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# Focus: Breaking Up Is Hard to Understand

Physics 8, 9
A new experimental technique probes the dissociation of a single molecule in water and reveals more complexities than expected.

A compound dissolves when water molecules disrupt its chemical bonds and break it apart into ions. New research addresses a seemingly straightforward question: How many water molecules does this process require? The experiments suggest a tentative answer for hydrochloric acid ( $HCl$): between five and six. But the researchers also found an alternative interpretation of their data in which increasing the number of water molecules causes them to rearrange around the $HCl$, rather than breaking it apart. Either way, they expect their new technique to help experimenters better understand the interactions of water with dissolving molecules.

To study the dissociation of molecules, researchers have observed their behavior in tiny droplets of just a handful of water molecules. There are important differences between such small clusters and a true aqueous environment, but the ability to perform detailed computer simulations of the clusters makes it possible to compare theory and experiment, says Vitaly Kresin of the University of Southern California in Los Angeles.

However, detecting when a molecule breaks apart is far from easy. Kresin and his colleagues came up with the idea that dissociation of a single molecule in a nanocluster would cause measurable changes in the cluster’s electric dipole moment. The dipole moment is a measure of the separation of positive and negative charges in an overall neutral object. The team reasoned that dissociation of the molecule would cause its ions to separate and water molecules (which have an intrinsic dipole moment) to rearrange themselves, leading to a redistribution of charge in the cluster as a whole. They chose to study $HCl$, which has often been used as a test-case for the dissociation of acids in general.

The technique for measuring electric dipole moments of droplets came from earlier work by Kresin and other co-workers [1]. They create a stream of nanodroplets by sending water vapor through a tiny nozzle. The droplets then pass through a region of non-uniform electric field in which their deflections are a measure of their dipole moments. A movable slit selects a particular amount of deflection, and the droplets are then ionized by an electron beam to prepare them for a mass spectrometer (which requires charged particles). The mass spectrometer’s mass measurement provides the number of water molecules in each cluster.

Injecting $HCl$ into the water vapor would create a fraction of droplets containing a molecule of the acid. But Kresin had found in earlier experiments that ionizing such droplets knocked out the chlorine ion, making it impossible to tell which droplets had carried an acid molecule and which had not. So the team substituted deuterium, using DCl instead of $HCl$. Even after the chlorine ion was ejected, droplets that had originally contained DCl would be slightly heavier than pure water droplets, a difference the mass spectrometer could detect. So the team could identify deuterium “doped” droplets and measure how their dipole moment depended on their size.

Looking at acid-containing droplets of three to nine water molecules, they found a distinct change in dipole moment when the number rose from five to six. This is consistent, Kresin says, with molecular simulations suggesting that about four water molecules should suffice to dissociate $HCl$. But it’s also possible that a change in dipole moment could come about through a rearrangement of water molecules around an $HCl$ molecule that remained intact. So Kresin and his colleagues undertook some theoretical studies to sort through the possibilities.

When HCl dissociates in water, the free proton ( $H+$) attaches itself to a water molecule, creating $H3O+$, but at modest temperatures the proton has enough energy to hop from one water molecule to another. In addition, $H3O+$ ions have complex rotational and vibrational motions and can switch among these states by quantum tunneling. Kresin and his colleagues performed molecular simulations that examined the influence of these processes on the dipole moment.

The team concluded that it is unclear whether $HCl$ dissociation or reconfiguration of the nanoclusters was responsible for the measured change in electric dipole moment. Nevertheless, Francesco Paisani of the University of California at San Diego says that the work is significant for emphasizing the need to include both thermal and quantum effects in order to understand how experiments can probe cluster structure. It would be extremely interesting, he adds, to study how the results vary with temperature, which would offer further insight into the cluster chemistry.

This research is published in Physical Review Letters.

–David Lindley

David Lindley is a freelance writer in Alexandria, Virginia, and author of Uncertainty: Einstein, Heisenberg, Bohr, and the Struggle for the Soul of Science (Doubleday, 2007).

## References

1. R. Moro, R. Rabinovitch, C. Xia, and V. V. Kresin, “Electric Dipole Moments of Water Clusters from a Beam Deflection Measurement,” Phys. Rev. Lett. 97, 123401 (2006)

## Subject Areas

Physical Chemistry

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# Linear Indepedence of Egienvectors and Jordon blocks

by crazygrey
Tags: blocks, egienvectors, indepedence, jordon, linear
 Share this thread:
 P: 7 Hi everyone, I had couple questions: 1) If I want to proof that egienvectors are linearly indpendent by induction, how do so? I do understand that I can start with a dimension of 1 and assume v1 to be a non zero vector so hence a linear indepedent, what do I do after that for other cases? 2) What the power of jordon block? for example (J)^k general case Thanks
 Sci Advisor HW Helper P: 9,396 1) Since eigenvectors have no reason to be linearly indepedendent then you cannot do this. Perhaps you should check the wording of the question? There are some extra hypotheses you've omitted. So what are they and how can you use them? 2) What have you done? Any examples? Have you tried computing some powers of jordan blocks? If not why not? If you just do it (try putting ones on the diagonal), you should be able to spot some patterns and then formulate an easy to prove conjecture.
 P: 7 1) Basically the idea is that having different eigenvlaues will results in independent eigenvectors. I just want to prove this thoerm. Let v=[v1 v2 v3.......vn] set of eigenvectors with distinct eigenvalues w1,w2,w3,......,wn. By induction: if n=1, v1 has to be linearly independent since it is a non zero vector. if n=2, a1v1+a2v2=0------1) Multiply 1) by A knowing that Av=wv we have a1w1v1+a2w2v2=0 ----2) Multiply 1) by w1 a1w1v1+a2w1v2=0 ----3) 2)-3) a2(w2-w1)v2=0 , thus a2=0 I want to prove that this holds for all eigenvectors, how do I do that? 2) Knowing that Jordan Block is a representation of a square matrix A with repeated eigenvalues. I understand the pattern of the jordan block where the repeated eigenvalues are allocated in the diagonal and 1's in the superdiagonal. I know if I square the jordan block, the 1s starts shifting location, but I'm unable to obtain a general forumla of (J)^k knowing that A=(q)^-1*J*q where q is the set of eigenvectors. I hope I clarified what I meant. Appreicate your help
 Sci Advisor HW Helper P: 9,396 Linear Indepedence of Egienvectors and Jordon blocks 1) You cannot prove it holds for all eigenvectors because it is trivially false. (if v is an eigenvector so is 2v). 2) Sorry, but this is quite an easy question once you guess the formula and I strongly urge you to try a couple of examples, like working out the first few powers of [1 1 0] [0 1 1] [0 0 1] because the numbers tou will see are very very well known and will show you what to do. You can even do that in your head. In fact doing [k 1 0] [0 k 1] [0 0 k] is probably just as easy and instructive. If you just do this you will learn a lot more than being told the answer.
 P: 7 1) I wanted to prove that all eigenvectors in the vector space are linearly independent, not a multiplication of it with a scalar. If I have a set of vectors [v1,v2,v3,--vn] 1,2,3,..,n are indices...this set of vectors belong to a vector space , and I want to prove that all of them are linearly independent... 2) Thanks I will work it through ...
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,363
 Quote by crazygrey 1) I wanted to prove that all eigenvectors in the vector space are linearly independent, not a multiplication of it with a scalar. If I have a set of vectors [v1,v2,v3,--vn] 1,2,3,..,n are indices...this set of vectors belong to a vector space , and I want to prove that all of them are linearly independent...
You still haven't understood. "All eigenvectors" includes both an eigenvector v and any multiple of it so that's not what you want to say. You can't prove that "all eigenvectors are linearly independent"- it's not true. You can show that eigenvectors corresponding to distinct eigenvalues are independent. But then it doesn't follow that you will have a "complete set of eigenvectors", that is, a basis consisting of eigenvectors". If such a basis exists, then writing the linear operator as a matrix in that basis gives a diagonal matrix- and not all matrices are "diagonalizable"- that's why you need the "Jordan Normal form".
 Sci Advisor HW Helper P: 9,453 define a generalized eigenvector for a matrix A to be a vector v such that some power of A annihilates v. i.e. A^r v = 0 for some r > 0. then if the characteristic polynomial of A is (X-a1)(^r1) (X-a2)^r2...(X-as)^rs, then A has r1 generalized eigenvectors for the value a1, r2 generalized eigenvectors for the value a2, etc... if there are exactly t1 actual eigenvectors for a1, and t2 for a2, ...etc, then in the jordan form for A, there are exactly t1 jordan blocks for a1, t2 jordan blocks for a2,.etc.... the size of these blocks is determiend in a slightly complicated way by the dimensions of the kernels of the operators (A-a)^i for all i,.... when ri = ti for all i, the matrix A is diagonalizable, i.e. if every generalized eigenvector is an actual eigenvector, i.e. if every vector annihilated by a power of (A-a) is annihilated by just (A-a) itself.
 Sci Advisor HW Helper P: 9,453 you might want to read my 14 page treatment of all of linear algebra from scratch through jordan and artional forms on my webpage, with proofs. the indepedence of generalized eigenspaces with disticnct eigenvectors depends on the eucldiean algorithm as follows: let V(i) = the subspace of vector annihilated by (t-ci)^ri, where (t-ci)^ri is a factor in the minimal polynomiAL FOR THE OPERATOR T. Then map the product of the V(i) into the original space V by addition, i.e. an m tuple (v1,...,vm) goes to their sum v1+...+vm in V. The independence of these different VI IS INJECTIVITY OF THIS MAP. So consider the product Qi of all factors (t-cj^rj for j ≠ i. These Qj are relatively prime so by euclids aLGORITHM THERE EXIST polynomials Pi such that summation PiQi = 1. If v1+...+vm = 0, then each vi is dependent on the others. Since Qi annihilates all vj except vi, and every other Qj annihilates vi, applying summation PiQi to vi annihilates it. i.e. every term except PiQi annihilTES Vi, and that etrm annihilates all the other vj, but vi can be written in terms of thise vj, so Qi also annihilates vi. but by the equation summation PiQi = 1, the right side equals vi. hence every vi = 0. this does it. a simialr arguemnt shows surjectivity, so V acually decomposes as a direct product of those generalized eigenspaces. this argument can be made to look more elementary, but no more understandable, and is probably given in your booka s an inductive argument.
 P: 7 Really appreicate your help, that was very helpful . Thanks
 Sci Advisor HW Helper P: 9,453 wow, how kind of you to say so.

 Related Discussions Introductory Physics Homework 1 Calculus & Beyond Homework 8 Calculus & Beyond Homework 3 Introductory Physics Homework 11 Introductory Physics Homework 9

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# Anorexia Nervosa and Appetite

Anorexia nervosa (AN) is clearly a disorder of under-eating, but the question of whether appetite is impaired remains controversial and poorly researched. Studies employing subjective assessment have consistently reported reduced hunger and desire to eat and enhanced satiety and sensation of fullness in people with AN (e.g. Halmi & Sunday, 1991; Robinson, 1989). Furthermore, the subjective reward value of food is reduced (Drewnowski et al., 1987; Sunday & Halmi, 1990), and the rate of eating is slow (Halmi & Sunday, 1991).

