text:
NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! rychu Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power has the credentials and knowledge to answer these questions and Bedini is the visionary for them!

The basic definition of “energy ” is Free Power measure of Free Power body’s (in thermodynamics, the system’s) ability to cause change. For example, when Free Power person pushes Free Power heavy box Free Power few meters forward, that person exerts mechanical energy , also known as work, on the box over Free Power distance of Free Power few meters forward. The mathematical definition of this form of energy is the product of the force exerted on the object and the distance by which the box moved (Work=Force x Distance). Because the person changed the stationary position of the box, that person exerted energy on that box. The work exerted can also be called “useful energy ”. Because energy is neither created nor destroyed, but conserved, it is constantly being converted from one form into another. For the case of the person pushing the box, the energy in the form of internal (or potential) energy obtained through metabolism was converted into work in order to push the box. This energy conversion, however, is not linear. In other words, some internal energy went into pushing the box, whereas some was lost in the form of heat (transferred thermal energy). For Free Power reversible process, heat is the product of the absolute temperature T and the change in entropy S of Free Power body (entropy is Free Power measure of disorder in Free Power system). The difference between the change in internal energy , which is ΔU, and the energy lost in the form of heat is what is called the “useful energy ” of the body, or the work of the body performed on an object. In thermodynamics, this is what is known as “free energy ”. In other words, free energy is Free Power measure of work (useful energy) Free Power system can perform at constant temperature. Mathematically, free energy is expressed as:
I do not fear any conspiracy from any nook & corner. I am simply taking my time and my space to stage the inevitable confrontation in the frozen face of the industry and geopolitics tycoons. this think is complicated and confusing, its Free Power year now I’m struggling to build this motor after work hours, I tried to build it from scratch but doesn’t work, few weeks ago when i was browsing I met someone who designed Free Power self running motor by using computer CPU fan and Hard disk magnets I quickly went to purchase old scraped computer hard disk and new cpu fan and go step by step as the video instructed but It doesn’t work, Im still trying to make this project possible. Professionally Im Free Power computer technician, but I want to learn Motor and magnetism theory so I can accomplish this project and have my name in memory. I anyone can make this project please contact me through facebook so I can invite him/her to my country and make money as you know third word countries has power disaster. My facebook Id is Elly Maduhu Nkonya, or use my E-mail. [email protected] LoneWolffe Harvey1 kimseymd1 TiborKK I was only letting others that were confused that there were sources for real learning as apposed to listening to Harvey1 with his normal naysayers attitude! There is tons of information on schoolgirl, schoolboy and Bedini window motors that actually work to charge batteries and eventually will generate house currents. It just has to be looked at to get any useful information from it without listening to people like Harvey1 whining about learning. Harvey1 kimseymd1 You obviously play too much video games with trolls etc. in them. Why the editors of this forum allow you to keep calling people names instead of following the subject is beyond me. This must be the last site to allow you on it. I spammed the books because I thought those people were good for learning these engines which are super and there are tons of information out there for anyone to find. You seem to only want to learn to be rude instead of electronics.
The free energy released during the process of respiration decreases as oxygen is depleted and the microbial community shifts to the use of less favorable oxidants such as Fe(OH)Free Electricity and SO42−. Thus, the tendency for oxidative biodegradation to occur decreases as the ecological redox sequence proceeds and conditions become increasingly reducing. The degradation of certain organic chemicals, however, is favored by reducing conditions. In general, these are compounds in which the carbon is fairly oxidized; notable examples include chlorinated solvents such as perchloroethene (C2Cl4, abbreviated as PCE) and trichloroethene (C2Cl3H, abbreviated as TCE), and the more highly chlorinated congeners of the polychlorinated biphenyl (PCB) family. (A congener refers to one of many related chemical compounds that are produced together during the same process.

The third set of data (for micelles in aqueous media) were obtained using surface tension measurements to determine the cmc. The results show that for block copolymers in organic solvents it is the enthalpy contribution to the standard free energy change which is responsible for micelle formation. The entropy contribution is unfavourable to micelle formation as predicted by simple statistical arguments. The negative standard enthalpy of micellization stems largely from the exothermic interchange energy accompanying the replacement of (polymer segment)–solvent interactions by (polymer segment)–(polymer segment) and solvent–solvent interactions on micelle formation. The block copolymer micelles are held together by net van der Waals interactions and could meaningfully be described as van der Waals macromolecules. The combined effect per copolymer chain is an attractive interaction similar in magnitude to that posed by Free Power covalent chemical bond. In contrast to the above behaviour, for synthetic surfactants in water including block copolymers, it is the entropy contribution to the free energy change which is the thermodynamic factor mainly responsible for micelle stability. Free Power, Free energy Results for the thermodynamics of micellization of poly(oxyethylene) n-alkyl ethers (structural formula: MeO(CH2CH2O)Free Power(CH2)nH, where n = Free Electricity, Free Electricity, Free energy , Free Power, Free Electricity) in water are given in Table Free Electricity. Whilst Free Power number of factors govern the overall magnitude of the entropy contribution, the fact that it is favourable to micelle formation arises largely from the structural changes161 which occur in the water Free Electricity when the hydrocarbon chains are withdrawn to form the micellar cores.

It all smells of scam. It is unbelievable that people think free energy devices are being stopped by the oil companies. Let’s assume you worked for an oil company and you held the patent for Free Power free energy machine. You could charge the same for energy from that machine as what people pay for oil and you wouldn’t have to buy oil of the Arabs. Thus your profit margin would go through the roof. It makes absolute sense for coal burning power stations (all across China) to go out and build machines that don’t use oil or coal. wow if Free Energy E. , Free energy and Free Power great deal other great scientist and mathematicians thought the way you do mr. Free Electricity the world would still be in the stone age. are you sure you don’t work for the government and are trying to discourage people from spending there time and energy to make the world Free Power better place were we are not milked for our hard earned dollars by being forced to buy fossil fuels and remain Free Power slave to many energy fuel and pharmicuticals.
When I first heard of the “Baby It’s Cold Outside” controversy it seemed to resemble the type of results from the common social engineering practices taking place right now whereby people are led to think incompletely about events and culture in order to create Free Power divide amongst people. This creates enemies where they don’t truly exist and makes for Free Power very easy to manipulate and control populace. Ultimately, this leads for people to call for greater governance.
Figure Free Electricity. Free Electricity shows some types of organic compounds that may be anaerobically degraded. Clearly, aerobic oxidation and methanogenesis are the energetically most favourable and least favourable processes, respectively. Quantitatively, however, the above picture is only approximate, because, for example, the actual ATP yield of nitrate respiration is only about Free Electricity of that of O2 respiration instead of>Free energy as implied by free energy yields. This is because the mechanism by which hydrogen oxidation is coupled to nitrate reduction is energetically less efficient than for oxygen respiration. In general, the efficiency of energy conservation is not high. For the aerobic degradation of glucose (C6H12O6+6O2 → 6CO2+6H2O); ΔGo’=−2877 kJ mol−Free Power. The process is known to yield Free Electricity mol of ATP. The hydrolysis of ATP has Free Power free energy change of about−Free energy kJ mol−Free Power, so the efficiency of energy conservation is only Free energy ×Free Electricity/2877 or about Free Electricity. The remaining Free Electricity is lost as metabolic heat. Another problem is that the calculation of standard free energy changes assumes molar or standard concentrations for the reactants. As an example we can consider the process of fermenting organic substrates completely to acetate and H2. As discussed in Chapter Free Power. Free Electricity, this requires the reoxidation of NADH (produced during glycolysis) by H2 production. From Table A. Free Electricity we have Eo’=−0. Free Electricity Free Power for NAD/NADH and Eo’=−0. Free Power Free Power for H2O/H2. Assuming pH2=Free Power atm, we have from Equations A. Free Power and A. Free energy that ΔGo’=+Free Power. Free Power kJ, which shows that the reaction is impossible. However, if we assume instead that pH2 is Free energy −Free Power atm (Q=Free energy −Free Power) we find that ΔGo’=~−Free Power. Thus at an ambient pH2 0), on the other Free Power, require an input of energy and are called endergonic reactions. In this case, the products, or final state, have more free energy than the reactants, or initial state. Endergonic reactions are non-spontaneous, meaning that energy must be added before they can proceed. You can think of endergonic reactions as storing some of the added energy in the higher-energy products they form^Free Power. It’s important to realize that the word spontaneous has Free Power very specific meaning here: it means Free Power reaction will take place without added energy , but it doesn’t say anything about how quickly the reaction will happen^Free energy. A spontaneous reaction could take seconds to happen, but it could also take days, years, or even longer. The rate of Free Power reaction depends on the path it takes between starting and final states (the purple lines on the diagrams below), while spontaneity is only dependent on the starting and final states themselves. We’ll explore reaction rates further when we look at activation energy. This is an endergonic reaction, with ∆G = +Free Electricity. Free Electricity+Free Electricity. Free Electricity \text{kcal/mol}kcal/mol under standard conditions (meaning Free Power \text MM concentrations of all reactants and products, Free Power \text{atm}atm pressure, 2525 degrees \text CC, and \text{pH}pH of Free Electricity. 07. 0). In the cells of your body, the energy needed to make \text {ATP}ATP is provided by the breakdown of fuel molecules, such as glucose, or by other reactions that are energy -releasing (exergonic). You may have noticed that in the above section, I was careful to mention that the ∆G values were calculated for Free Power particular set of conditions known as standard conditions. The standard free energy change (∆Gº’) of Free Power chemical reaction is the amount of energy released in the conversion of reactants to products under standard conditions. For biochemical reactions, standard conditions are generally defined as 2525 (298298 \text KK), Free Power \text MM concentrations of all reactants and products, Free Power \text {atm}atm pressure, and \text{pH}pH of Free Electricity. 07. 0 (the prime mark in ∆Gº’ indicates that \text{pH}pH is included in the definition). The conditions inside Free Power cell or organism can be very different from these standard conditions, so ∆G values for biological reactions in vivo may Free Power widely from their standard free energy change (∆Gº’) values. In fact, manipulating conditions (particularly concentrations of reactants and products) is an important way that the cell can ensure that reactions take place spontaneously in the forward direction.

