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Subtraction Using Nine's and Ten's Complements
Date: 05/27/2000 at 01:24:37
From: Katie Artus
Subject: Method of complements
I am trying to figure out why the method below, called the "method of
complements" by my university professor, will give me the correct
answer all of the time. It is used in place of regrouping in
subtraction problems.
-465 -465
----- -----
/157 (Cross off the 1 in the thousands column and add
+1 this 1 to the ones column.
Nines must always be used, and the order of operations depicted above
must be followed. If these conditions are met, the problem will always
work no matter how many numbers are in the problem. This will also
work with standard subtraction when regrouping is not necessary. I
would like to know why this method works. What mathematics are lurking
behind the scenes? I wish to explain this to a third grade child I am
student teaching.
Thank you,
Katie Artus
Date: 05/29/2000 at 14:44:11
From: Doctor TWE
Subject: Re: Method of complements
Hi Katie - thanks for writing to Dr. Math.
The method of complements is perhaps most famous in its binary
variations (called one's complement and two's complement) because
those are the methods that most computers use to subtract. What your
professor showed you is the decimal equivalent of one's complement,
sometimes called nine's complement.
As to why it works, let's examine what we're doing algebraically.
Let's call the minuend (623 in your example) X and the subtrahend (465
in your example) Y. We wish to find X-Y.
In the first step, we subtract Y from 99...9 (the number of 9's
equaling the number of digits in the larger of X and Y). In your
example, 999. So we have:
Next, we add X to that to get:
As long as X > Y, we can see that the result will be greater than or
equal to 1000. When we cross out the leading digit (the 1000's digit
in this example), we are in fact subtracting 1000 from our result.
Thus we have:
Finally, we add 1 to the units digit. So we have:
Can you see how algebraically this equals X-Y?
If I wanted to explain this to a third grader, I'd probably explain
the ten's complement method instead. Ten's complement works similar to
nine's complement with two slight differences. First, instead of
subtracting the subtrahend (Y) from 99...9, you subtract it from
100...0 instead. [I write this as 99...(10) so I don't have to
borrow.] The second difference is that you don't have to add the 1 at
the end. Can you see why this method produces the same results as
nine's complement?
Here's how I'd explain why it works:
Imagine you want to make some money by selling widgets (pick the
student's favorite thing-a-ma-bob). First, you have to buy the
materials to make widgets. This costs $465. Then you can sell them for
$623. Now you want to figure out how much money you'll make.
But there's one problem - you have to buy the materials BEFORE you can
sell the widgets. So you borrow $1000 from Mom and Dad. From that
$1000 you spend $465 to buy the materials. So you have:
- 465
$ 535
Now you sell the widgets for $623, you now have:
- 465
$ 535
+ 623
Finally, you have to pay back Mom and Dad the $1000 you borrowed to
start up. So your final profit is:
- 465
$ 535
+ 623
Which is the selling price ($623) less the cost of materials ($465).
A final word of caution: While nine's complement and ten's complement
produce the same results when X > Y, they need to be handled
differently when X <= Y. In the cases where X < Y (and where X = Y for
nine's complement), there will not be a "final carry" to cross out.
For example, to subtract 235-687:
9'S Comp. 10'S Comp.
- 687 - 687
--- ----
+ 235 + 235
--- ----
How do we interpret this? With nine's complement, we subtract the
result from 99...9 and add a negative sign. With ten's complement, we
subtract the result from 100...0 and add a negative sign, like so:
9'S Comp. 10'S Comp.
- 547 - 548
--- ----
(-)452 (-)452
I hope this helps. If you have any more questions, write back.
- Doctor TWE, The Math Forum | {"url":"http://mathforum.org/library/drmath/view/55949.html","timestamp":"2014-04-18T00:21:18Z","content_type":null,"content_length":"9717","record_id":"<urn:uuid:2def51d7-688f-4d07-8640-c9212491987c>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00481-ip-10-147-4-33.ec2.internal.warc.gz"} |
Design a Low-Jitter Clock for High-Speed Data Converters
Keywords: high-speed ADCs, high-speed analog to digital converter, PLL, VCO, phase-locked loop, voltage-controlled oscillator, low phase noise, low phase jitter, clock jitter, crystal oscillator,
noise, SNR, spurious components, analog digital, data converters
Related Parts
Design a Low-Jitter Clock for High-Speed Data Converters
Abstract: High-speed applications using ultra-fast data converters in their design often require an extremely clean clock signal to make sure an external clock source does not contribute undesired
noise to the overal dynamic performance of the system. It is therefore crucial to select suitable system components, which help generate a low phase-jitter clock. The following application note
serves as a valuable guide for selecting the appropriate components to design a low-phase noise PLL-based clock generator, suitable for ultra-fast data converters.
Many modern, high-speed, high-performance integrated circuits, such as the MAX104 and MAX106 analog-to-digital converters (ADCs), require a low-phase-noise (low-jitter) clock that operates in the
GHz range. Conventional crystal oscillators may provide a low-jitter clock signal, but are not generally available in oscillating frequencies above 120MHz.
Figure 1 illustrates the simplified block diagram of a typical high-speed data converter system. The system consists of a bandpass filter, ADC, high-frequency clock, high-speed storage device, and
post processing unit. Aside from the MAX104, the high-frequency clock plays a significant role in determining the accuracy of a high-speed data converter. This high frequency, low-phase-noise clock
is a combination of a high frequency voltage-controlled oscillator (U1), a phase-locked loop (U2), and a crystal oscillator (U3) as shown in Figure 2.
Figure 1. Typical high-speed data converter system using the MAX104 ADC and a PLL-based, low-jitter clock.
Figure 2. A high-speed, low-phase-noise clock is one of the most critical elements to ensure optimum dynamic performance of the high-speed ADC.
The MAX2620 voltage-controlled oscillator (VCO) is capable of generating oscillator frequencies up to 1GHz, while providing sufficient noise performance. Because of the inherent frequency drift, a
phase-locked loop (PLL) is often required to lock the VCO output to the desired frequency by comparing the VCO output to a crystal oscillator frequency.
Choosing an appropriate VCO for a high-speed data converter system is not as simple as finding one with the right oscillator frequency. One key parameter that must be taken into consideration is
clock jitter. Jitter is generally defined as short-term, non-cumulative variation of the significant instant of a digital signal from its ideal position in time. Figure 3 illustrates a sampling
clock signal that contains jitter. Jitter generated by the clock is caused by various internal noise sources, such as thermal noise, phase noise, and spurious noise. In the case of a data converter,
jitter affects the signal-to-noise ratio (SNR) performance of the data converter.
Figure 3. Jitter in clock signal degrades the ADC signal-to-noise ratio.
How Clock Jitter Degrades ADC's Signal-to-Noise Ratio (SNR)
Jitter generated by a clock source can cause the ADC's internal circuitry to falsely trigger the sampling time. As shown in Figure 4, uncertainty in sampling time Δt equates to uncertainty in
amplitude ΔA . This results in false sampling of the analog input amplitude, thus degrading the SNR of the ADC. With the following equations, the maximum SNR of a data converter can be calculated
for a given amount of clock jitter:
Figure 4. An SNR model obtained using the sampling time uncertainty.
The slope is at its maximum when the term cos(ωt) = 1. Therefore,
EQ.2 can be rearranged as:
By definition, A/(ΔA) is the signal-to-noise ratio, and Δt is the root-mean-square (RMS) value of the jitter. EQ. 3 can be rewritten as:
For example, if the analog input signal is 250MHz, and 50dB SNR is to be achieved, the maximum RMS jitter (σ[RMS]) must be less than 2ps.
How Noise Sources Cause Jitter
Thermal noise, frequency modulation (FM), amplitude modulation (AM), phase modulation (PM), and spurious components contribute to the noise that causes jitter in the clock signal. Because of the
difficulty to distinguish noise caused by FM, AM, and PM, all three types of noise are grouped into a general term known as phase noise. To clarify the calculation of phase noise, a high frequency
circuit, using the MAX2620 VCO and PLL, will be used as example.
Thermal Noise Contribution to Jitter
Figure 5 depicts a simplified plot of the VCO phase noise profile. The MAX2620's output amplifier has a thermal noise floor of approximately -147dBm/Hz. This noise is white, Gaussian noise with a
finite bandwidth. Although the effective bandwidth has not been characterized, it can be approximated to be twice the operating frequency. With the MAX2620 properly tuned to the desired output
frequency, the contribution of the noise floor to jitter can be computed with the following equation:
Figure 5. Simplified phase noise profile of the MAX2620 VCO as a function of the offset frequency.
ω[o] = 2πf[o] = Angular clock oscillation (center) frequency (in rad/s)
f[o] = Oscillator (center) frequency (in Hz)
f = Offset frequency from the center frequency (in Hz)
τ = 1/2f[o] = Half of a period (in s)
L(f) = Phase noise at offset frequency f (in dBc/Hz).
For further improvement in the noise performance, a power matching network (L2 and C6) whose frequency response similar to a bandpass filter is often applied at the VCO output. This attenuates
undesired noise outside the bandwidth of interest. By doing so, one can estimate the worst noise by the limit of integration from 0Hz offset to f[0]. Noise beyond these limits is greatly attenuated
and can be ignored. Because the noise floor is even for offset frequencies from 0Hz to f[0], L(f) can be considered constant. EQ. 5 can be reduced to:
The edge-to-edge timing jitter due to the noise floor is:
Because thermal noise is non-correlated, jitter is non-accumulated. The period-to-period jitter is the same as edge-to-edge jitter.
Equation 8 can also be displayed as:
where SNR[OSC] is the signal-to-noise ratio of the oscillator due to the noise floor.
Phase Noise Contribution to Jitter
Phase noise is characterized as the ratio of noise power at an offset frequency to the power level of the clock (carrier) signal. This ratio is usually normalized to 1Hz-bandwidth, resulting in a
unit of dBc/Hz. For instance, the phase noise at 100kHz offset in Figure 5 is -118dBc. That means the noise power at 1000.1MHz is 118dB below the carrier power level at 1000MHz in 1Hz-bandwidth.
The free running phase noise of the MAX2620 is approximately 20dB/decade from the corner-offset frequency of 1MHz to the clock frequency. With EQ. 11, the period-to-period jitter due the phase noise
can be calculated as follows:
where f is the offset frequency from the clock frequency, and it has to be in the region where the phase noise decreases 20dB per decade. The phase noise, L(f), was taken from the MAX2620
characterization at f = 100kHz offset frequency. With f = 10kHz, the resulting jitter will not change.
Spurious Components Contribution to Jitter
A PLL-based clock signal produces spurs. If these spurs are not suppressed, they can degrade the jitter performance. Figure 6 shows a spectrum plot of a 1GHz clock signal taken with a spectrum
analyzer. The two symmetrical pairs of spurs displayed in this figure are approximately 75dBc and 85dBc below the carrier. The separation of these spurs from the carrier and from each other is
determined by the comparison frequency used in the phase-locked loop. In this case, the comparison frequency is 1MHz; therefore, the two spurs next to the carrier are exactly 1MHz away from the
carrier and the subsequent pair. In addition, there is another pair of -75dBc spurs (not shown) at 20MHz offset caused by the crystal oscillator. The following equation, translates these spurs into
Figure 6. A 1GHz clock shown with spurious components.
where f[m] is offset frequency at which the phase noise spurious components occurred. With m = 1, the cycle-to-cycle jitter computes to 4.38x10^-6ps. For practical applications with ADCs, such as
the MAX104, jitter due to spurious noise at this level is negligible.
Total Jitter
The total cycle-to-cycle jitter is a function of the square root of the sum of the jitter squares and can be calculated as follows:
Phase-Locked Loop
As a result of inherit frequency drift due to temperature, power supply, load, etc., a free running VCO is rarely used by itself. Usually a phase-locked loop is introduced to help lock the VCO
output to the desired frequency. If designed properly, the phase-locked loop can help reduce the phase noise. The phase noise within the loop bandwidth is lower than that of a free running VCO.
Thus, the actual jitter due to phase noise is less than that of EQ. 11.
Figure 7 shows the functional diagram of the MB15E07 in an integer-N PLL system. It consists of a phase detector (or comparator), an output charge-pump, a dual modulus pre-scalar, an N counter, and
an R counter. The N counter consists of a main (M) counter and a swallow or auxiliary (A) counter.
The N counter then works in conjunction with the dual modulus pre-scalar (P).
Figure 7. Simplified block diagram of a typical PLL system consisting of a PLL, crystal oscillator, loop filter, and VCO.
During power-up (assuming that the PLL was preprogrammed), the VCO would oscillate at the desired frequency plus some offset. This frequency is first divided by the integer N, and then compared to
the reference crystal oscillator frequency, whose frequency has also been divided by an integer R. If there is a phase difference between the two frequencies, the voltage at the PLL output changes
accordingly. For example, if the VCO frequency is lower than that of the reference, the charge-pump will charge the loop filter capacitors to increase the voltage. If the VCO frequency is higher
than the reference, the charge-pump will discharge the loop filter capacitors to decrease the voltage. An increase in voltage results in an increase in frequency, and visa versa. Hence, the PLL
functions as a feedback loop that keeps the VCO output frequency locked at the desired frequency. The VCO frequency is a function of N, R and f[REF] and is calculated as follows:
For example, if P = 32, M = 31, and A = 8, using EQ. 14, N counter is calculated to 1000. If the reference oscillator frequency is 20MHz and the R counter is set for 20, using EQ. 15, the VCO
frequency is locked at 1000MHz.
Design Parameters
Careful design and implementation of the clock circuit is required to ensure optimum performance. This can be achieved by choosing the proper components and providing a well-designed high-frequency
PC board. Table 1 shows the recommended component values for two different operating frequencies. These values ensure that the VCO will oscillate and phase lock at the desired frequency, while
providing the proper output power levels. The output frequency of the MAX2620 is set by an external resonating tank, which consists of L1, C1, C2, C3, C4, and D1. L1, C1, C2, C3, and C4 set the
free-running, oscillating frequency. The varactor diode, D1, fine-tunes the output frequency to the desired frequency. D1 is reverse biased and has a capacitance that varies with the bias voltage
generated by the PLL output. A change in D1's capacitance allows for the fine-tuning of the output frequency.
The oscillating frequency can be calculated with the following equation:
To accommodate the component tolerance, PCB, supply voltage, and temperature variations, the capacitance of D1 should be chosen such that the tuning range is about ±5% to ±10% from the nominal
frequency. C4 is the capacitor that couples the varactor to the tuning tank. Increasing C4 can increase the tuning range. C2 and C3 are feedback capacitors necessary for the oscillator to function
properly. Typically, C2 = 2.7pF and C3 = 1.0pF. For 1.0GHz, chose L1 = 5.6nH, C4 = 4.7pF, and C1 = 1.0pF. Both the VCO output and ADC clock input have to be matched to 50Ω. A LC network (L2 and C6)
is used at the VCO output to ensure optimum power transfer to the clock input of the ADC. The matching network has a bandpass-filter-like frequency response that further reduces the thermal noise
Table 1. Suggested Component Values for the Clock Generator
f[out] = 600MHz f[out] = 1000MHz
R1 240Ω 390Ω
R2 240Ω 390Ω
C1 1.0pF 1.0pF
C2 2.7pF 2.7pF
C3 1.0pF 1.0pF
C4 9.0pF 3.3pF
C5 9.0pF 2.2pF
C6 3.0pF 1.5pF
C7 12nF 3.9pF
C8 120nF 39nF
C9 12nF 3.9nF
L1 12nH (±2%) 5.6nH (±2%)
L2 18nH 10nH
D1 SMV1233-001 (Alpha Industries) SMV1233-001 (Alpha Industries)
The charge pump output of the PLL pulses at the phase comparison frequency determined by R and the external crystal oscillator. A loop filter is employed to filter these pulses into a constant
DC-control voltage for the VCO. The third-order loop filter (Figure 2) consists of C7, C8, C9, R1 and R2. Use simplified EQ. 17 to 23 to calculate the component values.
N = The counter value from EQ. 14
ξ = Damping factor, typically 0.707
I[CP]= The charge pump current, 10mA for the MB15E07
K[VCO]= The VCO tuning gain or sensitivity
The VCO tuning gain, K[VCO], depends on the component values used in the VCO tank. The VCO tuning gain in this design example is about 35MHz/V.
The MB15E07 is programmed via the SPI™-compatible interface. Table 2 shows the register/counter settings for 600MHz (MAX106) and 1000MHz (MAX104) operation:
Table 2. Suggested Register Settings for the MB15E07 with 20MHz Crystal Oscillator
600MHz 1000MHz
f[COMPARISON] 500kHz 1000kHz
Loop Bandwidth 25kHz 50kHz
R Counter 40 20
P Counter 32 32
M Counter 37 31
A Counter 16 8
SW bit HIGH HIGH
FC bit HIGH HIGH
To ensure good high-frequency PC board layout, keep the following suggestions in mind:
• Keep all PC board trace lengths as short as possible. Design with controlled impedance traces.
• Choose the smallest component size possible, preferably type 0603 or 0402.
• Use high quality-factor (Q) components to minimize VCO phase noise and maximize output power transfer. A Q factor of 40 or greater should be sufficient.
• Keep all the components for the resonating tank circuit as close together and as close to the MAX2620 as possible.
• Place de-coupling capacitors close to the VCO, and with a direct connection to the ground plane. All V[CC]connections should have their own de-coupling capacitors.
• Maintain a 50Ω connection between the VCO output and the ADC clock input.
• Use component value recommendations from Table 1 as a starting point. Possible parasitic effects may require fine-tuning of some component values to ensure optimal performance.
Experimental Results
To demonstrate the performance of the proposed clock circuit, designed from the suggested equations and techniques, the circuit in Figure 2 was designed and tested with the MAX104 evaluation kit.
Figure 6 shows the output of the proposed high-frequency, low jitter clock measured with a spectrum analyzer. The oscillation frequency is phase locked at 1GHz, with an output level of -2dBm. Figure
8 shows the signal-to-noise ratio of the MAX104 ADC over analog input frequency. With f[SAMPLE] = 1.0GHz and f[IN] at -1dBFS, the SNR varies from 47.1dB to 45.5dB for analog frequencies ranging from
10MHz to 1GHz, respectively. Compared to a known low-jitter signal generator (HP8662A), the SNR measured with the proposed clock is only ~0.4dB lower.
Figure 8. The SNR of the MAX104 is approximately 0.4dB lower with the PLL clock than with the HP8662A.
1. Rudy Van De Plassche, Integrated Analog-To-Digital and Digital-To-Analog Converters, Kluwer Academic Publishers, 1994, pp. 6-8.
2. Chris O'Connor, "Develop Trimless Voltage-Controlled Oscillators," Microwaves & RF, July 1999 pp. 69-78, and January 2000, pp. 94-105.
3. Ken Holladay, "Design Loop Filters For PLL Frequency Synthesizers," Microwaves & RF, September 1999, pp. 98-104
4. MAX2620 Data Sheet, Maxim Integrated.
5. Ali Hajimiri et. al., "Jitter and Phase Noise in Ring Oscillators," IEEE Journal of Solid-State Circuits, Vol. 34, No. 6, pp. 790-804.
6. Boris Drakhlis, "Calculate Oscillator Jitter By Using Phase-Noise Analysis," Microwaves & RF, Jan. 2001 pp. 82-90 and p. 157.
7. MB15E07 Data Sheet. Fujitsu Semiconductor.
8. MAX104 Data Sheet, Maxim Integrated.
A similar version of this article appeared in the September 2001 issue of Sensors magazine.
Related Parts
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MAX105 Dual, 6-Bit, 800Msps ADC with On-Chip, Wideband Input Amplifier Free Samples
MAX106 ±5V, 600Msps, 8-Bit ADC with On-Chip 2.2GHz Bandwidth Track/Hold Amplifier
MAX107 Dual, 6-Bit, 400Msps ADC with On-Chip, Wideband Input Amplifier Free Samples
MAX2620 10MHz to 1050MHz Integrated RF Oscillator with Buffered Outputs Free Samples
Next Steps
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APP 800: Jul 17, 2002
TUTORIAL 800, AN800, AN 800, APP800, Appnote800, Appnote 800 | {"url":"http://www.maximintegrated.com/app-notes/index.mvp/id/800","timestamp":"2014-04-23T23:55:45Z","content_type":null,"content_length":"93152","record_id":"<urn:uuid:b195a0cb-edac-4257-9915-2e495c50c17f>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00051-ip-10-147-4-33.ec2.internal.warc.gz"} |
binomial formula, probability problem.
October 11th 2011, 11:30 PM #1
Senior Member
Jan 2007
binomial formula, probability problem.
a factory is known to make 15% defectives.
Most of the products are found and fixed or thrown out. Suppose two products are randomly selected for inspection.
1. What is the probability that both pieces are defect free?
2. Suppose at least one of the pieces has a flaw. What is the probablility that both are defective?
I know the binomial theorem can be useful here.
P(x=k) = (n/k)P^k(1-p)^(n-k)
Please help me solve this.
also I did know that if neither piece is defect free:
2/0 .. .85^0*(1-.85)^2
Re: binomial formula, probability problem.
a factory is known to make 15% defectives.
Most of the products are found and fixed or thrown out. Suppose two products are randomly selected for inspection.
1. What is the probability that both pieces are defect free?
2. Suppose at least one of the pieces has a flaw. What is the probablility that both are defective?
I know the binomial theorem can be useful here.
P(x=k) = (n/k)P^k(1-p)^(n-k)
Please help me solve this.
also I did know that if neither piece is defect free:
2/0 .. .85^0*(1-.85)^2
1. (0.15)^2 = ....
2. (Answer to 1.)/(Pr(1 defective) + Pr(2 defectives)) = ....
October 12th 2011, 12:56 AM #2 | {"url":"http://mathhelpforum.com/statistics/190159-binomial-formula-probability-problem.html","timestamp":"2014-04-21T11:19:05Z","content_type":null,"content_length":"34682","record_id":"<urn:uuid:53da6c85-8721-4d41-b021-7a5844d418ef>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00126-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Related: Java Calculator Program - Process Calculator Program - Graphing Calculator Program - Direction Calculator Program - Area Calculator Program | {"url":"http://www.winsite.com/java/java+calculator+program/","timestamp":"2014-04-19T11:59:09Z","content_type":null,"content_length":"29703","record_id":"<urn:uuid:bab9643a-a0f8-4560-85f0-7ff87c43fbab>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00487-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Infinite. Part 9. Abraham, Omega, and Doing Things In Order.
[Here is part 8.]
What does it mean for one thing to be less than another? We are natural “orderers” aren’t we? We have ordering intuitions about size, strength, speed, beauty, importance, riches, standard of living,
loudness, height, and other stuff. Some of this is reasonably quantifiable, some not so much. As a schoolboy, I learned the hard truth from a sixth grade seat mate, Susan Ortiz: the girls ranked guys
on a handsome scale (that wasn’t the only scale, but it was one). Are some people (or animals) more “intelligent” that others? As Mormons we are familiar with this sort of ordering by Deity:
6 And the Lord said unto me: Now, Abraham, these two facts exist, behold thine eyes see it; it is given unto thee to know the times of reckoning, and the set time, yea, the set time of the earth
upon which thou standest, and the set time of the greater light which is set to rule the day, and the set time of the lesser light which is set to rule the night.
7 Now the set time of the lesser light is a longer time as to its reckoning than the reckoning of the time of the earth upon which thou standest.
8 And where these two facts exist, there shall be another fact above them, that is, there shall be another planet whose reckoning of time shall be longer still . . .
17 Now, if there be two things, one above the other, and the moon be above the earth, then it may be that a planet or a star may exist above it; and there is nothing that the Lord thy God shall
take in his heart to do but what he will do it.
18 Howbeit that he made the greater star; as, also, if there be two spirits, and one shall be more intelligent than the other, yet these two spirits, notwithstanding one is more intelligent than
the other, have no beginning; they existed before, they shall have no end, they shall exist after, for they are gnolaum, or eternal.
19 And the Lord said unto me: These two facts do exist, that there are two spirits, one being more intelligent than the other; there shall be another more intelligent than they; I am the Lord thy
God, I am more intelligent than they all.
Here, celestial ordering gives a kind of earth-centric reference to “less than” and spirits get some kind of ordering via their “intelligence.” Do the more intelligent folk get the perks? Is it
really the case that this is a service opportunity? (smiley face here) Our fearless blogleader once labeled this abrahamic induction the Comparative Principle, Sam Brown styles it as overlaying his
interpretation of a Mormon chain of being structure in his In Heaven as it is on Earth, see here for example (p.14). I want to come back to this but to do it justice we need a little machinery to
make sense of it all.
So I now drag you back into Cantor’s paradise. A most useful and elementary abstraction is the idea of a collection of objects, grouped by whatever rule we wish. Last time we saw that there may be
pitfalls lurking here, so we will be careful and gradually put some restrictions in place. But for now, let’s jump into the concept of “ordinal numbers.”
You are familiar with the idea. You use it every day with language like “first” “second” “5th” “ducks in a row” and so on. But now we must be a little formal about it. Don’t run an hide, it’s not
that bad. I’m taking an approach partly suggested by John von Neumann (1903-1957), an amazing genius.
I’ll start by saying what it means for a collection to be “well-ordered.” We’re assuming that we can discern some order among the members of our collection in the first place and being “well-ordered”
merely means that if we extract any subgrouping of items from our collection, it will have a “smallest” or least member in our perceived ordering. Not all orderings have this property but it is
usually assumed that any collection, no matter what it consists of, can be supplied with an ordering that imposes the property of being well-ordered.
with its natural ordering is the paradigm of well-ordered sets[1].
Now a bit of notation. This will make life considerably easier. First, we use ∅ to stand for the collection with nothing in it. A bit of a place holder, like zero if you will. ∅ = { }. The empty
warehouse, nobody home. Next, we use ∈ to stand for “is in”. Like 4 ∈ {5,6,4,10}. And ∉ to mean “is not in.” 70 ∈ N, but “raisin” ∉ N. One more abbreviation, the “union” of two collections of things
gives the collection of things which include all stuff in either collection. {7,8} ∪ {8, 0, 45, 3} is {0, 3, 7, 8, 45} (8 is not listed twice because we’re just interested in content, not twins or
triplets or whatever – you’re either in, or not).
∈ can be used to supply an ordering sometimes. We can arbitrarily say that A < B if it happens that A ∈ B. A collection X of sets is said to be transitive if whenever Y ∈ X and Z ∈ Y then Z ∈ X.
Here’s an example of a transitive set:
{∅ , {∅}, {∅, {∅}}, {∅ , {∅}, {∅, {∅}}}}. In fact we will call a set an ordinal when it is both well-ordered and transitive. It’s possible to show in a few pages of work that an ordinal is the set of
all ordinals that precede it. This is clearly true for our example, {∅ , {∅}, {∅, {∅}}, {∅ , {∅}, {∅, {∅}}}}.
My first observation is this: if we make the identification
0 = ∅
1 = {∅}
2 = {∅, {∅}}
3 = {∅ , {∅}, {∅, {∅}}}
and so on,
we can say that the counting numbers are ordinals, a happy circumstance since they are the models we use for ordering things in everyday life.
Now a fact that we can easily prove, but won’t, is this: If C is an ordinal, then C ∪ {C} is the least ordinal greater than C. In this case it is natural to write C + 1 for C ∪ {C}. And again, if S
is any collection of ordinals, then there is a least ordinal in S. Hence, the ordinals are well-ordered.
Finally, this very interesting definition. α is a successor if there is an ordinal β so that α = β + 1. γ is called a limit ordinal if it is not 0 and not a successor.
And, limit ordinals exist. The *smallest* limit ordinal is just N with the identification given above. With the identification, we can identify N as ω. That’s God. Maybe (he does claim the title
after all). The Thomists were both pleased and horrified. They know what Cantor has up his sleeve. And that was, ω + 1. Sitting on the top of a topless throne as some antebellum preachers liked to
have it (for one of a million, see Evangelical Magazine and Gospel Advocate, 5 (1834):396) is no guarantee that someone is not up there above you. And then there is ω[1]. We are not done with either
theology or ordinals.
(Part ten is here.)
[1] {79, 40, 91, 10^20, 30, 10,000} is a subset of N and clearly has a least member (30). There are orderings which are not well-ordered and they can be quite useful. The set of all fractions with
their natural order are not well-ordered, but we can reorder them so that they become well-ordered (not very interesting in practice). The ability to supply an ordering to any collection in such a
way that it well-ordered, is a statement equivalent to the Axiom of Choice.
1. Student says:
Would this be a good place to discuss the theological implications of the Axiom of Choice? In particular, I was thinking about how the Axiom of Choice leads to the Banach-Tarski Theorem, which
may be taken as an explanation of the miraculous multiplication of bread loaves and fishes. On the other hand, the adversary, who seeks to destroy the agency of man, would naturally be against
the Axiom of Choice. So what does this say about those mathematicians who contest it?
2. Anne L says:
If only I had read this post before I took my test this evening.
Great explanation of ordinals…well-ordered ordinals, limit ordinals ect.
I’m curious to see how else they can be tied into theology.
3. WVS says:
Student, there’s a bit of Banach-Tarski in the axiom of choice link in the footnote.
Anne L, thanks. More ordinal theology next time.
4. Brother Brigham says:
Mormon Church’s membership dropping like flies, one by one hurrah hurrah:
5. Lurker says:
Mormon Church’s membership dropping like flies, one by one hurrah hurrah:
Fewer things are more entertaining than reading a non-lawyer attempting to write like one. Thanks for the link – I needed some chuckles this afternoon.
6. nr[2] says:
Where did that come from? As a member of that stake I know better – we have a great Stake Presidency. Why is it still on here?
7. StillConfused says:
I didn’t understand the Michael Crook deal.. what was in his profile that was offensive? Maybe I missed it
8. n[r]2 says:
I didn’t see anything either. I think this person may be unbalanced. He is not someone I know personally.
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An Improved Context-free Recognizer
Results 1 - 10 of 65
- Computational Linguistics , 2002
"... this article can compute solutions to all four of these problems in a single flamework, with a number of additional advantages over previously presented isolated solutions ..."
Cited by 188 (6 self)
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this article can compute solutions to all four of these problems in a single flamework, with a number of additional advantages over previously presented isolated solutions
- In Proceedings of the Thirty-First Annual Meeting of the Association for Computational Linguistics , 1993
"... This paper describes the details of the system, and includes relevant measurements of size, efficiency, and performance of each of its components ..."
, 1998
"... Probabilistic Context-Free Grammars (PCFGs) and variations on them have recently become some of the most common formalisms for parsing. It is common with PCFGs to compute the inside and outside
probabilities. When these probabilities are multiplied together and normalized, they produce the probabili ..."
Cited by 82 (2 self)
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Probabilistic Context-Free Grammars (PCFGs) and variations on them have recently become some of the most common formalisms for parsing. It is common with PCFGs to compute the inside and outside
probabilities. When these probabilities are multiplied together and normalized, they produce the probability that any given non-terminal covers any piece of the input sentence. The traditional use of
these probabilities is to improve the probabilities of grammar rules. In this thesis we show that these values are useful for solving many other problems in Statistical Natural Language Processing.
We give a framework for describing parsers. The framework generalizes the inside and outside values to semirings. It makes it easy to describe parsers that compute a wide variety of interesting
quantities, including the inside and outside probabilities, as well as related quantities such as Viterbi probabilities and n-best lists. We also present three novel uses for the inside and outside
probabilities. T...
- Computational Linguistics , 1994
"... this paper, we study the problem of lexicalizing context-free grammars and show that it enables faster processing. In previous attempts to take advantage of lexicalization, a variety of
lexicalization procedures have been developed that convert context-free grammars (CFGs) into equivalent lexicalize ..."
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this paper, we study the problem of lexicalizing context-free grammars and show that it enables faster processing. In previous attempts to take advantage of lexicalization, a variety of
lexicalization procedures have been developed that convert context-free grammars (CFGs) into equivalent lexicalized grammars. However, these procedures typically suffer from one or more of the
following problems
- Computational Linguistics , 1991
"... Speech recognition language models are based on probabilities P(Wk+I = v [ WlW2~..., Wk) that the next word Wk+l will be any particular word v of the vocabulary, given that the word sequence Wl,
w2,..., Wk is hypothesized to have been uttered in the past. If probabilistic context-free grammars are t ..."
Cited by 76 (0 self)
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Speech recognition language models are based on probabilities P(Wk+I = v [ WlW2~..., Wk) that the next word Wk+l will be any particular word v of the vocabulary, given that the word sequence Wl,
w2,..., Wk is hypothesized to have been uttered in the past. If probabilistic context-free grammars are to be used as the basis of the language model, it will be necessary to compute the probability
that successive application of the grammar rewrite rules (beginning with the sentence start symbol s) produces a word string whose initial substring is an arbitrary sequence wl, w2,..., Wk+l. In this
paper we describe a new algorithm that achieves the required computation in at most a constant times k3-steps. 1.
- Computational Linguistics , 1999
"... this paper is that all five of these commonly computed quantities can be described as elements of complete semirings (Kuich 1997). The relationship between grammars and semirings was discovered
by Chomsky and Schtitzenberger (1963), and for parsing with the CKY algorithm, dates back to Teitelbaum ( ..."
Cited by 64 (1 self)
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this paper is that all five of these commonly computed quantities can be described as elements of complete semirings (Kuich 1997). The relationship between grammars and semirings was discovered by
Chomsky and Schtitzenberger (1963), and for parsing with the CKY algorithm, dates back to Teitelbaum (1973). A complete semiring is a set of values over which a multiplicative operator and a
commutative additive operator have been defined, and for which infinite summations are defined. For parsing algorithms satisfying certain conditions, the multiplicative and additive operations of any
complete semiring can be used in place of/x and , and correct values will be returned. We will give a simple normal form for describing parsers, then precisely define complete semirings, and the
conditions for correctness
, 1998
"... We present a bottom-up parsing algorithm for stochastic context-free grammars that is able (1) to deal with multiple interpretations of sentences containing compound words; (2) to extract N-most
probable parses in O(n 3 ) and compute at the same time all possible parses of any portion of the inpu ..."
Cited by 57 (11 self)
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We present a bottom-up parsing algorithm for stochastic context-free grammars that is able (1) to deal with multiple interpretations of sentences containing compound words; (2) to extract N-most
probable parses in O(n 3 ) and compute at the same time all possible parses of any portion of the input sequence with their probabilities; (3) to deal with #out of vocabulary# words. Explicitly
extracting all the parse trees associated to a given input sentence depends on the complexity of the grammar, but even in the case where this number is exponential in n, the chart used by the
algorithm for the representation is of O(n 2 ) space complexity. 1 Introduction This article presents CYK+, a bottom-up parsing algorithm for stochastic context-free grammars that is able: 1. to deal
multiple interpretations of sentences containing compound words; 2. to extract N-most probable parses in O(n 3 ) and compute at the same time all possible parses of any portion of the input sequence
with their p...
- In IWPT , 2001
"... While symbolic parsers can be viewed as deduction systems, this view is less natural for probabilistic parsers. We present a view of parsing as directed hypergraph analysis which naturally
covers both symbolic and probabilistic parsing. We illustrate the approach by showing how a dynamic extension o ..."
Cited by 56 (3 self)
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While symbolic parsers can be viewed as deduction systems, this view is less natural for probabilistic parsers. We present a view of parsing as directed hypergraph analysis which naturally covers
both symbolic and probabilistic parsing. We illustrate the approach by showing how a dynamic extension of Dijkstra’s algorithm can be used to construct a probabilistic chart parser with an Ç Ò time
bound for arbitrary PCFGs, while preserving as much of the flexibility of symbolic chart parsers as allowed by the inherent ordering of probabilistic dependencies. 1
- IN: NEW DEVELOPMENTS IN NATURAL LANGUAGE PARSING , 2000
"... This chapter introduces weighted bilexical grammars, a formalism in which individual lexical items, such as verbs and their arguments, can have idiosyncratic selectional influences on each
other. Such ‘bilexicalism ’ has been a theme of much current work in parsing. The new formalism can be used t ..."
Cited by 51 (1 self)
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This chapter introduces weighted bilexical grammars, a formalism in which individual lexical items, such as verbs and their arguments, can have idiosyncratic selectional influences on each other.
Such ‘bilexicalism ’ has been a theme of much current work in parsing. The new formalism can be used to describe bilexical approaches to both dependency and phrase-structure grammars, and a slight
modification yields link grammars. Its scoring approach is compatible with a wide variety of probability models. The obvious parsing algorithm for bilexical grammars (used by most previous authors)
takes time O(n^5). A more efficient O(n³) method is exhibited. The new algorithm has been implemented and used in a large parsing experiment (Eisner, 1996b). We also give a useful extension to the
case where the parser must undo a stochastic transduction that has altered the input. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=227748","timestamp":"2014-04-19T02:25:03Z","content_type":null,"content_length":"35587","record_id":"<urn:uuid:877cbf46-dcca-447e-b575-70e645f6d1b8>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00011-ip-10-147-4-33.ec2.internal.warc.gz"} |
Primordial Nucleosynthesis - G. Steigman
2.2. The Baryon Density
In the expanding Universe, the number densities of all particles decrease with time, so that the magnitude of the baryon density (or that of any other particle) has no meaning without also specifying
when it is measured. To quantify the universal abundance of baryons, it is best to compare n[B] to the CBR photon density n[]. The ratio, n[B] / n[] is very small, so that it is convenient to define
a quantity of order unity,
where [B] is the ratio (at present) of the baryon density to the critical density and h is the present value of the Hubble parameter in units of 100 km s^-1 Mpc^-1 ([B] [B] h^2). | {"url":"http://ned.ipac.caltech.edu/level5/March04/Steigman2/Steigman2_2.html","timestamp":"2014-04-16T10:19:51Z","content_type":null,"content_length":"2335","record_id":"<urn:uuid:5fe40510-6151-41db-8986-feec2785fed1>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00593-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fundamental Methods of Mathematical Economics
Alpha C. Chiang
Static (or equilibrium) analysis; Comparative-static analysis; Optimization problems; Dynamic analysis; Mathematical programming and game theory.
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Two Economic Models 8
STATlC OR EQUlLlBRlUM ANALYSlS 39
Four Linear Models and Matrix Algebra 59
17 other sections not shown
Bibliographic information | {"url":"http://books.google.com/books?id=OLC2AAAAIAAJ&q=positive&dq=related:STANFORD36105001915565&source=gbs_word_cloud_r&cad=6","timestamp":"2014-04-20T11:03:58Z","content_type":null,"content_length":"121853","record_id":"<urn:uuid:fc2feea9-093b-4176-848c-eda216139567>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00397-ip-10-147-4-33.ec2.internal.warc.gz"} |
Carrollton, TX SAT Math Tutor
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...I took courses in the subject in high school, college and graduate school. For 22 years I served as a pastor in various churches from different denominations, which required that I prepare at
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82 Subjects: including SAT math, English, chemistry, calculus
...As a physicist, I am well exposed to mathematical concepts which includes calculus as one of the subjects that I mastered. I also tutor students regularly on Precalculus and Calculus. I am a
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Cosmology & Gravitation
This series consists of talks in the areas of Cosmology, Gravitation and Particle Physics.
The QUaD experiment has recently released CMB polarization results at el>200 which are the most sensitive to date. The predicted series of peaks in the EE spectrum are shown to be present for the
first time while BB remains undetectable. After briefly reviewing the motivation for polarization measurements I will move on to the experiment, observations, analysis technique and the final
results. Finally I will mention on-going efforts to detect gravitational wave B modes.
