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# Highlights from the Meeting

I had a nice time at the Evolution meeting in Utah--this was my first Evolution meeting.  There are some presentations that I attended that have stuck in my mind and I thought I should mention them here.

I attended a phylogenetics workshop on the first day.  Joe Felsenstine (U. of Washington) gave a talk that segued from a historical overview to some current problems in phylogenetics.  He gives entertaining talks that sometimes border on the cynical but are humorous.  I especially liked his funding sources in the acknowledgements that included the Felsenstine foundation who's slogan is "instead of painting the house."  On the more serious side he brought up the issue of inference of horizontal gene flow versus linage sorting in prokaryotes and archaea.  There are several instances of horizontal gene transfer where DNA sequence has moved from one species to another in bacteria, but how many of these are actually "incongruent" lineage sorting in bacteria with very large ancestral species population sizes?  On the other hand, rates of recombination in bacterial genomes are actually quite low, so if selective sweeps are frequent in bacteria (a later talk by R. Lenski is very relevant to this), perhaps effective populations sizes are indeed small and these are true horizontal transfers?  He also brought up the possibility of looking at QTLs/heritability across multiple species to have power to get at what is under selection in correlated traits.

Brant Faircloth (UCLA) talked about using ultraconserved elements (UCEs) to reconstruct various phylogenies including helping to resolve the placement of turtles among birds and reptiles (which turns out to be an unresolved problem).  There is a temptation to sequence and compare whole genomes of a range of species, especially since sequencing has become so cheap, but in the end a lot of the data is thrown out because it is hard to align, etc.  There are various ways to focus down on smaller parts of the genome for comparison such as genome-wide exon (exome) sequencing from mRNA (exons are typically more conserved across species), DNA enrichment by sequence capture (seqcap) of exome DNA with tethered oligonucleotide probes, and finding variants around restriction endonuclease sites across the genome with RadSeq and RadTags.  Another approach is to focus on regions that are highly conserved across a wide range of species (UCEs), there is a website dedicated to using these (http://ultraconserved.org/); these UCEs seem to contain a lot of binding sites for transcription factors that are deeply shared across metazoans (Ryu et al. 2012).

Later in the same workshop, Stacey Smith (University of Nebraska-Lincoln) gave one of the clearest, well organized, and informative (to the novice to phylogenetic inference) talks I have seen.  (Many presentations I have seen on this topic have been cryptic and hard to follow, which often reflects a lack of detailed understanding by the people presenting.)  She has also made her slides available online (link: http://www.iochroma.info/links) and said that anyone can use her graphics in their own presentations if they want!  She went over parsimony, maximum likelihood, and Bayesian inference for discrete and continuous traits, the inference of relative rates of change and ancestral states and ways to visualize and interpret the results.  Again, she gave an excellent presentation!

In a special address, Richard Lenski (Michigan S.U.) talked about some new results from his long term (25 year) artificial evolution experiment in E. coli.  The experiment is now up to 50,000 generations and is still going.  He has kept samples of the bacteria over the generations and can compare them to each other.  One overall result is that fitness has been increasing over time from new mutations.  Early in the experiment he thought the curve of fitness increase was hyperbolic, which asymptotically approaches a limit, but with more generations he showed that the change in fitness fits a power law curve significantly better, which does not have a limit.  The implication is that fitness will continue increasing forever and will never reach a peak.

The image below compares an example hyperbloic curve, tanh(x), to a power law curve, (x/1.5)^0.7.  The two curves may be similar early on, but diverge by a greater amount as time becomes large.

He also talked about the implications of one of his replicates that evolved citric acid metabolism.  They screened a huge number of cells and did not see this occur again independently.  But in the line where it did occur, if they went back to the generations just before this appeared, it did evolve again repeatedly (if I understood correctly what he was saying).  There was also a great deal of evolutionary fine-tuning of the novel citric acid metabolism after it appeared.  This demonstrated the steps of potentiation (setting the stage for a particular adaptation to be able to occur), actualization (the final mutant step that results in the new function), and refinement (additional changes to improve the system).  They are working on nailing down the precise mutational steps that have happened in this lineage and a gene duplication (in citrate transport) is involved.  He also discussed several lines of reasoning that this population of E. coli can be considered to have evolved into a new bacterial species--one of the defining features of E. coli is that it does not use citric acid as a carbon source, and intermediates have reduced fitness.

In other talks, Mingzi Xu talked about honest signals of fitness in male dragon flies.  The dragon flies she works on use mie scattering of light (which also makes clouds white) from fat deposits on their wings to give them white bands that females are attracted to.  The males don't seem to be able to cheat and produce a large white band if they are smaller and have less resources because the fat residue uses a lot of energy to create.  She is going on to estimate heritability of the trait among offspring, etc.

Carl Bergstrom talked about "Timing of antimicrobial use influences the evolution of antimicrobial resistance during disease epidemics." Missing antibiotic doses extends the time it takes to get rid of infections--perhaps in a very predictable way--and missing doses early in a program is far worse that missing them later in a program in terms of the possibility of antibiotic resistance arising. I think this is because there are many more microbe cells present early in the program that could mutate to resistant varieties.

William Soto talked about "Adaptive Radiation of Vibrio fischeri During the Free-living Phase and Subsequent Consequences for Squid Host Colonization." Vibrio genus bacteria in Hawai'i are something I have become more interested in lately so I attended this to talk to get some more background information. Bioluminescent Vibrio fischeri are both free living in seawater and symbiotic with nocturnal bobtail squid; however, there may be different morphologies that are differentially adaptive in colonizing versus free living stages.

Benjamin Parker talked about immune response in aphids.  They can generate winged and unwinged aphids that are genetically identical but develop differently and test how well they fight off infections.  There is a cost to producing wings and the winged aphids also do not stop infections as well as the unwinged aphids; the immune response genes are not expressed as highly, etc.  There are some parallels with costs of immunity in bumble bees.

Tyler Hether showed that the effects of mutations depend on the ways that genes interact (epistasis).  The pattern depends on the type of regulation, positive or negative.  This distorts the state space that can be traveled by a sequence via mutations so that change in some directions is faster than others as a function of the type of interaction among genes--the gene interaction network has an effect on the direction of evolvability.  In general networks constrain adaptation; the space to transverse by mutations to get between states was larger with epistasis than the null model of no interaction between genes.  This possibly relates to issues of phenotypic robustness despite mutational change in the underlying genes.

Benjamin Liebeskind gave a great presentation titled "Sodium Channels and the Origin(s) of Animal Behavior."  This is a very difficult question to approach: what are the origins of animal behavior.  Animals use sodium channels (similar to calcium channels), which also exist in protists, to create charge potentials and propagate signals along neurons.  By looking at the gene sequences among species he found that these sodium channels predate animals who inherited them from common ancestors with modern protists.  The channels started off as an "EEEE" holoenzyme, which evolved to a heterogeneous "DEEA" then a "DEKA" sodium channel holoenzyme.  Interestingly there has been convergent evolution in cnidarians and bilaterians in sodium channel evolution and possibly origins of complex behavior?

Then I gave a presentation on engineering underdominance to transform wild populations (link to PDF of the slides).  It was much more applied than the other talks but I still got some nice feedback from people that saw my presentation.

M. Slatkin (UCB) talked about inferring the number of genes underlying a complex trait (like stature that is influenced by many genes and the environment) from a very subtle signal in the covariance of the trait among parents and offspring.  It requires huge sample sizes to work, on the order of thousands; however, these kinds of sample sizes exist for many traits.  For example, each year I have my genetics class report height for themselves and their parents, if they can, to illustrate quantitative genetics (regression/heritability); this is a sample of about 200 each year so by the end of this fall I will have a total of about 600 parent offspring trios for a single trait.  Perhaps the sample will be large enough in the coming years to use his approach with the students data to illustrate estimating the number of genes affecting the trait to the class?

Rebecca Chong (CSU) talked about rates of evolution in rearranged mitochondrial genomes.  If I understood correctly, there is an acceleration of substitutions in genomes that have been rearranged, but this can not be explained by changes in mutation rate due to their position (single stranded DNA is more susceptible to mutation and the process of replication of the mitochonrial genome makes some areas single stranded longer than others) nor can it be explained by simple relaxation of selection (based on dN/dS ratios).  So the question is, what is driving the changes in these genomes?

There was also a presentation followed by a question and answer session by program officers from NSF to explain more about applying for funding from NSF and some issues with the new application procedure.  I appreciated them doing this and I learned some things, but honestly I did not get a lot out of it because I already read all the instructions, and follow them, for the application.  Long story short--and they did not say this--we need to do a better job of petitioning congress (mailed letters, not emails, are the best way to do this) to provide funding to NSF.  (I also received some comments back on my last application, but I will write about these in a later post.)

One talk that I missed that I really wanted to see was "The convivial origins of life on the Earth: A cooperative network of RNA replicators" by Niles Lehman (PSU).  I have had a side interest in RNA evolution for a long time, but I couldn't be in two places at once...  There were also a couple presentations about evolving robots that might have been fun to see, but the scheduling was bad for me.

Finally, there was a talk by Joan Strassmann (WU) on the discovery of bacterial farming in social amoebas.  I liked the talk, but toward the end of the meeting things were starting to blur together.  In a similar vein, I saw a presentation about strategies to minimize infections (undershooting versus overshooting a minimal infection threshold) during flu outbreaks--there is an interesting social (public versus private interest) conflict dynamic involved--but I can't remember now who gave the talk, sorry.

There were also poster sessions and one that jumped out at me was a poster by David Gokhman (HUJ) "Reconstructing the DNA Methylation Map of Extinct Human Species."  Simplistically put, DNA methylation can "turn" genes "on" or "off."  These are interesting because this process can explain a lot of things in genetics including trans-generational environmental effects on gene expression.  There are ancient genomes that are being sequenced and Gokhman et al. used a mutation bias between methylated and unmethylated C's in the DNA sequence.  (Methylated C's are deanimated to T's, unmethylated C's are deanimated to U's and lost from the sequence.)  The Denisovan genome (lived until 30,000 years ago in Siberia and likely S. to S.E. Asia) was studied.  Ancient methylated DNA sequences are, according to this mutation bias, expected to have higher C-to-T ratios and this can be compared to known methylation patterns in modern humans--the two are correlated. However they found 41 genes (e.g., EBP, HOXD9, UPF3B, NBEA, MAB21L1, miR-291-2, etc.) that are inferred to be differently methylated between the Denisovan and modern humans--these are genes affecting things like limb development and psychiatric disorders when they are disrupted.