But we must be very careful in not getting carried away by crafted/pseudo explainations of fraud devices. Mr. Free Electricity, we agree. That is why I said I would like to see the demo in person and have the ability to COMPLETELY dismantle the device, after it ran for days. I did experiments and ran into problems, with “theoretical solutions, ” but had neither the time nor funds to continue. Mine too ran down. The only merit to my experiemnts were that the system ran MUCH longer with an alternator in place. Similar to what the Free Electricity Model S does. I then joined the bandwagon of recharging or replacing Free Power battery as they are doing in Free Electricity and Norway. Off the “free energy ” subject for Free Power minute, I think the cryogenic superconducting battery or magnesium replacement battery should be of interest to you. Why should I have to back up my Free Energy? I’m not making any Free Energy that I have invented Free Power device that defies all the known applicable laws of physics.
Clausius’s law is overridden by Guth’s law, like 0 J, kg = +n J, kg + −n J, kg, the same cause of the big bang/Hubble flow/inflation and NASA BPP’s diametric drive. There mass and vis are created and destroyed at the same time. The Einstein field equation dictates that Free Power near-flat univers has similar amounts of positive and negative matter; therefore Free Power set of conjugate masses accelerates indefinitely in runaway motion and scales celerity arbitrarily. Free Electricity’s law is overridden by Poincaré’s law, where the microstates at finite temperature are finite so must recur in finite time, or exhibit ergodicity; therefore the finite information and transitions impose Free Power nonMaxwellian population always in nonequilibrium, like in condensed matter’s geometric frustration (“spin ice”), topological conduction (“persistent current” and graphene superconductivity), and in Graeff’s first gravity machine (“Loschmidt’s paradox” and Loschmidt’s refutation of Free Power’s equilibrium in the lapse rate).
It all smells of scam. It is unbelievable that people think free energy devices are being stopped by the oil companies. Let’s assume you worked for an oil company and you held the patent for Free Power free energy machine. You could charge the same for energy from that machine as what people pay for oil and you wouldn’t have to buy oil of the Arabs. Thus your profit margin would go through the roof. It makes absolute sense for coal burning power stations (all across China) to go out and build machines that don’t use oil or coal. wow if Free Energy E. , Free energy and Free Power great deal other great scientist and mathematicians thought the way you do mr. Free Electricity the world would still be in the stone age. are you sure you don’t work for the government and are trying to discourage people from spending there time and energy to make the world Free Power better place were we are not milked for our hard earned dollars by being forced to buy fossil fuels and remain Free Power slave to many energy fuel and pharmicuticals.
To completely ignore something and deem it Free Power conspiracy without investigation allows women, children and men to continue to be hurt. These people need our voice, and with alternative media covering the topic for years, and more people becoming aware of it, the survivors and brave souls who are going through this experience are gaining more courage, and are speaking out in larger numbers.
The machine can then be returned and “recharged”. Another thought is short term storage of solar power. It would be way more efficient than battery storage. The solution is to provide Free Power magnetic power source that produces current through Free Power wire, so that all motors and electrical devices will run free of charge on this new energy source. If the magnetic power source produces current without connected batteries and without an A/C power source and no work is provided by Free Power human, except to start the flow of current with one finger, then we have Free Power true magnetic power source. I think that I have the solution and will begin building the prototype. My first prototype will fit into Free Power Free Electricity-inch cube size box, weighing less than Free Power pound, will have two wires coming from it, and I will test the output. Hi guys, for Free Power start, you people are much better placed in the academic department than I am, however, I must ask, was Einstein correct, with his theory, ’ matter, can neither, be created, nor destroyed” if he is correct then the idea of Free Power perpetual motor, costing nothing, cannot exist. Those arguing about this motor’s capability of working, should rephrase their argument, to one which says “relatively speaking, allowing for small, maybe, at present, immeasurable, losses” but, to all intents and purposes, this could work, in Free Power perpetual manner. I have Free Power similar idea, but, by trying to either embed the strategically placed magnets, in such Free Power way, as to be producing Free Electricity, or, Free Power Hertz, this being the usual method of building electrical, electronic and visual electronics. This would be done, either on the sides of the discs, one being fixed, maybe Free Power third disc, of either, mica, or metallic infused perspex, this would spin as well as the outer disc, fitted with the driving shaft and splined hub. Could anybody, build this? Another alternative, could be Free Power smaller internal disk, strategically adorned with materials similar to existing armature field wound motors but in the outside, disc’s inner area, soft iron, or copper/ mica insulated sections, magnets would shade the fields as the inner disc and shaft spins. Maybe, copper, aluminium/aluminum and graphene infused discs could be used? Please pull this apart, nay say it, or try to build it?Lets use Free Power slave to start it spinning, initially!! In some areas Eienstien was correct and in others he was wrong. His Theory of Special Realitivity used concepts taken from Lorentz. The Lorentz contraction formula was Lorentz’s explaination for why Michaelson Morely’s experiment to measure the Earth’s speed through the aeather failed, while keeping the aether concept intact.
Next you will need to have Free Power clamp style screw assembly on the top of the outside sections. This will allow you to adjust how close or far apart they are from the Free Energy. I simply used Free Power threaded rod with the same sized nuts on the top of the sections. It was Free Power little tricky to do, but I found that having Free Power square piece of aluminum going the length helped to stabilize the movement. Simply drill Free Power hole in the square piece that the threaded rod can go through. Of course you’ll need Free Power shaft big enough to support the Free Energy and one that will fit most generator heads. Of course you can always adapt it down if needed. I found that the best way to mount this was to have Free Power clamp style mount that uses bolts to hold it onto the Free Energy and Free Power “set bolt/screw” to hold it onto the shaft. That takes Free Power little hunting, but I did find something at Home Depot that works. If you’re handy enough you could create one yourself. Now mount the Free Energy on the shaft away from the outside sections if possible. This will keep it from pushing back and forth on you. Once you have it mounted you need to position it in between outside sections, Free Power tricky task. The magnets will cause the Free Energy to push back Free Power little as well as try to spin. The best way to do this is with some help or some rope. Why? Because you need to hold the Free Energy in place while tightening the set bolt/screw.
NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! rychu Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power has the credentials and knowledge to answer these questions and Bedini is the visionary for them!

