text:
# Biotechnology

### Introduction

Biotechnology refers to technology used to manipulate DNA. The procedures are often referred to as genetic engineering.

DNA is the genetic material of all living organisms and all organisms use the same genetic code. Genes from one kind of organism can be transcribed and translated when put into another kind of organism.

For example, human and other genes are routinely put into bacteria in order to synthesize products for medical treatment and commercial use. Human insulin, human growth hormone, and vaccines are produced by bacteria.

Recombinant DNA refers to DNA from two different sources.  Individuals that receive genes from other species are transgenic.

### Viruses

Viruses contain genetic material but are not living. Host cells are required for their reproduction.

#### Structure

Viruses are composed of an inner nucleic acid core (genetic material) and an outer protein coat (capsid).

Viruses that infect animals have an outer envelope (membrane) that is derived from the cell membrane of the host cell may surround the capsid.

The genetic material in some viruses is DNA; in others it is RNA.

#### Reproduction

When viral genetic material enters a cell, it is replicated, transcribed (mRNA is produced) and translated (proteins are produced from the mRNA) by the host cell. By this process, the host cell uses the genetic instructions in the virus to make more viruses.

viral DNA $$\rightarrow$$ mRNA $$\rightarrow$$ protein

If the viral genetic material is RNA, a DNA copy must first be made before transcription and translation can occur. The DNA copy of the viral RNA is called cDNA.

viral RNA $$\rightarrow$$ cDNA $$\rightarrow$$ mRNA ? protein

#### Bacteriophages

Bacteriophages are viruses that infect bacteria. They are not surrounded by a membrane as the animal-infecting viruses discussed above.

The virus attaches to the bacteria cell, a viral enzyme digests away a part of the wall and its DNA enters the host cell.

Inside the host cell, the viral DNA is transcribed, translated, and replicated. Translation produces protein coats and the enzymes needed in the construction of new virus particles.

The viral DNA is replicated.

The protein coats and DNA are assembled into new viral particles.

The host cell wall to ruptures releasing the newly formed viruses.

Upon entering the cell, the viral DNA may instead, become integrated into the bacterial DNA. It is replicated along with the host DNA when the host reproduces. Eventually, it will become transcribed and translated as discussed above.

#### DNA Animal Viruses

Animal viruses usually do not kill the cell.

The viruses have a membranous outer envelope with spikes.

The virus enters the cell by endocytosis.

Nucleic acid is released inside the cell. It is transcribed and translated to produce more viral DNA, protein coats and spikes.

The virus acquires its membrane when it is released from the cell by budding (exocytosis).

The host cell is not necessarily killed.

#### Retroviruses

The genetic material of retroviruses is RNA. The retrovirus carries an enzyme called reverse transcriptase, which is capable of creating a DNA copy of the viral RNA.

The new DNA produced from the RNA template is called cDNA.

DNA synthesis follows the production of cDNA to produce a double-helix.

The DNA then becomes incorporated into the host DNA.

The new viruses escape the host cell by budding.

The AIDS virus (HIV) is an example of a retrovirus.

### Recombinant DNA Technology

#### Vectors

Vectors are DNA used to transfer genes into a host cell.

A vector must be capable of self-replicating inside a cell.

Marker genes can be used to determine if the gene has been taken up.

Marker genes must have some distinguishable characteristic. For example if you put a gene that enables an ampicillin resistance on the same vector as the same vector as the gene for human insulin production, then any bacteria that grow on an ampicillin plate will be able to produce insulin.

##### Plasmids

The host bacterium takes up the plasmid, which includes the foreign gene.

When the bacteria reproduces, the plasmids are also reproduced. The gene is cloned.

Shuttle vectors are plasmids that are capable of existing in several different species. They are useful when transferring genes to multicellular organisms.

##### Viruses

Viruses are the vectors of choice for animal cells.

They can accept larger amounts of DNA than plasmids.

When the virus reproduces within the animal cell, it also reproduces the foreign gene that it carries. The gene is therefore cloned.

The DNA of some retroviruses becomes integrated into the host chromosome.

#### Restriction enzymes

Restriction enzymes were discovered in bacteria. Bacteria use them as a defense mechanism to cut up the DNA of viruses or other bacteria.

Hundreds of different restriction enzymes have been isolated. Each one cuts DNA at a specific base sequence. For example, EcoRI always cuts DNA at GAATTC as indicated below.

The sequence GAATTC appears three times in the DNA strand below. As a result, the strand is cut into four pieces.

Other restriction enzymes cut at different sites, some examples are listed below.

 Enzyme Cutting Site Bam HI GGATCC Hae III GGCC Pst I CTGCAG Hinf I GANTC
##### Sticky Ends

Fragments of DNA that has been cut with restriction enzymes have unpaired nucleotides at the ends called sticky ends. All of the fragments will have the same sticky ends. The sticky ends have complimentary bases, so they could rejoin.

If the vector and the gene to be cloned are both cut with the same restriction enzyme, they will both have complimentary sticky ends.

#### Making Recombinant DNA

To make recombinant DNA, restriction enzymes are used to cut DNA from two sources such as the that of a vector and a gene to be cloned. If the vector and the gene to be cloned are both cut with the same restriction enzyme, they will both have complimentary sticky ends (see above).

After cutting, the two samples of DNA are mixed. Some of the fragments from one species will stick to those of the other because they both have the same sticky ends.

DNA ligase is used to seal the fragments.

Bacteria are capable of taking up DNA from their environment. This process is called transformation. CaCl2 and a procedure called heat shock are used to make E. coli cells more permeable so that they take up the modified plasmids more readily.

### Genomic Libraries

A genome is all of the genes in a particular organism. Bacteria or virus vectors can be used to store fragments of the DNA from a species.

The DNA is cut up into fragments and the different fragments are inserted into bacteria or viruses. The collection of bacteria or viruses is called a genomic library.

#### Finding Genes in a Gene Library

##### Blue-White Screening Method

The plasmid used contains a gene for ampicillin resistance. The transformed bacteria are grown on a medium that contains ampicillin. Any bacteria that grow have been transformed because the bacteria cannot grow unless they are ampicillin-resistant.

The plasmid also contains a gene for the production of b-galactosidase. The b-galactosidase gene contains a region that can be cut with a number of different restriction enzymes. Genes inserted into this site will inactivate the b-galactosidase gene because it has been cut and the new gene has been inserted within it. A recombinant plasmid therefore will not produce b-galactosidase.

The transformed bacteria are grown on a medium that contains X-gal, a substrate for b-galactosidase. Colonies that use X-gal as a food source produce a blue compound. Colonies that have received a gene cannot use X-gal and appear white.

Bacteria that have foreign genes, therefore, will grow (resistant to ampicillin) and appear white (unable to produce b-galactosidase).

The blue-white screening method described above selects for bacteria that have any gene. Radioactive probes can be used to find colonies that have specific genes.

Probes are short, single-stranded segments of DNA whose base sequence matches part of the gene in question. It is not necessary to match the entire gene, just a small fragment.

The cells are lysed and the DNA is denatured by treating it with an alkaline solution. When the probe is mixed with the denatured DNA, it will form a double-helix in the area where the gene has complimentary bases.

If the probe is radioactive or fluorescent, it can be visualized. The gene can then be isolated or cloned as needed.

It may be possible to see the chromosome and the location on the chromosome while viewing under a microscope.

Autoradiography is a process in which film is used to show the area of the vector where the probe has attached. This area is the gene in question.

##### Obtaining Probes

Probes can be synthesized using a DNA synthesis machine. This requires a knowledge of at least a part of the sequence of the nucleotides in the gene to be probed so that a complimentary fragment can be created. If this is unknown, it may be deduced by knowing the sequence of amino acids in the protein that the gene codes. Amino acids often have more than one spelling,so it may be necessary to create several different alternative probes.

In some cases, a similar gene which has already been isolated from another organism can serve as a probe.

### Inserting DNA Into Cells

#### Eukaryotic Genes

Eukaryotic genes contain introns but bacteria do not contain the necessary enzymes to remove introns, so eukaryotic genes that are inserted into bacteria must be inserted without introns.

#### Making Intron-Free DNA:

The DNA of eukaryotes is extremely long, containing many thousands of genes. It is often not possible to find specific genes in the DNA. Artificial genes can be made, however, using mRNA as a template.

In order to synthesize a gene, one must first obtain some mRNA produced by the gene in question. Recall that the introns of mature mRNA have already been removed. Reverse transcriptase (from retroviruses, see discussion above) is used to produce a DNA copy of the mRNA. This copy is called cDNA.

#### Expression Vectors

The promoter and ribosome binding site codes in Eukaryotic DNA are likely to be different than those used by the host organism. Plasmids called expression vectors have been created that have a promoter and ribosome binding site that can be recognized by E. coli. These sites are adjacent to a restriction cutting site so that any gene inserted into the plasmid will be transcribed and translated by E. coli.