There were also some displays related to teaching resources...(listed briefly here):

There were no presentations Monday afternoon, so I went on an outing to the top of a nearby mountain (11,000 ft elevation) and took this picture of myself at arms length with my phone's camera (yes, that is snow in the background, in June).  In the valley in the distance are the Sandy and West Jordan suburbs of Salt Lake City.

# chemfig: Chemistry in WordPress

I just learned about adding chemical graphics in wordpress using the QuickLaTeX plugin and the chemfig package (links: QuickLaTeX, chemfig and chemfig).

Here is my first attempt, starting with something simple: molecular hydrogen.

This code gives:

And now for a double bond: molecular oxygen.

Putting these together we get water.

However a water molecule actually has an angle of about 104.5 degrees. The angle below is raised 75.5 degrees because it would normally be 180 degrees between the hydrogens, .

The oxygen has a higher affinity for electrons (is highly electronegative) and develops a negative charge while the electrons pulled away from the hydrogen give them positive charges from the proton nucleus.

These charges result in hydrogen bonds between water molecules, which allows water to be a solid and a liquid at higher temperatures than is usual for a molecule of its size.  This also is responsible for the expansion of ice compared to the liquid form (the molecules take up more space to arrange themselves with optimal hydrogen bonding).

And here is an actual reaction (the bond angles in the water come from :

Switching gears with carbon, here is methane.

Actually methane is organized into a tetrahedron with approximately 109.5 degree angles between the hydrogen atoms (, , ). Projecting above and below the plane of the image can be indicated with Cram-style bonds.

Now for cyclic hydrocarbons! Benzene is:

And as is typically done for hydrocarbons, we can leave out the hydrogen atoms and just show the carbon backbone:

Finally, the delocalized pi-bonds can be illustrated as a ring which better reflects the aromatic p molecular orbitals (these give benzene and similar aromatic hydrocarbons enhanced molecular stability):

Glucose! (the most common form at least) I actually had help on this one; I started to put it together than found it already assembled in Section 11.3.4 P. 37 of the ChemFig Documentation.  I just modified it slightly:

Finally, let me have a shot (double entendre) at putting together caffeine.  First the hydrocarbon shorthand:

Then the full version with all the atoms labeled:

# GMO Forensics with PCR

The Polymerase Chain Reaction (PCR) is a very powerful technology invented in 1983 by Kary Mullis.  We use it routinely today in molecular genetics labs and it is easy to take it for granted.  What makes this reaction particularly significant is not only that it uses a enzyme catalyst that remains unchanged while driving the reaction, but also that it results in an exponential amplification of a defined DNA sequence.  Human intuition tends to underestimate the power of exponential amplifications.  Casually imagining that you get paid a penny on the first day of a job and that your pay doubles each day (two pennies on the second day, four pennies on the third day) seems like it would take a long time to make much money--at the end of the week you are only up to $1.28. However, you would have well over 10 million dollars ( pennies) by the end of the month! Kary Mullis was only given a$10,000 bonus for inventing PCR by Cetus, the company he worked for; but in 1993 he was finally awarded the nobel prize in chemistry for his invention.

PCR works like this on DNA.  A stretch of DNA is designated with a set of primers (short single stranded DNA segments that can be synthesized).  A DNA polymerase enzyme attaches to one end of the primers and synthesizes a new DNA strand in one direction matching the original template DNA.  There are cycles of heating ("melting" the DNA by breaking hydrogen bonds between the strands), annealing of the primers to single stranded DNA, and extension of the new strands by the polymerase.  Importantly, the newly synthesized DNA strand can be used as a template in the next round, so, in theory, the amount of DNA between the primers is doubled each round.  If you do this for 30, 35, or even 40 cycles, you can amplify enough DNA to work with and sequence from even a single starting molecule.

Here in Hawai'i there is currently a GMO food labeling debate.  Unlike the EU, crops that are genetically modified do not have to be indicated on food labels in the US.  This fits into the larger GMO crop debate and is something that the students in my class are very interested in learning about.  In fact, just after I arrived here in August 2011 there were newspaper headlines about acres of transgenic crops destroyed on the big island and rewards for information (link).  This led me to thinking about a lab activity that would capture their interest--detecting GMO food ingredients by PCR.

Almost all GMO crops use the same strong constitutive promoter (highly expressed and always on because of a high affinity for RNA polymerase) from the cauliflower mosaic virus (CMV35S) to drive expression of the inserted gene and a "nos" (from nopaline synthase) terminator from the Agrobacterium tumefaciens Ti plasmid to stop transcription of the gene on the opposite side.  Primers to detect these parts of the genetic insert are already published so I ordered some to try out.

The image below is an overview of a 9+ kbp (over nine thousand base pairs) genetic modification inserted into transgenic papayas to combat the papaya ringspot virus (the virus devastated the papaya industry here in Hawai'i over the last 50 years).  The type of insert varies for each application, but as you will see later on, there is a reason I picked the GM papaya to illustrate.

And below I've indicated the positions where three sets of primers are predicted to anneal: one set in the nos terminator and two sets in the CMV promoter indicated by the green triangles.

Below I have zoomed in a bit to see more detail; the sequence is so large it has to be wrapped on the screen so moving off to the right comes back on the next row down on the left.  In light green promoters are indicated, protein coding genes are in yellow and terminator sequences in orange.

Below is a closer picture of part of the CMV promoter with "forward" and "reverse" primers. This promoter is used to drive expression of a coat protein from the ringspot virus, to prime the papaya's immune response to the virus--this particular genetic modification is often credited with saving the papaya industry here in Hawai'i.  The particular primer pair I tried as of this moment was the nos pair and one pair in the CMV promoter; I have not tried the second CMV pair yet.

Now it's time to go out and get some samples to try them out on.  In the US the big three are soybeans, corn and cotton (the image below is from here).

I couldn't think of a source of cotton with enough DNA to easily test, so I focused on corn, soybeans and--here in Hawai'i--papaya.

At the local grocery store I found "unlabeled" corn from the US:

"Green Giant Nibblers" also grown in the US:

And "GloriAnn Sweet Corn" a product of Mexico:

The only soybeans I found were all products of China.  "Shirakiku Edamame":

"WelPac Shelled Edamame":

And "Safeway Kitchens Shelled Edamame":

For Papayas I found "Diamond Head Papaya":

Anonymous "Papaya" from a different store:

And "Solo" variety papaya from a farmer's market:

So we have three of each of corn, soybean and papaya samples.  After extracting DNA samples, running the PCR and running a gel to view the results, this is what I found:

To the left are the CMV primers and to the right the NOS primer pairs.  Three lanes in each showed up positive for amplification of the corresponding DNA.  The positive bands are faint so I indicated them with arrows in the image below.

These primers amplify a short DNA sequence which does not stain with much dye to see on the gel.  The CMV promoter primers worked "better" i.e. are easier to see.  If you look closely you can also see a fainter band just below the positive samples that is "primer dimer".  This happens when the primers anneal to each other and amplify an even shorter sequence from just the primers.  Primers are designed in pairs to minimize this effect however.

In the gel below I tried different combinations of the CMV and NOS primers together in the same PCR reaction to try to amplify the larger sequence of DNA between them.  In this case these primers were not designed to work together and the primer dimer is much stronger at the bottom of the lanes.  There are three lanes that gave a large (did not move as far down the gel) band in the upper right (CMVf, NOSr).  The PCR in the second lane in from the top left didn't seem to work at all, this could be due to a PCR inhibitor that was co-extracted with the DNA.

So what were the three positive samples?  They were the three papaya samples, two from the store and one from the farmer's market.  To nail this down a bit more I sequenced the DNA that was amplified to make sure it is what I suspected.

Above, the short sequences from the three samples (gray bands in the lower part) are mapped to the papaya insert as a reference sequence.  As expected they map to the CMV promoter.  Below I zoomed in some more on a section so you can see the DNA sequence, it is a match to the promoter sequence.

Above, two matches to the "right" promoter and below one match to the "left" promoter.

Below is the longer sequence I recovered from the CMV-NOS primer pairs.

Zooming in more below you can see (hopefully) it is a match to the uidA gene in the genetic modification insert.  (The colors in the trace file correspond to the different colored bases in the reference sequence at the top.)

uidA is a gene from E. coli and produces the enzyme beta-D-Glucuronidase.  When plant tissues are treated with a bromine containing compound (X-Gluc) that is cleaved by beta-D-Glucuronidase and stains the tissues blue.  This provides visual confirmation that the gene in the insert is being expressed (similar to the X-Gal reporter system in E. coli) and is called the GUS reporter system.  The point being here, this is not a gene sequence (with components originating from a virus, CMV, and two different bacteria, NOS and uidA) that would be present in non-genetically modified papaya, but is exactly what we expect in ringspot virus resistant genetically modified papaya.

Let me back up a moment at this point and put a disclaimer in here.  I did this informally in the lab to test for use as a teaching tool in my genetics class in the fall.  There is a chance of a false positive here from DNA contamination between samples.  As I said at the beginning, PCR is so powerful it can amplify a single DNA molecule so this is something to always keep in mind.  There are ways to address this with multiple independent extractions in different labs, amplification from different parts of the insert between different samples, looking for unique genetic differences between the samples (if they exist), etc. but I am not going to go into that detail here.

Also, while we are on a cautionary topic, the CMV promoter is from the cauliflower mosaic virus.  It is possible that cauliflower and related brassica may show up positive even if they are not genetically modified from natural CMV infection.  Agrobacterium infects certain plants naturally and even E. coli can contaminate fruits and vegetables from contaminated water sources (which can be a serious health problem).  But you would not expect to find the CMV - uidA combination above.  That is proof that these papaya (or at least one if we are being hyper-cautious) are transgenic genetically modified.

I was surprised that the "solo" variety from the farmer's market showed up positive for the ringspot virus insert.  This is not supposed to be a genetically modified variety.  However, if you do a search for the terms "papaya gmo seed contaminated" you can find that it is quite common for papaya varieties, solo included, to contain the genetic modification here in Hawai'i, unknown to the growers, sellers and buyers using them.