Some authors argue that these findings reflect tight cognitive control of normal appetite (Palmer, 2000). However, relative to healthy comparison women, those with AN show reduced salivation (LeGoff et al., 1988) and a heightened autonomic response to food (Leonard et al., 1998). Images of food elicit fear and disgust (Ellison et al., 1998). These objective data suggest that appetite may indeed be impaired in AN (Pinel et al., 2000), although some capacity to respond to hunger and satiety cues clearly remains (Cugini et al., 1998; Rolls et al., 1992).

A predisposition to leanness may be a risk factor for AN (Hebebrand & Remschmidt, 1995), supporting the notion that heritable risk for the disorder may be exerted through the biological systems regulating appetite and weight.

Peripheral Signals in AN

In acute AN, basal and fasting insulin and IGF-1 concentrations are reduced (e.g. Alderdice et al., 1985; Argente et al., 1997; de Rosa et al., 1983; Fukuda et al., 1999; Stoving et al., 1999a) while growth hormone (GH) (de Rosa et al., 1983; Stoving et al., 1999b) and cortisol concentrations are elevated (Stoving et al., 1999c for review). These findings are consistent with the need to promote gluconeogenesis in the starving state and resolve with weight gain (Casper et al., 1988; Casper, 1996; Golden et al., 1994), suggesting that they arise as an adaptive response to prolonged negative energy balance. Similarly, altered thyroid function in acute AN does not differ from that associated with malnutrition (de Rosa et al., 1983) and normalises with weight restoration (e.g. Komaki et al., 1992). The endocrine response to a meal may be abnormal in AN. For example, the insulin response to a meal is delayed and reduced (Alderdice et al., 1985; Blickle et al., 1984).

This may be attributable to delayed absorption and, in chronic cases, impaired pancreatic beta cell function (Nozaki et al., 1994). Insulin sensitivity appears to vary greatly between individuals with AN (Kiriike et al., 1990) and there may be a tendency towards increased glucose utilisation rather than storage (Franssila-Kallunki et al., 1991). Blunting of the glucose and insulin response to a meal persists even after full recovery from AN, when there is also evidence of an abnormal cortisol response to the meal (Ward et al., 1998). It remains unclear whether these findings reflect vulnerability factors for the disorder, or a scar of the illness. Interestingly, the biology of restricting and binge–purge subtypes of AN may differ significantly. For example, relative to the binge–purge subtype, the restricting subtype is associated with higher basal and growth hormone-releasing hormone stimulated GH concentrations, lower basal IGF-1 (Brambilla et al., 2001) and elevated cortisol release (Connan et al., 2003a).

Enhanced satiety resulting from elevated basal and post-prandial CCK release (e.g. Brambilla et al., 1995a; Fujimoto et al., 1997) might contribute to the aetiology of AN. However, this finding has not been consistently replicated (e.g. Pirke et al., 1994) and, when present, resolves with partial weight gain (e.g. Tamai et al., 1993). Furthermore, the satiating effect of CCK may be reduced by the impairment of 5HT function associated with AN (Stallone et al., 1989). Pancreatic polypeptide release is elevated in response to a meal during the acute illness (Alderdice et al., 1985; Fujimoto et al., 1997) although the significance of this finding is unclear. Although a wealth of studies have examined a possible role for leptin in the aetiology of AN, little evidence of functional abnormality has been unearthed. During the acute illness, leptin concentrations are low (Casanueva et al., 1997; Herpertz et al., 2000; Monteleone et al., 2000a) but proportional to BMI and fat mass (Ferron et al., 1997; Grinspoon et al., 1996; Monteleone et al., 2000a), except at extremely low weight (Casanueva et al., 1997; Pauly et al., 2000). The diurnal rhythm of leptin and its relationship with insulin and IGF-1 release are preserved, but the temporal relationship with cortisol is disrupted until weight is restored (Eckert et al., 1998; Herpertz et al., 1998, 2000).Interestingly, there is a dissociation between BMI and leptin levels during weight gain (Hebebrand et al., 1997; Mantzoros et al., 1997) which could contribute to difficulty achieving full weight restoration with rapid refeeding. Similarly, leptin levels are elevated above that expected for BMI in those with the purging subtype of AN (Mehler et al., 1999). Elevation of the CSF : plasma leptin ratio in acute AN (Mantzoros et al., 1997) suggests that central levels may be relatively elevated, perhaps contributing to impairment of appetite. However, normalisation following full recovery from AN (Mantzoros et al., 1997) indicates that altered leptin dynamics are a state-related phenomena. In keeping with this hypothesis, leptin levels are predictive of amenorrhoea in eating-disordered patients (Kopp et al., 1997) and restoration of normal levels is necessary, but not sufficient, for restoration of menses (Audi et al., 1998).

Central Regulation of Appetite and Weight in AN

Far less is known about the central mechanisms regulating appetite and weight in AN. Data examining the role of hypothalamic peptides in the eating disorders are limited to estimations of plasma and CSF levels because of the inherent difficulty in measuring regional brain peptide activity in vivo. In terms of orexigenic networks of the hypothalamus, NPY levels are elevated in the CSF of those with AN (Kaye et al., 1990), consistent with a reduced leptin signal from the periphery and an orexigenic response to starvation. Levels normalise with full recovery (Gendall et al., 1999; Kaye et al., 1990), at which time CSF polypeptide Y (PPY) levels are also reported normal (Gendall et al., 1999). The NPY Y5 receptor mediates the orexigenic effects of NPY (Schaffhauser et al., 1997) and there is no evidence for an association between receptor gene polymorphisms and AN (Rosenkranz et al., 1998). Galanin has been measured only in peripheral plasma, in which levels do not differ from normal (Baranowska et al., 1997). There are no data currently available regarding orexin, AgRP, MCH or ghrelin function in AN, but all are potential candidates for an aetiological role. Plasma β-endorphin (β-EP) levels are reduced in acute AN (Baranowska et al., 1997).

Circadian rhythmicity is lost and there is some evidence of disrupted regulation of POMCderived peptide secretion (Brambilla et al., 1991). CSF levels appear to be reduced, but only in the most severely underweight patients (Kaye et al., 1987a). Although β-EP stimulates feeding, there is some evidence that infusion of an opioid antagonist improves weight gain during refeeding in AN (Moore et al., 1981). This may reflect blockade of central motivation and reward systems, or more direct effects on the hypothalamic regulation of feeding.

Again, little is known about the functional activity of anorexigenic peptides, such as MSH and CART, in AN. One recent study found elevated plasma CART concentrations in those with the acute disorder, and normal concentrations in women who had fully recovered (Sarah Stanley, personal communication). This finding must be interpreted with caution, because recent data suggests that CART may have orexigenic properties in at least some areas of the hypothalamus (Abbott et al., 2001).

Studies ofHPAaxis activity inANsuggest that an impairment of feedback inhibition at the level of the hypothalamus gives rise to elevatedCRHactivity. This is turn overrides feedback inhibition at the level of the pituitary, giving rise to the HPA axis hyperactivity associated with AN (Kaye et al., 1987b; Licinio et al., 1996). Elevated concentrations of CRH in the CSF (Kaye et al. 1987b) and blunting of the ACTH and β-EP responses to exogenous CRH (Brambilla et al., 1996; Hotta et al., 1986) are consistent with this hypothesis. Although HPA axis AVP activity is up-regulated in the context of chronic stress in animals and depression in humans, there is no evidence of elevated AVP activity in acute AN (Connan et al., 2003b, 2003c). This is significant becauseAVP modulatesCRHrelease and sensitivity of the axis to feedback inhibition via altered glucocorticoid receptor activity (Felt et al., 1984; Plotsky et al., 1984). An abnormal HPA axis response to chronic stress may therefore contribute to persistently elevated CRH activity in AN. In addition to the inhibitory effect on appetite and feeding, elevated CRH activity has widespread behavioural and physiological effects, which include many of the features of AN, such as: increased locomotor activity; cardiovascular changes; reduced social and sexual behaviour; impaired sleep; and increased anxiety behaviours (Dunn & Berridge, 1990). There is some evidence to suggest that a heightened cortisol response to stress and a blunted cortisol response to a meal may persist even after full recovery (Connan 2003a; Ward et al., 1998). Subtle abnormalities of HPA axis regulation might therefore contribute to susceptibility to AN.


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1.8

# The anatomy of websites and code

Modern web browsers are amazingly powerful computer programs that can ‘understand’ a wide variety of inputs. Of the possible inputs, on this course we identify three kinds

• HTML code
• CSS style sheet code
• programming code, e.g. JAVASCRIPT

Of these, HTML code is essential for a browser to display a web page. This includes the text that is displayed on web pages and tags to do particular things such as display an image or link to another page. The HTML code for a web page is stored in files which have the .HTML file extension.

The web page you saw in the last step in Figure 1.(a) has text of different sizes and colour. These are set by a style sheet, rather than directly by HTML code. The style sheet is usually an external file which tells the browser how text and other things should be rendered. Technically they are called cascading style sheets and they have the .CSS file extension. Usually cascading style sheets play no active part in the Google Analytics process of collecting and analysing usage data.

Embedding program code such as Javascript into web pages is essential for user interaction on many websites, but is beyond the scope of this short course.

The HTML code that generated this web page is shown below.

Figure 1. The code to create the webpage in Step 1.3 Figure 1(a)

The portion of the HTML code highlighted in orange at the beginning informs the browser that this is a document of type HTML and the language to be used is English. The orange highlighted line at the end tells the browser that the file is complete. The first line, that begins with <!-- and ends with --> is a comment. It doesn’t do anything but gives information that makes the code easier to understand.

The section highlighted in blue does not display anything, but gives the browser information on how to process the page. The line <meta charset= “UTF-8”> establishes the default character set to be used, and Web Analytics gives the title of the web page. The next line links a style sheet file called WE1_Stylesheet.css to this web page. The style sheet determines the size and colour of the text.

The section highlighted in green generates the elements seen on the webpage. The first three lines beginning with <a> and ending with </a> display the image called WE1_DA.RE_logo.png and links it to the http://dare-project.eu website. The other lines of text in the green highlighted area display the text, links and images you saw in Step 1.6.

Google Analytics works by generating a few lines of tracking code that the website owner inserts into the part of the webpage highlighted in blue. This code does not display anything, but it allows Google Analytics to collect information behind the scenes, as you will see.


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# Volcano Watch — A lava lake is revealed within Halemaumau's new vent

September 11, 2008

"So, how far down is the lava?"

This is perhaps the most common question directed to HVO scientists since the new vent opened in Halemaumau Crater this past March. With its persistently voluminous plume, nighttime glow and occasional explosions, the vent has been the subject of many recent "Volcano Watch" articles, each addressing a unique aspect of the ongoing activity.

But none of the articles, thus far, has focused on the Big Question, in large part because—until recently—we have only had a rough idea of how deep the lava surface was below the vent rim.

Before last week, views into the vent during our routine helicopter overflights did not provide any clear views of a lava surface, so we had to rely on geophysical and geochemical tools to infer the depth to the lava. Our best guess before last week was that the lava was indeed very shallow, on the order of several hundred meters.