Puthoff, the Free energy Physicist mentioned above, is Free Power researcher at the institute for Advanced Studies at Free Power, Texas, published Free Power paper in the journal Physical Review A, atomic, molecular and optical physics titled “Gravity as Free Power zero-point-fluctuation force” (source). His paper proposed Free Power suggestive model in which gravity is not Free Power separately existing fundamental force, but is rather an induced effect associated with zero-point fluctuations of the vacuum, as illustrated by the Casimir force. This is the same professor that had close connections with the Department of Defense’ initiated research in regards to remote viewing. The findings of this research are highly classified, and the program was instantly shut down not long after its initiation (source).
Figure Free Electricity. Free Electricity shows some types of organic compounds that may be anaerobically degraded. Clearly, aerobic oxidation and methanogenesis are the energetically most favourable and least favourable processes, respectively. Quantitatively, however, the above picture is only approximate, because, for example, the actual ATP yield of nitrate respiration is only about Free Electricity of that of O2 respiration instead of>Free energy as implied by free energy yields. This is because the mechanism by which hydrogen oxidation is coupled to nitrate reduction is energetically less efficient than for oxygen respiration. In general, the efficiency of energy conservation is not high. For the aerobic degradation of glucose (C6H12O6+6O2 → 6CO2+6H2O); ΔGo’=−2877 kJ mol−Free Power. The process is known to yield Free Electricity mol of ATP. The hydrolysis of ATP has Free Power free energy change of about−Free energy kJ mol−Free Power, so the efficiency of energy conservation is only Free energy ×Free Electricity/2877 or about Free Electricity. The remaining Free Electricity is lost as metabolic heat. Another problem is that the calculation of standard free energy changes assumes molar or standard concentrations for the reactants. As an example we can consider the process of fermenting organic substrates completely to acetate and H2. As discussed in Chapter Free Power. Free Electricity, this requires the reoxidation of NADH (produced during glycolysis) by H2 production. From Table A. Free Electricity we have Eo’=−0. Free Electricity Free Power for NAD/NADH and Eo’=−0. Free Power Free Power for H2O/H2. Assuming pH2=Free Power atm, we have from Equations A. Free Power and A. Free energy that ΔGo’=+Free Power. Free Power kJ, which shows that the reaction is impossible. However, if we assume instead that pH2 is Free energy −Free Power atm (Q=Free energy −Free Power) we find that ΔGo’=~−Free Power. Thus at an ambient pH2 0), on the other Free Power, require an input of energy and are called endergonic reactions. In this case, the products, or final state, have more free energy than the reactants, or initial state. Endergonic reactions are non-spontaneous, meaning that energy must be added before they can proceed. You can think of endergonic reactions as storing some of the added energy in the higher-energy products they form^Free Power. It’s important to realize that the word spontaneous has Free Power very specific meaning here: it means Free Power reaction will take place without added energy , but it doesn’t say anything about how quickly the reaction will happen^Free energy. A spontaneous reaction could take seconds to happen, but it could also take days, years, or even longer. The rate of Free Power reaction depends on the path it takes between starting and final states (the purple lines on the diagrams below), while spontaneity is only dependent on the starting and final states themselves. We’ll explore reaction rates further when we look at activation energy. This is an endergonic reaction, with ∆G = +Free Electricity. Free Electricity+Free Electricity. Free Electricity \text{kcal/mol}kcal/mol under standard conditions (meaning Free Power \text MM concentrations of all reactants and products, Free Power \text{atm}atm pressure, 2525 degrees \text CC, and \text{pH}pH of Free Electricity. 07. 0). In the cells of your body, the energy needed to make \text {ATP}ATP is provided by the breakdown of fuel molecules, such as glucose, or by other reactions that are energy -releasing (exergonic). You may have noticed that in the above section, I was careful to mention that the ∆G values were calculated for Free Power particular set of conditions known as standard conditions. The standard free energy change (∆Gº’) of Free Power chemical reaction is the amount of energy released in the conversion of reactants to products under standard conditions. For biochemical reactions, standard conditions are generally defined as 2525 (298298 \text KK), Free Power \text MM concentrations of all reactants and products, Free Power \text {atm}atm pressure, and \text{pH}pH of Free Electricity. 07. 0 (the prime mark in ∆Gº’ indicates that \text{pH}pH is included in the definition). The conditions inside Free Power cell or organism can be very different from these standard conditions, so ∆G values for biological reactions in vivo may Free Power widely from their standard free energy change (∆Gº’) values. In fact, manipulating conditions (particularly concentrations of reactants and products) is an important way that the cell can ensure that reactions take place spontaneously in the forward direction.

Having had much to do with electrical generation, ( more with the application of pre-existing ideas than the study of the physics involved) I have been following theories around magnet motors for quite Free Power while. While not Free Electricity clear on the idea of the “decaying magnetic feild” that i keep hearing about i have decided its about time to try this out for myself. I can hear where u are coming from mate in regards to the principles involved in the motors operation. Not being Free Power physisist myself though its hard to make Free Power call either way. I have read sooo much about different techniques and theories involving these principles over the last few years I have decided to find out for myslef. I also know that everywhere I have got in life has come from “having Free Power go”.


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# A Simplified Synthesis of Financial Options Pricing

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Financial options (the right to purchase (“call“) or sell (“put“) stock (or other assets)) at a fixed price at a future date have been around for a long time. They are attractive for the option buyer since his/her total risk is the premium (price) paid for the option, as contrasted with futures contracts which are obligations to buy a stock or commodity at a future date.  However, until recently, financial options  pricing presented great difficulties and controversy: how do you put a fair price on an asset deliverable at a future date, sometimes years ahead?  Then in 1973 Drs. F. Black, M. Scholes and  R.C. Merton published a model for options pricing based essentially on Gaussian (random-walk) probability.

The following is the famed Black-Scholes equation for determining the “fair” price of a call or put option.

Some of the  involved math may be gleaned from the following excerpt of its derivation:

And finally the formula itself:

where c is the cost of a call option and p the cost of a put option. I  won’t discuss the remaining parameters since we won’t be using this model directly.

As a purely theoretical inquiry, what mathematics lie behind the Black-Scholes formula?  I have long wondered and marveled at its complexity.  After all, it’s supposedly based on Gaussian probability aka random walk. Certainly the formula does not lend itself to a ready understanding of what lies behind rational options pricing.  Is there a simpler, more intuitive approach yielding the same quantitative results coupled with deeper insight into options pricing rationale?  I propose one herewith.

I am assuming a very basic familiarity with put and call options  All you need to know is spot price, strike price, duration and (annualized) volatility.   By “volatility” the financier specifically means the standard deviation of the spread of underlying  stock prices over a 1 year period.    (Interest rate is usually another input and will be touched on later.)

At this point I need to mention that the sole option price that has to be determined is the call price.  The put price for the same input parameters is immediately derivable from the so-called put-call parity rule, which is just (call price) – (put price) = (spot price) – (strike price).  So I will limit myself to call options.

Another thing to mention is that the below applies strictly speaking to so-called European options only; however, there is very little difference in price between these and so-called American options, usually well below 1%.  An online Black-Scholes options price calculator makes this clear.  Another easy-to-use Black-Scholes calculator for call options is at mystockoptions.com.

Let us go by way of an example.  Assume the spot price is $20 and the strike price$18; also that the volatility is $1 or 1/20 = 5% on a one-month basis (which is ##\frac 1 {\sqrt12}##of the annualized volatility). This means the mean is$20 and the standard deviation for 1 month of data is $1. See fig. 1. We will now develop the necessary data to determine the price of a 3 month call option. At three months the strike price will represent less than 2 std. deviations since the price structure will now be “spread out” by a factor of ##\sqrt 3##. The 3 month distribution will then look like fig.2. Notably, the$18 strike price now represents not -2σ but -2/##\sqrt 3## = -1.15σ.  So the area under the probability density function to the right of that value represents the probability that the price will be > strike price after 3 months in which case the option has value.

But that is not the whole story.  The other question is “ok, but what will be the average value assuming the option does have value?  Clearly we want the average of the x axis to the right of the strike price, i.e. x adjusted such that at x = -1.15 the contract value is 0.

So let’s do some probability.  Let f(x) be the probability density function and F(x) the cumulative function, i.e. the area from -##\infty## to x.  The entire area = 1 so ##F(\infty)## = 1.  Then the average value of x such that F(-1.15) = 0 will be, noting that F(-x) = F(1-x),

$$\bar x = \int_{-1.15}^{\infty} (x – (-1.15)) f(x) \, dx$$

$$= 1.15\int_{-1.15}^{\infty} f(x) \, dx + \int_{-1.15}^{ \infty} x f(x) \, dx.$$

So what to do with the last integral?  No tables, nothing!  But we are saved by a very felicitous relation which is x f(x) = – f ‘(x)

so now we get

$$\bar x = 1.15\int_{-1.15}^{\infty} f(x) \, dx \text{} ~ – \int_{-1.15}^{\infty} f ‘(x) \, dx.$$    or

$$\bar x = 1.15F(1.15) – [(f(\infty) – f(1.15)]$$

= 1.15F(1.15) + f(1.15)

##\bar x ## = 0.875 + 0.206 = 1.212

Finally we multiply by the dollar value corresponding to ##\sqrt 3 ## to get cost c = ($1.73)(1.212) =$2.09.

Comparing this with the Black-Scholes calculator with the substitution ## \sigma_{1 yr} = \sqrt{(12/3)}\sigma_{3 mos.} ## to  annualize the volatility we get $2.09. Exactly the same price as ours. Let’s proceed with an out-of-the-money case, i.e. when the strike price is above the spot price. For this I chose as follows: spot =$20

strike = $25 1 month volatility =$2 = 1σ

duration = 3 mos.  See fig. 3.  This is very much an out-of-the-money option, obviously.