# U.S. Supreme Court unanimous in prohibiting patents on naturally occurring human genes

A victory in the courts!  Finally the law is catching up with reality on gene patents.

A basic fact about patenting is that naturally occurring substances cannot be patented.  I cannot patent water, gold or air and charge you a royalty if you use them.  There has to be an "invention" involved where you have created something new as a prerequisite for patenting.

At the center of this current debate is Myriad Genetics and their patents of BRCA1 and BRCA2.  Having mutant alleles at BRCA1 or 2 elevates a woman's lifetime risk of breast cancer.  The patent allowed Myriad to legally prevent people from testing if they had these alleles unless they bought a \$3,000+ testing kit from Myriad.  These are alleles that are a biological part of the people seeking to be tested.  Myriad does not own these people, how can they have a patent on part of their genome that the people naturally inherited from their parents?

I am fine with patenting genetic constructs that are novel and involve creativity in their design--constructs that do not already exist in nature.  I am also involved in a gene patent application based on our work to engineer underdominance via haploinsufficiency (M 10021/RN Max-Plank-Society for the Advancement of Science).  These kinds of patents can be potentially valuable to the institutes that own them and help drive research into new areas.

However, patents on naturally occurring substances can seriously inhibit medicine and research (see Paradise et al. 2005 and references therin).  Also, not having humanitarian and research exemptions can be a problem.  Imagine that you are developing a new line of research in the lab and then the university grants an exclusive license to a company from a patent based on your research.  Suddenly you are not allowed to continue your own research without permission from the company...  (see Andrews et al. 2006).

Incidentally, I am listed as an "inventor" on a human gene patent application (USPTO App. #20080220429) that stemmed from my postdoc work on lactase persistence in East Africa. I did not initiate patenting of the alleles; I was unaware a patent application had been drafted when the work was published; and transferring ownership of the patent to the University of Maryland was a term of my previous employment.  However, there is often confusion by the public that research scientists are responsible for gene patents rather than the companies and people that employ them.  After it was published I found out about an article in South Africa (Jordan 2009) criticizing me personally for this patent application.  (It also contained a number of other misconceptions, for example, I have never been affiliated with the University of Pennsylvania and I have no record of the author ever attempting to contact me for comment.)

When I first heard about the ACLU's challenge to human gene patents in early 2010 I contacted them and offered to help in any way I could--if my unique position as an "inventor" of gene patents and personal opposition to the granting of gene patents was useful.  They wrote back to me and we had a brief correspondence but in the end they did not take me up on my offer.  However, clearly they did not need any help and in the end were successful with today's court ruling.

# Basic Hardy-Weinberg and Probability

Everything in genetics starts with mutations, but once we have mutations to study, work with and think about, what follows?  One direction is thinking about the dynamics of these gene differences (alleles) in large populations over time.  In 1922 R. A. Fisher compared this to the study of gases in physics.  The trajectories of the individual molecules are too complex to keep track of individually, but when a large number are considered as a group, individual differences average out and certain measurable and predictable properties arise like the relationship between temperature, pressure and volume.  (The kinetic theory of gases and the ideal gas law.)

An allele is at some frequency in a population.  The frequency has to be a fraction between zero and one (or equal to zero or one).  We can keep track of the frequency with .  For example, if the allele is at 50% frequency we can write .  Most species we think about are diploids and have two copies of most genes.  For simplicity let's say there are only two alleles in a population ( and , for the moment we are not worrying about which one might be designated a mutant or wildtype) and that the population is very large, so that all possible combinations are present no matter how rare.  Let's also say  is the frequency of the  allele.  If we pick a diploid individual in the population and pick one gene copy, what is the probability it is an  allele?  The probability is simply the frequency of the allele in the population, which is equal to ; .

A related question is, what is the probability that both alleles found in an individual are ?  The simplest assumption is that choosing the two alleles is independent; i.e. if one allele is an  this doesn't affect the probability that the second allele is or is not an .  So we are asking what is the probability the first allele is  and the second allele is .  This is the logical intersect .  One way to think about this is that within the group where the first allele is , which is a frequency of , the fraction that has a second allele of  is also  had to be drawn twice and the chance of this is  for the first copy and  within that fraction for the second copy:  is also the expected frequency of  homozygotes (two copies of the same allele) in the population (probabilities and frequencies work both ways).

What about the frequency of the  allele?  Since we are only dealing with two alleles in the population, and the result of all possible outcomes must sum to one, 100%, the frequency/probability of the second allele is the probability it is not the first allele, .  (I like to use the  symbol for not because other not symbols can be ambiguous in general contexts.)  So the probability of drawing two  alleles is  .

This introduces the "and" and "not" rules in probability.  If events are independent, this and that, the probability of the combined outcome is found by multiplying the frequency of the individual events.  If we are talking about the opposite of an event, not that but everything else, the probability (complement) is found by subtracting from 1 (100%).  There is also an "or" rule that comes up quite frequently and that we will use next.  If two events are mutually exclusive, this or that occurred, then the combined probability is found by adding the two individual probabilities together.

So, what is the frequency of heterozygotes, where individuals have one of each allele,  and .  Based on what I wrote above you might at first think we should multiply the allele frequencies together, , after all, if choosing the alleles is independent then the first one does not affect the choice of the second.  This is right but not completely right.  The trick that comes up here is that there are two ways to be a heterozygote.  The first allele chosen could be an  and the second allele an  or vice versa, the first allele  was an  and the second an .  This may seem arbitrary; however, a natural way to keep track of the two outcomes to visualize this is the keep track of which allele comes from which parent.  The  could have come from an organisms father and  from the mother, or  was from the father and  from the mother.  These two events are mutually exclusive, either one happened or the other (they are not independent, if you are a heterozygote then getting an  from your mother means the  allele had to have come from your father).  In set theory this is the logical union, , of the two outcomes (and we are keeping track of the order of events), .  This is calculated by adding the two mutually exclusive outcomes together, .

Just for fun, let's substitute in all the logic symbols.

Then substitute in standard arithmetic symbols and  for the probability of .

 is equal to  so these can be added together by multiplying one by two.

Above is a plot to illustrate.  If  then the probability of drawing the corresponding allele first is , (blue in the "First" bar above).  Within that class of 40% the probability of drawing the same allele again is 40% of 40% or 16% ("Second"  allele above).  The two types of heterozygotes can be combined (yellow in the "Genotype" bar).  So if  is the frequency of  alleles then we expect 16%  homozygotes, 48%  heterozygotes, and 36%  homozygotes.  Here is another plot with .

As an allele becomes rare its corresponding homozygote becomes very rare.  Also, rare alleles are most often found in heterozygote form (which makes sense, if you are rare you are most often paired with something else).

OK, so now we have all possible outcomes.  If  is the frequency of the  allele (and there are only two alleles in the population), the frequency of  homozygotes is expected to be ; the frequency of ,  heterozygotes is ; and the frequency of  homozygotes is .  You may still be suspicious about multiplying the  heterozygotes by two, so to check this mathematically the frequency of all possible outcomes must sum to one, if we have done everything correctly (although this doesn't prove we are correct, there are ways to make mistakes that also sum to one, but if it does not sum to one it proves that this is incorrect).  First of all the allele  and  must equal one when added together. It is easy to see that  cancels out, so .   Adding the genotype frequencies gives ; this can be factored to .  As we just saw, .  So  and .

If we had not multiplied by two in the heterozygote term we would have had

This is not equal to one (except for the special case where  is zero), so not multiplying the heterozygote term by two is incorrect. Also, notice that we end up with one minus half of the heterozygotes (), which also makes sense, half of the heterozygotes are missing by not multiplying  by two.

Also, we can see that the genotype frequencies  are the binomial expansion of , which is another way of saying that we are combining alleles in pairs (from the allele frequencies in the fathers and mothers in the population).  To illustrate this lets make the  frequency equal to  to save space ().

If we let the sides of this square represent parental allele frequencies and an "m" subscript represents the allele frequencies in males while "f" represents females, then the areas inside the square give the relative proportions of offspring genotypes.  (Notice there are two types of heterozygotes but only one way to get each homozygote.)  It is often assumed that allele frequencies are equal between males and females but this does not have to be the case.  In the plot above .

The plot above gives the relative genotype frequencies expected as a function of .  At each point on this we can plot the corresponding square as in the plots below.

So, what can we do with this?  Well, for example, in the EU approximately 1 out of 2,500 people (link) are born with cystic fibrosis (CF) which can cause, among other complications, life-threatening lung infections in affected individuals. CF is caused by recessive alleles at a single gene, CFTR.  We can infer that these affected individuals are homozygotes and have two copies of the allele(s) that result in CF.  What fraction of people in the EU are carriers and have one copy of the disease causing allele but are unaffected because it is recessive?  Well, assuming Hardy-Weinberg genotype frequencies, we can set .  Taking the square root gives an allele frequency of .  Using this frequency estimate the fraction of heterozygote carriers in the population is .  (As a rule of thumb, the frequency of carriers of rare alleles is about twice the allele frequency.)  In other words about four percent, or one out of 25 people in the EU, are expected to be carriers of an allele that results in CF when homozygous--a surprisingly high number.


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# How do you find the vertex of -20(y+5)=(x+9)^2?

Sep 30, 2017

$\text{vertex } = \left(- 9 , - 5\right)$

#### Explanation:

$\text{using the translated form of the equation of a parabola}$

•color(white)(x)(x-h)^2==4p(y-k)

$\text{where the vertex } = \left(h , k\right)$

${\left(x + 9\right)}^{2} = - 20 \left(y + 5\right) \text{ is in this form }$

$\text{with "h=-9" and } k = - 5$

$\Rightarrow \textcolor{m a \ge n t a}{\text{vertex }} = \left(- 9 , - 5\right)$

format: plain


text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > KEYWORD > LOW DEGREE TEST:
Reports tagged with Low degree test:
TR04-046 | 4th June 2004

#### Robust Locally Testable Codes and Products of Codes

We continue the investigation of locally testable codes, i.e.,
error-correcting codes for whom membership of a given word in the
code can be tested probabilistically by examining it in very few
locations. We give two general results on local testability:
First, motivated by the recently proposed notion of robust
probabilistically ... more >>>

TR05-038 | 10th April 2005
Ran Raz

#### Quantum Information and the PCP Theorem

We show how to encode $2^n$ (classical) bits $a_1,...,a_{2^n}$
by a single quantum state $|\Psi \rangle$ of size $O(n)$ qubits,
such that:
for any constant $k$ and any $i_1,...,i_k \in \{1,...,2^n\}$,
the values of the bits $a_{i_1},...,a_{i_k}$ can be retrieved
from $|\Psi \rangle$ by a one-round Arthur-Merlin interactive ... more >>>

TR05-086 | 14th August 2005
Dana Moshkovitz, Ran Raz

#### Sub-Constant Error Low Degree Test of Almost Linear Size

Revisions: 1

Given a function f:F^m \rightarrow F over a finite
field F, a low degree tester tests its proximity to
an m-variate polynomial of total degree at most d
A providing the supposed restrictions of f to
affine subspaces of ... more >>>

ISSN 1433-8092 | Imprint


text:
# Quark Matter 2018

13-19 May 2018
Venice, Italy
Europe/Zurich timezone
The organisers warmly thank all participants for such a lively QM2018! See you in China in 2019!