#### Shuttle Vectors

Shuttle vectors are plasmids that can be propagated in two different species because they have two origins of replication.

This is done because it is easier to clone genes in some species such as E. coli. After the cloning procedures are done, the plasmid can be transferred to a different species for expression of the cloned DNA.

#### Bacterial Artificial Chromosomes

Artificial vectors have been developed that can be used for inserting large segments of DNA. These vectors are called bacterial artificial chromosomes.

#### Yeast Artificial Chromosomes

Yeast artificial chromosomes can hold 150-1000 kb pairs. The chromosomes can grow in bacteria and yeast.

The has a ring form that is used in bacteria. It can be cleaved to produce a linear form with a centromere that replicates in yeast.

#### Transgenic Plants

The only plasmid that plant cells take up is the Ti (tumor-inducing) plasmid from the bacterium Agrobacterium. The plasmid transferred by this bacterium causes plants to form a gall. A wide variety of plant cells will take up the plasmid and move it into a chromosome. Scientists have been able to remove the gall-forming genes and insert other genes into the plasmid. As a result, this plasmid has enabled genetic engineering of a large number of plants.

#### Transgenic Animals

Animal cells generally will not take up plasmids. Other methods such as microinjection must be used.

One method has been developed where animal eggs are placed in a mixer with needle-like fragments of silicon carbide. The needles make holes in the cells, allowing DNA to enter. Using this procedure, eggs from fish and several agricultural species have been given the gene for bovine growth hormone, producing larger individuals.

#### Viruses

Viruses may serve as vectors for transferring DNA into eukaryotic cells. After the DNA enters the cell, it must become inserted into the host chromosome.

#### Other Methods of Delivering DNA

Electroporation involves using an electric current to create pores in the cell wall and plasma membrane for DNA to enter.

It is difficult to create transgenic plants because the cell wall prevents entry of DNA. One solution is to remove the cell wall. These cells (called protoplasts) are then placed in a liquid with foreign DNA. Electroporation is used to make small, temporary holes in the membrane so DNA can pass in.

A gene gun propels small gold pellets coated with DNA. The pellets penetrate the cell wall and plasma membrane and enter the cell to deliver their DNA.

### Polymerase Chain Reaction (PCR)

The polymerase chain reaction can be used to make many copies of small pieces of DNA. Because techniques in biotechnology usually require large amounts of DNA (many copies), PCR has allowed much of the biotechnology development that we have seen in recent years.

#### Materials and Procedure

##### Materials needed

The procedure requires primers, DNA polymerase, and nucleotides.

Primers are short chains of about 20 nucleotides that are complimentary to a region in the DNA to be amplified. They are needed because the enzyme that copies the DNA (DNA polymerase) cannot start the process unless it has already been started.

DNA polymerase from the thermophile Thermus aquaticus is used because this species thrives at temperatures that are near boiling. It's DNA polymerase (called Taq polymerase) is stable at relatively high temperatures and functions optimally at 70 degrees C. This is important because high temperatures will be used to separate the strands of the double helix.

Nucleotides are needed because DNA is composed of nucleotide "building blocks".

##### Procedure

The DNA in question is heated to approximately 95 degrees C to separate the two strands of the double helix.

After the strands are separated, the DNA is cooled to about 50 degrees and the primers attach.

The temperature is raised to approximately 70 degrees C. This is the optimal temperature for Taq polymerase. The polymerase attaches to and copies the strand.

The solution is then heated and cooled again as described above at regular intervals. Each time it is cycled through this heating and cooling procedure, the DNA replication process repeats itself and the amount of DNA produced is doubled.

### DNA Fingerprinting (RFLP Analysis)

In RFLP analysis, the DNA of an organism is cut up into fragments using restriction enzymes. A large number of short fragments of DNA will be produced.

Restriction enzymes always cut at the same base sequence. Because no two individuals have identical DNA, no two individuals will have the same length fragments. For example, the enzyme EcoRI always cuts DNA at the sequence GAATTC. Different people are going to have different numbers of this particular sequence and will therefore have different fragment lengths. In addition, some of them will be at different locations on the chromosome.

Electrophoresis is a technique used to separate the DNA fragments according to their size. They are placed on a sheet of gelatin and an electric current is applied to the sheet. DNA is charged and will move in an electric field toward the positive pole.

In the diagram below, holes (wells) in the gelatin can be seen. DNA samples placed in these wells will migrate through the gelatin toward the + side after an electric current is applied.

The smallest fragments will move the fastest because they are able to move through the pores in the gelatin faster. Bands will be produced on the gelatin where the fragments accumulate. The shortest fragments will accumulate near one end of the gelatin and the longer, slower-moving ones will remain near the other end.

In the diagram below, four samples of DNA were placed on the gelatin. After an electric current was applied for a period of time, the fragments separated. Notice that sample D on the right does not match the other three samples.

The DNA bands must be stained to make them visible. Ethidium bromide-stained DNA will fluoresce when illuminated with UV light.

PCR techniques are used to produce sufficient quantities of DNA for this technique.

##### Southern Blotting

Electrophoresis separates DNA by size but does not provide any information about the nucleotide sequence in the DNA. A technique called Southern blotting is used to search for specific base sequences in DNA that has been separated by electrophoresis. It is useful in identifying pathogens and genetic diseases.

This technique uses a probe that has a complimentary base sequence to a portion of the DNA of interest. The probe is labeled with radioactivity, a fluorescent dye, or a colored dye so that it can be visualized.

First, the DNA to be searched is digested using restriction enzymes and separated using gel electrophoresis as described above.

The fragments on the gelatin are transferred to a filter by blotting.

The filter is treated with an alkaline solution to separate the strands of DNA.

The probe is added to the filter paper and then rinsed off. Because the probe has a base sequence that is complimentary to the DNA of interest, it will only attach to the DNA of interest, the rest will rinse off.

The DNA fragment containing a sequence that is complimentary to the probe can be seen because the probe has been labeled.

##### Uses for Electrophoresis

Electrophoresis requires a large amount of DNA so it is often used in conjunction with PCR discussed above. Some uses are identification of diseased genes including oncogenes, identification of viral infections, determining family relationships among individuals, and identifying tissue found at a crime scene.

For example, suppose that this procedure is used to identify cells found at a crime scene. Samples A and B (above) came from the scene of the crime and samples C and D came from two different suspects. What can you conclude?

Some genetic diseases that can be identified using this procedure are Sickle Cell disease, Huntington’s disease, Duchenne muscular dystrophy.

Taxonomists can use this technique to explore evolutionary relationships. Individuals of the same species, while not identical, will be more similar than individuals of different species.

The procedure for sequencing and mapping DNA requires RFLP analysis.

### Gene Therapy

Gene therapy uses technology to change the genetic composition of a cell.

#### Ex vivo

Ex vivo methods are done outside the organism. Cells are removed, treated and returned to the individual.

Retroviruses are often used as the vector. The retroviruses contain recombinant RNA which includes the gene to be added. Once in the cell, the enzyme reverse transcriptase makes a DNA copy of the RNA.

Currently, there are more than 100 clinical trials of this technique.

##### Example of ex vivo gene therapy

This procedure has been used to treat severe combined immunodeficiency syndrome (SCID). People with this disease are susceptible to infections because their white blood cells do not produce an enzyme needed by their immune systems. This disease has been treated in two different ways. In a short-term solution, the white blood cells were removed and infected with a retrovirus that carried the needed gene. After the cells were replaced, many of the cells contained the gene. White blood cells, however, are short-lived and a long-term solution is to apply this technique to the cells that produce the white blood cells (called stem cells).

#### In vivo

In vivo gene therapy treats cells in the individual without removing them.

Retroviruses can be used to introduce genes directly into the body.

Synthetic carriers like liposomes can also be used to carry genes. Liposomes are microscopic lipid vesicles that are readily taken up by cells. If they are coated with DNA, the DNA is also taken up.

#### Transgenic Bacteria

##### Protein products

Genes that code for the desired protein can be inserted into plasmids and the plasmids are used to transform cells. Many useful human proteins are now synthesized by transgenic bacteria. Some of these are listed below.

Human growth hormone is used to treat dwarfism. It previously took the pituitary glands from over 50 cadavers to make one dose.

Human Insulin is used to treat diabetes.  Insulin was previously obtained from the pancreas of slaughtered cattle and pigs. It sometimes caused allergic reactions.

Tissue plasminogen activator dissolves blood clots in heart attack victims.

Clotting factor VIII is one of several factors needed for blood to clot. Most cases of hemophilia are due to the absence of this factor.

Human lung surfactant is used in premature infants with respiratory distress syndrome.

Atrial natriuretic hormone can be used to treat hypertension.

Bovine growth hormone (bGH) increases milk production in cows by about 10%.