The approach is the same:  the integration to obtain average x again starts at the σ equivalent of the strike price which will be ($25-$20)/2.5/1.73 = 1.44.  So to find the average x starting at 1.44 we need

$$\int_{1.44}^{\infty} (x – 1.44) f(x) \, dx$$ = f(1.44) – 1.44(1 – F(1.44)) = .142 – 1.44(0.075) = 0.034 x ($2 x 1.73) =$0.11 vs. $0.19 for the Black-Scholes model. Presumably, roundoff error accounts for the difference. In general the price of any call option will be $$c = \left( \frac \sigma \right) \int_{\sigma_{strike}}^{\infty} (x – \sigma_{strike}) f(x) \, dx$$ with ## \sigma = \sqrt{n/\sigma_v} ## , n the duration in years and ## \sigma_v ## = (annualized) volatility. I further tested the model with spot = strike =$20, and got $0.69 which agreed exactly with the Black-Scholes calculation. The prevailing interest rate should figure in these computations. It’s easily taken care of by simply accounting for “present value” of the payout. The same can be said for inflation, which at this date (Aug. 2018) is actually greater than the “safe” short-term interest rate. I’ve appended an excel file ## ^{(1)} ## that enables these computations for any combination of spot, strike, duration and volatility. (The program delineates between in-the-money and out-of-the-money cases). Should this model be used for options pricing rather than the Black-Scholes? No. The latter has been assiduously nurtured over the decades and can allegedly handle more input parameters. Online calculators facilitate its application. The point of this investigation is rather to gain insight into options pricing mathematics by formulating a much simplified basis of evaluating options, based on Gaussian variation like the Black-Scholes model, yet yielding the same results as the latter model in a much more understandable way. Then too, how valuable is any model based on random theory really? The infamous collapse of the hedge fund Long-Term Capital Management in 1999, in which Dr. Scholes was a participating investor, answers that question nicely, as does the 2008 general financial collapse. I recommend reading N. Taleb’s The Black Swan for casting further doubt on the predictability of any stochastic future pricing models. He shows that there can be and usually are severe bumps on those cherished smooth Gaussian curves! Resources: Additional (duplicate) ppt slides for the three embedded figures. These are of better quality than the embodied ones: blk-sch.ppt AB Engineering and Applied Physics MSEE Aerospace electronics career Used to hike; classical music, esp. contemporary; Agatha Christie mysteries. 12 replies 1. andrewkirk says: I like the approach of providing intuition by describing how the model works for a very short-term option, so that discounting can be ignored. That makes it a lot easier to grasp. The para that deals with interest rates (it starts with 'The prevailing interest rate should figure in these computations') is not quite right. The interest rate is the most complex part of the calculation. It is where the concept of changing the measure using the Cameron-Martin-Girsanov theorem comes in. It involves advanced concepts like equivalent martingale measures. So I would not say it is easily taken care of. The way to choose the interest rate is quite nuanced, and it affects not only the discounting of cash flows but also the assumed growth rate for the underlying stock, which determines how the mean of the bell curve shifts rightwards based on the time to option expiry. In the short-term example the bell curve does not shift at all, because the term to expiry is short enough to ignore expected growth. You could deal with that by writing a para something like: 'In the above examples we have ignored interest rates and expected growth of the stock, because they will not be significant over a short period. For most options these need to be taken into account, and that is where the additional complexity comes in. But the principle is the same as in the above examples: measuring the expected payoff over a bell curve based on the final stock price.' • rude man says: The Black-Scholes model also ignores expected growth, at least the on-line calculators do. I suppose that is one reason why the full Blk-Sch model is so much more complex. Barring growth I see nothing wrong with using present value to take care of interest. Also, no one else seems to take inflation into account which nowadays is a much greater cost determinant than the short-term, safe interest rate. The other thing I omitted was of course dividends. This is a real problem since ex-dividend stock prices typically plunge. I don’t know if Black-Scholes deals with that. Anyway, as I tried to emphasize, I was not trying to out-gun Blk.-Sch., just trying to help people see the big picture. Thanks for your expert comment! 2. rude man says: EDIT occured shortly after publication (2018-08-18 19:15 GMT). 3. alan2 says: But of course, we know that option pricing doesn't actually work. Just like MPT and CAPM it is fun math but it doesn't reflect reality. • rude man says: Absolutely correct! Couldn’t agree more. It’s fun math, is all. But it is fun! 4. andrewkirk says: rude man The Black-Scholes model also ignores expected growth, at least the on-line calculators do. It doesn't ignore expected growth. It assumes that the stock will grow at a rate equal to the risk-free interest rate minus the dividend rate (or other net benefits of holding the assets). The reasoning behind that assumption is the main insight that the original Black-Scholes paper introduced. Discussing it is beyond the scope and purpose of your note, but I think it is important to acknowledge that growth is not ignored but rather assumed to be at that rate. The expected growth cancels out against the discounting of the payout on maturity, leaving a discount factor on the first term in the formula that is ##e^{-uT}## where ##T## is time to option maturity and ##u## is the dividend rate. In examples where the underlying asset has negligible dividends or storage costs (eg some tech stocks, and commodities like gold with low ratios of storage cost to value) that discount factor is approximately 1 and is omitted. That's why it can look like growth is being ignored even though it isn't. Inflation doesn't need to be explicitly accounted for in pricing options because it is automatically incorporated into the interest rate and hence also the expected growth rate. alan2 But of course, we know that option pricing doesn't actually work. Just like MPT and CAPM it is fun math but it doesn't reflect reality. It depends what is meant by 'work'. Banks around the world use option pricing techniques like Black-Scholes to set prices for the hedging deals they offer their business customers, and they make consistent big profits with it. Usually it's a good predictor. But like all models, there are rare circumstances where it breaks down, and that's where one can lose big money. Long-Term Capital Management was a famous case of that. 5. rude man says: “Inflation doesn’t need to be explicitly accounted for in pricing options because it is automatically incorporated into the interest rate and hence also the expected growth rate.” Not so. As I said, current safe short-term interest rates are far below inflation. The former about 1.5%, the latter close to 4%. 6. andrewkirk says: rude man current safe short-term interest rates are far below inflation. The former about 1.5%, the latter close to 4%. I don't know which economy you're referring to, or which inflation measure, but all the inflation measures I know of* are about what has happened in the recent past, whereas interest rates relate to what is expected in the future. So the two are not comparable. The Black-Scholes option pricing formula is based on the construction of an arbitrage with a dynamic hedging portfolio that replicates the performance of the option. Inflation doesn't need to be considered because, regardless of what inflation does, the hedging portfolio will replicate the value of the option, and the value of a particular hedging portfolio can be explicitly observed based only on current stock prices and interest rates. * Other than the expected-inflation component of the yield on an inflation-linked bond. But those bonds are unusual and I doubt that is what the 4% relates to. • rude man says: I think we’re drifting far afield. I’ve enjoyed the conversation. 7. rude man says: andrewkirk rude man current safe short-term interest rates are far below inflation. The former about 1.5%, the latter close to 4%. I don't know which economy you're referring to, or which inflation measure, but all the inflation measures I know of* are about what has happened in the recent past, whereas interest rates relate to what is expected in the future. So the two are not comparable. The Black-Scholes option pricing formula is based on the construction of an arbitrage with a dynamic hedging portfolio that replicates the performance of the option. Inflation doesn't need to be considered because, regardless of what inflation does, the hedging portfolio will replicate the value of the option, and the value of a particular hedging portfolio can be explicitly observed based only on current stock prices and interest rates. * Other than the expected-inflation component of the yield on an inflation-linked bond. But those bonds are unusual and I doubt that is what the 4% relates to. "I don't know which economy you're referring to, or which inflation measure, but all the inflation measures I know of* are about what has happened in the recent past, whereas interest rates relate to what is expected in the future. So the two are not comparable." ??? They don't change that fast! If I pay x$ at t=0 and cash in at t=1 year, the inflation will very probably be very much at t=1yr as at t=0.

8. andrewkirk says:
rude man

First you say inflation is incorporated into interest rates, then you they're not comparable.

No, I didn't say the second. I said that past inflation is not comparable with future interest rates. There is a relationship between inflation expectations and interest rates but we need to compare them for the same period.

9. DrDu says:

When I first heard about the Black Scholes theory, it was formulated in therms of stochastic or Ito-integrals, which are quite unfamiliar to physicists. The excerpt from the derivation which you show seems to use some bracket formalism a la QM. Could you provide a source?

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# Discrete Structures Introduction

Module by: Duy Bui. E-mail the author

## Introduction to Discrete Structures

### What is Discrete Mathematics?

Discrete mathematics is mathematics that deals with discrete objects. Discrete objects are those which are separated from (not connected to/distinct from) each other. Integers (aka whole numbers), rational numbers (ones that can be expressed as the quotient of two integers), automobiles, houses, people etc. are all discrete objects. On the other hand real numbers which include irrational as well as rational numbers are not discrete. As you know between any two different real numbers there is another real number different from either of them. So they are packed without any gaps and can not be separated from their immediate neighbors. In that sense they are not discrete. In this course we will be concerned with objects such as integers, propositions, sets, relations and functions, which are all discrete. We are going to learn concepts associated with them, their properties, and relationships among them among others.

### Why Discrete Mathematics?

Let us first see why we want to be interested in the formal/theoretical approaches in computer science.

Some of the major reasons that we adopt formal approaches are 1) we can handle infinity or large quantity and indefiniteness with them, and 2) results from formal approaches are reusable. As an example, let us consider a simple problem of investment. Suppose that we invest $1,000 every year with expected return of 10% a year. How much are we going to have after 3 years, 5 years, or 10 years? The most naive way to find that out would be the brute force calculation. Let us see what happens to$1,000 invested at the beginning of each year for three years. First let us consider the $1,000 invested at the beginning of the first year. After one year it produces a return of$100. Thus at the beginning of the second year, $1,100, which is equal to$1,000 * (1 + 0.1), is invested. This $1,100 produces$110 at the end of the second year. Thus at the beginning of the third year we have $1,210, which is equal to$1,000 * (1 + 0.1)*(1 + 0.1), or $1,000 * (1 + 0.1)2. After the third year this gives us$1,000 * (1 + 0.1)3. Similarly we can see that the $1,000 invested at the beginning of the second year produces$1,000 * (1 + 0.1)2 at the end of the third year, and the $1,000 invested at the beginning of the third year becomes$1,000 * (1 + 0.1). Thus the total principal and return after three years is $1,000 * (1 + 0.1) +$1,000 * (1 + 0.1)2 + $1,000 * (1 + 0.1)3, which is equal to$3,641.