## Studies of jet grooming and recursive splittings in pp and PbPb collisions with ALICE

15 May 2018, 12:10
20m
Sala Perla, 1st Floor (Palazzo del Casinò)

### Sala Perla, 1st Floor

#### Palazzo del Casinò

Parallel Talk Jet modifications and high-pT hadrons

### Speaker

Harry Arthur Andrews (University of Birmingham (GB))

### Description

Hard splittings in the evolution of a jet may be modified by the presence of a dense strongly interacting medium. Grooming procedures can be used to isolate such hard components of a jet and allows one to focus on the two subjets resulting from a sufficiently hard partonic splitting. The modification of these splittings in medium could highlight the role of jet induced medium response as well as potential single hard scatterings (higher-twist) and multiple soft medium induced radiation (BDMPS).
Measurements of the symmetry parameter ($z_g$) and angular separation of such subjets are reported as measured with the ALICE Detector in pp and PbPb collisions at $\sqrt{s} = 7$ TeV and $\sqrt{s_{NN}} = 2.76$ TeV respectively. Results are compared to predictions using Monte Carlo generators. The use of recursive splittings and their mappings to identify interesting regions of phase space will also be discussed with comparisons made between Monte Carlo generators and data in pp and PbPb collisions.

Collaboration ALICE Experiment Presenter name already specified


text:
# Question #1c2bd

Apr 27, 2016

$\text{4.4 J}$

#### Explanation:

Before the force is applied we know

$\text{momentum" = "mass" times "velocity" = "10 kg m/s}$

Since mass in the above is $\text{5 kg}$, then velocity = ${\text{2 ms}}^{- 1}$

We can now work out kinetic energy, KE:

$K E = \frac{1}{2} m {v}^{2} = \frac{1}{2} \cdot 5 \cdot {2}^{2} = \text{10 J}$

The force of $\text{0.2 N}$ is now applied in the direction of travel for $10$ seconds. Newton's second law can be expressed as "the force applied equals the rate of change of momentum". OR

$F = \frac{m v - \text{mu}}{t} = m \cdot \frac{v - u}{t}$

Where $v$ is the final velocity after applying force, and $u$ is the initial velocity ( ${\text{2 ms}}^{- 1}$ in this case from above)

So

$0.2 = 5 \cdot \frac{\left(v - 2\right)}{10}$

rearranging helps us find $v$

$2 = 5 \cdot \left(v - 2\right)$

$\frac{2}{5} = v - 2$

$v = \frac{12}{5} {\text{ ms}}^{- 1}$

Hence final kinetic energy, $K E$, is

$K E = \frac{1}{2} m {v}^{2} = \frac{1}{2} \cdot 5 \cdot {\left(\frac{12}{5}\right)}^{2} = \text{14.4 J}$

So the overall change in $K E$ is

${\Delta}_{\text{KE" = 14.4 - 10 = "4.4 J}}$


text:
## January 6, 2020

### Entanglement for Laymen

I’ve been asked, innumerable times, to explain quantum entanglement to some lay audience. Most of the elementary explanations that I have seen (heck, maybe all of them) fail to draw any meaningful distinction between “entanglement” and mere “(classical) correlation.”

This drives me up the wall, so each time I am asked, I strive to come up with an elementary explanation of the difference. Rather than keep reinventing the wheel, let me herewith record my latest attempt.

“Entanglement” is a bit tricky to explain, versus “correlation” — which has a perfectly classical interpretation.

Say I tear a page of paper in two, crumple up the two pieces into balls and (at random) hand one to Adam and the other to Betty. They then go their separate ways and — sometime later — Adam unfolds his piece of paper. There’s a 50% chance that he got the top half, and 50% that he got the bottom half. But if he got the top half, we know for certain that Betty got the bottom half (and vice versa).

That’s correlation.

In this regard, the entangled state behaves exactly the same way. What distinguishes the entangled state from the merely correlated is something that doesn’t have a classical analogue. So let me shift from pieces of paper to photons.

You’re probably familiar with the polaroid filters in good sunglasses. They absorb light polarized along the horizontal axis, but transmit light polarized along the vertical axis.

Say, instead of crumpled pieces of paper, I send Adam and Betty a pair of photons.

In the correlated state, one photon is polarized horizontally, and one photon is polarized vertically, and there’s a 50% chance that Adam got the first while Betty got the second and a 50% chance that it’s the other way around.

Adam and Betty send their photons through polaroid filters, both aligned vertically. If Adam’s photon makes it through the filter, we can be certain that Betty’s gets absorbed and vice versa. Same is true if they both align their filters horizontally.

Say Adam aligns his filter horizontally, while Betty aligns hers vertically. Then either both photons make it though (with 50% probability) or both get absorbed (also with 50% probability).

All of the above statements are also true in the entangled state.

The tricky thing, the thing that makes the entangled state different from the correlated state, is what happens if both Adam and Betty align their filters at a 45° angle. Now there’s a 50% chance that Adam’s photon makes it through his filter, and a 50% chance that Betty’s photon makes it through her filter.

(You can check this yourself, if you’re willing to sacrifice an old pair of sunglasses. Polarize a beam of light with one sunglass lens, and view it through the other sunglass lens. As you rotate the second lens, the intensity varies from 100% (when the lenses are aligned) to 0 (when they are at 90°). The intensity is 50% when the second lens is at 45°.)

So what is the probability that both Adam and Betty’s photons make it through? Well, if there’s a 50% chance that his made it through and a 50% chance that hers made it through, then you might surmise that there’s a 25% chance that both made it through.

That’s indeed the correct answer in the correlated state.

In fact, in the correlated state, each of the 4 possible outcomes (both photons made it through, Adam’s made it through but Betty’s got absorbed, Adam’s got absorbed but Betty’s made it through or both got absorbed) has a 25% chance of taking place.

But, in the entangled state, things are different.

In the entangled state, the probability that both photons made it through is 50% – the same as the probability that one made it through. In other words, if Adam’s photon made it through the 45° filter, then we can be certain that Betty’s made it through. And if Adam’s was absorbed, so was Betty’s. There’s zero chance that one of their photons made it through while the other got absorbed.

Unfortunately, while it’s fairly easy to create the correlated state with classical tools (polaroid filters, half-silvered mirrors, …), creating the entangled state requires some quantum mechanical ingredients. So you’ll just have to believe me that quantum mechanics allows for a state of two photons with all of the aforementioned properties.

Sorry if this explanation was a bit convoluted; I told you that entanglement is subtle…

Posted by distler at January 6, 2020 2:30 PM

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3185

### Re: Entanglement for Laymen

Thank you for this post. Can you please tell me how this last result comes from a state such as (|up>|down> - |down>|up>)/sqrt(2) ?
If Adam’s photon goes through, does the state collapse to something? How does it imply that Betty’s photon must also go through? I’m just trying to understand within the context of the above state and the axioms if quantum mechanics.

Posted by: Dave on January 7, 2020 8:27 PM | Permalink | Reply to this

### Re: Entanglement for Laymen

We are indeed studying an entangled state in a product of two-state systems. But we are measuring the operators $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\otimes\mathbb{1}$ and $\mathbb{1}\otimes\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ (in the basis where $|\text{up}\rangle=\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)$ and $|\text{down}\rangle=\left(\begin{smallmatrix}0\\1\end{smallmatrix}\right)$).

I am sure you can construct the corresponding projection operators and compute the probabilities for the various outcomes of those measurements.

Posted by: Jacques Distler on January 9, 2020 5:56 PM | Permalink | PGP Sig | Reply to this

### Re: Entanglement for Laymen

Why did my question get deleted?

Posted by: Dave on January 9, 2020 12:35 PM | Permalink | Reply to this

### Re: Entanglement for Laymen

It was swept up with a bunch of spam. Restored from backup now.

Posted by: Jacques Distler on January 9, 2020 6:04 PM | Permalink | PGP Sig | Reply to this

Post a New Comment


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# Scientific Notation

The easiest way to write the very large and very small numbers is possible due to the scientific notation. For example, the number 2500000000000000000000 is too large and writing it multiple times requires a short-hand notation called scientific notation.

The scientific notation involves the smallest number as possible (between 1 and 10) multiplied by (using the '$\times$' sign) the power of 10. The above number is represented in scientific notation as $2.5\times {{10}^{21}}$. You can also write the number as $250\times {{10}^{19}}$ but it's going to remove its name, the short-hand notation!

Generally you use the smallest number as 2.5 which is then multiplied by the appropriate power of 10. When you do the real multiplication between the smallest number and the power of 10, you obtain your number.

Another example is for small numbers. Consider 0.00000000000000000000453 and this can be written in the scientific notation as $4.53\times {{10}^{-23}}$. Scientific notations are frequently used in calculations with large or small numbers in physics.