##### Vaccines

Vaccines were previously made by killing or weakening a virus or bacteria and then injecting it. Its surface proteins caused an immune reaction. Occasionally, these vaccines would make people ill. Using biotechnology, some of the proteins of the pathogen can be made by cloning the gene that codes for them. Vaccines containing the protein component of pathogens are called subunit vaccines. They are sufficient to stimulate the immune system but are incapable of causing an infection.

Vaccines for hepatitis B and human papilloma virus (Gardasil) are now produced using biotechnology.

Hundreds of vaccines are currently are being developed.

Vaccines for hoof-and-mouth disease and scours (a form of dysentery) have been developed for farm animals.

##### Other Uses of Recombinant Bacteria

Bacteria have been produced that inhibit the formation of ice crystals. These bacteria have been released onto crop plants to protect them from frost damage.

A bacteria species that normally colonize corn roots have been given a gene that enables it to produce an insect-killing toxin (pesticide).

Bacteria are being developed that do a better job at breaking down oil. These may be useful to help clean up oil spills.

Bacteria have been developed that are capable of removing some kinds of toxins from the air and water.

Bacteria have been engineered to extract metals from low-grade ore (bioleaching). This technique is currently being tested.

#### Transgenic plants

Bt-toxin is an insecticide naturally produced by the bacterium Bacillus thuringiensis. This toxin is toxic only to insects because it must bind to receptors in their gut to function. Several different kinds of crop plants have been genetically engineered to produce this toxin.

Agricultural plants have been developed that are resistant to the effects of the herbicide RoundupTM. This herbicide kills non-resistant plants and is biodegradable.

Presently, there are approximately 50 types of genetically engineered plants that resist insects, viruses, and herbicides.

Rice has been engineered to produce B-carotene and increased levels of iron.

A weed called mouse-eared cress has been designed to produce a biodegradable plastic called polyhydroxubutrate (PHB).

##### Possibilities for the Future

In the future, biotechnology may be able to improve crop yields and produce plants that contain all of the amino acids required for human consumption.

If plants could be produced that can fix atmospheric nitrogen, they would require considerably less fertilizer.

Plants engineered to grow under harsh environmental conditions could allow wastelands to be more productive.

#### Transgenic Animals

Pharmaceutical companies are developing techniques to produce chemicals using animals. The drug is produced in the milk of females. For example, goats have been developed to produce antithrombin III, used to prevent blood clots. Clinical trials of this drug will begin soon.

Lactoferrin is added to infant formula to transport iron and to prevent bacterial infections in the gastrointestinal tract. A transgenic bull has been produced that carries a gene for the production of human lactoferrin. Females will produce milk with lactoferrin.

A sheep has been bioengineered to produce tPA (tissue plasminogen activator) in her milk. Tisue plasminogen activator is a "clot-busting" drug that is often administered to people who have suffered a heart attack or stroke.

A pig has been produced that can produce human hemoglobin. Artificial blood may soon be a reality.

#### Cloning Mammals

Cloning animals refers to producing offspring that are genetically identical to the animal being cloned. This process has been done by removing the nucleus of an egg and replacing it with a diploid nucleus from the organism to be cloned. The egg is then treated so that it begins dividing. It is placed in the uterus of a host animal where it continues to grow.

### The Human Genome Project

The Human Genome Project is a massive, government-funded project whose goal is to determine the base sequence of all of the human chromosomes.

It would take 200 volumes of 1,000 pages each just to list the letters of the bases.

Knowledge of the sequences of all genes making it easier to study, diagnose, and treat many kinds of human genetic diseases. Information on gene sequences is important in the development of gene therapy techniques.

The information is expected to lead to a better understanding of genetic systems, and ultimately answers to mysteries surrounding such topics as gene regulation and cancer.

### Safety and Ethical Issues

Harmful organisms may be accidentally produced. For example a genetically-engineered organism may be pathogenic or capable of causing ecological destruction.

To minimize this risk, most laboratories work only with microorganisms that have been modified so that they cannot survive in nature in case they are accidentally released. Organisms that are intended to be released into the environment may be engineered with genes that will eventually kill them.

Ethical questions have been raised over whether we should modify the genes of humans. Treating a person with a severe disease may seem to be appropriate use but it must be decided what a "severe disease" is. Treating a person's somatic (body) cells alters only that person but treating the person's germ (sex) cells alters future generations.

There is little legislation on the use of genetic screening and information produced by screening. There are presently few guidelines in place that protect an individual's privacy and limit access to screening information. For example, potential employers or insurance companies might find screening information useful and require employees or clients to undergo screening.

The technology is increasing the ability to diagnose genetic diseases prenatally, adding new complexity to the abortion controversy.

Genetic screening and gene therapy are expensive and may be unavailable to the poor.

Biological weapons could be created using biotechnology.


text:
# Make an hyperref link to current chapter/section

I would like a way to make an hyperref link to the beginning of the current section or chapter, that has the same effect of using \label{section} and \ref{section} with hyperref on, but without having to use \label.

Here is an MWE:

\documentclass[10pt,a5paper]{report}
\usepackage{fontspec}
\usepackage{lipsum}

\begin{document}
\chapter{Chapter 1}
\lipsum[1-2]

\section{Section 1}
\lipsum[3-5]

\section{Section 2}
\lipsum[6]
\lipsum[6]
\end{document}

• What is the current section and what is wrong with \label? – user31729 Sep 29 '15 at 12:47
• The "current" section is the section where the link is. Since I would use this function quite a lot, it would be nicer not to have to label them each time. – HcN Sep 29 '15 at 12:51
• Anyway, this means auto-labelling somewhat. How do you want to refer to the current section. The current section 'moves' through the document when being viewed. – user31729 Sep 29 '15 at 12:53
• See the edited question if it is clearer. – HcN Sep 29 '15 at 12:58
• The table of contents already has links to all the chapters etc. The names can be found in the aux file. – John Kormylo Sep 29 '15 at 13:46

Something like this?

\documentclass[10pt,a5paper]{report}
\usepackage{xcolor}
\usepackage{fontspec}
\usepackage{lipsum}

\begin{document}
\chapter{Chapter 1}\hypertarget{chapter::\theHchapter}{}%
\lipsum[1-2]

\section{Section 1}
\lipsum[3-5]

\section{Section 2}\hypertarget{section::\theHsection}{}%
\lipsum[6]
\lipsum[6]

\chapter{Chapter 1}\hypertarget{chapter::\theHchapter}{}%
\lipsum[1-2]

\section{Section 1}
\lipsum[3-5]

\section{Section 2}\hypertarget{section::\theHsection}{}%
\lipsum[6]

• Perfect. \hyperlink{section.\arabic{chapter}.\arabic{section}}{Link to current section.} does exactly what I want. Many thanks to you two. – HcN Sep 29 '15 at 14:09

format: plain


text:
## Elementary Algebra

Published by Cengage Learning

# Chapter 1 - Some Basic Concepts of Arithmetic and Algebra - 1.3 - Integers: Addition and Subtraction - Problem Set 1.3 - Page 19: 6

#### Answer

$-11$

#### Work Step by Step

We have -5 on the number line, and we have to add -6. So, on the line we have to move six units to the left, and we arrive at -11

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Divide by three, multiply by two

time limit per test

1 second

memory limit per test

256 megabytes

input

standard input

output

standard output

Polycarp likes to play with numbers. He takes some integer number

, writes it down on the board, and then performs with it

operations of the two kinds:

• divide the number

by ( must be divisible by

• );
• multiply the number

by

• .

After each operation, Polycarp writes down the result on the board and replaces

by the result. So there will be

numbers on the board after all.

You are given a sequence of length

— the numbers that Polycarp wrote down. This sequence is given in arbitrary order, i.e. the order of the sequence can mismatch the order of the numbers written on the board.

Your problem is to rearrange (reorder) elements of this sequence in such a way that it can match possible Polycarp’s game in the order of the numbers written on the board. I.e. each next number will be exactly two times of the previous number or exactly one third of previous number.

It is guaranteed that the answer exists.

Input

The first line of the input contatins an integer number

() — the number of the elements in the sequence. The second line of the input contains integer numbers (

) — rearranged (reordered) sequence that Polycarp can wrote down on the board.

Output

Print

integer numbers — rearranged (reordered) input sequence that can be the sequence that Polycarp could write down on the board.

It is guaranteed that the answer exists.

Examples

Input

Copy
6
4 8 6 3 12 9

Output

Copy
9 3 6 12 4 8

Input

Copy
4
42 28 84 126

Output

Copy
126 42 84 28

Input

Copy
2
1000000000000000000 3000000000000000000

Output

Copy
3000000000000000000 1000000000000000000
Note

In the first example the given sequence can be rearranged in the following way:

. It can match possible Polycarp’s game which started with .