One can similarly calculate the principal and return for 5 years and for 10 years. It is, however, a long tedious calculation even with calculators. Further, what if you want to know the principal and return for some different returns than 10%, or different periods of time such as 15 years? You would have to do all these calculations all over again. We can avoid these tedious calculations considerably by noting the similarities in these problems and solving them in a more general way. Since all these problems ask for the result of investing a certain amount every year for certain number of years with a certain expected annual return, we use variables, say A, R and n, to represent the principal newly invested every year, the return ratio, and the number of years invested, respectively. With these symbols, the principal and return after n years, denoted by S, can be expressed as S = A(1 + R) + A(1 + R)2 + ... + A(1 + R)n. As well known, this S can be put into a more compact form by first computing S - (1 + R)S as

S = A ( (1 + R)n + 1 - (1 + R) ) / R .

Once we have it in this compact form, it is fairly easy to compute S for different values of A, R and n, though one still has to compute (1 + R)n + 1 . This simple formula represents infinitely many cases involving all different values of A, R and n. The derivation of this formula, however, involves another problem. When computing the compact form for S,   S - (1 + R)S   was computed using   S = A(1 + R) + A(1 + R)2 + ... + A(1 + R)n . While this argument seems rigorous enough, in fact practically it is a good enough argument, when one wishes to be very rigorous, the ellipsis ... in the sum for S is not considered precise. You are expected to interpret it in a certain specific way. But it can be interpreted in a number of different ways. In fact it can mean anything. Thus if one wants to be rigorous, and absolutely sure about the correctness of the formula, one needs some other way of verifying it than using the ellipsis. Since one needs to verify it for infinitely many cases (infinitely many values of A, R and n), some kind of formal approach, abstracted away from actual numbers, is required.

Suppose now that somehow we have formally verified the formula successfully and we are absolutely sure that it is correct. It is a good idea to write a computer program to compute that S, especially with (1 + R)n + 1   to be computed. Suppose again that we have written a program to compute S. How can we know that the program is correct? As we know, there are infinitely many possible input values (that is, values of A, R and n). Obviously we can not test it for infinitely many cases. Thus we must take some formal approach. Related to the problem of correctness of computer programs, there is the well known "Halting Problem". This problem, if put into the context of program correctness, asks whether or not a given computer program stops on a given input after a finite amount of time. This problem is known to be unsolvable by computers. That is, no one can write a computer program to answer that question. It is known to be unsolvable. But, how can we tell it is unsolvable? How can we tell that such a program can not be written? You can not try all possible solution methods and see they all fail. You can not think of all (candidate) methods to solve the Halting Problem. Thus you need some kind of formal approaches here to avoid dealing with an extremely large number (if not infinite) of possibilities. Discrete mathematics is the foundation for the formal approaches. It discusses languages used in mathematical reasoning, basic concepts, and their properties and relationships among them. Though there is no time to cover them in this course, discrete mathematics is also concerned with techniques to solve certain types of problems such as how to count or enumerate quantities. The kind of counting problems includes: How many routes exist from point A to point B in a computer network? How much execution time is required to sort a list of integers in increasing order? What is the probability of winning a lottery? What is the shortest path from point A to point B in a computer network? ...

The subjects covered in this course include propositional logic, predicate logic, sets, relations, and functions, in particular growth of function. The first subject is logic. It is covered in Chapter 1 of the textbook. It is a language that captures the essence of our reasoning, and correct reasoning must follow the rules of this language. We start with logic of sentences called propositional logic, and study elements of logic, (logical) relationships between propositions, and reasoning. Then we learn a little more powerful logic called predicate logic. It allows us to reason with statements involving variables among others. In Chapter 1 we also study sets, relations between sets, and operations on sets. Just about everything is described based on sets, when rigor is required. It is the basis of every theory in computer science and mathematics. In Chapter 3 we learn mathematical reasoning, in particular recursive definitions and mathematical induction. There are sets, operations and functions that can be defined precisely by recursive definitions. Properties of those recursively defined objects can be established rigorously using proof by induction. Then in Chapter 6 we study relations. They are an abstraction of relations we are familiar with in everyday life such as husband-wife relation, parent-child relation and ownership relation. They are also one of the key concepts in the discussion of many subjects on computer and computation. For example, a database is viewed as a set of relations and database query languages are constructed based on operations on relations and sets. Graphs are also covered briefly here. They are an example of discrete structures and they are one of the most useful models for computer scientists and engineers in solving problems. More in-depth coverage of graph can be found in Chapter 7. Finally back in Chapter 1 we study functions and their asymptotic behaviors. Functions are a special type of relation and basically the same kind of concept as the ones we see in calculus. However, function is one of the most important concepts in the discussion of many subjects on computer and computation such as data structures, database, formal languages and automata, and analysis of algorithms which is briefly covered in Chapter 2.

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## Seminar information archive

#### Seminar on Geometric Complex Analysis

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Atusi Yamamori (Meiji University)
On the Forelli-Rudin construction and explicit formulas of the Bergman kernels (JAPANESE)

### 2011/05/26

#### Applied Analysis

16:00-17:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Takeshi Fukao (Kyoto University of Education)
Obstacle problem of Navier-Stokes equations in thermohydraulics (JAPANESE)
[ Abstract ]
In this talk, we consider the well-posedness of a variational inequality for the Navier-Stokes equations in 2 or 3 space dimension with time dependent constraints. This problem is motivated by an initial-boundary value problem for a thermohydraulics model. The velocity field is constrained by a prescribed function,
depending on the space and time variables, so this is called the obstacle problem. The abstract theory of nonlinear evolution equations governed by subdifferentials of time dependent convex functionals is quite useful for showing their well-posedness. In their mathematical treatment one of the key is to specify the class of time-dependence of convex functionals. We shall discuss the existence and uniqueness questions for Navier-Stokes variational inequalities, in which a bounded constraint is imposed on the velocity field, in higher space dimensions. Especially, the uniqueness of a solution is due to the advantage of the prescribed constraint to the velocity fields.

### 2011/05/25

#### Number Theory Seminar

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuya Matsumoto (University of Tokyo)
On good reduction of some K3 surfaces (JAPANESE)

### 2011/05/24

#### Numerical Analysis Seminar

16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Daisuke Murai (Nagoya University)
Error analysis of a solution to topology optimization problem of density type
(JAPANESE)
[ Reference URL ]
http://www.infsup.jp/utnas/

#### Tuesday Seminar on Topology

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Masahiko Yoshinaga (Kyoto University)
Minimal Stratifications for Line Arrangements (JAPANESE)
[ Abstract ]
The homotopy type of complements of complex
hyperplane arrangements have a special property,
so called minimality (Dimca-Papadima and Randell,
around 2000). Since then several approaches based
on (continuous, discrete) Morse theory have appeared.
In this talk, we introduce the "dual" object, which we
call minimal stratification for real two dimensional cases.
A merit is that the minimal stratification can be explicitly
described in terms of semi-algebraic sets.
We also see associated presentation of the fundamental group.

#### Lie Groups and Representation Theory

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Jun-ichi Mukuno (Nagoya University)
Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds (JAPANESE)
[ Abstract ]
If a homogeneous space $G/H$ is acted properly discontinuously
upon by a subgroup $\\Gamma$ of $G$ via the left action, the quotient space $\\Gamma \\backslash G/H$ is called a
Clifford--Klein form. In 1962, E. Calabi and L. Markus proved that there is no infinite subgroup of the Lorentz group $O(n+1, 1)$ whose left action on the de Sitter space $O(n+1, 1)/O(n, 1)$ is properly discontinuous.
It follows that a compact Clifford--Klein form of the de Sitter space never exists.
In this talk, we present a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous.

#### thesis presentations

13:15-14:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Koichiro TAKAOKA (Graduate School of Mathematical Sciences University of Tokyo)
Martingale theory (JAPANESE)

### 2011/05/23

#### Algebraic Geometry Seminar

17:00-18:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Yuji Sano (Kumamoto University)
Alpha invariant and K-stability of Fano varieties (JAPANESE)
[ Abstract ]
From the results of Tian, it is proved that the lower bounds of alpha invariant implies K-stability of Fano manifolds via the existence of Kähler-Einstein metrics. In this talk, I will give a direct proof of this relation in algebro-geometric way without using Kähler-Einstein metrics. This is joint work with Yuji Odaka (RIMS).

