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# Which limit for matsubara frequency sum?

in the context of a simple toy problem for Feynman path integrals, I consider a two-site Hubbard model for spinless fermions. I expand the path integral to first order in the interaction $V$, which means that I have to compute an "average" of the type

$$\left\langle \bar \psi_1(\omega_n) \bar \psi_2(\omega_m) \psi_2(\omega_p) \psi_1(\omega_q) \right\rangle$$ with respect to the non-interacting action. Since the Grassmann numbers for sites $1$ and $2$ are different numbers, this average immediately factors, and I'm left with two-variable averages, which give me just the non-interacting Matsubara Green's functions. But now I have to calculate a Matsubara sum of the type

$$\frac{1}{\beta} \sum_n \frac{1}{i\omega_n - \epsilon}$$ where $\epsilon$ is just a constant.

From searching in the literature, I know that contour integration is the way to go, but unfortunately this sum is ambiguous and I have to introduce a factor $e^{-i\omega_n \tau}$ with $\tau \rightarrow 0^+$ or $\tau \rightarrow 0^-$, and the result of the contour integration will be quite different depending on which limit I take.

My question now is, how do I know what limit to take? And is there an easier way of carrying out the sum over frequency?

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Yes, for this particular Matsubara sum you mentioned, taking different limits will lead to two results differed by 1. This is because the summation in consideration does not converge, which can be seen from the following integral (the continuous limit of the sum) considering the ultraviolet divergence (the large $\omega$ behavior) $$\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\omega\frac{1}{i\omega-\epsilon}\sim\frac{1}{2\pi i}\int_{-\infty}^{\infty}\mathrm{d}\omega\frac{1}{\omega}\rightarrow\infty.$$ Asking for the result of an essentially divergent sum will not even lead to a definite answer. Introducing the factor $e^{i\omega_n\tau}$ is to control the convergence of the sum, but the result will then depend on how one chooses to control the convergence, i.e. $\tau\rightarrow0^+$ (controlling the left-half complex plane) or $0^-$ (controlling the right-half complex plane).
The two results will be different by a shift of 1. Even this difference "1" has a physical meaning, reflecting the ambiguity in defining Fermion particle number. Because the summation $$\frac{1}{\beta}\sum_n\frac{1}{i\omega_n-\epsilon}=G(\tau=0)=-\mathcal{T}_{\tau}\langle\psi(\tau=0)\bar{\psi}\rangle$$ corresponds to the Green's function at imaginary time 0, which physically means to count the number of Fermions on the energy level $\epsilon$. Here $\mathcal{T}_{\tau}$ stands for the time ordering operator. However $\tau=0$ can be either understood as $\tau\rightarrow0^+$ or $\tau\rightarrow0^-$ (corresponding to the two ways to control the convergence), which becomes very critical here because the time ordering operator will then order the two cases differently: $$-\mathcal{T}_{\tau}\langle\psi(\tau=0^+)\bar{\psi}\rangle = -\langle\psi\bar{\psi}\rangle,$$ $$-\mathcal{T}_{\tau}\langle\psi(\tau=0^-)\bar{\psi}\rangle = \langle\bar{\psi}\psi\rangle.$$ According to the anticommutation relation of the Fermion operators, we have $\bar{\psi}\psi=1-\psi\bar{\psi}$, that is why it is expected that the two results will differ by a constant 1. In fact $\bar{\psi}\psi$ counts the number of particles, while $\psi\bar{\psi}$ counts the number of holes. It becomes a problem whether to define the Fermion number (with respect to the vacuum state) as the particle number or as the negative of the hole number. This depends on our definition of the vacuum state: whether we treat the vacuum as no particle state or as a state filled with particles (like the concept of Fermi sea). We know both choices are acceptable, just as we can either define electron as a particle in the empty space or as a hole in the anti-electron Fermi sea. Because of this ambiguity, one needs to specify the choice when the $\tau\rightarrow0$ limit is taken. In conclusion, both ways of taking the limit are legitimate, and the difference in the results is just a matter of the definition of Fermion vacuum state.
Similar discrepancy also exists in the Bosonic case, which is again related to the definition of Boson vacuum state. In general, such discrepancy roots in the divergence of the Matsubara summation in consideration. If the summation itself converges to a definite result, then the result will be unique no matter which limit is chosen to take. For example, if one tries to calculate $$\frac{1}{\beta}\sum_n\frac{1}{(i\omega_n-\epsilon)^2},$$ the result will always be $\beta/4 \mathrm{sech}^2 (\beta\epsilon/2)$ (for Fermionic case) no matter $\tau\rightarrow0^+$ or $0^-$.
Okay, I think that'll help me. So if I want to compute a certain average such as $\langle \bar \psi \psi\rangle$, because here the $\bar \psi$ is first I want to take the limit $\tau \rightarrow 0^-$ to preserve that ordering? –  Lagerbaer Feb 10 '12 at 16:11


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# Seeking a way out of speeding ticket

I had one accident in 2003, and i believe 3-5 speeding tickets since then.

the last speeding ticket i got was 2:10 am doing 85 in a 65. My charge was "reckless driving by speeding" the amount to pay is 191 dollars. I have the money order written out, but my father pays for insurance and this would disappoint me if i had to tell him.

the way it is rigged up is that the car i drive and the insurance is in his name, i am just a happenstance driver so the insurance company does not know i exist. so far on his policy i have gotten 2 or 3 speeding tickets, my memory is not so good.

i want to know if this ticket will affect my fathers insurance rates, like i said his bill is clean.

also, if i do decide to go to court (my license is from pennsylvania, i live in north carolina, and the court is in virginia 2 hours away) is there any chance the charge will be thrown out? Someone was tailing me and i could say that i wanted to get away from him because he wouldnt leave and i was getting nervous - partially true btw. or should i just pay the ticket because there is no way the charge will be dropped?

thanks

Pengwuino
Gold Member
With 3-5 past speeding tickets, you won't get it thrown out most likely. They probably won't even believe your story.

Does Pennsylvania not have the "Traffic School" option?

Here in California, anyone who gets a ticket can avoid getting it on their permanent record, and by extension can avoid the insurance company finding out about it, by attending a "Traffic School". This is about eight hours of class time going over all the traffic laws and rules. You still pay the fine, but there are no insurance consequences.

Evo
Mentor
Here if you pay twice the ticket price, they drop the ticket, it doesn't go on your record. I had two of the evo child's tickets erased that way. It's all about the money.

Pengwuino
Gold Member
zoobyshoe said:
Does Pennsylvania not have the "Traffic School" option?

Here in California, anyone who gets a ticket can avoid getting it on their permanent record, and by extension can avoid the insurance company finding out about it, by attending a "Traffic School". This is about eight hours of class time going over all the traffic laws and rules. You still pay the fine, but there are no insurance consequences.

Yah but your not allowed to do it if you've had a ticket in the 18 months prior to the ticket in question's issuance.

Evo's thing is crazy. Thats not fair... I wish i could have just skipped traffic school and payed twice for my speeding ticket. Wonder if its the same for red light running there.... its a $360 ticket here for red light running! Evo Mentor Pengwuino said: Yah but your not allowed to do it if you've had a ticket in the 18 months prior to the ticket in question's issuance. Evo's thing is crazy. Thats not fair... I wish i could have just skipped traffic school and payed twice for my speeding ticket. Wonder if its the same for red light running there.... its a$360 ticket here for red light running!
Check with the traffic court where you live, you might be surprised to find out they have the same deal. I think there might be a limit to how many you can pay off within a two year period.

interesting, my last ticket was most likely only 3 months ago. (summer month). ill have to call the courthouse on monday, but for driving school you have to have an NC license and i dont.

jma2001
Gold Member
Check this site. It is worth fighting, the ticket may not be dropped but most people who fight will be able to get the fines/points reduced.

http://www.motorists.org/issues/tickets/index.html [Broken]

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Pengwuino said:
Yah but your not allowed to do it if you've had a ticket in the 18 months prior to the ticket in question's issuance.
No, I'm pretty sure you can only attend traffic school once every 18 months. That is: regardless of how many tickets you have, you're only allowed to make one disappear every 18 months with traffic school.

jma2001
Gold Member
zoobyshoe said:
Here in California, anyone who gets a ticket can avoid getting it on their permanent record, and by extension can avoid the insurance company finding out about it, by attending a "Traffic School".
Here in California? And all this time I've been thinking you were from New Zealand! Don't know how I got that idea in my head.

Pengwuino
Gold Member
zoobyshoe said:
No, I'm pretty sure you can only attend traffic school once every 18 months. That is: regardless of how many tickets you have, you're only allowed to make one disappear every 18 months with traffic school.

jma2001 said:
Here in California? And all this time I've been thinking you were from New Zealand! Don't know how I got that idea in my head.
New Zealand? No. I grew up in New Hampshire, though. They're both New.

i love bureaucracy and red tape is my favorite

The lawyers for virginia cost more than my ticket, and driving there in gas will cost more than i have probably. i honestly was trying to lose the guy behind me, but i doubt if its enough to get it reduced. blast....signing away the money order as i speek.

devil-fire
iv read about a couple of things you can do that usualy work out. first is delay as long as you can, do anything and put it back as long as you can each and every time. if they officer dosnt show, you win and hes more inclined not to show if he dosnt remember you. the other benifit from delay is you can question the officer about the circumstances of the incident and "i cant remember"s play in your favor. dress well when you show to court. the other thing is to ask about when the equipment used to measure your speed what last calibrated. apparently these things have to be recalibrated annualy or something because there is a margin of error that emerges.

iv never been to court to fight a speeding ticket so i dont know how well the above actualy works, the place i got this stuff off of said it works great. you might be able to check on google for finding ways to beat tickets

Evo
Mentor
devil-fire said:
iv read about a couple of things you can do that usualy work out. first is delay as long as you can, do anything and put it back as long as you can each and every time. if they officer dosnt show, you win and hes more inclined not to show if he dosnt remember you. the other benifit from delay is you can question the officer about the circumstances of the incident and "i cant remember"s play in your favor. dress well when you show to court. the other thing is to ask about when the equipment used to measure your speed what last calibrated. apparently these things have to be recalibrated annualy or something because there is a margin of error that emerges.

iv never been to court to fight a speeding ticket so i dont know how well the above actualy works, the place i got this stuff off of said it works great. you might be able to check on google for finding ways to beat tickets
You can usually only delay traffic court once.