#include <iostream>
#include <bits/stdc++.h>
using namespace std;
long long int h_3(long long int a)//3的个数
{
long long int cnt=0;
while(a%3==0)
{
cnt++;
a=a/3;
}
return cnt;
}
bool cmp(long long int a,long long int b)
{
long long aa,bb;
aa=h_3(a);
bb=h_3(b);
if(aa==bb)
{
return a<b;
}
else return aa>bb;
}
int main()
{
int n;
long long int a[110];
cin>>n;
for(int i=0;i<n;i++)
{
cin>>a[i];
}
sort(a,a+n,cmp);
for(int i=0;i<n;i++)
{
if(!i)cout<<a[i];
else
{
cout<<" "<<a[i];
}
}
cout<<endl;
return 0;
}

#include <iostream>
#include <bits/stdc++.h>
using namespace std;

int main()
{
long long int n,a[110],book[110];
cin>>n;
memset(book,0,sizeof(book));
for(int i=0;i<n;i++)
{
cin>>a[i];
}
long long int d,f;
for(int i=0;i<n;i++)
{
int f=0;
for(int j=0;j<n;j++)
{
if(j==i)continue;
if(a[j]*2==a[i])
{
f=1;
break;
}
if(a[j]%3==0)
{
if(a[j]/3==a[i])
{
f=1;
break;
}
}
}
if(f==0)
{
d=i;
break;
}
}
cout<<a[d];
book[d]=1;
for(int i=1;i<n;i++)
{
for(int j=0;j<n;j++)
{
if(book[j])continue;
if(a[d]*2==a[j]||(a[d]%3==0&&a[d]/3==a[j]))
{
book[j]=1;
d=j;
cout<<" "<<a[j];
break;
}
}
}
cout<<endl;
return 0;
}

#include<bits/stdc++.h>
using namespace std;
long long int a[105],vis[105],f[105],flag=0,top,n;
int j(long long int x,long long int y)
{
if(x*2==y)return 1;
else if(x%3==0&&x/3==y)return 1;
else return 0;
}
void dfs(int x)
{
if(top==n-1&&!flag)
{
flag=1;
for(int i=0;i<=n-1;i++)
{
if(i==n-1)cout<<f[i]<<endl;
else cout<<f[i]<<" ";
}
return;
}
for(int i=1;i<=n;i++)
{
if(!vis[i]&&j(a[x],a[i])==1)
{
top++;
f[top]=a[i];
vis[i]=1;
dfs(i);
top--;
vis[i]=0;
}
}
}
int main()
{
cin>>n;
for(int i=1;i<=n;i++)
{
cin>>a[i];
}
for(int i=1;i<=n;i++)
{
memset(vis,0,sizeof(vis));
vis[i]=1;
top=-1;
top++;
f[top]=a[i];
dfs(i);
if(flag)break;
}
return 0;
}


text:
# Math Help - limit and graph

1. ## limit and graph

1>
Let f(x)=|x-1|+x-1
(a) sketch the graph of f(x)
(b) find lim x-> -2f(x)
(c) find lim x->2 f(x)

2> Find a cutoff distance δ such that if the distance from x to 1 is less than δ and x>1, then f(x) =3x-1. approximates 2 with an error of at most 1/100

2. Originally Posted by bobby77
1>
Let f(x)=|x-1|+x-1
(a) sketch the graph of f(x)
(b) find lim x-> -2f(x)
(c) find lim x->2 f(x)
The curve is below.
Note if your graphing software does not have the absolute value function you can graph y=sqrt(x^2) that is a nice trick to replace it.

To find the limits is very easy.
Note that,
f1=|x| is continous everywhere,
f2=x-1 is continous everywhere,
Thus,
f1 o f2=|x-1| is continous everywhere, (by composition rule).
Thus,
(f1 o f2)+f2=|x-1|+x-1 is continus everywhere, (by addition rule of continous functions).
Thus,
to find the limits of f=|x-1|+x-1
All we need to is evaluate the function at those points.

3. Originally Posted by bobby77

2> Find a cutoff distance δ such that if the distance from x to 1 is less than δ and x>1, then f(x) =3x-1. approximates 2 with an error of at most 1/100
Put x=1+x. x>0.

Then f(x)=2+3u, and if the error here is <=1/100 ,we have

f(x)-2<=1/100,

or:

3u<=1/100,

u<=1/300

So if 0<(x-1)<δ,and 0<=δ<=1/300 then f(x) =3x-1. approximates 2 with an error
of at most 1/100.

RonL


text:
#### Category : laravel-6

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text:
Sparser Bipartite graphs?

Maximal Planar Bipartite graphs are sparser than maximal planar graphs. For which other classes of graphs are maximal Bipartite members sparser than arbitrary maximal members.

Let $\mathcal{C}$ be a class of graphs and let $\mathcal{B}$ be the class of bipartite graphs. We say that $\mathcal{C}$ is biparted if $\rho(\mathcal{C}) > \rho(\mathcal{B}\cap\mathcal{C})$ (where $\rho(\mathcal{C})$ is defined below) i.e. the asymptotic density of arbitrary graphs in the class is larger than the asymptotic density of the bipartite graphs in the same class.

Examples of biparted graphs are:

• Planar Graphs ($\rho(\mathcal{C}) = 6 > 4 = \rho(\mathcal{C}\cap \mathcal{B})$)
• Graphs of finite genus (same parameters as above)
• Subclasses of planar graphs e.g. outerplanar graphs

My question is:

Are there any other natural classes of biparted graphs?

For a graph $G$, let $\rho(G) = \frac{2|E(G)|}{|V(G)|}$ be the average degree of a vertex. We say that a class of graphs $\mathcal{C}$ is maximally $\rho$-sparse - written as $\rho = \rho(\mathcal{C})$ i.e. the maximum of average density (with maximum taken over all graphs from $\mathcal{C}$ with sufficiently many vertices) approaches $\rho$.

Note: Planar graphs can be proved to be biparted because Euler formula holds. The same is the case with bounded genus graphs. What about bounded tree-width graphs, graphs excluding a finite set of fixed minors,...?

For an extreme example, chordal graphs can have as many as $\binom{n}{2}$ edges but chordal graphs that happen to also be bipartite can have only $n-1$ edges (they are forests). Or even more extremely, consider complete graphs versus (complete $\cap$ bipartite) graphs. But perhaps it makes sense to restrict your problem only to classes of graphs that are naturally sparse, as the planar and minor-closed graph families are.
It is not true that all minor-closed families are biparted; in particular, the graphs of treewidth $k$ are not. An $n$-vertex graph with treewidth $k$ has at most $\binom{k}{2}+k(n-k)$ edges, but (for large $n$ and small $k$) the complete bipartite graph $K_{n-k,k}$ has treewidth $k$ and is almost as dense with $k(n-k)$ edges.
One can extend the bounds from planar graphs to $k$-apex graphs (graphs such that you can delete $k$ vertices and get a planar graph), and their generalization to $(g,k)$-apex graphs for bounded genus $g$: these graphs have at most $(k+3)n+O(g)$ edges but their bipartite subclasses have at most $(k+2)n+O(g)$.
Moving beyond minor-closed properties but continuing with graphs related to planar graphs, the 1-planar graphs (graphs that can be drawn with at most one crossing per edge) can have at most $4n-8$ edges, but bipartite 1-planar graphs can have at most $3n-6$. See Czap and Hudák, "On drawings and decompositions of 1-planar graphs", Elect. J. Comb. 2013. And the graphs with book thickness $k$ have at most $(k+1)n-O(1)$ edges ($2n-3$ for the first page and $n-3$ for each subsequent page), but when they are bipartite the constant factor is lower because the first page is outerplanar. I suspect that similar things happen with other types of almost-planar graph.
• Sorry for not being clear. You are correct that chordal Bipartite graphs are not a subclass of Chordal graphs. My question was if we slightly modify the definition of biparted to compare Chordal and Chordal Bipartite densities then will there be a difference. But here again (at least for bounded tree width) the question does not have a positive answer, as the case of $K_{n-k,k}$ shows. So you were correct on all fronts. – SamiD May 23 '15 at 22:05


text:
DEPRECATION WARNING

This documentation is not using the current rendering mechanism and will be deleted by December 31st, 2020. The extension maintainer should switch to the new system. Details on how to use the rendering mechanism can be found here.

# Batch processing images¶

This extension provides a scheduler task to batch process uploaded images in the directories you normally watch for image upload (see General settings).

It is particularly useful if you let users upload images outside of TYPO3 (e.g., using FTP), thus bypassing upload post-processing to automatically resize them according to your rule sets (see figure Configuration form of a scheduler task to batch resize images).

Important

As the scheduler task will process each directory found in your rule sets, make sure to exclude directories you normally do not show to your editors (e.g., fileadmin/templates) which may contain large image assets that should never be processed by this extension.


text:
## anonymous 5 years ago 1/3x^ = 2x+1

1. anonymous

What's your x (on the left-hand side) to the power of?