## Scientific Notation Rules

The scientific notation is expressed in the form $a \times 10^n$ where $a$ is the coefficient and $n$ in $\times 10^n$ (power of 10) is the exponent. The coefficient is the number between 1 and 10, that is $1 < a < 10$ and you can also include 1 ($1 \geq a < 10$) but 1 is not generally used (instead of writing 1, it's easier to write in power of 10 notation). The scientific notation is the way to write very large and very small numbers in practice and it is applied to positive numbers only. The rules to convert a number into scientific notation are:

1. First thing is we determine the coefficient. Convert the number into greater than 1 and smaller than 10 by placing the decimal point at appropriate location (only one nonzero number exists to the left of the decimal point), and remove any trailing or leading zeros. For example, if 3453000 is the number, convert it to 3.453.
2. Move either to the right or to the left (depending on the number) across each digit to the new decimal location and the the number places moved will be the exponent. In 3453000, we move from the right end and number of places we move to our new location is 6, so 6 will be the exponent.
3. If the original number is less than 1 (x < 1), the exponent is negative and if it is greater than or equal to 10 (x $\geq$ 10), the exponent is positive. If it is between 1 and 10 including 1 (1 $\geq$ x < 10), the exponent is zero. In 3453000, the exponent is positive.
4. Alternatively you can say the rule number 3 as, if you move to the right, the exponent is negative and if you move to the left, the exponent is positive. And if you do not move at all, the exponent is zero but you do not need to express such number in scientific notation. Note that the scientific notation is the way to express very small and very large numbers easily.
5. Add the $\times 10^n$ part.

The above rules are more elaborated in the examples given below.

### Scientific Notation Example 1

You have a number 0.00000026365 and you want to write this number in scientific notation. First convert this number to greater than 1 and smaller than 10. To do that you you just need to add a decimal point between 2 and 6. The new number is 2.6365. The final step is to count the number of steps (places) we need to move to the right from the old decimal location to the new location as shown in Figure below.

The exponent is the negative of the number of steps (number of places) we moved to the right of decimal point to our new location. Hence the number in scientific notation is $2.6365 \times 10^{-7}$. The figure above explains this more clearly.

### Scientific Notation Example 2

Now you have a large number 3424300000 and you want to express this number in scientific notation. To convert this number to a number smaller than 10 and greater than 1 you just need to add decimal point between 3 and 4 and the number without leading zeroes becomes 3.4243.

Now you got the new location of decimal point. Simply move to the left from the right end of the number to the new decimal location. Note that this is a whole number and the decimal point is understood to be at the right end (3424300000.). Count the number of digits you moved across and that number will be exponent. So the number in scientific notation is $3.4243 \times 10^{9}$.

## Removing Scientific Notation

Now we convert numbers already in scientific notation to their original form. You can follow some easy steps to successfully convert the number in scientific notation back to normal form. Here are the rules.

1. If the exponent is positive, move to the right the number of decimal places expressed in the exponent.
2. If the exponent is negative, move to the left the number of decimal places expressed in the exponent.
3. If there are not enough digits to move across, add zeros in the empty spaces.
4. You do not need the $\times$ 10 anymore and remove it.

This is quiet easy. The exponent tells you the number of decimal places to move. All you have to do is move either to the right or to the left across digits. If there is no digit to move across, add zero in the empty place until you complete.

### Removing Scientific Notation Example 1

We consider a number 3.456 $\times$ 10$^7$ and convert it to original number without scientific notation. The exponent is 7 so we move 7 steps to the right of the current decimal location. The figure shows you the way to move.

After moving across three digits, there are no more digits to move but we add 0's in empty places and you get the original number, 34560000.

### Removing Scientific Notation Example 2

Let's consider a small number with negative exponent, $7.312 \times 10^{-5}$. Now you move to the left of decimal location 7 times. Here moving means we are taking the decimal point to the new location. So the number without scientific notation is .00007312 or 0.00007312 (the zero before the decimal point is optional).

Let's look at the addition, subtraction, multiplication and division of numbers in scientific notation. The arithmetic with numbers in scientific notation is similar to the arithmetic of numbers without scientific notation. One difference is that the rules of exponent applies with scientific notation.

## Addition in Scientific Notation

The addition in scientific notation can be done by following very simple rules:

1. Change all numbers to the same power of 10.
2. Add the coefficients and put the common power of 10 as $\times 10^n$.
3. Convert to scientific notation again if there is not only one nonzero number to the left of decimal point. If you need to do this, change or add the exponents again (apply exponents rule).

### Scientific Notation Addition Example

You have two numbers $2.4 \times 10^3$ and $5.71 \times 10^5$. To add these two numbers easily, you need to change all numbers to the common power of 10. You can change exponent of any number. Here we change the exponent in $5.71 \times 10^5$ to 3 and it is $571 \times 10^3$ (note the decimal point moved two places to the right).

\begin{align*} 2.4 \times 10^3 + 5.71 \times 10^5 \\ 2.4 \times 10^3 + 571 \times 10^3 \\ (2.4 + 571) \times 10^3 \\ 573.4 \times 10^3 \\ 5.734 \times 10^2 \times 10^3\\ 5.734 \times 10^{2+3} \\ 5.734 \times 10^5 \end{align*}

Now simply add coefficients, that is 2.4 + 571 and put the power 10, so the number after addition is $573.4 \times 10^3$. The final step is to convert this number to the scientific notation. Note that the coefficient must be greater than 1 and smaller than 10 in scientific notation. So the number in scientific notation after the addition is $5.734 \times 10^5$.

You do not need to convert the final number into scientific notation again if you have changed exponent in $2.4 \times 10^3$ to 5, so it is a good idea to convert smaller exponent to greater exponent. After subtracting the two exponents 5 - 3 you get 2 and the 2 to the power of 10 is 100. So 2.4 needs to be divided by 100 or the decimal point needs to be moved two places to the left, and that gives 0.024. Now we have the same exponent in both numbers. Just add 0.024 + 5.71 which gives 5.734 and the result is $5.734 \times 10^5$.

\begin{align*} 2.4 \times 10^3 + 5.71 \times 10^5 \\ 0.024 \times 10^3 + 5.71 \times 10^5 \\ (0.024 + 5.71) \times 10^5 \\ 5.734 \times 10^5 \\ \end{align*}

## Multiplication in Scientific Notation

Multiplication of numbers in scientific notation is easy. Simply multiply the coefficients and add the exponents. If necessary, change the coefficient to number greater than 1 and smaller than 10 again.

### Multiplication in Scientific Notation Example

Here we have two numbers $7.23 \times 10^{34}$ and $1.31 \times 10^{11}$. When you multiply these two numbers, you multiply the coefficients, that is $7.23 \times 1.31 = 9.4713$. Then all exponents are added, so the exponent on the result of multiplication is $11+34 = 45$. The final result after the multiplication is $9.4713 \times 10^{45}$ or the process is shown below:

$(7.23 \times 10^{34}) \times (1.31 \times 10^{11}) \\ 7.23 \times 1.31 \times 10^{34} \times 10^{11} \\ 9.4713 \times 10^{34 + 11}\\ 9.4713 \times 10^{45}$

Apply the exponents rule and voila! If the coefficient in the result is greater than 10 convert that number to greater than 1 and smaller than 10 by changing the decimal location and add the exponents again.

## Division in Scientific Notation

The division of two scientific numbers is similar to multiplication but in this case we divide coefficients and subtract the exponents.

### Division in Scientific Notation Example

You have two numbers $1.03075 \times 10^{17}$ and $2.5 \times 10^5$ . To divide these numbers we divide 1.03075 by 2.5 first, that is 1.03075/2.5 = 0.4123. Then we subtract the exponents of these numbers, that is 17 - 5 = 12 and the exponent on the result of division is 12.

Note that the number 0.4123 is less than 1, so we make this number greater than 1 and smaller than 10. To do that the decimal point goes between 4 and 1 and the number of steps we moved to the right across the digits to our new location is subtracted from the exponent of 10. So, The final exponent of 10 is $12 - 1 = 11$ and the number is 4.123. So the result is $4.123 \times 10^{11}$. Or mathematically,

\begin{align*} \frac{1.03075 \times 10^{17}}{2.5 \times 10^5} &= \frac{1.03075}{2.5} \times 10^{17 - 5} \\ &= 0.4123 \times 10^{12} = 4.123 \times 10^{-1} \times 10^{12} \\ &= 4.123 \times 10^{-1+12} = 4.123 \times 10^{11} \end{align*}

## Significant Figures in Scientific Notation

How to determine the significant figures of very large and very small numbers? One benefit of scientific notation is you can easily express the number in the correct number significant figures.

For example, you are not sure that this number 17100000000000 has two, three or five significant figures. If this number has two significant figures, this number can be expressed in scientific notation as $1.7 \times 10^{13}$. If this number has five significant figures, it can be expressed in scientific notation as $1.7100 \times 10^{13}$.

## How to Do Scientific Notation Using Calculators?

All scientific calculators allow you to express numbers in scientific notation and do calculation. The buttons to express numbers in scientific notation in calculators look like EXP, EE, $\times 10^{n}$ etc. The button depends on the make and model of your calculator but the function is the same in all calculators.

For example, in some calculators if you want to write $1.71 \times 10^{13}$ in scientific notation you write 1.71E13 using the button EXP or EE in the display screen. The button EXP or EE display E or e in calculator screen which represents the exponent. Other buttons such as $\times 10^n$ or $\times 10^x$ etc allow you to add exponent directly in the exponent form including the $\times 10$. Explore a little bit in your calculator and you'll be easily able to do calculations involving scientific notation.


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# Question on the Projection Theorem

My question concerns the second sentence in the last paragraph i.e. 'The projection theorem guarantees that there is at least one solution of $\phi_n$ of (2.3.9)'. If a vector is uniquely defined by $\phi_n$ how is it that there can be more than one solution of $\phi_n$? Also, how does the singularity of $\Gamma_n$ imply that there are infinitely many solutions for $\phi_n$?

