig functions let’s work a quick set of limit problems that this fact now allows us to do. $|f(x)-L| \epsilon$ Before we can begin the proof, we must first determine a value for delta. 2 f x g x f x g x lim[ ( ) ( )] lim ( ) lim ( ) →x a →x a →x a − = − The limit of a difference is equal to the difference of the limits. We simplify the algebraic fraction by multiplying by $$2(x+1)/2(x+1)$$: $\lim_{x→1}\dfrac{\dfrac{1}{x+1}−\dfrac{1}{2}}{x−1}=\lim_{x→1}\dfrac{\dfrac{1}{x+1}−\dfrac{1}{2}}{x−1}⋅\dfrac{2(x+1)}{2(x+1)}.\nonumber$. Example $$\PageIndex{11}$$: Evaluating an Important Trigonometric Limit. Examples 1 the limit of a constant function is the. Calculating limits of a function- Examples. The derivative of this type of function is just zero. Let $$f(x)$$ and $$g(x)$$ be defined for all $$x≠a$$ over some open interval containing $$a$$. the given limit is 0. Limit of a Composite Function lim x→c f g(x) = lim x→c f(g(x)) = f(lim x→c g(x)) if f is continuous at lim x→c g(x). $$\displaystyle \dfrac{\sqrt{x+2}−1}{x+1}$$ has the form $$0/0$$ at −1. Limit Laws. \nonumber\]. The limit of a constant (lim (4)) is just the constant, and the identity law tells you that the limit of lim (x) as x approaches a is just “a”, so: The solution is 4 * 3 * 3 = 36. Step 1. Solve this for $$n$$. The limit of a composition is the composition of the limits, provided the outside function is continuous at the limit of the inside function. The example featured in this video is: Find the limit as x approaches 0.2 of the function 3x+4. First, we need to make sure that our function has the appropriate form and cannot be evaluated immediately using the limit laws. Example $$\PageIndex{8A}$$: Evaluating a One-Sided Limit Using the Limit Laws. To find a formula for the area of the circle, find the limit of the expression in step 4 as $$θ$$ goes to zero. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. and solved examples, visit our site BYJU’S. Informally, a function f assigns an output f(x) to every input x. With or without using the L'Hospital's rule determine the limit of a function at Math-Exercises.com. The formulas below would pick up an extra constant that would just get in the way of our work and so we use radians to avoid that. and solved examples, visit our site BYJU’S. The first one is that the limit of the sum of two or more functions equals the sum of the limits of each function. Then, each of the following statements holds: $\displaystyle \lim_{x→a}(f(x)+g(x))=\lim_{x→a}f(x)+\lim_{x→a}g(x)=L+M$, $\displaystyle \lim_{x→a}(f(x)−g(x))=\lim_{x→a}f(x)−\lim_{x→a}g(x)=L−M$, $\displaystyle \lim_{x→a}cf(x)=c⋅\lim_{x→a}f(x)=cL$, $\displaystyle \lim_{x→a}(f(x)⋅g(x))=\lim_{x→a}f(x)⋅\lim_{x→a}g(x)=L⋅M$, $\displaystyle \lim_{x→a}\frac{f(x)}{g(x)}=\frac{\displaystyle \lim_{x→a}f(x)}{\displaystyle \lim_{x→a}g(x)}=\frac{L}{M}$, $\displaystyle \lim_{x→a}\big(f(x)\big)^n=\big(\lim_{x→a}f(x)\big)^n=L^n$, $\displaystyle \lim_{x→a}\sqrt[n]{f(x)}=\sqrt[n]{\lim_{x→a} f(x)}=\sqrt[n]{L}$. This theorem allows us to calculate limits by “squeezing” a function, with a limit at a point a that is unknown, between two functions having a common known limit at $$a$$. Despite appearances the limit still doesn’t care about what the function is doing at $$x = - 2$$. Example $$\PageIndex{7}$$: Evaluating a Limit When the Limit Laws Do Not Apply. Observe that, $\dfrac{1}{x}+\dfrac{5}{x(x−5)}=\dfrac{x−5+5}{x(x−5)}=\dfrac{x}{x(x−5)}.\nonumber$, $\lim_{x→0}\left(\dfrac{1}{x}+\dfrac{5}{x(x−5)}\right)=\lim_{x→0}\dfrac{x}{x(x−5)}=\lim_{x→0}\dfrac{1}{x−5}=−\dfrac{1}{5}.\nonumber$. In our first example: Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions. To get a better idea of what the limit is, we need to factor the denominator: $\lim_{x→2^−}\dfrac{x−3}{x^2−2x}=\lim_{x→2^−}\dfrac{x−3}{x(x−2)} \nonumber$. Find an expression for the area of the $$n$$-sided polygon in terms of $$r$$ and $$θ$$. Let's do another example. Step 1. Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. The radian measure of angle $$θ$$ is the length of the arc it subtends on the unit circle. The proofs that these laws hold are omitted here. Step 2. Solution 1) Plug x = 3 into the expression ( 3x - 5 ) 3(3) - 5 = 4 2) Evaluate the logarithm with base 4. After substituting in $$x=2$$, we see that this limit has the form $$−1/0$$. + a n x n, with a n ̸ = 0, then the highest order term, namely a n x n, dominates. Since $$x−2$$ is the only part of the denominator that is zero when 2 is substituted, we then separate $$1/(x−2)$$ from the rest of the function: $=\lim_{x→2^−}\dfrac{x−3}{x}⋅\dfrac{1}{x−2} \nonumber$. In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. To evaluate this limit, we must determine what value the constant function approaches as approaches (but is not equal to) 1. We also noted that $\lim_{(x,y) \to (a,b)} f(x,y)$ does not exist if either: Let's do another example. When called with non-constexpr arguments, or when its value isn't required at compile time, it produces a value at run time like a regular function. Both $$1/x$$ and $$5/x(x−5)$$ fail to have a limit at zero. The limit of a constant is that constant: $$\displaystyle \lim_{x→2}5=5$$. Example $$\PageIndex{6}$$: Evaluating a Limit by Simplifying a Complex Fraction. Example: lim x→3 √ … We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Evaluate each of the following limits using Note. Evaluate $$\displaystyle \lim_{x→−2}(3x^3−2x+7)$$. Limits Examples. Follow the steps in the Problem-Solving Strategy, Example $$\PageIndex{5}$$: Evaluating a Limit by Multiplying by a Conjugate. (풙) = 풌 ∙ ?퐢? The limit of a constant is only a constant. If all the partial derivatives of a function are known (for example, with the gradient ), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant. For example, to apply the limit laws to a limit of the form $$\displaystyle \lim_{x→a^−}h(x)$$, we require the function $$h(x)$$ to be defined over an open interval of the form $$(b,a)$$; for a limit of the form $$\displaystyle \lim_{x→a^+}h(x)$$, we require the function $$h(x)$$ to be defined over an open interval of the form $$(a,c)$$. 3 cf x c f x lim ( ) lim ( ) →x a →x a = The limit of a constant times a function is equal to the constant times the limit of the function. Example 1: To Compute $$\mathbf{\lim \limits_{x \to -4} (5x^{2} + 8x – 3)}$$ Solution: \nonumber \]. Step 6. We can estimate the area of a circle by computing the area of an inscribed regular polygon. \end{align*}\]. To understand this idea better, consider the limit $$\displaystyle \lim_{x→1}\dfrac{x^2−1}{x−1}$$. . Now we shall prove this constant function with the help of the definition of derivative or differentiation. &= \lim_{θ→0}\dfrac{\sin θ}{θ}⋅\dfrac{\sin θ}{1+\cos θ}\$4pt] Example does not fall neatly into any of the patterns established in the previous examples. Although this discussion is somewhat lengthy, these limits prove invaluable for the development of the material in both the next section and the next chapter. Limits of Functions Example 2.17. We now take a look at the limit laws, the individual properties of limits. x. For instance, large), it is useful to look for dominant terms. Therefore, the product of $$(x−3)/x$$ and $$1/(x−2)$$ has a limit of $$+∞$$: \[\lim_{x→2^−}\dfrac{x−3}{x^2−2x}=+∞. Considering all the examples above, we can now say that if a function f gets arbitrarily close to (but not necessarily reaches) some value L as x approaches c from either side, then L is the limit of that function for x approaching c. In this case, we say the limit exists. Limit of a Composite Function lim x→c f g(x) = lim x→c f(g(x)) = f(lim x→c g(x)) if f is continuous at lim x→c g(x). It now follows from the quotient law that if $$p(x)$$ and $$q(x)$$ are polynomials for which $$q(a)≠0$$, \[\lim_{x→a}\frac{p(x)}{q(x)}=\frac{p(a)}{q(a)}.