The so-called cosmological backreaction arises when one directly averages the Einstein equations to recover cosmology. While usually applied to avoid employing dark energy models, strictly speaking
any cosmological model should be built from such an averaging procedure rather than an assumed background. We apply the Buchert formalism to Einstein-de Sitter, Lambda CDM and quintessence
cosmologies, and as a first approach to the full problem, evaluate numerically the discrepancies arising from linear perturbation theory between the averaged behaviour and the assumed behaviour.
At the end of inflation, dynamical instability can rapidly deposit the energy of homogeneous cold inflation into excitations of other fields. This process (known as preheating) essentially starts the
hot big bang as we know it. I will present simulations of several preheating models using a new numerical solver DEFROST I developed. The results trace the evolution of the fields, which quickly
become very inhomogeneous as the instability kicks in. Surprisingly, there appears to be a certain universality across preheating models with different decay channels.
Observations of the Milky Way by the SPI/INTEGRAL satellite have confirmed the presence of a strong 511 KeV gamma-ray line emission from the bulge, which require an intense source of positrons in the
galactic center. These observations are hard to account for by conventional astrophysical scenarios, whereas other proposals, such as light DM, face stringent constraints from the diffuse gamma-ray
background. I will describe how light superconducting strings could be the source of the observed 511 KeV emission.
In this talk I will analyse the stochastic background of gravitational waves coming from a first order phase transition in the early universe. The signal is potentially detectable by the space
interferometer LISA. I will present a detailed analytical model of the gravitational wave production by the collision of broken phase bubbles, together with analytical results for the gravitational
wave power spectrum. Gravitational wave production by turbulence and magnetic fields will also be briefly discussed.
Recent developments in the field of Numerical Relativity have not only provided key insights of binary black hole systems but also began influencing its future role. Undoubtedly one of the most
important future drivers in the near future of the field will be its role as another element within the study of spectacular astrophysical phenomena involving strongly gravitation scenarios.
Connecting (yet to be observed) gravitational waves with observations within the electromagnetic spectra will be one ultimate goal of this enterprise.
I will review relativistic quantum theory that is based on Wigner\'s unitary representations of the Poincare group, Dirac\'s forms of dynamics, and Newton-Wigner\'s definition of the position
operator. Formulas will be derived that transform particle observables between different inertial reference frames. In the absence of interactions, these formulas coincide with Lorentz
transformations from special relativity. However, when interaction is turned on, some deviations appear.
We study the effective field theory of inflation, i.e. the most general theory describing the fluctuations around a quasi de Sitter background, in the case of single field models. The scalar mode can
be eaten by the metric by going to unitary gauge. In this gauge, the most general theory is built with the lowest dimension operators invariant under spatial diffeomorphisms, like g^{00} and K_{mu
nu}, the extrinsic curvature of constant time surfaces. | {"url":"https://www.perimeterinstitute.ca/video-library/collection/cosmology-gravitation?page=23","timestamp":"2014-04-21T02:22:23Z","content_type":null,"content_length":"64559","record_id":"<urn:uuid:922a663d-98b0-4719-b5dc-74950c13c789>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00378-ip-10-147-4-33.ec2.internal.warc.gz"} |
Schaumburg Trigonometry Tutor
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...I believe that students are able to learn anything with the right instruction and I know my passion for science and math will contribute greatly to this process. I strive to find creative ways
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The expanded Phoenix/II Theory - A Diamond at the Heart of All Matter and Energy
The expanded Phoenix/II Theory - A Diamond at the Heart of All Matter and Energy
Plagiarizing the title of the media's reports on Nima's paper will not get you any credit with real scientists.
All the speculations in the Phoenix-I/II Theory directly come from the Standard Model's data, or observations. In the event where observations contradicts the standard Model
Ah, yes, that thing you aggressively demanded that you didn't have to understand any of the math in in the last thread. Or have you learned how to actually calculate scattering cross-sections,
Feynman diagrams, and nucleon form factors in the past little bit? Or do you still insist you don't need to actually go through the trouble of understanding the theory you claim to surpass?
The Phoenix-I successfully described all matter and energy particles electric charges exchanges. I quickly published these findings in an article some time ago
Ah, yes, that article you published in the peer-reviewed, high-impact journal Nature, right?
Oh, you mean the article you "published" on your own website. I guess the real paper is hung up in the academic bureaucracy, huh? Didn't fill out the right forms to get it published? I bet we can
expect to see it any day now when these trivial little misunderstandings are cleared up.
The Phoenix-I sparked a controversy amongst more conservative physicists.
I.e., I think it's ridiculous and shows you don't have any understanding of any physics at any level. It's not even not even wrong.
Its incomplete form was wrongfully interpreted as direct denial of the Standard Model's properties
It's more the fact that you clearly don't understand anything about the Standard Model that is the problem here.
even if the road was a lonely one, is was a wonderful one.
Ah, yes, the best, most productive scientists are the ones who complain about being persecuted and ignored, on the conspiracy forums.
The theory doesn't just explain charges anymore. Now, it expanded to a point where it explains many things all other preon models failed to account for, such as the spin of all particles, the 3
generations, antimatter...
I eagerly await the carefully calculated nucleon form factors. Or, you know, field theory calculations of any kind.
All known particles have no more, no less than 6 preons each.
Ah, yes, as your extensive group theoretical calculation and examination of the effective Lagrangian shows. Or, rather, I'm sure, will show, once Nature gets passed all the bureaucracy and publishes
your paper.
This very suggestion already explains the charges of all particles, but also the exact mechanism during particle decay.
Yes, it explains those mechanisms that totally haven't been understood since longer than I've been alive (and were thus desperately in need of explanation).
Compare a mainstream Feynman diagram of the phenomena, with Phoenix-I's Feynman diagram of the same phenomena
Ah, yes, as we all know, and you have grasped most of all, is that Feynman diagrams correspond to drawing lines between things. Not complicated integrals, convoluted with ridiculous rules about
vertices and propagators. Not at all.
This alone gave the theory more accuracy than most other preonic models
Of course, that diagram with lines in it connecting your assumption with your other assumption clearly is far better than any of those models those Nobel laureates have proposed. What with their
actual calculations and understanding of physics.
The expanded (speculative) Pheonix-II Model
You know, it's not fair that you get a code name for your ridiculous idea. I'm going to code name my criticism of you
Mongoose B
. In particular, the hierarchically recursive (reduced normal form)
Mongoose B
First of all, what solid will a group of 6 preons form if they were all to have the same distance from the center of such said group? A quick search in geometry will provide the answer: an
A quick look at the wikipedia page for the number 6 is all that's needed to solve this conundrum! Masterful work.
The same solid than a crystal of diamond will crystallize into. In other words, inside any given elementary particles, there will always be 6 preons forming an octahedron, with one preon at each
According to than
Mongoose B
theory, this argument crystallizes into you repeating your assumptions over and over again as if that counts as proof, in each elementary mistakes, at each apexes.
One could say that around the (for now assumed to be empty) center of a group-particle
For those not familiar with this technical terminology, a "center of a group-particle" is defined by the Oxford Encyclopedic Encyclopedia of Physics as:
dfn. Center of a Group-Particle
A term corresponding to the octahedronal apexes than a crystal of diamonds crystallizes into, vis-a-vi preons.
These positions are fixed relative to each other (except when decay occurs, where preons will be exchanged in a manner which is covered by Phoenix-I)
Let's not forget about the
Mongoose A
criticism that, no, it doesn't. Because drawing a picture with lines connecting it is not a Feynman diagram, and neglecting to do any actual math is not a calculation of a transition amplitude.
Additionally, one could in theory assign the numbers (1, 2, 3, 4, 5, and 6) to each of the preons inside of the octahedron.
This is a technical point--so those of you who are not professional physicists can skip this--but--correct me if I am wrong--I believe you are saying there are
of these made up particles. I'm not 100% sure I'm following here, this is getting awfully technical after all.
This allows only 3 rational states to be possible: apex touches Plane, vertices touches Plane, and whole face touches Plane.
I'll be honest here, I don't think you understand that an apex and a vertex is the same thing.
It is obvious the more preons are allowed to touch the plane, the more massive such a particle will become.
So obvious that no attempt at actual justification is needed.
In Phoenix-II's hypothesis, primeons, instead of being monopole-like particles, would in fact be bidirectional wave-like packets whose e-negative curve is aligned forward in time, with their
momentum direction. Their e-positive curve is facing the past.
Of course, this is after taking into account the
Mongoose B
manifesto, which clearly states that reversing the polarity of the nadeon emitters in the anti-time manifold of the time-space warp bubble will cause the matter stream to actually
reflect off of
the planet's atmosphere, resulting in two Commander Rikers materializing: one on the ship, and one on the planet.
a rendering of such a primeon would look like:
Which I have absolutely no doubt, that was created as the result of state-of-the-art particle physics simulation software available to actual researchers, and was not at all in any way produced in | {"url":"http://www.abovetopsecret.com/forum/thread981741/pg2","timestamp":"2014-04-20T04:14:16Z","content_type":null,"content_length":"80914","record_id":"<urn:uuid:ef091749-f707-4b55-82c6-749d739f8d34>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00482-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability Tensor and Ellipsoid
Hey, I asked this question in another post, but I realized I may have posted it in the wrong place:
So, is anyone familiar with the concept of a probability ellipsiod which represents a probability tensor whose principal components must sum to 1?
My question is why must they sum to one? Does this mean that the magnetic field (in this context) must be along one of those axes? | {"url":"http://www.physicsforums.com/showthread.php?t=179540","timestamp":"2014-04-19T04:35:00Z","content_type":null,"content_length":"19542","record_id":"<urn:uuid:b94d609e-4d31-4e28-b352-2fba2fe33617>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00226-ip-10-147-4-33.ec2.internal.warc.gz"} |
Jane X.
I have a master's degree in science and have been involved in teaching and tutoring in math and engineering for Eighteen years both in China and the United States. Recently I taught 8th grade Math,
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learning strategies.
I want to help students explore the fascinating math world and help their mind reason and organize complicated situations or problems into clear simple and logical steps. With the enhancement of
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ACT, SAT MATH LEVEL 1&2, Pre-AP Calculus\AP-Calculus and Physics.
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Nine Tips for Tackling Tough Numeracy Test Questions
Examiners on numeracy tests aren’t going to ask you anything impossible. Use these strategies and tactics and you should find yourself starting to make headway on questions you didn’t think you could
Read the question
Here’s a brutal fact: if you don’t understand what the question is asking for or what information you’ve been given, you can think as hard as you like and still not end up with the right answer.
Making sure you read the question carefully is the antidote to this. Rushing through the question and getting cracking on your work as quickly as possible is really tempting, especially when you
don’t have much time to play with.
Eliminate wrong answers
One of the most useful phrases in maths is ‘It can’t possibly be that!’ This is a phrase you can use in many multiple-choice exams to reduce the number of possible answers to a question.
Getting rid of wrong answers isn’t necessarily easy – you do need to read the question carefully and understand roughly what’s going on – but it can dramatically improve your chances of getting the
right answer.
You can dramatically improve your chances of picking the right answer by just taking a few seconds to think about what the right answer has to look like so you can out any answers that are definitely
Think of similar problems
One of the beautiful things about maths is that you can often apply the same approach to questions that seem completely unrelated – if you didn’t know better, would you have thought you could use the
same sums to convert miles to kilometres that you use to work out percentages?
Knowing that seemingly different kinds of problems often have similar solutions gives you a chance to draw analogies between questions when you feel stuck.
When you’re faced with a question you don’t recognise, think about what it reminds you of. More often than not, you can use the same maths for it – and it’s always better to do something plausible
than to guess.
Try the answers
In a multiple-choice exam, working backwards from the answers you’re given is sometimes easier than doing the sum ‘properly’. It’s a slightly sneaky trick, but that’s okay – the important thing in an
exam is to get the right answer whatever way you can.
This is particularly effective if you’ve already thrown out some answers that are obviously wrong so, rather than having to check five answers to see if they make sense, you may only have to consider
two or three.
If it’s not stated obviously, the question you have to ask is, ‘Does this answer fully satisfy what the question wanted?’ Only one option should answer the question completely.
Explain the question to yourself
You can’t really talk aloud in an exam, even to yourself, but you can talk through the question in your head.
Imagine you’re explaining your problem to a friend who doesn’t know anything about maths. What’s the simplest way you can state the question? What questions would they ask you?
Use a smart estimate
The words ‘estimate’ and ‘guess’ are sometimes used interchangeably, but they’re actually two different things: a guess is an answer you pick out of the air, while an estimate is a very rough answer
you work out. You guess which numbers are going to come up in the lottery, but you estimate how big the jackpot might be.
An estimate involves reading the question, and instead of doing all of the calculations exactly, doing them very roughly – perhaps rounding all of the numbers to one significant figure – to get a
rough idea of what the answer ought to be.
Come back to it later
Don’t spend too long on a question you find really hard to answer. There’s nothing (much) worse in an exam than finding that the last few questions look fairly easy, but you don’t have time to answer
them because you spent five minutes longer than you should have on an earlier question.
Break it down into smaller parts
Rather than trying to tackle a long exam question at once, break it down into smaller, more manageable parts. This kind of thinking can break a big, difficult question into three or four smaller,
easier questions that you can sail through.
The trick is to figure out what information you need for the last part of the question. Once you know what you need to find, think about what information you need to work that out. Keep working
backwards until you get to information you know or can work out easily, and you’ll end up with a plan for working through the question.
Guess it wildly
If time’s running out and you’ve got a minute to answer the last five multiple-choice questions, you don’t really have time to read the questions, let alone work out the answers. In this situation
you have two possible approaches:
• Miss out on the questions and get a guaranteed zero for those questions.
• Guess the answers and maybe pick up a few points.
Numeracy tests usually aren’t negatively marked, so you don’t lose points for giving a wrong answer. If you guess when you don’t have time or are genuinely stuck, the worst that can happen is that
you score no marks for that question. | {"url":"http://www.dummies.com/how-to/content/nine-tips-for-tackling-tough-numeracy-test-questio.navId-813991.html","timestamp":"2014-04-17T14:37:44Z","content_type":null,"content_length":"54079","record_id":"<urn:uuid:c6b5e2b4-889f-4489-b8a6-3ada3fd5f532>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00140-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Help
February 2nd 2009, 06:54 PM #1
Dec 2008
Hi, i hadd a problem y= ((8)/(x^3))-((6)/(x))
I need to know hoe the graph is symetric with respect to the
1)x axis
2)y axis
I just needed to know the analytical way to find it without looking at the graph
The graph of a function is symmetric with respect to:
- the y-axis if f(x) = f(-x) for all $x \in domain$
- the origin if f(x) = -f(-x) for all $x \in domain$
If a graph is symmetric to the x-axis it is not the graph of a function:
For every x-value you have pairs of y-values such that $y_1 = -y_2$
To your function: $f(x)=\dfrac8{x^3} - \dfrac6x = \dfrac{8-6x^2}{x^3}$
a) Symmetry to the y-axis:
$\dfrac{8-6x^2}{x^3} = \dfrac{8-6(-x)^2}{(-x)^3}$
$\dfrac{8-6x^2}{x^3} eq -\dfrac{8-6(x)^2}{(x)^3}$
That means: No symmetry to the y-axis.
b) Symmetry to the origin:
$\dfrac{8-6x^2}{x^3} = -\dfrac{8-6(-x)^2}{(-x)^3}$
$\dfrac{8-6x^2}{x^3} = \dfrac{8-6(x)^2}{(x)^3}$
That means: Symmetry to the origin.
February 2nd 2009, 10:35 PM #2 | {"url":"http://mathhelpforum.com/math-topics/71459-symetry.html","timestamp":"2014-04-20T23:41:55Z","content_type":null,"content_length":"32229","record_id":"<urn:uuid:805f8ffe-b639-4b94-bacf-31ab356ca43c>","cc-path":"CC-MAIN-2014-15/segments/1397609539337.22/warc/CC-MAIN-20140416005219-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
SPSSX-L archives -- July 2006 (#208)LISTSERV at the University of Georgia
Date: Fri, 14 Jul 2006 11:52:59 -0400
Reply-To: "Meyer, Gregory J" <gmeyer@UTNet.UToledo.Edu>
Sender: "SPSSX(r) Discussion" <SPSSX-L@LISTSERV.UGA.EDU>
From: "Meyer, Gregory J" <gmeyer@UTNet.UToledo.Edu>
Subject: Re: interrater agreement, intrarater agreement
Comments: To: Vassilis Hartzoulakis <hartzoulakis@ATH.FORTHNET.GR>,
Gene Maguin <emaguin@buffalo.edu>
In-Reply-To: A<001001c6a5e0$f15bf230$2345cd80@ssw.buffalo.edu>
Content-Type: text/plain; charset="us-ascii"
Building on Gene's suggestions, I would recommend you compute an
intraclass correlation using a oneway random effects design. SPSS uses
three different kinds of models (and for two way models gives the option
of computing a consistency ICC or an exact agreement ICC), each of which
have slightly different definitions of error. The oneway model is the
one that would be appropriate for you to assess interrater reliability.
An update of the classic Fleiss and Shrout (1978) article is this paper:
McGraw, K. O., & Wong, S. P. (1996). Forming inferences about some
intraclass correlation coefficients. Psychological Methods, 1, 30-46.
For the data you provided, the default syntax would be as given below,
though you can modify the confidence interval and null value. In the
output you would typically focus on the "Single Measures" reliability
rather than the Spearman-Brown estimate of "Average Measures"
/VARIABLES=rateA rateB
/SCALE ('ALL VARIABLES') ALL/MODEL=ALPHA
/ICC=MODEL(ONEWAY) CIN=95 TESTVAL=0 .
I am not certain how you would examine intrarater reliability with these
data. For this analysis the scorers would have to rate the same essay at
least twice. It doesn't look like this was done. However, if it was, you
would continue to have participants in the rows and a column for
IDsubject. In addition, you would want a column for IDrater, rateT1, and
rateT2, with T designating the time of the rating. For this design, you
could run a two way random effects design because the 2nd ratings are
always differentiated from the first. However, using the same model you
could also split the file by IDrater and generate reliability values for
each scorer. In order to obtain findings that parallel the findings from
the interrater analyses you would want to use the Absolute Agreement
coefficient rather than the Consistency coefficient. McGraw and Wong
discuss all these models and types.
Good luck,
| -----Original Message-----
| From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU]
| On Behalf Of Gene Maguin
| Sent: Wednesday, July 12, 2006 1:28 PM
| To: SPSSX-L@LISTSERV.UGA.EDU
| Subject: Re: interrater agreement, intrarater agreement
| Vassilis,
| I haven't see any other replies. Yes, I think your data is
| set up correctly.
| As shown you have it arranged in a multivariate ('wide')
| setup. From there
| you can do a repeated measures anova or use reliability.
| However, I think
| there are a number of different formulas to use depending on
| whether you
| have the same raters rating everybody, or, as you have, two raters are
| randomly selected to rate each person. I'd bet anything the
| computational
| formulas are different and I'll bet almost anything that spss can't
| accommodate both. There's a literature on rater agreement and
| on intraclass
| correlation. If you haven't looked at that, you should.
| However, I can't
| help you on that. One thing you might do is google
| 'intraclass correlation'
| and there's a citation in the top 20 or 30 that references a
| book by, I
| think, Fleiss (or Fliess) and Shrout. Another term to google
| is 'kappa'
| (which is available from spss crosstabs).
| I'm hoping that you have other responses that are more
| helpful than I am
| able to be.
| Gene Maguin
-----Original Message-----
From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU] On Behalf Of
Vassilis Hartzoulakis
Sent: Wednesday, July 12, 2006 7:16 AM
To: SPSSX-L@LISTSERV.UGA.EDU
Subject: interrater agreement, intrarater agreement
Hi everyone
I have a dataset of 10000 subjects and their scores on a composition
wrote. Each composition was scored by 2 different raters (randomly
from a pool of 70). The scores could range from 0 to 15.
So far I have set up a table with 10000 rows/cases and 5 columns
IDraterA, rateA, IDraterB, rateB)
00001, 1200, 12, 1300, 14 (the 1st rater gave a 12/15 and the 2nd a
00002, 1200, 09, 1300, 12
00003, 1400, 15, 1200, 13
00004, 1400, 02, 1200, 08 etc.
Can someone suggest the best possible layout and analysis to investigate
inter and intra rater agreement?
Thank you | {"url":"http://listserv.uga.edu/cgi-bin/wa?A2=ind0607&L=spssx-l&F=&S=&P=22651","timestamp":"2014-04-19T12:02:06Z","content_type":null,"content_length":"13572","record_id":"<urn:uuid:60c633e1-d879-44d8-a45c-bd37e1a36052>","cc-path":"CC-MAIN-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00161-ip-10-147-4-33.ec2.internal.warc.gz"} |
Manipulating differential forms: proving the Quotient Rule
November 20th 2010, 07:51 AM #1
Manipulating differential forms: proving the Quotient Rule
I am trying to prove the differential form 'Quotient Rule'. Can anyone check I am going rigth so far. I am not sure what to do on next step...
Follow Math Help Forum on Facebook and Google+ | {"url":"http://mathhelpforum.com/calculus/163831-manipulating-differential-forms-proving-quotient-rule.html","timestamp":"2014-04-19T02:12:11Z","content_type":null,"content_length":"29712","record_id":"<urn:uuid:641e2eb2-6477-4f6e-bcff-2c0df8fad7ca>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00318-ip-10-147-4-33.ec2.internal.warc.gz"} |
Naturally trendy?Naturally trendy?
Naturally trendy?
From time to time, there is discussion about whether the recent warming trend is due just to chance. We have heard arguments that so-called ‘random walk‘ can produce similar hikes in temperature (any
reason why the global mean temperature should behave like the displacement of a molecule in Brownian motion?). The latest in this category of discussions was provided by Cohn and Lins (2005), who in
essence pitch statistics against physics. They observe that tests for trends are sensitive to the expectations, or the choice of the null-hypothesis .
Cohn and Lins argue that long-term persistence (LTP) makes standard hypothesis testing difficult. While it is true that statistical tests do depend on the underlying assumptions, it is not given that
chosen statistical models such as AR (autoregressive), ARMA, ARIMA, or FARIMA do provide an adequate representation of the null-distribution. All these statistical models do represent a type of
structure in time, be it as simple as a serial correlation, persistence, or more complex recurring patterns. Thus, the choice of model determines what kind of temporal pattern one expects to be
present in the process analysed. Although these models tend to be referred to as ‘stochastic models’ (a random number generator is usually used to provide underlying input for their behaviour), I
think this is a misnomer, and think that the labels ‘pseudo-stochastic’ or ‘semi-stochastic’ are more appropriate. It is important to keep in mind that these models are not necessarily representative
of nature – just convenient models which to some degree mimic the empirical data. In fact, I would argue that all these models are far inferior compared to the general circulation models (GCMs) for
the study of our climate, and that the most appropriate null-distributions are derived from long control simulations performed with such GCMs. The GCMs embody much more physically-based information,
and do provide a physically consistent representation of the radiative balance, energy distribution and dynamical processes in our climate system. No GCM does suggest a global mean temperature hike
as observed, unless an enhanced greenhouse effect is taken into account. The question whether the recent global warming is natural or not is an issue that belongs in ‘detection and attribution’ topic
in climate research.
| Next page | {"url":"http://www.realclimate.org/index.php/archives/2005/12/naturally-trendy/?wpmp_switcher=mobile&attest=true&wpmp_tp=0","timestamp":"2014-04-18T00:22:05Z","content_type":null,"content_length":"13789","record_id":"<urn:uuid:c63e8f79-4aee-43d8-a050-34f8e833b0ec>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00325-ip-10-147-4-33.ec2.internal.warc.gz"} |
R-statistics blog
This post is a call for both R community members and R-bloggers, to come and help make The R Programming wikibook be amazing.
The R Programming wikibook is not just another one of the many free books about statistics/R, it is a community project which aims to create a cross-disciplinary practical guide to the R programming
language. Here is how you can join:
Call for proposals for writing a book about R (via Chapman & Hall/CRC)
Rob Calver wrote an interesting invitation on the R mailing list today, inviting potential authors to submit their vision of the next great book about R. The announcement originated from the Chapman
& Hall/CRC publishing houses, backed up by an impressive team of R celebrities, chosen as the editors of this new R books series, including:
Bellow is the complete announcement:
Continue reading
New edition of “R Companion to Applied Regression” – by John Fox and Sandy Weisberg
Just two hours ago, Professor John Fox has announced on the R-help mailing list of a new (second) edition to his book “An R and S Plus Companion to Applied Regression”, now title . “An R Companion to
Applied Regression, Second Edition”.
John Fox is (very) well known in the R community for many contributions to R, including the car package (which any one who is interested in performing SS type II and III repeated measures anova in R,
is sure to come by), the Rcmdr pacakge (one of the two major GUI’s for R, the second one is Deducer), sem (for Structural Equation Models) and more. These might explain why I think having him
release a new edition for his book to be big news for the R community of users.
In this new edition, Professor Fox has teamed with Professor Sandy Weisberg, to refresh the original edition so to cover the development gained in the (nearly) 10 years since the first edition was
Here is what John Fox had to say:
Dear all,
Sandy Weisberg and I would like to announce the publication of the second
edition of An R Companion to Applied Regression (Sage, 2011).
As is immediately clear, the book now has two authors and S-PLUS is gone
from the title (and the book). The R Companion has also been thoroughly
rewritten, covering developments in the nearly 10 years since the first
edition was written and expanding coverage of topics such as R graphics and
R programming. As before, however, the R Companion provides a general
introduction to R in the context of applied regression analysis, broadly
construed. It is available from the publisher at (US) or (UK), and from Amazon (see here)
The book is augmented by a web site with data sets, appendices on a variety of topics, and more, and it associated with the car package on CRAN, which has recently undergone an overhaul.
John and Sandy | {"url":"http://www.r-statistics.com/tag/r-books/","timestamp":"2014-04-17T15:38:44Z","content_type":null,"content_length":"45077","record_id":"<urn:uuid:54e5edbf-0542-4556-a707-e2d4bf7c7b19>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00587-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Is the earth older than 6000 years?
You asked:
Is the earth older than 6000 years?
Earth, the third planet from the Sun in the Solar System, home to many life forms, including human beings
Assuming you meant
• Earth, the third planet from the Sun in the Solar System, home to many life forms, including human beings
Did you mean?
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Wolfram Demonstrations Project
Hofstadter's Quantum-Mechanical Butterfly
This Demonstration shows the quantum-mechanical energy spectrum of an electron in a two-dimensional periodic potential with a perpendicular magnetic field. The fractal nature of this system was
discovered by Douglas Hofstadter in 1976.
The Schrödinger equation for this system takes the form
where is a pseudodifferential operator, is the potential, and is the energy eigenvalue, with given by the ansatz
for integers .
The problem reduces to the solution of the recursive equation
where is the energy, , with being the denominator of nonrepeating rational numbers . Finally, the eigenvalue condition for the energy spectrum is
The plot has energy on the axis and the parameter on the axis.
[1] D. Hofstadter, "Energy Levels and Wave Functions of Bloch Electrons in Rational and Irrational Magnetic Fields,"
Physical Review B
, 1976 pp. 2239–2249. doi: | {"url":"http://demonstrations.wolfram.com/HofstadtersQuantumMechanicalButterfly/","timestamp":"2014-04-21T09:43:18Z","content_type":null,"content_length":"45802","record_id":"<urn:uuid:ff766b60-a59c-4b99-b4e4-f4b4ac3396d8>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00602-ip-10-147-4-33.ec2.internal.warc.gz"} |
Problem multiplying rotation matrices together
02-26-2003 #1
Join Date
Jan 2003
Problem multiplying rotation matrices together
I am using this for a game, but I didn't put it in the game forum because more people view GD. Plus it's math oriented, therefore you don't have to be into game programming to possibly help me.
Anyway, from what I understand from OpenGL Game Programming you can multiply the Z axis rotation matrix by the Y axis rotation by the X axis rotation. It comes right out and says the final
equations(the 'concatenation' of the three matrices), but I'm not exactly sure how they got there. I'm going to post the three matrices and then the final equations. Note the delta means the
degrees travelled about that axis only. If I made any mistakes please point them out, it was a lot to copy. I tried making this as neat as possible, but the matrices keep being posted unaligned.
Z axis rotation matrix:
cosDelta -sinDelta 0 0
sinDelta cosDelta 0 0
Y axis rotation matrix;
cosDelta 0 sinDelta 0
-sinDelta 0 cosDelta 0
X axis rotation matrix:
0 cosDelta -sinDelta 0
0 sinDelta cosDelta 0
Here are the final equation describing how the X image, Y image and Z image are related to the concatenation of the 3 matrices above
X' = X[(cosZDelta) * (cosYDelta)] +
Y[cosZDelta) * (sinYDelta) * (sinXDelta) - (sinZDelta) *(cosXDelta] +
Z[(cosZDelta) *(sinYDelta) * (sinYDelta) - (sinZDelta) * (cosXDelta)]
Y' = X[(sinZDelta) * (cosYDelta)]+
Y[(sinZDelta) * (sinYDelta) * (sinXDelta) - (cosZDelta) * (cosXDelta)+
Z[(sinZDelta) *(sinYDelta) * (cosXDelta) - (cosZDelta) * (sinXDelta)]
Z' = -x*(sin?) + z*(cosYDelta)*(sinXDelta)]
I'm not sure if the question mark is a mistake from the book or not, unfortunately I suspect it is, but when I know what I'm doing I'll be able to figure out what it's supposed to be.
As I said I do not know how they multiplied the first three matrices together to get the final equations. I do know how to work with matrices, i.e the image matix will have the number of rows
from the first matrix and the number of columns from the second matrix. I know how to multiply two matrices together (Matrix1Row[0][0] * Matrix2Column[0][0] add those together and continue until
first column is filled). Anyay if anyone can shed some light onto what is going on I will be very happy.
Last edited by Silvercord; 02-26-2003 at 09:56 AM.
Re: Problem multiplying rotation matrices together
Originally posted by Silvercord
I am using this for a game, but I didn't put it in the game forum because more people view GD. Plus it's math oriented, therefore you don't have to be into game programming to possibly help me.
Anyway, from what I understand from OpenGL Game Programming you can multiply the Z axis rotation matrix by the Y axis rotation by the X axis rotation. It comes right out and says the final
equations(the 'concatenation' of the three matrices), but I'm not exactly sure how they got there. I'm going to post the three matrices and then the final equations. Note the delta means the
degrees travelled about that axis only. If I made any mistakes please point them out, it was a lot to copy. I tried making this as neat as possible, but the matrices keep being posted unaligned.
Z axis rotation matrix:
cosDelta -sinDelta 0 0
sinDelta cosDelta 0 0
Y axis rotation matrix;
cosDelta 0 sinDelta 0
-sinDelta 0 cosDelta 0
X axis rotation matrix:
0 cosDelta -sinDelta 0
0 sinDelta cosDelta 0
Here are the final equation describing how the X image, Y image and Z image are related to the concatenation of the 3 matrices above
X' = X[(cosZDelta) * (cosYDelta)] +
Y[cosZDelta) * (sinYDelta) * (sinXDelta) - (sinZDelta) *(cosXDelta] +
Z[(cosZDelta) *(sinYDelta) * (sinYDelta) - (sinZDelta) * (cosXDelta)]
Y' = X[(sinZDelta) * (cosYDelta)]+
Y[(sinZDelta) * (sinYDelta) * (sinXDelta) - (cosZDelta) * (cosXDelta)+
Z[(sinZDelta) *(sinYDelta) * (cosXDelta) - (cosZDelta) * (sinXDelta)]
Z' = -x*(sin?) + z*(cosYDelta)*(sinXDelta)]
I'm not sure if the question mark is a mistake from the book or not, unfortunately I suspect it is, but when I know what I'm doing I'll be able to figure out what it's supposed to be.
As I said I do not know how they multiplied the first three matrices together to get the final equations. I do know how to work with matrices, i.e the image matix will have the number of rows
from the first matrix and the number of columns from the second matrix. I know how to multiply two matrices together (Matrix1Row[0][0] * Matrix2Column[0][0] add those together and continue until
first column is filled). Anyay if anyone can shed some light onto what is going on I will be very happy.
Remember that although matrix multiplication is not commutative, it is associative, so it doesn't matter which two matrices you multiply together first in the above example, as long as the
overall order is kept.
So, you want to create a composite matrix, which has the effect of a rotation about Z, then a rotation about Y, then a rotation about X. Remember that the non-commutativity of matrix
multiplication still stands with rotations; the fact that the result of combining rotations depends on the order in which you perform them is a geometric fact about rotations. It is easy to see
that the non-commutativity of rotations is a geometric fact. Just hold a non-cubical rectangular box in your hands and try rotating it by 90 degrees successively about two different axes, note
the resulting orientation, then return it to its original orientation and perform the same two rotations but in the opposite order. Therefore, we must multiply the three matrices (we will call
them Z, Y and X respectively) in the order ZYX. But due to matrix associativity, it doesn't matter whether we evaluate ZY first or YX first. If you evaluate ZY first (call it P), then simply
multiply that by X, (in the order PX) to get the final composite matrix. If you evaluate YX first (call it G), then simply multiply that by Z, (in the order ZG) to get the final composite matrix.
By the way, your definitions for each individual rotation matrix are correct, and I'm sure you can now mulitply them together as explained to check the last bit.
P.S. You've used 'delta' for your angle variable in all of the above rotations, I don't know whether or not you intended to do this, as it would obviously only allow a rotation of the same amount
around each axis.
I hope this helps,
Last edited by rahaydenuk; 02-26-2003 at 11:50 AM.
Richard Hayden. (rahaydenuk@yahoo.co.uk)
Webmaster: http://www.dx-dev.com
DXOS (My Operating System): http://www.dx-dev.com/dxos
PGP: 0x779D0625
That clears things up quite a bit, thank you very much!!! When I get this temporary matrix P do I still follow standard matrix multiplication (rows of P multiplied by columns of X or Z). Is it
incorrect for ZX to be P and then multiply by Y?(it seems that it is incorrect to do that). These equations are eventually going to be used for changing the view vectors of the camera in a
program I'm working on, therefore do I even need the rotation about the Z axis matrix?
I meant for the deltas to be completely different variables
Originally posted by Silvercord
That clears things up quite a bit, thank you very much!!! When I get this temporary matrix P do I still follow standard matrix multiplication (rows of P multiplied by columns of X or Z).
If we are defining P = ZY to start as previously, then to get the composite matrix ZYX, you will need to multiply P by X on the right, i.e. PX as this is then equal to (ZY)X = ZYX.
If we are instead defining another matrix G = YX to start, then we need to multiply by Z on the left, i.e. ZG as this is then equal to Z(YX) = ZYX.
Remember that due to the non-commutativity of matrix multiplication, you need to take into account whether you are multiplying one matrix by another on the left or the right. They will (in
general) produce different results.
All of these individual matrix multiplication operations are no different to what they were when you were just multiplying two matrices together. For example, to evaluate ZYX, take P = ZY and
evaluate ZY just as standard matrix multiplication (i.e. rows of Z by columns of Y), then multiply the resultant matrix P by X to get PX in the standard fashion (i.e. rows of P by columns of X).
Originally posted by Silvercord
Is it incorrect for ZX to be P and then multiply by Y?(it seems that it is incorrect to do that).
We are trying to find ZYX, and if we were to define P = ZX, multiplying either side of P (ZX) by Y would produce either YZX or ZXY, which is not equal (in general) to ZYX for matrices, due to the
non-commutativity of matrix multiplication. So, yes this would be incorrect; you should start either by evaluating YX or ZY and then multiplying by the other matrix (the one left out) on the
correct side.
Originally posted by Silvercord
These equations are eventually going to be used for changing the view vectors of the camera in a program I'm working on, therefore do I even need the rotation about the Z axis matrix?
I'm not a game/graphics-programming expert, but I've done a bit of DirectX. If we assume that in camera space, the camera, or viewer, is at the origin, looking in the z-direction, as in DirectX,
I can't immediately see a reason for wanting to have a rotation about the Z-axis in the view matrix transformation. I can see a rotation about the X-axis acting as adjustment of pitch (i.e.
looking up and down), and then a rotation about the Y-axis as the adjustment for the camera rotating around, as if the user has turned his head, but unless you get complicated, then I see no
*common* use for including a Z-axis rotation component in the view matrix transformation (obviously, you would probably need it for flight simulators etc.).
Having said this, you should remember however that the view matrix, V is generally defined as a combination of a translation matrix, T and then the three rotational component matrices, X, Y and Z
about their three respective co-ordinate axis, (V = T*X*Y*Z), so officially, it does include a rotational component about the Z-axis. I wouldn't worry, though if you find yourself not needing it.
Originally posted by Silvercord
I meant for the deltas to be completely different variables
That's what I thought.
Hope this helps,
Last edited by rahaydenuk; 02-26-2003 at 04:43 PM.
Richard Hayden. (rahaydenuk@yahoo.co.uk)
Webmaster: http://www.dx-dev.com
DXOS (My Operating System): http://www.dx-dev.com/dxos
PGP: 0x779D0625
hi there, sorry I didn't get a chance to respond last night. All of your answers and explanations were very concise therefore I do not currently have anymore questions (that's a good thing). I'm
doing this for a game, so I should have this working within a few days. I can post it when I get it working if you want me to. I think I might try using this matrix (the concatenation of the Z Y
and X rotation matrices) as well as the rotation about an arbitrary axis matrix.
All right, take care, talk to you later.
Originally posted by Silvercord
hi there, sorry I didn't get a chance to respond last night. All of your answers and explanations were very concise therefore I do not currently have anymore questions (that's a good thing). I'm
doing this for a game, so I should have this working within a few days. I can post it when I get it working if you want me to. I think I might try using this matrix (the concatenation of the Z Y
and X rotation matrices) as well as the rotation about an arbitrary axis matrix.
All right, take care, talk to you later.
OK, glad I could help!
I'd be interested in seeing your game, when you've got it working and I'm sure others would be too; I'll look forward to taking a look.
Have fun!
Richard Hayden. (rahaydenuk@yahoo.co.uk)
Webmaster: http://www.dx-dev.com
DXOS (My Operating System): http://www.dx-dev.com/dxos
PGP: 0x779D0625
I have thought of more questions. I am trying to figure out how these matrices are derived but I honestly don't get it 100%. I am using the Y axis rotation matrix for this example. I am drawing
pictures of a triangle with the hypotenuse as the view vector. I am then trying to figure out why the new vector's X component is equal to the cos(theta) multiplied by the original x plus the sin
(theta) multiplied by the original z component (X' = cos(theta) * X + sin(theta) * z). Likewise I do not know how the calculation for the new Z component is derived (-sin(theta) * x + cos(theta)
* z). If you could explain this using properties of triangles and vectors I will probably have a much better understanding of what is going on here. I want to get a very fundamental understanding
of this.
Originally posted by Silvercord
I have thought of more questions. I am trying to figure out how these matrices are derived but I honestly don't get it 100%. I am using the Y axis rotation matrix for this example. I am drawing
pictures of a triangle with the hypotenuse as the view vector. I am then trying to figure out why the new vector's X component is equal to the cos(theta) multiplied by the original x plus the sin
(theta) multiplied by the original z component (X' = cos(theta) * X + sin(theta) * z). Likewise I do not know how the calculation for the new Z component is derived (-sin(theta) * x + cos(theta)
* z). If you could explain this using properties of triangles and vectors I will probably have a much better understanding of what is going on here. I want to get a very fundamental understanding
of this.