• The key is in the next sentence: "If $\Gamma_n$ is singular..." In that case, the vector $\phi_n$ is not uniquely defined. The reason is that you, in effect, have fewer than $n$ independent vectors that have been formed into $n$ linear combinations that make up $\Gamma_n$. In that case, your coefficients aren't unique either; if $x_1 + x_2 = x_3$, then $\beta_1, \beta_2$, and $\beta_3$ can take on lots of different values while retaining the same value of $x_1\beta_1 + x_2\beta_2 + x_3\beta_3$. – jbowman Apr 28 '18 at 16:39
• Thanks a lot @jbowman! So what does that mean geometrically? Surely there should only be one point in a linear subspace $\mathcal{M}$ where an element $x \in \mathcal{H}$ (where $\mathcal{H}$ is a Hilbert Space) can project to? – Vykta Wakandigara Apr 28 '18 at 16:49

Let's consider a simple example, one where $\Gamma_2$ has two columns $[\gamma_1, \gamma_2]$ that are identical: $\gamma_1 = \gamma_2$. In this case, any projection is necessarily on to the space spanned by, let us choose for concreteness, $\gamma_1$, although it in some sense appears to be onto a space spanned by two vectors.
Let us assume the best projection onto $\gamma_1$ is $\beta \gamma_1$. In the space $[\gamma_1, \gamma_2]$, though, we have that any $\beta_1, \beta_2$ such that $\beta_1\gamma_1 + \beta_2\gamma_2 = \beta\gamma_1$ will be an equally good projection. Since (in our special case) $\gamma_1 = \gamma_2$, all we need is that $\beta_1 + \beta_2 = \beta$, and there are an infinite number of $\beta_1, \beta_2$ combinations such that this will hold.
More generally, you're projecting onto a $k < n$ dimensional space, but you have $n$ vectors in that space to choose from when defining that projection. So there are clearly multiple ways of defining the projection, all of which are identical in terms of the end result.
• A plane cuts through the hypercube and you project onto the plane (when you think you're projecting onto the hypercube, if you don't know that $\Gamma$ is less than full rank.) Now, there is a line in the hypercube that is a) perpendicular to the plane, and b) points in the direction of the original point. Any point on this line is a) in the hypercube and b) indistinguishable from any other point, because they all get projected to the same point on the plane. At least that's how I envision it, but I'm not the best visual thinker around. – jbowman May 3 '18 at 18:16


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# Meaning of 'hue" in seaborn [closed]

I know what "hue" does a little

I'm studying kaggle and the image is about bike sharing demand analysis

What I want to know is

shouldn't the second images's sum of each point's y axis be the first images's point's y axis??

You would expect that to be the case, however, by default seaborn.pointplot uses the average estimator to calculate the number for each hour. So the numbers you are seeing on the y-axis is the average number of bikes shared for each hour. Since the number of bikes shared is not equal for category workingday=0 and workingday=1 the two averages for those categories do not add up. Using estimator=sum for seaborn.pointplot does give the expected results.


text:
## Sunday, December 18, 2016 ... //

### Czech L-159 jets destroy Daesh info center, arms depots

Production of ALCAs was resumed

A year ago, I wrote that 15 L-159 ALCA jets that were previously labeled "redundant" by the Czech Air Force were sold to the Iraqi Air Force. L-159 is a cute successor of L-59 (and therefore an older L-39 Albatros) – a training and combat aircraft.

According to Wikipedia, the price should be around $10 million a piece but Iraq got it for$2 million. Well, that's very cheap, I think – we sort of subsidized the fight against Daesh, I think. (We're planning to donate them some extra ammunition, too.) It seems that the officials of the Iraqi Air Force love it – the most beloved things are sometimes gotten cheaply and easily. It seems that when they buy another package – the production of L-159 got resumed in Aero Vodochody (with a Honeywell engine) – it will be for $10 million or more, not$2 million.

For decades, these toys – which are also able to seriously fight – were slow sellers of a sort. And the users of them were largely "learning to fly". Well, it has changed in recent months.

For a month or two, the Iraqi Air Force was using their new L-159 toys to fight the Caliphate. The Twitter account of the Iraqi Air Force often informed us about some happy surprises that the 115th Squadron – which has these Czech jets – has prepared for the Caliphate.

Finally, yesterday, official Iranian media (in English) and Iraqi media (in Arabic) informed us about some particular achievements of those L-159 jets:

Earlier on Saturday, Aero L-159 ALCA fighter jets of the Iraqi Air Force pounded a Daesh communication center in the central Bab al-Toub neighborhood of Mosul, in addition to an arms depot in al-Darnaj village.
There were several arms and ammunition depots pounded and mentioned in those articles but it's not clear which of those were targeted by L-159 and which by F-16.

It's sort of exciting to see that these toys are finally making some difference. These jets are small enough and sort of "minimal" to achieve certain things – but that may be ideal to actually do things in a conflict that is not too high-tech in most cases. The pilots know that they're only responsible for a toy they got for $2 million. So they may probably afford to do various things with them. My belief is that they may get much closer to the enemy, too. My understanding is that only 12 of the 15 jets are meant to be used to fight – while the remaining 3 were sold to be disassembled to have some spare components for repairs. Only a part of the jets has been delivered and four of them are flying every day, an additional batch will arrive next week; corresponding 4 extra Iraqi pilots are being trained in Pardubice, CZ. The standard practice is for an ALCA to carry one 500-pound bomb on each wing (so far just "dumb", not guided, bombs). The fly at more than 1-kilometer altitude during the fights – a flight video 30 meters above the ground (in Sweden) is somewhere in the comments below. General Faris Hasan Falah, deputy commander of the Iraqi air force: Our commanders specifically ask for L-159 because they can aim the targets well and they can be well operated above the ground. Czechia has been an important enough producer of weapons. For example, when the Third Reich was conquering the Western Continental Europe, it was primarily relying on tanks produced here in Pilsen (Škoda Works). Assad's father has bought lots of the Czechoslovak weapons in the 1970s and 1980s – and has created quite some debt that has already been repaid. That's different from the Cuban debt to Czechia. The Cubans want to repay$276 million in rum, we were just told. It's a lot of rum. I am not sure whether even our president would be capable of drinking so much rum – and he's a world leader in similar disciplines. ;-)

To sell weapons may bring some ethical dilemmas. I, for one, think that they are heavily exaggerated. The people over there have to solve some disagreements and they are likely to kill or otherwise hurt the foes in one way or another. If someone is killed by a jet, is it so much worse than when his throat is cut? I am not so sure. To buy jets means to have a certain advantage. When such possibilities exist, it gives an advantage to the side of the conflict that has more money.

I don't claim that it's a rule but I believe that there is a positive correlation between the side's financial wealth and its being a more humane and civilized side of the conflict. I surely do think that that the Iraqi government is one of the most civilized sides in this collection of ongoing conflicts. But I do appreciate that in a year or several years, I could be sorry for this current attitude of mine. The current Iraqi government is probably increasingly close to Iran and with some dangerous Iranian policies – especially the so far verbal attacks against Israel – these Shiites could very well turn into the "most obvious villains" of the Middle East in a foreseeable future.

But the Czech maker of the aircraft can't really be held responsible for all such conceivable future mishaps, I think. They're doing some politically impartial engineering and business and that's the primary angle that gives me some moderate pride because I am Czech. Like science, the aircraft may be used for more noble goals or less noble goals and it's primary the user who gives the ethical signs to science and technology. Also, while something may go wrong with Iraq, I think it's very unlikely that something will become great about Daesh and I will be sorry that they will have been pounded by the jets.

And there's one more point I wanted to make in the morning but I forgot: I think it's way more natural to sell these folks – like the Iraqi government – some technology and train them than to join the fights with our own armies. At the end, I do think that Daesh is primarily their problem, it is their war, not ours.

Good luck to them.


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# How many AR markers should I use to get the object pose?

I'd like to get the object pose based on the attached AR marker to the object. I believe I only need one tag to get the 6D pose. However, I sometimes see more than one AR markers are used in the research papers or the demos by Boston Dynamics (https://boygeniusreport.files.wordpress.com/2016/02/screenshot-76.png) or by the team at Amazon Robotics Challenge.

Do I need to use more than one marker? If so, how can I calculate the 6D pose from more than one marker's information?

• Correct, you can get the information you need with one ArUco marker. We have performed automatic landing of multirotors using a large single marker. If you would like to utilize a marker map with rigidly defined 3D geometry, you can just use the library, which uses OpenCV's solvePnPRansac under the hood. If you want to use disparate markers like in the Boston Dynamics image, you will have to do some sort of averaging, which is simple for translation but requires more finesse for the orientation. – Parker Lusk Sep 13 '17 at 4:02
• I'm using solvePnP. Do you think solvePnPRansac is better for the pose estimation from an ArUco marker rather than solvePnP? – kangaroo Sep 13 '17 at 19:59
• It depends on your geometry and exactly what you are trying to do. Use solvePnPRansac if you have noisy points / possibly incorrect 2d to 3d correspondences. – Parker Lusk Sep 13 '17 at 23:54


text:
### Book Store

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CBSE Gujarat Board Haryana Board

### Previous Year Papers

Download the PDF Question Papers Free for off line practice and view the Solutions online.
Currently only available for.
Class 10 Class 12
A compound is formed by two elements M and N. The element N forms ccp and atoms of M occupy 1/3 of tetrahedral voids. What is the formula of the compound?

The number of tetrahedral voids formed is equal to twice the number of atoms of element N and only 1/3rd of these are occupied by the atoms of element M. Hence the ratio of number of atom of M and N is 2 x (1/3) : 1 or 2 : 3. So the formula of the compound is M2 N3.
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Refractive index of a solid is observed to have the same value along all directions. Comment on the nature of this solid. Would it show clevage property?

Refractive index of a solid which have the same value along all directions are isotropic in nature. It would not show cleavage property.
1055 Views . 11 Shares

Classify the following as amorphous or crystalline solids: Polyurethane, naphthalene, benzoic acid, teflon, potassium nitrate, cellophane, polyvinyl chloride, fibre glass, copper.

Amorphous solids: Polyurethane, teflon, cellophane, polyvinyal chloride, fibre glass.
Crystalline solids: naphthlene, benzoic acid, potassium nitrate, copper.
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Why is glass considred a super-cooled liquid?

Glass is amorphous solids and  have a tendency to flow, though very slowly. Therefore, these are called pseudo solids or super cooled liquids .
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Why are solids rigid?

Solids are rigid because the internuclear distance between the  molecules of a solid are very less and closely packed and their positions are fixed due to the strong forces of attraction between them.
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Why do solids have a definite volume?

A solid have a definite volume because the inter molecular distance  between its molecules are fixed.
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Do a good deed today
Refer a friend to Zigya


text:
# Looking for Something? Just #find It!

Searching through nested data structures can be downright messy. Even saying “nested data structures” sounds complicated. Luckily, Ruby has some powerful Enumerable methods that can simplify our lives (or at least our code).

Let’s say you’ve been programming for hours and suddenly realize your stomach is growling at you (why does this happen so often?). Here is how you might search for some delicious food, if you happened to be searching inside an array filled with hashes. Just humor me for the sake of example.