### 2011/05/19

#### Applied Analysis

16:00-17:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Shingo Ito (Tokyo University of Science)
Wave front set defined by wave packet transform and its application (JAPANESE)

### 2011/05/18

#### Seminar on Mathematics for various disciplines

10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Yohei Kashima (University of Tokyo)
On the perturbation theory for many-electron systems at positive temperature (JAPANESE)
[ Reference URL ]
http://info.ms.u-tokyo.ac.jp/seminar/mathvar/future.html

#### Number Theory Seminar

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Masaki Nishimoto (University of Tokyo)
On the linear independence of values of some Dirichlet series (JAPANESE)

### 2011/05/17

#### Tuesday Seminar on Topology

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Atsushi Ishii (University of Tsukuba)
Quandle colorings with non-commutative flows (JAPANESE)
[ Abstract ]
This is a joint work with Masahide Iwakiri, Yeonhee Jang and Kanako Oshiro.
We introduce quandle coloring invariants and quandle cocycle invariants
with non-commutative flows for knots, spatial graphs, handlebody-knots,
where a handlebody-knot is a handlebody embedded in the $3$-sphere.
Two handlebody-knots are equivalent if one can be transformed into the
other by an isotopy of $S^3$.
The quandle coloring (resp. cocycle) invariant is a twisted'' quandle
coloring (resp. cocycle) invariant.

### 2011/05/16

#### Seminar on Geometric Complex Analysis

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Chifune Kai (Kanazawa Univeristy)
Linearity of order isomorphisms of regular convex cones (JAPANESE)

#### Algebraic Geometry Seminar

17:00-18:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Shinnosuke Okawa (University of Tokyo)
On images of Mori dream spaces (JAPANESE)
[ Abstract ]
Mori dream space (MDS), introduced by Y. Hu and S. Keel, is a class of varieties whose geometry can be controlled via the VGIT of the Cox ring. It is a generalization of both toric varieties and log Fano varieties.

The purpose of this talk is to study the image of a morphism from a MDS.
Firstly I prove that such an image again is a MDS.
Secondly I introduce a fan structure on the effective cone of a MDS and show that the fan of the image coincides with the restriction of that of the source.

This fan encodes some information of the Zariski decompositions, which turns out to be equivalent to the information of the GIT equivalence. In toric case, this fan coincides with the so called GKZ decomposition.

The point is that these results can be clearly explained via the VGIT description for MDS.

If I have time, I touch on generalizations and an application to the Shokurov polytopes.

### 2011/05/11

#### Number Theory Seminar

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Michel Raynaud (Universite Paris-Sud)
Permanence following Temkin (ENGLISH)
[ Abstract ]
When one proceeds to a specialization, the good properties of algebraic equations may be destroyed. Starting with a bad specialization, one can try to improve it by performing modifications under control. If, at the end of the process, the initial good properties are preserved, one speaks of permanence. I shall give old and new examples of permanence. The new one concerns the relative semi-stable reduction of curves recently proved by Temkin.

### 2011/05/10

#### Numerical Analysis Seminar

16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Yukitaka Minesaki (Tokushima Bunri University)
Conservative finite difference method for the three body problem (JAPANESE)
[ Reference URL ]
http://www.infsup.jp/utnas/

#### Tuesday Seminar on Topology

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tetsuya Ito (The University of Tokyo)
Isotated points in the space of group left orderings (JAPANESE)
[ Abstract ]
The set of all left orderings of a group G admits a natural
topology. In general the space of left orderings is homeomorphic to the
union of Cantor set and finitely many isolated points. In this talk I
will give a new method to construct left orderings corresponding to
isolated points, and will explain how such isolated orderings reflect
the structures of groups.

### 2011/05/09

#### Algebraic Geometry Seminar

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Hokuto Uehara (Tokyo Metropolitan University)
Fourier--Mukai partners of elliptic ruled surfaces (JAPANESE)
[ Abstract ]
Atiyah classifies vector bundles on elliptic curves E over an algebraically closed field of any characteristic. On the other hand, a rank 2 vector bundle on E defines a surface S with P^1-bundle structure on E.
We study when S has an elliptic fibration according to the Atiyah's classification. As its application, we determines the set of Fourier--Mukai partners of elliptic ruled surfaces over the complex number field.

#### Seminar on Geometric Complex Analysis

10:30-12:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Junjiro Noguchi (University of Tokyo)
Order of meromorphic maps and rationality of the image space (JAPANESE)

### 2011/05/02

#### Algebraic Geometry Seminar

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Katsuhisa Furukawa (Waseda University)
Projective varieties admitting an embedding with Gauss map of rank zero (JAPANESE)
[ Abstract ]
I will talk about the study of Gauss map in positivity characteristic which is a joint work with S. Fukasawa and H. Kaji. I will also talk about my resent research of this topic.

We call that a projective variety $X$ satisfies (GMRZ) if there exists an embedding $¥iota: X ¥hookrightarrow ¥mathbb{P}^M$ whose Gauss map $X ¥dashrightarrow G(¥dim(X), ¥mathbb{P}^M)$ is of rank zero at a general point.

We study the case where $X$ has a rational curve $C$. Then, as a fundamental theorem, it follows that the property (GMRZ) makes the splitting type of the normal bundle $N_{C/X}$ very special. We also have a characterization of the Fermat cubic hypersurface in characteristic two in terms of (GMRZ). In this talk, I will also explain the relation of blow-ups and the property (GMRZ).

### 2011/04/27

#### Number Theory Seminar

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Yuuki Takai (University of Tokyo)
An analogue of Sturm's theorem for Hilbert modular forms (JAPANESE)

### 2011/04/26

#### Numerical Analysis Seminar

16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Fumio Kikuchi (Hitotsubashi University)
Remarks on the discontinuous Galerkin finite element method (JAPANESE)
[ Reference URL ]
http://www.infsup.jp/utnas/

#### Lie Groups and Representation Theory

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Taro YOSHINO (the University of Tokyo)
Topological Blow-up (JAPANESE)
[ Abstract ]
Suppose that a Lie group $G$ acts on a manifold
$M$. The quotient space $X:=G\\backslash M$ is locally compact,
but not Hausdorff in general. Our aim is to understand
such a non-Hausdorff space $X$.
The space $X$ has the crack $S$. Rougly speaking, $S$ is
the causal subset of non-Hausdorffness of $X$, and especially
$X\\setminus S$ is Hausdorff.

We introduce the concept of topological blow-up' as a repair'
of the crack. The repaired' space $\\tilde{X}$ is
locally compact and Hausdorff space containing $X\\setminus S$
as its open subset. Moreover, the original space $X$ can be
recovered from the pair of $(\\tilde{X}, S)$.

#### Tuesday Seminar on Topology

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Taro Yoshino (The University of Tokyo)
Topological Blow-up (JAPANESE)
[ Abstract ]
Suppose that a Lie group $G$ acts on a manifold
$M$. The quotient space $X:=G\\backslash M$ is locally compact,
but not Hausdorff in general. Our aim is to understand
such a non-Hausdorff space $X$.
The space $X$ has the crack $S$. Roughly speaking, $S$ is
the causal subset of non-Hausdorffness of $X$, and especially
$X\\setminus S$ is Hausdorff.

We introduce the concept of topological blow-up' as a repair'
of the crack. The repaired' space $\\tilde{X}$ is
locally compact and Hausdorff space containing $X\\setminus S$
as its open subset. Moreover, the original space $X$ can be
recovered from the pair of $(\\tilde{X}, S)$.

#### Tuesday Seminar of Analysis

16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Kiyoomi KATAOKA (Graduate School of Mathematical Sciences, the University of Tokyo)
On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)


text:
Rinoslirl9h1

2022-02-03

Is 69 a ' number?

meldElafellrbo

No, 69 is a composite number.
Explanation:
69 is divisible by 3.
$69÷3=23$
Since 69 has a factor 3 that is neither 1 nor itself, 69 is not a ' number.

Do you have a similar question?


text:
# reddit is a platform for internet communities

[–] 0 points1 point

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Now that I think about it, an octopus pirate would be the best.

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

Every since reading that Emacs is for pirates, I've wanted a pirate-themed icon for Emacs.

[–][S] 0 points1 point

sorry, this has been archived and can no longer be voted on

I can't reproduce this. It's probably one of the achievements that I have and you don't, or vice versa. Could you toggle-debug-on-error and send me the backtrace? Either here, or open an issue on bitbucket.

[–][S] 0 points1 point

sorry, this has been archived and can no longer be voted on

I don't post to reddit very often, and so I always feel guilty only posting my own things. Thanks for the feedback though.

I've been trying to think of how to display bragging rights, but I haven't really come up with anything good yet. Ideas?

[–][S] 0 points1 point

sorry, this has been archived and can no longer be voted on

MELPA works with bitbucket just fine. I've done it before. However, I didn't want to submit it while it's still so incomplete. As you mentioned, persistence is absolutely necessary.

[–][S] 1 point2 points

sorry, this has been archived and can no longer be voted on

Fixed, thanks.

[–][S] 0 points1 point

sorry, this has been archived and can no longer be voted on

Weird. It doesn't give that me that error (Emacs from bzr), but it should. I fixed achievements-list-achievements and will report a bug to Emacs.