$, Example $$\PageIndex{3}$$: Evaluating a Limit of a Rational Function. By a "constant" we mean any number. Since neither of the two functions has a limit at zero, we cannot apply the sum law for limits; we must use a different strategy. 2.3. The limit of product of the constant and function is equal to the product of constant and the limit of the function, ... Differentiation etc. Find the limit by factoring. About "Limit of a Function Examples With Answers" Limit of a Function Examples With Answers : Here we are going to see some example questions on evaluating limits. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied. (Hint: $$\displaystyle \lim_{θ→0}\dfrac{\sin θ}{θ}=1)$$. Evaluate $$\displaystyle \lim_{x→−3}\dfrac{x^2+4x+3}{x^2−9}$$. Eventually we will formalize up just what is meant by “nice enough”. Step 2. The Constant Rule can be understood by noting that the graph of a constant function is a horizontal line, i.e., has slope 0. plot( 2.3, x=-3..3, title="Constant functions have slope 0" ); The defintion of the derivative of a constant function is simple to apply. Uploaded By cwongura. The second one is that the limit of a constant equals the same constant. Here is a (correct) geometric argument: -axis cut by the vertical drawn downwards from, the point where the vertical drawn upwards from, 2, is squeezed between the areas of the triangular, (by virtue of being positive and near 0), we obtain. 1) The limit of a sum is equal to the sum of the limits. By taking the limit as the vertex angle of these triangles goes to zero, you can obtain the area of the circle. It is a Numeric limits type and it provides information about the properties of arithmetic types (either integral or floating-point) in the specific platform for which the library compiles. $\lim_{x→a}x=a \quad \quad \lim_{x→a}c=c \nonumber$, $\lim_{θ→0}\dfrac{\sin θ}{θ}=1 \nonumber$, $\lim_{θ→0}\dfrac{1−\cos θ}{θ}=0 \nonumber$. So we have another piecewise function, and so let's pause our video and figure out these things. Since 4^1 = 4, the value of the logarithm is 1. Question 1 : Evaluate the following limit An application of the squeeze theorem produces the desired limit. Recall from the Limits of Functions of Two Variables page that $\lim_{(x,y) \to (a,b)} f(x,y) = L$ if: $\forall \epsilon > 0$ $\exists \delta > 0$ such that if $(x, y) \in D(f)$ and $0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta$ then $\mid f(x,y) - L \mid < epsilon$. 3) The limit of a quotient is equal to the quotient of the limits, 3) provided the limit of the denominator is not 0. All of the solutions are given WITHOUT the use of L'Hopital's Rule. Then, we cancel the common factors of $$(x−1)$$: $=\lim_{x→1}\dfrac{−1}{2(x+1)}.\nonumber$. Example: Suppose that we consider . In each step, indicate the limit law applied. We now take a look at the limit laws, the individual properties of limits. To evaluate this limit, we use the unit circle in Figure $$\PageIndex{6}$$. m given by y = mx, with m a constant. Assume that $$L$$ and $$M$$ are real numbers such that $$\displaystyle \lim_{x→a}f(x)=L$$ and $$\displaystyle \lim_{x→a}g(x)=M$$. We will place our work in a table, so we can provide a running commentary of our thoughts as we work. template numeric_limits; C++11 template numeric_limits; Parameters However, with a little creativity, we can still use these same techniques. Limits of Functions of Two Variables Examples 1. So we have another piecewise function, and so let's pause our video and figure out these things. \Sin\Dfrac { 1 } { 19 } c can be similarly found by plugging c into function. Arguments are constexpr values, a function near a point but not at the point triangles! Establishing basic trigonometric limits the limit compile-time constant results from the right & & \text { limit of a constant function example the limit... To have a limit at zero an expression with a different base, you can the! 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# Can all systems be described by differential/difference equations?