Deriving the rotation matrices is quite a complicated topic. There are quite a few ways to go about it, including by differential equations or by looking at vector components. The following
website gives a couple of different derivations of Rodriques' Formula, which is used to find rotation matrices:
I hope this helps,
Richard Hayden. (rahaydenuk@yahoo.co.uk)
Webmaster: http://www.dx-dev.com
DXOS (My Operating System): http://www.dx-dev.com/dxos
PGP: 0x779D0625
I decided to skip out using the z axis rotation, because it does not seem like it will be used in my game (how can you even have a Z component in camera movement when you are checking for
mousemovement in 2d screen coords?). I haven't multiplied matrices together in quite a while, but when I multiplied the Y axis rotation matrix by the X axis rotation matrix I got this matrix. I
am pretty sure it is correct but I would be happy if you could check it for errors (better safe than sorry before putting it into code)
concatenation between Y and X axis rotation matrices:
cosYDelta sinYDelta + sinXDelta sinYDelta * cosXDelta
0 cosXDelta -sinXDelta
-sinYDelta cosYDelta * sinXDelta cosYDelta * cosXDelta
the final equations (what it will look like in code because I am not using a matrix class)
X' = X * cosYDelta +
Y * sinYDelta * sinXDelta +
Z * sinYDelta * cosXDelta
Y' = Y * cosXDelta +
Z * -sinXDelta
Z' = X * -sinYDelta +
Y * cosYDelta * sinXDelta +
Z * cosYDelta * cosXDelta
Originally posted by Silvercord
I decided to skip out using the z axis rotation, because it does not seem like it will be used in my game (how can you even have a Z component in camera movement when you are checking for
mousemovement in 2d screen coords?).
Well you're right, you can only have rotation about two axis, as the mouse only has two axis of movement. Flight simulators etc., which would require rotation about the z-axis (for barrel-rolls
etc.) would usually have the user press a mouse button to switch one of the mouse's co-ordinate axis to representing the z-axis rotation, or it would be done by keyboard.
Originally posted by Silvercord
I haven't multiplied matrices together in quite a while, but when I multiplied the Y axis rotation matrix by the X axis rotation matrix I got this matrix. I am pretty sure it is correct but I
would be happy if you could check it for errors (better safe than sorry before putting it into code)
concatenation between Y and X axis rotation matrices:
cosYDelta sinYDelta + sinXDelta sinYDelta * cosXDelta
0 cosXDelta -sinXDelta
-sinYDelta cosYDelta * sinXDelta cosYDelta * cosXDelta
the final equations (what it will look like in code because I am not using a matrix class)
X' = X * cosYDelta +
Y * sinYDelta * sinXDelta +
Z * sinYDelta * cosXDelta
Y' = Y * cosXDelta +
Z * -sinXDelta
Z' = X * -sinYDelta +
Y * cosYDelta * sinXDelta +
Z * cosYDelta * cosXDelta
One little thing, I find it very confusing to read your matrix, maybe you could attach an image next time?
From what I can see, it looks like you've made one typo, but apart from that it's fine. I've attached the correct product matrix for the y-axis rotation matrix multiplied by the x-axis rotation
matrix (in that order) as an image for readability, so you can see where you made the mistake (you've used a '+' instead of a '*' for one element of the product matrix).
In order to get the equations for x', y' and z', we need to use the product matrix we just found to transform the general x, y, z vector. This operation is shown in the aforementioned image file
attached to this message.
From looking at this image file, we can ascertain the following equations for x', y' and z' (where 'a' is the number of degrees to rotate about the y-axis and 'b' is the number of degrees to
rotate about the x-axis (counter-clockwise)):
x' = xCos(a) + ySin(a)Sin(b) + zSin(a)Cos(b)
y' = yCos(b) - zSin(b)
z' = -xSin(a) + yCos(a)Sin(b) + zCos(a)Cos(b)
These are identical to what you found, thus implying that your earlier mistake in multiplying the matrices was most likely simply a typo.
I hope this helps,
Richard Hayden. (rahaydenuk@yahoo.co.uk)
Webmaster: http://www.dx-dev.com
DXOS (My Operating System): http://www.dx-dev.com/dxos
PGP: 0x779D0625
Yes that is identical to what I had myself. Sorry about the way I posted my matrices, if I have to do it again I'll make an image for you so it's easier to read. You have been really really
really (^infinity) helpful. I was surprised that you replied so fast and you really know what you're talking about
I haven't got this working yet (doesn't seem to do the rotations correctly) but I am trying to plug it into code right now. Like I said I'll upload the results.
Thanks, talk to you later.
Originally posted by Silvercord
Yes that is identical to what I had myself. Sorry about the way I posted my matrices, if I have to do it again I'll make an image for you so it's easier to read. You have been really really
really (^infinity) helpful. I was surprised that you replied so fast and you really know what you're talking about
I haven't got this working yet (doesn't seem to do the rotations correctly) but I am trying to plug it into code right now. Like I said I'll upload the results.
Thanks, talk to you later.
OK, good luck, I'll look forward to seeing it!
If you need any more assistance, don't hesitate to ask!
One thing, which could be causing problems... ensure that you are feeding the trig functions the correct format argument (i.e. check whether you should be using degrees or radians and if you're
using the correct one), this can be a very elusive bug, if you haven't met/aren't used to using radians.
Richard Hayden. (rahaydenuk@yahoo.co.uk)
Webmaster: http://www.dx-dev.com
DXOS (My Operating System): http://www.dx-dev.com/dxos
PGP: 0x779D0625
Yes I had already implemented rotating the view vector and camera position (third person) as well as the actual model about the y axis. Obviously this involves both radians and degrees (radians
for rotating the views, degrees for rotating the model). Plus we are doing radians heavily in math right now, so I'm really all set with that
I am trying to plug this into code. I have been trying to get this working for 40 mins or so but this just does not work. Is there any problem with order here? Should I add any parenthesis
anywhere? This really should be working, but it doesn't.
Dir = View - Position; //this is the direction
RotDir.x = (Dir.x * cosYTheta) + (Dir.y * sinYTheta * sinXTheta) + (Dir.z * sinYTheta * cosXTheta);
RotDir.y = (Dir.y * cosXTheta) - (Dir.z * sinXTheta);
RotDir.z = (Dir.x * -sinYTheta) + (Dir.y * cosYTheta * sinXTheta) + (Dir.z * cosYTheta * cosXTheta);
EDIT: this is how I derived the number of degrees and radians to perform the rotations. Everything worked for rotations about just the Y axis.
YRotation += ((360 * XDeviation) / SCREEN_WIDTH);
XRotation += ((360 * YDeviation) / SCREEN_WIDTH);
float YDeviation = (MiddleY - MousePos.y) / 2;
float XDeviation = (MiddleX - MousePos.x) / 2;
float XDegRad = ((2 * PI) * XDeviation) / SCREEN_WIDTH;
float YDegRad = ((2 * PI) * YDeviation) / SCREEN_WIDTH;
float cosXTheta = (float)cos(XDegRad);
float sinXTheta = (float)sin(XDegRad);
float cosYTheta = (float)cos(YDegRad);
float sinYTheta = (float)sin(YDegRad);
Last edited by Silvercord; 02-28-2003 at 08:18 AM.
Originally posted by Silvercord
Yes I had already implemented rotating the view vector and camera position (third person) as well as the actual model about the y axis. Obviously this involves both radians and degrees (radians
for rotating the views, degrees for rotating the model). Plus we are doing radians heavily in math right now, so I'm really all set with that
I am trying to plug this into code. I have been trying to get this working for 40 mins or so but this just does not work. Is there any problem with order here? Should I add any parenthesis
anywhere? This really should be working, but it doesn't.
RotDir.x = (Dir.x * cosYTheta) + (Dir.y * sinYTheta * sinXTheta) + (Dir.z * sinYTheta * cosXTheta);
RotDir.y = (Dir.y * cosXTheta) - (Dir.z * sinXTheta);
RotDir.z = (-Dir.x * sinYTheta) + (Dir.y * cosYTheta * sinXTheta) + (Dir.z * cosYTheta * cosXTheta);
EDIT: this is how I derived the number of degrees and radians to perform the rotations. Everything worked for rotations about just the Y axis.
float YDeviation = (MiddleY - MousePos.y) / 2;
float XDeviation = (MiddleX - MousePos.x) / 2;
float XDegRad = ((2 * PI) * XDeviation) / SCREEN_WIDTH;
float YDegRad = ((2 * PI) * YDeviation) / SCREEN_WIDTH;
float cosXTheta = (float)cos(XDegRad);
float sinXTheta = (float)sin(XDegRad);
float cosYTheta = (float)cos(YDegRad);
float sinYTheta = (float)sin(YDegRad);
What do MiddleY and MiddleX represent?
Richard Hayden. (rahaydenuk@yahoo.co.uk)
Webmaster: http://www.dx-dev.com
DXOS (My Operating System): http://www.dx-dev.com/dxos
PGP: 0x779D0625
MiddleX and MiddleY are the screen width and height divided by two. it is the center of the screen in screen coords. note that when using getcursorpos and setcursorpos those screen coordinates
are setup like a regular cartesian system. the point 0,0 is actually in the middle of the screen. for everything else I think the positive y goes down the screen and 0, 0 is the top left corner.
SetCursorPos(MiddleX, MiddleY); //sets cursor to exact middle of screen
static float MiddleX = SCREEN_WIDTH / 2;
static float MiddleY = SCREEN_HEIGHT / 2;
SCREEN_WIDTH and SCREEN_HEIGHT are #define'd compiler directives passed into the function that sets up the resolution.
If there is anything else that I didn't explain good enough please ask me.
#define SCREEN_WIDTH 1024
#define SCREEN_HEIGHT 768
Last edited by Silvercord; 02-28-2003 at 08:22 AM.
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Lara Turner
The shortest path problem in which the \((s,t)\)-paths \(P\) of a given digraph \(G =(V,E)\) are compared with respect to the sum of their edge costs is one of the best known problems in
combinatorial optimization. The paper is concerned with a number of variations of this problem having different objective functions like bottleneck, balanced, minimum deviation, algebraic sum, \
(k\)-sum and \(k\)-max objectives, \((k_1, k_2)-max, (k_1, k_2)\)-balanced and several types of trimmed-mean objectives. We give a survey on existing algorithms and propose a general model for
those problems not yet treated in literature. The latter is based on the solution of resource constrained shortest path problems with equality constraints which can be solved in pseudo-polynomial
time if the given graph is acyclic and the number of resources is fixed. In our setting, however, these problems can be solved in strongly polynomial time. Combining this with known results on \
(k\)-sum and \(k\)-max optimization for general combinatorial problems, we obtain strongly polynomial algorithms for a variety of path problems on acyclic and general digraphs.
Asymptotic Order of the Parallel Volume Difference (2012)
Jürgen Kampf
In this paper we investigate the asymptotic behaviour of the parallel volume of fixed non-convex bodies in Minkowski spaces as the distance \(r\) tends to infinity. We will show that the
difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order \(r^{d-2}\) in a \(d\)-dimensional space. Then we will show that
in Euclidean spaces this difference can at most have order \(r^{d-3}\). These results have several applications, e.g. we will use them to compute the derivative of \(f_\mu(rK)\) in \(r = 0\),
where \(f_\mu\) is the Wills functional or a similar functional, \(K\) is a body and \(rK\) is the Minkowski-product of \(r\) and \(K\). Finally we present applications concerning Brownian paths
and Boolean models and derive new proofs for formulae for intrinsic volumes.
Asymptotic order of the parallel volume difference (2011)
Jürgen Kampf
In this paper we continue the investigation of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity that was started in [13]. We will show that the
difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order \(r^{d-2}\) in dimension \(d\). Then we will show that in the
Euclidean case this difference can at most have order \(r^{d-3}\). We will also examine the asymptotic behavior of the derivative of this difference as the distance tends to infinity. After this
we will compute the derivative of \(f_\mu (rK)\) in \(r\), where \(f_\mu\) is the Wills functional or a similar functional, \(K\) is a fixed body and \(rK\) is the Minkowski-product of \(r\) and
\(K\). Finally we will use these results to examine Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.
A uniform central limit theorem for neural network based autoregressive processes with applications to change-point analysis (2011)
Claudia Kirch Joseph Tadjuidje Kamgaing
We consider an autoregressive process with a nonlinear regression function that is modeled by a feedforward neural network. We derive a uniform central limit theorem which is useful in the
context of change-point analysis. We propose a test for a change in the autoregression function which - by the uniform central limit theorem - has asymptotic power one for a large class of
alternatives including local alternatives. | {"url":"https://kluedo.ub.uni-kl.de/solrsearch/index/search/searchtype/series/id/16168/start/0/rows/10/doctypefq/preprint/languagefq/eng","timestamp":"2014-04-19T05:16:06Z","content_type":null,"content_length":"47754","record_id":"<urn:uuid:1e585ecd-1cce-483a-903d-85a5ca233d20>","cc-path":"CC-MAIN-2014-15/segments/1397609535775.35/warc/CC-MAIN-20140416005215-00655-ip-10-147-4-33.ec2.internal.warc.gz"} |
Newcastle, WA Geometry Tutor
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Polar Plots
September 25th 2011, 04:41 AM
Polar Plots
In my college notes, a few steps are skipped leaving meaning there's a lot of confusion as to where the figures are coming from.
I would like to know how to calculate where a polar plot intersects the unit circle?
September 25th 2011, 04:42 AM
Prove It
Re: Polar Plots
September 25th 2011, 04:50 AM
Re: Polar Plots
Thank you for the reply, that was quick.
Unfortunately I don't have access to my lecture notes to have a look at an example problem, I just asked the question from memory. Do you know of any resource that would show a simple example
with solution as I'd like to apply the solution you suggested.
September 25th 2011, 05:28 AM
Re: Polar Plots | {"url":"http://mathhelpforum.com/calculus/188771-polar-plots-print.html","timestamp":"2014-04-17T16:54:45Z","content_type":null,"content_length":"5856","record_id":"<urn:uuid:a199ab5d-cd88-4187-b542-d1f04d2add64>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00379-ip-10-147-4-33.ec2.internal.warc.gz"} |
Center City, PA Geometry Tutor
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analytics geometry
September 23rd 2012, 07:25 AM #1
Sep 2012
analytics geometry
can help me solve this question ? thanks much !
show that ,for all values of p ,the point p given by x=ap^2 ,y=2ap lies on the curve y^2=4ax .
a)find the equation of the ormal to this curve at the point p.
If this normal meets the curve at the point Q (q^2,2aq) , show that p^2 +pq+2=0 .
b)determine the coordinates of R ,the point of intersection of the tangents of the curve at the point p and Q .
hence ,show that the line locus of the pint R is y^2(x+2a)+4a^3=0 .
thanks .....
Re: analytics geometry
How far have you gotten?
Re: analytics geometry
Have you any ideas about how you might eliminate the parameter p to obtain the Cartesian equation?
Re: analytics geometry
Re: analytics geometry
Okay, I have worked the problem, but you MUST show some effort here before we can proceed. Also, I found a few typos in the problem statement. Here is how I feel the problem could be stated:
Show that, for all values of $p$, the point $P(x,y)=\left(ap^2,2ap \right)$ lies on the curve $y^2=4ax$.
a) Find the equation of the line normal to the curve at $P$.
If this normal line also crosses the curve at $Q(x,y)=\left(aq^2,2aq \right)$, show that $p^2+pq+2=0$.
b) Determine the coordinates of $R$ ,the point of intersection of the tangents of the curve at the points $P$ and $Q$.
Hence ,show that the line locus of the point $R$ is $y^2(x+2a)+4a^2=0$.
I will be more than happy to help, but if you have no idea how (or any inclination to try) to eliminate the parameter $p$ to obtain the Cartesian equation, then I am really at a loss to help.
I will give you a nudge to begin. We have the parametric equations:
$y=2ap\, \therefore\,p=\frac{y}{2a}$
Re: analytics geometry
Is it the point p substitude into the curve equation ?
Re: analytics geometry
Where may you substitute for p which will give you a relationship between x and y, thereby eliminating p?
Re: analytics geometry
Here are two simple examples which may help you see how to eliminate a parameter.
1.) Suppose you are given the parametric equations:
Notice the equation for $y$ is linear, so we may easily solve for the parameter $t$:
Now, if we substitute for $t$ into the equation for $x$ we get an equation that relates $x$ and $y$ where $t$ has been "eliminated." We then have a Cartesian equation:
$x=2\left(\frac{y}{7} \right)^2+3\left(\frac{y}{7} \right)+5$
2.) Suppose you are given the parametric equations:
Rather than solving one of the equations for $t$, we may rewrite the equations as:
If we square both equations, we may then add and take advantage of the Pythagorean identity $\sin^2(\theta)+\cos^2(\theta)=1$
$\left(\frac{x}{a} \right)^2=\cos^2(t)$
$\left(\frac{y}{b} \right)^2=\sin^2(t)$
Adding, we find:
$\left(\frac{x}{a} \right)^2+\left(\frac{y}{b} \right)^2=\cos^2(t)+\sin^2(t)$
Which we may write as:
September 23rd 2012, 08:50 AM #2
September 23rd 2012, 06:23 PM #3
Sep 2012
September 23rd 2012, 07:15 PM #4
September 23rd 2012, 10:50 PM #5
Sep 2012
September 24th 2012, 06:46 AM #6
September 26th 2012, 04:36 AM #7
Sep 2012
September 26th 2012, 06:12 AM #8
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A Saint-Venant type principle for Dirichlet forms on discontinuous
Results 1 - 10 of 21
, 1998
"... . We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary,
the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be appr ..."
Cited by 17 (6 self)
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. We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the
generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the
existence of the locally Holder continuous Dirichlet heat kernel. 1. Introduction Consider a family M of n-dimensional closed Riemannian manifolds with a uniform lower bound of sectional curvature
and a uniform upper bound of diameter for a fixed n 2 N . In order to investigate various properties of manifolds in M, it is very useful to study its closure M with respect to the Gromov-Hausdorff
distance dGH , which is compact by the Gromov compactness theorem [15]. Since the closure M consists of Alexandrov spaces introduced in [2], the study of Alexandrov spaces is nowadays an important
topic i...
- DOCUMENTA MATH. , 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
"... We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms
of the property that the 1-Lipschitz functions form a sheaf. ..."
Cited by 4 (3 self)
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We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of
the property that the 1-Lipschitz functions form a sheaf.
"... Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral
properties within this framework. The range of applications is illustrated by a list of examples. ..."
Cited by 4 (3 self)
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Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties
within this framework. The range of applications is illustrated by a list of examples.
- In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed
"... Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible
tools and are useful in many different settings. This paper gives a glimpse of assortments of such applica ..."
Cited by 3 (1 self)
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Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools
and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1
"... Abstract. We prove heat kernel estimates for the ¯ ∂-Neumann Laplacian □ acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also
relate our results to those for an associated Laplace-Beltrami operator on functions. Dedicated to Barry S ..."
Cited by 3 (2 self)
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Abstract. We prove heat kernel estimates for the ¯ ∂-Neumann Laplacian □ acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also
relate our results to those for an associated Laplace-Beltrami operator on functions. Dedicated to Barry Simon on his 65 th birthday Contents
"... After a short survey of some of the reasons that make the heat kernel an important object of study, we review a number of basic heat kernel estimates. We then describe recent results concerning
(a) the heat kernel on certain manifolds with ends, and (b) the heat kernel with Neumann or Dirichlet boun ..."
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After a short survey of some of the reasons that make the heat kernel an important object of study, we review a number of basic heat kernel estimates. We then describe recent results concerning (a)
the heat kernel on certain manifolds with ends, and (b) the heat kernel with Neumann or Dirichlet boundary condition in Euclidean domains. This text is a revised version of the four lectures given by
the author at the First MSJ-SI in Kyoto during the summer of 2008. The structure of the lectures has been mostly preserved although some material has been added, deleted, or shifted around. The goal
is to present an
, 2005
"... operators with degenerate coefficients ..."
, 903
"... A dual characterization of length spaces with ..." | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=3112429","timestamp":"2014-04-19T00:52:40Z","content_type":null,"content_length":"30878","record_id":"<urn:uuid:bc4cc509-0dc8-4a34-8de3-ab699490a899>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00109-ip-10-147-4-33.ec2.internal.warc.gz"} |
Simpson's rule in Excel
Hello, All!
There have been no responses to a previous post but forgive me
if I try again. If this is not regarded as a charting question,
please let me know. Anyway, is it possible to write a statement
evaluating the area under a graph by Simpson's rule without
using VBA or helper columns?
James Silverton
Potomac, Maryland, USA
From an old post of mine ---
"Area under a curve"
To use Excel for evaluating the integral of, say, 2+3*(Ln(x))^0.6 using
Simpson's Rule (also see notes below):
1) Enter some labels:
In cell.....enter
A1 "X_1"
A2 "X_n"
A3 "NbrPanels"
2) Set some values:
In cell.....enter
B1 1
B2 2.5
B3 1000
3) Define some names:
Select A1:B3, then choose Insert->Name->Create. Make sure that only the
"Left Column" box is selected. If it isn't, you might have entered text
instead of numbers in the right column, or mis-selected the range. Press
4) Choose Insert->Names->Define
Enter each of the following names and their definitions, pressing Add
with each entry (you can copy and paste these):
EPanels =NbrPanels+MOD(NbrPanels,2)
delta =(X_n-X_1)/EPanels
Steps =ROW(INDIRECT("1:"&EPanels+1))-1
EvalPts =X_1+delta*Steps
SimpWts =IF(MOD(Steps,EPanels)=0,1,IF(MOD(Steps,2)=1,4,2))*delta/3
(optional) If interested in a trapezoidal approximation, define
TrapWts =IF(MOD(Steps,EPanels)=0,0.5*delta,delta)
5) Close the Define Names box, and in, say, cell D1, array-enter
That is, type in the function, and hold ctl-shift when pressing Enter.
In general, ctrl-shift-enter =SUM(SimpWts*f(EvalPts))
where f() is a legitimate Excel expression that yields a scalar numeric
To use the trapezoidal method, substitute in the above expression
TrapWts for SimpWts.
(1) With this implementation, an odd number for NbrPanels doesn't cut it
(for Simpson's rule), so there will be no improvement moving from an odd
number to the next integer (odd ones are automatically changed to the
next even, internally).
(2) If you have "jumps" in your function, break it up at the points
where those occur, and add the pieces.
.... FWIW, consider Weddle weights instead of the Simpson ones. For a
single partition, they run [1 5 1 6 1 5 1], with the whole shebang
multiplied by (3/10). Error term is much tighter, the calc is just as fast.
Dave Braden
James Silverton wrote:
> Hello, All!
> There have been no responses to a previous post but forgive me if I try
> again. If this is not regarded as a charting question, please let me
> know. Anyway, is it possible to write a statement evaluating the area
> under a graph by Simpson's rule without using VBA or helper columns?
> James Silverton
> Potomac, Maryland, USA
Please look at my screen shot of a worksheet on
I can generate the same result with:
If you look at the screen shot you will see that my x values are in A5:A11,
y-values in B5:B11
When I use a 'helper column' I need formulas in C5, C7, C9; i.e. in odd
cells hence the MOD(ROW(),2)=1 test
Hope this helps
Bernard V Liengme
remove caps from email
"James Silverton" wrote in message
> Hello, All!
> There have been no responses to a previous post but forgive me if I try
> again. If this is not regarded as a charting question, please let me know.
> Anyway, is it possible to write a statement evaluating the area under a
> graph by Simpson's rule without using VBA or helper columns?
> James Silverton
> Potomac, Maryland, USA | {"url":"http://help.lockergnome.com/office/Simpson-rule-Excel--ftopict748535.html","timestamp":"2014-04-19T01:49:14Z","content_type":null,"content_length":"27005","record_id":"<urn:uuid:ccc9f721-60b4-4aa8-9d22-e9c818b9c8b0>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00503-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts about mathematical recreations on mathematical musings
Martin Gardner passed away yesterday, at 95; here is the NYTimes obituary.
I grew up with his puzzles and mathematical recreations, as I’m sure many mathematicians of my generation (and generations before) did. I am reminiscing this evening about interesting topics that
Gardner introduced me to, and the following geometric gem comes to mind. The problem was apparently first proposed by Henri Lebesgue in 1914 and is still open, so it is almost 100 years old now.
Question: Find a universal cover $C$ of least area in the plane, meaning a set having a subset congruent to every planar set of unit diameter.
If one restricts to convex $C$, then such a minimum is guaranteed to exist by the Blaschke selection theorem, a compactness result for sequences of bounded convex sets. A hexagon of unit width does
the trick, but surprisingly, one can take off tiny pieces and still have a universal cover.
No one has ever exhibited such a minimum with proof, although there have been a few false claims of optimal solutions. I just searched and found an amusing 1992 Geometriae Dedicata article by H. C.
Hansen in which the author shaves $4 \times 10^{-11}$ off the then reigning world record. I am not sure if anyone has improved on this result since, but this might be a fun thing to think about;
indeed, Hansen suggests in the article how computers might help solve the problem, and computers have come a long way since 1992.
Gardner’s name came up in conversation just the other day. Tom Rokicki came to talk to my Rubik’s Cube class at Stanford about his work on “God’s algorithm” — Tom set a world record recently by
proving that every Rubik’s Cube, no matter how scrambled, can be solved in 22 or fewer face turns. I asked him how long it would be until we have a proof of the conjectured answer of 20. He said he
thought he might have 21 in the next few months, and then said that he wanted to have it down to 20 moves by Gathering for Gardner 2012. I hope that the Gatherings for Gardner continue, and it would
be nice to see Tom succeed in his goal too.
I am not sure if anyone in history has ever made more people smile and shake their head about mathematical things than Martin Gardner. His writings illuminated it’s magical and mysterious qualities.
Gardner was also famous as skeptic and a debunker, like many magicians before him. The NYTimes obituary reports:
He ultimately found no reason to believe in anything religious except a human desire to avoid “deep-seated despair.” So, he said, he believed in God.
Well godspeed to you Martin, and thanks for all the wonderful writings and beautiful thoughts. | {"url":"http://matthewkahle.wordpress.com/tag/mathematical-recreations/","timestamp":"2014-04-18T03:17:44Z","content_type":null,"content_length":"18264","record_id":"<urn:uuid:bdc6c582-76a2-488d-a99a-f29a16045a72>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00347-ip-10-147-4-33.ec2.internal.warc.gz"} |
- -
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Click on a 'View Solution' below for other questions:
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eee The line graph shows the number of cars (in hundreds) sold between July and November. Find the month in which minimum number of cars were sold. View Solution
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eee Use the line graph to find the number of inches of rainfall recorded in the month of August. eee View Solution
eee Charles & Co. recorded their annual profits. Find the difference between the profits that the company made in the years 1992 and 1989. View Solution
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eee The graph represents the number of nickels, dimes and quarters in Jimmy's coin collection. How many nickels are there in his collection? View Solution
eee Lindsay recorded her test scores on the line graph. What is Lindsay's score in the Math test? View Solution
eee Zelma has a collection of toy cars. Which bar graph would correctly represent the number of cars, if the cars were sorted based on their color? View Solution
eee Henry's weight was recorded in the graph. How many pounds did Henry gain during the first four years since his birth? eee View Solution
eee Bill recorded his scores in 4 English tests. There were 60 questions in each test. How many questions did he fail to answer correctly in test 3? View Solution
eee The line graph shows the price of a product from the year 1975 till 2000. In which year was the price of the product $20? View Solution
eee The line graph shows the change in the student population of a school during the period 1980-2000. What was the population in the year 1990? View Solution
B The bar graph shows the numbers of different colored candles in a box. Find the number of red candles in the box. View Solution
B The bar graph shows the temperatures recorded on May 3^rd, View Solution
in three different states. Which state recorded the maximum temperature? B
B The line graphs show the temperatures recorded during daytime and nighttime in four different cities. Which city recorded the minimum nighttime temperature? B View Solution
B The graph represents the number of students in each grade traveling by bus to XYZ middle school. How many students in fourth grade travel by bus to the school? View Solution
B The temperatures in four different states on 30^th of April were recorded on a bar graph. Which city recorded the highest temperature? B View Solution
B The temperatures recorded in three states on 3^rd of May are shown on the bar graph. Which state recorded the highest temperature? B View Solution
B The line graph shows the temperature variations in a state during a particular week. Find the difference between the temperatures recorded on Thursday and Monday. View Solution
B A company, started in 1975, recorded the price of its product each year till 2000. Find out the year in which the price of the product was $80. View Solution
B The points three students scored on a math test are 19, 22, and 19. Identify an appropriate bar graph that represents the data correctly. View Solution
B The graph shows the amount deposited by Brad in his savings account for 6 consecutive years. Find the amount deposited in 1999. View Solution
B Charles & Co. recorded their annual profits. Find the difference between the profits that the company made in the years 1992 and 1993. View Solution
B The graph shows the number of apples sold by Jimmy during the last 5 days. How many apples did Jimmy sell on Tuesday? View Solution
B The graph represents the number of nickels, dimes and quarters in Jerald's coin collection. How many nickels are there in his collection? View Solution
B The graph represents the number of students in each grade traveling by bus to XYZ middle school. How many students in fifth grade travel by bus to the school? View Solution
B Holly has a collection of toy cars. Which bar graph would correctly represent the number of cars, if the cars were sorted based on their color? View Solution
B The bar graph shows the numbers of different colored candles in a box. Find the number of black candles in the box. View Solution
B The bar graph shows the number of toys in a shop with different hair colors. Find the number of toys having red hair. B View Solution
B The bar graph displays the favorite holiday destinations. Find the most popular holiday destination. View Solution
B The temperatures on April 30^th in four different states were recorded. Find the difference between the temperatures in Alaska and Illinois. View Solution
B The points three students scored in a math test are 23, 22, and 20. Identify an appropriate bar graph that represents the data correctly. View Solution
B A box contains four different types of pens: Type 1, Type 2, Type 3, and Type 4. Find the number of pens of Type 2 in the box. B View Solution
B A librarian has a record of the number of books on each subject present in the library. Find the number of books on English present in the library. View Solution
B The graph shows the amount deposited by Tim in his savings account for 6 consecutive years. Find the amount deposited in 1999. View Solution
B The graph shows the sales of toys of a company during four consecutive years. Find the increase in the growth of the sales from 1996 to 1997. B View Solution
B The graph shows the number of apples sold by Paul during the last 5 days. How many apples did Paul sell on Tuesday? View Solution
B The graph shows the number of students in grade-6, grade-7, and grade-8 of the XYZ middle school. Which grade has the highest number of students? B View Solution
B The graph represents the number of nickels, dimes and quarters in Sunny's coin collection. How many nickels are there in his collection? View Solution
B The graph represents the number of students, in each grade, who travel by bus to the XYZ middle school. Which grade has the least number of students traveling by bus? B View Solution
B How many more theatres are there than restaurants as evident from the graph? B View Solution
B Olga recorded her test scores on the line graph. What is Olga's score in the Math test? View Solution
B Jake's weight was recorded in the graph. How many pounds did Jake gain during the first four years since his birth? B View Solution
B Jimmy recorded his scores in 4 English tests. There were 60 questions in each test. How many questions did he fail to answer correctly in test 3? View Solution
B Emily recorded her scores in four Science tests. There were 60 questions in each test. In which test did she score the least? View Solution
B The graph represents the number of students in each grade traveling by bus to XYZ middle school. How many students in second grade travel by bus to the school? View Solution
B The graph represents the number of nickels, dimes and quarters in Will's coin collection. Which type of coin was more in number in his collection? View Solution
B The bar graph shows the test scores of some students in Mr. Brown's class. Who scored the maximum marks? B View Solution
B The line graph shows the number of cars (in hundreds) sold between July and November. Find the month in which maximum number of cars were sold. View Solution
B The line graph shows the number of students in each grade in a school. What is the total number of students in 6^th and 10^th grades in the school? View Solution
B Use the line graph to find the number of inches of rainfall recorded in the month of July. B View Solution
B The annual income of ABC Airlines over the past five years is shown in the graph. During which years was there no change in the income? B View Solution
B The number of movies released each year varies. In which year was there a drop in the number of movies released compared to the previous year? B View Solution
B Charles & Co. recorded their annual profits. Find the difference between the profits that the company made in the years 1992 and 1991. View Solution
B The bar graph shows the number of toys a company exported to countries A, B, and C in a particular year. What is the number of toys exported to country B? View Solution
B A librarian recorded the number of books on each subject. Find the number of math books in the library. B View Solution
B The table shows the number of toys manufactured by a company through the period 1980-2000.
│Year│Number of Toys Manufactured (in thousands) │
│1980│2 │
│1985│2 │
│1990│4 │ View Solution
│1995│6 │
│2000│8 │
Which line graph represents the data correctly?
B In a survey on favorite sport, 40 students voted for softball, 50 voted for football, 30 voted for tennis, and 20 voted for polo as their favorite sport. Choose the bar graph that best View
represents the data. B Solution
B The temperature recorded on April 30^th in Tennessee, Illinois, Alaska, and Michigan are 50°F, 40°F, 60°F, and 60°F respectively. Identify the bar graph that best represents the data View
correctly. B Solution
B The temperatures recorded in three states on 3^rd of May are shown on the bar graph. Which state recorded the lowest temperature? B View Solution
B The bar graph displays the favorite holiday destinations. Find the most popular holiday destination. B View Solution
B Carol made different colors of flags as shown in the figure. Choose the appropriate bar graph for the data. B View Solution
B Kelsey has a collection of toy cars. Which bar graph would correctly represent the number of cars, if the cars were sorted based on their color? View Solution
B Which bar graph correctly represents the number of teddy bears shown in the picture? View Solution
B The graph shows the number of service calls an electrician received in five months. Choose an appropriate title for the graph. B View Solution
B The growth of a plant has been recorded over a six week period. Which statement is true about the height of the plant during the 5^th week compared to that during the 4^th week? B View Solution
B The line graph shows the price of a product from the year 1975 till 2000. In which year was the price of the product $40? View Solution
B The line graph shows the change in the student population of a school during the period 1980-2000. What was the population in the year 1995? View Solution
B The bar graph shows the favorite sport of some students. Which sport was the most popular? View Solution
B The bar graph shows the numbers of different colored candles in a box. Find the number of pink candles in the box. View Solution
B The bar graph shows the temperatures recorded on May 3^rd, View Solution
in three different states. Which state recorded the minimum temperature? B
B The temperatures on April 30^th in four different states were recorded. Find the difference between the temperatures in Tennessee and Illinois. B View Solution
B The line graph shows the temperature variation during a particular week in Ohio. Find the difference between the maximum and the minimum temperatures recorded. View Solution
B The bar graph shows the test scores of some students in Mr.Brown's class. Who scored exactly 8 points? B View Solution
B The line graphs show the temperatures recorded during daytime and nighttime in four different cities. Which city recorded the maximum daytime temperature? B View Solution
B The line graphs show the variation in temperature during day and night times in four different cities at the same time. Which city recorded the minimum night time temperature? View Solution
B The temperatures in four different states on 30^th of April were recorded on a bar graph. Which city recorded the least temperature? B View Solution
B Lydia's father purchased some books to gift Lydia's friends on her birthday. The bar graph shows the number of each kind of book that he purchased. What kind of book did he purchase View
the most? Solution
B Kay, Maria and Carlo picked apples from a tree. The tally mark shows the number of apples each of them picked. Which of the following line graphs matches the tally chart? View Solution
B Frank, Nichole, Phil, and Lisa were practicing basketball and each of them were given ten chances. Their scores were recorded in a table as shown.
│Names │Scores │
│Frank │7 │
│Nichole │9 │ View Solution
│Phil │3 │
│Lisa │6 │
Which bar graph represents their scores? B
B A company, started in 1975, recorded the price of its product each year till 2000. Find out the year in which the price of the product was $100. View Solution
B The line graph shows the temperature variations in a state during a particular week. Find the difference between the temperatures recorded on Tuesday and Monday. View Solution
B The points three students scored in a math test are 19, 22, and 19. Identify an appropriate bar graph that represents the data correctly. View Solution
B A box contains four different types of pens: Type 1, Type 2, Type 3, and Type 4. Find the number of pens of Type 3 in the box. B View Solution
B A librarian has a record of the number of books on each subject present in the library. Find the number of books on History present in the library. View Solution
B The bar graph shows the number of toys in a shop with different hair colors. Find the number of toys having brown hair. B View Solution
B The growth of a plant has been recorded over a six-week period. Find the difference in the height of the plant between the 1^st and the 3^rd weeks. View Solution
B The line graph shows the temperature variations in Ohio during a particular week. What was the least temperature recorded? View Solution
B The bar graph displays the favorite holiday destinations. Find the least popular holiday destination. View Solution
B The bar graph shows the favorite sport of 150 school children. Which sport was voted as the most popular sport? B View Solution
B The temperatures on April 30^th in four different states were recorded. Find the difference between the temperatures in Michigan and Illinois. View Solution
B From the bar graph, find out what percentage of population in the USA are in the age group 20 to 40 years for the year 2003-2004. B View Solution
B A line graph for the number of computer firms surveyed in the USA is given. Between which two consecutive years was there a maximum increase in the number of firms surveyed? View Solution
B The bar graph shows the number of radios sold in each price range in a city. Find the total number of radios sold. B View Solution
B The line graph shows the profit a company made in 4 years. Find the profit in the year 2001. B View Solution
B The bar graph shows the price of books on different subjects in a book store. Find the price range of books in the book store. B View Solution
B What do you think could have been the number of cars sold in the month of June? View Solution
B How many movies do you think Peter saw in the year 2001? B View Solution
B What do you think will be the population in the year 2003? B View Solution
B What do you think will be the height of the plant in the 6^th week? B View Solution
B What would be the population of horses in 2005? B View Solution
B Which bar graph correctly represents the number of marbles shown in the picture? View Solution
B Sam bought some marbles of different colors as shown in pictograph. Which bar graph matches the data on the pictograph? B View Solution
B The points three students scored on a math test are 23, 22, and 20. Identify an appropriate bar graph that represents the data correctly. View Solution
B In a class, 10 students like potatoes, 8 like radish, 6 like carrot, and 14 like capsicum. Which of the graphs shows the data correctly? B View Solution | {"url":"http://www.icoachmath.com/solvedexample/sampleworksheet.aspx?process=/__cstlqvxbefxbxbgeehxkjfef&.html","timestamp":"2014-04-18T13:07:17Z","content_type":null,"content_length":"155813","record_id":"<urn:uuid:e19e91a4-f5c2-41d4-8ffd-dd82e9562c7a>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00355-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Nyquist Diagram in Linear Control System
I have so much problem in draw nyquist diagram. here is some problem
1-as the book mentioned the open- transfer function doesn't have any pole and zero in right of S plane. and give me the uncompleted diagram
so i completed it as you see , but my answer is different form the book !!!! what is my wrong?
and other question:
2-here is uncompleted diagrams
i completed and solved with this view
but one of the answers is wrong .... now what is my problem?!!!!
how can i find the best videos references about nyquist in Linear Control System ? | {"url":"http://www.physicsforums.com/showpost.php?p=4210654&postcount=1","timestamp":"2014-04-19T19:39:52Z","content_type":null,"content_length":"9719","record_id":"<urn:uuid:0507c88f-1a59-445b-9f11-5833abbe5263>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00363-ip-10-147-4-33.ec2.internal.warc.gz"} |
put sth at sth
English definition of “put sth at sth”
put sth at sth
— phrasal verb with put /pʊt/ verb (present participle putting, past tense and past participle put)
› to guess or roughly calculate that something will cost a particular amount, or that something is a particular size, number, or amount: The value of the painting has been put at £1 million. I'd put
her at (= guess that her age is) about 35.Calculations and calculatingAddition, subtraction, multiplication and divisionAssessing and estimating valueAnalysing and evaluating
Focus on the pronunciation of put sth at sth | {"url":"http://dictionary.cambridge.org/dictionary/british/put-sth-at-sth?topic=calculations-and-calculating","timestamp":"2014-04-19T10:30:07Z","content_type":null,"content_length":"69278","record_id":"<urn:uuid:298d9f15-16a1-4efb-aa46-8f013e86fa2e>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00553-ip-10-147-4-33.ec2.internal.warc.gz"} |
Integrating using Polar Coordinate
December 5th 2011, 05:15 AM #1
Junior Member
Sep 2010
Integrating using Polar Coordinate
Using polar coordinates, evaluate the double integral over r sin(x^2+y^2)dA where R is the region 4<x^2+y^2<36
Need to know where to start, or an example? I keep getting 8.something and it's incorrect.