Here is our restaurants array:

So, we want to get the name of a restaurant whose :rating is "delicious". We could do something like this:

If we call this method on our restaurants hash, it will return the value “The GoGo Grill”. Sounds good to me. But in order to get there we needed a return inside an if statement inside an each loop inside another each loop. Not pretty.

There is a better way. The find method! Behold:

Just one line of code seems to do the same thing as its ugly predecessor. Let’s look at it more closely. Here is what find does:

• It passes each element of the object on which it was called to a block.
• It returns the first element for which the block evaluates to true.
• If none of the elements return true for the given block, it returns nil.

As it turns out, this nil value could cause some problems for our find_delicious_food method. If there are no restaurants in the hash with a :rating of "delicious", our code breaks. We would get an error, because Ruby cannot make any sense out of nil[:name].

Here is the method refactored to avoid that error:

Sure, it may not be as cute and little as our one-line method, but it will never break and it’s still a lot prettier than loops inside loops.

So when you are searching for that one thing you need inside of some nasty nesting, just find it!


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# Integration of Sinx

Here you will learn proof of integration of sinx or sin x and examples based on it.

Let’s begin –

## Integration of Sinx or Sin x

The integration of sinx is -cosx + C

where C is the integration constant.

i.e. $$\int$$ (sinx) dx = -cos x + C

Proof :

We will prove this formula using differentiation,

Let $$d\over dx$$(-cos x + C) = -$$d\over dx$$ cos x + $$d\over dx$$ C

Using differentiation formula,

$$d\over dx$$ cos x = -sin x and differentation of constant is 0.

$$\implies$$ $$d\over dx$$(-cos x + C) = -$$d\over dx$$ cos x + $$d\over dx$$ C

$$\implies$$ $$d\over dx$$(-cos x + C) = -(-sin x) + 0

We can also write it as,

sin x = $$d\over dx$$(-cos x + C)

Now, integrating on both sides,

$$\int$$ sin x = $$\int$$ $$d\over dx$$(-cos x + C)

We know that integration and differentiation both are reciprocals of each other, so in right hand side expression they cancel each other and we get,

Hence, $$\int$$ sin x = -cos x + C

Example : Prove that $$\int$$ sin (ax + b) = $$-1\over a$$ cos(ax + b) + C.

Solution : We have,

I = $$\int$$ sin (ax + b) dx

Let ax + b = t, Then , d(ax + b) = dt $$\implies$$ adx = dt

$$\implies$$ dx = $$1\over a$$ dt

Putting ax + b = t and  dx = $$1\over a$$ dt , we get

$$\int$$ sin (ax + b) dx = $$1\over a$$ $$\int$$ sin t dt

= $$-1\over a$$ cos t + C

= $$-1\over a$$ cos(ax + b) + C

Hence, $$\int$$ sin (ax + b) = $$-1\over a$$ cos(ax + b) + C

### Related Questions

What is the Differentiation of sin x ?

What is the Integration of Sin Inverse x ?

What is the Differentiation of sin inverse x ?


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# Asymmetric Beta

What is Asymmetric Beta?

Asymmetric is imbalance. Asymmetric is more of one thing, less of another.

An example of asymmetric is more profit than loss or more reward than risk. The payoff is non-linear.

The beta of an investment is a measure of the systematic or volatility because of exposure to market movements as opposed to individual security factors.

• Beta is the tendency of a security’s returns to respond to trends the market.
• Beta is also a measure of relative strength or relative momentum.

For example, if we are looking at the beta of a stock and assume the S&P 500 stock index is “the market” if the stock has a beta of 1.5 it means the price has moved 50% more than “the market”.

So, an Asymmetric Beta must mean the beta (movement relative to the market) is asymmetric (imbalanced).

Asymmetric Beta is a position taken with a non-linear asymmetric payoff. That is, the beta is asymmetric (more or less than 1) and more than symmetry (a beta of 1).

To see asymmetric investment returns, visit ASYMMETRY®

Asymmetric Payoff

Asymmetric Investing

Asymmetric Risk


text:
Creating speed more than light?

1. Aug 9, 2015

Auror

If we make a great voltage difference and let electricity flow , electron's kinetic energy will be equal to $eV$ . For some value of V (we wish to give) , won't the velocity be over speed of light ? What's wrong with such approach of thinking ?

Again , if we increase temperature of a system of gas , average speed,rms speed will increase and a some temperature it should cross speed of light . What's the bug in such idea ?

2. Aug 9, 2015

Orodruin

Staff Emeritus
You cannot accelerate an electron to the speed of light no matter how much energy you put in, the relativistic formula for total energy is $E = \gamma m$, where $\gamma$ is the gamma factor.

3. Aug 9, 2015

Borg

The problem is that you can't just add velocities in a simplistic manner to get something to go faster than light. It's more complex than that as you can see from the Wiki article on Special Relativity. It's a very long article but that's what it takes to understand why your not thinking about it correctly.

4. Aug 9, 2015

Staff: Mentor

The electron's KE is $KE = \frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2$. This goes to infinity as v goes to c.

5. Aug 9, 2015

bcrowell

Staff Emeritus
There are a nice educational paper and accompanying video in which exactly the experiment you describe was carried out:

Bertozzi, "Speed and kinetic energy of relativistic electrons," Am. J. Phys. 32 (1964) 551, http://www.scribd.com/doc/258743358/Bertozzi-Speed-and-kinetic-energy-of-relativistic-electrons-Am-J-Phys-32-1964-551 [Broken]

If you look at figure 3 in the paper, you'll see that the velocity approached c but never surpassed it, even when Newton's laws would have predicted it to go many times faster than c.

Last edited by a moderator: May 7, 2017
6. Aug 9, 2015

soothsayer

Simply put, your eV formula only applies for non relativistic electrons. As the electrons become relativistic, the rules change significantly. This is the same reason you can't just look at $\frac{1}{2}mv^2$ and say "if I put in enough energy, my velocity could exceed c!"

7. Aug 9, 2015

Orodruin

Staff Emeritus
The kinetic energy will still be eV. It will just not be enough to accelerate the electron to light speed or above because of how relativistic mechanics work.

8. Aug 9, 2015

soothsayer

Oh, you're right of course. My bad. I was somehow thinking it was a velocity equation and not energy...

9. Aug 10, 2015

Markus Hanke

This approach is based on the assumption that speeds ( kinetic energies ) add linearly; in Special Relativity however this is no longer true as you approach the speed of light. The faster you go, the less effect adding more energy will have, so it gets increasingly difficult to add to your speed. The speed of light itself is never reached - it would require an infinite amount of energy to get there, which is of course not possible. As others have pointed out, relativistic kinematics are based on hyperbolic geometry, not Euclidean geometry, so they work differently from what we might have learned back in high school.

10. Aug 10, 2015

Staff: Mentor

Kinetic energies do add linearly. It's just speeds that don't.

11. Aug 11, 2015

Markus Hanke

Apologies, you are right ! My bad
Thanks for pointing it out.


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Alcubierre Drive - Clarification on relativistic effects

On the Wikipedia article on the Alcubierre drive, it says:

Since the ship is not moving within this bubble, but carried along as the region itself moves, conventional relativistic effects such as time dilation do not apply in the way they would in the case of a ship moving at high velocity through flat spacetime relative to other objects.

And...

Also, this method of travel does not actually involve moving faster than light in a local sense, since a light beam within the bubble would still always move faster than the ship; it is only "faster than light" in the sense that, thanks to the contraction of the space in front of it, the ship could reach its destination faster than a light beam restricted to travelling outside the warp bubble.

I'm confused about the statement "conventional relativistic effects such as time dilation do not apply".

Say Bob lives on Earth, and Jill lives on a planet in Andromeda, and we'll say for the sake of argument that they're stationary. If I were to travel from Bob to Jill using an Alcubierre drive such that the journey would take me, say, 1 week from my reference frame... how long would Jill have to wait from her reference frame? Do the time dilation effects cancel out altogether? Would she only wait 1 week?

Spacetime can dynamically evolve in a way which apparently violates special relativity. A good example is how galaxies move out with a velocity v = Hd, the Hubble rule, where v = c = Hr_h at the de Sitter horizon (approximately) and the red shift is z = 1. For z > 1 galaxies are frame dragged outwards at a speed greater than light. Similarly an observer entering a black hole passes through the horizon and proceeds inwards at v > c by the frame dragging by radial Killing vectors.

The Alcubierre warp drive is a little spacetime gadget which compresses distances between points of space in a region ahead of the direction of motion and correspondingly expands the distance between points in a leeward region. If distances between points in a forwards region are compressed by a factor of 10 this serves as a “warp factor” which as I remember is $w~=~1~+~ln(c)$, so a compression of 10 is a warp factor 3.3. The effect of this compression is to reduce the effective distance traveled on a frame which is commoved with the so called warp bubble. This compression of space is given by $g_{tt}$ $=~1~-~vf(r)$.

Of course as it turns out this requires exotic matter with $T^{00}~<~0$, which makes it problematic. Universe is also a sort of warp drive, but this is not due to a violation of the weak energy condition $T^{00}~\ge~0$. Inflationary pressure is due to positive energy. The gravity field is due to the quantum vacuum, and this defines an effective stress-energy tensor $T^{ab}$ with components $T^{00}~=~const*\rho$, for $\rho$ energy density, and $T^{ij}~=~const*pu^iu^j$, for $i$ and $j$ running over spatial coordinates $u^i$ velocity and $p$ pressure density. For the de Sitter spacetime the energy density and pressure satisfies a state $p~=~w*\rho$ where $w~=~-1$. So the pressure in effect is what is stretching out space and frame dragging galaxies with it. There is no need for a negative energy density or exotic matter.

Negative energy density or negative mass fields have serious pathologies. Principally since they are due to quantum mechanics the negative eigen-energy states have no lower bound. This then means the vacuum for these fields is unstable and would descend to ever lower energy levels and produce a vast amount of quanta or radiation. I don’t believe this happens. The Alcubierre warp drive then has a serious departure between local laws of physics and global ones, which is not apparent in the universe or de Sitter spacetime. The Alcubierre warp drive is then important as a gadget, along with wormholes as related things, to understand how nature prevents closed timelike curves and related processes.

The question was asked about the redshift factor and the cosmological horizon. This requires a bit more than a comment post. On a stationary coordinate region of the de Sitter spacetime $g_{tt}~=~1~-~\Lambda r^2/3$. This metric term is zero for $r~=~\sqrt{3/\Lambda}$, which is the distance to the cosmological horizon.