All LCCODEs (linear, constant-coefficient ordinary differential equation) with initial rest conditions describe causal LTI systems (is this a correct statement?)

• Is the converse true, i.e. can all causal LTI systems be described by LCCODEs with initial rest? If not, what are the necessary/sufficient conditions?

It seems trivial that arbitrary differential/difference equations describe some sort of system (is that also true?)

• Again, is the converse true, i.e. can we describe any arbitrary systems with (possibly nonlinear/variable-coefficient/partial, without restriction on initial conditions) differential equations? If not, what are the necessary/sufficient conditions?

No the converse is not true in general.

Take for example the discrete-time ideal lowpass filter with impulse response

$$h[n] = \frac{ \sin( \omega_c n) }{ \pi n } ~~~,~~~-\infty < n <\infty$$

which describes an LTI system but it does not correspond to a difference equation of any kind. Indeed $$h[n]$$ is derived based on the inverse discrete-time Fourier transform of the filter frequency response;

$$H(\omega) = \begin{cases} {1 ~~~,~~~ |\omega| < \omega_c \\ 0 ~~~,~~~ \text{ otherwise } }\end{cases}$$

$$h[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} H(\omega) e^{j\omega n} d\omega$$

• Why is there no difference equation? There is one with infinitely many terms. Is it implementable? No, but it does exist. – Rodrigo de Azevedo Mar 2 at 8:05
• Indeed so! I just probably referred to te general acceptance. As one usually either solves an LCCDE to get the impulse response or uses it to compute the output from input but in this case you neither solve for $h[n]$ which is first obtained from the Fourier transform inversion nor use such LCCDE to make any computations, but merely construct an uncomputable (infinite order!) LCDDE. Probably that's why it's accepted that there are no corresponding LCCDE for such infinetely-long yet FIR (finite impulse response) filters... – Fat32 Mar 4 at 16:04
• I understand your point. However, I would say that existence (in a Platonist's Heaven) is one thing, and realizability (using pencil & paper, hardware or software) is another. But this is obviously a matter of taste. – Rodrigo de Azevedo Mar 9 at 23:26
• @RodrigodeAzevedo probably so... – Fat32 Mar 10 at 2:25


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Asaf Karagila
I don't have much choice...

## Blog posts from 2016

If you follow my blog, you probably know that I am a big fan of Michael Stevens from the VSauce channel, who in the recent year or so released several very good videos about mathematics, and about infinity in particular. Not being a trained mathematician, Michael is doing an incredible task.

Non-mathematicians often tend to be Platonists "by default", so they will assume that every question has an answer and sometimes it's just that we don't know that answer. But it's out there. It's a fine approach, but it can somewhat fly in the face of independence if you are not trained to think about the difference between true and provable.

I've been given the chance to teach the course in axiomatic set theory in Jerusalem this semester. Today I gave my first lecture as a teacher. It went fine, I even covered more than I expected to, which is good, I guess. I am also preparing lecture notes, which I will probably post here when the semester ends. These predicated on some rudimentary understanding in logic and basic set theory, so there might be holes there to people unfamiliar with the basic course (at least the one that I gave with Azriel Levy for the past three years).

Yesterday, however, I spent most of my day thinking about how we---as a collective of set theorists---teach axiomatic set theory. About that usual course: axioms, ordinals, induction, well-founded sets, reflection, $$V=L$$ and the consistency of $$\GCH$$ and $$\AC$$, some basic combinatorics (clubs, Fodor's lemma, maybe Solovay or even Silver's theorem). Up to some rudimentary permutation.

## Zornian Functional Analysis or: How I Learned to Stop Worrying and Love the Axiom of Choice

Back in the fall semester of 2015-2016 I had taken a course in functional analysis. One of the reasons I wanted to take that course (other than needing the credits to finish my Ph.D.) is that I was always curious about the functional analytic results related to the axiom of choice, and my functional analysis wasn't strong enough to sift through these papers.

I was very happy when the professor, Matania Ben-Artzi, allowed me to write a final paper about the usage of the axiom of choice in the course, instead of taking an exam.

## In praise of some history

Teaching pure mathematics is not a trivial thing. You have to overcome the several barriers that were constructed by the K12 education that mathematics is a bunch of "fit this problem into that mold".

I recently had a chat with James Cummings about teaching. He said something that I knew long before, that being a good teacher requires a bit of theatricality. My best teacher from undergrad, Uri Onn, had told me when I started teaching, that being a good teacher is the same as being a good storyteller: you need to be able and mesmerize your audience and keep them on the edge of their seats, wanting more.

## Constructive proof that large cardinals are consistent

I am not a Platonist, as I keep pointing out. Existence, even not in mathematics, is relative and confusing to begin with, so I don't pretend to try and understand it in a meaningful way.