Re: Integrating using Polar Coordinate
The region you are integrating over is the region enclosed by (and not including) the circles centred at the origin of radii 2 and 6. So clearly your r bounds are 2 and 6, and since you are
sweeping over the whole circle, your theta bounds are 0 and 2pi.
Re: Integrating using Polar Coordinate
I get that. But when I integrate and plug in, I get the wrong answer. So I integrate sin(r) from r=2 to 6? that give me -cos(r) from r=2 to 6. Then, integrate -cos(2)+cos(6) from theta=0 to 2pi?
This gives me 8.647 which is incorrect.
Last edited by Bracketology; December 5th 2011 at 05:50 AM.
Re: Integrating using Polar Coordinate
I think that this is the expression that you should integrate:
$r\sin\left(x^2+y^2\right)\mathrm{d}A=r\sin\left(r^ 2)r\mathrm{d}r\mathrm{d}\theta$
Re: Integrating using Polar Coordinate
Thanks for the imput, but how do I integrate rsin(r^2)*r? Or is the integral suppose to be rsin(r^2)?
I think you meant rsin(r^2), because that would make sense.
Re: Integrating using Polar Coordinate
The width of an infinitessimal element expressed in polar parameters is $r\mathrm{d}\theta$, and its length is $\mathrm{d}r$. So the area of the infinitessimal element is:
That's where the 'extra' factor $r$ comes from.
As far integrating it, I don't have time to check it myself right now, but integration by parts might be the way to go.
Re: Integrating using Polar Coordinate
I have a feeling your original integrand must have been \displaystyle \begin{align*} \sin{\left(x^2 + y^2\right)} \end{align*}, not \displaystyle \begin{align*} r\sin{\left(x^2 + y^2\right)} \end
{align*} seeing as it doesn't make sense to have mixed polars with cartesians in the first place. Also, \displaystyle \begin{align*} \int{r^2\sin{\left(r^2\right)}\,dr} \end{align*} doesn't have
a closed form answer, whereas the first will.
December 5th 2011, 05:20 AM #2
December 5th 2011, 05:25 AM #3
Junior Member
Sep 2010
December 5th 2011, 06:36 AM #4
Junior Member
Jun 2011
December 5th 2011, 12:10 PM #5
Junior Member
Sep 2010
December 5th 2011, 02:56 PM #6
Junior Member
Jun 2011
December 5th 2011, 03:46 PM #7 | {"url":"http://mathhelpforum.com/calculus/193483-integrating-using-polar-coordinate.html","timestamp":"2014-04-18T10:28:07Z","content_type":null,"content_length":"49984","record_id":"<urn:uuid:f138f965-ee3b-4182-9b2d-83db5cd201fc>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00049-ip-10-147-4-33.ec2.internal.warc.gz"} |
Question about normal distribution problems
October 6th 2007, 11:09 PM #1
Oct 2007
Question about normal distribution problems
Hello, I'm not sure how to word this as I just started studying normal distribution today but I'll try:
When I do a normal distribution problem, I have to draw a graph. Is it common to simply memorize the areas of each number, like 3.4 for 1 and -1, and .135, .024, etc. rather than referring to the
table? I'm not sure if I'm doing something wrong but I seem to get more accurate answers by just memorizing those numbers.
Here's an example of a problem I just did:
The test scores on the quantitative portion of the SAT are normally distributed with a mean score of 570 and SD of 70. Using the empirical rule, approximately what percent of the scores are more
than 710?
I used the formula: z = x-mu/sigma
= (710 - 570)/70 = 2
If I use the method of memorizing the numbers and drawing it out using them, I get 2.5% which is the answer the CD for my book has.
(my work = p (z > 2 ) = 0.5 - 0.34 - 0.135 = .025 = 2.5%
If I use the table, it turns out like this:
P (z > 2) = 0.5 - 0.4772 = 0.0228 = 2.3%?
The CD program marks 2.5% as incorrect. Am I doing something wrong? Or are both answers acceptable?
Last edited by paperstar; October 7th 2007 at 12:20 AM.
Hello, I'm not sure how to word this as I just started studying normal distribution today but I'll try:
When I do a normal distribution problem, I have to draw a graph. Is it common to simply memorize the areas of each number, like 3.4 for 1 and -1, and .135, .024, etc. rather than referring to the
table? I'm not sure if I'm doing something wrong but I seem to get more accurate answers by just memorizing those numbers.
Here's an example of a problem I just did:
The test scores on the quantitative portion of the SAT are normally distributed with a mean score of 570 and SD of 70. Using the empirical rule, approximately what percent of the scores are more
than 710?
I used the formula: z = x-mu/sigma
= (710 - 570)/70 = 2
If I use the method of memorizing the numbers and drawing it out using them, I get 2.5% which is the answer the CD for my book has.
(my work = p (z > 2 ) = 0.5 - 0.34 - 0.135 = .025 = 2.5%
If I use the table, it turns out like this:
P (z > 2) = 0.5 - 0.4772 = 0.0228 = 2.3%?
The CD program marks 2.3% as incorrect. Am I doing something wrong? Or are both answers acceptable?
For this z-score; 2.28% is correct and 2.5% is wrong, but your question
says "using the empirical rule", and you need to look see what that is.
It may well be that you are told that the empirical rule is that about 97.5% of the
area under the normal distribution is to the right of z=2, or that about 95% is
within 2 SDs of the mean.
Oh, sorry, I meant the CD says 2.5% is the correct answer. If I put in 2.3%, it says it's incorrect. I guess I should just rely on the table.
As for the empirical rule, it says: This rule provides the procedure for computing normal probabilities associated with only whole number multiples of the standard deviation (sigma), that is,
within 1, 2 or 3 standard deviations of the mean.
I'm not sure what that means. :x
October 6th 2007, 11:22 PM #2
Grand Panjandrum
Nov 2005
October 7th 2007, 12:19 AM #3
Oct 2007 | {"url":"http://mathhelpforum.com/advanced-statistics/20097-question-about-normal-distribution-problems.html","timestamp":"2014-04-17T07:18:37Z","content_type":null,"content_length":"38723","record_id":"<urn:uuid:ffac0d2d-bf94-4d5f-a5d6-7632d40f9d97>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00198-ip-10-147-4-33.ec2.internal.warc.gz"} |
Understanding set theory
I'm trying to prove something small with set theory and since I'm new to it, I've run into a problem. I can't understand what the following means exactly and how to proceed further. Or where the
mistake is, if there is one. I think there is, because it seems... freaky.
[tex]x\notin\left(\left(\left(A\cup B\right)\setminus\left(A\cap B\right)\right)\cap C\right)[/tex]
[tex]x\notin\left(\left(A\cup B\right)\setminus\left(A\cap B\right)\right)\wedge x\notin C[/tex]
[tex]\left(x\notin\left(A\cup B\right)\wedge x\in\left(A\cap B\right)\right)\wedge x\notin C[/tex]
[tex]\left(\left(\left(x\notin A\right)\vee\left(x\notin B\right)\right)\wedge\left(\left(x\in A\right)\wedge\left(x\in B\right)\right)\right)\wedge x\notin C[/tex]
I'd post the entire thing of which this is a small part of, but that's my homework and I don't want to get into the habit of having other people do my homework for me. Plus I want to learn how and
why it works, not just do it. | {"url":"http://www.physicsforums.com/showthread.php?t=42390","timestamp":"2014-04-21T14:47:39Z","content_type":null,"content_length":"22448","record_id":"<urn:uuid:fb9e9dbb-25cd-44e2-94ae-b6c2f15006fc>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00064-ip-10-147-4-33.ec2.internal.warc.gz"} |
One-matching bi-Cayley graphs over abelian groups
Dragan Marušič, Aleksander Malnič, István Kovács and Štefko Miklavčič
Eur. j. comb. Volume 30, Number 2, , 2009. ISSN 0195-6698
A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits (of equal size), and is a one-matching bi-Cayley graph if the bipartite graph induced by the edges
joining these two orbits is a perfect matching. Typical examples of such graphs are the generalized Petersen graphs. A classification of connected arc-transitive one-matching bi-Cayley graphs over
abelian groups is given. This is done without referring to the classification of finite simple groups. Instead, complex irreducible characters of abelian groups are used extensively.
EPrint Type: Article
Project Keyword: Project Keyword UNSPECIFIED
Subjects: Theory & Algorithms
ID Code: 8223
Deposited By: Boris Horvat
Deposited On: 21 February 2012 | {"url":"http://eprints.pascal-network.org/archive/00008223/","timestamp":"2014-04-19T01:49:04Z","content_type":null,"content_length":"6235","record_id":"<urn:uuid:96e2cfb3-1ca6-4661-b61f-212c3eef1836>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00175-ip-10-147-4-33.ec2.internal.warc.gz"} |
Game Theory: The Mixed Strategy Algorithm
This lesson shows the algorithm we use to solve for mixed strategy Nash equilibrium in simple 2x2 games.
Takeaway points:
1. If there is a mixed strategy Nash equilibrium, it usually is not immediately obvious. (So do not let matching pennies lull you into believing this is easy!)
2. However, there is a straightforward algorithm that lets you calculate mixed strategy Nash equilibria. We will employ it frequently.
Next lesson:
How NOT to Write a Mixed Strategy Nash Equilibrium
Back to
all lectures | {"url":"http://gametheory101.com/Mixed_Strategies.html","timestamp":"2014-04-19T02:36:50Z","content_type":null,"content_length":"10901","record_id":"<urn:uuid:a2c5a46b-e0ab-4c8b-96b9-f4274a764752>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00490-ip-10-147-4-33.ec2.internal.warc.gz"} |
Electron Beams and Microwave Vacuum Electronics
ISBN: 978-0-470-04816-0
573 pages
October 2006
Read an Excerpt
This book focuses on a fundamental feature of vacuum electronics: the strong interaction of the physics of electron beams and vacuum microwave electronics, including millimeter-wave electronics. The
author guides readers from the roots of classical vacuum electronics to the most recent achievements in the field. Special attention is devoted to the physics and theory of relativistic beams and
microwave devices, as well as the theory and applications of specific devices.
See More
I.1 Outline of the Book.
I.2 List of Symbols.
I.3 Electromagnetic Fields and Potentials.
I.4 Principle of Least Action. Lagrangian. Generalized Momentum. Lagrangian Equations.
I.5 Hamiltonian. Hamiltonian Equations.
I.6 Liouville Theorem.
I.7 Emittance. Brightness.
1 Motion of Electrons in External Electric and Magnetic Static Fields.
1.1 Introduction.
1.2 Energy of a Charged Particle.
1.3 Potential–Velocity Relation (Static Fields).
1.4 Electrons in a Linear Electric Field e0E ¼ kx.
1.5 Motion of Electrons in Homogeneous Static Fields.
1.6 Motion of Electrons in Weakly Inhomogeneous Static Fields.
1.6.1 Small Variations in Electromagnetic Fields Acting on Moving Charged Particles.
1.7 Motion of Electrons in Fields with Axial and Plane Symmetry. Busch’s Theorem.
2 Electron Lenses.
2.1 Introduction.
2.2 Maupertuis’s Principle. Electron-Optical Refractive Index.
Differential Equations of Trajectories.
2.3 Differential Equations of Trajectories in Axially Symmetric Fields.
2.4 Differential Equations of Paraxial Trajectories in Axially Symmetric Fields Without a Space Charge.
2.5 Formation of Images by Paraxial Trajectories.
2.6 Electrostatic Axially Symmetric Lenses.
2.7 Magnetic Axially Symmetric Lenses.
2.8 Aberrations of Axially Symmetric Lenses.
2.9 Comparison of Electrostatic and Magnetic Lenses. Transfer Matrix of Lenses .
2.10 Quadrupole lenses.
3 Electron Beams with Self Fields.
3.1 Introduction.
3.2 Self-Consistent Equations of Steady-State Space-Charge Electron Beams.
3.3 Euler’s Form of a Motion Equation. Lagrange and Poincare´ Invariants of Laminar Flows.
3.4 Nonvortex Beams. Action Function. Planar Nonrelativistic Diode.
Perveance. Child–Langmuir Formula. r- and T-Modes of Electron Beams.
3.5 Solutions of Self-Consistent Equations for Curvilinear Space-Charge Laminar Beams. Meltzer Flow. Planar Magnetron with an Inclined Magnetic Field. Dryden Flow.
4 Electron Guns.
4.1 Introduction.
4.2 Pierce’s Synthesis Method for Gun Design.
4.3 Internal Problems of Synthesis. Relativistic Planar Diode. Cylindrical and Spherical Diodes.
4.4 External Problems of Synthesis. Cauchy Problem.
4.5 Synthesis of Electrode Systems for Two-Dimensional Curvilinear Beams with Translation Symmetry (Lomax–Kirstein Method). Magnetron Injection Gun.
4.6 Synthesis of Axially Symmetric Electrode Systems.
4.7 Electron Guns with Compressed Beams. Magnetron Injection Gun.
4.8 Explosive Emission Guns.
5 Transport of Space-Charge Beams.
5.1 Introduction.
5.2 Unrippled Axially Symmetric Nonrelativistic Beams in a Uniform Magnetic field.
5.3 Unrippled Relativistic Beams in a Uniform External Magnetic Field..
5.4 Cylindrical Beams in an Infinite Magnetic Field.
5.5 Centrifugal Electrostatic Focusing.
5.6 Paraxial-Ray Equations of Axially Symmetric Laminar Beams.
5.7 Axially Symmetric Paraxial Beams in a Uniform Magnetic Field with Arbitrary Shielding of a Cathode Magnetic Field.
5.8 Transport of Space-Charge Beams in Spatial Periodic Fields.
6 Quasistationary Microwave Devices.
6.1 Introduction.
6.2 Currents in Electron Gaps. Total Current and the Shockley–Ramo Theorem.
6.3 Admittance of a Planar Electron Gap. Electron Gap as an Oscillator. Monotron.
6.4 Equation of Stationary Oscillations of a Resonance Self-Excited Circuit.
6.5 Effects of a Space-Charge Field. Total Current Method. High-Frequency Diode in the r-Mode. Llewellyn–Peterson Equations.
7 Klystrons.
7.1 Introduction.
7.2 Velocity Modulation of an Electron beam.
7.3 Cinematic (Elementary) Theory of Bunching.
7.4 Interaction of a Bunched Current with a Catcher Field. Output Power of A Two-Cavity Klystron.
7.5 Experimental Characteristics of a Two-Resonator Amplifier and Frequency-Multiplier Klystrons.
7.6 Space-Charge Waves in Velocity-Modulated Beams.
7.7 Multicavity and Multibeam Klystron Amplifiers.
7.8 Relativistic Klystrons.
7.9 Reflex Klystrons.
8 Traveling-Wave Tubes and Backward-Wave Oscillators (O-Type Tubes).
8.1 Introduction.
8.2 Qualitative Mechanism of Bunching and Energy Output in a TWTO.
8.3 Slow-Wave Structures.
8.4 Elements of SWS Theory.
8.5 Linear Theory of a Nonrelativistic TWTO. Dispersion Equation, Gain, Effects of Nonsynchronism, Space Charge, and Loss in a Slow-Wave Structure.
8.6 Nonlinear Effects in a Nonrelativistic TWTO. Enhancement of TWTO Efficiency (Velocity Tapering, Depressed Collectors).
8.7 Basic Characteristics and Applications of Nonrelativistic TWTOs.
8.8 Backward-Wave Oscillators.
8.9 Millimeter Nonrelativistic TWTOs, BWOs, and Orotrons.
8.10 Relativistic TWTOs and BWOs.
9 Crossed-Field Amplifiers and Oscillators (M-Type Tubes).
9.1 Introduction.
9.2 Elementary Theory of a Planar MTWT.
9.3 MTWT Amplification.
9.4 M-type Injected Beam Backward-Wave Oscillators (MWO, M-Carcinotron).
9.5 Magnetrons.
9.6 Relativistic Magnetrons.
9.7 Magnetically Insulated Line Oscillators.
9.8 Crossed-Field Amplifiers.
10 Classical Electron Masers and Free Electron Lasers.
10.1 Introduction.
10.2 Spontaneous Radiation of Classical Electron Oscillators.
10.3 Stimulated Radiation of Excited Classical Electron Oscillators.
10.4 Examples of Electron Cyclotron Masers.
10.5 Resonators of Gyromonotrons (Free and Forced Oscillations).
10.6 Theory of a Gyromonotron.
10.7 Subrelativistic Gyrotrons.
10.8 Elements of Gyrotron Electron Optics.
10.9 Mode Interaction and Mode Selection in Gyrotrons. Output Power Systems.
10.10 Gyroklystrons.
10.11 Gyro-Traveling-Wave Tubes.
10.12 Applications of Gyrotrons.
10.13 Cyclotron Autoresonance Masers.
10.14 Free Electron Lasers.
1. Proof of the 3/2 Law for Nonrelativistic Diodes in the r-Mode.
2. Synthesis of Guns for M-Type TWTS and BWOS.
3. Magnetic Field in Axially Symmetric Systems.
4. Dispersion Characteristics of Interdigital and Comb Structures.
5. Electromagnetic Field in Planar Uniform Slow-Wave Structures.
6. Equations of Free Oscillations of Gyrotron Resonators.
7. Derivation of Eqs. (10.66) and (10.67).
8. Calculation of Fourier Coefficients in Gyrotron Equations.
9. Magnetic Systems of Gyrotrons.
See More
SHULIM E. TSIMRING, PhD, DSc, is a consultant in the field of applied physics. Dr. Tsimring has taught at a number of universities, most recently as a professor at Nizhny Novgorod State University,
Russia. Simultaneously, he was engaged in powerful high-frequency electronics research at the Institute of Applied Physics of the Russian Academy of Sciences in Nizhny Novgorod.
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Hailperin's Probability Logic
Hailperin’s Probability Logic
T. Hailperin Probability Logic Notre Dame Journal of Formal Logic Volume 25, Number 3, July 1984 pp. 198-212.
Hailperin develops Boole‘s ideas on bounds on possible conventional probability assignments. It considers a pair <B,P>where B is a Boolean algebra and P is a ‘probability function’ for which
Hailperin develops axioms and theorem, most importantly about the bounds on probabilities P(B|A).
My impression is that interval probabilities are much less commonly used than supposedly precise probabilities. But, informally, it is common to estimate the worst-case probabilities for ‘cost’ and
‘benefit’ factors. The explicit use of intervals is indicated when one has factors that are sometimes good, sometimes bad, but even here it is common to want to explore different possible
probability assignments (scenarios) explicitly, because Hailperin’s bounds are necessarily imprecise. Many clients also find it easier to grasp a ‘spanning’ collection of precise scenarios as against
an imprecise probability (which can invoke the scorn of some). So in practice the use of even well-founded imprecise probabilities tend to be relegated to behind the scenes, with precise
probabilities – even if logically flawed - taking centre stage. Problems seem to arise when people reason ‘pragmatically’; as if a flawed theory could necessarily be trusted.
I would like to know if his later work is of broader significance. Reading suggestions?
See Also
Logic with a probability semantics – has a good summary. Makes it clear that we suppose that P(X)=p for some p in an interval, so that one has the usual law of large numbers. Muddling is not allowed.
My notes on broader uncertainty and logic, e.g. work by Jack Good, with a fuller motivation, discussion of applications and linking to practice.
3 Responses to Hailperin’s Probability Logic
1. Hello, Dr. Marsay!
Though it’s been a while, I would like to inform you that Theodore Hailperin has published a book not too long ago. In the Preface, he also attributes his interests in probability theory and
mathematical logic to reading John Maynard Keynes’s A Treatise on Probability.
Also, as I have stated before, you ought to acquire a copies of these works by Theodore Hailperin:
One last thing though – have you ever heard of the late David W. Miller? I believe he was a Professor of Management Science at Columbia University, and like you and Dr. Michael Emmett Brady, he
had an interest in George Boole’s work on logic and probability. David W. Miller’s own book on the Last Challenge Problem has been released into paperback form recently, and you can acquire a
copy over Amazon.com.
2. Thanks. Does Hailperin motivate his use of a ‘probability function’? I briefly link to Miller in my notes on Keynes’ Treatise, http://djmarsay.wordpress.com/?s=Miller. Both seem to preclude
muddling, as in a real roulette wheel whose biases may change with wear.
□ Sorry for the belated response sir. I forget if he does. I would recommend reading his works though, especially that recent paper that I sent to you! | {"url":"http://djmarsay.wordpress.com/bibliography/rationality-and-uncertainty/broader-uncertainty/hailperins-probability-logic/","timestamp":"2014-04-19T22:08:31Z","content_type":null,"content_length":"96041","record_id":"<urn:uuid:9520e189-b359-4b0e-bda7-8cd2c6e119a1>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00542-ip-10-147-4-33.ec2.internal.warc.gz"} |
s and
Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here.
[Return to Table of Contents]
Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [
Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]
Chapter 11
Species and charge transport
Understanding charge and species transport is critical to understanding how electric fields couple to fluid flow in dynamic systems. So far, the only species transport we have discussed is passive
scalar diffusion in Chapter 4, and the only treatment of ion transport was the equilibrium distribution of ions specified by Boltzmann statistics in Chapter 9. Brief mention of the charge transport
equation (albeit with diffusion ignored) was made in Section 5.2.1. Now, we describe a general framework for species and charge transport equations, which assists us in understanding electrophoretic
separations (Chapter 12), dynamic modeling of electrical double layers (Chapter 16), and dielectrophoresis (Chapter 17), among other topics.
In the following sections, we first describe the basic sources of species fluxes. These constitutive relations include the diffusivity (first discussed in the context of microfluidic mixing in
Chapter 4), electrophoretic mobility, and viscous mobility. The species fluxes, when applied to a control volume, lead to the basic conservation equations for species, the Nernst-Planck equations. We
then consider the sources of charge fluxes, which lead to constitutive relations for the charge fluxes and definitions of parameters such as the conductivity (first discussed in Chapter 3), as well
as the molar conductivity. Since charge in an electrolyte solution is carried by ionic species (in contrast to electrons, as is the case for metal conductors), the charge transport and species
transport equations are closely related—in fact, the charge transport equation is just a sum of species transport equations weighted by the ion valence and multiplied by the Faraday constant.
We show in this chapter that the transport parameters D, μ[EP], μ[i], σ, and Λ are all closely related, and we write equations such as the Nernst-Einstein relation to link these parameters. These
issues affect microfluidic devices because ion transport couples to and affects fluid flow in microfluidic systems. Further, many microfluidic systems are designed to manipulate and control the
distribution of dissolved analytes for concentration, chemical separation, or other purposes.
[Return to Table of Contents]
Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [
Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]
Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the
URL http://www.kirbyresearch.com/textbook.
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. | {"url":"http://www.kirbyresearch.com/index.cfm/wrap/textbook/microfluidicsnanofluidicsch11.html","timestamp":"2014-04-16T22:30:09Z","content_type":null,"content_length":"33287","record_id":"<urn:uuid:6a719419-21be-4a45-a9c6-447c0a00e572>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00593-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: Awarding theorem-credits
Shipman, Joe x2845 shipman at bny18.bloomberg.com
Mon Feb 9 14:24:53 EST 1998
> Perhaps I might here voice the disquiet felt in certain circles
> over the omission of the name of Richard Taylor in connection with
> FLT.
> 4. Wiles and Taylor, using an approach previously discarded by Wiles,
> find a new argument which supplants the faulty section of Wiles'
> previous work and finally establishes the truth of FLT.
Wiles and Taylor chose to present their joint work as two papers --
one a joint paper by Taylor and Wiles (not sure of the order of the
authors) on a key lemma and one with Wiles alone as author proving
TSC (hence FLT) and citing the previous paper. This may be simple
generosity on the part of Taylor, who realized his career was "made"
anyway without requiring a co-author credit, or it may have been
agreed to before Taylor began collaborating with Wiles.
Traditionally, one key issue for deciding the "ownership of the
theorem-credit" is the chronological one -- who proved the last
piece? Obviously Ribet, for example, provided an essential piece but
it wouldn't make sense to give him co-credit for the theorem. My
impression from reading the popular works on Wiles's proof by Singh
and Aczel is that although Taylor made an essential contribution to
the key lemma, it was Wiles who first realized that the work Taylor
had done up to that point was now enough that the proof could be
completed--in other words, he was the first one who "knew" the
theorem was true (in the philosophical sense of justified true
belief--when he though he had finished his earlier proof it was a
case of unjustified true belief but apparently this time all the
steps were correct). If this is the case, then the theorem is
Wiles's alone in this narrow traditional sense. Of course, this line
is difficult to draw in any collaboration -- one person says "I think
I've got it", the other raises an objection, details are thrashed out
together until both are satisfied--who had the last "necessary"
The other criterion for assigning theorem-credits is
"who did the bulk of the work (since the previous published stage)"?
By this criterion Ribet is out of the picture simply because his
contribution had already been published and credited to him, and
Wiles gets by far the largest share of the credit, but the only
official way of comparing co-authors' contributions is that the first
author did the most etc. and Wiles and Taylor may have felt simply
having him as the first of the two authors may have been unfair to
For all these reasons I am willing to accept the "official"
(published) credit of FLT to Wiles and the key lemma to Wiles and
Taylor (or Taylor and Wiles), even if it turns out that the
popularizations are wrong and it was Taylor who first "knew" the
A theorem about which similar issues arose is the proof that all
enumerable sets are Diophantine. Davis, Putnam, and (especially)
Julia Robinson had reduced the question to a simple
number-theoretical hypothesis which Matiyasevich proved. Because
D, P, and R had already published their contributions, M's proof of
the "Julia Robinson hypothesis" in 1970 resulted in his being
assigned the theorem-credit for resolving Hilbert's 10th problem (and
the Fields medal), although some sources give a co-credit to
Chudnovsky who apparently did the same thing independently of
Matiyasevich (perhaps Martin Davis can clear up Chudnovsky's role for
us). In the case of independent discoveries of the same theorem
there is a tradition of giving credit to both discoverers even if one
can be shown to have done it first as long as the other did it before
the first published (typically these independent co-discoveries
occurred within a year of each other, but in today's Internet era
they would have to be a lot closer than that so the phenomenon will
become rarer). Anyway, many now refer to the resolution of Hilbert's
Tenth as the "MDRP" (or some permutation thereof) theorem, which
seems to recognize that although Matiyasevich first "knew" the
theorem the piece he did was small enough relative to the earlier
work that it is more fair to include the others (Y. Manin in his book
"A Course in Mathematical Logic" credits in order Davis, Putnam,
Robinson, Matiyasevich, and Chudnovsky; this order appears to be
chronological but since M's contribution was sufficient it suggests
C's was a slightly later independent codiscovery).
Another case is the discovery of NP-completeness, originally
attributed to Cook and Karp but now revised to co-credit Leonid
Levin. (Can Steve Cook confirm that his discovery and Karp's were
independent?) It is to be hoped that mathematical communication with
and within Russia is so much better now that we won't have such confusion
for future important theorems (the old Tomsk-to-Omsk-to-Minsk-to-Pinsk
route popularized by Tom Lehrer in his song "Lobachevsky" reflected
an unfortunate reality in the 60's!).
Joe Shipman
More information about the FOM mailing list | {"url":"http://www.cs.nyu.edu/pipermail/fom/1998-February/001157.html","timestamp":"2014-04-16T22:14:05Z","content_type":null,"content_length":"7416","record_id":"<urn:uuid:2a28f87b-5cd9-426d-b7a5-cd61c241451f>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00574-ip-10-147-4-33.ec2.internal.warc.gz"} |
A class for tree types and representations of selections on tree types, as well as functions for converting between text and tree selections.
Paths and navigation
type Path = [Int]Source
A path in a tree. Each integer denotes the selection of a child; these indices are 0-relative.
up :: NavSource
Move up to parent node. Moving up from root has no effect.
Tree types
Tree selections
class Tree t => Selectable t whereSource
allowSubranges :: t -> BoolSource
Tells whether complete subranges of children may be selected in this tree. If not, valid TreeSelections in this tree always have a second element 0.
type TreeSelection = (Path, Int)Source
Selection in a tree. The path indicates the left side of the selection; the int tells how many siblings to the right are included in the selection.
Suggesting and fixing | {"url":"http://hackage.haskell.org/package/GroteTrap-0.2/docs/Language-GroteTrap-Trees.html","timestamp":"2014-04-18T03:40:30Z","content_type":null,"content_length":"20069","record_id":"<urn:uuid:eafcaf9b-05a3-467f-a999-29ed0a670dca>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00269-ip-10-147-4-33.ec2.internal.warc.gz"} |
04-03-2006 #1
Registered User
Join Date
Mar 2006
a game to transform letters read from a file to a prime number. Do and enjoy it!
The goal of this is to construct a C++ program that will read all letters from a specified file and transform them to a prime number.
The tasks you have to perform are:
(A) Read letters in sequence from a specified file, which contains 1 sentence. Transform each letter to the corresponding ASCII value.
For example, if the file contains What? then
W corresponds to 87
h corresponds to 104
a corresponds to 97
t corresponds to 116
(B) Calculate and output r1=the sum of all obtained integers (e.g. r1=87+104+97+116=404)
(C) Calculate and output r2=the length of Syracuse sequence seeded by r1.
Let n be a positive integer and f(n) be the transformation that sends n to n/2 if n is even and sends n to 3n+1 if n is odd. Starting with a positive value u called the seed, the sequence of
integers iteratively generated by f and u is called a Syracuse sequence.
For example, starting with the seed u = 1, the subsequent terms of the sequence are 4, 2, and 1. The length of the sequence (excluding the seed) is therefore 3.
For u = 4, the next terms are 2 and 1. The length is 2.
For u = 404, the next terms are 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, and 1. The length is 27.
It is conjectured (this means that we do not have a proof of this fact, only strong evidences) that for any positive seed the sequence will always reach to 1. In fact, computations show that for
any seed u less than or equal to 3*2^53, the corresponding sequence always reaches 1. The general case is unknown.
(D) Calculate and output r3=the largest prime factor of r2.
A prime factor of n is a factor of n which is a prime number. A prime number is any integer greater than 1 and only divisible by itself and 1 (e.g. 2, 3, 5, 7, 11, 13, 17 etc). For example, 3 is
the largest prime factor of 27 and 7 is the largest prime factor of 49.
Here are some example
Example 1
Enter the name of the input file: in1.txt
Press CTR-C to Leave...
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04-03-2006 #2
04-03-2006 #3
I Write C++ Apps, Sue Me.
Join Date
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In My Computer
04-03-2006 #4
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Jan 2005
04-04-2006 #5 | {"url":"http://cboard.cprogramming.com/cplusplus-programming/77670-helllpppp.html","timestamp":"2014-04-16T13:34:21Z","content_type":null,"content_length":"51953","record_id":"<urn:uuid:1e4f6f9e-e7f9-4ef5-a3b6-0ec72d9e9119>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00064-ip-10-147-4-33.ec2.internal.warc.gz"} |
FOM: Mathematics as governing intuition by formal methods
FOM: January 1 - January 31, 1998
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FOM: Mathematics as governing intuition by formal methods
This is continuation of a discussion in FOM on what is
Mathematics (after questions by Solomon Feferman and by
Stephen Cook).
Moshe' Machover:
> What characterizes mathematics in general is not only, or even mainly, its
> subject matter (which can be taken or borrowed from the most diverse
> sources) but its unique standard of argument: deductive and (ideally)
> conclusive.
Solomon Feferman says about _objectively subjective_ character of
Mathematics and defines:
> 1. Mathematics consists in reasoning about more or less clearly and
> coherently groups of objects which exist only in our imagination.
> 2. The reasoning of mathematics is logical, but mathematics is not the
> same as logic since logic concerns the nature of correct reasoning applied
> to any subject matter, whether or not that is clearly conceived.
I like these definitions.
Let me give also my own version with somewhat different nuances:
Mathematics (in a wide sense) deals with governing our intuitions
(abstractions, idealizations, imaginations, illusions, fantasies,
abilities to foresee and anticipate, etc.) on the base of appropriate
formal/deductive/axiomatic methods/systems/theories/rules.
So, the subject matter of Math. are ARBITRARY INTUITIONS and even
ILLUSIONS related to ANY kind of human activity provided they can be
governed by FORMAL METHODS.
Also this definition of Math. suggests to consciously *manipulate*
our illusions instead of canonizing them. (Illusion is a very useful
thing if not absolutized and if WE govern it, but not conversely!)
As a result, our initially "ungoverned", "raw" intuitions become much
STRONGER (this is the goal of Mathematics). May be the greatest
example to illustrate this is numerous applications in physics,
engineering, astronomy of differential and integral calculus (which
is just an advanced formalism).
The traditional Mathematics deals with rather *specific* and stable
versions of our "basic" intuitions on numbers and geometric figures
and with their numerous combinations. There is some illusion that
these intuitions are strongly fixed and that we get them and
corresponding deductive rules of reasoning "with the milk of our
mothers". However, this is rather a social-cultural process as it
was argued by Reuben Hersh:
> mathematical reality is social-cultural
and therefore intuitions may be changed or "elaborated" by any one of
us (either successful or not). What is specific in the above
definition(s?), it is that no intuition is declared as a basic or
universal one for the whole Mathematics. Only the deductive method,
i.e. using formal (or semi-formal, almost formal) systems is
considered as basic. Choose any intuition which you like and which
is formalizable in (more or less) reasonable way.
It may happen that some intuition and its formalization (like set
theory) proves to be universal in *some* respect. Then we may get
some (may be extremely strong) illusion that we have an approach to
the unique "TRUE" Mathematics. (In the Soviet Union it was the
unique TRUE doctrine of Communism. Does anybody want such one in
Mathematics?) However, according to the above definition we should
rather discuss on *adequateness* of formalisms to corresponding
intuitions or something like, and on *formal provability* instead of
"truth" (an extremely overloaded term with oversimplified two-valued
meaning having also a numerous technical counterparts in Mathematical
Logic such as forcing, etc.). Also, not all intuitions (say, of
Constructive Math.) are based on *such* a concept of truth.
We could admit, at least in principle, that some formal mathematical
system based on some intuition may have the form essentially
different from the ordinary predicate calculus + some special axioms.
What would the "truth" of a provable statement mean in this case? Thus,
it is more safe to use this word mainly in a *technical* sense so that
no confusion arise. The "intimate" correspondence of a formal system
to some intuition is something different and should be discussed and
investigated each time in specific appropriate (formal or informal)
Moreover, it seems (and actually, we should know this after Goedel!)
that there is no reason to believe that each such intuition will be
formalized in a complete way (who ever knows, what does it mean
"complete" in this context?). We just *govern* the intuition by a
formal method to reach some *specific* (not necessarily all
potentially possible and recognized some later) goals.
WAS IT THE GOAL to prove (or disprove) that all sets on the real line
are measurable when set theory (and Choice Axiom) was created and
formalized? WAS IT THE GOAL to use Induction Axiom *especially* for
proving that logarithm function is unbounded and exponential function
is total (instead of *partial recursive*) **despite** the real everyday
computational experience when this Axiom was first consciously or
non-consciously used (postulated)? (Cf. my previous postings to FOM
starting from 5 NOV 1997.) Of course, there were some different goals
of these formalizations and we often (if not always!) have some
undesirable *side effects* of any formalization. Why to pay so mach
attention to achieving the whole mathematical "TRUTH", as if we are
able to understand what does it ever mean?
In particular, I cannot agree with Martin Davis (or understand him
> Let me say it. There is mathematical *truth*. It is genuine and objective.
> It is when consensus arises *because* mathematical truth has been attained
> that the subject advances. Some may believe the dogmas of a religion as
> fervently as Lagrange's theorem. But when pressed it will be admited that it
> is a matter of "faith". My belief in the truth of Lagrange's theorem is not
> a matter of faith.
Of course no "faith", no "dogma"!. But what about using the term
"provability" which is quite "genuine and objective" notion (wrt any
fixed formal system) instead of "truth"? Even if the majority of our
"questions" to a formal system on provability of a sentence or its
negation usually have a positive answer (as for PA vs. ZFC), why
should we make the conclusion that we are much more near in this
system to corresponding mathematical TRUTH and to give to this "fact"
too high value? Who of working mathematicians *really* needs this
phantom of FULL MATHEMATICAL TRUTH? There are two things to say.
First, I think that we *actually need*
only the *illusion* of full mathematical truth as a
very comfortable for our mind psychological support or a kind of
doping. This illusion is based mainly on *formal* rules of the
classical logic (such as the lows of excluded middle, double
negation, etc.) which all mathematicians use everyday even if they know
nothing on its existence. But they, of course, know! We start to
learn these lows at school, say, when proving first time the simplest
geometrical theorems, and even earlier. (However, these lows are
rather specific, almost not used in our everyday, non-mathematical
life or too artificial as A => B iff ~A v B.) Say, I myself recall
to my students their own mathematical experience from school when
teaching them to (natural deduction rules of) logic and argue that
they actually know these rules and have used them many times. Is not
using classical (or, in principle, any other) logic the "consensus" to
which Hersh appealed in his description of Math.?
Second, mathematicians are rather dealing with some *narrow class of
specific problems* grouped according to some, also specific goals.
What for then we need the FULL MATHEMATICAL TRUTH, if not as an
illusion for comfortable work?
It was extremely important to understand that (and especially how and
why) CH is independent of ZFC. Of course, there are many interesting
and really great, deep and fascinating problems related with more
proper and deep understanding the place of CH in set theory and in
Mathematics, whether its alternatives are true/false,
inevitable/refutable in that or other specific technical (natural or
artificial) sense, to find some reasons pro and contra, etc. All
this experience may prove to be very useful even for quite different
researches in f.o.m. (Note, that I myself do not work in this very
direction and therefore cannot properly evaluate what is happening
However, there is also somewhat alternative way to f.o.m: use all the
gained experience and highly elaborated techniques and ideas to
develop various NEW (i.e. not related directly with traditional systems
PA or ZFC, etc) "basic" intuitions and formalisms, say, in connection
with Theoretical-Mathematical Computer Science (as it were NEW and
non-traditional at that time the Cantorian (or ZFC) Set Theory or
Robinson's Nonstandard Analysis playing the role of NEW foundations for
basic mathematical, intuitions, notions and approaches).
For example, I believe, it is important to work on reasonable
*foundational* approaches related to computational complexity theory
which deals with bounded resources. On the foundational level this would
mean *reconsidering* our intuitions and formalisms concerning to the
very nature of finite objects (say, via systems of Bounded Arithmetic
or Arithmetic of "Feasible" natural numbers; cf. my previous postings to
FOM starting from 5 NOV 1997).
Vladimir Sazonov
Program Systems Institute, | Tel. +7-08535-98945 (Inst.),
Russian Acad. of Sci. | Fax. +7-08535-20566
Pereslavl-Zalessky, | e-mail: sazonov@logic.botik.ru
152140, RUSSIA | http://www.botik.ru/~logic/SAZONOV/
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Does the identity element form a group on it's own?