As noted in the August 21 "Volcano Watch", precise locations of seismic tremor bursts indicated the tremor source (and thus perhaps the lava surface) to be 100-200 m (330-660 ft) deep.

But on Friday morning, September 5, the view that HVO scientists have been waiting months to see finally materialized, and uncertainty regarding the depth to the lava surface was finally put to rest. During our routine overflight, we were able to clearly see a 50-m (160 ft)-diameter lava lake about 100 m (330 ft) below the overhanging vent rim on the floor of Halemaumau.

Vigorous bubbling, sloshing and roiling disrupted the lava surface as globs of spatter were thrown onto the conduit walls, creating an awe-inspiring spectacle. Finally, we could see the lava lake we'd only been able to imagine during the preceding months.

Besides elation and awe, we also felt vindicated that two of our assessments regarding subsurface conditions were sound. First, the view confirmed that the lava surface was very shallow, as geophysical data suggested. Second, while hovering over the vent, we observed a cycle of activity that confirmed that the seismic tremor bursts we've been recording for several months were due to a process called "gas-pistoning," as we have speculated over the past few months.

The term "gas-pistoning" was introduced to the volcanological lexicon by HVO scientists observing activity during the Mauna Ulu eruption of Kīlauea in the late 1960s. During sustained lava pond activity within the vent, the partially crusted lava surface would slowly rise toward the vent rim. When it reached its maximum height, spattering along the pond margins commenced and expanded until the whole pond surface was involved.

The lava level would then drop as lava drained back into the conduit. This activity, repeated in a cyclic manner, was interpreted as gas accumulating underneath the lava surface, leading to the surface rise, followed by gas release during the spattering, which caused the lava level to drop.

Gas-pistoning has been commonly observed within Puu Oo's crater, as well, and for several months, it was present within the perched lava channel active in late 2007. Now, after seeing it within Halemaumau, it appears clear that gas-pistoning is a common process at Kīlauea.

Our views into the vent on September 5 revealed gas-pistoning at Halemaumau. When we arrived at the vent, we heard by radio from the observatory that we were in an interval between tremor bursts. The lava surface at that point was moderately active, with a discontinuous crust and numerous scattered bubbling sources. Over the radio, we heard that a tremor burst was starting, and as we watched, the lava surface began to bubble and slosh more violently, until the whole lake surface was overturning and becoming a single roiling fountain as gas gushed more vigorously from the vent. A few minutes later, when the tremor burst and roiling had died down, we could see that the lava level had dropped several meters (yards), owing to all the escaped gas.

For now, Pele chooses to keep the lava surface out of sight from Jaggar Overlook, but, nonetheless, visitors during the past week have gotten some clear—though indirect—views of the gas-pistoning. Come to Jaggar at night and watch for the vent to darken as the lava crusts over, followed several minutes later by a dramatic increase in vent brightness as the pond surface is disrupted by intense spattering beneath the floor of Halemaumau Crater.

Frequently, gas-pistoning is replaced by steady tremor and vent glow that can last for days or weeks. Why the gas-pistoning comes and goes at the lava lake is not yet clear, demonstrating that—although the lava lake has been revealed—there are plenty of mysteries left to ponder.

————————————————————————————————————————————————————————————————

### Volcano Activity Update

Kīlauea Volcano continues to be active. A vent in Halemaumau Crater is erupting elevated amounts of sulfur dioxide gas and very small amounts of ash. Resulting high concentrations of sulfur dioxide in downwind air have closed the south part of Kīlauea caldera and produced occasional air quality alerts in more distant areas, such as Pahala and communities adjacent to Hawaii Volcanoes National Park, during kona wind periods. On September 5, scientists observed a 50-m (160 ft)-diameter lava lake about 100 m (330 ft) below the vent rim on the floor of Halemaumau; the lava cannot be seen from the rim of Halemaumau Crater or Jaggar Museum Overlook. There have been several small ash-emission events from the vent, lasting only minutes, in the last week.

Puu Ōō continues to produce sulfur dioxide at even higher rates than the vent in Halemaumau Crater. Trade winds tend to pool these emissions along the West Hawaii coast, while Kona winds blow these emissions into communities to the north, such as Mountain View, Volcano, and Hilo.

Lava continues to erupt from fissure D of the July 21, 2007, eruption and flows toward the ocean through a well-established lava tube. No significant incandescence was observed from the fissure to the top of the pali above the abandoned Royal Gardens subdivision or on the coastal plain, indicating little or no new surface flows in the past week. Lava continues to flow into the ocean at Waikupanaha.

Be aware that active lava deltas can collapse at any time, potentially generating large explosions. This may be especially true during times of rapidly changing lava supply conditions. Do not venture onto the lava deltas. Even the intervening beaches are susceptible to large waves generated during delta collapse; avoid these beaches. In addition, steam plumes rising from ocean entries are highly acidic and laced with glass particles. Check Civil Defense Web site or call 961-8093 for viewing hours.

Mauna Loa is not erupting. Two earthquakes were located beneath the summit this past week. Continuing extension between locations spanning the summit indicates slow inflation of the volcano.

One earthquake beneath Hawaii Island was reported felt within the past week. A magnitude-3.6 earthquake occurred at 5:34 a.m., H.s.t., on Sunday, September 7, 2008, and was located 5 km (3 miles) west of Pahala, at a depth of 35 km (22 miles).

Visit our Web site for daily Kīlauea eruption updates, a summary of volcanic events over the past year, and nearly real-time Hawaii earthquake information. Kīlauea daily update summaries are also available by phone at (808) 967-8862. Questions can be emailed to askHVO@usgs.gov. skip past bottom navigational bar


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## April 11, 2018

### Torsion: Graph Magnitude Homology Meets Combinatorial Topology

#### Posted by Simon Willerton

As I explained in my previous post, magnitude homology categorifies the magnitude of graphs. There are two questions that will jump out to seasoned students of homology.

• Are there two graphs which have the same magnitude but different homology groups?
• Is there a graph with torsion in its homology groups?

Both of these were stated as open questions by Richard Hepworth and me in our paper as we were unable to answer them, despite thinking about them a fair bit. However, recently both of these have been answered positively!

The first question has been answered positively by Yuzhou Gu in a reply to my post. Well, this is essentially answered, in the sense that he has given two graphs both of which we know (provably) the magnitude of, one of which we know (provably) the magnitude homology groups of and the other of which we can compute the magnitude homology of using mine and James Cranch’s SageMath software. So this just requires verification that the program result is correct! I have no doubt that it is correct though.

The second question on the existence of torsion is what I want to concentrate on in this post. This question has been answered positively by Ryuki Kaneta and Masahiko Yoshinaga in their paper

It is a consequence of what they prove in their paper that the graph below has $2$-torsion in its magnitude homology; SageMath has drawn it as a directed graph, but you can ignore the arrows. (Click on it to see a bigger version.)

In their paper they prove that if you have a finite triangulation $T$ of an $m$-dimensional manifold $M$ then you can construct a graph $G((T)$ so that the reduced homology groups of $M$ embed in the magnitude homology groups of $G((T)$:

$\widetilde{\mathrm{H}}_i(M)\hookrightarrow MH_{i+2, m+2}( G(T)) \,\,\,\, \text{for }\,\,0\le i \le m.$

Following the suggestion in their paper, I’ve taken a minimal triangulation $T_0$ of the real projective plane $\mathbb{R} P^2$ and used that to construct the above graph. As we know $\mathrm{H}_1(\mathbb{R} P^2)=\mathbb{Z}/2\mathbb{Z}$, we know that there is $2$-torsion in $MH_{3,4}(G({T_0}))$.

In the rest of this post I’ll explain the construction of the graph and show explicitly how to give a $2$-torsion class in $MH_{3,4}(G({T_0}))$. I’ll illustrate the theory of Kaneta and Yoshinaga by working through a specific example. Barycentric subdivision plays a key role!

Posted at 4:15 PM UTC | Permalink | Followups (2)

## April 5, 2018

### Magnitude Homology Reading Seminar, II

#### Posted by Simon Willerton

guest post by Scott Balchin

Following on from Simon’s introductory post, this is the second installment regarding the reading group at Sheffield on magnitude homology, and the first installment which looks at the paper of Leinster and Shulman. In this post, we will be discussing the concept of magnitude for enriched categories.

The idea of magnitude is to capture the essence of size of a (finite) enriched category. By changing the ambient enrichment, this magnitude carries different meanings. For example, when we enrich over the monoidal category $[0,\infty ]$ we capture metric space invariants, while changing the enrichment to $\{ \text {true},\text {false}\}$ we capture poset invariants.

We will introduce the concept of magnitude via the use of zeta functions of enriched categories, which depend on the choice of a size function for the underlying enriching category. Then, we describe magnitude in a more general way using the theory of weightings. The latter will have the advantage that it is invariant under equivalence of categories, a highly desirable property.

What is presented here is taken almost verbatim from Section 2 of Leinster and Shulman’s Magnitude homology of enriched categories and metric spaces. It is, however, enhanced using comments from various other papers and, of course, multiple $n$-Café posts.

## April 2, 2018

### Dynamical Systems and Their Steady States

#### Posted by John Baez

guest post by Maru Sarazola

Now that we know how to use decorated cospans to represent open networks, the Applied Category Theory Seminar has turned its attention to open reaction networks (aka Petri nets) and the dynamical systems associated to them.

In A Compositional Framework for Reaction Networks (summarized in this very blog by John Baez not too long ago), authors John Baez and Blake Pollard put Fong’s results to good use and define cospan categories $\mathbf{RxNet}$ and $\mathbf{Dynam}$ of (open) reaction networks and (open) dynamical systems. Once this is done, the main goal of the paper is to show that the mapping that associates to an open reaction network its corresponding dynamical system is compositional, as is the mapping that takes an open dynamical system to the relation that holds between its constituents in steady state. In other words, they show that the study of the whole can be done through the study of the parts.

I would like to place the focus on dynamical systems and the study of their steady states, taking a closer look at this correspondence called “black-boxing”, and comparing it to previous related work done by David Spivak.

Posted at 8:49 PM UTC | Permalink | Followups (1)


text:
# Defining different bounds for each element of a matrix of parameters

\beta is a matrix of regression coefficient of dimension N\times K where N is the number of outcomes and K the number of covariates (so that \beta \cdot x, where x is the vector of covariates of length K, gives a vector of length N). Some coefficients in the matrix \beta are constrained according to other coefficients and I would like to find a way to declare these constraints directly with the matrix \beta, and not one by one, as both N and K are high.
As a simple example, let’s consider N=2, K=3 and the following constraints in \beta:=(\beta_{i,j})_{i\leq N,j\leq K}:
\beta_{1,1}>\beta_{1,2}
\beta_{1,1}>\beta_{1,3}
\beta_{2,3}<\beta_{2,1}
\beta_{2,3}<\beta_{2,2}

One way to write them would be to use real variables for each of the 6 entries:

parameters {
real beta_11;
real <upper = beta_11> beta_12;
real <upper = beta_11> beta_13;
real beta_23;
real <lower = beta_23> beta_21;
real <lower = beta_23> beta_22;
}

However, as N and K are large, it would be better to declare \beta as a matrix. However, I’m not sure whether it is possible to assign some constraint/bound for each element of the matrix. Coming back to my example, I would like to do something like:

parameters {
real beta[N,K];
for(i in 1:N){
for(j in 1:K){
//define lower and upper bound for coefficient beta[i,j]
}
}
}

Could anyone think of such way of defining bounds or should the definition of bounds necessarily be at the same place as variable declaration?