The red shift factor can be considered as the expansion of a local volume of space, where photons that enter and leave this “box” can be thought of as a standing wave of photons. The expansion factor is then given by the scale factor for the expansion of the box $$z~=~\frac{a(t_0)}{a(t)}~-~1$$ The dynamics for the scale factor is given by the FLRW metric $$\Big(\frac{\dot a}{a}\Big)^2~=~\frac{8\pi G\rho}{3}$$ for $k~=~0$. The left hand side is the Hubble factor, which is constant in space but not time. Writing the $\Lambda g_{ab}~=~8\pi GT_{ab}$ as a vacuum energy and $\rho~=~T_{00}$ we get $$\Big(\frac{\dot a}{a}\Big)^2~=~H^2~=~\frac{\Lambda}{3}$$ the evolution of the scale factor with time is then $$a(t)~=~\sqrt{3/\Lambda}e^{\sqrt{\Lambda /3}t}.$$ Hence the ratio is $a(t)/a(t_0)~=~ e^{\sqrt{\Lambda /3}(t-t_0)}$.The expansion is this exponential function, which is Taylor expanded to give to first order the ratio above $$a(t)/a(t_0)~\simeq~1~+~H(t_0)(t_0-t)~=~1~+~H(t_0)(d-d_0)/c$$ which gives the Hubble rule. $z~=~a(t)/a(t_0)~-~1$. It is clear that from the general expression that $a(t)$ can grow to an arbitrarily large value, and so can $z$. On the cosmological horizon for $d~-~d_0~=~r_h~=~\sqrt{3/\Lambda}$ we have $z~=~1$.

Looking beyond the cosmological horizon $r_h~\simeq~10^{10}$ly is similar to an observer in a black hole looking outside to the exterior world outside the black hole horizon. People get confused into thinking the cosmological horizon is a black membrane similar to that on a black hole. Anything which we do observe beyond the horizon we can never send a signal to, just as a person in a black hole can see the exterior world and can never send a message out.

• Isn't $z=\infty$ the redshift of the horizon? $1+z = \frac{\lambda_{now}}{\lambda_{then}}$ Apr 19, 2011 at 13:18
• May I ask what you do for a living Lawrence B. Crowell? Jun 16, 2014 at 14:21
• So, with all this mathematically gobbilty-gook you've written (that I barely comprehend) are you saying in terms of outer space, gravity is such a weak force that it wouldn't take "much" energy for the warp drive to expand/contract space? Similar to the way cleaning vacuums create very high suction by only generating a very small amount of negative pressure? Jun 16, 2014 at 14:26
• While you exposé of the physics behind the Alcubierre Drive is interesting, I am still curious about the answer to the actual question: How much time will the person travelling from Bob to Jill experience compared to the amount of time experienced by Jill (assuming the traveller is moving at non-relativistic speeds within the bubble)? Jul 12, 2017 at 20:02

Relativistic effects happen when you travel near light speed through space-time. With the Alcubierre drive, you don't travel through space-time, but remain stationary and the space-time around you is warped in a way that brings you closer to your destination. So if it only took a week for you, it would only take a week for your observers.

It'd be like if you put two objects on a sheet, and instead of moving them across the sheet, you scrunched up the part of the sheet between them--moving them closer together even though they both remained stationary relative to the sheet.

• 2 things about your 2nd paragraph: 2nd, if you move objects around on a sheet, there's no reason time dilation would enter into the situation unless we're talking about a very large sheet, and very large speeds. 2nd, I'm not really able to visualise the "scrunching". Assuming the sheet is elastic, you could move a region of the sheet closer to a point. In that case, there would be compressed space in between, and expanded behind, but the only way the region can stay there is if the distortions stay. Only solution I can visualise would be to cut the region, and place it at the destination? Apr 19, 2011 at 22:51
• its not a particularily enlightening analogy in this case.. usually sci-fi movies about wormholes try to use the "folded paper with a pen through it"-trick when they explain how to go from A to B in space by topology though :) Apr 20, 2011 at 0:01
• I was just excited to see a question on this site that I could at least partially answer :) Apr 21, 2011 at 2:31

I'm confused about the statement "conventional relativistic effects such as time dilation do not apply".

The Wikipedia section: "Alcubierre Metric" has been updated since your question was asked, but I suspect it's no clearer to you than what you quoted:

The Alcubierre metric defines the warp-drive spacetime. It is a Lorentzian
manifold that, if interpreted in the context of general relativity, allows
a warp bubble to appear in previously flat spacetime and move away effectively
faster than lightspeed. The interior of the bubble is an inertial reference
frame and inhabitants suffer no proper acceleration. This method of transport
does not involve objects in motion at speeds faster than light with respect
to the contents of the warp bubble; that is, a light beam within the warp
bubble would still always move faster than the ship. Because objects within
the bubble are not moving (locally) faster than light, the mathematical
formulation of the Alcubierre metric is consistent with the conventional
claims of the laws of relativity (namely, that an object with mass cannot
attain or exceed the speed of light) and conventional relativistic effects
such as time dilation would not apply as they would with conventional motion
at near-light speeds.

The oversimplified answer to that point is: "The space moves, it's movement does not dilate your time, only your own movement dilates your time.".

Note the statement above: "... inhabitants suffer no proper acceleration".

See also Miguel Alcubierre's paper (link below), on page 6:

To prove that the trajectory of the spaceship is indeed a timelike curve, regardless of the value of $v_s(t)$, we substitute $x = x_s(t)$ in the metric (8). It is then easy to see that for the spaceship's trajectory we will have: $$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad d=dt .\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (13)$$ This implies not only that the spaceship moves on a timelike curve, but also that its proper time is equal to coordinate time. Since coordinate time is also equal to the proper time of distant observers in the at region, we conclude that the spaceship suffers no time dilation as it moves. It is also straightforward to prove that the spaceship moves on a geodesic. This means that even though the coordinate acceleration can be an arbitrary function of time, the proper acceleration along the spaceship's path will always be zero.

So the Drive itself causes no time dilation for the occupants of the 'warp field bubble'.

Say Bob lives on Earth, and Jill lives on a planet in Andromeda, and we'll say for the sake of argument that they're stationary. If I were to travel from Bob to Jill using an Alcubierre drive such that the journey would take me, say, 1 week from my reference frame... how long would Jill have to wait from her reference frame? Do the time dilation effects cancel out altogether? Would she only wait 1 week?

For more info about the Alcubierre drive see Harold White's paper at NASA Warp Field Mechanics 101 (.PDF) or Miguel Alcubierre's paper at arXiv.org The warp drive: hyper-fast travel within general relativity.

From what I understand of the mathematical contrivance ...

From the Alcubierre paper:

• Page 2: "... one can actually make such a round trip in an arbitrarily short time as measured by an observer that remained at rest ...".

• Page 3: "... one can use an expansion of spacetime to move away from some object at an arbitrarily large speed. In the same way, one can use a contraction of spacetime to approach an object at any speed. This is the basis of the model for hyperfast space travel that I wish to present here: create a local distortion of spacetime that will produce an expansion behind the spaceship, and an opposite contraction ahead of it. In this way, the spaceship will be pushed away from the Earth and pulled towards a distant star by spacetime itself. One can then invert the process to come back to Earth, taking an arbitrarily small time to complete the round trip.".

• Page 7: Note 3 indicates, from a practical standpoint (if there's such a thing with an impractical device), that you'll want to move some distance from your starting point (Earth) and discontinue use of the drive some distance before arriving at your destination (the Andromeda Gallery, 2.54 ± 0.11 Mly away); failure to do so would disturb the spacetime of the endpoints.

The implication of that is that your motion will cause an extremely tiny time dilation that you might wish to disregard; but if I don't mention it someone may comment about the omission. The faster you travel to the origin's standoff location the sooner you can go to warp, which in turn increases the dilation; but still only by a relatively small amount. Upon arrival you'll want to approach Jill's location at a reasonable speed and landing or docking will take an additional unknown amount of time.

If you traveled 0.10 $c$ you would save (60 * 60) - (0.995 * 60 * 60) = 18 second per hour, and at 0.866 $c$ you'd save 30 minutes per hour, accelerating to that speed is going to take some time.

• Page 8: So it will only take: $$\tau \simeq T \simeq 2 \sqrt{\frac Da}$$

"It is now clear that $T$ can be made as small as we want by increasing the value of $a$. Since a round trip will only take twice as long, we find that we can be back at Earth after an arbitrarily small proper time, both from the point of view of the spaceship and from the point of view of the people on Earth (or at the destination). The spaceship will then be able to travel much faster than the speed of light. However, as we have seen, it will always remain on a timelike trajectory, that is, inside its local light-cone: light itself is also being pushed by the distortion of spacetime.".

So setup time is the only consideration, after obtaining exotic matter and building the ship and engines.

... how long would Jill have to wait from her reference frame? Do the time dilation effects cancel out altogether? Would she only wait 1 week?

If you get into position prior to contacting Jill you'll arrive in her vicinity well before she gets the message. If you drag debris from across the galaxy and impact your destination with them she may be ticked off when you arrive (if you survive the journey). It won't be a week and your unlikely to have an hour difference in your clocks. You'll need to determine standoff distance and how fast you can get there to be any more precise than that.

There's a detailed technical explanation in those documents. I provided my additional answer because there are comments to the effect that your question had not been answered.

String theory (and all other theories involving hidden dimensions) predict that gravity and electromagnetism unify in hidden dimensions and that the hidden dimensions are indetectible because of their small size. It does also predict that sufficiently short-waved photons, with wavelengths shorter than the size of the hidden dimensions, can enter them. Producing ultra-short photons can thus manipulate gravity, with revolutionizing space travel applications such as cheap anti-gravity launches. The problem that it would require high energy can be practically solved by concentrating several laser beams on a nanoparticle, heating it to locally extreme temperatures. An Alcubierre metric can be created by ejecting multiple nanoparticles from the craft and then beam perfectly timed laser beams on them (fire at the most distant first so that they are hit simultaneously), so each nanoparticle contributes a slower than light effect but together add up to faster than light, creating no discrete event horizon and thus no Hawking radiation.

• @ Martin J Sallberg - Can you provide any citations supporting your answer? Aug 1, 2011 at 10:48