Always contest your tickets. Always. Especially in Canada, here in BC half the time the police man doesn't even show up, so the judge has no witness, no choice but to throw it out. (well - it used to be that way, they're fixing it up now)

Evo said:
You can usually only delay traffic court once.
Well you could always accidently lose your passport when travelling over seas. Maybe you get delayed at customs for a few days, maybe you miss your court date. Maybe they have no choice but to delay again.

Evo
Mentor
Smurf said:
Well you could always accidently lose your passport when travelling over seas. Maybe you get delayed at customs for a few days, maybe you miss your court date. Maybe they have no choice but to delay again.
You'll have a warrant for your arrest waiting for you. They don't mess around.

Here if you can't take the time to go to court, an attorney will take care of everything for only \$100, plus the double fine. (you learn a lot when you're the parent of teenagers :grumpy: ) It amazing what they can "fix". I :!!) my attorney.

Pengwuino
Gold Member
Evo said:
You'll have a warrant for your arrest waiting for you. They don't mess around.

Same here in CA. A parking ticket will be the least of your problem if you think you can get "delayed". They give you like 4 months to appear here so they wouldn't buy any of this "delayed" crap.

Moonbear
Staff Emeritus

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# Lightning and nuclear fusion

I'm going to be brief, I just saw a Discovery Channel show that showed a lot of interesting phenomena around lightning (like elves, how cool is that(!)), and got me wondering.

1) Thinking of lightning as a purely mechanical phenomenon, I would think the elves are the "other side of the momentum balance". What I mean is this: somewhere in some cloud formation, an event happens that triggers a lightning flash. This means a ton of electrons (and other, associated particles) that start a very fast journey to the surface of the Earth. Their momentum must be balanced by particles going in the opposite direction, hence elves. Am I right?

2) Taking this further, is it possible that the "trigger" for the lightning flash could be atomic/molecular fusion forced by extremely high electrical fields, that may only exist for a nanosecond, causing a ton of energy to be transferred to the electrons around the atoms/molecules, and we have lightning. I would then think of the energy necessary to force the fusion to occur as a quantum fluctuation as in $$\Delta E \Delta t \leq \hbar$$

3) Is every arc caused by a particle collision, somewhat like in a particle accelerator? It seems logical to me: there are more than enough particles in the air to collide with, and all the light could well be some form of Brehm or Cherenkov radiation.

The question I seem to be asking is where to find good scientific theory/information about lightning. Some say we know a lot about it, but I haven't found any good papers explaining it. I have a pretty good background in physics (1st year Master student) and am not afraid of serious literature. Thanks!

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Why the downvote? –  rubenvb May 27 '11 at 22:34
rubenvb, can you provide any reference for your suggestion that nuclear fusion is involved in lightning? Also what do you mean by molecular fusion? The existing answers haven't addressed this part, I guess because it's not clear where this idea is coming from. (I am not the one who downvoted.) –  Dan Brumleve May 28 '11 at 5:21
have a look at this article which correlates gamma rays with lightning. news.nationalgeographic.com/news/2007/10/… also aip.org/pnu/2007/split/841-2.html –  anna v May 28 '11 at 7:18
@Dan: the idea comes from the dark corners of my mind, and the comment made in the documentary about the fact that the clouds producing the lightning should not be able to produce enough energy to account for the energy in a lightning bolt –  rubenvb May 28 '11 at 10:16
""the clouds producing the lightning should not be able to produce enough energy to account for the energy in a lightning bolt "" Ahh, what is a "lightning bolt" ? –  Georg May 28 '11 at 12:35

no, that is not how momentum conservation works; the cloud gain a counter moment briefly which is transferred later to the earth, triggering vibrational modes but making net earth momentum conserved. no need to reach a balance. It might be plausible however, that positive ions travel in the opposite direction than electrons to balance the same electric potential, so the elves are probably these bursts of plasma ions moving upward

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Correct, no momentum considerations here. Elves as accelerated plasma ions is correct. –  Bill Slugg May 28 '11 at 3:52


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Review

# Cancer Genome Landscapes

See allHide authors and affiliations

Science  29 Mar 2013:
Vol. 339, Issue 6127, pp. 1546-1558
DOI: 10.1126/science.1235122

## Abstract

Over the past decade, comprehensive sequencing efforts have revealed the genomic landscapes of common forms of human cancer. For most cancer types, this landscape consists of a small number of “mountains” (genes altered in a high percentage of tumors) and a much larger number of “hills” (genes altered infrequently). To date, these studies have revealed ~140 genes that, when altered by intragenic mutations, can promote or “drive” tumorigenesis. A typical tumor contains two to eight of these “driver gene” mutations; the remaining mutations are passengers that confer no selective growth advantage. Driver genes can be classified into 12 signaling pathways that regulate three core cellular processes: cell fate, cell survival, and genome maintenance. A better understanding of these pathways is one of the most pressing needs in basic cancer research. Even now, however, our knowledge of cancer genomes is sufficient to guide the development of more effective approaches for reducing cancer morbidity and mortality.

...

Ten years ago, the idea that all of the genes altered in cancer could be identified at base-pair resolution would have seemed like science fiction. Today, such genome-wide analysis, through sequencing of the exome (click hlighlighted terms for definitions used in this Review) or of the whole genome, is routine.

### Challenges

One challenge of genome-based medicine in oncology is already apparent from the opportunities described above: All of the clinically approved drugs that target the products of genetically altered genes are directed against kinases. One reason for this is that kinases are relatively easy to target with small molecules and have been extensively studied at the biochemical, structural, and physiologic levels (129). But another reason has far deeper ramifications. The vast majority of drugs on the market today, for cancer or other diseases, inhibit the actions of their protein targets. This inhibition occurs because the drugs interfere with the protein’s enzymatic activity (such as the phosphorylation catalyzed by kinases) or with the binding of the protein to a small ligand (such as with G protein–coupled receptors). Only 31 of the oncogenes listed in tables S2 and S3 have enzymatic activities that are targetable in this manner. Many others participate in protein complexes, involving large interfaces and numerous weak interactions. Inhibiting the function of such proteins with small drugs is notoriously difficult because small compounds can only inhibit one of these interactions (130, 131).

.

### Highlights

1. Most human cancers are caused by two to eight sequential alterations that develop over the course of 20 to 30 years.

2. Each of these alterations directly or indirectly increases the ratio of cell birth to cell death; that is, each alteration causes a selective growth advantage to the cell in which it resides.

3. The evidence to date suggests that there are ~140 genes whose intragenic mutations contribute to cancer (so-called Mut-driver genes). There are probably other genes (Epi-driver genes) that are altered by epigenetic mechanisms and cause a selective growth advantage, but the definitive identification of these genes has been challenging.

4. The known driver genes function through a dozen signaling pathways that regulate three core cellular processes: cell fate determination, cell survival, and genome maintenance.

5. Every individual tumor, even of the same histopathologic subtype as another tumor, is distinct with respect to its genetic alterations, but the pathways affected in different tumors are similar.

6. Genetic heterogeneity among the cells of an individual tumor always exists and can impact the response to therapeutics.

7. In the future, the most appropriate management plan for a patient with cancer will be informed by an assessment of the components of the patient’s germline genome and the genome of his or her tumor.

8. The information from cancer genome studies can also be exploited to improve methods for prevention and early detection of cancer, which will be essential to reduce cancer morbidity and mortality.

Though one can at least imagine the development of drugs that inhibit nonenzymatic protein functions, the second challenge evident from table S2 poses even greater difficulties: A large fraction of the Mut-driver genes encode tumor suppressors. Drugs generally interfere with protein function; they cannot, in general, replace the function of defective genes such as those resulting from mutations in tumor suppressor genes. Unfortunately, tumor suppressor gene–inactivating mutations predominate over oncogene-activating mutations in the most common solid tumors: Few individual tumors contain more than one oncogene mutation (Fig. 5).

The relatively small number of oncogene mutations in tumors is important in light of the intrametastatic heterogeneity described earlier. To circumvent the inevitable development of resistance to targeted therapies, it will likely be necessary to treat patients with two or more drugs. The probability that a single cancer cell within a large metastatic lesion will be resistant to two agents that target two independent pathways is exponentially less than the probability that the cell will be resistant to a single agent. However, if the cancer cell does not contain more than one targetable genetic alteration (i.e., an oncogene mutation), then this combination strategy is not feasible.