No; you have to write them out as scalars in this case.


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# Synopsis: A question of size

Just how small is the proton?

The charge radius of the proton is one of nature’s fundamental parameters. Its currently accepted CODATA (Committee on Data for Science and Technology) value, $0.8768×{10}^{-15}\phantom{\rule{0.333em}{0ex}}\text{m}$, has been determined primarily by measurements of the hydrogen Lamb shift and, to lesser accuracy, by electron-proton scattering experiments. This value has recently been called into question by a research team at the Paul Scherrer Institut (PSI) in Villigen, Switzerland. By measuring the Lamb shift in muonic hydrogen, these researchers obtained a value of $0.8418×{10}^{-15}\phantom{\rule{0.333em}{0ex}}\text{m}$ for the charge radius, five standard deviations below the CODATA value.

In a paper appearing in Physical Review Letters, the A$1$ Collaboration has determined the electric and magnetic form factors of the proton with higher statistics and precision than previously known, using the Mainz (Germany) electron accelerator MAMI (Mainz Microtron) to measure the electron-proton elastic-scattering cross section. Both form factors show structure at ${\text{Q}}^{2}$$\approx$ ${\text{m}}_{\pi }^{2}$ that may indicate the influence of the proton’s pion cloud. But, in addition, the collaboration’s extracted value for the charge radius agrees completely with the CODATA value. The discrepancy between “electron-based” measurements and the recent PSI “muon-based” measurement thus remains a puzzle. – Jerome Malenfant

### Announcements

More Announcements »

## Subject Areas

Particles and Fields

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## Next Synopsis

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A new value for the Higgs boson mass will allow stronger tests of the standard model and of theories about the Universe’s stability. Read More »

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### Synopsis: IceCube Neutrinos Pass Flavor Test

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text:
# Extracting entropy of exotic quasiparticles from conductance measurements

Cheolhee Han, Andrew K. Mitchell, Zubair Iftikhar, Yaakov Kleeorin, Anne Anthore, Frédéric Pierre, Yigal Meir, Eran Sela

Research output: Working paper/PreprintPreprint

## Abstract

Fractional entropy is a signature of nonlocal degrees of freedom, such as Majorana zero modes or more exotic non-Abelian anyons. However, in the mesoscopic quantum systems hosting these states direct entropy measurements remain highly challenging. Here we introduce a self-consistent framework to extract entropy from existing experimental conductance measurements. Applying this approach to conductance measurement through a metallic quantum dot displaying `charge' Kondo effects, we extract the fractional entropy in the two- and three-channel Kondo effects, corresponding to Majorana and Fibonacci zero modes, respectively. In these cases, the entropy extracted from the experimental data falls on a theoretical curve displaying scaling towards $k_{\rm{B}}\log \sqrt{2}$ or $k_{\rm{B}}\log \frac{1+\sqrt{5}}{2}$ entropy, respectively. Our protocol provides a general route towards a direct extraction of entropy from charge measurements in mesoscopic quantum systems.
Original language English GB Published - 29 Aug 2021

## Keywords

• cond-mat.mes-hall
• cond-mat.str-el

## Fingerprint

Dive into the research topics of 'Extracting entropy of exotic quasiparticles from conductance measurements'. Together they form a unique fingerprint.


text:
# Algebraic independence for values of integral curves

Fonseca, T

23 March 2019

## Journal:

Algebra and Number Theory

## Last Updated:

2021-06-20T08:36:14.57+01:00

3

13

## DOI:

10.2140/ant.2019.13.643

643-694

## abstract:

We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasi-projective variety over Q that are integral curves of some algebraic vector field (defined over Q). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields. This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series E2, E4, E6. The main technical improvement in our approach is the replacement of a rather restrictive hypothesis of polynomial growth on Taylor coefficients by a geometric notion of moderate growth formulated in terms of Value Distribution Theory.

963704

Submitted

Journal Article


text:
## Conceptual Physics (12th Edition)

The decibel scale is logarithmic, based upon powers of 10. Adding 10 dB means the intensity is multiplied by a factor of 10. In other words, when the intensity of sound is increased by a factor of 10, the intensity level increases by 10 dB. A sound of 40 dB has an intensity 10,000 times greater than 0 dB, the threshold of hearing. A sound of 40 dB corresponds to an intensity of $10^{-8} \frac{W}{m^{2}}$, which is $10^{4} = 10,000$ times greater than the intensity of a 0 dB sound, $10^{-12} \frac{W}{m^{2}}$. This is discussed on page 394, and shown in Table 21.1.


text:
# List Of Cancelled Trains Today 9 Ausgust 2022 – Check Full List

List Of Cancelled Trains Today  – Check Updated List Here – The Indian Railways is the fourth largest railway network in the world by size and caters to the needs of millions of people. As of March 2019, Indian Railways runs 13,523 passenger trains daily to various destinations. Due to some reasons like bad weather, blockage on track, railway authorities cancel some trains daily and the list of the same will be displayed online for the convenience of passengers. In the olden days, passengers had to call the railway helpline to know the status of the train. Due to digitization, all the information about trains becomes available in a few clicks. eTrainStatus, one of the popular train enquiry service providers, ranks first in providing passengers with a list of cancelled trains and saving their precious time.

Cancellation of train service is the last call taken by the Indian Railways. Depending on the situation on the route, the authorities either divert or partially cancel most of the trains. In the section of eTrainStatus cancelled trains, the list of trains is updated daily and passengers can know it before they start preparing for the journey. During this lockdown, both Indian Railways and IRCTC have cancelled 98% of the trains and are running just 300 trains to different destinations.

Once you see the status of the train you booked as cancelled, you may first have concerns about the return of the money you paid for the ticket. Don’t worry, we’ll provide you with cancellation and refund rules that can be helpful. PM Modi Schemes List

## List Of Cancelled Trains Today  9 August 2022

 01605 , 01606 , 01607 , 01608 , 01609 , 01610 , 03085 , 03086 , 03087 , 03094 , 03311 , 03312 , 03341 , 03342 , 03371 , 03372 , 03502 , 03549 , 03591 , 03592 , 03607 , 03608 , 03657 , 03658 , 04026 , 04084 , 04098 , 04424 , 04432 , 04454 , 04462 , 04509 , 04510 , 04601 , 04602 , 04647 , 04648 , 04685 , 04686 , 04699 , 04700 , 04769 , 04775 , 04776 , 04910 , 04946 , 04990 , 05035 , 05036 , 05040 , 06407 , 06408 , 06845 , 06846 , 06977 , 06980 , 07520 , 07906 , 07907 , 09108 , 09109 , 09110 , 09113 , 09175 , 09176 , 09483 , 10101 , 10102 , 11018 , 11027 , 11028 , 11139 , 11140 , 11421 , 11422 , 11754 , 12101 , 12102 , 12115 , 12116 , 12129 , 12130 , 12157 , 12158 , 12169 , 12170 , 12767 , 12771 , 12809 , 12810 , 12833 , 12834 , 12855 , 12856 , 14214 , 14805 , 15081 , 15777 , 15778 , 16217 , 16381 , 16382 , 17031 , 17032 , 18029 , 18030 , 18109 , 18110 , 18238 , 18239 , 18240 , 20823 , 20843 , 20948 , 20949 , 22159 , 22160 , 22179 , 22648 , 22717 , 22718 , 22815 , 22846 , 22920 , 31411 , 31414 , 31423 , 31432 , 31711 , 31712 , 33657 , 33658 , 36033 , 36034 , 37211 , 37216 , 37246 , 37247 , 37253 , 37256 , 37305 , 37306 , 37307 , 37308 , 37319 , 37327 , 37330 , 37338 , 37343 , 37348 , 37411 , 37412 , 37415 , 37416 , 37611 , 37614 , 37657 , 37658 , 37731 , 37732 , 37741 , 37746 , 37782 , 37783 , 37785 , 37786 , 37823 , 37834 , 47176 , 47187 , 47189 , 47190 , 47191 , 47192 , 47210 , 52544 , 52590 , 52591 , 52594

### How to check your station code: Step-by-step guide

• Visit the official website – irctchelp.in
• Click on the station name against the station code
• You will find the station code and save the details for further updates

#### Steps to check the full list of cancelled trains

• First Visit enquiry.indianrail.gov.in/mntes and select the date of journey
• Select Exceptional Trains on the top panel of the screen
• Click on Cancelled Trains option
• Select Fully or Partially option to see full list of trains with time, routes and other details.

For more information, the Railway authorities have urged the passengers to visit enquiry.indianrail.gov.in/mntes or NTES app to get the details of the actual arrival-departure of these cancelled trains.

#### How to Get Refund for Train Tickets Booked at Counter (Offline)?

Even for tickets booked through counters, Indian Railways automatically cancels tickets and passengers are asked to go to the counter and deposit the refund amount. However, in situations when passengers want to cancel their tickets, IRCTC has come up with a new facility for those who had booked offline tickets at the railway station’s reservation counter. It is not possible to visit the offline counter for any kind of enquiry during this lockdown. Hence, IRCTC has introduced an online portal called ‘Counter Ticket Cancellation’ for cancellation of offline or counter booked tickets.


text:
# NLP

• Token classification: classify each word into classes (remember token != work)

• Masked language modeling: fill in the blank

• Perplexity (exponentiated perplexity) or CrossEntropy, or more sophisticated metrics BLEU or ROUGE, since blank can be filled with multiple correct answer
• To deal with variable text length, either: zero pad to the longest, chunkate long text, or concat everything together with a split token indicate context transition (best)
• good for fine tuning encoder before addapting it to other tasks
• BERT's random masking: handeled by collator
• 80% of the time: Replace the word with the [MASK] token, e.g., my dog is hairy -> my dog is [MASK]
• 10% of the time: Replace the word with a random word, e.g., my dog is hairy -> my dog is apple. This will force model to generate proper contextual embedding for all tokens in the sequence, not only the [MASK] ones. This is consistent with the goal of finetuning.
• 10% of the time: Keep the word unchanged, e.g., my dog is hairy -> my dog is hairy. The purpose of this is to bias the representation towards the actual observed word. Otherwise, the model will only look at the context to define the word, and ever look at the word itself.
• Translation: BLEU score (geometric average of 1~4-gram where each gram is clipped precision of matching contiguous subsequence)

• BLEU Score: widely used and simple, but
• doesn't consider meaning
• doesn't consider sentence structure
• may not work for non-English
• hard to compare with different tokenizers
• SacreBLEU: built-in internal tokenizers for consistent comparison
• Tokenizer: must use different tokenizer (and padding rule) if language is different
• Summarization: ROUGE score

• ROUGE1-F1: ROUGE1 precision and recall of matching
• ROUGE1, ROUGE2: 1 and 2 gram version with recall only
• ROUGE-L: longest common sequence (might not be contiguous) in a sentence, capture sentence structure better. Averaged for each sentence.
• ROUGE-LSum: longest common sequence in the whole summary
• as_target_tokenizer: used when label needs different tokenizer than input
• Causal language modeling (regressive modeling): Perplexity and CrossEntropy

• Question answering: context with a question, mark answer in text. F1 Score. stride is number of overlapping token between 2 truncated sections (a truncated section might contain partial answer, in this case we mark as no answer, or no answer).