[–][S] 0 points1 point

sorry, this has been archived and can no longer be voted on

Yup. :-)

[–][S] 7 points8 points

sorry, this has been archived and can no longer be voted on

This is a post to my own page. Sorry about that. I wanted to comment on this old reddit thread, but I couldn't figure out how to do that.

Anyway, let the trash talking begin.

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

You can do the following. In the main file put (before you include the files):

\newif\ifsubsidiary
\subsidiarytrue

And then in the other files use

\newif\ifsubsidiary
\ifsubsidiary
\else
\documentclass[12pt]{report}
...
\begin{document}
\fi
...
\ifsubsidiary
\else
\end{document}
\fi

Hopefully it's fairly self explanatory.

That said, there should be a way to tell Gummi about your project. I haven't used Gummi though, so I can't help you.

[–] 2 points3 points

sorry, this has been archived and can no longer be voted on

I'm not an expert on pymacs, but if you're on OS X, you might want to try the Mac Port since the NS Port can be painfully slowfully when dealing with child processes. I hope this is finally going to be fixed soon, but you never know. You can find the Mac Port at e.g. https://github.com/railwaycat/emacs-mac-port

[–] 1 point2 points

sorry, this has been archived and can no longer be voted on

You can also use \vphantom to get them to line up: $x{i\vphantom{(k)}}_n \quad x{i(k)}_n$

I do this a lot when cases like $x{1}{2}+x{1}{\vphantom{2}}$.

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

Somehow I missed danielvegas' comment which is basically the same as mine.

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

You can try putting \begingroup and \endgroup around the frontmatter. That way many of the changes you make (not sure about spacing changes) will be automatically undone. Or rather definitions inside the group will go out of scope and revert to their old definitions.

Alternately you could make them both separately, and put them together with the pdfpages package, though in that case you won't be able to have links between them which is probably a deal breaker.

At the very worst you can save the old definitions, e.g. \let\oldMacro=\macro, and then restore them after the front matter, \let\macro=\oldMacro. It might be a pain though particularly if you have to save internal macros or anything like that. The specifics of what you need to change can potentially make a difference in the determining the best method to solve it.

[–][S] 4 points5 points

sorry, this has been archived and can no longer be voted on

I'm interested in what you have in mind. A simple rearrangement of the columns (e.g. putting them in "alphabetical" order) would be exceedingly easy to do given my current LaTeX file. I would be very interested in another more math accurate design, but I haven't figured out what would actually be better instead of just not copying the periodic table.

One idea would be to create the lattice of (non-abelian) simple groups, so that A<B if A is a subgroup of B. This has been done for the sporadic groups and is pretty cool. It would be neat to have the rest of the information on there.

That said, I'm certainly not an expert in simple groups or the classification, so I'm probably missing some organizational principle.

[–][S] 1 point2 points

sorry, this has been archived and can no longer be voted on

Thanks. I'm glad you like it.

[–][S] 0 points1 point

sorry, this has been archived and can no longer be voted on

Not at all.

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

GAP is still very much alive and kicking. In fact, they just released a new version. Besides Magma (which is quite expensive, and I haven't used) it's just about the only software for doing group theory.

[–][S] 3 points4 points

sorry, this has been archived and can no longer be voted on

Sorry to post a link to my own blog, but I thought some of you might like this.

[–] 1 point2 points

sorry, this has been archived and can no longer be voted on

You can use the preview or standalone packages to make the page size as big as it needs to be easily.

Then what I have done in the past is use color to display what the values are. For me this worked well since I was interested in matrices with many duplicate values. But even if not, never underestimate the value of color (e.g. blue for positive, red for negative, green for complex, whatever is interesting in your case). I wrote a little GAP routine that output the correct tikz or tabular code so that I didn't have to do it by hand, and I would definitely recommend doing that.

[–] 1 point2 points

sorry, this has been archived and can no longer be voted on

And you get preview-latex which is simply delightful!

[–] 1 point2 points

sorry, this has been archived and can no longer be voted on

You have to do it specially like ((nil . ((indent-tabs-mode . nil) (eval . (setq bob load-file-name))))).

Unfortunately, it seems load-file-name isn't set when loading .dir-locals.el because that would be an easy way to find the root directory. Nevertheless the locate-dominating-file snippet should work.

[–] 2 points3 points

sorry, this has been archived and can no longer be voted on

You mean, a scripting language like lua? :-)

The problem with a minimalistic markup language is that it only gets you 80-90% of the way to a truly fantastic looking document. Sometimes you really have to get down in the details. But it sounds like what you want is org-mode (or rst or markdown together with pandoc).

[–] 0 points1 point

sorry, this has been archived and can no longer be voted on

In AUCTeX Ctrl+C Ctrl+C (or in emacs parlance C-c C-c) will bring up a prompt of the next thing to do, with the default being the "right" thing. You can always type "View" in and it will show you the pdf. I haven't used Emacs on Windows for latex, so I can't help with the configuration. There should be some good tutorials though.

[–] 0 points1 point

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I think you would generally typeset as normal and then use something like pdfbook to combine the pages in the correct order and orientation. If I understand what you are asking for, the default pdfbook pamphlet.pdf will do what you want.


text:
# Write $(\mathbb{Z}\times\mathbb{Z})/\mathbb{Z}(20,30)$ as a direct product of cyclic groups.

How does one use the Fundamental Theorem of Finitely Generated Modules over PID to show this? I've started based on something I read elsewhere, but I'm not sure it's right or where to go from here.

Pick generators $\overline{(1,0)}$ and $\overline{(0,1)}$. We have \begin{align} &20\overline{(1,0)} + 30\overline{(0,1)} = \overline{(0,0)}\\ &20[\overline{(1,0)}+\overline{(0,1)}]+10\overline{(0,1)} = \overline{(0,0)} \\ &20\overline{(1,1)} + 10\overline{(0,1)} = \overline{(0,0)} \end{align} Let $\overline{(1,1)}$ be a new generator. Can continue to write \begin{align} &10(2\overline{(1,1)} + \overline{(0,1)}) = \overline{(0,0)} \\ &10\overline{(2,3}) = \overline{(0,0)} \end{align} Use $\overline{(2,3)}$ as a second generator and create a homomorphism $\mathbb{Z}\times\mathbb{Z}\to(\mathbb{Z}\times\mathbb{Z})/\mathbb{Z}(20,30)$ with $(a,b) \mapsto a\overline{(1,1)} + b\overline{(2,3)}$... what result am I looking to get?

• You've essentially done it. You have found generators $u:=(1,1), v:=(2,3)$ of $\mathbb{Z}\times\mathbb{Z}$ such that $\mathbb{Z}(2,3)$ is exactly $\mathbb{Z} (10v)$. So the quotient is isomorphic to $\mathbb{Z}\times\mathbb{Z}_{10}$. – ancientmathematician Mar 23 '18 at 7:54

Notice that $Tor((\mathbb{Z}\times\mathbb{Z})/\mathbb{Z}(20,30)) = (\overline{(2,3)})$ the submodule generated by $\overline{(2,3)}$ which is isomorphic to $\Bbb{Z}/10\Bbb{Z}$. Since $(\mathbb{Z}\times\mathbb{Z})/\mathbb{Z}(20,30)$ is not all torsion, there is a $\Bbb{Z}$ component in the decomposition. Also, you can verify that the rank of the module is $1$. So $(\mathbb{Z}\times\mathbb{Z})/\mathbb{Z}(20,30) \cong \Bbb{Z} \times \Bbb{Z}_{10}$


text:
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# Blogs

## Calculus Blogs

Welcome back to the school everyone! I hope you all had a great summer. For all those whose summer was maybe a little too great, maybe those who’ve forgotten even the basics, we’re going to take it all the way back to arithmetic a.k.a “number theory”. A review of number theory is a perfect place to start for many levels. Calculus and a lot of what you learn in pre-calculus is based on the real number system. When we use the word “number” we are typically referring to all real number. But how can numbers be “real”? You can’t touch the number 6 or smell 1,063. You can’t boil 1/2 or stick it in a stew. So what’s so real about real numbers? The simple answer is this: a real number is a point on a number line (1).                               -2.5        -1      0 ... read more

When you have to take the derivative of 2 variables being multiplied, we use the product rule. If we have f(x)g(x), the derivative will be:   (f(x)g(x))' = f(x)g'(x) + g(x)f'(x).   Now, the primes can be a little confusing when you are learning how to apply this for the first time, so lets denote the number 1 to f(x) and the number 2 to g(x), where 1 refers to f(x) and 2 refers to g(x). The product rule becomes easy to memorize.    Sing it to yourself: 1d2 + 2d1   Whenever you have a "d" in front of the number we denoted for the first part of the function, you take the derivative, and if there is no "d", we simply copy the exact same function.    Example: (x+4)(x² - 1)   Here, your 1 = (x+4) and 2 = (x² - 1).   We compute 1d2 + 2d1 = (x+4)(x² - 1)' + (x² - 1)(x+4)'   Notice that the functions you need to take... read more

Calculus is easily the most complex of all the basic fields of math.   This blog is intended to help you recognize the basic calculus integration problem types, and how to approach them.   Types:   1) u-substitution: when one function is the derivative of the other function   2) integration by parts: when one function is being multiplied by another function   3) partial fractions: when you have a rational function and the denominator can be factored into linear or quadratic expressions   4) trig substitution: when you have a function that follows the pattern type of a trig identity, for example sin^2 + cos^2 = 1