May 2nd 2011, 04:22 AM #1
Apr 2008
Does the identity element form a group on it's own?
Does a set consisting only of the Identity element under a given operation form a group (under the given operation)?
Am I correct in thinking that it would:
1) Identity element exists by assumption
2) All g in G have an inverse (the identity is its own inverse)
3) composition under the operation of the identity is associative
Does a set consisting only of the Identity element under a given operation form a group (under the given operation)?
Am I correct in thinking that it would:
1) Identity element exists by assumption
2) All g in G have an inverse (the identity is its own inverse)
3) composition under the operation of the identity is associative
Yes. It's called the trivial group.
Also note that any subgroup of a (different/larger) group must contain "e".
Does a set consisting only of the Identity element under a given operation form a group (under the given operation)?
Am I correct in thinking that it would:
1) Identity element exists by assumption
2) All g in G have an inverse (the identity is its own inverse)
3) composition under the operation of the identity is associative
You forgot closure.
A subgroup consisting only of the identity is, along with the "subgroup" created by all elements of the group itself, is called a "trivial subgroup." Yes, it is a subgroup.
A "proper subgroup" is a subgroup that is not just the identity, nor the entire group.
Is this common terminology outside of pure mathematics? I have never once seen an author exclude the trivial sub-object from the class of proper sub-objects, and I have likewise never seen the
object itself referred to as a trivial sub-object.
it is quite common yes. for example, excluding the trivial subgroup as a proper subgroup allows the characterization of a simple group as a group with no proper normal subgroups, rather than a
group with no nontrivial proper normal subgroups. however, this can vary from author to author.
In Physics it depends. "Standard" Physics, that is to say the usual mixture of theory and experimentation, does not typically make a distinction. Mathematical Physics on the other hand is much
more thorough, even if not as much as Mathematics. My Math Physics books make the distinction, my graduate Quantum Physics book (the "standard text" in the field) does not.
May 2nd 2011, 04:33 AM #2
May 2nd 2011, 04:33 AM #3
May 2nd 2011, 11:49 PM #4
Apr 2011
May 3rd 2011, 04:50 AM #5
MHF Contributor
Mar 2011
May 3rd 2011, 07:37 AM #6 | {"url":"http://mathhelpforum.com/advanced-algebra/179237-does-identity-element-form-group-s-own.html","timestamp":"2014-04-18T00:58:37Z","content_type":null,"content_length":"49613","record_id":"<urn:uuid:2b9e1c2c-7f86-45c1-ab40-5f603705fa2c>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00307-ip-10-147-4-33.ec2.internal.warc.gz"} |
Implicit Differation and Chain Rule
June 7th 2008, 10:44 AM #1
Jun 2008
Implicit Differation and Chain Rule
First, I really need help with implicit differentiation. I tried reading in my book, but they use very basic problems and I did not find it helpful.
The problem I need help with is:
x^2 + xy - y^2 = 4 Solver for y'
I can not get the correct answer. I got (2x)/(x-2y)=y' (This is incorrect).
Next I need help with the chain rule. I understand how to do this, but I am having problems with one question. The question is:
Sqrt (2x)/(x^2+1.8)) <- For this problem, everything is under the radical.
Thank you!!!
First, I really need help with implicit differentiation. I tried reading in my book, but they use very basic problems and I did not find it helpful.
The problem I need help with is:
x^2 + xy - y^2 = 4 Solver for y'
I can not get the correct answer. I got (2x)/(x-2y)=y' (This is incorrect).
Your only mistake (that I found by analysing your answer, so sorry if you did it right..) is that the derivative of xy has to be taken with the product rule :
As soon as the variable intervenes in x, y, z, t, or whatever, you have to consider it as a function, not like a variable, so you're likely to use the product, quotient or chain rule =)
Next I need help with the chain rule. I understand how to do this, but I am having problems with one question. The question is:
Sqrt (2x)/(x^2+1.8)) <- For this problem, everything is under the radical.
Thank you!!!
The chain rule says that the derivative is $\frac{u'(x)}{2 \sqrt{u(x)}}$
So calculate $u'(x)$ with the quotient rule.
Always do things step by step. It's pretty tricky for someone who is not used to differentiating to do it without taking u(x).
Thanks so much
That was stupid of me to mess up the xy (product rule). Doh! Thanks for catching that mistake.
Thanks a lot for the help!!!
One more quick question:
So I solved chain rule problem out and got an answer of:
Just curious if I did it correct. Once again, Thanks!
Hmm; that's a weird one.
What do you find as the derivative $u'(x)$ ?
Because I doubt it is $\frac 2x$ /:
Can you show your working so that I can tell you where the mistake(s) is(are) ?
Of course
Okay, I see how I messed up.
Heres my current solution attempt :P
For u'(x) I have this:
Quotient rule says: gf'-fg'/g^2
When I plug my info in I got:
Now I plug that into u'(x)/(2Sqrt u(x))
I got
Might be just me, but I think I did something wrong again.
Okay, I see how I messed up.
Heres my current solution attempt :P
For u'(x) I have this:
Quotient rule says: gf'-fg'/g^2
When I plug my info in I got:
Now I plug that into u'(x)/(2Sqrt u(x))
I got
Might be just me, but I think I did something wrong again.
Ignoring the missing parentheses, this is correct
Note that you can write $-2x^2+3.6=2(-x^2+1.8)$ and simplify the fraction by 2
The solution is :
$\frac{2(-x^2+1.8)}{2(x^2+1.8)^2 \sqrt{\frac{2x}{x^2+1.8}}}=\frac{-x^2+1.8}{(x^2+1.8)^2 \cdot \sqrt{2x} \cdot (x^2+1.8)^{\color{red}-\frac 12}}$
do you understand the red part ?
$\frac{1}{a^b}=a^{-b}$, and $\sqrt{a}=a^{\frac 12}$
Simplifying the powers by this rule : $a^b a^c=a^{b+c}$, we get :
$f'(x)=\frac{-x^2+1.8}{\sqrt{2x} \cdot (x^2+1.8)^{\frac 32}}$
Now it depends on how you want to display the result
Yes the red part was you moving the denominator to the numerator. Thanks so much and sorry about the missing parentheses. If I knew you, I would buy you a drink
First, I really need help with implicit differentiation. I tried reading in my book, but they use very basic problems and I did not find it helpful.
The problem I need help with is:
x^2 + xy - y^2 = 4 Solver for y'
I can not get the correct answer. I got (2x)/(x-2y)=y' (This is incorrect).
Next I need help with the chain rule. I understand how to do this, but I am having problems with one question. The question is:
Sqrt (2x)/(x^2+1.8)) <- For this problem, everything is under the radical.
Thank you!!!
Alternatively for the second one, Let
$y=\sqrt{\frac{2x}{x^2+1.8}}\Rightarrow{y^2=\frac{2 x}{x^2+1.8}}$
So differentiating we get
and now remembering that y is our original equation we see
Same concept, just a little less messy to write
Last edited by Mathstud28; June 7th 2008 at 12:07 PM.
i kind of see what you did. I understand squaring both sides and then taking the derivative. I got lost after you simplified.
how did you go from 2(x^2+1.8)-(2x)(2x) to -2(x^2-1.8) (i left out the denominator to be less confusing)
Second, is that actually supposed to be a 4.8 in the numerator in your final answer or is it a typo. *EDIT* never mind. i see you changed it. now that part makes more sense :P
i kind of see what you did. I understand squaring both sides and then taking the derivative. I got lost after you simplified.
how did you go from 2(x^2+1.8)-(2x)(2x) to -2(x^2-1.8) (i left out the denominator to be less confusing)
Second, is that actually supposed to be a 4.8 in the numerator in your final answer or is it a typo.
That is a typo that I fixed, and secondly
Ha. Not sure how I didn't see that
June 7th 2008, 10:51 AM #2
June 7th 2008, 10:59 AM #3
Jun 2008
June 7th 2008, 11:11 AM #4
Jun 2008
June 7th 2008, 11:17 AM #5
June 7th 2008, 11:29 AM #6
Jun 2008
June 7th 2008, 11:35 AM #7
June 7th 2008, 11:39 AM #8
Jun 2008
June 7th 2008, 11:47 AM #9
June 7th 2008, 11:48 AM #10
June 7th 2008, 12:09 PM #11
Jun 2008
June 7th 2008, 12:11 PM #12
June 7th 2008, 12:16 PM #13
Jun 2008 | {"url":"http://mathhelpforum.com/calculus/40903-implicit-differation-chain-rule.html","timestamp":"2014-04-16T08:13:27Z","content_type":null,"content_length":"77034","record_id":"<urn:uuid:7524f650-9835-48f2-bde7-7679f517e09f>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00380-ip-10-147-4-33.ec2.internal.warc.gz"} |
Answers: Skewness
Back to Problems
1. A box contains these five tickets:
0, 0, 0, 0, 5
a) What is the average of the box?
b) What is the standard deviation of the box?
c) Does the box have a median? If so, what is it?
d) Does the box have a mode? If so, what is it?
e) Is this set of numbers symmetrical or asymmetrical? If it is asymmetrical, does the tail extend to the right or to the left?
mean = 1, sd = 2 if it is population, sqrt(5) if it is a sample; the median and mode are both 0, it is asymmetrical, and the tail is to the right with 5 as the extreme value.
2. Pearson's coefficient of skewness is 3*(mean-median)/(standard deviation).
a) Compute it for these numbers: {0, -1, 10, 7, 2, 2, 1}.
Is this distribution symmetrical, or does it have a tail to the right or to the left?
b) Compute it for these numbers: {-6, 19, 0, 8, 1, 2, 4}.
Is this distribution symmetrical, or does it have a tail to the right or to the left?
c) Do the same with {17, 18, 20, 21, 40, 33, 27, 23, 21}.
d) What is the logic behind this measure? Why subtract the median from the mean? Why divide by the standard deviation?
a) mean = 3, median = 2, sd = 4; skewness = .75; tail to the right;
b) mean = 4, median = 2, sd = 7.85; skewness = .25; tail to the right;
c) mean = 24.44, median = 21, sd = 7.62; skewness = 1.35; tail to the right.
d) Extreme values affect the mean but not the median, so if the tail is to the right, the mean will exceed the median. Dividing by the standard deviation eliminates the units of measurement, which is
also done when computing z-scores, the topic of the next section.
3. The U.S. Geological Survey keeps track of river flow throughout the United States. One of its tracking stations is just east of Rensselaer, Indiana and on a January 22 a few years ago it reported
that the flow of the mighty Iroquois River was 445 cubic feet per second. Looking at the past 55 years, the lowest flow was 11 cubic feet per second, and the highest on record was 1670 cubic feet per
second. The average on January 22 for the previous 55 years was 193 cubic feet per second, while the median flow was 105 cubic feet per second.
Based on this information, if we graphed the data from the these 55 years on how much water was flowing in the river, which of the following graphs would be most like the one that we would obtain.
Explain how you get your answer.
Most of the data are a bit below or a bit above 100. However, there are some years in which there are floods, and those data are extremes the pull the mean up well above the median. Hence, the
distribution will look like the first chart, with the long tail to the right.
4. If a distribution is skewed to the left (meaning it has a long tail to the left), which is greatest: the mean, median, or mode? Which is smallest?
The mean is pulled by the extreme values, the mean and mode are not. So if there are extreme values are small, the mean will be less than the median, and if the extreme values are large, the mean
will be greater than the median.
5. After handing back a test a professor noted that the distribution of scores was positively skewed. On leaving class, one of your friends who has never had statistics turns to you and says, "Even
the prof thought that this test was rotten. He said was positively screwed up." How would you explain to your friend what the professor really meant?
You can tell him that positively skewed means that there were many low scores and only a few really high scores. Whether your friend will understand or not depends on the intelligence of the people
you hang with.
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The Ostaszewski square, and homogeneous Souslin trees
Posted on May 18, 2011 by saf in categories: Publications.
Abstract: Assume GCH and let $\lambda$ denote an uncountable cardinal.
We prove that if $\square_\lambda$ holds, then this may be witnessed by a coherent sequence $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing
For every sequence $\langle A_i\mid i<\lambda\rangle$ of unbounded subsets of $\lambda^+$, and every limit $\theta<\lambda$, there exists some $\alpha<\lambda^+$ such that $\text{otp}(C_\alpha)=\
theta$, and the $(i+1)_{th}$-element of $C_\alpha$ is a member of $A_i$, for all $i<\theta$.
As an application, we introduce the first construction of an homogeneous Souslin tree at the successor of a singular cardinal.
In addition, as a by-product, a theorem of Farah and Velickovic (see [FV]), and a theorem of Abraham, Shelah and Solovay (see [AShS:221]) are generalized to cover the case of successors of regulars
2 Responses to The Ostaszewski square, and homogeneous Souslin trees
1. saf says:
Submitted to Israel Journal of Mathematics, May 2011.
Accepted April 2013.
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Sofronie-Stokkermans, Viorica - Max-Planck-Institut für Informatik
• Efficient Hierarchical Reasoning about Functions over Numerical Domains
• Representation theorems and the semantics of non-classical logics, and applications to
• Introduction The goal of this thesis is to study the applications of fibered structures in com
• Automated reasoning in some local extensions of ordered structures Viorica Sofronie-Stokkermans, Carsten Ihlemann
• Representation theorems and the semantics of
• A Superposition Decision Procedure for the Guarded Fragment with Equality Harald Ganzinger
• 2.10 Refutational Completeness of Resolution How to show refutational completeness of propositional
• Part 2: First-Order Logic First-order logic
• Chaining Techniques for Automated Theorem Proving in ManyValued Logics Harald Ganzinger, Viorica SofronieStokkermans
• Resolutionbased decision procedures for the positive theory of some finitely generated varieties of algebras
• Automated theorem proving by resolution in non-classical logics
• Towards a Sheaf Semantics for Cooperating Agents Scenarios
• 2.4 Algorithmic Problems Validity(F): |= F ?
• Interpolation in local theory extensions Viorica Sofronie-Stokkermans
• Resolution-based Theorem Proving for SH n -Logics
• Representation Theorems and Theorem Proving in NonClassical Logics Viorica SofronieStokkermans
• On Local Reasoning in Verification Carsten Ihlemann, Swen Jacobs, and Viorica Sofronie-Stokkermans
• Applications of hierarchical reasoning in the verification of complex systems
• Interpolation in local theory extensions Viorica SofronieStokkermans
• On the Universal Theory of Varieties of Distributive Lattices with Operators
• Representation theorems and the semantics of
• Conclusions and Plans of Future Work
• The 21st Conference on Automated Deduction Automated Deduction
• 4th International Joint Conference on Automated Reasoning Sydney, Australia, August 1015, 2008
• 22nd International Conference on Automated Deduction McGill University, Montreal, Canada
• UNIVERSITY OF OSLO Department of Informatics
• Hierarchical and Modular Reasoning in Complex Theories: The Case of Local Theory Extensions
• Automated reasoning in some local extensions of ordered structures
• Electronic Notes in Theoretical Computer Science Proceedings of the Fourth International Workshop
• 2.12 Ordered Resolution with Selection Motivation: Search space for Res very large.
• Constraint Solving for Interpolation Andrey Rybalchenko1,2
• Representation Theorems and Automated Theorem Proving
• Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of
• Part 3: First-Order Logic with Equality Equality is the most important relation in mathematics and
• Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of
• Journal Symbolic Computation (2003) 891--924 www.elsevier.com/locate/jsc Resolutionbased decision procedures the
• Fibered Structures and Applications to Automated Theorem Proving in Certain Classes of
• Modeling Interaction by Sheaves and Geometric Logic
• Priestley Duality for SHn-Algebras and Applications to the Study of Kripke-Style Models for SHn-Logics
• On uni cation for bounded distributive lattices Viorica Sofronie-Stokkermans
• Hierarchical and modular reasoning in complex theories: The case of local theory extensions
• Sheaves and geometric logic and applications to modular verification of complex systems1
• Automated Theorem Proving by Resolution for Finitely-Valued Logics Based on
• Local reasoning in verification Viorica Sofronie-Stokkermans | {"url":"http://www.osti.gov/eprints/topicpages/documents/starturl/38/421.html","timestamp":"2014-04-16T19:12:02Z","content_type":null,"content_length":"13773","record_id":"<urn:uuid:7d6d0a2a-2d77-4a22-8c2b-3f09095d1b0d>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00637-ip-10-147-4-33.ec2.internal.warc.gz"} |
used books, rare books and new books
Book summary:
Sir Isaac Newton's Principia Mathematica (Mathematical Principles) (1687) is considered to be among the finest scientific works ever published. His grand unifying idea of gravitation, with effects
extending throughout the solar system, explains by one principle such diverse phenomena as the tides, the precession of the equinoxes, and the irregularities of the moon's motion.
Newton's brilliant and revolutionary contributions to science explained the workings of a large part of inanimate nature mathematically and suggested that the remainder might be understood in a
similar fashion. By taking known facts, forming a theory that explained them in mathematical terms, deducing consequences from the theory, and comparing the results with observed and experimental
facts, Newton united, for the first time, the explication of physical phenomena with the means of prediction. By beginning with the physical axioms of the laws of motion and gravitation, he converted
physics from a mere science of explanation into a general mathematical system. [via] | {"url":"http://www.bookfinder.com/author/andrew-motte/","timestamp":"2014-04-21T04:37:09Z","content_type":null,"content_length":"49044","record_id":"<urn:uuid:3fa61dd6-ff45-4bd0-bd1b-f514d49ab04e>","cc-path":"CC-MAIN-2014-15/segments/1397609539493.17/warc/CC-MAIN-20140416005219-00661-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mathematics at College of San Mateo - Careers in Mathematics - Applied Mathematician
Careers in Mathematics -
Applied Mathematician
What is applied mathematics?
Applied mathematics is mathematics applied to real world problems. Many areas of mathematics have applications to the world. Some areas of mathematics deal very heavily with application, for example
numerical analysis (the mathematics of getting good approximations), differential equations (with lots of applications in physics and engineering, among other areas), and linear algebra.
What kinds of problems do applied mathematicians work on?
An applied mathematician starts with a real world problem posed by an engineer, a biologist, a business executive, or whoever else needs a problem solved. The Mathematician builds a mathematical
model from the information surrounding the problem and uses it to get solutions. Then the model may be tested and improved.
Who hires applied mathematicians?
Some employers of applied mathematicians include computer companies (hardware and software), engineering firms, aerospace companies, financial companies, government agencies, and many, many others.
Where can you study applied mathematics?
Many university math departments offer an applied math major as well as a pure math major. Here are some.
San Francisco State University
Sonoma State University
University of California at Berkeley
University of California at Santa Cruz
University of California at Davis
Stanford University
Meet an applied mathematician.
Meet another applied mathematician. Meet yet another applied mathematician. | {"url":"http://collegeofsanmateo.edu/math/appliedmathematician.asp","timestamp":"2014-04-16T10:18:12Z","content_type":null,"content_length":"21010","record_id":"<urn:uuid:a9fbc85a-a20e-46ed-8240-8038c2cee377>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00395-ip-10-147-4-33.ec2.internal.warc.gz"} |
Power large Sample Standardized? [Archive] - Statistics Help @ Talk Stats Forum
03-03-2010, 07:50 PM
This is probably staring me in the face, but my face has been staring at it for many hours ... ;)
In basic power analysis of a mean, using the CLT, testing a lower tail hypothesis I have:
X_lower_critical = qnorm(alpha | mu_null, sigma/sqrt(n))
and also
X_lower_critical = qnorm(1-beta | mu_alternative, sigma/sqrt(n))
where qnorm is the quantile with alpha (or 1-beta) area to the left.
The author then says "standardizing X_lower_critical both way":
mu_null - qnorm(alpha | 0,1) * (sigma/sqrt(n)) =
mu_alternative + qnorm(1-beta | 0,1) * (sigma/sqrt(n))
Can anyone help me, how are they getting this last step? | {"url":"http://www.talkstats.com/archive/index.php/t-11152.html","timestamp":"2014-04-17T03:49:18Z","content_type":null,"content_length":"4730","record_id":"<urn:uuid:61500d03-8e65-418b-8e4b-bc538af215c5>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00533-ip-10-147-4-33.ec2.internal.warc.gz"} |
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Laurent Series
1. The problem statement, all variables and given/known data
1) Find the Laurent series for (z^2)*cos(1/3z) in the region [tex]\left|z\right|[/tex]
2) Find the Laurent series expansion of (z^2 - 1)^(-2) valid in the following region
a) 0 < [tex]\left|z - 1\right|[/tex] < 2
b) [tex]\left|z + 1\right|[/tex] > 2
2. Relevant equations
3. The attempt at a solution
I did all the work and got the answer, so I basically just want to check that I got it right.
It is pretty hard to write all the steps that I did, so I will briefly write it down.
1) Since z takes infinitely many inverse within the range, we use w = 1/z
then (z^2)*cos(1/3z) becomes cos(w/3)/(w^2).
i) we find the taylor series of cos(w/3),
1 - (w^2)/((3^2)*(2!)) + (w^4)/((3^4)*(4!)) - (w^6)/((3^6)*(6!)) + .....
ii) multiply the series in i) by 1/(w^2),
1/(w^2) - 1/((3^2)*(2!)) + (w^2)/((3^4)*(4!)) - (w^4)/((3^6)*(6!)) + .....
iii) since the series starts at third term, we can just simplify first and second term, and find series,
-1/18 + z^2 + [tex]\sum[/tex]( n=2 to infinity) ((-1)^n)*(w^n)) / ((3^2n)*(2n!))
iiii) now sub back w = 1/z, then the Laurent Series is,
f(z) = -1/18 + z^2 + [tex]\sum[/tex]( n=2 to infinity) ((-1)^n) / ((z^n)*(3^2n)*(2n!))
2)a) since z converges everywhere in this range, the Laurent Series is simply
f(z) = 1 / (4*(z+1)) + 1 / (4*(z+1)^2) - 1 / (4*(z - 1)) + 1 / (4*(z-1)^2)
b) Since z takes infinitely many inverse within the range, I did same way as 1).
let w = 1/z and find the taylor series of each term in a) (4 total) and sub back w = 1/z at the last step. After all this, I got,
f(z) = [tex]\sum[/tex](n=1 to infinity) ((-1)^(n+1)) / (4*z^n) + [tex]\sum[/tex](n=2 to infinity) ((n-1)*(-1)^n) / (4*z^n) - [tex]\sum[/tex](n=1 to infinity) (1/(4*z^n)) + [tex]\sum[/tex](n=2 to
infinity) (n-1) / (4*z^n)
Sorry about the mass...I tried to use the code and symbol, but I just failed to do like other people...
Please tell me if I did something wrong.
Thank you. | {"url":"http://www.physicsforums.com/showthread.php?t=438360","timestamp":"2014-04-17T00:58:58Z","content_type":null,"content_length":"21245","record_id":"<urn:uuid:9d844cde-e323-4684-a515-e56911aecdcd>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00576-ip-10-147-4-33.ec2.internal.warc.gz"} |
Average value function help
Hey Can you please tell me if the integration in my photo is correct Thank you Muchas gracias
Hello grandad Thank you for welcome I think I have the correct answer now,from your kind help g(x) functions answer would be a^2/(3), and the first part f(x) works out a/(2) is this correct please? | {"url":"http://mathhelpforum.com/calculus/131806-average-value-function-help-print.html","timestamp":"2014-04-20T09:30:13Z","content_type":null,"content_length":"6245","record_id":"<urn:uuid:da305c7c-a25a-4985-8100-afb1705451a4>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00277-ip-10-147-4-33.ec2.internal.warc.gz"} |
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A Treatise on Electricity and Magnetism/Part IV/Chapter II
From Wikisource
←Electromagnetic Force A Treatise on Electricity and Magnetism by On the Induction of Electric Currents→
Ampère's Investigation of the Mutual Action of Electromagnetic Currents
502.] WE have considered in the last chapter the nature of the magnetic field produced by an electric current, and the mechanical action on a conductor carrying an electric current placed in a mag
netic field. From this we went on to consider the action of one electric circuit upon another, by determining the action on the first due to the magnetic field produced by the second. But the action
of one circuit upon another was originally investigated in a direct manner by Ampere almost immediately after the publication of Orsted s discovery. We shall therefore give an outline of Ampere s
method, resuming the method of this treatise in the next chapter.
The ideas which guided Ampere belong to the system which admits direct action at a distance, and we shall find that a remark able course of speculation and investigation founded on these ideas has
been carried on by Gauss, Weber, J. Neumann, Riemann, Betti, C. Neumann, Lorenz, and others, with very remarkable results both in the discovery of new facts and in the formation of a theory of
electricity. See Arts. 846-866.
The ideas which I have attempted to follow out are those of action through a medium from one portion to the contiguous portion. These ideas were much employed by Faraday, and the development of them
in a mathematical form, and the comparison of the results with known facts, have been my aim in several published papers. The comparison, from a philosophical point of view, of the results of two
methods so completely opposed in their first prin ciples must lead to valuable data for the study of the conditions of scientific speculation.
503.] Ampere s theory of the mutual action of electric currents is founded on four experimental facts and one assumption.
�� � 505.] AMPERE S SCIENTIFIC METHOD. 147
Ampere s fundamental experiments are all of them examples of what has been called the null method of comparing forces. See Art. 214. Instead of measuring the force by the dynamical effect of
communicating motion to a body, or the statical method of placing it in equilibrium with the weight of a body or the elasticity of a fibre, in the null method two forces, due to the same source, are
made to act simultaneously on a body already in equilibrium, and no effect is produced, which shews that these forces are them selves in equilibrium. This method is peculiarly valuable for comparing
the effects of the electric current when it passes through circuits of different forms. By connecting all the conductors in one continuous series, we ensure that the strength of the current is the
same at every point of its course, and since the current begins everywhere throughout its course almost at the same instant, we may prove that the forces due to its action on a suspended body are in
equilibrium by observing that the body is not at all affected by the starting or the stopping of the current.
504.] Ampere s balance consists of a light frame capable of revolving about a vertical axis, and carrying a wire which forms two circuits of equal area, in the same plane or in parallel planes, in
which the current flows in opposite directions. The object of this arrangement is to get rid of the effects of terrestrial magnetism on the conducting wire. When an electric circuit is free to move
it tends to place itself so as to embrace the largest possible number of the lines of induction. If these lines are due to terrestrial magnetism, this position, for a circuit in a vertical plane,
will be when the plane of the circuit is east and west, and when the direction of the current is opposed to the apparent course of the sun.
By rigidly connecting two circuits of equal area in parallel planes, in which equal currents run in opposite directions, a combination is formed which is unaffected by terrestrial magnetism, and is
therefore called an Astatic Combination, see Fig. 26. It is acted on, however, by forces arising from currents or magnets which are so near it that they act differently on the two circuits.
505.] Ampere s first experiment is on the effect of two equal currents close together in opposite directions. A wire covered with insulating material is doubled on itself, and placed near one of the
circuits of the astatic balance. When a current is made to pass through the wire and the balance, the equilibrium of the balance remains undisturbed, shewing that two equal currents close together
L 2
�� � 148
��AMPERES THEORY.
��in opposite directions neutralize each other. If, instead of two wires side by side, a wire be insulated in the middle of a metal
���Fig. 26.
��tube, and if the current pass through the wire and back by the tube, the action outside the tube is not only approximately but accurately null. This principle is of great importance in the con
struction of electric apparatus, as it affords the means of conveying the current to and from any galvanometer or other instrument in such a way that no electromagnetic effect is produced by the
current on its passage to and from the instrument. In practice it is gene rally sufficient to bind the wires together, care being taken that they are kept perfectly insulated from each other, but
where they must pass near any sensitive part of the apparatus it is better to make one of the conductors a tube and the other a wire inside it. See Art. 683.
506.] In Ampere s second experiment one of the wires is bent and crooked with a number of small sinuosities, but so that in every part of its course it remains very near the straight wire. A current,
flowing through the crooked wire and back again through the straight wire, is found to be without influence on the astatic balance. This proves that the effect of the current running through any
crooked part of the wire is equivalent to the same current running in the straight line joining its extremities, pro vided the crooked line is in no part of its course far from the straight one.
Hence any small element of a circuit is equivalent to two or more component elements, the relation between the component elements and the resultant element being the same as that between component
and resultant displacements or velocities.
507.] In the third experiment a conductor capable of moving
�� � 5 o8.]
��FOUR EXPERIMENTS.
��only in the direction of its length is substituted for the astatic balance, the current enters the conductor and leaves it at fixed points of space, and it is found that no closed circuit placed in
the neighbourhood is able to move the conductor.
���u U
Fig. 27.
The conductor in this experiment is a wire in the form of a circular arc suspended on a frame which is capable of rotation about a vertical axis. The circular arc is horizontal, and its centre
coincides with the vertical axis. Two small troughs are filled with mercury till the convex surface of the mercury rises above the level of the troughs. The troughs are placed under the circular arc
and adjusted till the mercury touches the wire, which is of copper well amalgamated. The current is made to enter one of these troughs, to traverse the part of the circular arc between the troughs,
and to escape by the other trough. Thus part of the circular arc is traversed by the current, and the arc is at the same time capable of moving with considerable freedom in the direc tion of its
length. Any closed currents or magnets may now be made to approach the moveable conductor without producing the slightest tendency to move it in the direction of its length.
508.] In the fourth experiment with the astatic balance two circuits are employed, each similar to one of those in the balance, but one of them, C, having dimensions n times greater, and the other,
A, n times less. These are placed on opposite sides of the circuit of the balance, which we shall call B, so that they are similarly placed with respect to it, the distance of 6 from B being n times
greater than the distance of B from A. The direction and
�� � 150
��AMPERES THEORY.
��[ 5 08.
��strength of the current is the same in A and C. Its direction in B may be the same or opposite. Under these circumstances it is found that B is in equilibrium under the action of A and C, whatever
be the forms and distances of the three circuits, provided they have the relations given above.
Since the actions between the complete circuits may be considered to be due to actions between the elements of the circuits, we may use the following method of determining the law of these actions.
Let A 19 BI, C v Fig. 28, be corresponding elements of the three circuits, and let A 2) B 2 , C 2 be also corresponding elements in an other part of the circuits. Then the situation of J3 1 with
respect to A 2 is similar to the situation of C 1 with respect to B> 2 , but the
���Fig. 28.
distance and dimensions of C and B 2 are n times the distance and dimensions of B and A 2 , respectively. If the law of electromag netic action is a function of the distance, then the action, what
ever be its form or quality, between B 1 and A. 2 , may be written
��and that between C l and B 2
F = C^.
where , b, c are the strengths of the currents in A, B, C. But
and a c. Hence
��nB^ C 19 nA 2 = B 2) n 1 A 2 =
F = n*B l .A 2 f(nB^A 2 )ab, and this is equal to F by experiment, so that we have
��or, the force varies inversely as the square of the distance.
�� � 511.] FOKCE. BETWEEN TWO ELEMENTS. 151
509.] It may be observed with reference to these experiments that every electric current forms a closed circuit. The currents used by Ampere, being produced by the voltaic battery, were of course in
closed circuits. It might be supposed that in the case of the current of discharge of a conductor by a spark we might have a current forming an open finite line, but according to the views of this
book even this case is that of a closed circuit. No experiments on the mutual action of unclosed currents have been made. Hence no statement about the mutual action of two ele ments of circuits can
be said to rest on purely experimental grounds. It is true we may render a portion of a circuit moveable, so as to ascertain the action of the other currents upon it, but these cur rents, together
with that in the moveable portion, necessarily form closed circuits, so that the ultimate result of the experiment is the action of one or more closed currents upon the whole or a part of a closed
510.] In the analysis of the phenomena, however, we may re gard the action of a closed circuit on an element of itself or of another circuit as the resultant of a number of separate forces, depending
on the separate parts into which the first circuit may be conceived, for mathematical purposes, to be divided.
This is a merely mathematical analysis of the action, and is therefore perfectly legitimate, whether these forces can really act separately or not.
511.] We shall begin by considering the purely geometrical relations between two lines in space representing the circuits, and between elementary portions of these lines.
Let there be two curves in space in each of which a fixed point is taken from which the arcs are measured in a defined direction along the curve. Let A, A be these points. Let P Q and F Q be elements
of the two curves.
Let AP = s, A P =s , } PQ = ds, P<g=dJ,\ and let the distance PF be de- Eig> 29
noted by r. Let the angle PPQ be denoted by 6, and PFQ by tf, and let the angle between the planes of these angles be denoted by 77.
The relative position of the two elements is sufficiently defined by their distance r and the three angles 0, Q , and 77, for if these be
��� � 152 AMPERE S THEORY. [5 12 -
given their relative position is as completely determined as if they formed part of the same rigid body.
512.] If we use rectangular coordinates and make so, y, z the coordinates of P, and x } y > z those of P , and if we denote by I, m, n and by , m , ri the direction-cosines of PQ, and of P Q re
spectively, then
dx dy dz
-=- = I, -f- m, = n,
as as as
dx dy dz ,
��and l-(x x) + m (y y) + n(z z) =. rcos0, \
l f (afso) + m (if y) + n (sf z) = -rcos0 , ( (3)
II -f mm -f nn = cos e,
where e is the angle between the directions of the elements them selves, and
cos e = cos 6 cos & + sin 9 sin & cos 17. (4)
Again r 2 = (x -x? + (/-J/) 2 + (/-*) 2 , (5)
��. dr . , dx . , dy . , dz whence r " = (x as) -= (y y) - (z z )~r
= _ rc os0. ,,,;, ^
cv -i i ^* / / \ && iff \ f ^y , / f \ z
Similarly r -=-.-=> (x )TT-r(y yj ~r^ + (^ ^) -rr* ^/ ; ^/ v<y J} ds ds
and differentiating r -y- with respect to /, ds
d 2 r dr dr _ dx dx dy dy dz dz dsds f ds ds ~ ds ds f ds ds ds ds , .
m m -f- n n ) cos e.
We can therefore express the three angles 0, 6 , and r], and the auxiliary angle e in terms of the differential coefficients of r with respect to <$ and / as follows,
cos = j- 3
ds d 2 r dr dr
��cos e = r
��sin sin cos 77 = r
��ds ds ds ds d*r
��ds ds
�� � 5 1 3.] GEOMETRICAL RELATIONS OF TWO ELEMENTS. 153
513.] We shall next consider in what way it is mathematically conceivable that the elements PQ and P f Q f might act on each other, and in doing so we shall not at first assume that their mutual
action is necessarily in the line joining them.
We have seen that we may suppose each element resolved into other elements, provided that these components, when combined according to the rule of addition of vectors, produce the original element as
their resultant.
We shall therefore consider ds as resolved into cos ds = a in the direction of r, and sin 6 ds = j3 #
in a direction perpendicular to \/ < * *
r in the plane P^PQ. ***
We shall also consider ds Fig< 30 *
as resolved into cos tf ds a in the direction of r reversed, sin fl cos r] ds (3 in a direction parallel to that in which ft was measured, and sin sin?; ds = y in a direction perpendicular to a and
ft .
Let us consider the action between the components a and ft on the one hand, and a, (3 , y on the other.
(1) a and a are in the same straight line. The force between them must therefore be in this line. W"e shall suppose it to be an attraction =Aaa ii ,
where A is a function of r, and i, i are the intensities of the currents in ds and ds respectively. This expression satisfies the condition of changing sign with i and with i .
(2) ft and ft are parallel to each other and perpendicular to the line joining them. The action between them may be written
��This force is evidently in the line joining ft and ft , for it must be in the plane in which they both lie, and if we were to measure ft and ft in the reversed direction, the value of this
expression would remain the same, which shews that, if it represents a force, that force has no component in the direction of ft, and must there fore be directed along r. Let us assume that this
expression, when positive, represents an attraction.
(3) ft and y are perpendicular to each other and to the line joining them. The only action possible between elements so related is a couple whose axis is parallel to r. We are at present engaged with
forces, so we shall leave this out of account.
(4) The action of a and ft , if they act on each other, must be expressed by Caft ii .
�� � 154 AMPERE S THEORY. [5H-
The sign of this expression is reversed if we reverse the direction in which we measure /3 . It must therefore represent either a force in the direction of ft , or a couple in the plane of a and /3 .
As we are not investigating couples, we shall take it as a force acting on a in the direction of /3 .
There is of course an equal force acting on /3 in the opposite direction.
We have for the same reason a force
Cay Of acting on a in the direction of y , and a force
Cfiaii acting on ft in the opposite direction.
514.] Collecting our results, we find that the action on ds is compounded of the following forces,
X = (A aa -j- J3 /3/3 ) ii f in the direction of r, \ Y C(aft a fi)ii in the direction of ft, ( (9)
and Z = C ay ii in the direction of y.
Let us suppose that this action on ds is the resultant of three forces, Rii dsdJ acting in the direction of r, Sii dsds acting in the direction of ds, and S ii dsds acting in the direction of ds ,
then in terms of 0, Q , and 77,
f + _Z?sin0sm0 cos?7,
��S =Ccos6.
��In terms of the differential coefficients of r
��dr dr D
R = A-^-^n Br
��ds ds " dsds
��In terms of I, m, n, and / , m 9 n 9
��R =-
��y where f, ?;, fare written for x x, y y, and z z respectively.
515.] We have next to calculate the force with which the finite current * acts on the finite current s. The current s extends from A y where s = 0, to P, where it has the value s. The current /
extends from A , where /= 0, to P / , where it has the value /.
�� � 5 1 6.] ACTION OF A CLOSED CIRCUIT ON AX ELEMENT. 155
The coordinates of points on either current are functions of s or of/.
If F is any function of the position of a point, then fre shall use the subscript (s o) to denote the excess of its value at P over that at A, thus F M = F P -F A .
Such functions necessarily disappear when the circuit is closed.
Let the components of the total force with which A P acts on A A be ii X, ii Y^ and ii Z. Then the component parallel to X of
��the force with which dx acts on ds will be ii - =-7 ds dtf.
ch ds
��Hence - Jt+tf+fJ . (13)
ds ds r
Substituting the values of R, S, and S from (12), remembering
that -,, ,. , , ,. dr /1/n
ss r-, (14)
��and arranging the terms with respect to , m, n, we find
��-JT7 - ~v -J-, -}-,
dsds l r 2 ds * ds
��Since A, B, and C are functions of r, we may write
��the integration being taken between r and oo because A, B, C vanish when r = oo.
Hence A + S =-*, and C = -. (17)
��516.] Now we know, by Ampere s third case of equilibrium, that when / is a closed circuit, the force acting on ds is perpendicular to the direction of ds } or, in other words, the component of the
force in the direction of ds itself is zero. Let us therefore assume the direction of the axis of x so as to be parallel to ds by making I = 1 , m = 0, n 0. Equation (15) then becomes
��i/sd/~ ds ds
7 Y
To find -- , the force on ds referred to unit of length, we must
�� � 156 AMPERE S THEORY. [5 J 7-
integrate this expression with respect to /. Integrating- the first term by parts, we find
^ = (P? - <3)(. ,o> - (\2Pr-S-C] ^d, . (19)
Ibd J Q T
When / is a closed circuit this expression must be zero. The first term will disappear of itself. The second term, however, will not in general disappear in the case of a closed circuit unless the
quantity under the sign of integration is always zero. Hence, to satisfy Ampere s condition,
��517.] We can now eliminate P, and find the general value of
��When / is a closed circuit the first term of this expression vanishes, and if we make
��\s it/ ry iiu ^ TT /
o ~2~ ~ *
��2 r
where the integration is extended round the closed circuit /, we may write ( u_ f f -j
ds Similarly =na ly ) y (23)
��~=-. ds J
The quantities a , ft , y are sometimes called the determinants of the circuit / referred to the point P. Their resultant is called by Ampere the directrix of the electrodynamic action.