2. anonymous

2

3. anonymous

You need to remove any x's you have from the denominator. So, multiply everything through by 3x^2 (I said 3x^2 instead of just x^2 because we'd end up multiplying through by 3 anyway - may as well get it over and done with). So,

4. anonymous

$\frac{1}{3x^2}=2x+1 \rightarrow 1=3x^2(2x+1)=6x^3+3x^2$

5. anonymous

he may have meant (1/3)x^2...

6. anonymous

Is that what you meant, mac?

7. anonymous

that would sound more likely to be the homework assigned to an algebra class

8. anonymous

Because solving 6x^3+3x^2=1 algebraically is not nice.

9. anonymous

^not at all...

10. anonymous

$\frac{1}{3}x^2=2x+1 \rightarrow x^2=6x+3$ after multiplying through by 3. You have a quadratic to solve:$x^2-6x-3=0$

11. anonymous

ergh... that still wont give integer roots

12. anonymous

You can complete the square or use the quadratic formula.

13. anonymous

Can't help that...

14. anonymous

$x^2-6x-3=0 \rightarrow x=3 \pm 2\sqrt{3}$

15. anonymous

as for the 6xcubed i would use long division... or rather synthetic... is that the best way?

16. anonymous

Do you mean you would use long division or synthetic after finding a factor?

17. anonymous

i would use them to check if i can guess a factor lol

18. anonymous

Yes

19. anonymous

i wonder where mac went...

20. anonymous

The method for solving cubics is a pain, so I usually try to find one solution and, like you said, use long/synthetic division on the cubic and go from there. Else, numerical solution, like Newton-Raphson method.

21. anonymous

Mac's lurking...

22. anonymous

newton-raphson method? whats that?

23. anonymous

It's a method for finding solutions to equations when you can't do so algebraically (or can't be bothered). The only condition is that the function you're looking at has to be continuous (because you use calculus).

24. anonymous

im curious, as the derivative cant tell you solutions... can it??

25. anonymous

You're right, the derivative itself won't give you the solutions (unless you have a multiple root), but the method takes your guess, then uses calculus to find the equation of the tangent at the function value of your guess for x, then seeks out where the tangent cuts the x-axis. That point - where it cuts the x-axis - is closer to the root than your initial guess (usually...there are cases where it won't work). That new x-value is your new 'guess' which goes through the process again to yield a better 'guess' for x...you can keep going until you get to a level of accuracy you want.

26. anonymous

... so this will only work with f(x) with a degree 3 or more so that f'(x) is at least quadratic... and you find the tangent, then find the zero of that equation, and then use that in the original and find the tangent and use the zero of that equation and so on?

27. anonymous

The method will will work with any function that has at least one derivative. I'll try and find some information for you. It's a very useful tool to have, mainly because you end up having at your disposal a means to solve many practical problems that either don't have algebraic solutions, or where the solutions are a pain to get at and you don't need your answer to be exact.

28. anonymous

but then you will never get the exact zero, you will only get closer and closer to it? as the zero of a tangent line can never be equal to the value put into the original function for that tangent line unless you have a multiple root case ( as you said above)

29. anonymous

Check out the image - this is what's going on. http://en.wikipedia.org/wiki/Newton's_method

30. anonymous

thank you man

31. anonymous

You're welcome ;)

32. anonymous

what the heckkk that is freaking genius. damn newton -.-

33. anonymous

Yeah, I think it's his best thing outside of the calculus and Laws of Motion.

34. anonymous

when it comes to physics laws of that sort, id always put kepler over newton... idk why newton got all the fame for those, keplers were way better -.-

35. anonymous

Try it with something like finding the square root of 2 - it will find it pretty damn quick. $x=\sqrt{2} \rightarrow f(x)=x^2-2=0 \rightarrow f'(x)=2x$Then$x_{n+1}=x_n-\frac{f(x)}{f'(x)} \rightarrow x_{n+1}=x_n-\frac{x^2_n-2}{2x_n}$

36. anonymous

i didnt understand the second half of that lol

37. anonymous

I'm going to do an example.

38. anonymous

You have to find something to start with, so we make an educated guess for sqrt(2). Since $1<2<4 \rightarrow \sqrt{1}<\sqrt{2}<\sqrt{4} \iff 1 <\sqrt{2}<2$ as sqrt function is monotonic increasing (i.e. the direction of the inequality is preserved). So we have for an initial guess for x,$x \approx \frac{3}{2}$Then, a better 'guess' for x will be,$x_2 = \frac{3}{2}-\frac{\left( \frac{3}{2} \right)^2-2}{2\frac{3}{2}} = \frac{17}{12} \approx 1.416667$and the square root of 2 is about $1.414213562$

39. anonymous

You can see after one trial, it's pretty close.

40. anonymous

The closer you are to your answer in the first place, the faster it will converge. You could then plug 17/12 into the formula to get a better approximation, and continue.

41. anonymous

Next one would be 577/408 ~= 1.414215686

42. anonymous

dang, the formula, has you divide the tangent line function by the tangent line slope, why?

43. anonymous

Well, going back to the sqrt of 2, when you start out, you don't know what the sqrt of 2 is, but you know it should be something, so call that something, x. Then x =sqrt(2). If I square both sides and subtract 2 from both sides, I get x^2-2=0 So, finding x = sqrt(2) amounts to finding where on the x-axis x^2-2 will cut it. So I plot my parabola, and imagine the setup that's in that Wikipedia pic. Since the method finds where the graph cuts the x-axis, it will find that x where x^2-2=0; that is, the x that came from x=sqrt(2).

44. anonymous

yeah i get that, but i dont get how subtracting the tangent line formula divided by the slope of the tangent gets you the intersection of the tangent line?

45. anonymous

The reason for why we divide the function by the derivative is due to the derivation. The derivation can be done in either a geometrical setting (using that picture, say) or with Taylor series expansion on an arbitrary function. I'm sure proofs would be online. It's a pretty old theorem ;)

46. anonymous

oh wait i get it...

47. anonymous

You know, you can derive it yourself by analyzing each step in that pic. and doing what the mathematics would dictate in that situation.

48. anonymous

your taking y=mx+b and doing 0=mx+b so mx=-b so x=-b/m and you take that, which is the distance from the x value to the zero, and subtract it from your original point which will show you where on the x axis it is...

49. anonymous

nope that made no sense... ill figure this out, gimmy a sec

50. anonymous

You're half there.

51. anonymous

It's difficult to explain online. They need to put a drawing feature in this thing.

52. anonymous

lol they do, like tutorvista

53. anonymous

I think mac must think his thread's been hijacked.

54. anonymous

i think he left

55. anonymous

so youre taking x and subtracting y/y'(x)

56. anonymous

i still dont get why ><

57. anonymous

Hang on, I'll see if I can put a small proof together.

58. anonymous

how is the y value divided by the slope at that point, equal to the difference btween x and the intersection of the tangent line of x

59. anonymous

no you dont need to make a proof of it i just need to understand how y/y' is the difference between x and the zero of y'

60. anonymous

Wait, if I prove it using the geometrical route, you'll see...I'm just going to see if it's feasible (versus finding something online).

61. anonymous

ok :/

62. anonymous

Alright, writing it up isn't hard, but you'll see what's going on if you do a drawing like the one seen in Wikipedia.

63. anonymous

And go along with it...

64. anonymous

i can understand how it all works, but how does the drawing explain how the formula works?

65. anonymous

66. anonymous

Suppose x_1 is close to a root of the equation f(x)=0. Then the equation of the tangent at x=x_1 is$y-y_1=m(x-x_1)$that is,$y-f(x_1)=f'(x_1)(x-x_1)$This tangent crosses the x-axis at a point where x = x_2 and y = 0. Hence,$0-f(x_1)=f'(x_1)(x_2-x_1) \rightarrow x_2=x_1-\frac{f(x_1)}{f'(x_1)}$

67. anonymous

Then repeat...

68. anonymous

oh wow that makes sense...

69. anonymous

Happy now?

70. anonymous

yes :D thanks for spending time to teach me this stuff :O

71. anonymous

No worries - it's good you want to know :)

72. anonymous

its incredible that you can put up with teaching me like that, even alot of teachers would loose it pretty soon

73. anonymous

i honestly am really curious as to where mac went...

74. anonymous

he stopped observing a while ago even

75. anonymous

You learnt quickly - it wasn't a problem. You would have learnt quicker if the interface was easier (but it's not the designers' fault - where's the holography already?).

76. anonymous

Yeah, he probably got his answer and was all, "WTF is going on?!"

77. anonymous

whats after differential equations?

78. anonymous

Eh?

79. anonymous

like course wise, you take calc1 calc2 calc3 diff eq then what?

80. anonymous

Have no idea man, not American ;D

81. anonymous

but you studied in europe, so the course order must be nearly identicle, especially in advanced mathematics, so instead i should probably ask, what did you do (courses) for you degree and btw, you have a masters? phD?