Given the paucity of oncogene alterations in common solid tumors and these principles, can targeted therapeutic approaches ever be expected to induce long-term remissions, even cures, rather than the short-term remissions now being achieved? The saviors are pathways; every tumor suppressor gene inactivation is expected to result in the activation of some growth-promoting signal downstream of the pathway. An example is provided by PTEN mutations: Inactivation of the tumor suppressor gene PTEN results in activation of the AKT kinase (Fig. 8). Similarly, inactivation of the tumor suppressor gene CDKN2A results in activation of kinases, such as cyclin-dependent kinase 4, that promote cell cycle traverse (132). Furthermore, inactivation of tumor suppressor gene APC results in constitutive activity of oncogenes such as CTNNB1 and CMYC (133135).

We believe that greater knowledge of these pathways and the ways in which they function is the most pressing need in basic cancer research. Successful research on this topic should allow the development of agents that target, albeit indirectly, defective tumor suppressor genes. Indeed, there are already examples of such indirect targeting. Inactivating mutations of the tumor suppressor genes BRCA1 or BRCA2 lead to activation of downstream pathways required to repair DNA damage in the absence of BRCA function. Thus, cancer cells with defects in BRCA1 or BRCA2 are more susceptible to DNA damaging agents or to drugs that inhibit enzymes that facilitate the repair of DNA damage such as PARP [poly(adenosine diphosphate–ribose) polymerase] (136). PARP inhibitors have shown encouraging results in clinical trials when used in patients whose tumors have inactivating mutations of BRCA genes (137).

Further progress in this area will require more detailed information about the signaling pathways through which cancer genes function in human cancer cells, as well as in model organisms. One of the lessons of molecular biology over the past two decades is that pathway functions are different, depending on the organism, cell type, and precise genetic alterations in that cell (138). A pertinent example of this principle is provided by results of treatment with drugs inhibiting mutant BRAF kinase activity. In the majority of patients with melanomas harboring (V600E; V, Val; E, Glu) mutations in the BRAF gene, these drugs induce dramatic (though transient) remissions (108). But the same drugs have no therapeutic effect in colorectal cancer patients harboring the identical BRAF mutations (139). This observation has been attributed to the expression of EGFR, which occurs in some colorectal cancers but not in melanoma and is thought to circumvent the growth-inhibitory effects of the BRAF inhibitors. With this example in mind, no one should be surprised that a new drug that works well in an engineered tumor in mice fails in human trials; the organism is different, the cell type is usually different, and the precise genetic constitutions are always different. The converse of this statement—that a drug that fails in animal trials will not necessarily fail in human trials—has important practical consequences. In our view, if the biochemical and conceptual bases for a drug’s actions are solid and the drug is shown to be safe in animals, then a human trial may be warranted, even if it does not shrink tumors in mice.

### Genome-Based Medicines of the Future

Cancer genomes can also be exploited for the development of more effective immunotherapies. As noted above, typical solid tumors contain 30 to 70 mutations that alter the amino acid sequences of the proteins encoded by the affected genes. Each of these alterations is foreign to the immune system, as none have been encountered during embryonic or postnatal life. Therefore, these alterations, in principle, provide a “holy grail” for tumor immunology: truly tumor-specific antigens. These antigens could be incorporated into any of the numerous platforms that already exist for the immunotherapy of cancer. These include administration of vaccines containing the mutant peptide, viruses encoding the mutant peptides on their surfaces, dendritic cells presenting the mutated peptide, and antibodies or T cells with reactivity directed against the mutant peptides (140).

To realize these sorts of therapeutics, several conditions must be met. First, the mutant protein must be expressed. As cancer cells generally express about half of the proteins that are encoded by the human genome (141), this condition is not limiting. Second, as most proteins affected by mutations are intracellular, these mutations will not be visible to the immune system unless the mutant residue is presented in the context of a human leukocyte antigen (HLA) protein. Based on in silico analyses of binding affinities, it has been estimated that a typical breast or colorectal cancer contains 7 to 10 mutant proteins that can bind to an individual patient’s HLA type (142). These theoretical predictions have recently gained experimental support. Studies of mouse tumors have identified mutant genes and shown that the corresponding peptides can induce antitumor immunity when administered as vaccines (143). Moreover, clinical trials of brain cancer patients immunized against a mutant peptide have yielded encouraging results (144).

As with all cancer therapies that are attractive in concept, obstacles abound in practice. If a tumor expresses a mutant protein that is recognizable as foreign, why has the host immune system not eradicated that tumor already? Indeed, immunoediting in cancers has been shown to exist, resulting in the down-regulation or absence of mutant epitopes that should have, and perhaps did, elicit an immune response during tumor development (145, 146). Additionally, tumors can lose immunogenicity through a variety of genetic alterations, thereby precluding the presentation of epitopes that would otherwise be recognized as foreign (147). Though these theoretical limitations are disheartening, recent studies on immune regulation in humans portend cautious optimism (148, 149).

### Other Ways to Reduce Morbidity and Mortality Through Knowledge of Cancer Genomics

When we think about eradicating cancer, we generally think about curing advanced cases—those that cannot be cured by surgery alone because they have already metastasized. This is a curious way of thinking about this disease. When we think of cardiovascular or infectious diseases, we first consider ways to prevent them rather than drugs to cure their most advanced forms. Today, we are in no better position to cure polio or massive myocardial infarctions than we were a thousand years ago. But we can prevent these diseases entirely (vaccines), reduce incidence (dietary changes, statins), or mitigate severity (stents, thrombolytic agents) and thereby make a major impact on morbidity and mortality.

This focus on curing advanced cancers might have been reasonable 50 years ago, when the molecular pathogenesis of cancers was mysterious and when chemotherapeutic agents against advanced cancers were showing promise. But this mindset is no longer acceptable. We now know precisely what causes cancer: a sequential series of alterations in well-defined genes that alter the function of a limited number of pathways. Moreover, we know that this process takes decades to develop and that the incurable stage, metastasis, occurs only a few years before death. In other words, of the one million people that will die from cancer this year, the vast majority will die only because their cancers were not detected in the first 90% of the cancers’ lifetimes, when they were amenable to the surgeons’ scalpel.

This new knowledge of cancer (Box 2) has reinvigorated the search for cures for advanced cancers, but has not yet permeated other fields of applied cancer research. A common and limited set of driver genes and pathways is responsible for most common forms of cancer (table S2); these genes and pathways offer distinct potential for early diagnosis. The genes themselves, the proteins encoded by these genes, and the end products of their pathways are, in principle, detectable in many ways, including analyses of relevant body fluids, such as urine for genitourinary cancers, sputum for lung cancers, and stool for gastrointestinal cancers (150). Equally exciting are the possibilities afforded by molecular imaging, which not only indicate the presence of a cancer but also reveal its precise location and extent. Additionally, research into the relationship between particular environmental influences (diet and lifestyle) and the genetic alterations in cancer is sparse, despite its potential for preventative measures.

The reasons that society invests so much more in research on cures for advanced cancers than on prevention or early detection are complex. Economic issues play a part: New drugs are far more lucrative for industry than new tests, and large individual costs for treating patients with advanced disease have become acceptable, even in developing countries (151). From a technical standpoint, the development of new and improved methods for early detection and prevention will not be easy, but there is no reason to assume that it will be more difficult than the development of new therapies aimed at treating widely metastatic disease.

Our point is not that strenuous efforts to develop new therapies for advanced cancer patients should be abandoned. These will always be required, no matter our arsenal of early detection or preventative measures. Instead, we are suggesting that “plan A” should be prevention and early detection, and “plan B” (therapy for advanced cancers) should be necessary only when plan A fails. To make plan A viable, government and philanthropic organizations must dedicate a much greater fraction of their resources to this cause, with long-term considerations in mind. We believe that cancer deaths can be reduced by more than 75% in the coming decades (152), but that this reduction will only come about if greater efforts are made toward early detection and prevention.

## Glossary

Adenoma: A benign tumor composed of epithelial cells.

Alternative lengthening of telomeres (ALT): A process of maintaining telomeres independent of telomerase, the enzyme normally responsible for telomere replication.

Amplification: A genetic alteration producing a large number of copies of a small segment (less than a few megabases) of the genome.

Angiogenesis: the process of forming vascular conduits, including veins, arteries, and lymphatics.

Benign tumor: An abnormal proliferation of cells driven by at least one mutation in an oncogene or tumor suppressor gene. These cells are not invasive (i.e., they cannot penetrate the basement membrane lining them), which distinguishes them from malignant cells.

Carcinoma: A type of malignant tumor composed of epithelial cells.

Clonal mutation: A mutation that exists in the vast majority of the neoplastic cells within a tumor.

Driver gene mutation (driver): A mutation that directly or indirectly confers a selective growth advantage to the cell in which it occurs.

Driver gene: A gene that contains driver gene mutations (Mut-Driver gene) or is expressed aberrantly in a fashion that confers a selective growth advantage (Epi-Driver gene).

Epi-driver gene: A gene that is expressed aberrantly in cancers in a fashion that confers a selective growth advantage.

Epigenetic: Changes in gene expression or cellular phenotype caused by mechanisms other than changes in the DNA sequence.

Exome: The collection of exons in the human genome. Exome sequencing generally refers to the collection of exons that encode proteins.

Gatekeeper: A gene that, when mutated, initiates tumorigenesis. Examples include RB, mutations of which initiate retinoblastomas, and VHL, whose mutations initiate renal cell carcinomas.

Germline genome: An individual’s genome, as inherited from their parents.

Germline variants: Variations in sequences observed in different individuals. Two randomly chosen individuals differ by ~20,000 genetic variations distributed throughout the exome.