Transformers: traditionally, transformer means transformer module that is introduced in the original "Attention is all you need" paper. However, since this architecture cna be found in many models, we usually summarize those models by calling them "transformers"

Sequence to Sequence: a task, can be something like language translation or chat bot. Sequence to sequence is generally solved using an encoder and decoder. Encoder encodes every word into a sequence of embeddings in parallel where each word has context information associated with it. Decoder takes in the whole sequence of embeddings generated by the encoder as well as its own sequence of answer tokens to produce the next answer token.

• RNN Attention: instead sending encoded sequence once, we send many times for each next-token prediction to the decoder.

Encoders-only Architecture: ALBERT, BERT, DistilBERT, ELECTRA, RoBERTa. Can be used to predict masked word or sequence classification.

Decoder-only Architecture: CTRL, GPT, GPT-2, Transformer XL. The right word is masked compared to encoder. Can be used for sequence generation. If a generated sequence exceed model's maximum context size, model cannot remember first word it generated.

Encoder-decoder: BART, T5, Marian, mBART.

Loss: loss propagates through time. For RNN, if model predicts wrong, our next decoder input will have 50% chance of being ground truth instead of predicted token.

Tokenizers: tranlate word to numbers, a step before feeding into encoder. (Tokenizers are usually not trained using gradient descent, but rather deterministic rules based on heuristics. It will optimize for best word spliting based on data distribution.)

• types: different tokenization methods

• word based: split space. bad for unknown word (nobody actually use it)
• character based: split characters. bad for Chinese and lengthened input (nobody actually use it)
• subword: decompose tokenization to token and ##ization
• WordPiece: used by BERT and DistilBERT
• Unigram: used by XLNet and ALBERT
• Byte-Pair Encoding: used by GPT-2, RoBERTa
• loading: we load two parts, algorithm (store code that splits to subwords, and add special token) and vocabulary (store data like dictionary mappings that translate subword string to numbers).

• sequence batch: since we want to train in batches, model should support multiple inference. if length of sequences in a batch don't match, we zero-pad them and assign zero part with no attention. (also chunkate if our input is longer than what model expects) Batch might have different shape (might be slower on TPU, but faster on GPU).

• sentence pair: since many dataset are about properties of a pair or sentences, we often add a field to training data to indicate which sentence is the first and which is the second (in ordered manner).

Model:

• head: An additional component, usually made up of one or a few layers, to convert the transformer predictions to a task-specific output

Accelerate: HuggingFace's distributed training integrated with PyTorch

In order to better understand the role of [CLS] let's recall that BERT model has been trained on 2 main tasks:

Masked language modeling: some random words are masked with [MASK] token, the model learns to predict those words during training. For that task we need the [MASK] token.

Next sentence prediction: given 2 sentences, the model learns to predict if the 2nd sentence is the real sentence, which follows the 1st sentence. For this task, we need another token, output of which will tell us how likely the current sentence is the next sentence of the 1st sentence. And here comes the [CLS]. You can think about the output of [CLS] as a probability.

Now you may ask the question: can we instead of using [CLS]'s output just outputting a number (as probability)? Yes, we can do that if the task of predicting next sentence is a separate task. However, BERT has been trained on both tasks simultaneously. Organizing inputs and outputs in such a format (with both [MASK] and [CLS]) will help BERT to learn both tasks at the same time and boost its performance.

When it comes to classification task (e.g. sentiment classification), as mentioned in other answers, the output of [CLS] can be helpful because it contains BERT's understanding at the sentence-level.

Answer by hoang tran on stackoverflow.

Here're my understandings:

(1)[CLS] appears at the very beginning of each sentence, it has a fixed embedding and a fix positional embedding, thus this token contains no information itself. (2)However, the output of [CLS] is inferred by all other words in this sentence, so [CLS] contains all information in other words.

This makes [CLS] a good representation for sentence-level classification.

Model BPE WordPiece Unigram
Training Starts from a small vocabulary and learns rules to merge tokens Starts from a small vocabulary and learns rules to merge tokens Starts from a large vocabulary and learns rules to remove tokens
Training step Merges the tokens corresponding to the most common pair Merges the tokens corresponding to the pair with the best score based on the frequency of the pair, privileging pairs where each individual token is less frequent Removes all the tokens in the vocabulary that will minimize the loss computed on the whole corpus
Learns Merge rules and a vocabulary Just a vocabulary A vocabulary with a score for each token
Encoding Splits a word into characters and applies the merges learned during training Finds the longest subword starting from the beginning that is in the vocabulary, then does the same for the rest of the word Finds the most likely split into tokens, using the scores learned during training

Traditional RNN: process one token at a time

• encoder

• input: current word vector, previous word embedding
• output: current word embedding
• decoder

• input: last word embedding, or generated previous word embedding
• output: generated word vector, generated next word embedding

In traditional RNN, the entire sequence correspond to one backprop. However, in transformers, every token in the sequence correspond to one backprop. (With the assumption batch_size = 1). This also solves the issue of diminishing gradient.

This video is very good in terms of explaining Transformers.

Transformers: feed x (input) label to encoder, feed masked y (ground truth) label to decoder

• Positional Encoding: add data to establish ordering of sequence, since we no longer use RNN and process tokens in parallel. The position encoding is

• Masked Input: different than CV, we actually send ground truth to decoder, but mask out the things that we want the model to predict. This is so that we can train transformer in parallel (without using what was predicted).

• Attention: there are 3 input and 1 output. We dot product Key and Query vector to select matching key (and softmax exponentate for make pseudo differentiable selection), after dot product and softmax, you get a distribution with peat at selected area. (for first token, we copy one token to 3 inputs)

• Value: embeddings generated by encoder
• Key: "key" generated by encoder
• Query: "query" generated by decoder when looking at "what we generated so far"
• Multi-headed: Multiple attention heads in a single layer in a transformer is analogous to multiple kernels in a single layer in a CNN: they have the same architecture, and operate on the same feature-space, but since they are separate 'copies' with different sets of weights, they are hence 'free' to learn different functions.

Here is how you would implement and train transfromer from scratch with Pytorch.

Table of Content


text:
# Problem on the Cech Cohomology of a Sheaf over a Paracompact Space

I am reading Rotman, Introduction to Homological Algebra, and $\textbf{Theorem 6.88.}$ reads:

Let $\mathcal{F}$ be a sheaf over a paracompact space $X$. If $X$ has an open cover $\mathcal{U}$ with $dim(\mathcal{N}(\mathcal{U}))<n$, then $\check{H}^q(\mathcal{F})=\{0\}$ for all $q\geq n+1$.

The proof of this theorem refers to Swan, The Theory of Sheaves, p.109, but I don't have this book.

However, $\mathcal{U}=\{X\}$ itself is an open cover where $dim(\mathcal{N}(\mathcal{U}))\leq 0$, and then by the above theorem, $\check{H}^q(\mathcal{F})=\{0\}$ for all $q\geq 1$, and this is definitely not true. This example even appears in Rotman too; it's $\textbf{Example 6.78.}$ (iii).

I think there is something wrong, but I don't know how to fix it. I think maybe there needs some kind of condition for an open cover $\mathcal{U}$: for example, every intersection of distinct sets in $\mathcal{U}$ is contractible. I am not sure if this is true if I add this condition. Or if every open cover $\mathcal{V}$ has a refinement $\mathcal{U}$ with $dim(\mathcal{N}(\mathcal{U}))<n$, then I am quite sure the theorem is true, but this is way too easier than the original theorem, so I don't think this is what Rotman wanted to state.

• if $\dim\mathcal{N(U)}<n$, then $\check H^q(\mathcal{U,F})=0$ for all $q\geq n+1$.
• in particular, if there exists $\mathcal{U}$ is a good cover for $\mathcal{F}$ (that is one such that any finite intersections $V=U_1\cap...\cap U_n$ satisfies $H^q(V,\mathcal{F})=0$ for all $q>0$) such that $\dim\mathcal{N(U)}<n$ then $\check H^q(\mathcal{F})=0$ for all $q\geq n+1$.
• I am still in confusion. I think if the theorem is corrected in that way, it seems true. But the next theorem in Rotman's book claims that, as a corollary of this theorem, if X is compact, $H^p(X)=0$ for sufficiently large $p>n$. I am not sure that the corrected theorem implies this corollary. – J1U May 17 '18 at 1:01
• Well there exists a good cover, since $X$ is compact then you can take a finite sub-cover which will still be a good cover. – Roland May 17 '18 at 6:45


text:
# Math Help - velocity

1. ## velocity

In a computer simulation, a satellite orbits around Earth at a distance from the Earth's surface of miles. The orbit is circular, and one revolution around Earth takes 10.5 days. Assuming the radius of the Earth is 3960 miles, find the linear speed of the satellite. Express the answer in miles per hour to the nearest whole mile.

The radius is 13,960 I believe. I'm not sure what to do after that.. Thanks


text:
# Justifying Ampere's Law and Faradays Law with an experiment

1. Apr 1, 2016

### PhotonSSBM

1. The problem statement, all variables and given/known data
Given Ampere's Law and Faraday's Law (in differential or integral form fwiw) explain why it is easier to design an experiment to show that a changing magnetic field creates an electric field than it is to show a changing electric field creates a magnetic field. Justify your answer with numbers.

2. Relevant equations
$\nabla x \vec{B} = \mu_0\vec{J} + \mu_0\epsilon_0\frac{d\vec{E}}{dt}$
$\nabla x \vec{E} = \frac{-d\vec{B}}{dt}$

3. The attempt at a solution
So this was from an exam I just took. I beat my head against it for about 45 minutes and could only really come up with this:

For Faraday's law, it only requires that a changing magnetic field be present. This can be accomplished easily by simply moving a magnet around a conductor, since if an electric field is created, the electrons in the conductor will move, and we can see the effects of this in a coil of wire with a segment cut out. As the magnet moves sparks of current should be visible at that cut out.

For Ampere's Law however it is hard to create a changing electric field independent of current (where $\vec{J} = \vec{0}$). One would need to create a changing electric field that propagates through free space and measure the magnetic effects on objects as a result, which would have been challenging without tools that can properly control photons.

I honestly just didn't know how to describe the difficulties of Ampere's Law better than this. And I didn't have time to think of any numerical justifications for my arguments since the other 3 problems also took about 45 minutes a piece. How would you have gone about answering this differently?