82. anonymous

83. anonymous

Partial differential equations is usually end of the line for calculus. Then there's other stuff like Lebesgue integration (overtakes Riemann integral), and other stuff in algebra and geometry - the further you go, the more the areas start to link up. E.g. differential geometry, algebraic topology, etc. But for calculus, numerical and algebraic methods for solving partial differential equations is usually end of line. If you know how to solve (methods to solve) nth order, non-linear, inhomogeneous partial differential equations with non-constant coefficients, you're done!

84. anonymous

Since i did not understand any of nth order, non-linear, inhomogeneous partial differential equations with non-constant coefficients i guess im not done :D

85. anonymous

lol, keep going, though!

86. anonymous

do you know all that is taught of mathematics to the extent that research is your only option?

87. anonymous

No, you can do whatever you want with it. I teach as well. You can use it in industry and government, as in modelling for things...haven't really sold it, have I?

88. anonymous

well not in that sense, i meant like if you wanted to know more on math, your only option would be to study unsolved theorems or unproved research or do research on your own

89. anonymous

anddd man i need help on something: $\int\limits_{}^{} (\sin x - \cos x) \div (\sin x)$

90. anonymous

nevermind i got it

91. anonymous

Yes, you're right on your last point.

92. anonymous

thats pretty crazy, that you know everything O.o

93. anonymous

I don't know everything..!

94. anonymous

you know everything in math...

95. anonymous

No I don't ;)


text:
1. Jul 23, 2004

### Ian

If the suppsed Hawking Radiation truly exists then since all matter was supposedly concentrated into a singularity in the time before the big bang (if there was one); then this single black hole must have evaporated away according to the rules that govern Hawking Radiation.

To me this seems to say that theoretically there cannot have been both a big bang and Hawking Radiation - one does not exist, but which one?
There is a horrifying (if not hilarious) problem here: If Professor Hawking's math is correct (the guy is a brilliant with numbers) then much of 20th century theory must curl up and die!

Besides, what happens when a black hole evaporates through HR down to the mass of a proton? We have never observed proton decay, so why should a singularity of same mass as a proton decay by HR?

2. Jul 23, 2004

### Phobos

Staff Emeritus
A black hole singularity is different than the Big Bang singularity (an object embedded in spacetime vs. spacetime itself).

3. Jul 28, 2004

### Ian

Phobos,
Why is there a difference? Matter is matter, and space-time is space-time, the interface between the two must always remain constant.
To present a difference between a black hole and the big bang singularity is a theoretical dodge.

4. Jul 28, 2004

### devilmech

There is no theoretical dodge. The singularity of a black hole is fundamentally different from the big bang singularity in that the singularity of a black hole exists in space-time, which gives it a reference frame, and thus can be observed in theory. The big bang singularity had no reference frame, as the concept of space-time is a result of the big bang itself, and therefore there is no observable reference frame for the big bang singularity. You can effectively view a black hole as as a vector $$\vec{y}$$ with it's endpoint at some non-zero point on a vector $$\vec{x}$$, with $$\vec{x}$$ being the big bang singularity and having it's endpoint at the origin, or zero.

5. Jul 29, 2004

### Ian

If the big bang singularity had no reference frame, it simply cannot be observed in theory - i.e. it can never have happened.
You can't have it both ways!

If theory extrapolates backward to a big bang singularity of matter, it must also extrapolate backward to a 'space-time singularity' in conjunction with the matter singularity.

6. Jul 29, 2004

### turin

I cannot observe the birth of my parents. So they were never born?

7. Jul 29, 2004

### Olias

If one rewinds our Galaxy,(without reference to other far away current Galaxies), then there comes a defined Horizon, wherby there is extreme particle creation. This is centered around the Blackhole that we deemed to be at our Galactic core, in a time-reversed manner.

The compaction of our Galaxy to such a phase of Virtual pair creation, has in this sense a 'perfect' background for particle creation, which is on the scale of a tear in the fabric of space, (not to be confused with a tear in spacetime) the Big-Bang is the moment of our Galaxy Creation, the singularity centered at our 'current' Milky Way, has its own seperate Spacetime, far far away from other Galaxies.

If one rewinds Einstein Field Equations, then each Galaxy has seperate spacetimes, only Galaxies that can be rewound and actually collide at the same instant (within the same time reversed horizon of EM Vacuum space) would share a percieved same "Big-Bang" Universal Instant.

If one Rewinds all of the actual Universe, which is supposed to be current accepted version of events, then one cannot explain how some Glaxies will not merge into the same 'singularity' at the same given instant, the moment of Universe Creation.

Just look at Andromeda, it is currently Heading towards us, if one could stop it in its tracks along with our Milkyway, then rewind both Galaxies in a Time-Reversed fashion, their proximity to each other will actually seperate Exponentially, the farther back in time you rewind the Universe, the further Andromeda and the Milkyway will Seperate. How can their creation be at the same instant if they are moving away from each other in a time-reversal scenario?

If every single galaxy in the observable universe were actually moving away from each other, then I would conceed that any Time-reversal of the Big-Bang would infact produce a single instant of all Galaxies, and all Galaxies would share the same birthdates!

8. Jul 30, 2004

### marcus

Ian this doesnt make sense and appears to be based on a misconception about what a "singularity" is, misinformation about what cosmologists assume.

You say you live in the UK. Probably you pay taxes or members of your family do. Some of that tax money goes to support research in cosmology.
If you care at all about cosmology issues, why dont you find out what's going on?

A singularity is not something in nature. It is something that occurs in a human-built theory. It's just a point or region where the theory breaks down.
The two famous kinds of singularity are failures in General Relativity (the classical 1915 GR fails at the big bang and at black holes) and there is an on-going effort to fix these failures and get rid of the singularities in GR.
Other theories can have singularities too but the famous ones nowadays are flaws in GR.

Cosmologists do not assume that the BB began with a singularity. How could they, when such a thing does not happen in nature? It is just a flaw in theory that they are in process of eliminating. The current picture is more of a changeover from contraction to expansion----a reversal----the word "bounce" has been used----it is risky to use metaphors like "bounce" or "bang" but media and the journalistic imagination seem to demand some catchy metaphor. I prefer the plainer word "reversal" for the crossover from contraction to expansion believed to have happened at the time where there used to be a singularity in the theory.

---------
when you buy a shirt at the store it can have various kinds of defects
a seam might not have been sewn
a button may be missing etc.
singularities are flaws and there are different types
In Gen Rel the bb singularity is different kind from the bh one. Just because there is a BB singularity at the time of BB does not mean that this is the same thing as a BH! So one would not expect Hawking radiation! That is totally illogical.

You seemed to be arguing that way in your post. Like you buy the shirt and you find that a seam was not sewn and a sleeve is coming loose---so you conclude that therefore button must be missing.

It does not follow. (there might be a connection between the two types of singularity but we dont know that yet)

In any case remember that the flaws are in the theory, not nature, so we cant draw much in the way of conclusions from them until the experts fix the flaws and check the corrected theory out by observations or experiment.


text:
Back in January, I was up at my cabin in Silverton hoping to help my dad out with a temperature monitoring system – his method was to use a video camera to continuously monitor a built in temperature gage, and he would check the images every so often to track the temperature. Obviously, this kind of system is bulky and problematic because he has to check the image at a specific time to see if the temperature has crossed an alarming threshold.

Back in the winter, I wanted to set up a temperature system using wifi so that he could get alerts if the temperature went below a specific threshold, so he didn’t have to worry about continuously checking. I used an ESP32 Thing Plus to collect basic temperature data and built an app with Blynk to display it.

What if I wanted to view the data on a display inside the cabin, or maybe a low power processor that would only wake when the temperature threshold has been hit? It seemed to me that this project was actually better suited for MicroMod because it requires versatility. So, this project turned into my progression in building a temperature system with MicroMod, and seeing all the different ways I can expand on the project with endless flexibility!

Step one: actually collecting data!

The data logger carrier board is an obvious choice for this project becasue it’s low power and easy to connect peripheral sensor boards through Qwiic. And the SAMD51 is also an easy to use and economical processor to run the project. As is with all MicroMod projects, all that’s needed is to screw in the M.2 connector on the processor to the carrier, and that’s about all that’s needed for hardware assembly! On top of that, I’ve added a temperature sensor through Qwiic.

DEV-16829

DEV-16791

DEV-16401

DEV-16985


text:
WPILibC++  2020.3.2-60-g3011ebe
frc::TrajectoryConstraint Class Referenceabstract

An interface for defining user-defined velocity and acceleration constraints while generating trajectories. More...

#include <TrajectoryConstraint.h>

Inheritance diagram for frc::TrajectoryConstraint:

## Classes

struct  MinMax
Represents a minimum and maximum acceleration. More...