Human leukocyte antigen (HLA): A protein encoded by genes that determine an individual’s capacity to respond to specific antigens or reject transplants from other individuals.

Homozygous deletion: Deletion of both copies of a gene segment (the one inherited from the mother, as well as that inherited from the father).

Indel: A mutation due to small insertion or deletion of one or a few nucleotides.

Karyotype: Display of the chromosomes of a cell on a microscopic slide, used to evaluate changes in chromosome number as well as structural alterations of chromosomes.

Kinase: A protein that catalyzes the addition of phosphate groups to other molecules, such as proteins or lipids. These proteins are essential to nearly all signal transduction pathways.

Liquid tumors: Tumors composed of hematopoietic (blood) cells, such as leukemias. Though lymphomas generally form solid masses in lymph nodes, they are often classified as liquid tumors because of their derivation from hematopoietic cells and ability to travel through lymphatics.

Malignant tumor: An abnormal proliferation of cells driven by mutations in oncogenes or tumor suppressor genes that has already invaded their surrounding stroma. It is impossible to distinguish an isolated benign tumor cell from an isolated malignant tumor cell. This distinction can be made only through examination of tissue architecture.

Metastatic tumor: A malignant tumor that has migrated away from its primary site, such as to draining lymph nodes or another organ.

Methylation: Covalent addition of a methyl group to a protein, DNA, or other molecule.

Missense mutation: A single-nucleotide substitution (e.g., C to T) that results in an amino acid substitution (e.g., histidine to arginine).

Mut-driver gene: A gene that contains driver gene mutations.

Nonsense mutation: A single-nucleotide substitution (e.g., C to T) that results in the production of a stop codon.

Nonsynonymous mutation: A mutation that alters the encoded amino acid sequence of a protein. These include missense, nonsense, splice site, translation start, translation stop, and indel mutations.

Oncogene: A gene that, when activated by mutation, increases the selective growth advantage of the cell in which it resides.

Passenger mutation (passenger): A mutation that has no direct or indirect effect on the selective growth advantage of the cell in which it occurred.

Primary tumor: The original tumor at the site where tumor growth was initiated. This can be defined for solid tumors, but not for liquid tumors.

Promoter: A region within or near the gene that helps regulate its expression.

Rearrangement: A mutation that juxtaposes nucleotides that are normally separated, such as those on two different chromosomes.

Selective growth advantage (s): The difference between birth and death in a cell population. In normal adult cells in the absence of injury, s = 0.000000.

Self-renewing tissues: Tissues whose cells normally repopulate themselves, such as those lining the gastrointestinal or urogenital tracts, as well as blood cells.

Single-base substitution (SBS): A single-nucleotide substitution (e.g., C to T) relative to a reference sequence or, in the case of somatic mutations, relative to the germline genome of the person with a tumor.

Solid tumors: Tumors that form discrete masses, such as carcinomas or sarcomas.

Somatic mutations: Mutations that occur in any non–germ cell of the body after conception, such as those that initiate tumorigenesis.

Splice sites: Small regions of genes that are juxtaposed to the exons and direct exon splicing.

Stem cell: An immortal cell that can repopulate a particular cell type.

Subclonal mutation: A mutation that exists in only a subset of the neoplastic cells within a tumor.

Translocation: A specific type of rearrangement where regions from two nonhomologous chromosomes are joined.

Tumor suppressor gene: A gene that, when inactivated by mutation, increases the selective growth advantage of the cell in which it resides.

Untranslated regions: Regions within the exons at the 5′ and 3′ ends of the gene that do not encode amino acids.

## References and Notes

1. Acknowledgments: We thank M. Nowak and I. Bozic for critical reading of the manuscript, S. Gabelli for assisting with the production of Fig. 8, and A. Dixon, V. Ferranta, and E. Cook for artwork. This work was supported by The Virginia and D.K. Ludwig Fund for Cancer Research; The Lustgarten Foundation for Pancreatic Cancer Research; and NIH grants CA 43460, CA 47345, CA 62924, and CA 121113. All authors are Founding Scientific Advisors of Personal Genome Diagnostics (PGDx), a company focused on the identification of genetic alterations in human cancer for diagnostic and therapeutic purposes. All authors are also members of the Scientific Advisory Board of Inostics, a company that is developing technologies for the molecular diagnosis of cancer. All authors own stock in PGDx and Inostics. The terms of these arrangements are being managed by Johns Hopkins University, in accordance with their conflict-of-interest policies.
View Abstract


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# The average (arithmetic mean) of a set of n test scores is 8

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The average (arithmetic mean) of a set of n test scores is 8 [#permalink]  08 Mar 2020, 10:48
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The average (arithmetic mean) of a set of n test scores is 80. The average (arithmetic mean) of these n scores together with a score of 85 is 81.

 Quantity A Quantity B $$n$$ $$5$$

A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

Kudos for the right answer and explanation
[Reveal] Spoiler: OA

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Re: The average (arithmetic mean) of a set of n test scores is 8 [#permalink]  09 Mar 2020, 08:44
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Carcass wrote:
The average (arithmetic mean) of a set of n test scores is 80. The average (arithmetic mean) of these n scores together with a score of 85 is 81.

 Quantity A Quantity B $$n$$ $$5$$

The average (arithmetic mean) of a set of n test scores is 80.
We can write: (sum of all n test scores)/n = 80
Multiply both sides by n to get: sum of all n test scores = 80n

The average (arithmetic mean) of these n scores together with a score of 85 is 81.
Once we add the score of 85 to the other n scores, we have a total of n+1 scores
So, we can write: (sum of the n test scores + 85)/(n + 1) = 81
Substitute to get: (80n + 85)/(n + 1) = 81
Multiply both sides of the equation by (n + 1) to get: 80n + 85 = 81(n + 1)
Expand: 80n + 85 = 81n + 81
Subtract 81 from both sides: 80n + 4 = 81n
Subtract 80n from both sides: 4 = n

We get:
QUANTITY A: 4
QUANTITY B: 5

RELATED VIDEO FROM MY COURSE

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Re: The average (arithmetic mean) of a set of n test scores is 8 [#permalink]  06 Apr 2020, 13:18
would you define this question as one of the hard ones?
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Re: The average (arithmetic mean) of a set of n test scores is 8 [#permalink]  06 Apr 2020, 15:12
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mike1917 wrote:
would you define this question as one of the hard ones?

IMO, it's more of a medium question. I'd place it in the 150 to 155 range.

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Re: The average (arithmetic mean) of a set of n test scores is 8 [#permalink]  06 Apr 2020, 15:39
Expert's post
The tag medium rather than hard comes directly from the book.

Of course, at the time a question like this was pretty hard but over time became medium.

The level has decreased.

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Re: The average (arithmetic mean) of a set of n test scores is 8   [#permalink] 06 Apr 2020, 15:39
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# Revision history [back]

You must have forgotten the multiplication sign *. Quite obviously, $(c^2+d^2)$ was substituted for $a$ in the expression, thus you got $(c^2+d^2)^2 + b^2$. If you are using a recent version of Sage, you must have gotten a warning when evaluating the expression, saying that the correct form for expression substitution is

(a^2 + b^2)(a=(c^2 + d^2))

In Sage 6.4.1

sage: var('a,b')
(a, b)
sage: (a+b)(a+b)
/home/dfl/sage/local/lib/python2.7/site-packages/IPython/core/interactiveshell.py:2883: DeprecationWarning:
Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future
release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)
See http://trac.sagemath.org/5930 for details.
exec(code_obj, self.user_global_ns, self.user_ns)
a + 2*b
sage: (a+b)*(a+b)
(a + b)^2


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1. ## more trig

another form of cos(30deg-A)+sin(60deg+A)
sinx?

2. $\displaystyle 2\,\sin(\frac{1}{3}\,\pi +A)$

Reply if you need more explanations.

3. ## sorry stated the question wrong

the equations is cos(30(degrees)-A + sin(60(degrees) + A)

is one of the froms of this equation $\displaystyle sinx$

4. Please rephrase your entire question using proper and syntax-checked sentences and expressions.

No, that is not an equation. An equation contains the equal sign.

5. Is what you mean:

1. $\displaystyle (cos(30)-A) + sin(60 + A)$
2. $\displaystyle cos(30-A) + sin(60 + A)$
3. $\displaystyle cos(30-A + sin(60 + A))$

6. Originally Posted by Math Help
Is what you mean:

1. $\displaystyle (cos(30)-A) + sin(60 + A)$
2. $\displaystyle cos(30-A) + sin(60 + A)$
3. $\displaystyle cos(30-A + sin(60 + A))$
In the cases of 1 and 3, there are no better ways to reduce the number of terms. In the case of 2, I already gave the result.


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# Function Calls

Functions are packets of statements that can be used repeatedly. They can accept any number of inputs and can return any number of outputs.

The name of a function must be a valid identifier. If a function name is the same as a variable name, the variable name takes precedence. Similarly, if a user-defined function and built-in function have the same name, the user-defined function takes precedence.

Inputs are passed to a function using ()’s. They are separated by ,’s.
my_function(input1, input2, …, inputN)
If a function can be called without inputs, the ()’s are optional.
Inputs can be any valid expression.
sqrt(1+b/3*5^2) % assuming b is a valid variable

If the number of inputs passed to a function does not match the number expected by that function, an error occurs.