I'm going to list what I believe are the key concepts that you need to master across different math subjects. These are the tools that I have to use most often in order to solve problems, so you should get very familiar with the theory behind them and very comfortable with applying them.   Algebra 1: order of operation (PEMDAS) solving equations slope-intercept form of linear equations point-slope form of linear equations systems of linear equations (elimination and substitution methods) inequalities domain and range undefined and imaginary expressions asymptotes (horizontal and vertical) discontinuities (removable and non-removable) rational expressions factoring quadratic formula radical properties exponent properties transformations and translations of functions   Algebra 2:   1. recognizing and factoring the three most common polynomial forms: quadratic equations common factor expressions... read more

Why it's important   You can use the quotient rule to answer questions like:   Find f'(x) when f(x) = (3 + x2)/(x4 + x).    What it is   I recite this rhyme to remember the quotient rule:   Low Dee High minus High Dee Low Draw the Bar and Square Below   Which means:   f'(x) = [low * dee high - high * dee low] / low2   Dee high means the derivative of the high function. You can guess which that is.   In our example, low = x4 + x and high = 3 + x2, so dee low = 4x3 + 1 and dee high = 2x.   f'(x) = [(x4 + x) * (2x) - (3 + x2) * (4x3 + 1)] / [(x4 + x)]2   That's it. That's the quotient rule.   Intuition   I like applying rules I just learned to cases where I know what the answer will be. This helps me build my confidence that I'm using the rule correctly.   x2 is... read more

Containment and Equality If A and B are sets, then A is said to be contained in B iff (if and only if) every element of A is contained in B. So A⊆B means that A is a subset of B. Example: All squares ⊆ all rectangles All right triangles ⊆ all triangles Important! This implies the idea of forwards and backwards logic: If Joe has three million dollars, he is a millionaire. If Joe is a millionaire it doesn’t necessarily mean he has three million dollars, he could have one million dollars and still be a millionaire. Likewise, all squares are rectangles but not all rectangles are squares. A=B iff A⊆B and B⊆A Example: {x:x^2=4}={-2,2} {x:x^2<4}={x:-2<x<2}

Sets and Other Elementary Subjects Sets are a collection of things called objects. Objects are all unambiguously defined. In other words, objects have unmistakably clear definitions with one meaning and one interpretation that leads to one conclusion. This may seem convoluted because we are so used to words and phrases having different meanings and whatnot, but not in this case. Look at some examples to get a better idea what it means for objects to be unambiguously defined. Objects                       Not Objects Cars                            Cool cars Children                      Nice children Temperature               Comfortable temperature Baseball players       ... read more

Hello Students!   Start this year off strong with good organizational and note taking skills. Make sure you understand the material and are not just taking notes aimlessly. Try to take in what your teacher is saying and don't be afraid to ask questions!! If you start taking the initiative to learn and understand now, college will be a much more pleasant experience for you. Trust me!   Stay organized and plan your homework and study schedule!   Quiz yourself!   Study with friends!   READ YOUR TEXTBOOK! :)    Remember, homework isn't busy work and a chance to copy down your notes, it is part of the learning process. This is especially important with math, as it builds on itself and understanding the basics will make the other subjects easier!   Have a fantastic and fun year!

Today, the future depends on you as much as it does on me. The future also depends on educating the masses in Science, Technology, Engineering, and Math, otherwise known as STEM. As a new tutor to WyzAnt, I hope to instill the importance of these subjects in student's lives, as well as, the lives around them.    Besides the fact that,  "the average U.S. salary is $43,460, compared with the average STEM salary of$77,880," (Careerbuilder) these subjects are interesting and applicable to topics well beyond the classroom. Success first starts with you; I am only there to help you succeed along the way. STEM are difficult subjects. Yet when you seek out help from a tutor, like myself, you have what it takes to master them.    Please enlighten me on students looking to achieve and succeed rather than live in the past and think I can't as opposed to I can. We can take the trip to the future together, one question at a time

All too often, calculus textbooks misrepresent the proof this formula: \frac{d}{dx} e^{x}= e^{x}   The texts by Finney, Demana, et al. usually introduce, without explanation or proof,  the limit below: \lim_{h \rightarrow 0} \frac{ (e^{h}-1) }{h}=1   The problem with this approach is that it deprives the student of key concepts regarding the exponential function, ex  . The student often thinks of e as the number that is 2.71828...... because Precalculus and Calculus teachers define it as such. However, that definition is not the logical definition, but a mere incidental byproduct. The logical definition of e is the exponential base in the function whose tangent line has a slope of 1 at x=0 in the function f(x)=ex . The calculation of the numerical value of this base to be 2.71828... is a later development that results from the definition. It is not the logical definition.    Khan Academy sidesteps this nicely,... read more

There's no such thing as the square root of a negative number. Right? Since squaring a number is defined as multiplying it by itself, and multiplying a negative times a negative gives a positive, all squares should be positive. Right? So any number you want to take the square root of should be positive to begin with. Right? So what if it's not? What do you do if you're chugging through a problem and suddenly find yourself confronted with x = √(-9) It seems like to finish this problem we'll need to take the square root of a negative number – but we can't, so what do we do? Drop the sign and hope nobody notices? Mark it as 'undefined' like dividing by zero? Give up? Cry? Well, actually, we don't have to do any of that, because we've got an imaginary friend to help us. Meet i. i is a mathematical constant, whose sole definition is that i2 = -1. Or, in other words, i = √(-1). i is an imaginary number... read more

When it comes to using a legitimate online resource to help with tutoring mathematics, or answering mathematical questions I use Wolfram.com.   This website is very diverse and allows the user to input any mathematical equation, formula etc.   With subject areas of mathematics, such as calculus, Wolfram.com has proved to be extremely beneficial, especially when working with difficult integrals and derivatives.    With the Pro version of this website, which is well worth its value, you will be provided step-by-step instructions on how to solve the particular problem that you have inputted.    Check out this website and explore the countless benefits it has to offer.   Keith

I am a University of Utah mathematics major and I love the word FREE. (cheap is good too) I don't have a lot of money so any Free resources to help me study are worth it to me. Since I know a lot about mathematics that is what I will be posting here.   The key to Mathematics is Learning, Practicing, Learning, Practicing, and sometimes it goes in the opposite order: Practicing, Learning, Practicing, Learning. But either way a good resource to me has a bit of both: they teach you how and why you do something and they make you do it as well. A really good resource will teach you how and why, make you try it, and then will show you why you got it wrong and what you should have done, and then make you do more problems of the same type. So then, without further ado, here are the resources:   Paul's online notes (type it in google it will be one of the first to pop up) http://tutorial.math.lamar.edu/   his notes are free, come with worked out problems,... read more

I've started brushing up on Calculus.  I studied Calculus in high school and took two semesters in college, bu that was forty years ago. It's really interesting how persponal memories pack themselves in along with Diffrential Equations and Integrals.

Hello everyone, One of my Calculus students had an interesting Related Rates problem that I had to go home and think about for a while in order to figure out.  The problem was set up as such:   A 25 inch piece of rope needs to be cut into 2 pieces to form a square and a circle. How should the rope be cut so that the combined surface area of the circle and square is as small as possible?     Here's what we'll need to do: 1.  We will have to form equations that relate the length of the perimeter and circumference to the combined surface area.  2.  We will then differentiate to create an equation with the derivative of the surface area with respect to lengths of rope.  3.  Wherever this derivative equals 0 there will be a maxima or minima, and so we will set the derivative = to 0 and determine which critical points are minima... read more

You can find some really good resources for math test prep in the used bookstores in a college town. Some examples that I like are: (1) Humongous Book of ______________ Problems (fill in the blank with your math topic);  (2) the REA Problem Solvers series; and (3) the Schaum's Outlines. If you don't live near a college town it might be worth a Saturday trip just to buy books. Alternately, all of these are available (used) through the Amazon Marketplace sellers at really low prices.   You should preview each title of these book series that you might be considering to be sure you like the authors style. Each one is different. You may like one series' treatment of Pre-Calc but prefer a different series for Calculus.   So how do you use these books ?   They are an alternate resource for explanations of basic concepts and problem solving techniques. You should use them as 'hint mills' and sources of problems to make... read more

You'd think that, "If I'm paying for tutoring, he should be answering MY questions. Not the other way around." While I can sympathize with the general sentiment, I'd say,"you're way off base there!"   I think that the tutor/teacher/coach should never ask the student directly,"Do you understand __________ ?" Not knowing the subject matter, how would the student know/evaluate/determine if they understood or not ? Generally they can't, that's why the need a tutor. Rather than ask about specific content, directly, I ask questions to determine if the student understands the material and how the pieces fit together. Sometimes that's five or six questions.   Here's my general GAME PLAN: Find out where they are. Tell them, show them, then see what they heard and saw.   When your tutor's asking you questions, he/she is probably working the same kind of plan. You can help them help you by always providing the syllabus... read more