## Public Member Functions

TrajectoryConstraint (const TrajectoryConstraint &)=default

TrajectoryConstraintoperator= (const TrajectoryConstraint &)=default

TrajectoryConstraint (TrajectoryConstraint &&)=default

TrajectoryConstraintoperator= (TrajectoryConstraint &&)=default

virtual units::meters_per_second_t MaxVelocity (const Pose2d &pose, units::curvature_t curvature, units::meters_per_second_t velocity) const =0
Returns the max velocity given the current pose and curvature. More...

virtual MinMax MinMaxAcceleration (const Pose2d &pose, units::curvature_t curvature, units::meters_per_second_t speed) const =0
Returns the minimum and maximum allowable acceleration for the trajectory given pose, curvature, and speed. More...

## Detailed Description

An interface for defining user-defined velocity and acceleration constraints while generating trajectories.

## ◆ MaxVelocity()

 virtual units::meters_per_second_t frc::TrajectoryConstraint::MaxVelocity ( const Pose2d & pose, units::curvature_t curvature, units::meters_per_second_t velocity ) const
pure virtual

Returns the max velocity given the current pose and curvature.

Parameters
 pose The pose at the current point in the trajectory. curvature The curvature at the current point in the trajectory. velocity The velocity at the current point in the trajectory before constraints are applied.
Returns
The absolute maximum velocity.

## ◆ MinMaxAcceleration()

 virtual MinMax frc::TrajectoryConstraint::MinMaxAcceleration ( const Pose2d & pose, units::curvature_t curvature, units::meters_per_second_t speed ) const
pure virtual

Returns the minimum and maximum allowable acceleration for the trajectory given pose, curvature, and speed.

Parameters
 pose The pose at the current point in the trajectory. curvature The curvature at the current point in the trajectory. speed The speed at the current point in the trajectory.
Returns
The min and max acceleration bounds.

The documentation for this class was generated from the following file:


text:
1. ## Orthonormal basis

Let $\textbf{u}_1$ and $\textbf{u}_2$ form an orthonormal basis for $R^2$ and let $\textbf{u}$ be a unit vector in $R^2$. If $\textbf{u}^T\textbf{u}_1=1/2$, determine the value of $|\textbf{u}^T\textbf{u}_2|$

Can someone help me begin this problem, thank you!

2. Note that: $
u^t \cdot u_2 = \underbrace {\left\| {u^t } \right\| \cdot \left\| {u_2 } \right\|}_{ = 1} \cdot \cos \left( {\measuredangle u^t u_2 } \right)
$
(all of our vectors are unit vectors)

$\measuredangle u^t u_2$ is one of the angles between the vectors

$
u^t \cdot u_2 = \cos \left( {\measuredangle u^t u_2 } \right)
$

But we also know that: $
u^t \cdot u_1 = \cos \left( {\measuredangle u^t u_1 } \right) = \tfrac{1}
{2}
$
and $
u_1 \bot u_2
$
since it's a orthogonal basis

Now try to work with a bit of trigonometry and you'll get the answer


text:
Impact Factor: 3.9

## 2017 Vol. 34, No. 5

Display Method:
2017, 34(5): 571-575. doi: 10.1007/s00376-017-6329-6
Abstract:
2017, 34(5): 576-586. doi: 10.1007/s00376-017-6268-2
Abstract:
Anthropogenic climate change is expected to strengthen the vertical wind shears at aircraft cruising altitudes within the atmospheric jet streams. Such a strengthening would increase the prevalence of the shear instabilities that generate clear-air turbulence. Climate modelling studies have indicated that the amount of moderate-or-greater clear-air turbulence on transatlantic flight routes in winter will increase significantly in future as the climate changes. However, the individual responses of light, moderate, and severe clear-air turbulence have not previously been studied, despite their importance for aircraft operations. Here, we use climate model simulations to analyse the transatlantic wintertime clear-air turbulence response to climate change in five aviation-relevant turbulence strength categories. We find that the probability distributions for an ensemble of 21 clear-air turbulence diagnostics generally gain probability in their right-hand tails when the atmospheric carbon dioxide concentration is doubled. By converting the diagnostics into eddy dissipation rates, we find that the ensemble-average airspace volume containing light clear-air turbulence increases by 59% (with an intra-ensemble range of 43%-68%), light-to-moderate by 75% (39%-96%), moderate by 94% (37%-118%), moderate-to-severe by 127% (30%-170%), and severe by 149% (36%-188%). These results suggest that the prevalence of transatlantic wintertime clear-air turbulence will increase significantly in all aviation-relevant strength categories as the climate changes.
2017, 34(5): 587-598. doi: 10.1007/s00376-016-6088-9
Abstract:
Recent snow height measurements (2008-15) from nine automatic weather stations (AWSs) on the Ross Ice Shelf are used to examine the synoptic and seasonal variability in snow accumulation, and also to evaluate the performance of the Antarctic Mesoscale Prediction System (AMPS) for precipitation. The number of snow accumulation events varies from one station to another between 2008 and 2015, thus demonstrating geographic dependence. The interannual variability in snow accumulation is too high to determine its seasonality based on the current AWS observations with limited time coverage. Comparison between the AMPS and AWS snow height measurements show that approximately 28% of the AWS events are reproduced by AMPS. Furthermore, there are significant correlations between AMPS and AWS coincident event sizes at five stations (p<0.05). This finding suggests that AMPS has a certain ability to represent actual precipitation events.
2017, 34(5): 599-612. doi: 10.1007/s00376-016-6212-x
Abstract:
Based on basic equations in isobaric coordinates and the quasi-geostrophic balance, an eddy-flux form budget equation of the divergent wind has been derived. This newly derived budget equation has evident physical significance. It can show the intensity of a weather system, the variation of its flow pattern, and the feedback effects from smaller-scale systems (eddy flows). The usefulness of this new budget equation is examined by calculating budgets for the strong divergent-wind centers associated with the South Asian high, and the strong divergence centers over the Tibetan Plateau, during summer (June-August) 2010. The results indicate that the South Asian high significantly interacts with eddy flows. Compared with effects from the mean flow (background circulation), the eddy flows' feedback influences are of greater importance in determining the flow pattern of the South Asian high. Although the positive divergence centers over the Tibetan Plateau intensify through different mechanisms, certain similarities are also obvious. First, the effects from mean flow are dominant in the rapid intensification process of the positive divergence center. Second, an intense offsetting mechanism exists between the effects associated with the eddy flows' horizontal component and the effects related to the eddy flows' convection activities, which weakens the total effects of the eddy flows significantly. Finally, compared with the effects associated with the convection activities of the mean flow, the accumulated effects of the eddy flows' convection activities may be more favorable for the enhancement of the positive-divergence centers.
2017, 34(5): 613-622. doi: 10.1007/s00376-016-6139-2
Abstract:
A new parameterization of canopy asymmetry factor on phase function, which is dependent on the leaf normal distribution and leaf reflection/transmission, is derived. This new parameterization is much more accurate than the existing scheme. In addition, the new solutions for both the diffuse and direct radiation can be obtained using the Eddington approximation. It is found that the direct radiation can be described as a function of the diffuse radiation. This new approach offers a substantial improvement in accuracy, as compared with the hemispheric constant method, for both isotropic and anisotropic cases. Given the analytical nature of the solution and its high accuracy, we recommend the new parameterization for application in land surface radiation modeling.
2017, 34(5): 623-634. doi: 10.1007/s00376-016-6111-1
Abstract:
Remote sensing of atmospheric water vapor using global positioning system (GPS) data has become an effective tool in meteorology, weather forecasting and climate research. This paper presents the estimation of precipitable water (PW) from GPS observations and meteorological data in Algeria, over three stations located at Algiers, Bechar and Tamanrasset. The objective of this study is to analyze the sensitivity of the GPS PW estimates for the three sites to the weighted mean temperature (T m), obtained separately from two types of T m-T s regression [one general, and one developed specifically for Algeria (T s stands for surface temperature)], and calculated directly from ERA-Interim data. The results show that the differences in T m are of the order of 18 K, producing differences of 2.01 mm in the final evaluation of PW. A good agreement is found between GPS-PW and PW calculated from radiosondes, with a small mean difference with Vaisala radiosondes. A comparison between GPS and ERA-Interim shows a large difference (4 mm) in the highlands region. This difference is possibly due to the topography. These first results are encouraging, in particular for meteorological applications in this region, with good hope to extend our dataset analysis to a more complete, nationwide coverage over Algeria.
2017, 34(5): 635-649. doi: 10.1007/s00376-016-6194-8
Abstract:
Surface energy balance and the partitioning of sensible heat flux (SHF) and latent heat flux (LHF) play key roles in land-atmosphere feedback. However, the lack of long-term observations of surface energy fluxes, not to mention spatially extensive ones, limits our understanding of how the surface energy distribution has responded to a warming climate over recent decades (1979-2009) at the national scale in China. Using four state-of-the-art reanalysis products with long-term surface energy outputs, we identified robust changes in surface energy partitioning, defined by the Bowen ratio (BR = SHF/LHF), over different climate regimes in China. Over the past three decades, the net radiation showed an increasing trend over almost the whole of China. The increase in available radiative energy flux, however, was balanced by differential partitioning of surface turbulent fluxes, determined by local hydrological conditions. In semi-arid areas, such as Northeast China, the radiative energy was transferred largely into SHF. A severe deficiency in near-surface and soil moistures led to a significant decreasing trend in LHF. The combined effect of increased SHF and decreased LHF resulted in significant upward trends in the BR and surface warming over Northeast China. In contrast, in the wet monsoon regions, such as southern China, increased downward net radiation favored a rise in LHF rather than in SHF, leading to a significant decreasing trend in the BR. Meanwhile, the increased LHF partly cancelled out the surface warming. The warming trend in southern China was smaller than that in Northeast China. In addition to impacts on heat-related events, the changes in the BR also reflected recent cases of extreme drought in China. Our results indicate that information regarding the BR may be valuable for drought monitoring, especially in regions prone to such conditions.
2017, 34(5): 650-662. doi: 10.1007/s00376-016-6152-5
Abstract:
Time series of MODIS land surface temperature (T s) and normalized difference vegetation index (NDVI) products, combined with digital elevation model (DEM) and meteorological data from 2001 to 2012, were used to map the spatial distribution of monthly mean air temperature over the Northern Tibetan Plateau (NTP). A time series analysis and a regression analysis of monthly mean land surface temperature (T s) and air temperature (T a) were conducted using ordinary linear regression (OLR) and geographical weighted regression (GWR). The analyses showed that GWR, which considers MODIS T s, NDVI and elevation as independent variables, yielded much better results [R Adj2>0.79; root-mean-square error (RMSE) = 0.51°C-1.12°C] associated with estimating T a compared to those from OLR (R Adj2=0.40-0.78; RMSE = 1.60°C-4.38°C). In addition, some characteristics of the spatial distribution of monthly T a and the difference between the surface and air temperature (T d) are as follows. According to the analysis of the 0°C and 10°C isothermals, T a values over the NTP at elevations of 4000-5000 m were greater than 10°C in the summer (from May to October), and T a values at an elevation of 3200 m dropped below 0°C in the winter (from November to April). T a exhibited an increasing trend from northwest to southeast. Except in the southeastern area of the NTP, T d values in other areas were all larger than 0°C in the winter.
2017, 34(5): 663-678. doi: 10.1007/s00376-016-6175-y
Abstract:
A modified Bowen ratio (BRm), the sign of which is determined by the direction of the surface sensible heat flux, was used to represent the major divisions in climate across the globe, and the usefulness of this approach was evaluated. Five reanalysis datasets and the results of an offline land surface model were investigated. We divided the global continents into five major BRm zones using the climatological means of the sensible and latent heat fluxes during the period 1980-2010: extremely cold, extremely wet, semi-wet, semi-arid and extremely arid. These zones had BRm ranges of ($-\infty$, 0), (0, 0.5), (0.5, 2), (2, 10) and (10, $+\infty$), respectively. The climatological mean distribution of the Bowen ratio zones corresponded well with the Köppen-like climate classification, and it reflected well the seasonal variation for each subdivision of climate classification. The features of climate change over the mean climatological BRm zones were also investigated. In addition to giving a map-like classification of climate, the BRm also reflects temporal variations in different climatic zones based on land surface processes. An investigation of the coverage of the BRm zones showed that the extremely wet and extremely arid regions expanded, whereas a reduction in area was seen for the semi-wet and semi-arid regions in boreal spring during the period 1980-2010. This indicates that the arid regions may have become drier and the wet regions wetter over this period of time.
2017, 34(5): 679-684. doi: 10.1007/s00376-016-6184-x
Abstract:
The impact of La Niña on the winter Arctic stratosphere has thus far been an ambiguous topic of research. Contradictory results have been reported depending on the La Niña events considered. This study shows that this is mainly due to the decadal variation of La Niña's impact on the winter Arctic stratosphere since the late 1970s. Specifically, during the period 1951-78, the tropospheric La Niña teleconnection exhibits a typical negative Pacific-North America pattern, which strongly inhibits the propagation of the planetary waves from the extratropical troposphere to the stratosphere, and leads to a significantly strengthened stratospheric polar vortex. In contrast, during 1979-2015, the La Niña teleconnection shifts eastwards, with an anomalous high concentrated in the northeastern Pacific. The destructive interference of the La Niña teleconnection with climatological stationary waves seen in the earlier period reduces greatly, which prevents the drastic reduction of planetary wave activities in the extratropical stratosphere. Correspondingly, the stratospheric response shows a less disturbed stratospheric polar vortex in winter.