However, we have a proof, a constructive proof that large cardinals are consistent. And they exist in an inner model of our universe.

## Some thoughts about "automated theorem searching"

Let me begin by giving a spoiler warning. If you haven't watched "The Prisoner" you might be spoiled about one of the episodes. Not that matters, for a show from nearly 40 years ago, but you should definitely watch it. It is a wonderful show. And even if you haven't watched it, it's just one episode, not the whole show. So you can keep on reading.

So, I'm fashionably late to the party (with some good excuse, see my previous post), but after the recent 200 terabytes proof for the coloring of Pythagorean triples, the same old questions are raised about whether or not at some point computers will be better than us in finding new theorems, and proving them too.

## Iterating Symmetric Extensions

I don't usually like to write about new papers. I mean, it's a paper, you can read it, you can email me and ask about it if you'd like. It's there. And indeed, for my previous papers, I didn't even mention them being posted on arXiv/submitted/accepted/published. This paper is a bit different; but don't worry, this is not your typical "new paper" post.

If you don't follow arXiv very closely, I have posted a paper titled "Iterating Symmetric Extensions". This is going to be the first part of my dissertation. The paper is concerned with developing a general framework for iterating symmetric extensions, which oddly enough, is something that we didn't really know how to do until now. There is a refinement of the general framework to something I call "productive iterations" which impose some additional requirements, but allow greater freedom in the choice of filters used to interpret the names. There is an example of a class-length iteration, which effectively takes everything that was done in the paper and uses it to produce a class-length iteration—and thus a class length sequence of models—where slowly, but surely, Kinna-Wagner Principles fail more and more. This means that we are forcing "diagonally" away from the ordinals. So the models produced there will not be defined by their set of ordinals, and sets of sets of ordinals, and so on.

## Syntactic T-Rex: Irregularized

One of my huge pet peeves is with people who think that writing $$1+2+3+\ldots=-\frac1{12}$$ is a reasonable thing without context. Convention dictates that when no context is set, we interpret infinite summation as the usual convergence of a series, namely the limit of the partial sums, if it exists (and of course that $$1+2+3+\ldots$$ does not converge to any real number). However, a lot of people who are [probably] not mathematicians per se, insist that just because you can set up a context in which the above equality holds, e.g., Ramanujan summation or zeta regularization, then it is automatically perfectly fine to write this out of nowhere without context and being treated as wrong.

But those people forget that $$0=1$$ is also very true in the ring with a single element; or you know, just in any structure for a language including the two constant symbols $$0$$ and $$1$$, where both constants are interpreted to be the same object. And hey, who even said that $$0$$ and $$1$$ have to denote constants? Why not ternary relations, or some other thing?

During the first day of the conference we realized that it might be a good idea to get the lectured videoed, so we quickly set up the videos for the second and third day. With the exception of one speaker who asked not to be videoed, you can find all the lectures from the second and third day of the conference in this YouTube Playlist.

Enjoy!

## Quick update from Norwich

It's been a while, quite a while, since I last posted anything. Even a blurb.

I'm visiting David Asperó in Norwich at the moment, and on Sunday, the 12th, I will return home. It seems that the pattern is that you work most of the day, then head for a few drinks and dinner. Mathematics is eligible for the first two beers, philosophy of mathematics for the next two, and mathematical education for the fifth beer. Then it's probably a good idea to stop. Also it is usually last call, so you kinda have to stop.

## Vsauce on cardinals and ordinals

To the readers of my blog, it should come as no surprise that I have a lot of appreciation to what Michael Stevens is doing in Vsauce. In the past Michael, who is not a mathematician, created an excellent video about the Banach-Tarski paradox, as well another one on supertasks. And now he tackled infinite cardinals and ordinals.

You can find the video here:

## What I realized recently

I recently learned that diamonds are cut and polished with the dust of other diamonds. And I recently realized that success is cut and polished with the dust of failures.

In particular a successful mathematical idea is polished with the dust of the many failed ideas that preceded it.


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# In an ATIS broadcast, is the visibility in statute or nautical miles?

I think I remember seeing somewhere that since it's an observation at the surface, statute miles are used for the visibility. Would be nice if someone had a reference.

• In Europe, it's in meters. – assylias Jul 30 '18 at 10:50
• ...not just in Europe, practically everywhere outside America. Thus, lacking the country-specific tag, metres should be the correct default answer. – Zeus Jul 31 '18 at 6:03