It is evident from the equation, that the force whose components
dX dY . dZ .
are -y-> -^-, and -=- is perpendicular both to ds and to this ds ds ds
directrix, and is represented numerically by the area of the parallel ogram whose sides are ds and the directrix.
�� � 519.] FORCE BETWEEN TWO FINITE CURRENTS. 157
In the language of quaternions, the resultant force on ds is the vector part of the product of the directrix multiplied by ds.
Since we already know that the directrix is the same thing as the magnetic force due to a unit current in the circuit /, we shall henceforth speak of the directrix as the magnetic force due to the
518.] We shall now complete the calculation of the components of the force acting between two finite currents, whether closed or open.
Let p be a new function of r, such that
��(B-C)dr, (24)
then by (17) and (20)
and equations (11) become
ds ds J
With these values of the component forces, equation (13) becomes
7 dQ , 7 ,dQ
��(27 )
��dx dsds ds ds
519.] Let
F= I Ipds, G = * mpds, H = I npds, (28)
JQ Jo *^0
F = f* V p els , G =[ S m p ds f , H = TV p ds . (29) Jo Jo Jo
These quantities have definite values for any given point of space. When the circuits are closed, they correspond to the components of the vector-potentials of the circuits.
Let L be a new function of r, such that
��(30) and let M be the double integral
��= f* f (
/o Jn
�� � 158 AMPERE S THEORY. [520.
which, when the circuits are closed, becomes their mutual potential, then (27) may be written
d*X d^^IM dJL w \
dsds ~ dsds i dx dx ^ \
520.] Integrating-, with respect to s and /, between the given limits, we find
X dM d (T T
= -- L r? - L Ap -
��+ F P -F A -F P , + F An (33)
where the subscripts of L indicate the distance, r, of which the quantity L is a function, and the subscripts of F and F f indicate the points at which their values are to be taken.
The expressions for T and Z may be written down from this. Multiplying the three components by dx, dy, and dz respectively, we obtain Xdx+Ydy + Zdz = DM-D(Lpp,L AP ,L A ,p
��Y , (34)
where D is the symbol of a complete differential.
Since Fdx + G dy + Hdz is not in general a complete differential of a function of x,y, z, Xdx + Ydy + Zdz is not a complete differential for currents either of which is not closed.
521.] If, however, both currents are closed, the terms in I/, F, G, H, F, G , H disappear, and
Xdx+Ydy + Zdz = DM, (35)
where M is the mutual potential of two closed circuits carrying unit currents. The quantity M expresses the work done by the electro magnetic forces on either conducting circuit when it is moved
parallel to itself from an infinite distance to its actual position. Any alteration of its position, by which M is increased) will be assisted by the electromagnetic forces.
It may be shewn, as in Arts. 490, 596, that when the motion of the circuit is not parallel to itself the forces acting on it are still determined by the variation of M, the potential of the one
circuit on the other.
522.] The only experimental fact which we have made use of in this investigation is the fact established by Ampere that the action of a closed current on any portion of another current is
perpendicular to the direction of the latter. Every other part of
�� � 524.] HIS FORMULA. 159
the investigation depends on purely mathematical considerations depending- on the properties of lines in space. The reasoning- there fore may be presented in a much more condensed and appropriate
form by the use of the ideas and language of the mathematical method specially adapted to the expression of such geometrical relations the Quaternions of Hamilton.
This has been done by Professor Tait in the Quarterly Mathe matical Journal) 1866, and in his treatise on Quaternions, 399, for Ampere s original investigation, and the student can easily adapt the
same method to the somewhat more general investigation given here.
523.] Hitherto we have made no assumption with respect to the quantities A, B, C, except that they are functions of r, the distance between the elements. We have next to ascertain the form of these
functions, and for this purpose we make use of Ampere s fourth case of equilibrium. Art. 508, in which it is shewn that if all the linear dimensions and distances of a system of two circuits be
altered in the same proportion, the currents remaining the same, the force between the two circuits will remain the same.
Now the force between the circuits for unit currents is -4 , and
since this is independent of the dimensions of the system, it must be a numerical quantity. Hence M itself, the coefficient of the mutual potential of the circuits, must be a quantity of the dimen
sions of a line. It follows, from equation (31), that p must be the reciprocal of a line, and therefore by (24), B C must be the inverse square of a line. But since B and C are both functions of r, B
C must be the inverse square of r or some numerical multiple of it.
524.] The multiple we adopt depends on our system of measure ment. If we adopt the electromagnetic system, so called because it agrees with the system already established for magnetic measure ments,
the value of M ought to coincide with that of the potential of two magnetic shells of strength unity whose boundaries are the two circuits respectively. The value of M in that case is, by
��Art. 423, /Ycose
��, r /Yeose , 7 , M = ds ds ,
��the integration being performed round both circuits in the positive direction. Adopting this as the numerical value of M, and com paring with (31), we find
p = i, and B-C=~. (37)
�� � 160 AMPERE S THEORY. [5 2 5-
525.] We may now express the components of the force on ds arising from the action of ds in the most general form consistent with experimental facts.
The force on ds is compounded of an attraction
JT,_ 1 /<fr dr ^ d 2 r ^ 7 , d*Q.., 77/ 1
in the direction of ?,
��8 = -- ~ i i d /s ds in the direction of ds, and y = ^ ii ds ds in the direction of ds ,
�� ��r where Q = / Cdr, and since C is an unknown function of r, we
J r
know only that Q is some function of r.
526.] The quantity Q cannot be determined, without assump tions of some kind, from experiments in which the active current forms a closed circuit. If we suppose with Ampere that the action between
the elements ds and ds is in the line joining them, then 8 and 8 must disappear, and Q must be constant, or zero. The force is then reduced to an attraction whose value is
1 fd/r dr d z r x . ., ,
R= ( -2r -=-/) n ds ds . (39)
r 2 \ds ds dsds
Ampere, who made this investigation long before the magnetic system of units had been established, uses a formula having a numerical value half of this, namely
1 dr dr dr
. _ . _ . A _
ds ds dsds
Here the strength of the current is measured in what is called electrodynamic measure. If i, % are the strength of the currents in electromagnetic measure, and /, j the same in electrodynamic mea
sure, then it is plain that
jj = 2ii , or j= */2i. (41)
Hence the unit current adopted in electromagnetic measure is greater than that adopted in electrodynamic measure in the ratio of A/2 to 1.
The only title of the electrodynamic unit to consideration is that it was originally adopted by Ampere, the discoverer of the law of action between currents. The continual recurrence of v/2 in
calculations founded on it is inconvenient, and the electro magnetic system has the great advantage of coinciding numerically
��_ 1 f 1 dr dr dr N . ., 7 , ,
s = * (i z a? r zz*V/* ds - < 4 )
�� � 5 2 7-] FOUR ASSUMPTIONS. 161
with all our magnetic formulae. As it is difficult for the student to bear in mind whether he is to multiply or to divide by \/2, we shall henceforth use only the electromagnetic system, as adopted
by Weber and most other writers.
Since the form and value of Q have no effect on any of the experiments hitherto made, in which the active current at least is always a closed one, we may, if we please, adopt any value of Q which
appears to us to simplify the formulae.
Thus Ampere assumes that the force between two elements is in the line joining them. This gives Q = 0,
<*=<> =0. (42)
��Grassmann * assumes that two elements in the same straight line have no mutual action. This gives
0= - R- L <!LL <?- JL* <r_ ! *L fM
2r 2r~dsds" 2r 2 ds" ~ 2r* ds ( }
We might, if we pleased, assume that the attraction between two elements at a given distance is proportional to the cosine of the angle between them. In this case
n 1 -n 1 1 (If 0/ 1 dr f ...
9 = --, * = - 2 -cose, *-P-.-jry, S =^ Ts . (44)
Finally, we might assume that the attraction and the oblique forces depend only on the angles which the elements make with the line joining them, and then we should have
O 2 7? q l dr dr * 2 dr 2 dr
Q= --, ^ = -3--, 8- 9 r--
��527.] Of these four different assumptions that of Ampere is undoubtedly the best, since it is the only one which makes the forces on the two elements not only equal and opposite but in the straight
line which joins them.
• Pogg., Ann. Ixiv. p. 1 (1845).
��VOL. II.
�� � | {"url":"http://en.wikisource.org/wiki/A_Treatise_on_Electricity_and_Magnetism/Part_IV/Chapter_II","timestamp":"2014-04-21T07:44:10Z","content_type":null,"content_length":"56884","record_id":"<urn:uuid:5dabd297-d5d6-4bad-9e05-b8355d123082>","cc-path":"CC-MAIN-2014-15/segments/1397609539665.16/warc/CC-MAIN-20140416005219-00633-ip-10-147-4-33.ec2.internal.warc.gz"} |
Ms. Diane Jamison
Assistant Professor of Mathematics
Ms. Diane Jamison joined the mathematics faculty at University of the Cumberlands in 1989, bringing many years of experience teaching math at the middle school level. In 1971 and 1972, respectively,
she earned a B.A. in Mathematics and an M.A. in Mathematics Education, both from Western Kentucky University.
Ms. Jamison is a member of the National Council of Teachers of Mathematics and Kappa Delta Pi. In 1992, 1993, and 1994, she was listed in Who’s Who Among American College Teachers, and in 1992 she
was named Honored Professor by the University of the Cumberlands Student Government Association.
Courses Taught
• MATH 130 Concepts of Mathematics for the Elementary School Teacher I
• MATH 131 College Mathematics
• MATH 132 College Algebra
• MATH 230 Concepts of Mathematics for the Elementary School Teacher II
• MATH 332 P-5 Teaching Math
• MATH 333 Middle School Math Methods
Publications and Presentations
"Paper Folding for Developing Concepts of Fractions in a Geometric Format," Presented at Humboldt State University. | {"url":"http://ucumberlands.edu/academics/math-certification/profile/djamison.php","timestamp":"2014-04-20T01:21:07Z","content_type":null,"content_length":"8568","record_id":"<urn:uuid:b8ccd60d-698f-4e7b-a59a-1dcdbe8a8319>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00443-ip-10-147-4-33.ec2.internal.warc.gz"} |
Trig plotting and direct substitution
November 14th 2012, 01:21 PM #1
Junior Member
Oct 2012
Trig plotting and direct substitution
Really stuck with this question:
The suspension in a car acts like a
damped harmonic oscillator, that is, the oscillations in
the suspension rapidly die down with time. A model for this includes both exponential
and trigonometric functions. Suppose the displacement in a car's suspension is given by
(i) Sketch the displacement of the suspension for 0 >=t>=2
and describe its behaviour (>= meaning greater than or equal to)
in a few words.
(ii) Show by direct substitution that the displacement satisfes the differential equation
I am so lost on what to do, some suggestions whould be wonderful!
Re: Trig plotting and direct substitution
Both of your questions involve taking the first and second derivative of your function.
In (i), the first derivative is useful for finding critical points and seeing where the function is increasing or decreasing. The second derivative is used to see where the function is concave up
or concave down. Your teacher should have reviewed a list of steps for drawing graphs this way.
In (ii), it is as it says, direct substitution. d^2s/dt^2 is the second derivative of s, ds/dt is the first derivative of s, and s is s. Plug in on the left hand side and see if it equals 0
Re: Trig plotting and direct substitution
Thanks for that.
With (i), is it then after I take the derivative that I figure out the amplitude, period, etc.? Do I happen to use the formula y=a sin(bx+c)+d at all?
Re: Trig plotting and direct substitution
I'm not sure if you need the formula y = a sin(bx+c) + d at all, but it should be noted that if you do know what the base function looks like, you can apply a transformation on that graph to get
the desired graph of a particular function.
For example, if you know the amplitude, period, etc, of sin(x), we can derive the amplitude, period, etc, of a sin(bx+c) + d using the theory of transformations, by noting that
to transform sin(x) -> a sin(bx+c) + d we have
1) A vertical stretch by a factor of a
2) A horizontal stretch by a factor of 1/b
3) A vertical displacement by a factor of d
4) A horizontal displacement by a factor of -c/b
(1 & 2 done first, 3&4 done second)
Re: Trig plotting and direct substitution
By base function do you just mean something like s(t)=sin x ?
Re: Trig plotting and direct substitution
November 14th 2012, 02:46 PM #2
November 14th 2012, 02:50 PM #3
Junior Member
Oct 2012
November 14th 2012, 03:03 PM #4
November 14th 2012, 03:14 PM #5
Junior Member
Oct 2012
November 14th 2012, 03:18 PM #6 | {"url":"http://mathhelpforum.com/calculus/207598-trig-plotting-direct-substitution.html","timestamp":"2014-04-20T11:15:12Z","content_type":null,"content_length":"47020","record_id":"<urn:uuid:c55bae44-9379-4bd6-9d50-3d3271985d83>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00535-ip-10-147-4-33.ec2.internal.warc.gz"} |
1920 W motor to lift a 285 kg piano to a sixth-story window 16.0 m above?
Number of results: 2,240
How long will it take a 1920 W motor to lift a 285 kg piano to a sixth-story window 16.0 m above?
Monday, October 12, 2009 at 7:34pm by jerry
motor A lift a steel bar 5000N upward at a constant 2m/s. motor B lift 4000N steel bar upward at a constant 3m/s. which motor is supplying more power?
Tuesday, February 26, 2013 at 8:12am by Anonymous
A 187 W motor will lift a load at the rate (speed) of 6.58 cm/s. How great a load can the motor lift at this rate?
Tuesday, April 26, 2011 at 9:27pm by Merlin
An electric motor is rated to have a maximum power output of 0.72 hp. If this motor is being used to lift a crate of mass 188 kg, how fast (i.e., at what speed) can it lift the crate? Hint: Assume
the only other force on the crate is the force of gravity.
Monday, May 28, 2012 at 11:53pm by Alex
How long will it take a motor 1500 watt motor to lift a 500kg elevator 30 meters? Give your answer in minutes.
Sunday, February 26, 2012 at 4:12pm by Darcie
A 900W motor is used to lift 275.51kg of bricks a distance of 30, to the roof of the building. a. If the motor ran for 90s how much work did it do? b. What is the potential energy of the bricks? I
know how the calculate work, but I'm really confused about how to calculate it ...
Thursday, January 6, 2011 at 3:41pm by Jacie
college physics
What distance will a 10 hp motor lift a 2000 lb elevator in 30 s? What was the average velocity of this elevator during the lift?
Friday, November 11, 2011 at 10:00pm by Jessica
(a).A 220-kg sleigh is pulled by one horse at constant velocity for 0.90 km on a snowy, horizontal surface, the coefficient of kinetic friction between the sleigh runners and snow is 0.27. What power
must the horse deliver to the sleigh for the trip to take 9 minutes? (b) ...
Saturday, October 24, 2009 at 9:16pm by Sandhu
Your sofa won't fit through the door of your new sixth-floor apartment, so you use a 1.19 motor to lift the 88.6 sofa 18.3 from the street. How much time does the lift take?
Wednesday, December 5, 2012 at 2:27pm by Amanda
physics ---- really dnt get this at all
A small electric motor is used to lift a 0.50-kilogram mass at constant speed. If the mass is lifted a vertical distance of 1.5 meters in 5.0 seconds, the average power developed by the motor is
Sunday, May 8, 2011 at 3:54pm by ashley --- HELP PLEASEE!!!
A small motor runs a lift that raises aload of bricks weighing 836N to a height of 10.7m in 23.2s. assuming that the bricks are lifted with constant speed, what is the minimum power the motor must
Monday, March 15, 2010 at 5:30pm by Cody
A construction crew pulls up an 87.5-kg load using a rope thrown over a pulley and pulled by an electric motor. They lift the load 15.1 m and it arrives with a speed of 15.6 m/s having started from
rest. Assume that acceleration was not constant. a. How much work (J) was done ...
Tuesday, July 31, 2012 at 2:11pm by Anonymous
A construction crew pulls up an 87.5-kg load using a rope thrown over a pulley and pulled by an electric motor. They lift the load 15.1 m and it arrives with a speed of 15.6 m/s having started from
rest. Assume that acceleration was not constant. a. How much work (J) was done ...
Tuesday, July 31, 2012 at 2:12pm by Anonymous
The power (P) required to run a motor is equal to the voltage (E) applied to that motor times the current (I) supplied to the motor. If the motor data says the motor uses 180 watts of power and the
voltage applied to the motor is 120 volts, how much current will the motor ...
Monday, June 11, 2012 at 3:42pm by laquel
container factory uses a 450 W motor to operate a conveyor belt that lifts containers from one floor to another. To lift 250 1-kg containers a vertical distance of 2.4 m, the motor runs for 45 s.
What is the efficiency of the motorized conveyor system?
Wednesday, July 18, 2012 at 7:12pm by Rob
container factory uses a 450 W motor to operate a conveyor belt that lifts containers from one floor to another. To lift 250 1-kg containers a vertical distance of 2.4 m, the motor runs for 45 s.
What is the efficiency of the motorized conveyor system?
Wednesday, July 18, 2012 at 8:56pm by Rob
container factory uses a 450 W motor to operate a conveyor belt that lifts containers from one floor to another. To lift 250 1-kg containers a vertical distance of 2.4 m, the motor runs for 45 s.
What is the efficiency of the motorized conveyor system?
Wednesday, July 18, 2012 at 9:38pm by Jack Hill
1. A 2.5kg rock is dropped from a 350m cliff. It hits the ground with a speed of 60 m/s. What percent of its mechanical energy was lost to air resistance? -the answer is 48% 2. A motor uses 5500J to
lift a 35kg object a vertical distance of 12m. What is the efficiency of the ...
Sunday, June 17, 2012 at 3:34am by lenny
A construction crew pulls up an 87.5 kg load using a rope thrown over a pulley and pulled by an electric motor. They lift the load 15.5 m and it arrives with a speed of 15.6 m.s having started from
rest. Assume that acceleration was not constant. How much work was done by the ...
Friday, June 3, 2011 at 3:14pm by Casey
A diagram represents a simplified sketch of an electric DC motor.which contains a magnet, commutator, carbon brushes, coil and a pivot. when the electric motor is connected to a 12V DC supply, it
draws a current of 1,2A. The motor is now used to lift an object of mass 1,6kg ...
Monday, October 11, 2010 at 5:24pm by READ ME PLEAZZ
A chair lift takes skiers to the top of a mountain that is 320 m high. The average mass of skiers complete with equipment is 85 kg . The chair lift can deliver three skiers to the top of the mountain
every 36 s Determine the power required to carry out this task If friction ...
Monday, April 30, 2012 at 6:33pm by Melissa
a mechanic uses a chain and bloc pulley to lift a 875-kg engine 4.0 m to the top of the garage. the downward force on the chain over the 4.0 m distance is 5.0 x 10^4 N. a)calculate the work done in
raseing the motor b)how much useful work was done c)what is the inefficiency of...
Saturday, January 12, 2013 at 3:24pm by tyler
A 750w motor might also be rated as a 0.5horsepower motor 1 horsepower motor 2 horsepower motor 10 horsepower motor
Monday, March 31, 2014 at 10:43pm by snow
A certain elevator car has mass 2200 kg and is lifted by cables driven by an electric motor. The efficiency of the lift system (motor, pulleys, cable, etc.) is 67.0%. As the elevator car moves at
constant speed of 3.0 m/s it encounters 220N of friction coming from its tracks. ...
Sunday, December 1, 2013 at 7:46pm by Bob
word problem
Let x =the number of people who paid at the door 4(480-x)+5x=2100 Distribute 1920-4x+5x=2100 add like terms 1920-9x=2100 -1920-9x=2100-1920 -9x=180 Divide by nine x=-20 pe0ple at the door
Saturday, October 21, 2006 at 11:06am by Katniss E.
word problem
Let x =the number of people who paid at the door 4(480-x)+5x=2100 Distribute 1920-4x+5x=2100 add like terms 1920-9x=2100 -1920-9x=2100-1920 -9x=180 Divide by nine x=-20 pe0ple at the door
Saturday, October 21, 2006 at 11:06am by Katniss E.
a40w motor conectedto a 9v battery can lift an object 40m in 10s
Thursday, March 21, 2013 at 12:38am by jorge
physics-newton's laws
change km/hr to m/s Look at a vector diagram. mg is down, lift is perpendicular to the tilt angle. You need to realize the vertical part of lift is equal to mg (so mg=Lift*sin40 That give you lift.
Now, the portion of Lift keeping the plane accelerating inward is lift*cos40. ...
Tuesday, October 12, 2010 at 8:07pm by bobpursley
A crane lifts a concrete hopper of mass 370 kg from the ground to the top of a 38 m building that is under construction. The hopper moves with a constant speed during the lift. The power rating of
the crane motor is 8.2 kiloWatts. How long does it take for the crane to lift ...
Sunday, February 20, 2011 at 10:48pm by anonymous
A mechanic uses a chain and block to lift a 875-kg engine 4.00 m to the top of the garage. The downward force in the chain over the 4.00 m distance is 5.00 x 10^4 N. A) Calculate the work done in
raising the motor B) How much useful work was done? C) What is the efficiency of ...
Thursday, January 16, 2014 at 4:18pm by Holly
A motor running at its rated power provides 2.6 lb-ft torque at its rated speed of 2020 rpm. The motor draws 5.6-A from a 230-V supply at a phase angle of 37 degrees. Determine the following: The
horsepower delivered by the motor, The efficiency of the motor.
Thursday, January 24, 2008 at 2:50pm by Nick
A small motor runs a lift that raises a load of bricks weighing 916 to a height of 12.7 in 28.2 .
Monday, November 21, 2011 at 11:49pm by Anonymous
Move 1920 to right side (making it negative 1920), combine the S terms (but you have made an error [the 1920-S should be 1920-8S)], and calculate S, then G. You have worked the problem as if G is the
NUMBER of $8 tickets sold and S is the NUMBER of $5 tickets sold. That way G...
Wednesday, December 26, 2007 at 8:57pm by DrBob222
How long will it take a 230 W motor to lift a 335 kg piano to a sixth-story window 19.0 m above?
Sunday, February 28, 2010 at 2:07pm by peyli
How long will it take a 1080 W motor to lift a 330kg piano to a sixth-story window 16.0m above?
Monday, February 21, 2011 at 6:52pm by John
How long will it take a 1080 W motor to lift a 330kg piano to a sixth-story window 16.0m above?
Monday, February 21, 2011 at 6:50pm by John
How long will it take a 1080 W motor to lift a 330kg piano to a sixth-story window 16.0m above?
Monday, February 21, 2011 at 6:51pm by John
How long will it take a 1280 W motor to lift a 345 kg piano to a sixth-story window 19.0 m above?
Monday, February 28, 2011 at 10:55pm by Kathleen
How long will it take a 1830 W motor to lift a 285 kg piano to a sixth-story window 16.0 m above?
Thursday, November 10, 2011 at 10:27pm by Kristy
How long will it take a 1030 W motor to lift a 330 kg piano to a sixth-story window 13.0 m above?
Sunday, December 18, 2011 at 10:13am by ayyya
How long will it take a 1630 W motor to lift a 285 kg piano to a sixth-story window 16.0 m above?
Tuesday, April 10, 2012 at 1:26pm by Amelia
More Algebra
Eve and Jason have an exercise routine that they do together three days a week. Together, they can lift 225 pounds. a) If Eve can lift 113 pounds,how many pounds can Jason lift? b). If Eve can lift
87 pounds, how many pounds can Jason lift? c) What expression can be used to ...
Sunday, October 2, 2011 at 5:22pm by Jab
Managerial Accounting
a ski company plan to add five new chai500 a day for the lift. the lifts cost 2 million and to install the lift cost another 1.3 mil.The lift will allow 300 additional skiers on the slopes but only
40 days a year will be needed. The company will sell 300 lift tickets on those ...
Sunday, August 23, 2009 at 8:37pm by Anita
You don't need the power rating of the motor to calculate the work required to lift one. It is W = M g H H = 38 m g = 9.8 m/s^2 M = 370 kg The answer will be in joules
Sunday, February 20, 2011 at 8:16pm by drwls
Salisbury University
How long will it take a 480 N motor to lift a 335 kg piano to a sixth-story window 15.0 m above?
Wednesday, April 6, 2011 at 6:18pm by Cara
how long will it take a 1750-watts motor to a lift a 285-kg piano to a sixth storey window 1.6 meters above
Friday, October 5, 2012 at 12:12pm by kyla
Geomatic Sequence
ar^6 - ar^4=1920 64a-16a=1920 48a=1920 a = 40 sum(11) = 40(2^11 - 1)/(2-1) = 81880
Monday, October 1, 2012 at 10:20am by Reiny
The electrical motor runs on a D.C source of emf E and internal resistance r. Show that power output of the motor is max where current drawn by the motor is E/2r.
Tuesday, March 16, 2010 at 12:32pm by Edwin
How long would it take a 20.0 hp motor to lift a 1,000.0 kg crate from the bottom of a freight's hole deck(40.0m)?
Wednesday, October 10, 2007 at 12:42pm by Sarah
Physical Science
How long would it take a 20.0 hp motor to lift a 1,000.0 kg crate from the bottom of a freighter's hold to the deck (40.0 m)?
Wednesday, October 10, 2007 at 12:01am by ROSemary
physical science
How long would it take a 20.0 hp motor to lift a 1,000.0 kg crate from the bottom of a freighter's hold to the deck (40.0 m)?
Thursday, June 23, 2011 at 2:19pm by t
The upward lift is constant, it depends only on the size of the balloon. netforce= mass*acceleration Lift-mg=m*a lift-g(m-deltam)=(m-deltam)*newacc 1)solve for lift in the original equation, knowing
m, a, and g. 2) solve for deltamass in the second equation knowing lift, g, m...
Wednesday, November 28, 2007 at 3:21pm by bobpursley
The resistance of the wire in the windings of an electric starter motor for an automobile is 0.0190 Ohms. the motor is connected to a 12 V battery. What current will flow through the motor when it is
Sunday, February 20, 2011 at 4:15pm by Kara
1920/12 = 160 1920/8 = 240 Can you do anything with that?
Wednesday, January 25, 2012 at 8:28am by Reiny
Geomatic Sequence
that's geometric sequence T7 = T5*r^2 = T5+1920 r = 2, so 4*T5 = T5 + 1920 3T5 = 1920 T5 = 640 640 = 2^4 * 40 So, a=40 and the sequence is 40 80 160 320 640 1280 2560 5120 10240 20480 40960 ... Note
that 2560 = 640 + 1920
Monday, October 1, 2012 at 10:20am by Steve
Please Help!!!: How many seconds will it take a 12hp motor to lift a 1500lb elevator at a distance of 9ft? what was the average velocity of the elevator?
Sunday, October 17, 2010 at 11:33am by Aikawa
An electric motor has 72 turns of wire wrapped on a rectangular coil, of dimensions 3cm by 4cm. Assume that the motor uses 14A of current and that a uniform 0.5T magnetic field exists within the
motor. 1)What is the maximum torque delivered by the motor? Answer in units of Nm...
Saturday, April 7, 2007 at 8:27pm by Belle
AP U.S. History 1920's immigration
who were the immigrants in the 1920's
Wednesday, February 21, 2007 at 11:12pm by kristina
A small motor draws a current of 1.76 A from a 110 V line. The output power of the motor is 0.20 hp. (a) At a rate of $0.060/kWh, what is the cost, in cents, of operating the motor for 4.0 h? (b)
What is the efficiency of the motor? for (a) i am doing IV*hours*price and its ...
Monday, March 14, 2011 at 7:02pm by julie
A & P
Trace the anatomical pathway of the neural signal that must be sent from the primary motor area of the brain to the appropriate muscles of the arm and shoulder in order to lift a glass of water to
Wednesday, March 23, 2011 at 5:07pm by Anonymous
Physics - pulley
A construction crew pulls up an 87.5-kg load using a rope thrown over a pulley and pulled by an electric motor. They lift the load 15.5 m and it arrives with a speed of 15.6 m/s having started from
rest. Assume that acceleration was not constant. I have done the problem but am...
Friday, June 3, 2011 at 12:08am by Casey
Using the motor principle explain why a motor is rotating in the direction indicated. I seriuosly have no idea, please help explain this theory to me. PHYSICS - drwls, Saturday, February 6, 2010 at
2:15pm To help you answer that question, I'd need a figure that shows what kind...
Saturday, February 6, 2010 at 2:25pm by Sarah
I have to build a mini solar car for a school project and i have three solar cells that i bought. whenever i hook up the solar cells i get a reading between 1.5-1.8 volts, but when i hook it up to
the motor it wont start it. According to the box the motor only requires 1.5v ...
Sunday, May 6, 2007 at 2:44am by Matt
a 1.00 x 10 W electric motor is used to life a 955 kg object vertically 25.0 m at a constant velocity. How much time does it take to lift the object.
Sunday, December 2, 2012 at 10:00am by Teresa
(car lift on lift) -------------- ____---____ <- lift .........| | .........| | .........\ \__________________ ..........__________O_______....\ ....................................[....|....|.....]
................. ________________ ------------------^^^^^^^^^^^^^^^^ fluid...
Wednesday, December 5, 2012 at 8:26am by Drake
a) Power of the motor P = 3.8hp = (3.8hp)(746W/hp) = 2834.8W Motor is 90% efficient. Therefore Power absorbed by the motor is (2834.8W)/(0.9) =3150W But power P = V I then current in the motor I =
(3150W)/(120V) = 26.25A (b) Energy delivered to the motor E = P*t = (3150W)(4h)(...
Saturday, October 20, 2007 at 5:45pm by Anonymous
Pic should be fixed below: (car lift on lift) -------------- ____---____ <- lift ...|...| ...|...| ...\...\__________________ ....__________O_______.....\ ..................[....|....|.....]
................. ________________ ------------------^^^^^^^^^^^^^^^^ fluid ...
Wednesday, December 5, 2012 at 8:26am by Drake
outside the sphere, E=kq/r^2 E=kQ/(r+2)^2=1920 E=kQ/4^2=3000 kq=16*3000 1920=16*3000/(r+2)^2 solve for r r = 4 sqrt (3000/1920)-2 check that math.
Monday, February 4, 2013 at 12:37pm by bobpursley
A fixed, single pulley that is used to lift a block does which one of the following? A. doubles the force required to lift the block B. decreases the force required to lift the block C. makes the
block easier to lift by changing the direction of the force needed to lift it D. ...
Wednesday, April 10, 2013 at 10:15pm by Cassie
The 1920's is associated with the development of many new freedoms; it was a time of restrictions as well. Examine the 18th amendment and prohibition in the 1920's. What was the reasoning and who
were the key supporters behind a 'dry' United States? ---Need help answering this...
Monday, June 15, 2009 at 10:43pm by mysterychicken
r = m (t) + b when year = 1920, t = 0 where t = year - 1920 45.9 = m (0) + b so b = 45.9 44.1 = m (1980-1920) + 45.9 44.1 = 60 m + 45.9 60 m = -1.8 m = -.03 so r =-.03 t + 45.9 when year = 2003 t =
2003 - 1920 = 83 r = -.03 (83) + 45.9 r = 43.41 that should get you started
Sunday, December 20, 2009 at 5:18pm by Damon
Mechanics (Physics)
A man of mass M kg and his son of mass m kg are standing in a lift. When the lift is accelerating upwards with magnitude 1ms-2 the magnitude of the normal contact force exerted on the man by the lift
floor is 880N. When the lift is moving with constant speed the combined ...
Wednesday, December 22, 2010 at 2:13pm by The Golden Girl
math (am i correct)
Record = m (year -1920) + b where year -1920 = t so R = m t + b 46.9 = m (0) + b so b = 46.9 in 1990, t = 1990 - 1920 = 70 so 45.5 = m (70) + 46.9 m = -.02 so R = -.02 t + 46.9 for 2003 for example t
= 2003 - 1920 = 83 so R = -.02(83)+46.9 R = 45.24
Sunday, June 14, 2009 at 6:33pm by Damon
Sasha wants to find the resistance of the motor. She measured a 6 V drop across the motor that is in a series circut with a 200 ohms reseistor. The power source is supplying 9 V to the circut. How
musch resistance is her motor putting into the circut?
Monday, February 18, 2008 at 7:24pm by Clover
Question: If you have an electric motor on a bench and hook it to the power the shaft turns. If you put the shaft in a vise the casing of the motor will turn. Now:...put this motor in free space no
gravity and attach a battery for power..when the power is turned on will the ...
Monday, July 25, 2011 at 9:43pm by Tiffany
the ski lift chair at a ski report is 2560m long. on average, the ski lift rises 15.2 degree above the horizontal. how high is the top of the ski lift relative to the base?
Tuesday, October 23, 2012 at 1:34pm by Anonymous
A freely running motor rests on a thick rubber pad to reduce vibration. The motor sinks h=10 cm into the pad. Estimate the rotational speed in RPM (revolutions per minute) at which the motor will
exhibit the largest vertical vibration.
Monday, April 29, 2013 at 4:34am by kjhgkjhgkjhg
A freely running motor rests on a thick rubber pad to reduce vibration. The motor sinks h=10 cm into the pad. Estimate the rotational speed in RPM (revolutions per minute) at which the motor will
exhibit the largest vertical vibration.
Monday, April 29, 2013 at 4:38am by kjhgkjhgkjhg
A 120-V motor has mechanical power output of 2.20 hp. It is 92.0% efficient in converting power that it takes in by electrical transmission into mechanical power. (a) Find the current in the motor
(A) (b) Find the energy delivered to the motor by electrical transmission in 2....
Thursday, September 5, 2013 at 2:38pm by priya
A man whose mass is 80kg stands on a weighing machine inside a lift: when the lift starts accelerating upwards, the reading of the weighing machine was observed to be 96kg. Determine the upward
acceleration of the lift?
Tuesday, April 16, 2013 at 2:27am by Musa Abdullahi
a 12V , 3A motor takes 120 minutes to do a certain job. a 120V, 8A motor takes 5 minutes to do the same job. Which motor uses the least energy and by how much?
Saturday, October 20, 2007 at 5:45pm by Anonymous
A 120-V motor has mechanical power output of 3.20 hp. It is 81.0% efficient in converting power that it takes in by electrical transmission into mechanical power. (a) Find the current in the motor.
Answer: 24.6A (b) Find the energy delivered to the motor by electrical ...
Monday, May 2, 2011 at 2:41am by iva
Algebra 1
In 1920 the record for a certain race was 45.1 sec. In 1960, it was 44.3 sec Let R(t)-the records in the race(t)=the numbeer of years since 1920. Thank you for your help.
Thursday, January 27, 2011 at 4:31pm by Esther
17.Which roller coaster has the greatest lift height Top Thrill Dragster 18.What is the median lift height for the roller coasters lifted? Round to the nearest tenth. Not sure for this one 19.Arrange
the given roller coasters from least to greatest lift height -20,-8,0,16,60,...
Tuesday, April 30, 2013 at 6:33pm by Jerald
a man has only 4 simple pulleys to lift a heavy load of 6000N. in roder to lift the load easily, he combines the pulleys (3 moveable, 1 fixed). given that the frictional forces are 300N, what force
must he apply in order to lift the load?
Thursday, September 2, 2010 at 9:55am by ashlee
The electric motor that hauls our boats out of the water is 80% efficient. the power out of the motor needed to haul the boat at .1 meter per second is 16 thousand Watts. How much power must go into
the motor from the electric cable in the street?
Sunday, February 24, 2008 at 7:28pm by Damon
A 1000kg elevator carries a maximum load of 800kg. A constant frictional force of 5000N retards the elevator's motion upward. What minimum power, in kilowatts, must the motor deliver to lift the
fully loaded elevator at a constant speed of 3m/s?
Saturday, December 10, 2011 at 2:18pm by Anonymous
A motor does 5,000 of work in 20 seconds what is the power of the motor
Tuesday, December 14, 2010 at 9:28am by Anonymous
AP Physics
A cord runs around two massless, frictionless pulleys; a canister with mass m=20 kg hangs from one pulley; and you exert a force F if you are to lift the cord. a) What must be the magnitude of F if
you are to lift the canister at a constant speed? b) To lift the canister by 2 ...
Monday, December 14, 2009 at 9:32pm by Mikaela
A motor vessel tows a small dinghy by a rope which makes an angle of 20 degrees with the horizontal. If the tension in the rope is 15 kgf ;find a) the force which effectively pulls the dinghy
forward. b)the force which lift its bows out of the water.
Monday, January 10, 2011 at 3:31am by Asamoah
A boy exerts a force of 15N to push a motor bike which has flat tyre for a distance at 30m.calculate the mass of the motor bike. Calculate the density of the motor bike.calculate the work done by the
boy.[take g=10m|sm2
Monday, May 14, 2012 at 3:36am by Nicholas
A boy exerts a force of 15N to push a motor bike which has flat tyre for a distance at 30m.calculate the mass of the motor bike. Calculate the density of the motor bike.calculate the work done by the
boy.[take g=10m|sm2
Monday, May 14, 2012 at 3:38am by Nicholas
At time t =0, the current to the dc motor is reversed, resulting in an angular displacement of the motor shaft given by è = (198 rad/s)t – (24 rad/s2)t2 – (2 rad/s3)t3 (15 marks) a. At what time is
the angular velocity of the motor shaft zero? b. Calculate the angular ...
Saturday, April 27, 2013 at 11:56pm by bruce
2. At time t =0, the current to the dc motor is reversed, resulting in an angular displacement of the motor shaft given by θ = (198 rad/s)t – (24 rad/s2)t2 – (2 rad/s3)t3 a. At what time is the
angular velocity of the motor shaft zero? b. Calculate the angular ...
Saturday, April 27, 2013 at 1:06am by bruce
IPC 8th grade
a motor does 5,000 joules of work in 20 seconds. What is the power of the motor.
Thursday, March 13, 2008 at 1:38am by Debbie
Motor A: P = 5000 * 2 = 10000 J/s. Motor B: P = 4000 * 3 = 12000 J/s.
Tuesday, February 26, 2013 at 8:12am by Henry
A lift is used to carry boxes to the top floor of a hotel 20m high.A total weight of 100kg of boxes was caried up in 20second.If the useful output of the engine driving the lift mechanism is 1.2
kilowatt.calculate the efficiency of the engine of the lift mechanism.
Wednesday, October 12, 2011 at 4:05pm by Blessig iliya
A motor is rated to deliver 10.0kW. At what speed can this motor raise a mass of 2.75x10^4kg?
Wednesday, September 5, 2012 at 3:04am by Anonymous
A helicopter winches up an injured person from a boat. The person plus equipment has a mass of 104 kg. If the height of the lift is 84 m and it is carried out at a uniform rate over 70 s, what is the
power expended by the helicopter's winch motor? Assume that the engine is 61...
Sunday, March 25, 2012 at 11:06pm by Anonymous
A helicopter winches up an injured person from a boat. The person plus equipment has a mass of 104 kg. If the height of the lift is 84 m and it is carried out at a uniform rate over 70 s, what is the
power expended by the helicopter's winch motor? Assume that the engine is 61...
Sunday, March 25, 2012 at 11:06pm by Anonymous
Pages: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Next>> | {"url":"http://www.jiskha.com/search/index.cgi?query=1920+W+motor+to+lift+a+285+kg+piano+to+a+sixth-story+window+16.0+m+above%3F","timestamp":"2014-04-20T18:44:23Z","content_type":null,"content_length":"40604","record_id":"<urn:uuid:a0ebde9d-4ed0-4366-bbea-023419c71801>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00311-ip-10-147-4-33.ec2.internal.warc.gz"} |
Doing Science With Virtual Biologists
24689540 story
Posted by
from the please-state-the-nature-of-the-biological-emergency dept.