If a function returns a single value, this return value can be used in other expressions.
sum(sqrt(1)+sqrt(2))
It can also be assigned to a variable.
output = sqrt(1)
If a function returns multiple values, each return value must be assigned to a distinct variable. The output variables are grouped using [ ]’s and separated by ,’s (or optionally spaces).
[out1, out2] = f(in1, in2)
function [len,first] = info(x)
len = length(x);
first = x(1);
end

[a,b] = info([4,8,2,9])

Unlike single return function calls, multi-return function calls cannot be directly combined with other operators.

If a function returns more outputs than are assigned, the remainder is ignored. If a function returns fewer outputs than are assigned, an error occurs.

Each function has its own rules as to what constitutes valid inputs. OML language supports an alternate function syntax. String arguments can be passed to a function without using ()’s and by using spaces instead of commas to separate them. These two calls are equivalent:
myfunc('hello','goodbye')
myfunc hello goodbye

OML language has many built-in functions that are directly callable. These will be cataloged in a separate document. You can also define you own functions (see Function Definitions).


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# Output Functions for Optimization Toolbox™

### What Is an Output Function?

For some problems, you might want output from an optimization algorithm at each iteration. For example, you might want to find the sequence of points that the algorithm computes and plot those points. To do so, create an output function that the optimization function calls at each iteration. See Output Function and Plot Function Syntax for details and syntax.

This example uses the solver-based approach for output functions. For the problem-based approach, see Output Function for Problem-Based Optimization.

Generally, the solvers that employ an output function can take nonlinear functions as inputs. You can determine which solvers can use an output function by looking in the Options section of function reference pages.

### Use an Output Function

This example shows how to use an output function to monitor the `fmincon` solution process for solving a constrained nonlinear optimization problem. At the end of each `fmincon` iteration, the output function does the following:

• Plot the current point.

• Store the current point and its corresponding objective function value in a variable named `history`, and store the current search direction in a variable named `searchdir`. The search direction is a vector that points in the direction from the current point to the next one.

Additionally, to make the history available outside of the `fmincon` function, perform the optimization inside a nested function that calls `fmincon` and returns the output function variables. For more information about this method of passing information, see Passing Extra Parameters. The `runfmincon` helper function at the end of this example contains the nested function call.

### Objective and Constraint Functions

The problem is to minimize the function

`$f\left(x\right)=\mathrm{exp}\left({x}_{1}\right)\left(4{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}+2{x}_{2}+1\right)$`

subject to the nonlinear inequality constraints

`$\begin{array}{l}{x}_{1}+{x}_{2}-{x}_{1}{x}_{2}\ge 3/2\\ {x}_{1}{x}_{2}\ge -10.\end{array}$`

The `objfun` function nested in `runfmincon` implements the objective function. The `confun` function nested in `runfmincon` implements the constraint function.

### Call Solver

To obtain the solution to the problem and see the history of the `fmincon` iterations, call the `runfmincon` function.

`[xsol,fval,history,searchdir] = runfmincon;`
``` Max Line search Directional First-order Iter F-count f(x) constraint steplength derivative optimality Procedure 0 3 1.8394 0.5 Infeasible start point 1 6 1.85127 -0.09197 1 0.109 0.778 2 9 0.300167 9.33 1 -0.117 0.313 Hessian modified twice 3 12 0.529835 0.9209 1 0.12 0.232 4 16 0.186965 -1.517 0.5 -0.224 0.13 5 19 0.0729085 0.3313 1 -0.121 0.054 6 22 0.0353323 -0.03303 1 -0.0542 0.0271 7 25 0.0235566 0.003184 1 -0.0271 0.00587 8 28 0.0235504 9.035e-08 1 -0.0146 8.51e-07 Active inequalities (to within options.ConstraintTolerance = 1e-06): lower upper ineqlin ineqnonlin 1 2 ```

```Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. ```

The output function creates a plot of the points evaluated by `fmincon`. Each point is labeled by its iteration number. The optimal point occurs at the eighth iteration. The last two points in the sequence are so close that they overlap.

The output `history` is a structure that contains two fields.

`disp(history)`
``` x: [9x2 double] fval: [9x1 double] ```

The `fval` field in `history` contains the objective function values corresponding to the sequence of points `fmincon` computes.

`disp(history.fval)`
``` 1.8394 1.8513 0.3002 0.5298 0.1870 0.0729 0.0353 0.0236 0.0236 ```

These same values are displayed in the iterative output in the column with the header `f(x)`.

The `x` field of `history` contains the sequence of points that `fmincon` computes.

`disp(history.x)`
``` -1.0000 1.0000 -1.3679 1.2500 -5.5708 3.4699 -4.8000 2.2752 -6.7054 1.2618 -8.0679 1.0186 -9.0230 1.0532 -9.5471 1.0471 -9.5474 1.0474 ```

The `searchdir` output contains the search directions for `fmincon` at each iteration. The search direction is a vector pointing from the point computed at the current iteration to the point computed at the next iteration.

`disp(searchdir)`
``` -0.3679 0.2500 -4.2029 2.2199 0.7708 -1.1947 -3.8108 -2.0268 -1.3625 -0.2432 -0.9552 0.0346 -0.5241 -0.0061 -0.0003 0.0003 ```