text:
# Random variables invariant under almost automorphisms.

Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A_1\to\Omega/A_2$ is almost automorphism, if $P(A_1)=P(A_2)=0$, $f(A)$ is measurable if and only if $A$ is, and $P(f(A))=P(A)$ in this case. An almost automorphism preserves a (real-valued) random variable (r.v.) $X$ if $X(\omega)=X(f(\omega))$ for almost all $\omega$.

The question is: Assume that every almost automorphism preserving $Y$ preserves also $X$. Does this imply that $X$ is $Y$-measurable, that is $X=g(Y)$ a.s. for some $g:R \to R$?

-

Fix for convenience $\Omega = [0,1)$ with Lebesgue measure $\mu$. We exhibit a family of continuum-many Borel functions $X_\alpha \colon [0,1) \to \mathbb{R}$, indexed by $\alpha \in (0, 1/2)$, such that

• Each $X_\alpha$ has only trivial almost automorphisms, in the sense that each almost automorphism $f$ preserving $X_\alpha$ has $\mu$-null support (i.e., the set $\{r : r \neq f(r)\}$ is $\mu$-null).

• When $\alpha \neq \beta$ we have that $X_\alpha$ is not in your sense $X_\beta$-measurable.

In particular we get a negative answer to the question.

For each $\alpha \in (0,1/2)$ put $I_\alpha = [0,\alpha)$ and $J_\alpha = [0,1) \setminus I_\alpha = [\alpha, 1)$. Let $i_\alpha \colon [0,1) \to [0,1)$ be the obvious fixed-point-free involution flipping $I_\alpha$ and $J_\alpha$. Note that since $\alpha < 1/2$ this involution distorts $\mu$ in the following sense: any $\mu$-positive Borel set $A$ contains a $\mu$-positive Borel set $A' \subseteq A$ such that $\mu(A') \neq \mu(i_\alpha[A'])$. In particular, no restriction of $i_\alpha$ to a set of $\mu$-positive measure can be a $\mu$-preserving map.

Now define $X_\alpha$ by

• $X_\alpha(r) = r$ if $r \in I_\alpha$,

• $X_\alpha(r) = i_\alpha(r)$ if $r \in J_\alpha$.

Note that $X_\alpha(r) = X_\alpha(s)$ iff $r = s$ or $r = i_\alpha(s)$. It follows that any almost automorphism preserving $X_\alpha$ is almost everywhere some restriction of $i_\alpha$, but by the previous paragraph it must in fact be almost everywhere the identity. So each $X_\alpha$ has only trivial almost automorphisms.

Finally, suppose that $\alpha < \beta$ in $(0,1/2)$. Put $R = [\alpha, \beta)$. Note that for each $r \in R \subseteq J_\alpha$ we have $i_\beta(r) \in J_\alpha$ and in particular $X_\alpha(r) \neq X_\alpha(i_\beta(r))$. Of course by construction we have $X_\beta(r) = X_\beta(i_\beta(r))$. So if $g \colon \mathbb{R} \to \mathbb{R}$ is any function, for each $r\in R$ we must have at least one of $g(X_\beta(r)) \neq X_\alpha(r)$ or $g(X_\beta(i_\beta(r)) \neq X_\alpha(i_\beta(r))$ and in particular $g \circ X_\beta$ does not agree $\mu$-a.e. with $X_\alpha$. In other words, $X_\alpha$ is not $X_\beta$-measurable.

A similar argument using $i_\alpha$ shows that $X_\beta$ is not $X_\alpha$-measurable.

-
Thank you very much! –  Bogdan Apr 9 '13 at 12:34
You're welcome! I'll delete the disclaimer as I suppose it is no longer relevant. –  Clinton Conley Apr 10 '13 at 19:10


text:
Find the area of the shaded region.

Alright. So all I can tell from this is that the radius of the circle is 2.5
Also, the area of the square is 25.
So subtract the area of the circle from the area of the square and there you go. Only problem is I have no idea how to find the area of the circle with the given info. I know A= .5r^2theta

Is theta 90 degrees?

It doesn't seem as though I have enough info. What am I missing? Thanks