2. Apr 1, 2016

### Simon Bridge

Hard to change the electric field without moving charges about yes.


text:
# Missing direction information in reactions.dat file

The reactions.dat files list various known reactions. For each reaction, a list of "LEFT" and "RIGHT" compounds is provided. Many reactions also specify their REACTION-DIRECTION which can be one of the following: 'LEFT-TO-RIGHT', 'RIGHT-TO-LEFT', 'PHYSIOL-RIGHT-TO-LEFT', 'PHYSIOL-LEFT-TO-RIGHT', 'REVERSIBLE', 'IRREVERSIBLE-LEFT-TO-RIGHT'

The problem is that many reactions don't have this field. Is it safe to assume that when REACTION-DIRECTION isn't listed, the direction is left-to-right? .

edit retag close merge delete

Sort by » oldest newest most voted

No, if no reaction direction is listed, it is not safe to make any assumption about it. In general, reactions are defined in a direction (i.e. the left and right slots) according to EC conventions, which may have little bearing on the physiological direction. When we specify a reaction direction, it is either because our curators have specifically curated that direction based on the literature or on chemical knowledge, or because the software can infer a direction based on the direction it goes within some pathway or when catalyzed by some enzyme. If we don't have the information to make an inference, and a curator has not specifically examined the reaction, we don't assign one. This doesn't necessarily mean the direction is unknown -- some of the reactions imported from the EC have comments that make the direction clear, but that info is not in a form that is interpretable by our software.

more


text:
# Can future observation of the living partner influence the past decayed state in entangled neutral K-mesons?

Jose Bernabeu, Antonio Di Domenico

Received Date: 20th February 20

The Einstein-Podolsky-Rosen (EPR) entanglement of a neutral K-meson (kaon) pair enjoys the peculiar quantum behaviour of mixing, charge-parity ${\cal CP}$ violation (${\cal C}$ - charge conjugation, ${\cal P}$ - parity) and two non-orthogonal eigenstates of definite time evolution $K_L$ and $K_S$ with very different lifetimes. The dynamics of this strange entanglement', experimentally accessible at the $\phi$-factory, makes possible the search for novel phenomena not accessible in any other system, i.e. at interference times and at decoherence times of the single partners after their disentanglement, in order to unveil the nature of the correlation between the two neutral kaons. Until now, the studies have been concentrated on the single kaon intensity distribution between the two decay times $t_1$ and $t_2$ with $\Delta t = t_2 - t_1 > 0$. Here we show that the entire two single time distributions, before each decay, are physical in both senses: "from past to future", leading to the state of the living partner at time $t_2$ from the observation of the first decay channel at $t_1$, and "from future to past", leading to the past state of the decayed kaon at time $t_1$ from the observation of the second decay channel at $t_2$. Our results thus demonstrate an affirmative answer to the title: it does, the past decayed state depends on the result of the future measurement for the living partner. This novel effect is truly observable through the first decay time distributions. Besides the implications for quantum physics for all $\Delta t$, at large decoherence times the resulting first decayed state is always $K_S$, providing a genuine tag of this kaon state in the decoherence region, and a unique and important experimental tool to prepare a pure $K_S$ beam.

Read in full at arXiv.

This is an abstract of a preprint hosted on an independent third party site. It has not been peer reviewed but is currently under consideration at Nature Communications.

### Nature Communications

Nature Research, Springer Nature


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Implementing CD serial key system

I am trying to create a system where to unlock the application one needs to enter a serial code. I have read many articles on the theme but there are two problems bugging me. One is, If I have a keygen and I can generate as many keys as I like then anyone else can do that too, but solving this by requiring a long "private key" there occurs another problem.

Usually serial codes are supposed to be short and easy to type but even the shortest Elliptical curve signature algorithm has a signature size of 160-bits which could not possibly fit into the 100 bits I have. Is it possible to use a threshold approach?

I have seen many similar questions (please do not mark as duplicate) but none have answered my main question as to how can you treat the serial code as a signed document if the length of signature is at least 60 bits more than the maximum allowable size of a serial code?

Could you please give a deeper insight on my question and bare with my lack of knowledge in many areas? Thank you.

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What do you mean by "use a threshold approach"? $\;$ –  Ricky Demer Jul 28 '14 at 19:33
@RickyDemer It seems that the threshold was more about splitting the signature in multiple parts. I don't think that counts as threshold-crypto, so I removed the tag. –  Maarten Bodewes Jul 28 '14 at 19:46
100-bits is about 17 characters in base64. A Microsoft style (base32) format key with 5x5 characters gives 125 bits, extending that to 6x5 gives 150-bits (Symantec style). A reformatted base32 key is also more resilient to incorrect keys, since you can treat "O" and "0" as the same value, etc. You may want to separate the "product code" and "license key" to gain more bits for the sig. –  Richie Frame Jul 28 '14 at 19:55
@RickyDemer, I meant doing something like having a serial key and an inbuilt program share two shares that return a secret used to unlock the program –  user2332868 Jul 29 '14 at 20:05

If you want $N$ serial numbers, your serial numbers will have to use $n$ bits for uniqueness, where $n = \log_2 N$. So if you have 100 bits to use for the serial, you could use 20 to get about a million serials and have 80 bits to use for a cryptographic MAC or signature.

Now there are two approaches, the symmetric and the asymmetric.

• In the symmetric approach, you have a key $k$ and calculate a $b$-bit MAC for each $n$-bit ID: $S = I||MAC_k(I)$.

This is very easy and allows an arbitrary split between $b$ and $n$ by truncating a longer MAC if needed. However, if the program can be easily studied, you may find the key and be able to use that for a key generator.

With $b = 80$ this is quite secure as long as the key remains secret. (Within the realm of plausibility if the adversary is dedicated, but very unlikely against a random CD key.)

• In the asymmetric approach, you have a key-pair $(s, p)$ and calculate a signature for the ID that can be verified with the public key.

Here you would need a public key system with exactly $b$-bit signatures. OTOH, it would prevent even someone who can find the public key from creating serials... but wouldn't prevent someone from overwriting the public key with their own.

However, with $b = 80$ you'd need an 80-bit elliptic curve, which is not secure (nor would a 100-bit one be). It's only approximately as secure as a 40-bit symmetric key would be, and can be broken relatively easily without special hardware.

Therefore, if you need only 100-bit serials, I would recommend the former route. It's just security through obscurity, but the alternative isn't secure either.

There is a workaround to make an asymmetric scheme work, but then you need to shift bits from the "signatures" into the "public key".

For each ID you could create an $80$-bit MAC (or a random number, but a MAC allows you to recover serials), but instead of including the key in your program, you'd include hashes for each such MAC. E.g. $H(MAC_k(I))$ for some preimage resistance hash function $H$. The program would verify a serial $I|M$ by calculating $H(M)$, then comparing that to the hash it has for $I$. With $2^{20}$ serials, you'd need to store 10MB of "public key" to verify the serials with $80$-bit security.

Now the enterprising attacker could still overwrite (one of) those hashes to whatever they like... so it's not necessarily much better than the one where you just stored the key. (This is always the case with DRM, unless you involve some kind of hardware module.)

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+1 you totally answered my question. And I'm not so paranoid as to assume some hacker is going to be so determined as to write a different public key xD. Anyway, thanks. –  user2332868 Jul 29 '14 at 22:01


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Combinatorial interpretation of ${i\choose n}$, where $i^2=-1$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:52:59Z http://mathoverflow.net/feeds/question/116336 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116336/combinatorial-interpretation-of-i-choose-n-where-i2-1 Combinatorial interpretation of ${i\choose n}$, where $i^2=-1$ Richard Stanley 2012-12-14T01:43:49Z 2012-12-15T23:41:19Z
At MIT all departments have numbers, and math is 18. Last year MIT math majors produced a tee shirt that said ${i\choose 18}$ ("I choose 18") on the front, and on the back $$\frac{34376687+1499084559i}{14485008384}.$$ With the more natural denominator $18!$ this is $$\frac{15194495654000+662595375078000i}{18!}.$$ This suggests the question: for any $n\geq 1$ find a "nice" combinatorial interpretation of the real and imaginary parts of $i(i-1)(i-2)\cdots (i-n+1)=f_n+ig_n$. It is easy to express $f_n$ and $g_n$ as certain alternating sums of Stirling numbers of the first kind, but I don't consider this "nice." The $g_n$'s seem to alternate in sign beginning with $n=5$. The $f_n$'s alternate in sign up to $n=17$ and then seem to alternate in sign beginning with $n=18$. It is curious that $i(i-1)(i-2)(i-3)=-10$, a real number. One could ask the same question with $i$ replaced by any Gaussian integer $a+bi$. One can also ask about the asymptotic rate of growth of $f_n$ and $g_n$. Clearly $f_n^2+g_n^2\sim C\cdot (n-1)!^2$, so one would expect $f_n$ and $g_n$ to be roughly of the size of $(n-1)!$.
 http://mathoverflow.net/questions/116336/combinatorial-interpretation-of-i-choose-n-where-i2-1/116346#116346 Answer by Eric Naslund for Combinatorial interpretation of ${i\choose n}$, where $i^2=-1$ Eric Naslund 2012-12-14T04:14:07Z 2012-12-15T23:41:19Z
Asymptotics: Lets look at the quantity

$$S(n)=(-1)^{n}(n+1)\binom{i}{n+1}=i\prod_{k=1}^{n+1}\left(1-\frac{i}{k}\right).$$ It's just your binomial coefficient above with the $(-1)^{n+1}$ factored in, and an extra $n+1$ so it factors nicely as a product.

Claim: We have that

$$S(n)=\sqrt{\frac{\sinh{\pi}}{\pi}}e^{iC_{0}}e^{-i\log n}\left(1+O\left(\frac{1}{n}\right)\right),$$ where

$$C_{0}=\frac{-\pi}{2}-1+\int_0^\infty \frac{\{x\}}{1+x^2}dx =\arg(\Gamma(i))\approx -1.872.$$

In particular, the angle moves around the circle like $\log n$.

Application to your question: The above claim shows that $$f_{n+1} = (-1)^{n+1} n! \sqrt{\frac{\sinh{\pi}}{\pi}}\cos(-\log n+C_0)\left(1+O\left(\frac{1}{n}\right)\right)$$ and $$g_{n+1} \sim (-1)^{n+1} n! \sqrt{\frac{\sinh{\pi}}{\pi}} \sin(-\log n+C_0)\left(1+O\left(\frac{1}{n}\right)\right).$$

In particular, the ratio $g_n/f_n$ can be made arbitrarily large or small.