### Helper Functions

The following code creates the `runfmincon` function, which contains the `outfun` output function, `objfun` objective function, and `confun` nonlinear constraint function as nested functions.

```function [xsol,fval,history,searchdir] = runfmincon % Set up shared variables with outfun history.x = []; history.fval = []; searchdir = []; % Call optimization x0 = [-1 1]; options = optimoptions(@fmincon,'OutputFcn',@outfun,... 'Display','iter','Algorithm','active-set'); [xsol,fval] = fmincon(@objfun,x0,[],[],[],[],[],[],@confun,options); function stop = outfun(x,optimValues,state) stop = false; switch state case 'init' hold on case 'iter' % Concatenate current point and objective function % value with history. x must be a row vector. history.fval = [history.fval; optimValues.fval]; history.x = [history.x; x]; % Concatenate current search direction with % searchdir. searchdir = [searchdir;... optimValues.searchdirection']; plot(x(1),x(2),'o'); % Label points with iteration number and add title. % Add .15 to x(1) to separate label from plotted 'o'. text(x(1)+.15,x(2),... num2str(optimValues.iteration)); title('Sequence of Points Computed by fmincon'); case 'done' hold off otherwise end end function f = objfun(x) f = exp(x(1))*(4*x(1)^2 + 2*x(2)^2 + 4*x(1)*x(2) +... 2*x(2) + 1); end function [c, ceq] = confun(x) % Nonlinear inequality constraints c = [1.5 + x(1)*x(2) - x(1) - x(2); -x(1)*x(2) - 10]; % Nonlinear equality constraints ceq = []; end end```


text:
# Let f:[0,4pi]->[0,pi] be defined by f(x)=cos^-1(cos x). The number of points x in[0,4pi] 4satisfying the equation f(x)=(10-x)/10 is

Updated On: 17-04-2022

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# 3.14: Where Does the N - 1 Come from?

If we know $$\mu$$ and we have a set of $$N$$ data points, the best estimate we can make of the variance is

$\sigma^2=\int^{u_{max}}_{u_{min}}{\left(u-\mu \right)}^2\left(\frac{df}{du}\right)du \approx \sum^N_{i=1}{\left(u_i-\mu \right)}^2\left(\frac{1}{N}\right)$

We have said that if we must use $$\overline{u}$$ to approximate the mean, the best estimate of $$\sigma^2$$, usually denoted $$s^2$$, is

$estimated\ \sigma^2=s^2 =\sum^N_{i=1}{\left(u_i-\overline{u}\right)}^2\left(\frac{1}{N-1}\right)$

The use of $$N-1$$, rather than $$N$$, in the denominator is distinctly non-intuitive; so much so that this equation often causes great irritation. Let us see how this equation comes about.

Suppose that we have a distribution whose mean is $$\mu$$ and variance is $$\sigma^2$$. Suppose that we draw $$N$$ values of the random variable, $$u$$, from the distribution. We want to think about the expected value of $${\left(u-\mu \right)}^2$$. Let us write $$\left(u-\mu \right)$$ as

$\left(u-\mu \right)=\left(u-\overline{u}\right)+\left(\overline{u}-\mu \right).$

Squaring this gives

${\left(u-\mu \right)}^2={\left(u-\overline{u}\right)}^2+{\left(\overline{u}-\mu \right)}^2+2\left(u-\overline{u}\right)\left(\overline{u}-\mu \right).$

From our definition of expected value, we can write:

$\begin{array}{l} \text{Expected value of } \left(u-\mu \right)^2= \\ ~~~~ =expected\ value\ of\ \ {\left(u-\overline{u}\right)}^2 \\ \ \ \ \ +expected\ value\ of\ {\left(\overline{u}-\mu \right)}^2 \\ \ \ \ \ +expected\ value\ of\ 2\left(u-\overline{u}\right)\left(\overline{u}-\mu \right) \end{array}$

From our discussion above, we can recognize each of these expected values:

• The expected value of $${\left(u-\mu \right)}^2$$ is the variance of the original distribution, which is $$\sigma^2$$. Since this is a definition, it is exact.
• The best possible estimate of the expected value of $${\left(u-\overline{u}\right)}^2$$ is $\sum^N_{i=1}{{\left(u_i-\overline{u}\right)}^2\left(\frac{1}{N}\right)}$
• The expected value of $${\left(\overline{u}-\mu \right)}^2$$ is the expected value of the variance of averages of $$N$$ random variables drawn from the original distribution. That is, the expected value of $${\left(\overline{u}-\mu \right)}^2$$ is what we would get if we repeatedly drew $$N$$ values from the original distribution, computed the average of each set of $$N$$ values, and then found the variance of this new distribution of average values. By the central limit theorem, this variance is $${\sigma^2}/{N}$$. Thus, the expected value of $${\left(\overline{u}-\mu \right)}^2$$ is exactly $${\sigma^2}/{N}$$.
• Since $$\left(\overline{u}-\mu \right)$$ is constant, the expected value of $$2\left(u-\overline{u}\right)\left(\overline{u}-\mu \right)$$ is $2\left(\overline{u}-\mu \right)\left[\frac{1}{N}\sum^N_{i=1}{\left(u_i-\overline{u}\right)}\right]$ which is equal to zero, because $\sum^N_{i=1}{\left(u_i-\overline{u}\right)} = \left(\sum^N_{i=1}{u_i}\right)-N\overline{u}=0$ by the definition of $$\overline{u}$$.

Substituting, our expression for the expected value of $${\left(u-\mu \right)}^2$$ becomes:

$\sigma^2\approx \sum^N_{i=1} \left(u_i-\overline{u}\right)^2\left(\frac{1}{N}\right)+\frac{\sigma^2}{N}$

so that

$\sigma^2\left(1-\frac{1}{N}\right)=\sigma^2\left(\frac{N-1}{N}\right)\approx \sum^N_{i=1} \frac{\left(u_i-\overline{u}\right)^2}{N}$

and

$\sigma^2 \approx \sum^N_{i=1} \frac{\left(u_i-\overline{u}\right)^2}{N-1}$

That is, as originally stated, when we must use $$\overline{u}$$ rather than the true mean, $$\mu$$, in the sum of squared differences, the best possible estimate of $$\sigma^2$$, usually denoted $$s^2$$, is obtained by dividing by $$N-1$$, rather than by $$N$$.


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# Behnke-Stein theorem

###### Theorem (Behnke-Stein).

Suppose that $G_{k}\subset{\mathbb{C}}^{n}$ are domains of holomorphy such that $G_{k}\subset G_{j}$ whenever $k. Then $\bigcup_{k=1}^{\infty}G_{k}$ is a domain of holomorphy.

This is related to the fact that an increasing union of pseudoconvex domains is pseudoconvex and so it can be proven using that fact and the solution of the Levi problem. Though historically this theorem was in fact used to solve the Levi problem.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Behnke-Stein theorem BehnkeSteinTheorem 2013-03-22 14:31:13 2013-03-22 14:31:13 jirka (4157) jirka (4157) 5 jirka (4157) Theorem msc 32T05 increasing union of domains of holomorphy is a domain of holomorphy


text:
Aloe Protocol
Search…
🚀
Creating a Pool
Step-by-step guide to creating your first Blend Vault
The information provided by Aloe Labs, Inc. (“we,” “us” or “our”) on docs.aloe.capital (the “Site”) is for general informational purposes only. All information on the Site is provided in good faith, however we make no representation or warranty of any kind, express or implied, regarding the accuracy, adequacy, validity, reliability, availability or completeness of any information on the Site.
Under no circumstance shall we have any liability to you for any loss or damage of any kind incurred as a result of the use of the site or reliance on any information provided on the site. Your use of the site and your reliance on any information on the site is solely at your own risk.

Getting Started

Aloe Blend is decentralized and permissionless, so anyone can deploy a pool. Unlike some other "permissionless" protocols, pool creators have no special access or control. Once a pool is deployed, all users are treated equally.

Prerequisites

• ~1 ETH you're willing to part with
• the Aloe Blend repository, cloned and yarn installed or extreme confidence in your Etherscan skills
• the address of the underlying Uniswap V3 pair
Deployment costs are dependent on gas prices, as well as the cardinality of the underlying Uniswap V3 oracle. It's also cheaper to use existing silos than it is to deploy new ones.

Preparing the Uniswap Pair

When a Blend pool is first deployed, it'll create a very wide liquidity position in the underlying Uniswap V3 pair. Once Blend detects that 10% (or less) of earned yield is sufficient to compensate bots for rebalance()ing, it will start scaling position width based on implied volatility. This implied volatility is computed from Uniswap oracle data, so you'll need to make sure the Uniswap pair has sufficient oracle cardinality.
We recommend increasing the Uniswap oracle cardinality before creating a Blend pool, but it's not strictly necessary. In most cases, it'll take a few weeks for the contract's internal trackers to equilibriate. Only then will it start calculating implied volatility. If you haven't increased cardinality by then, rebalance() transactions will fail until you (or someone else) remembers to do so.
To increase Uniswap oracle cardinality, follow these steps:
1. 1.
Open scripts/poke_oracle.js and scroll to the bottom. Write/modify lines to call increaseOracleCardinality(poolAddress, gasPrice) where poolAddress is the address of the Uniswap pair and gasPrice is expressed in wei (so 100 gwei would be entered as 100e9)
2. 2.
By default, the oracle cardinality is increased by 300. Assuming every block contains at least 1 transaction that interacts with the pair and a block is mined every 13 seconds, this corresponds to
$\frac{300*13}{3600}=1.08$
hours of oracle data. Blend requires at least 1 hour and will use up to 24 hours if available. Just be careful to provide enough gas when increasing cardinality, and spread your calls out over multiple blocks if necessary.
3. 3.
Once you're satisfied that everything looks correct, open a terminal in the repository's root directory and run npx hardhat run scripts/poke_oracle.js --network mainnet. Assuming you've configured your .env file correctly, this will send a transaction from the DEPLOYER account.

Preparing the Volatility Oracle

Note: This step is pointless if the Uniswap Pair's cardinality is too low.
Similar to the Factory contract, the VolatilityOracle is independent of the Blend pool itself. As such, you have to initialize it separately. To do so, follow these steps:
1. 1.
Visit the Volatility Oracle contract page on Etherscan
2. 2.
Connect your wallet and select cacheMetadataFor
3. 3.
Enter the address of the Uniswap pair you've chosen and click "Write"
4. 4.
After that transaction is confirmed, select estimate24H, enter the same address as before, and again click "Write"
5. 5.
That's it!

Choosing Silos

Every Blend pool is associated with two silos: silo0 and silo1. When Blend detects that liquidity wouldn't be productive in Uniswap, it instead deposits it to these silos. token0 is placed in silo0 and token1 is placed in silo1. It is crucial that the silos can actually handle their corresponding ERC20s.
You can read more about existing silos here and the code is available here. If you want to deploy a new instance of something like FuseFTokenSilo or implement a new silo altogether, we assume you know what you're doing. Feel free to reach out on Discord or open a PR on GitHub so that we can document your work.

Creating the Pool

This is the easy part. Open another terminal in the Aloe Blend repository's root directory and run:
npx hardhat run scripts/deploy_pair.js --network mainnet
This script will prompt you for information as needed, validate inputs, and verify whether the underlying Uniswap pair and Volatility Oracle are ready to go. If everything looks good, a new Blend pool will be deployed and the script will spit out another command you can run to verify the new contract on Etherscan.
Example:
Example output of the deploy_pair.js script


text:
Corpus ID: 232478810

# Relationship between renormalized values of shuffle type and of harmonic type of multiple zeta functions

@inproceedings{Komiyama2021RelationshipBR,
title={Relationship between renormalized values of shuffle type and of harmonic type of multiple zeta functions},
author={Nao Komiyama},
year={2021}
}
In this paper, we settle the problem posed by Singer which is on a comparison problem between the renormalized values of shuffle type and harmonic type of multiple zeta functions.

#### References

SHOWING 1-10 OF 14 REFERENCES
Analytic continuation of multiple zeta-functions and their values at non-positive integers
• Mathematics
• 2001
Analytic continuation of the multiple zeta-function is established by a simple application of the Euler-Maclaurin summation formula. Multiple zeta values at non-positive integers are defined andExpand
An equivalence between desingularized and renormalized values of multiple zeta functions at negative integers
It is known that the special values of multiple zeta functions at non-positive arguments are indeterminate in most cases due to the occurrences of infinitely many singularities. In order to give aExpand
RENORMALISATION GROUP FOR MULTIPLE ZETA VALUES
• Mathematics
• 2015
Calculating multiple zeta values at arguments of any sign in a way that is compatible with both the quasi-shuffle product as well as meromorphic continuation, is commonly referred to as theExpand
The Hopf algebra of (q)multiple polylogarithms with non-positive arguments
• Mathematics
• 2015
We consider multiple polylogarithms in a single variable at non-positive integers. Defining a connected graded Hopf algebra, we apply Connes' and Kreimer's algebraic Birkhoff decomposition toExpand
Nested sums of symbols and renormalized multiple zeta values
• Mathematics
• 2010
We define discrete nested sums over integer points for symbols on the real line, which obey stuffle relations whenever they converge. They relate to Chen integrals of symbols via the Euler-MacLaurinExpand
q-Analogues of Multiple Zeta Values and Their Application in Renormalization
In this paper we report on recent developments on q-analogues of multiple zeta values (MZVs), which are power series in a formal parameter q that reduce to classical MZVs in the limit \(q\rightarrowExpand
Renormalization of multiple zeta values
• Mathematics, Physics
• 2006
Abstract Multiple zeta values (MZVs) in the usual sense are the special values of multiple variable zeta functions at positive integers. Their extensive studies are important in both mathematics andExpand
Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem
• Physics, Mathematics
• 2000
Abstract:This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedureExpand
Quasi-Shuffle Products
Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which weExpand
Renormalization of q - regularised multiple zeta values
• Lett . Math . Phys .
• 2016


text:
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July  2015, 11(3): 921-932. doi: 10.3934/jimo.2015.11.921

Clustering based polyhedral conic functions algorithm in classification

 1 Department of Industrial Engineering, Faculty of Engineering, Anadolu University, Eskisehir, 26555, Turkey 2 Vitra, Eczacibasi Yapi Gerecleri, 11300 Bilecik, Turkey

Received  October 2013 Revised  June 2014 Published  October 2014

In this study, a new algorithm based on polyhedral conic functions (PCFs) is developed to solve multi-class supervised data classification problems. The $k$ PCFs are constructed for each class in order to separate it from the rest of the data set. The $k$-means algorithm is applied to find vertices of PCFs and then a linear programming model is solved to calculate the parameters of each PCF. The separating functions for each class are obtained as a pointwise minimum of the PCFs. A class label is assigned to the test point according to its minimum value over all separating functions. In order to demonstrate the performance of the proposed algorithm, it is applied to solve classification problems in publicly available data sets. The comparative results with some mainstream classifiers are presented.
Citation: Gurkan Ozturk, Mehmet Tahir Ciftci. Clustering based polyhedral conic functions algorithm in classification. Journal of Industrial & Management Optimization, 2015, 11 (3) : 921-932. doi: 10.3934/jimo.2015.11.921
References:
 [1] A. Astorino and M. Gaudioso, Polyhedral separability through successive LP,, Journal of Optimization Theory and Applications, 112 (2002), 265. doi: 10.1023/A:1013649822153. [2] A. Astorino, M. Gaudioso and A. Seeger, Conic separation of finite sets. i: The homogeneous case,, Journal of Convex Analysis, 21 (2014), 001. [3] K. Bache and M. Lichman, UCI machine learning repository, 2013., URL , (). [4] A. M. Bagirov, Max-min separability,, Optimization Methods and Software, 20 (2005), 277. doi: 10.1080/10556780512331318263. [5] A. M. Bagirov and J. Ugon, Supervised data classification via max-min separability,, Applied Optimization, 99 (2005), 175. doi: 10.1007/0-387-26771-9_6. [6] A. M. Bagirov, M. Ghosh and D. Webb, A derivative-free method for linearly constrained nonsmooth optimization,, Journal of Industrial and Management Optimization, 2 (2006), 319. [7] A. M. Bagirov, J. Ugon, D. Webb, G. Ozturk and R. Kasimbeyli, A novel piecewise linear classifier based on polyhedral conic and max-min separabilities,, TOP, 21 (2013), 3. doi: 10.1007/s11750-011-0241-5. [8] C. J. C. Burges, A tutorial on support vector machines for pattern recognition,, Data Mining and Knowledge Discovery, 2 (1998), 121. [9] R. N. Gasimov and G. Ozturk, Separation via polihedral conic functions,, Optimization Methods and Software, 21 (2006), 527. doi: 10.1080/10556780600723252. [10] M. Hall, E. Frank, G. Holmes, B. Pfahringer, P. Reutemann and I. H. Witten, The weka data mining software: An update,, SIGKDD Explorations, 11 (2009), 10. doi: 10.1145/1656274.1656278. [11] R. Kasimbeyli, Radial epiderivatives and set-valued optimization,, Optimization, 58 (2009), 521. doi: 10.1080/02331930902928310. [12] R. Kasimbeyli, A nonlinear cone separation theorem and scalarization in nonconvex vector optimization,, SIAM J. on Optimization, 20 (2009), 1591. doi: 10.1137/070694089. [13] R. Kasimbeyli and M. Mammadov, On weak subdifferentials, directional derivatives, and radial epiderivatives for nonconvex functions,, SIAM Journal on Optimization, 20 (2009), 841. doi: 10.1137/080738106. [14] G. Ozturk, A New Mathematical Programming Approach to Solve Classification Problems,, PhD thesis, 6 (2007). [15] R. Rosenthal, GAMS: A User's Guide,, GAMS Development Corporation, (2013). [16] K. Schittkowski, Optimal parameter selection in support vector machines,, Journal of Industrial and Management Optimization, 1 (2005), 465. doi: 10.3934/jimo.2005.1.465.

show all references

References:
 [1] A. Astorino and M. Gaudioso, Polyhedral separability through successive LP,, Journal of Optimization Theory and Applications, 112 (2002), 265. doi: 10.1023/A:1013649822153. [2] A. Astorino, M. Gaudioso and A. Seeger, Conic separation of finite sets. i: The homogeneous case,, Journal of Convex Analysis, 21 (2014), 001. [3] K. Bache and M. Lichman, UCI machine learning repository, 2013., URL , (). [4] A. M. Bagirov, Max-min separability,, Optimization Methods and Software, 20 (2005), 277. doi: 10.1080/10556780512331318263. [5] A. M. Bagirov and J. Ugon, Supervised data classification via max-min separability,, Applied Optimization, 99 (2005), 175. doi: 10.1007/0-387-26771-9_6. [6] A. M. Bagirov, M. Ghosh and D. Webb, A derivative-free method for linearly constrained nonsmooth optimization,, Journal of Industrial and Management Optimization, 2 (2006), 319. [7] A. M. Bagirov, J. Ugon, D. Webb, G. Ozturk and R. Kasimbeyli, A novel piecewise linear classifier based on polyhedral conic and max-min separabilities,, TOP, 21 (2013), 3. doi: 10.1007/s11750-011-0241-5. [8] C. J. C. Burges, A tutorial on support vector machines for pattern recognition,, Data Mining and Knowledge Discovery, 2 (1998), 121. [9] R. N. Gasimov and G. Ozturk, Separation via polihedral conic functions,, Optimization Methods and Software, 21 (2006), 527. doi: 10.1080/10556780600723252. [10] M. Hall, E. Frank, G. Holmes, B. Pfahringer, P. Reutemann and I. H. Witten, The weka data mining software: An update,, SIGKDD Explorations, 11 (2009), 10. doi: 10.1145/1656274.1656278. [11] R. Kasimbeyli, Radial epiderivatives and set-valued optimization,, Optimization, 58 (2009), 521. doi: 10.1080/02331930902928310. [12] R. Kasimbeyli, A nonlinear cone separation theorem and scalarization in nonconvex vector optimization,, SIAM J. on Optimization, 20 (2009), 1591. doi: 10.1137/070694089. [13] R. Kasimbeyli and M. Mammadov, On weak subdifferentials, directional derivatives, and radial epiderivatives for nonconvex functions,, SIAM Journal on Optimization, 20 (2009), 841. doi: 10.1137/080738106. [14] G. Ozturk, A New Mathematical Programming Approach to Solve Classification Problems,, PhD thesis, 6 (2007). [15] R. Rosenthal, GAMS: A User's Guide,, GAMS Development Corporation, (2013). [16] K. Schittkowski, Optimal parameter selection in support vector machines,, Journal of Industrial and Management Optimization, 1 (2005), 465. doi: 10.3934/jimo.2005.1.465.
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2017 Impact Factor: 0.994


text:
# Common Format Features¶

Certain format specifications are shared between the bottle and CTD WHP-exchange files. All WHP-exchange text files MUST be UTF-8 encoded. Unix style line endings, LINE FEED (U+000A), SHOULD be used, DOS line endings, CARRIAGE RETURN (U+000D) followed by LINE FEED (U+000A), MAY be used, other line endings SHOULD NOT be used.

Note

UTF-8 was chosen as the encoding for WHP-Exchange files because it is backwards compatable with ASCII. Valid ASCII files are also valid UTF-8 files. UTF-8 allows for the full range of unicode points to display non ASCII text. Non ASCII text should only be encountered in the comment lines of an enchnage file.

Warning

Be careful if editing or creating files on Windows as the default text encoduing is UTF-16. UTF-16 is not compatable with UTF-8 or ASCII.

For both CTD and Bottle files the first rows must be the following and in the presented order:

1. A REQUIRED File Identification Stamp
2. Optional comment line(s)

## File Identification Stamp¶

The first line of a WHP-exchange file contains the file identifier and a creation stamp seperated by a COMMA (U+002C) ,. The file itendifier will be either BOTTLE in the case of water samples or CTD in the case of a CTD profile. The creation stamp contains information on when the file was created and who created it.

A bottle file identifier will look like:

BOTTLE,20140716CCHSIOSCD

A CTD file identifier will look like:

CTD,20140716CCHSIOSCD

Note

If while attempting to read a WHP-exchange file and the first line does not start with BOTTLE or CTD an attempt to read the rest of the file will likely fail. When writing a WHP-exchange format reader, always check if this identification stamp is present and has a valid value.

The creation stamp contians the following information:

1. 20140716CCHSIOSCD: A date stamp in the from of YYYYMMDD (ISO 8601)