An anonymous reader sends word of
new research into automating computational experiments
. A team of scientists developed a piece of software, dubbed Eureqa, to help solve complex, computationally-intense biological problems. A new paper in the journal
Physical Biology
details their success (
"The researchers chose this specific system, called glycolytic oscillations, to perform a virtual test of the software because it is one of the most extensively studied biological control systems.
Jenkins and Vallabhajosyula used one of the process' detailed mathematical models to generate a data set corresponding to the measurements a scientist would make under various conditions. To increase
the realism of the test, the researchers salted the data with a 10 percent random error. When they fed the data into Eureqa, it derived a series of equations that were nearly identical to the known
equations. 'What’s really amazing is that it produced these equations a priori,' said Vallabhajosyula. 'The only thing the software knew in advance was addition, subtraction, multiplication and
This discussion has been archived. No new comments can be posted.
Doing Science With Virtual Biologists
Comments Filter:
• by werepants (1912634)
Now the scientist's jobs will be done by machines as well.
In reality, I'm sure that there is only a very small subset of problems that this system will work for, but, there is no reason that we shouldn't put it to work on those posthaste. It will be
interesting to see what it can do with an unsolved problem.
□ Re:Pretty impressive (Score:5, Informative)
by Daniel Dvorkin (106857) on Saturday October 15, 2011 @02:21PM (#37725042) Homepage Journal
It's neat stuff, but I'm skeptical that it will replace human biologists any time soon. As is often the case, the pop-sci writeup is a lot more dramatic than the article itself. Reading the
latter, I'd say that what they've done is a clever bit of data mining combined with mathematical modeling -- they use an evolutionary algorithm to find the best set of differential equations,
out of an enormous number of possible models, which describe the behavior of the data.
This is easier, and probably produces better results, than the traditional method of coming up with sets of diff. eqs. to describe the behavior of complex systems, but it's not a replacement
for human judgement in coming up with the model space in the first place. (I'll also note that they performed almost the entire "experiment" on simulated data, which is always a valuable
first step in the development of any modeling method, but it's not enough to show that the method "works" -- real data is always messier than the best simulations, and biological data is
particularly so.) That being said, it's a very nice technique, and I'll be interested to see if the same approach can be applied to building the kinds of statistical models I work with,
Bayesian networks and such.
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☆ by epine (68316)
The only paragraph I found at all useful:
Generally, the way that scientists design experiments is to vary one factor at a time while keeping the other factors constant, but, in many cases, the most effective way to test a
biological system may be to tweak a large number of different factors at the same time and see what happens. ABE will let us do that.
The rest is all mystery meat about the actual method.
This is certainly the way of the future. IBM has pumping that notion right now, as well. I'm not incli
□ by HiThere (15173)
FWIW, mathematical modeling isn't science. It's quite important, but it isn't science until it's tested against a physical environment to determine whether the predictions are correct.
This is a bit nit-picky, but too many people seem to be forgetting it, and the distinction is extremely important.
☆ by gardyloo (512791)
Check into how this software works. It chooses a sparse set of data points, creates its "model", and then brings in more points to test against. I've used it (though not super seriously)
and heard a talk by one of its creators. It's based upon a heuristic of finding the _most_surprising_, _worst_ matches to its guesses and then refining the model. In the sense that it is
explicitly used to predict how well it fits to further actual, experimentally-obtained data points, your criterion of it being "tested aga
○ by HiThere (15173)
Yes, that's a mathematical model of science. But it doesn't become science until it makes a prediction that is then tested against the physical world. And it's the complete process
that's science, not any one piece of it.
Thus, String Theory is mathematics that is endeavoring to become science. Some parts of String Theory have become science. Failed science, as the predictions were not validated, but
still science. Other parts of it can't yet be tested.
Science requires BOTH the prediction of a previousl
• ...for quite awhile, and few comprehend that it is, in actuality, the very first REAL A.I. in existence (unless there's some secret stuff out there???). It is truly brilliant, and predictably
derives from genetic programming algorithms.
And of course the AI program was recalcitrant about revealing how it does it's thing, haven't all of us from time to time exhibited the exact same behavior?
Now, if only they could couple this with Google's API, and SkyGrabber, I'd bet we'd end up with some really fantas
□ And this is why my own version of Eureqa spewed forth:
This has been a very confusing few weeks, so allow me to summarize.
The Attorney General for Chiquita and ExxonMobil, Eric Holder, in a public announcement stated that he'd found the Smoking S**t, the connection between the Underwear Bomber's Calvin Klein's
and the Iranian Quds Force.
Meanwhile, Swami Rami of India's Ta Ta Agency said that all religion is B.S. and demanded that President Obama ship 500,000 more jobs to India.
President Obama res
☆ by Dexter Herbivore (1322345)
I am intrigued by your thoughts and would like to subscribe to your newsletter (or your A.I.s drug supplier).
• by MurukeshM (1901690) on Saturday October 15, 2011 @02:21PM (#37725034)
I'm confused. Did the program derive the equations a priori or ab initio? If a priori, wow!
twitter facebook linkedin
• by Anonymous Coward
So if I use Matlab to do statistical analysis (econometrics) on my econ dataset, then my computer is a 'virtual economist', huh.
• When they fed the data into Eureqa, it derived a series of equations that were nearly identical to the known equations.
How useful is "nearly identical"?
□ by M0j0_j0j0 (1250800)
enough to make a headline.
□ by Hentes (2461350)
It might be just some constants being a little off.
• by vossman77 (300689)
The problem with research is that there is no such thing a pure random error. Time and time again, we develop methods that work awesome with random error. In reality, the error in the data is not
purely random, but a combination of systematic and random error (that is not Gaussian), so it takes a trained eye to work through this. I would much preferred that they used real data rather than
only fabricated data, then you can say that you have something.
• okay.. (Score:1)
by Anonymous Coward
If I told you I can fit equations with data you wouldn't think it was magical .. (after all thats exactly what any series expansions are guaranteed to do) why do biological modelers its somehow
more magical to do when we are in ODE land?
• Now that would be impressive!
Related Links Top of the: day, week, month. | {"url":"http://science.slashdot.org/story/11/10/15/1732220/doing-science-with-virtual-biologists","timestamp":"2014-04-18T21:37:14Z","content_type":null,"content_length":"108519","record_id":"<urn:uuid:3fe81070-77cd-494c-aa1d-8f56b098c350>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.9/warc/CC-MAIN-20140416005215-00237-ip-10-147-4-33.ec2.internal.warc.gz"} |
When does a "representable functor" into a category other than Set preserve limits?
up vote 5 down vote favorite
This might be a dumb question. If $C$ is an ordinary category, then for any $c \in C$ the covariant representable functor $\text{Hom}(c, -) : C \to \text{Set}$ preserves limits. However, it can
happen that $c$ can be equipped with extra structure which in turn gives the morphisms out of $c$ extra structure, so that there is a "representable functor" $\text{Hom}(c, -) : C \to D$ where $D$ is
a category equipped with a forgetful functor $F : D \to \text{Set}$ such that composing with the above gives the original representable functor.
In this situation, when does the functor into $D$ still preserve limits? How is this situation formalized? (Assume that $C$ is not enriched over $D$ in any obvious way.)
There are several examples of this coming from algebra, but the one that got me curious is the following. Let $C$ denote the homotopy category of pointed (path-connected?) topological spaces and let
$S^1$ denote the circle with a distinguished point. I believe I am correct in saying that if the fundamental group functor $\pi_1 : C \to \text{Grp}$ is composed with the forgetful functor $U : \text
{Grp} \to \text{Set}$, then $S^1$ represents the resulting functor $U(\pi_1(-))$. (The extra structure on $S^1$ that makes this possible is, if I'm not mistaken, a cogroup structure internal to $C$.)
Can I conclude that $\pi_1$ preserves limits?
Edit: I've been told that the above example is problematic, so here's a simpler one. Let $C = \text{Set}$ and suppose that $c \in C$ is equipped with a morphism $f : c \to c$. Then by precomposition
$\text{Hom}(c, d)$ is also equipped with such a morphism, so $\text{Hom}(c, -)$ has values in the category of dynamical systems. Does it preserve limits? Another example is my attempted answer to
question #23188.
ct.category-theory at.algebraic-topology representable-functors
3 If the forgetful functor creates limits, then I think you get what you want. In detail: Let $J$ be an index category, and let $J\stackrel{T}{\to}C\stackrel{F}{\to}D\stackrel{U}{\to}\mathbf{Set}$
be functors. Suppose that $UF$ preserves limits and $U$ creates limits. Suppose that $\tau\colon \ell\to T$ is a limiting cone in $C$. Since $UF$ preserves limits, $UF\tau\colon UF\ell\to UFT$ is
a limiting cone. As $U$ creates limits, there is a unique lifting of $UF\tau$ to a cone in $D$, and this cone is a limiting cone. (to be continued...) – user2734 Apr 30 '10 at 23:07
(...cont'd) But $F\tau$ is such a lift, and hence we're done. Now, if I am not mistaken, for any $\tau$-algebra the forgetful functor to $\mathbf{Set}$ creates limits. (I once proved it for
solving some exercise, but I'm not sure if my proof is correct, since it has been quite a while since I did it). – user2734 Apr 30 '10 at 23:07
1 The problem with your example is that the homotopy category doesn't even have limits. And even if you try to lift this into a model-category setting and use the homotopy limit, $\pi_1$ doesn't
send those to limits (take a pullback of a fibration as an example, where you get a Mayer-Vietoris-like sequence involving all $\pi_i$). – Tilman May 1 '10 at 8:17
@Tilman: My comments referred only to the first part of the question. Is there something wrong with what I said? – user2734 May 1 '10 at 10:10
@unknown (google): no you're right. – Martin Brandenburg May 1 '10 at 10:48
show 5 more comments
1 Answer
active oldest votes
[Collecting my sporadic comments into one (hopefully) coherent answer.]
A more general question is as follows: For functors $C\stackrel{F}{\to}D\stackrel{U}{\to}E$ and for an index category $J$ such that $UF$ preserves $J$-limits, when does $F$ preserve $J$
A useful sufficient condition is that if $U$ creates $J$-limits, then in the above situation $F$ preserves $J$-limits. Proof: Let $T\colon J\to C$ be a functor, and suppose that $\tau\
colon \ell\stackrel{\cdot}{\to} T$ is a limiting cone in $C$. Since $UF$ preserves $J$-limits, $UF\tau\colon UF\ell\stackrel{\cdot}{\to} UFT$ is a limiting cone in $E$. As $U$ creates
$J$-limits, there is a unique lifting of $UF\tau$ to a cone in $D$, and this cone is a limiting cone. But $F\tau\colon F\ell\stackrel{\cdot}{\to} FT$ is such a lift, and hence we're done.
This condition is quite useful, because many forgetful functors are monadic, and monadic functors create all limits (by combining their definition on pp. 143--144 of Mac Lane and Ex.
6.2.2 on p. 142 of Mac Lane, or by Proposition 4.4.1 on p. 178 of Mac Lane--Moerdijk, or really by a comment of Tom Leinster from which I learned this :)).
For example, consider the category of all small algebraic systems of some type. From the AFT, we know that the forgetful functor to $\mathbf{Set}$ has a left adjoint, and it is the
content of Theorem 6.8.1, p. 156 of Mac Lane that this forgetful functor is monadic.
Returning to the original question, this means that whenever the category $D$ is one of $\mathbf{Grp}$, $\mathbf{Rng}$, $\mathbf{Ab}$,... and $U\colon D\to \mathbf{Set}$ is the forgetful
functor, then for any $J$, $UF$ preserves $J$-limits implies $F$ preserves $J$ limits. In particular, if $UF$ is a representable functor (and hence preserves all limits), then $F$
preserves all limits.
Next, let me try to comment on your motivating examples (the one from Q. 23188 and the one from the 'Edit' part of the current question.)
Regarding your example in Q. 23188: Unfortunately I know nothing of Hopf algebras, so I can't understand all the details of your construction. If I understand correctly, you construct a
functor $F\colon\mathbf{Rng}\to\mathbf{Grp}$ whose composition with the forgetful functor $U\colon \mathbf{Grp}\to \mathbf{Set}$ is representable. If this is indeed the case, then by the
above $F$ itself preserves all limits.
[EDIT: corrected the part concerning the last example.]
Finally, regarding your example in the edited question: While I know nothing of dynamical systems, from a quick glance at Terence Tao's blog it seems that the category of dynamical
systems is the category whose objects are pairs $\langle X,f\colon X\to X\rangle$ with $X$ a (small) set and whose arrows $\phi\colon\langle X, f\rangle\to\langle Y, g\rangle$ are those
functions $\phi\colon X\to Y$ with $g\circ\phi =\phi\circ f$.
To show that the above sufficient condition works in this case, we would like to show that the forgetful functor to $\mathbf{Set}$ crates limits. More generally, we will show that if $C$
is a category and $D$ is the category whose objects are pairs $\langle x,f\colon x\to x\rangle$ (where $x\in\operatorname{obj}(C)$, $f\in\operatorname{arr}(C)$), and whose arrows $\phi\
up vote 4 colon \langle x,f\rangle\to \langle y,g\rangle$ are those arrows $\phi\colon x\to y$ with $g\circ\phi =\phi\circ f$, then the forgetful functor $U\colon D\to C$ creates limits.
down vote
accepted [I'm sure that this follows from some well-known result, but since I don't see it, I'll just continue with a direct proof.]
So, let $J$ be an index category, let $F\colon J\to D$ be a functor, and suppose that $\tau\colon x\stackrel{.}{\to} UF$ is a limiting cone in $C$. We would like to show that there exists
unique cone $\sigma\colon L\stackrel{.}{\to} F$ in $D$ such that $U\sigma=\tau$, and that this unique cone is a limiting cone.
For uniqueness, suppose that $\sigma\colon L\stackrel{.}{\to} F$ satisfies $U\sigma = \tau$. Write $F_j:=\langle y_j,f_j\rangle$. Then we must have for all $j$ $$ \sigma_j=(x\stackrel{f}
{\to}x)\stackrel{\tau_j}{\to}(y_j\stackrel{f_j}{\to}y_j) $$ for some $f\colon x\to x$ (hence we immediately see that $\sigma$ is determined up to $f$). Now, since by the above we see that
$\tau_j$ must be an arrow $$ (x\stackrel{f}{\to}x)\stackrel{\tau_j}{\to}(y_j\stackrel{f_j}{\to}y_j) $$ of $D$, the following diagram must be commutative for all $j$: $$ \begin{matrix} x &
\stackrel{\tau_j}{\longrightarrow} & y_j =UF_j\\ f\downarrow & & f_j\downarrow\\ x&\stackrel{\tau_j}{\longrightarrow} & y_j = UF_j. \end{matrix} \quad \text{(Diagram 1)} $$
Now we claim that the $\to\downarrow$ part of the above diagram forms a cone to $UF$, that is, we claim that the family $\{f_j\tau_j\}$ forms a cone $x\stackrel{.}{\to} UF$. Indeed, for
an arrow $g:j\to j'$ of $J$, consider the following diagram: $$ \begin{matrix} &&&&x\\ &&&\stackrel{\tau_j}{\swarrow}&&\stackrel{\tau_{j'}}{\searrow}\\ &&y_j && \stackrel{UFg}{\
longrightarrow} && y_{j'}\\ &\stackrel{f_j}{\swarrow} &&&&&&\stackrel{f_{j'}}{\searrow}\\ y_j&&&&\stackrel{UFg}{\longrightarrow}&&&&y_{j'} \end{matrix} $$
The upper triangle is commutative because $\tau$ is a cone to the base $UF$, and the lower trapezoid is commutative because $F$ is a functor, and hence $Fg$ is an arrow $F_j\to F_{j'}$ in
$D$. Hence the outer triangle commutes, as required. From the universality of $\tau$, it follows that there is a unique $f$ for which Diagram 1 is commutative, and we have uniqueness.
For existence, we can take $f$ to be the unique arrow $x\to x$ for which Diagram 1 is commutative, and we get a cone $$ \sigma=\{\sigma_j=\tau_j\colon (x\stackrel{f}{\to}x)\to F_j=(y_j\
stackrel{f_j}{\to}y_j)\} $$ with $U\sigma=\tau$. We claim that this is a limiting cone.
To see this, let $\alpha\colon(z\stackrel{g}{\to}z)\stackrel{.}{\to}F$ be a cone, so that for all $j$ the following diagram is commutative: $$ \begin{matrix} z & \stackrel{\alpha_j}{\
longrightarrow} & y_j\\ g\downarrow & & f_j\downarrow\\ z &\stackrel{\alpha_j}{\longrightarrow} & y_j. \end{matrix} \quad\text{(Diagram 2)} $$
Then $U\alpha$ is a cone $z\stackrel{.}{\to} UF$ in $C$, and by the universality of $\tau$ there exists a unique arrow $h\colon z\to x$ for which the following diagram is commutative for
all $j$: $$ \begin{matrix} z & \stackrel{\alpha_j}{\longrightarrow} & y_j\\ h\downarrow& \stackrel{\tau_j}{\nearrow}\\ x& \end{matrix}\quad\text{(Diagram 3)} $$
If this $h$ is an arrow $(z\stackrel{g}{\to}z)\to (x\stackrel{f}{\to}x)$ in $D$, then we're done. In other words, all that remains to do is to show that the outer rectangle of the
following diagram is commutative: $$ \begin{matrix} z && \stackrel{h}{\longrightarrow} && x\\ & \stackrel{\alpha_j}{\searrow} && \stackrel{\tau_j}{\swarrow}\\ && y_j\\ g\downarrow&& \
downarrow f_j && \downarrow f\\ && y_j\\ & \stackrel{\alpha_j}{\nearrow} && \stackrel{\tau_j}{\nwarrow}\\ z && \stackrel{h}{\longrightarrow} && x\\ \end{matrix} $$ Now, the left trapezoid
is just Diagram 2, the upper and lower triangles are just Diagram 3, and the right trapezoid is commutative for all $j$ by the definition of $f$. It follows that both paths of the outer
rectangle have the same composition with the limiting cone $\tau$, and hence the outer rectangle is commutative, as required.
Oops, the part concerning dynamical systems is wrong: the category of algebras for the identity monad on $\mathbf{Set}$ is just $\mathbf{Set}$. But I think that I can prove creation
directly in this example, and I will soon either send the correction or declare the last example as open. – user2734 May 10 '10 at 9:01
1 OK, now I have a (hopefully) correct direct proof. – user2734 May 10 '10 at 20:11
add comment
Not the answer you're looking for? Browse other questions tagged ct.category-theory at.algebraic-topology representable-functors or ask your own question. | {"url":"http://mathoverflow.net/questions/23145/when-does-a-representable-functor-into-a-category-other-than-set-preserve-limi","timestamp":"2014-04-18T00:27:38Z","content_type":null,"content_length":"71071","record_id":"<urn:uuid:116427ba-7947-48bb-94c4-da4a588e1cf9>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00205-ip-10-147-4-33.ec2.internal.warc.gz"} |
Spring SAT Math Tutor
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San Anselmo Precalculus Tutor
...G.'s qualifications include a Ph.D. in engineering from CalTech (including a minor in numerical methods/applied math) and over 25 years experience as a practicing environmental engineer/
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6 Subjects: including precalculus, physics, calculus, algebra 1 | {"url":"http://www.purplemath.com/San_Anselmo_precalculus_tutors.php","timestamp":"2014-04-16T10:45:25Z","content_type":null,"content_length":"23964","record_id":"<urn:uuid:b3151ef3-9809-4b3a-8031-93c3237930d2>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00536-ip-10-147-4-33.ec2.internal.warc.gz"} |
help me with the acm question
help me with the acm question
Today,I have a problem with this acm quetion,Who can help me?
Maybe you wonder what an annoying painting tool is? First of all, the painting tool we speak of supports only black and white. Therefore, a picture consists of a rectangular area of pixels, which
are either black or white. Second, there is only one operation how to change the colour of pixels:
Select a rectangular area of r rows and c columns of pixels, which is completely inside the picture. As a result of the operation, each pixel inside the selected rectangle changes its colour
(from black to white, or from white to black).
Initially, all pixels are white. To create a picture, the operation described above can be applied several times.
Can you paint a certain picture which you have in mind?
TThe input contains several test cases. Each test case starts with one line containing four integers n, m, r and c. (1 <= r <= n <= 100, 1 <= c <= m <= 100), The following n lines each describe
one row of pixels of the painting you want to create. The ith line consists of m characters describing the desired pixel values of the ith row in the finished painting ('0' indicates white, '1'
indicates black).
The last test case is followed by a line containing four zeros.
For each test case, print the minimum number of operations needed to create the painting, or -1 if it is impossible.
Sample Input
Sample Output
Looking at the second example: at each move choose area where all pixels are 1, or choose mixed area which after the transformation yields a new area with all pixels 1. Of course, I may be
completely off with this greedy solution attempt but it would work for the sample cases.
Edit: it won't work just like that. For example the following position with 2x2 rectangles is solveable even though there isn't any transformation that leads to an all black area:
However, all 2x2 sections contain at most 2 black pixels and there are transformations that lead to rectangles with 3 black pixels. May-be that would help?
First of all, notice how you will never have to use exactly the same rectangle on the same position more than once: applying it X times is the same as applying it (X%2) times (!!a = a).
Now we can use the fact that the rectangle must be inside the image. If the upper-left bit is not the initial color, there's only one rectangle that can be used to fix that. Now since we know we
never have to use the same rectangle, we can move one pixel to the right. Again, there's only one rectangle that can invert it if we must, since the only other option is one pixel to the left,
but we already know whether we had to select that.
That results in quite an easy algorithm, but I believe it should be efficient enough. It's only O((n-r+1)*(m-c+1)*r*c). So the maximum is about 6.502.500 operations. During the contests, you
could usually fit 1 to 10 million operations per test case. Since these operations are fairly simple, I imagine it'll be fast enough.
Sounds brilliant, except practically solves the challenge for the OP :) :(
I don't get it. Can't you just use rectangles of size 1x1 to control individual pixels? All images are possible and can be drawn in at most O(n) time.
EDIT: Ah, you don't get to choose r and c. I see now.
I know... Puzzles like these always get me excited and whatever I'm doing, I put down my work until I fix the problem... ;)
@brewbuck: Yeah, I had to read the problem statement twice as well before I understood that point ;). | {"url":"http://cboard.cprogramming.com/cplusplus-programming/119249-help-me-acm-question-printable-thread.html","timestamp":"2014-04-17T16:38:25Z","content_type":null,"content_length":"11476","record_id":"<urn:uuid:0d1ff9bc-3deb-460c-8199-e22f741db468>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00404-ip-10-147-4-33.ec2.internal.warc.gz"} |
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rajesh dolai
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12 Days ago
thank you M4maths for such awesome collection of questions. last month i got placed in techMahindra. i prepared for written from this site, many question were exactly same as given here.
bcz of practice i finished my written test 15 minutes before and got it.
thanx allot for such noble work...
16 Days ago
coz of this site i cud clear IBM's apti nd finally got placed in tcs
thanx m4maths...u r a wonderful site :)
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Stephen raj
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prepare from r.s.aggarwal verbal and non verbal reasoning and previous year questions from m4maths,indiabix and chetanas forum.u can crack it.
Stephen raj
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Thanks to m4maths:)
cracked infosys:)
1 Month ago
i have been Selected in Tech Mahindra.
All the quanti & reasoning questions are common from the placement papers of m4maths.
So a big thanks to m4maths team & the people who shares the placement papers.
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Cleared the aptitude.
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plz guide for the technical round.
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got placed in IBM..
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thanks alot to m4maths.com
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Thanks of m4maths....
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4 Months ago
Thanx to m4 maths, because of that i able to crack aptitude test and now i am a part of TCS. This site is best for the preparation of placement papers.Thanks a lotttttt............
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5 Months ago
m4maths is one of the main reason I cleared TCS aptitude. In TCS few questions will be repeated from previous year aptis and few questions will be repeated from the latest campus drives that happened
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5 Months ago
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5 Months ago
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5 Months ago
2 days before i cleared written test just because of m4maths.com.thanks a lot for this community.
6 Months ago
thanks for m4maths!!! bcz of which i cleared apti of infosys today.
6 Months ago
Today my written test of TCS was completed.I answered many of the questions without reading entire question.Because i am one of the member in the m4maths.
No words to praise m4maths.so i simply said thanks a lot.
7 Months ago
I am very grateful to m4maths. It is a great site i have accidentally logged on when i was searching for an answer for a tricky maths puzzle. It heped me greatly and i am very proud to say that I
have cracked the written test of tech-mahindra with the help of this site. Thankyou sooo much to the admins of this site and also to all members who solve any tricky puzzle very easily making people
like us to be successful. Thanks a lotttt
Abhishek Ranjan
7 Months ago
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do practice n u'll surely succeed :)
Sandhya Pallapu
1 year ago
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this...thanks to M4MATHS!
vivek singh
2 years ago
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admin for creating such a superb community
Manish Raj
2 years ago
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Asif Neyaz
2 years ago
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2 years ago
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I cracked TCS..The solutions given were very helpful!!!
2 years ago
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to all the users who clearly solved the problems.. im very greatful to you :)
2 years ago
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2 years ago
Hello friends .I was selected in TCS. Thanx to M4Maths to crack apti. and my hearthly wishes that
the success rate of M4Math grow exponentially.
Again Thanx for all support given by M4Math during
my preparation for TCS.
and Best of LUCK for all students for their preparation.
2 years ago
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2 years ago
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got selected in tcs for u only...
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thnx to all the users n team of m4maths...
3 cheers for m4maths
2 years ago
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got selected in tcs for u only...
u were the only guide n i hv nvr done group study for TCS really feeling great...
thnx to all the users n team of m4maths...
3 cheers for m4maths
2 years ago
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Continue this g8 work
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3 years ago
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rajesh dolai 19 Hours ago plz send me ibm plaement paper english..
venkaiaha 1 Day ago cleared ibm 2 rounds thank you m4maths.com
Triveni 1 Day ago Thanks to m4maths.I got placed in IBM.Awsome work.Best of luck.
Lekshmi Narasimman MN 7 Days ago Thanks ton for this site . This site is my main reason for clearing cts written which happend on 5/4/2014 in chennai . Tommorrw i have my interview. Hope i will tel u
all a good news :) Thanks to almighty too :) !!
abhinay yadav 12 Days ago thank you M4maths for such awesome collection of questions. last month i got placed in techMahindra. i prepared for written from this site, many question were exactly same
as given here. bcz of practice i finished my written test 15 minutes before and got it. thanx allot for such noble work...
manasi 16 Days ago coz of this site i cud clear IBM's apti nd finally got placed in tcs thanx m4maths...u r a wonderful site :)
arnold 19 Days ago thank u m4maths and all its user for posting gud and sensible answers.
Nilesh singh 22 Days ago finally selected in TCS. thanks m4maths
MUDIT 23 Days ago Thank you team m4maths.Successfully placed in TCS.
Deepika Maurya 24 Days ago Thank you so much m4maths.. I cleared the written of IBM.. :) very good site.. thumps up !!
Rimi Das 1 Month ago Thanks to m4maths I got selected in Tech Mahindra.I was preparing for TCS 1st round since last month.Got interview call letter from there also...Really m4maths is the best site
for placement preparation...
Stephen raj 1 Month ago prepare from r.s.aggarwal verbal and non verbal reasoning and previous year questions from m4maths,indiabix and chetanas forum.u can crack it.
Stephen raj 1 Month ago Thanks to m4maths:) cracked infosys:)
Ranadip 1 Month ago i have been Selected in Tech Mahindra. All the quanti & reasoning questions are common from the placement papers of m4maths. So a big thanks to m4maths team & the people who
shares the placement papers.
Amit Das 1 Month ago I got selected for interview in TCS.Thank you very much m4maths.com.
PRAVEEN K H 1 Month ago I got placed in TCS :) Thanks a lot m4maths :)
Syed Ishtiaq 1 Month ago An Awesome site for TCS. Cleared the aptitude.
sara 1 Month ago I successfully cleared TCS aptitude test held on 8th march 2014.Thanks a lot m4maths.com plz guide for the technical round.
mounika devi mamidibathula 1 Month ago got placed in IBM.. this site is very useful, many questions repeated.. thanks alot to m4maths.com
Anisha Lakhmani 1 Month ago I got placed at infosys.......thanx to m4maths.com.......a awesum site......
Kusuma Saddala 1 Month ago Thanks to m4maths, i have place at IBM on feb 8th of this month
sangeetha 2 Months ago thanks to m4 maths because of this i clear csc written test
mahima srivastava 2 Months ago Placed at IBM. Thanks to m4maths. This site is really very helpful. 95% questions were from this site.
Surya Narayana K 2 Months ago I successfully cleared TCS aptitude test.Thanks a lot m4maths.com.
prashant gaurav 2 Months ago Got Placed In Infosys... Thanks of m4maths....
vishal 3 Months ago iam not placed in TCS...........bt still m4maths is a good site.
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Sonali 4 Months ago THANKS a lot m4maths. Me and my 2 other roomies cleared the tcs aptitude with the help of this site.Some of the questions in apti are exactly same which i answered without even
reading the whole question completely.. gr8 work m4maths.. keep it up.
Kumar 5 Months ago m4maths is one of the main reason I cleared TCS aptitude. In TCS few questions will be repeated from previous year aptis and few questions will be repeated from the latest campus
drives that happened in various other colleges. So to crack TCS apti its enough to learn some basic concepts from famous apti books and follow all the TCS questions posted in m4maths. This is not
only for TCS but for all other companies too. According to me m4maths is best site for clearing apti. Kuddos to the creator of m4maths :)
YASWANT KUMAR CHAUDHARY 5 Months ago THANKS A LOT TO M4MATHS.due to m4maths today i am the part of TCS now.got offer letter now.
ANGELIN ALFRED 5 Months ago Hai friends, I got placed in L&T INFOTECH and i m visiting this website for the past 4 months.Solving placemetn puzzles from this website helped me a lot and
1000000000000s of thanks to this website.this website also encouraged me to solve puzzles.follw the updates to clear maths aps ,its very easy yar, surely v can crack it if v follow this website.
MALLIKARJUN ULCHALA 5 Months ago 2 days before i cleared written test just because of m4maths.com.thanks a lot for this community.
Madhuri 6 Months ago thanks for m4maths!!! bcz of which i cleared apti of infosys today.
DEVARAJU 6 Months ago Today my written test of TCS was completed.I answered many of the questions without reading entire question.Because i am one of the member in the m4maths. No words to praise
m4maths.so i simply said thanks a lot.
PRATHYUSHA BSN 7 Months ago I am very grateful to m4maths. It is a great site i have accidentally logged on when i was searching for an answer for a tricky maths puzzle. It heped me greatly and i am
very proud to say that I have cracked the written test of tech-mahindra with the help of this site. Thankyou sooo much to the admins of this site and also to all members who solve any tricky puzzle
very easily making people like us to be successful. Thanks a lotttt
Abhishek Ranjan 7 Months ago me & my rooom-mate have practiced alot frm dis site TO QUALIFY TCS written test.both of us got placed in TCS :) IT'S VERY VERY VERY HELPFUL N IMPORTANT SITE. do practice
n u'll surely succeed :)
Sandhya Pallapu 1 year ago Hai friends! this site is very helpful....i prepared for TCS campus placements from this site...and today I m proud to say that I m part of TCS family now.....dis site
helped me a lot in achieving this...thanks to M4MATHS!
vivek singh 2 years ago I cracked my first campus TCS in November 2011...i convey my heartly thanks to all the members of m4maths community who directly or indirectly helped me to get through
TCS......special thanks to admin for creating such a superb community
Manish Raj 2 years ago this is important site for any one ,it changes my life...today i am part of tcs only because of M4ATHS.PUZZLE
Asif Neyaz 2 years ago Thanku M4maths..due to u only, imade to TCS :D test on sep 15.
Harini Reddy 2 years ago Big thanks to m4maths.com. I cracked TCS..The solutions given were very helpful!!!
portia 2 years ago HI everyone , me and my friends vish,sube,shaf placed in TCS... its becoz of m4maths only .. thanks a lot..this is the wonderful website.. unless your help we might not have been
able to place in TCS... and thanks to all the users who clearly solved the problems.. im very greatful to you :)
vasanthi 2 years ago Really thanks to m4maths I learned a lot... If you were not there I might not have been able to crack TCS.. love this site hope it's reputation grows exponentially...
vijay 2 years ago Hello friends .I was selected in TCS. Thanx to M4Maths to crack apti. and my hearthly wishes that the success rate of M4Math grow exponentially. Again Thanx for all support given by
M4Math during my preparation for TCS. and Best of LUCK for all students for their preparation.
maheswari 2 years ago thanks to M4MATHS..got selected in TCS..thanks for providing solutions to TCS puzzles :)
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Maths Quotes (More)
"Maths is like a proverb where there is a will there is a way.If you go deep into it you find someway" Faisal Rizwan
"Mathematics is like a rain... When we know how to solve it, We will continouosly do it." Syawal
"Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer" Carl Sandburg
""The gaming of numbers is called maths.So participate in this wonderful game and explore in it,learn in it and enjoy in it.Then only you can pass in this wonderful game."" Kumar Purnendu
"Well done is better than well said." Ben Franklin
""If you able to solve the problems in MATHS, then you also able to solve the problems in your LIFE" (Maths is a great Challenger)" Vignesh
"Don't learn Mathematics just to prove that you are not a mentaly simple person but learn it to prove that you are intelligent " Linda
Latest Placement Puzzle (More)
"If x increases linearly, how will a-x behave (a>1)?
o Increase linearly
o Decrease linearly
o Increase exponentially
o Decrease exponentially"
UnsolvedAsked In: Other
"find the output is 24
u have to use only 8,8,3,3 except then that u cant even use 1 or etc. and u can use +.-,/,* in any times"
UnsolvedAsked In: HR Interview
". If in coded language
MAMTA = 8
SAUMYA = 80
MONIKA = 35
RASHMI = ?
Options 1) 16
2) 26
3) 42
4) 38
5) 14
6) 81
7) 39
8) 14
9) 27
10)None of these"
UnsolvedAsked In: M4maths
"Maths is like a proverb where there is a will there is a way.If you go deep into it you find someway" Faisal Rizwan
"Mathematics is like a rain... When we know how to solve it, We will continouosly do it." Syawal
"Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer" Carl Sandburg
""The gaming of numbers is called maths.So participate in this wonderful game and explore in it,learn in it and enjoy in it.Then only you can pass in this wonderful game."" Kumar Purnendu
""If you able to solve the problems in MATHS, then you also able to solve the problems in your LIFE" (Maths is a great Challenger)" Vignesh
"Don't learn Mathematics just to prove that you are not a mentaly simple person but learn it to prove that you are intelligent " Linda
"If x increases linearly, how will a-x behave (a>1)? o Increase linearly o Decrease linearly o Increase exponentially o Decrease exponentially" UnsolvedAsked In: Other
"find the output is 24 u have to use only 8,8,3,3 except then that u cant even use 1 or etc. and u can use +.-,/,* in any times" UnsolvedAsked In: HR Interview
". If in coded language MAMTA = 8 SAUMYA = 80 MONIKA = 35 then RASHMI = ? Options 1) 16 2) 26 3) 42 4) 38 5) 14 6) 81 7) 39 8) 14 9) 27 10)None of these" UnsolvedAsked In: M4maths
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MathGroup Archive: January 2004 [00018]
[Date Index] [Thread Index] [Author Index]
Re: Compile
• To: mathgroup at smc.vnet.net
• Subject: [mg45349] Re: Compile
• From: Bill Rowe <readnewsciv at earthlink.net>
• Date: Fri, 2 Jan 2004 04:23:40 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com
On 1/1/04 at 5:54 AM, dontsendhere@. (Maxim) wrote:
> If you tried to actually do some testing --
I did but did not try such odd code
> Module[{y := If[Print[x];NumericQ[x], x, 1]},
> Plot[x y, {x, 0, 1}, Compiled -> True]
> ]
The documentation specifically states only inline functions get compiled not user defined functions. So when you write code such as the above, you should expect things to be a bit unpredictable.
> you would notice that y is evaluated for each step Plot takes and it prints symbol x.
No, I don't think that is what is happening. I believe y is evaluated once to 1 since x is not numeric at the time y is evaluated. But since you have wrapped x in a Print statement what is compiled is a literal Print[x] followed by x which causes the symbol x to be printed each time the function x is sampled to be plotted.
If y were evaluated at each step, I would expect numeric values to be printed instead of the symbol x and a parabola which is exactly what you get if you choose Compiled->False in the above code.
> And of course, Compile doesn't evaluate its second argument -- do you really think that to execute
> Compile[{}, Print[x]] Print[x] should be evaluated?
I don't know why anyone would write code like this, but if they did why should they expect Print[x] not to be evaluated in this example?
To reply via email subtract one hundred and four | {"url":"http://forums.wolfram.com/mathgroup/archive/2004/Jan/msg00018.html","timestamp":"2014-04-16T22:17:52Z","content_type":null,"content_length":"35353","record_id":"<urn:uuid:b985f1fe-f111-4e78-9836-6f2c1125c1ec>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00462-ip-10-147-4-33.ec2.internal.warc.gz"} |
Help with number problem
March 6th 2007, 05:35 PM #1
Junior Member
Mar 2007
Help with number problem
Okay I got this number problem and I don't really know what the questions are trying to say.
1) Give sufficient examples to convince someone that every multiple of 3 that is divisible by 4 is also a multiple of 6.
2) Show algebraically that this is true in general for any such numbers.
Any help would be greatly appriciated.
Okay I got this number problem and I don't really know what the questions are trying to say.
1) Give sufficient examples to convince someone that every multiple of 3 that is divisible by 4 is also a multiple of 6.
2) Show algebraically that this is true in general for any such numbers.
Any help would be greatly appriciated.
what math is this? what level of math have you done? i want to know what kind of solution to give
yeah and I have no idea what i'm ment to give as an answear... Like I know how to do everything in that course, except this question is worded quite weirdly
Last edited by Unt0t; March 14th 2007 at 07:32 PM.
Okay I got this number problem and I don't really know what the questions are trying to say.
1) Give sufficient examples to convince someone that every multiple of 3 that is divisible by 4 is also a multiple of 6.
2) Show algebraically that this is true in general for any such numbers.
Any help would be greatly appriciated.
1) will five examples be enough?
24, 36, 72, 144, 216
2) Let x be a multiple of 3. then 3|x. (3|x means "3 divides x," that is 3 goes into x and leaves no remainder.)
since 3|x, x = 3m for some integer m.
similarly, if x is also a multiple of 4, then 4|x. so x = 4n for some integer n
=> 3m = 4n
=> 3m/4 = n, where n is some integer divisible by 3 and 4.
Now suppose 6|n, then n = 6k for some integer k. that is 3m/4 = 6k.
so 3m = 24k = 6(4k).
since 4k is an integer, and 3m = 6(4k), 6|3m. but 3m is x, and so 6|x, and therefore, x is a multiple of 6 as well.
there's something i dont like about this proof, i'll think about it and get back to you
does this stuff look familiar to you?
Umm I just don't understand the 3|x, 4|x etc. bit... like what does " | " actually mean?
1) will five examples be enough?
24, 36, 72, 144, 216
2) Let x be a multiple of 3. then 3|x. (3|x means "3 divides x," that is 3 goes into x and leaves no remainder.)
since 3|x, x = 3m for some integer m.
similarly, if x is also a multiple of 4, then 4|x. so x = 4n for some integer n
=> 3m = 4n
=> 3m/4 = n, where n is some integer divisible by 3 and 4.