Proof of the claim: We first note that the size is

$$\sqrt{\prod_{k=1}^{n}\left(1+\frac{1}{k^{2}}\right)}=\sqrt{\prod_{k=1}^{\infty}\left(1+\frac{1}{k^{2}}\right)}+O\left(\frac{1}{n}\right).$$ To evaluate this product, recall the Weierstrass product for the Gamma function $$\left(\Gamma(z)\right)^{-1}=ze^{\gamma z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k^{2}}\right)e^{-\frac{z}{r}}.$$ From this it follows that $$\frac{1}{|\Gamma(i)|^{2}}=\frac{1}{\Gamma(i)\Gamma(-i)}=\prod_{k=1}^{\infty}\left(1+\frac{1}{k^{2}}\right).$$ Using the identity $$\Gamma(x)\Gamma(-x)=-\frac{\pi}{x\sin\left(\pi x\right)},$$ we now have that $$\frac{1}{\Gamma(i)\Gamma(-i)}=\frac{-i\sin(i\pi)}{\pi}=\frac{\sinh(\pi)}{\pi},$$ which gives rise to the $\sqrt{\frac{\sinh(\pi)}{\pi}}$ term. Moving on to the evaluation of the angle, by looking at each triangle, and noting that the argument is additive when multiplied, we get that the argument equals

$$-\sum_{k=1}^{n}\tan^{-1}\left(\frac{1}{k}\right).$$

The negative sign arises since we are working in the fourth quadrant. By looking at the Taylor series for $\tan^{-1}$ we see that the above is $\log n+O(1)$, however, I would like to compute this argument more precisely, and obtain the constant. Lets compare our $\tan^{-1}$ series to the harmonic series. Rewriting things in terms of a Riemann Stieltjes integral, and using summation by parts, we have that

$$\sum_{k=1}^{n}\tan^{-1}\left(\frac{1}{k}\right)=\int_{0}^{n}\tan^{-1}\left(\frac{1}{x}\right)d\left[x\right]=[n]\tan^{-1}(1/n)\int_{0}^{n}\frac{\left[x\right]}{1+x^{2}}dx.$$

Pulling out the main term with the identity $[x]=x-\{x\}$, the above equals

$$\int_{0}^{n}\frac{x}{1+x^{2}}dx-\int_{0}^{n}\frac{\{x\}}{1+x^{2}}dx.$$

Since the first integral evaluates to $\frac{1}{2}\log(1+x^2)$, we have that $$\sum_{k=1}^{n}\tan^{-1}\left(\frac{1}{k}\right)=\log n +1-\int_0^\infty \frac{\{x\}}{1+x^2}dx +O\left(\frac{1}{n}\right).$$

Acknowledgements: I would like to thank Noam Elkies for pointing out that $$\prod_{k=1}^\infty \sqrt{1+\frac{1}{k^2}}=\frac{1}{|\Gamma(i)|}=\sqrt{\frac{\sinh(\pi)}{\pi}}$$ in the comments.

Edit: Fixed the constants appearing. Interestingly $$\Gamma(i)=\sqrt{\frac{\pi}{\sinh{\pi}}}\exp\left(i\left(\frac{-\pi}{2}-1+\int_0^\infty \frac{\{x\}}{1+x^2}dx \right)\right).$$
 http://mathoverflow.net/questions/116336/combinatorial-interpretation-of-i-choose-n-where-i2-1/116422#116422 Answer by Noam D. Elkies for Combinatorial interpretation of ${i\choose n}$, where $i^2=-1$ Noam D. Elkies 2012-12-15T02:14:01Z 2012-12-15T02:14:01Z
There is no need to reinvent the wheel by estimating $\prod_{k<n}(1+\frac1{k^2})$. The asymptotic formula for $f_n + i g_n$ follows readily from Stirling's approximation (as I already noted in my comment to the original question), and indeed the same is true for the asymptotics as $n \rightarrow \infty$ of $w \choose n$ for any $w \in {\bf C}$; the answer is simply  $$\phantom{*0000000000000000000} {w \choose n} = \Bigl(1+O\bigl(\frac1n\bigr)\Bigr) \frac{(-1)^n}{\Gamma(-w)} n^{-w-1} \phantom{0000000000000000000}(*)$$  (and the $O(1/n)$ can be refined to an asymptotic series in powers of $1/n$). Note that this gives zero precisely for the values $w=0,1,2,3,\ldots$ for which $-w$ is a pole of $\Gamma$, which are also the $w$ for which ${w \choose n} = 0$ for sufficiently large $n$. For $w=i$, we recover the observed behavior: $\Gamma(i)$ is a complex number of absolute value $(\pi / \sinh \pi)^{1/2}$ [in general $$|\Gamma(it)| = (\Gamma(it)\Gamma(-it))^{-1/2} = \left(\frac \pi {t \phantom. \sinh \pi t} \right)^{1/2}$$ for real $t \neq 0$], and $n^{-w-1}$ is a complex number of absolute value $1/n$ that goes once around the origin when $n$ increases by a factor $e^{2\pi}$. Thus each of $\lbrace f_n \rbrace$ and $\lbrace g_n \rbrace$ alternates in sign outside an infinite sequence of exceptions that's asymptotically a geometric sequence with common ratio $e^\pi$.

To prove (*), write $${w \choose n} = \frac{(-1)^n}{n!} \prod_{k=0}^{n-1} (k-w) = \frac{(-1)^n}{n!} \frac{\Gamma(n-w)}{\Gamma(-w)} = \frac{(-1)^n}{n\Gamma(-w)} \frac{\Gamma(n-w)}{\Gamma(n)}.$$ Now we understand $(-1)^n/n$, and the factor $1 / \Gamma(-w)$ is constant, so we're left with $\Gamma(n-w) / \Gamma(n)$. We apply the following form of Stirling's formula: there exists a constant $\varpi>0$ (known to equal $2\pi$, but we shall not need this) such that $$\Gamma(z) = \bigl(1 + O(|z|^{-1}\bigr) z^z e^{-z} \sqrt{\varpi/z}$$ holds as $|z| \rightarrow \infty$ in the right half-plane, where $z^z = \exp (z \log z)$ and $\sqrt{\varpi/z}$ are defined using the principal branches of $\log z$ and $\sqrt z$. This gives $$\frac{\Gamma(n-w)}{\Gamma(n)} = \Bigl(1+O\bigl(\frac1n\bigr)\Bigr) \frac{(n-w)^{n-w} e^{-(n-w)} (\varpi/(n-w))^{1/2}} {n^n e^{-n} (\varpi/n)^{1/2}}$$ $$= \Bigl(1+O\bigl(\frac1n\bigr)\Bigr) \frac{(n-w)^{n-w}}{n^n} e^w \left(1-\frac{w}{n}\right)^{-1/2}.$$ Now the last factor is $1 + O(1/n)$; the factor $e^w$ is constant; and $$(n-w)^{n-w} = (n-w)^{-w} (n-w)^n = \Bigl(1+O\bigl(\frac1n\bigr)\Bigr) n^{-w} (n-w)^n.$$ So we're left with $$\frac{\Gamma(n-w)}{\Gamma(n)} = \Bigl(1+O\bigl(\frac1n\bigr)\Bigr) n^{-w} e^{-w} \left(1 - \frac{w}{n}\right)^n = \Bigl(1+O\bigl(\frac1n\bigr)\Bigr) n^{-w}.$$ This completes the proof of (*) (and the cancellation in the last step leads me to suspect that even this use of Stirling is more complicated than necessary).


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# How can I show that $\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1\end{pmatrix}$?

Well, the original task was to figure out what the following expression evaluates to for any $n$.

$$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^{\large n}$$

By trying out different values of $n$, I found the pattern: $$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^{\large n} = \begin{pmatrix} 1 & n \\ 0 & 1\end{pmatrix}$$

But I have yet to figure out how to prove it algebraically.

Suggestions?

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Use induction on $n$.

(1) Prove the base case (trivial), perhaps even establish the case for $n = 2$ (two base cases here are not necessary, but as you found, it helps reveal the pattern.)

(2) Then assume it holds for $n = k$.

(3) Finally, show that from this assumption, it holds for $n = k+1$.

You've established the base case(s). Now, (2) assume the inductive hypothesis (IH) $$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^k = \begin{pmatrix} 1 & k \\ 0 & 1\end{pmatrix}.$$

Then, $$\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}^{k + 1} = \begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^k \quad \overset{IH}{=} \quad \begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix} \begin{pmatrix} 1 & k \\ 0 & 1\end{pmatrix}=\quad\cdots$$

I think you can take it from here!

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This was really helpful, thanks! –  Jimmy C Sep 20 '13 at 16:18
You're welcome, Jimmy! –  amWhy Sep 20 '13 at 16:19

The matrix $$N=\begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}$$ is nilpotent with index 2 of nilpotency: $N^2=0$ so by the binomial formula we have

$$\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}^n=(I_2+N)^n=\sum_{k=0}^n {n\choose k}N^k={n\choose 0}I_2+{n\choose 1}N=I_2+nN=\begin{pmatrix} 1& n\\ 0 & 1 \end{pmatrix}$$

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Very nice approach +1 –  Amzoti Sep 20 '13 at 15:24
Thank you for your proof! Unfortunately I'm somewhat too early into my linear algebra career to really understand your approach. :) –  Jimmy C Sep 20 '13 at 16:16
@JimmyC You are welcome:) –  Sami Ben Romdhane Sep 20 '13 at 16:25
@JimmyC he took the matrix and split it into the identity matrix which has the ones diagonally and that matrix N that he had defined. Then by applying Newton's binomial formula... –  Module Dec 15 '13 at 18:22
Get $30$ at the beginning of the day. :) –  B. S. Dec 20 '13 at 7:45

Hint:

Use Mathematical Induction.

-

Geometrically, your matrix represents a shear transformation that preserves the horizontal direction and shifts the vertical direction by the horizontal direction. What happens geometrically if you apply the same shear transformation $n$ times?

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This is good explanation and intuition about what it does, but it doesn't seem to offer any information in the way of a proof. –  rschwieb Sep 20 '13 at 15:00
I guess it depends on how formal you want a proof to be. In my opinion, geometry can be enough for a proof, and e.g. I'd argue you don't have to use trig identities to "prove" the formula for powers of a 2-d rotation matrix rotating by angle $\theta$. I think you can even argue the other way around, and e.g. "prove" angle addition trig identities by taking a product of rotation matrices. –  user2566092 Sep 20 '13 at 15:44

Powers of matrices occur in solving recurrence relations.

If you write $$\begin{pmatrix} x_{n+1} \\ y_{n+1} \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix} \begin{pmatrix} x_n \\ y_n \end{pmatrix}$$ then clearly $$\begin{pmatrix} x_{n} \\ y_{n} \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^n \begin{pmatrix} x_0 \\ y_0 \end{pmatrix}$$ and also $$x_{n+1}=x_n+y_n, \qquad y_{n+1}=y_n$$ from which you get $$x_n = x_0 + n\ y_0, \qquad y_n = y_0$$ The first column of $A^n$ is given by taking $x_0=1$ and $y_0=0$, and so is $(1 \ 0)^T$.

The second column is given by taking $x_0=0$ and $y_0=1$, and so is $(n \ 1)^T$.

-

The statement is true for an arbitrary complex number $n$, with matrix logarithm. All the $\log$s below refer to the natural logarithm.