2. 20140716CCHSIOSCD: The division (or group) of the instituion that wrote the file, typically three characters. The CCHDO uses CCH as the division.
3. 20140716CCHSIOSCD: The instituion that the group is associated with, typically three characters. The CCHDO is locaded at the Scripps Instituion of Oceanography, thus SIO is used.
4. 20140716CCHSIOSCD: The initials of the person who wrote the file, typically three characters. Use only code points U+0041 to U+005A and for the initials. In this example, SCD.

Warning

Do not rely on the creation stamp to be the same legnth in every WHP-exchange file. While all the same elemnts will be present, their lengths may vary.

## Optional Comment Lines¶

After the File Identification Stamp any number of comment line, including none may appear. Comment lines start with a NUMBER SIGN (U+0023) #. Comment lines typically contain information about the file history and will often contain data citation information.

An example:

# This is one line of comments

An example of the begining of a file, including the File Identification Stamp:

BOTTLE,20140716CCHSIOSCD
# This is a comment line
# BOTTLE,20130215CCHSIOSCD

Note

Notice that an older File Identification Stamp is in a comment line. This is a convention often used by the CCHDO to record when changes were made to files

Warning

Comments may contain UTF-8 encoded code points above U+007F, especially in proper names that may be present with data citation information. If writing your own WHP-exchange reader, ensure that it can handle code points above U+007F or have it skip comment lines without trying to read them.

## Parameter and Unit Lines¶

Warning

After any format specific headers, the parameter and unit lines are next. The parameter names are first, units are second.

Parameter names are COMMA (U+002C) , seperated values that define the columns the exchange file will contain. The names must be unique, capitalized, contain no empty fields, and not end with a trailing comma. The parameter names must contain only code points in the range U+0021 to U+007E except a COMMA (U+002C) ,. A trailing comma, or a comma that occurs at the end of the line with nothing else after it, MUST NOT be included on the parameter line. Certain parameter names, or parameter combinations, are required to be present. See the respective sections on Required Bottle Parameters and CTD required headers for information specific to each format.

The unit line contains information for the units of each parameter listed in the parameter line. The unit line, like the parameters, are comma seperated values. Like the parameter names, units must contain only code points in the range U+0021 to U+007E except a COMMA (U+002C) ,. A trailing comma MUST NOT be included in the unit line. Units may contain empty fields if the parameter has no units. Units for a paramter must be in the same column as that paramter, essentialy, the sname number of commas occur before the parameter name and its unit.

Warning

Parameter names and units MUST NOT contain commas as part of the name or unit. Commas are reserved for seperating the, names, units, and data into columns.

The parameter and unit lines of a CTD file might look like this:

CTDPRS,CTDPRS_FLAG_W,CTDTMP,CTDSAL,CTDOXY
DBAR,,ITS-90,PSS-78,UMOL/KG

Note the presence of quality flag column (suffixed with _FLAG_W) which has the corrisponding units of nothing denoted by two commas next to each other. For more information on quality flags, see the Quality Codes section. White space MUST have no meaning in the exchange format so it may be included for purly asthetic reasons. The parameter and units could very easially have looked like:

CTDPRS, CTDPRS_FLAG_W, CTDTMP, CTDSAL, CTDOXY
DBAR,              , ITS-90, PSS-78, UMOL/KG

Note

Some technical details for formatting the whitespace.

While not strictly requiered, parameter, units, and data lines may contain whitespace matching the length of the print format of the paramter. This is a convention followed by the CCHDO to ease reading of files by humans. Quality flag columns usually have a 1 character width which will often cause the parameter/units and data to not be aligned into pretty columns.

## Data Lines¶

The data lines occur directly after the unit line. Each line of data contains COMMA (U+002C) , seperated values of related data. Each data point of the data line may contain any combination of characters from U+0020 to U+007F except a COMMA (U+002C) ,. Like the Parameter and Unit Lines, a trailing comma MUST NOT be included at the end of each line. Data points for each parameter of the Parameter and Unit Lines must be in the same column as that paratemer, i.e. the same number of commas occur before the parameter label and the datum.

Numeric data which occurs on the data lines MUST only contain numbers, spaces, an optional decimal marker, and an optional negative sign. All whitespace within data lines has no symantic meaning. Integers may be represented as bare numerals with no decimal marker. All real numeric data (i.e. data that are real numbers) MUST be decimal and MUST represent their decimal mark using a FULL STOP (U+002E) .. For both negative real numbers and integers, prepend a HYPHEN-MINUS (U+002D) - to the numeric portion, positive real numbers MUST NOT be prefixed by a PLUS SIGN (U+002B) +.

The validity of each datum is determined by the parameter column in which it occurs. For example, the EXPOCODE column may contain any combination of letter, numbers, or symbols (except a comma). A CTDPRS column may only contain real decimal numbers (U+0030 to U+0039) using a FULL STOP (U+002E) . as the decimal mark.

Note

Parameters may have a different precision depending on how the measurement was made. The CCHDO maintains a list of parameter names which includes precisions for historic reasons. Previous versions of the Exchange format specification stated the CCHDO would pad “meaningless” zeros to the end of any data without enough precision. Newer software allows the CCHDO to keep the precision as reported, both less and more precise. For these and other reasons, a mix of precisions may occur in a column of data.

Always report the precision as measured.

Warning

The exchange format currently has no support for quoted strings within the parameter, unit, and data lines. This means it is not possible for any meaningful whitespace to be included.

After all datalines, the end of the data is indicated by a line containing only END_DATA. Here is a short example of what exchange data might look like:

2.0,2,  19.1840,  34.6935,    220.8
4.0,2,  19.1992,  34.6924,    220.7
6.0,2,  19.2002,  34.6922,    220.5
8.0,2,  19.2022,  34.6920,    220.5
END_DATA

## Post Data Content¶

After the END_DATA line, any additional content may be included without format restriction. Additional content after END_DATA MUST continue to be UTF-8 encoded.

## Examples¶

Full examples of data in exchange format are presented in their speciifc sections:


text:
# How to Multiply Radicals | Math Review

This video shows multiple examples of how to multiply radicals. You can multiply and divide radicals even if they are not alike. To multiply radicals, first multiply the coefficients, then the radicands. Once you’ve done this, simplify the answer if it is possible to do so.

You can multiply and divide radicals, even if they aren’t alike. To multiply radicals, first multiply the coefficients, and then do the same with the radicands. Finally, simplify if possible.

Let’s look at two examples. First, we have three square roots of 5 times two square roots of 11. We’re going to start by multiplying our coefficients. 3 times 2, and then we multiply our radicands. That’s the square root of 5 times 11. 3 times 2 is 6 square roots of 5 times 11 is 55. 55 is not a perfect square and it also doesn’t have any perfect square factors.

We can see because we’ve pretty much already factor treed it here, 55 is 5 times 11. Neither one of those are perfect squares either. This can’t be simplified. Therefore, 6 square roots of 55 is our answer. Let’s look at four square roots of 6 times the square root of 3. We’re going to start by multiplying our coefficients.

4 times- and the coefficient of the square root of 3 is just 1. 4 times 1 is 4. Then we have the square root of 6 times the square root of 3. We have four square roots of 6 times 3 is 18. 18 is not a perfect square, but it does have a perfect square factor. We’re going to simplify it.

4 square roots of- the perfect square factor of 18 is 9, so I’m going to rewrite it as 9 times 2, which means this is 4 times the square root of 9 times the square root of 2. We have 4 times the square root of 9. The square root of 9 is 3. This is really 4 times 3 times the square root of 2. When we multiply this all together, 4 times 3 is 12 times the square root of 2. Twelve square roots of 2.

728277

by Mometrix Test Preparation | Last Updated: August 15, 2019


text:
# Corollary of Hahn-Banach theorem regarding the norm

Let $E$ be a normed space. Then, for each $x \in E$ we can define its norm as $\|x\| = \sup \{|l(x)| : l \in E^∗, \|l\| ≤ 1 \}$.

Proof:

Let $x \in E$ and set $S = \sup\{|l(x)| : l \in E^∗, \|l\| ≤ 1 \} \leq \|x\|.$

On the one-dimensional subspace $F = \mathrm{span}(x)$ we define a functional $f$ by $f(\lambda x)= \lambda \|x\|$ for all $\lambda \in \mathbb{R}$ it follows that $f \in F^*$ with $\|f\| = 1$. Hence, by the Hahn-Banach Theorem, there exists some $l \in E^*$ with $l|_F\equiv f$ and $\|l\| = 1$. In particular this implies $|l(x)| = |f(x)| = \|x\|$. This implies that $S = \|X\|$ and that the supremum is attained.

Now, why is it, that there a $\leq$ sign in the set of wich we take the supremum, since the proof uses a functional that clearly has norm equal to 1?

• Why don't you prove that $$\sup \{|l(x)| : l \in E^∗, \|l\| ≤ 1 \} = \sup \{|l(x)| : l \in E^∗, \|l\| = 1 \}$$ then the answer to your question would be: we can use either definition, and in practice use whichever one is more convenient at the time. – GEdgar Jul 21 '17 at 12:50

It doesn't matter that there is a $\leq$ inside the set which we are taking the supremum over since the supremum is always achieved, i.e. $$\|x\|=\sup\{|l(x)|~:~l\in E^*,~\|l\|\leq 1\}=\max\{|l(x)|~:~l\in E^*,~\|l\|\leq 1\}.$$ Since you know exactly what the linear functional looks like and that it has $\|l\|=1$ you can replace the $\leq$ with an $=$ if you wanted!


text:
## Global convergence of a modified BFGS-type method for unconstrained non-convex minimization.(English)Zbl 1128.65040

Authors’ summary: To the unconstrained programming of a non-convex function, this article gives a modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm associated with the general line search model. The idea of the algorithm is to modify the approximate Hessian matrix for obtaining the descent direction and guaranteeing the efficiency of the new quasi-Newton iteration equation $$B_{k+1}s_k=y^*_k$$, where $$y^*_k$$ is the sum of $$y_k$$ and $$A_ks_k$$, and $$A_k$$ is some matrix. The global convergence properties of the algorithm associating with the general form of line search is proved.

### MSC:

 65K05 Numerical mathematical programming methods 90C26 Nonconvex programming, global optimization
Full Text:

### References:

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