For convenience, let's set $$A = \begin{pmatrix} 0 & a \\ 0 & 0 \end{pmatrix}.$$

We know that if $k \in \mathbb N$ and $k \ge 2$, then $A^k = O$. Therefore, $$\exp A = \sum_{k=0}^\infty A^k = I + A.$$

All eigenvalues of $A$ are zero, so $A$ is the principal logarithm. $$\log \left( I + A \right) = A.$$

This leads to $$\log \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$$

And $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^n = \exp \begin{pmatrix} 0 & n \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}.$$

-

$$\left(% \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right) = 1 + A \quad\mbox{where}\quad A = \left(% \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right)$$ Notice that $A^{2} = 0$. It means that a function ${\rm f}\left(A\right)$ is ${\it linear}$ in $A$. For instance, $\left(1 + \mu A\right)^{n} = \alpha + \beta A$. Also

$$nA\left(1 + \mu A\right)^{n -1} = \alpha' + \beta' A \quad\Longrightarrow\quad nA\left(\alpha + \beta A\right) = \left(1 + \mu A\right)\left(\alpha' + \beta' A\right) = \alpha' + \beta' A + \mu\alpha' A$$

$$\mbox{That's means}\quad 0 = \alpha'\,,\quad n\alpha = \beta' + \mu\alpha'= \beta'\,, \quad\mbox{with}\quad \left.\alpha\right\vert_{\mu\ =\ 0} = 1\ \mbox{and}\ \left.\beta\right\vert_{\mu\ =\ 0} = 0\$$

Then, $\alpha = 1$ and $\beta = n\mu$:

$$\begin{array}{|c|}\hline\\ \color{#ff0000}{\large\quad% \left(1 + \mu A\right)^{n} \color{#000000}{\ =\ } 1 + n\mu A \quad\color{#000000}{\Longrightarrow}\quad \left(1 + A\right)^{n} \color{#000000}{\ =\ } 1 + nA \quad} \\ \\ \hline \end{array}$$

Or 'easy way': $$\left(1 + A\right)^{n} = 1\ +\ \overbrace{\quad{n \choose 1}\quad}^{=\ n}\ A\ +\ \overbrace{\quad{n \choose 2}\,A^{2} + \cdots\quad}^{=\ 0}$$

-


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# Lindy effect

The Lindy effect is a theory that the future life expectancy of some non-perishable things like a technology or an idea is proportional to their current age, so that every additional period of survival implies a longer remaining life expectancy.[1] Where the Lindy effect applies, mortality rate decreases with time.

## History

The origin of the term can be traced to Albert Goldman and a 1964 article he had written in The New Republic titled "Lindy's Law".[2] The term Lindy refers to Lindy's delicatessen in New York, where comedians "foregather every night at Lindy's, where ... they conduct post-mortems on recent show business 'action'". In this article, Goldman describes a folkloric belief among New York City media observers that the amount of material comedians have is constant, and therefore, the frequency of output predicts how long their series will last:[3]

... the life expectancy of a television comedian is [inversely] proportional to the total amount of his exposure on the medium. If, pathetically deluded by hubris, he undertakes a regular weekly or even monthly program, his chances of survival beyond the first season are slight; but if he adopts the conservation of resources policy favored by these senescent philosophers of "the Business", and confines himself to "specials" and "guest shots", he may last to the age of Ed Wynn [d. age 79 in 1966 while still acting in movies]

Benoit Mandelbrot defined a different concept with the same name in his 1982 book The Fractal Geometry of Nature.[4] In Mandelbrot's version, comedians do not have a fixed amount of comedic material to spread over TV appearances, but rather, the more appearances they make, the more future appearances they are predicted to make: Mandelbrot expressed mathematically that for certain things bounded by the life of the producer, like human promise, future life expectancy is proportional to the past. He references Lindy's Law and a parable of the young poets' cemetery and then applies to researchers and their publications: "However long a person's past collected works, it will on the average continue for an equal additional amount. When it eventually stops, it breaks off at precisely half of its promise."

Nassim Taleb presented a version of Mandelbrot's idea in The Black Swan: The Impact of the Highly Improbable by extending it to a certain class of non-perishables where life expectancy can be expressed as power laws.

With human projects and ventures we have another story. These are often scalable, as I said in Chapter 3. With scalable variables ... you will witness the exact opposite effect. Let's say a project is expected to terminate in 79 days, the same expectation in days as the newborn female has in years. On the 79th day, if the project is not finished, it will be expected to take another 25 days to complete. But on the 90th day, if the project is still not completed, it should have about 58 days to go. On the 100th, it should have 89 days to go. On the 119th, it should have an extra 149 days. On day 600, if the project is not done, you will be expected to need an extra 1,590 days. As you see, the longer you wait, the longer you will be expected to wait.[5]

In Taleb's 2012 book Antifragile: Things That Gain from Disorder he for the first time explicitly referred to his idea as the Lindy Effect, removed the bounds of the life of the producer to include anything which doesn't have a natural upper bound, and incorporated it into his broader theory of the Antifragile.

If a book has been in print for forty years, I can expect it to be in print for another forty years. But, and that is the main difference, if it survives another decade, then it will be expected to be in print another fifty years. This, simply, as a rule, tells you why things that have been around for a long time are not "aging" like persons, but "aging" in reverse. Every year that passes without extinction doubles the additional life expectancy. This is an indicator of some robustness. The robustness of an item is proportional to its life! [6]

According to Taleb, Mandelbrot agreed with the expanded definition of the Lindy Effect: "I [Taleb] suggested the boundary perishable/nonperishable and he [Mandelbrot] agreed that the nonperishable would be power-law distributed while the perishable (the initial Lindy story) worked as a mere metaphor."[7]

## Pareto distribution

Lifetimes following the Pareto distribution (a power-law distribution) demonstrate the Lindy effect.[8][9] For example with the parameter ${\displaystyle \alpha =2}$, conditional on reaching an age of ${\displaystyle x>x_{\min {}}}$, the expected future lifetime is also ${\displaystyle x}$. In particular, initially the expected lifetime is ${\displaystyle 2x_{\min {}}}$ but if that point is reached then the expected future lifetime is also ${\displaystyle 2x_{\min {}}}$; if that point is reached making the total lifetime so far ${\displaystyle 4x_{\min {}}}$ then the expected future lifetime is ${\displaystyle 4x_{\min {}}}$; and so on.

More generally with proportionality rather than equality, given ${\displaystyle m>0}$ and using the parameter ${\displaystyle \alpha ={\frac {m}{m-1}}}$ in the Pareto distribution, conditional on reaching any age of ${\displaystyle x>x_{\min {}}}$, the expected future lifetime is ${\displaystyle (m-1)x}$. Example: for ${\displaystyle \alpha =2}$ or ${\displaystyle m=2}$ the expected future lifetime is ${\displaystyle x}$.

Lindy effect is connected to Pareto’s Law, to Zipf’s Law, and to socioeconomic inequality.[10]

## References

1. ^ Nassim Nicholas Taleb (2012). Antifragile: Things That Gain from Disorder. Random House. p. 514. ISBN 9781400067824.
2. ^ "Lindy's Law" (PDF). Connection.ebscohost.com. Retrieved 2017-05-30.
3. ^ "The simple rule that can help you predict the future". Retrieved 21 May 2020.
4. ^ Mandelbrot, B.B (1984). The fractal geometry of Nature. Freeman. p. 342. ISBN 9780716711865.
5. ^ Nassim Nicholas Taleb (2007). The Black Swan: The Impact of the Highly Improbable. Random House. p. 159. ISBN 9781588365835. Like many biological variables, life expectancy.
6. ^ Nassim Nicholas Taleb (2012). Antifragile: Things That Gain from Disorder. Random House. p. 318. ISBN 9780679645276. another forty years.
7. ^ Taleb, Nassim Nicholas (2012-11-27). Antifragile: Things That Gain from Disorder. ISBN 9780679645276.
8. ^ Cook, John (December 17, 2012). "The Lindy effect". John D. Cook. Retrieved May 29, 2017.
9. ^ Cook, John (December 19, 2012). "Beethoven, Beatles, and Beyoncé: more on the Lindy effect". John D. Cook. Retrieved May 29, 2017.
10. ^ Iddo Eliazar "Lindy’s Law", Physica A, 486 (2017) 797–805


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# The regular semisimple element in the algebraic group

Suppose some maximal torus $T$ of $G$ is $C_G(T)$, then the set of semisimple elements for which $C_G(s)$ is a torus contains a nonempty open set. Such element are called regular semisimple.

I want to know how to prove this claim and want to find some good examples for some common algebraic groups such as $\mathrm{SL}(n,K)$.

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We say that a set is Dedekind-finite if there is no proper subset of the same cardinality. Every finite set is Dedekind-finite, and assuming the axiom of choice the converse is also true. In fact even assuming countable choice is enough.

However, the assertion axiom "Every Dedekind-finite set is finite" is strictly weaker than the axiom of countable choice.

In fact the assertion axiom "The countable union of countable set is countable" is also weaker than the axiom of countable choice.

In the spirit of the above, there are several other propositions which imply are provable from the existence axiom of an infinite Dedekind-finite setcountable choice, all of which and are weaker than consistent with the assertion assumption "Every There exists an infinite Dedekind-finite" , one of these assertion (but do not imply it), for example ) is "Every infinite set can be split into two disjoint infinite sets".

Generally speaking, there are several "orthogonal" choice principles (and their obvious extensions, etc.) for example axiom of choice for families of size $\kappa$ is generally independent from choice principles like the Boolean Prime Ideal Theorem (and its consequences).

The generalized Kinna-Wagner principle, while related to the Boolean Prime Ideal theorem can be shown to hold even when BPIT fails badly (e.g. there exists an amorphous set which cannot be linearly ordered). This principle is also a good example for something independent from the "choice-related" choice principles (again, if there is an amorphous set countable choice fails badly).

2 added 671 characters in body

We say that a set is Dedekind-finite if there is no proper subset of the same cardinality. Every finite set is Dedekind-finite, and assuming the axiom of choice the converse is also true. In fact even assuming countable choice is enough.

However, the assertion "Every Dedekind-finite set is finite" is strictly weaker than the axiom of countable choice.

In fact the assertion "The countable union of countable set is countable" is also weaker than the axiom of countable choice.

In the spirit of the above, there are several other propositions which imply the existence of an infinite Dedekind-finite set, all of which are weaker than the assertion "Every Dedekind-finite", one of these assertion (for example) is "Every infinite set can be split into two disjoint infinite sets".

Generally speaking, there are several "orthogonal" choice principles (and their obvious extensions, etc.) for example axiom of choice for families of size $\kappa$ is generally independent from choice principles like the Boolean Prime Ideal Theorem (and its consequences).

The generalized Kinna-Wagner principle, while related to the Boolean Prime Ideal theorem can be shown to hold even when BPIT fails badly (e.g. there exists an amorphous set which cannot be linearly ordered). This principle is also a good example for something independent from the "choice-related" choice principles (again, if there is an amorphous set countable choice fails badly).

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We say that a set is Dedekind-finite if there is no proper subset of the same cardinality. Every finite set is Dedekind-finite, and assuming the axiom of choice the converse is also true. In fact even assuming countable choice is enough.

However, the assertion "Every Dedekind-finite set is finite" is strictly weaker than the axiom of countable choice.

In fact the assertion "The countable union of countable set is countable" is also weaker than the axiom of countable choice.

In the spirit of the above, there are several other propositions which imply the existence of an infinite Dedekind-finite set, all of which are weaker than the assertion "Every Dedekind-finite", one of these assertion (for example) is "Every infinite set can be split into two disjoint infinite sets".