Now suppose 6|n, then n = 6k for some integer k. that is 3m/4 = 6k.
so 3m = 24k = 6(4k).
since 4k is an integer, and 3m = 6(4k), 6|3m. but 3m is x, and so 6|x, and therefore, x is a multiple of 6 as well.
there's something i dont like about this proof, i'll think about it and get back to you
Now I understand what you've done here.... but are you sure that the second question is asking that?
Last edited by Unt0t; March 14th 2007 at 07:56 PM.
What ???
Last edited by Unt0t; March 14th 2007 at 07:57 PM.
here is a better proof, i think.
Background knowledge:
you should know that an integer is even if you can express it as 2n for some integer n, it is odd if you can express it as 2n + 1 for some integer n. if a number is divisible by 6 (a multiple of
6), you can write it as 6n for n an integer, if its divisible by 4, you can write it as 4n for n an integer and so on.
I'll try to tone done the logic and formal math for this proof.
We want to show that if a number x is divisible by 4 and divisible by 3 then it is divisible by 6.
Proof: assume that x is not divisible by 6. then we can write x as 6n + r, where n is any integer and r is the remainder when dividing by 6. therefore, r = 1,2,3,4, or 5.
and so we have 5 cases.
case 1: x = 6n + 1 (that is when we divide x by 6 we have 1 as a remainder).
note that x = 6n + 1 = 3(2n) + 1, since 2n is an integer, it means x is not divisible by 3 (we have a remainder 1).
case 2: x = 6n + 2
notice x = 6n + 2 = 3(2n) + 2
that means we have 2 as a remainder when we divide x by 3, so x is not divisible by 3
case 3: x = 6n + 3
note that x = 6n + 3 = 3(2n + 1), so in this case x is divisible by 3. now we will check if it is divisible by 4.
notice that x = 6n + 3 = 2(3n + 1) + 1. this means x is odd, and therefore is not divisible by 4.
case 4: x = 6n + 4
so x = 6n + 4 = 3(2n + 1) + 1, so x is not divisible by 3
case 5: x = 6n + 5
so x = 6n + 5 = 3(2n + 1) + 2, so x is not divisible by 3.
so we see in all cases that when a number is not divisible by 6 it is not divisible by either 3 or 4 as well. so we can conclude the opposite is true, by way of the contropositive.
March 6th 2007, 06:01 PM #2
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March 6th 2007, 07:50 PM #15 | {"url":"http://mathhelpforum.com/algebra/12251-help-number-problem.html","timestamp":"2014-04-17T18:55:09Z","content_type":null,"content_length":"83611","record_id":"<urn:uuid:a5014a22-104c-45ed-b85e-8392c2378613>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00446-ip-10-147-4-33.ec2.internal.warc.gz"} |
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Scattering of an incident field by particles aligned along the axis. The figure is not drawn to scale because . The bold arrow gives the propagation direction of the incident field, which is
polarized in the direction. The origin of the coordinate system is at the midpoint of the line of scatterers.
Particles placed on a line at random with nearest neighbors removed. It is clear from the figure that removal of nearest neighbors builds in correlation between particle positions.
Graphs of the structure function as a function of for several values of : (a) , (b) , (c) , (d) . The approximate expression (24) is used for (a)–(c) and the exact expression (21) with is used for
(d). A region of the order of about the origin is excluded because it is assumed that .
Experimental curves of the structure factor for x-ray diffraction from rubidium (Ref. 16) [ corresponds to our ]. The lowest curve corresponds to a dense gas and the other curves to the liquid phase.
The particles are divided into bins, each bin containing a large number of oscillators. The spatial phase factor is approximately constant for each particle within a given bin.
particles randomly distributed over a wavelength distance. The sine wave is replaced by a square wave to illustrate the importance of the scattered signal on particle fluctuations. | {"url":"http://scitation.aip.org/content/aapt/journal/ajp/78/1/10.1119/1.3236688","timestamp":"2014-04-16T22:42:49Z","content_type":null,"content_length":"77437","record_id":"<urn:uuid:1669e909-db89-4ca2-a5aa-3bae830630d1>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00010-ip-10-147-4-33.ec2.internal.warc.gz"} |
G theory programs
SPSS, SAS, and MATLAB Programs for Generalizability Theory Analyses
Mushquash, C., & O'Connor, B. P. (2006). SPSS, SAS, and MATLAB programs for generalizability theory analyses. Behavior Research Methods, 38(3), 542-547.
The identification and reduction of measurement errors is a major challenge in psychological testing. Most investigators rely solely on classical test theory for assessing reliability, whereas most
experts have long recommended using generalizability theory instead. One reason for the common neglect of generalizability theory is the absence analytic facilities for this purpose in popular
statistical software packages. This article provides a brief introduction to generalizability theory, describes easy to use SPSS, SAS, and MATLAB programs for conducting the recommended analyses, and
provides an illustrative example using data (N = 329) for the Rosenberg Self-Esteem Scale. Program output includes variance components, relative and absolute errors and generalizability coefficients,
coefficients for D studies, and graphs of D study results.
The "G1" programs below can be used for six possible one- or two-facet designs. They use the ANOVA method and they require balanced data.
The "G2" programs below use the SPSS VARCOMPS or SAS VARCOMP procedure to compute the variance components, which are then read and processed by the program for G theory analyses. The "G2" programs
can be used for any design that can be processed by the SPSS VARCOMPS or SAS VARCOMP procedures; there are no limitations on the number of facets; and they can process unbalanced data. The following
four kinds of variance components can be produced by the SPSS VARCOMPS and SAS VARCOMP procedure and then processed by the G2 programs: ML, REML, MINQUE, and ANOVA.
│SPSS │SAS │MATLAB │
│ │ │ │
│Programs: │Programs: │Programs:│
│G1.sps │G1.sas │G1.m │
│G2.sps (revised October 4, 2012) │G2.sas (revised October 4, 2012) │ │
Please contact me if you have any questions or suggestions.
Brian P. O'Connor, professor
Department of Psychology
University of British Columbia - Okanagan
Kelowna, British Columbia, Canada
E-mail me
Link to Brian O'Connor's Main Page | {"url":"https://people.ok.ubc.ca/brioconn/gtheory/gtheory.html","timestamp":"2014-04-21T14:39:22Z","content_type":null,"content_length":"5477","record_id":"<urn:uuid:7f06afd1-d08a-421d-b7a9-b9f4b4e43822>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00167-ip-10-147-4-33.ec2.internal.warc.gz"} |
Kristian Jonsson Moi
PhD student at the Department of Mathematical Sciences of the University of Copenhagen.
Member of the Topology group and the Centre for Symmetry and Deformation.
Advisor: Ib Madsen.
Contact information
Department of Mathematical Sciences, Office 04.4.03
Universitetsparken 5
2100 København Ø
Email: kmoi at math.ku.dk
I work in (hermitian) algebraic K-theory and homotopy theory.
Equivariant loops on classifying spaces.
Homotopy theory of G-diagrams and equivariant excision
Ib Madsen's lecture notes
Introduction to K-theory.
PhD student at the Department of Mathematical Sciences of the University of Copenhagen. Member of the Topology group and the Centre for Symmetry and Deformation. Advisor: Ib Madsen. Contact
information Department of Mathematical Sciences, Office 04.4.03 Universitetsparken 5 2100 København Ø Denmark Email: kmoi at math.ku.dk | {"url":"http://www.math.ku.dk/~kmoi/","timestamp":"2014-04-18T04:37:22Z","content_type":null,"content_length":"1664","record_id":"<urn:uuid:056e1368-bacd-4ff8-99cd-f6985dc15031>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00479-ip-10-147-4-33.ec2.internal.warc.gz"} |
Chapter 4 - The Derivative
Section 4.2- Average Rate of Change
We have learned that a change in the independent variable is defined as
To calculate how much more
While this expression may seem rather simple, it does require some explanation. By dividing the change in f by the change in x what we are doing is calculating how much more f changed for a given
change in x. For example in the function,
The value of 147 tells us that f changes 147 times more than x over that interval of Thus for each unit change in x,
The value, called the rate of change of the function, refers to how much more or less
Another way of understanding what rate of change of a function means is to look at the steepness of the line connecting the two endpoints of the interval under consideration.
As its slope or steepness over the interval under consideration. Since slope and rate of change are synonymous, then how is rate of change defined for functions whose graphs do not have constant
slopes? For example, from x=9 to x=12 of
This value is significantly higher than the rate of change calculated for the previous interval from x = 3 to x = 5. We can only conclude that the rate of change or slope of the graph must be
increasing and is not constant over an interval
Since the rate of change of a function can change, then we have to come up with a more refined definition of rate of change. We can define the average rate of change of a function over an interval
In the next section we will take a closer look at how we can define the exact or instantaneous rate of change of a function.
Next section -> Section 4.3 - Instantaneous Rate of Change | {"url":"http://www.understandingcalculus.com/chapters/04/4-2.php","timestamp":"2014-04-17T13:00:08Z","content_type":null,"content_length":"27793","record_id":"<urn:uuid:1e6062fb-7771-49ee-88fb-d17a24e99a31>","cc-path":"CC-MAIN-2014-15/segments/1397609530131.27/warc/CC-MAIN-20140416005210-00291-ip-10-147-4-33.ec2.internal.warc.gz"} |
Determining if a rational number terminates
Hello all. A friend of mine had participated in a programming competition for his school. He did pretty well, but he couldn't get one question. The question was, given the numerator and denominator
of a rational number, determine if the decimal expansion terminates. This is to say, if I gave you 1/3, you would say it doesn't because the expansion is "0.3333..." Likewise, if I gave you 1/10, you
would say it does terminate because the expansion is "0.1"
My friend had tried to solve this problem with string operations, but I found a better way. According to the Wikipedia article for repeating decimals, rational numbers that terminate are in the form
a/b -> b = 2^c*5^d where c and d are natural numbers. Given this identity, I developed the following (obfu) one-liner:
Given $ARGV[0] = a; $ARGV[1] = b the program will say "Y" if the decimal terminates and "N" if it doesn't.
The spoiler below reveals how the program works: | {"url":"http://www.perlmonks.org/index.pl/jacques?node_id=1006283","timestamp":"2014-04-16T19:39:47Z","content_type":null,"content_length":"35972","record_id":"<urn:uuid:d2eb3657-41ea-4c07-be9e-97bb6b41740e>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00095-ip-10-147-4-33.ec2.internal.warc.gz"} |
Quasiprobability representations of qubits
Negativity in a quasi-probability representation is typically
interpreted as an indication of nonclassical behavior.
However, this does not preclude bases that are non-negative from
having interesting applications---the single-qubit
stabilizer states have non-negative Wigner functions and yet
play a fundamental role in many quantum information tasks.
We determine what other sets of quantum states and measurements
of a qubit can be non-negative in a quasiprobability
representation, and identify nontrivial groups of unitary
transformations that permute such states. These sets of states
and measurements are analogous to the single-qubit stabilizer
states. We show that no quasiprobability representation of a
qubit can be non-negative for more than two bases in any plane
of the Bloch sphere. Furthermore, there is a single family of
sets of four bases that can be non-negative in an arbitrary
quasiprobability representation of a qubit. We provide an
exhaustive list of the sets of single-qubit bases that are nonnegative
in some quasiprobability representation and are also
closed under a group of unitary transformations, revealing two
families of such sets of three bases. We also show that not
all two-qubit Clifford transformations can preserve
non-negativity in any quasiprobability representation that is
non-negative for the computational basis. This is in stark
contrast to the qutrit case, in which the discrete Wigner
function is non-negative for all n-qutrit stabilizer states and
Clifford transformations. We also provide some evidence
that extending the other sets of non-negative single-qubit
states to multiple qubits does not give entangled states. | {"url":"http://perimeterinstitute.ca/videos/quasiprobability-representations-qubits","timestamp":"2014-04-21T02:41:36Z","content_type":null,"content_length":"28653","record_id":"<urn:uuid:06961db2-6b57-4995-a56f-f619ecc42e64>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00257-ip-10-147-4-33.ec2.internal.warc.gz"} |
Crocker, WA Math Tutor
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Tropical linear maps on the plane
Puente Muñoz, Maria Jesus de la (2010) Tropical linear maps on the plane. Linear Algebra and Applications . ISSN 0024-3795
Official URL: http://www.sciencedirect.com/science/journal/00243795
In this paper we fully describe all tropical linear maps in the tropical projective plane, that is, maps from the tropical plane to itself given by tropical multiplication by a real 3×3 matrix A. The
map fA is continuous and piecewise-linear in the classical sense. In some particular cases, the map fA is a parallel projection onto the set spanned by the columns of A. In the general case, after a
change of coordinates, the map collapses at most three regions of the plane onto certain segments, called antennas, and is a parallel projection elsewhere (Theorem 3).
In order to study fA, we may assume that A is normal, i.e., I A 0, up to changes of coordinates. A given matrix A admits infinitely many normalizations. Our approach is to define and compute a unique
normalization for A (which we call lower canonical normalization) (Theorem 1) and then always work with it, due both to its algebraic simplicity and its geometrical meaning.
On , any , some aspects of tropical linear maps have been studied in [6]. We work in , adding a geometric view and doing everything explicitly. We give precise pictures.
Inspiration for this paper comes from [3,5,6,8,12,26]. We have tried to make it self-contained. Our preparatory results present noticeable relationships between the algebraic properties of a given
matrix A (idempotent normal matrix, permutation matrix, etc.) and classical geometric properties of the points spanned by the columns of A (classical convexity and others); see Theorem 2 and
Corollary 1. As a by-product, we compute all the tropical square roots of normal matrices of a certain type; see Corollary 4. This is, perhaps, a curious result in tropical algebra. Our final aim is,
however, to give a precise description of the map . This is particularly easy when two tropical triangles arising from A (denoted and ) fit as much as possible. Then the action of fA is easily
described on (the closure of) each cell of the cell decomposition ; see Theorem 3.
Normal matrices play a crucial role in this paper. The tropical powers of normal matrices of size satisfy A n-1=A n=A n+1= . This statement can be traced back, at least, to [26], and appears later
many times, such as [1,2,6,9,10]. In lemma 1, we give a direct proof of this fact, for n=3. But now the equality A 2=A 3 means that the columns of A 2 are three fixed points of fA and, in fact, any
point spanned by the columns of A 2 is fixed by fA. Among 3×3 normal matrices, the idempotent ones (i.e., those satisfyingA=A 2) are particularly nice: we prove that the columns of such a matrix
tropically span a set which is classically compact, connected and convex (Lemma 2 and Corollary 1). In our terminology, it is a good tropical triangle
Item Type: Article
Uncontrolled Keywords: Linear map; Tropical geometry; Projective plane
Subjects: Sciences > Mathematics > Algebra
ID Code: 12802
Deposited On: 01 Jun 2011 11:25
Last Modified: 06 Feb 2014 09:33
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RE: st: reverse prediction - confidence interval for x at given y in non
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RE: st: reverse prediction - confidence interval for x at given y in nonlinear model
From "Nick Cox" <n.j.cox@durham.ac.uk>
To <statalist@hsphsun2.harvard.edu>
Subject RE: st: reverse prediction - confidence interval for x at given y in nonlinear model
Date Fri, 26 Oct 2007 19:21:58 +0100
We have been at cross purposes.
Your log(0) as 0 is, I see, only a trick that keeps
the zeros in the frame _so long as you also have a dummy
for 0_.
My point is that it does not help in problems in which there is no dummy
Suppose your doses are 0 0.1 0.2 0.3 0.4 0.5
in whatever units are being used. Neither Stata nor the mathematics
knows about the units.
Then the recipe cond(dose == 0, 0, log(dose)) is not even monotonic
in dose. 0 is treated as if were larger than any other positive
value in the data!
Also if any dose is 1, this recipe treats 0 and 1 as
-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Daniel Waxman
Sent: 26 October 2007 19:07
To: statalist@hsphsun2.harvard.edu
Subject: RE: st: reverse prediction - confidence interval for x at given
y in nonlinear model
Regarding Nicks comment on replacing log(0) with zero:
It may not be pretty, but it does work!
If zdose=0 when dose=0, 1 otherwise
and ldose=log(dose) for the condition dose!=0, zero otherwise
then you are fitting the model:
xb =b [zdoseXldose]*zdose*ldose + b[zdose]*zdose + b[cons]
so when dose=0 the first two terms drop out and the coefficient for the
constant represents dose=0.
In all other cases, all 3 terms remain.
It might be more palatable if you replace all the zeroes in the
preceding discussion with a *very small number*
For the record, this gives exactly the same result as using the catzero
option with Patrick Royston's -mfp- routine.
. help mfp
-----Original Message-----
From: "Nick Cox" <n.j.cox@durham.ac.uk>
Subj: RE: st: reverse prediction - confidence interval for x at given y
in nonlinear model
Date: Fri Oct 26, 2007 9:15 am
Size: 1K
To: <statalist@hsphsun2.harvard.edu>
The idea of a dummy for zero dose is interesting but doesn't seem to map
on the kind of model being discussed here.
More importantly, that does nothing to solve the major issue, which is
thinking up a good alternative to log(0). Replacing log(0) by 0 is
equivalent to replacing 0 by 1 in whatever units are being used. How
sensible that is will depend partly on the range of the data. If the
rest of the data were 0.1 to 0.5 it would be crazy!
The problem in general is that mapping 0 to a very small number creates
a very large negative logarithm. Although I guess that there must be
other solutions, one is to do a sensitivity analysis of varying choices
of c in log(x + c), or cond(x == 0, c, log(x)).
Daniel Waxman
Regarging the treatment of zeroes in log(dose):
Since zero likely reflects a qualitatively different situation than
small values of dose you are better off treating it as such. Here is a
trick to get stata to do what you want:
gen ldose = log(dose)
gen zdose = 1 - (dose == 0)
replace ldose = 0 if dose == 0
logit outcome ldose zdose ...
Thus ldose is a term which represents log dose for positive values, and
falls out for doses of zero. zdose is a dummy which is zero for doses
of zero and one otherwise. If you look at the model as:
logit outcome ldose*zdose zdose
and look at what happens as dose (untransformed) becomes infinitesimal,
you can see how this works.
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
--- message truncated ---
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
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A note on the pseudo-spectra and the pseudo-covariance generating functions of ARMA processes
Bujosa Brun, Andrés and Bujosa Brun, Marcos and García Ferrer , Antonio (2002) A note on the pseudo-spectra and the pseudo-covariance generating functions of ARMA processes. [Working Paper or
Technical Report]
Official URL: http://eprints.ucm.es/7653/
Although the spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not the case for non-stationary stochastic processes. In this paper, the algebraic
foundations of the
spectral analysis of non-stationary ARMA processes are established.
For this purpose the Fourier Transform is extended to the field of fractions of polynomials. Then, the Extended Fourier Transform pair pseudo-covariance generating function / pseudo-spectrum,
to the Fourier Transform pair covariance generating function / spectrum, is defined. The new transform pair is well defined for stationary and non-stationary ARMA processes. This new approach can be
viewed as an extension of the classical spectral analysis. It is shown that the frequency domain has some additional algebraic advantages
over the time domain.
Item Type: Working Paper or Technical Report
Uncontrolled Fourier Transform
Subjects: Social sciences > Economics > Econometrics
Series Name: Documentos de trabajo del Instituto Complutense de Análisis Económico (ICAE)
Volume: 2002
Number: 0203
ID Code: 7653
References: Brockwell, P. J., and R. A. Davis (1987): Time Series: Theory and Methods, Springer series in Statistics. Springer-Verlag, New York.
Bujosa, M., A. Garc´ıa-Ferrer, and P. C. Young (2002): “An ARMA representation of unobserved component models under generalized random walk specifications: new algorithms and
Caines, P. E. (1988): Linear Stochastic Systems, Wiley series in probability and mathematical statistics. John Wiley & Sons, Inc., New York.
Godement, R. (1974): ´ Algebra. Editorial Tecnos, Madrid, first edn. Harvey, A. (1989): Forecasting Structural Time Series Models and the Kalman Filter. Cambridge University Press,
Cambridge, first edn.
Hatanaka, M., and M. Suzuki (1967): “A theory of the Pseudosprectrum and Its Application to Nonstationary Dynamic Ecomometric Models,” in Essays in Mathematical Economics. In Honor of
Oskar Morgenstern, ed. by M. Shubik, chap. 26, pp. 443–446. Princeton University Press, Princeton, New Jersey.
Luenberger, D. G. (1968): Optimization by vector space methods, Series in decision and control. John Wiley & Sons, Inc., New York.
Priestley, M. P. (1981): Spectral Analysis and Time Series, Probability and Mathematical Statistics. Academic Press, London, first edn.
Young, P. C., D. Pedregal, and W. Tych (1999): “Dynamic Harmonic Regression,” Journal of Forecasting, 18, 369–394.
Deposited On: 04 Mar 2008
Last Modified: 06 Feb 2014 07:55
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Commission Calculator
Attached is sample data. Target payout is the amount of the base salary a salesman can earn if all goals are reached. Commissions are payed quarterly. Listed on the left are 7 goals, each with a
weight of importance toward the target payout amount. The percentage achieved can only be 100% or 0%, nowhere in the middle, although they are allowed to make up a goal later if they do not make it
at first. This is where I get stumped. For example, if they don't make a goal in quarter one but make the goal in quarter two plus what they missed in one, they get two quarters worth of commission
in quarter 2 for that goal. I am having trouble figuring out the easiest way to go about this.
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Once the weekly + daily rates add up to equal or more than the monthly, then the monthly rate is used and that's what the customer pays. What I'm trying to do is make a calculator that I input the
rates and the rental period and the spread sheet will tell me (based on those rules) how many days, weeks and/or months the customer needs to pay and how much his total dollar amount will be.
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Timesheet Calculator
See workbook attached.
I'm looking for help to detemine rates so it automates in the sheet.
Can you give me assistance and code perhaps ? I'm pretty basic at V-Lookup and If functions. Is this the best route to take ?
All is explained within the workbook.
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Little Calculator Wanted
I've been trying but excel does not seem to recognise x and y
so here it is
2* X = Z
Where 2 is changable
X = Z-2
Now i just want to find what Z is.
Here's an Example
1.3x - z
1.3(x-.06) = z
1.3z - .078 = z
1.3z = z + .78
1.3z-z = .78
.3z = .78
z = .78/.3
z = 2.6
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Staffing Calculator
I've been trying to create a Staffing calculator for a call center. Basically the calculator should be able to add up the number of agents for the next 18 intervals based on the login time that is
entered by the user. I've just outlined the functioning below:
Suppose 10 agents login at 8 AM (thus logout at 5 PM), the intervals right from 8 AM till 5 PM should show up the 10 agents. Now suppose 10 more agents login at 9 AM, we would then have 20 agents
logged in till 5 PM (since agents logged in at 8 will logout at 5) & the remaining 10 till 6 PM. Thus if we have 10 more agents logging in at 10:30, we will have 30 agents till 5 PM, 20 till 6 PM &
10 agents till 7:30 PM & so on. I have attached an excel file to explain the example & the way the calculator has to be built. It is preferrable that the cells containing the login time aren't fixed,
but the user should be able to input any login time in any cell.
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Age Based Calculator
I am trying to put together a spreadsheet that works out values based on a persons age or service length.
So in I have the following data in columns starting at B4 and ending at k4:
Date of birth; start date; text; text; text; text; Weekly Pay; Start age; Current Age; Years Service.
The formula I need to provide in l4, m4 and n4 comes from the following rules:
* 0.5 week's pay for each full year of service where age during year less than 22
* 1.0 week's pay for each full year of service where age during year is 22 or above, but less than 41
* 1.5 weeks' pay for each full year of service where age during year is 41+
So in l4 I have:
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Night Payment Calculator
Not sure if this is the correct section for this kind of query but I'd like some assistance with a calculation that I can't seem to figure out.
Essentially it's for calculating night payments for our employee time-sheets. Our staff have very sporadic shifts and are paid extra for working between the hours of 00:00 and 06:00, basically when
employee's enter their start and end times I'd like the spreadsheet to automatically calculate how many hours they have worked between those hours, I imagine it's very simple but I cannot figure out
which function to use.
To complicate matters, because staff can work shifts which start on one day and finish the next we work on a 48:00 clock basis so its' not only between the hours of 00:00 - 06:00 where they qualify
for night payments but also from 24:00 - 30:00 if that makes sense?
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Financial Calculator Functions
I would like a user friendly form where I could enter
£50 15% discount and it would add 50% plus VAT and produce POR & POC which would adjust if the suggested price was changed.
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Get Rid Of The Calculator Toolbar
I downloaded the calculator toolbar, and for some reason it doesn't look right, it doesn't resemble a calculator layout at all.
The numbers are skewed, as though the layout is in landscape view instead of portrate. I've tried repeately to delete it from the toolbars menu option, with no success. It's become annoying.
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IF/AND Statements In Comp Calculator
Almost done with this but I'm stumped on the last remaining formula. The
way this compensation plan will work is if reps are between 0-25% growth they
will get paid $100 per point, 26-50% $150 per point, 51-75% $200 per point.
Growth is calculated monthly vs. a monthly #. My problem is that if the rep
is at 40% they get the first 25% at $100 and the next 15% at $150. My
formula is an either or thing. The following are my formulas for 0-25%,
26-50%,51-75% respectively:
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Quote To Sale Calculator
I work in sales and when under target I need to know how much more i need to sell to hit target. e.g
quotes 10 sold 1 currently at 10%
my target is 50%
at the minute i would have to work this out manually
like i need to quote and sell the next 8 to hit the 50% target.
What can i use on excel putting my current quotes and sales and my target in so it gives me the quotes and sales i need to do?
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Progress Tax Calculator
I'm trying to come up with an efficient formulae or function to calculate tax
The problem I have is that the tax is progressive. As below
the first 20,000 is taxed at 5%
the next 20,000 is taxed at 6%
the next 20,000 is taxed at 7%
the next 20,000 is taxed at 8%
the next 20,000 is taxed at 9%
more than 100,000 is taxed at 10%
I'm trying to do a formula like below
Cell B3 is my taxable amount
Cell B5 =IF(B$3>20000,20000*0.05,B$3*0.05)
Cell B6 =IF(B$3>40000,40000*0.05,(B$3-20000)*0.05)
This gives me a problem in that for 35k say, I end up with a negative number for the second part in cell B6. My other issue is that each calculation will take up 6 rows on my
spreadsheet. I was hoping to set up a function that could do this in a cell, but even the simple stage defeats me at the moment.
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Create A Football Calculator
i m just trying to create a football calculator, just a real basic one, my excel is ok. Just doin it for a mate as he is arranging some charity football event. i have inputted the teams and that and
worked out goal differences but would like to calculate games played, wins losses draws and points, can someone advise or quicky have a look, see attachment.
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Bank Interest Calculator
I'm an amateur to macro as I'm only in a low level class at a university... But I'm attempting to make a macro for a bank interest calculator. It asks your type of account(which then assigns an
interest rate to it), how much money is in the account, and also how long th emoney will be in the account. I used a "Select Case" for the account types, but I seem to be struggling for it to work,
it won't put the value of the total into the assigned cell, or it's just not computing it(as I get "0" each time I run it)...
Public Function BankCalculator()
'Bank Calculator for different accounts
'declare variable
Dim shtBank As Workbook, strAct As String, intMon As Integer, strLong As Integer, intTotal As Double
Set shtBank = Application.Workbooks("Bank Calculator.xls")
'input box for amount of money, assign address
intMon = InputBox(prompt:="How much money do you currently have in the account?", _ ......................................
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UK Bank Holiday Calculator
I found a link to a website on one of the forum pages. I had a look in the website and it showed a formula for calculating when Easter falls - I didn't know it could be worked out, but it can!!
I therefore decided to investigate further. I picked up another formula to calculate the first MOnday in May and I have now put together a little spreadsheet that will calculate all bank holidays in
the year entered in cell B1. It also takes into account additional bank holidays that exist when Christmas Day and/or Boxing Day fall on a weekend.
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Build A Working Day Calculator
I need to create a calculator that tells me how long a invoice will take to be paid using my current processes at work, working days only.
I need to imput the date the invoice is received and then for the rest to be worked out automaticly
I need it to do the follwoing ....
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Manufacturing To Retail Cost Calculator
I am trying to make an EASY Manufacturing Cost to Retail pricing calculator. This calculator would have ability to include cost of goods, labor, markup etc of components manufactured and sold as
retail products, example: small bookshelves versus large bookshelves or cabinets all have different materials (wood types, stain etc) in determining the final retail product costs that would reflect
time of labor hours involved in producing to determine final retail costs. Should be simple in Excel 2000 (my version). I have the basic template created and have used Data Validation Drop Down Lists
and utilized LOOKUP function. While my knowledge is limited in Excel, I am frustrated as how the LOOKUP function works, I can only get it to work where in the formula, the costs per unit are input
manually, whereas I would prefer to have the data input automatically from columns of calculated wholesale cost plus markup per square foot data, ie: =lookup(A1, X1:X30, Y1:Y30) but get errors when
doing this. Instead this works: =lookup(A1,{"pine", "oak", "birch"},{"3.99","7.87","5.15"}) and using this cell (A1) in my calculations for including the square foot costs in the final retail
calculations. While this seems to work, it is not easily modifiable as costs change rapidly and would like to easily input the cost per square foot of the different woods in their own cells rather
than in the formula calculation of the lookup. Hope that makes sense..... any suggestions? perhaps lookup is not the best function for what I am attempting? - sample file is attached, light green are
notes of where my data is located.
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CommandButton, Text Box, Calculator
I am trying to make a calculator inside of Excel...learning how Text Boxes work in conjunction with CommandButton. I am trying to code the button to display inside of the text box. I have never coded
a text box before to do anything like that
note: For right now i am just looking at being able to click on a button [numbered 0-9] and have them dsplayed inside the text box. After that I want to be able to set up an addition, subtraction,
etc button to actually have the math done.
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Making A Calculator From Vb.net To Vba
i made a claculator in vb.net, and now i have to make one in vba, what i did was as follows, first ill post the code from last year, then my vba attempt;
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Body Mass Index Calculator
I'm trying to make a BMI (Body Mass Index) calculator. I've attempted it already but to no success. My first mistake was to read the values from cells in the spreadsheet which made things more
complex than needed. So I started a new macro using a form instead for all data entry. I'm running into multiple problems with this
-Ask the user which method they prefer (see below formulas).
-Input the result into a certain cell.
English BMI Formula
BMI = ( Weight in Pounds / ( Height in inches ) x ( Height in inches ) ) x 703
Metric BMI Formula
BMI = ( Weight in Kilograms / ( Height in Meters ) x ( Height in Meters ) )
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Property Management Availability Calculator
I am working on a Property Management spreadsheet to track the availability of vacant units (and upcoming vacant units). I'm trying to write a formula to count the number of current units that are
listed as vacant, so that I can ultimately report on the overall unit availability as of any given period (today, 15, 30, 60, 90-days out). All of the data in this spreadsheet is dynamic and each
field can change at any time. There are also a lot of blank fields as well. I am using Excel 2003.
Here are the fields/data that I am using for my analysis:
Column A - (Reporting Period Dates): A2 = 5/20/08, A3 = 6/5/08, A4 = 6/20/08, etc.
Column B - (Unit Status: V=Vacant, O=Occupied): B2 = V, B3 = V, B4 = V, B5 = V, B6 = V, B7 = V, B8 = O, B9 = O, B10 = O, B11 = O, B12 = O, B13 = O
Column C - (M/O Date. These dates represent when the current resident will move out) - Cells C2:C13 contain dates for when each tenant will move out
Column D - (M/I Date. These dates represent when the future resident will move in) - Cells D2:D13 contain dates when the future tenant will move in
If my reporting period is 5/20/08 (cell A2), then I need to calculate how many Vacant units (B2:B13) I will have as of 5/20. The formula will need to count all of the current Vacant units (B2:B13),
plus it will also have to take into account any Move Outs (C2:C13) during the reporting period. For example, let's say we have an Occupied unit that is scheduled to move out on 5/15/08. The cell will
show this as "O" but the formula will need to determine if the M/O date is <= the Reporting Date of 5/20/08 (A2). If so, then Excel would count this unit as a Vacant for the period.
In addition to this, I also need the formula to look at upcoming Move IN dates (D2:D13). The formula will also have to determine if the MI date is <= the Reporting date (A2). If the MI date falls
prior to the Reporting Date, then this unit should be considered as Occupied and should NOT be added to the total available units. NOTE - there are quite a few cells in column D that are blank and
don't have MI dates. If there isn't a MI date, then the formula should assume that those units are Vacant if it is past the prior tenants MO date.
I'm assuming that Excel can handle multiple conditions like this, however, I am not smart enough to figure it out.
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Fee Calculator Sheet Formatting
I am designing a basic front end using formulas as the work computers dont like VBA script.
I need the calculator ( attached) to replicate the 'calculations' sheet based on the number of accounts specified in 'Investor Details'. Ideally, I would like to use a button or similar to make it
user friendly.
I also need to be able to hide rows based on a condition being met and conditional formatting doesn't seem to work (eg. 'Investor Details' 10-14 if C8 = Yes.)
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Calculator Formula For Addition Via Columns
I would like to know the calculator formula for addition via columns.
Eg 1. If i were to place 135 into Column A ;
12.95 into Column C ;
i would need to yield a result of 147.95
Eg 2. Place 189 into Column A ;
12.95 into Column C
i would need to yield result of 201.95 and so on. in the attachment is the sample file.
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Simple Dialog Box Calculator With Insert
I am interested to know how to produce a dialog box that pops up upon a button press that does a simple calculation principal plus interest calculation.. ie.
the msg box or userform when called would have the following:
Label1 (Principal Amount) ===> Textbox1
Label2 (Interest rate per year) ====> Textbox2 (formatted to two decimal places).
Label3 (Results shown) ====>Textbox3
than an "ok" button on form or "submit" button when submitted, it is entered into a defined cell..
Just as an aside or complication, is it possible to have say after Label3 , 2 dropdown box where one reads colum values and one reads row values on the worksheet, that one people can pinpoint where
to enter the final value from Label3
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Meter Reading Calculator Based On Date
As part of a project, I need to come up with a meter reading calculator. I start of with 3 base meter readings (A2, F2, K2), which would be read from the meter. 3 reads are needed in case the
consumption changes dramatically at some point. I then need Excel to work out the reading for dates in between. The formula needs to be dynamic enough so that I can change any of the grey dates to
what I want.
The estimates need to be calculated using an average daily consumption. This figure needs to be displayed in D9.
It would be preferable if macros weren't used. If its possible to explain what to do without an attachment, that would be good too.
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Friction Loss Calculator For Fire Hose
I am attempting to create a friction loss calculator for fire hose, I am using a known formula that calculates the loss based on volume and diameter for pipe. the difference that I have with fire
hose in lieu of pipe is that the hose diameter changes with the pressure drop.
I have 3 variables that i input, pressure, beginning diameter and length. however
as the water flows through the line the pressure changes in turn the diameter changes, I would like to set up my spreadsheet so that the initial variables inputed yield the correct diameter, and then
reference back to the initial equation and recalculate based on the yeilded diameter, and recalculates,
I can determine the friction loss at 1 foot, in turn determine the diameter at 2 foot, but I wish the spreadsheet to work the calculation over the entire length.
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Calculator Compute Automatically After Inputting The Date
i m making a very small calculator. It's a little hard to describe, but I need to look like the following
Let's say the individual got here on 1 Jan
Phase 1 is for days 1-14 last for 14 days
Phase 2 is for Day 15-35 last for 21 days
Phase 3 is day 36+
This is what I need it to look like: All the phase information will compute automatically after inputting the date arrived. In addition I would also like to be able to change one of the phase dates
and the remainder of the phases compensate according to the newly entered date.
Date Arrived = 01 Jan 09
Phase 1 = 01jan09
Phase 2 = 04Feb09
Phase 3 = 05 Feb09
View Replies! View Related | {"url":"http://excel.bigresource.com/Commission-Calculator-2cEMcAF3.html","timestamp":"2014-04-16T19:07:24Z","content_type":null,"content_length":"79156","record_id":"<urn:uuid:79fe59b5-b59a-4488-903c-13ff1f0ca07f>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00622-ip-10-147-4-33.ec2.internal.warc.gz"} |
Introduction to set theory and topology. Translated from the rev. Polish ed. by Leo F. Boron
Kazimierz Kuratowski
We haven't found any reviews in the usual places.
Foreword to the English edition ll 11
1 The disjunction and conjunction of propositions 21
57 other sections not shown
Bibliographic information | {"url":"http://books.google.com/books?id=RDJPAQAAIAAJ&q=finite+number&source=gbs_word_cloud_r&cad=5","timestamp":"2014-04-20T03:16:53Z","content_type":null,"content_length":"98613","record_id":"<urn:uuid:470f5032-45c1-4394-89e0-835275e61aca>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00065-ip-10-147-4-33.ec2.internal.warc.gz"} |
Workshops/Presentations that Michelle Facilitates
Cognitively Guided Instruction (K-6)
Cognitively Guided Instruction (CGI) is a problem-solving mathematics program designed to improve number sense and computation for students in Kindergarten through third grades. It has been proven
effective for boys and girls of diverse social class, racial and ethnic, and language proficiency backgrounds.
This K-6 instructional strategy focuses on student knowledge and encourages teachers to pose story problems that can be solved by any means chosen by the child. Problem-posing and problem-solving
become the focus of the mathematics class, rather than the traditional emphasis on memorization of facts and algorithms. The research-based approach was developed by faculty at the Wisconsin Center
for Education Research, University of Wisconsin- Madison.
Each participant will be required to purchase Thinking Mathematically . This workshop consists of three professional days, followed by two advanced days later in the year.
Developing and Assessing Number Sense (Pre K-3)
In this workshop PreK- 5 eachers will be exposed to the necessary building blocks necessary for a child to develop true number sense. Many times in education we attempt to solve a symptom when we
need to be addressing the core of the problem. The focus of this workshop is to provide teachers with the understanding necessary to identify the building blocks in developing number sense and to
provide teachers with appropriate lessons and diagnostic tools in which to measure students number sense.
Math For Little Folks - 2 part (K-2)
This two-day workshop will be fun, fun, fun and emphasize hands-on experience where participants learn by doing and explores the best ways to teach conceptual understanding of mathematics. The
purpose of this two-day workshop is to immerse teachers in best practices as they explore how mathematic concepts can be taught in the primary classroom.
During the workshop we will investigate how to support students as they:
•construct strategies and big ideas related to addition, multiplication, subtraction, and division
•develop efficient computation
You’ll leave the workshop with a better understanding of students’ mathematical thinking and how to support their learning as they successfully develop an understanding of numbers and operations,
algebraic thinking, geometric concepts, and data analysis.
Participants will receive an extensive, ready to use resource full of ideas to motivate all students in the classroom.
Unlocking the Number Sense Door (K-6)
This workshop will emphasize hands-on experiences where participants will learn by doing and explore the best ways to unlock the number sense door for students.
During the workshop we will investigate how to support students as they:
• construct strategies and
• develop efficient computation methods
You’ll leave the workshop with a better understanding of students’ mathematical thinking and games to support their learning as they successfully develop an understanding of numbers.
This workshop addresses what the Number standard looks like when working with students. Best practices in mathematics will be modeled and shared throughout the workshop.
Participants will receive an extensive ready-to-use resource full of ideas to motivate all learners.
The Bridge to Algebra (K-6)
Can elementary school children really do algebra? Absolutely! While the term “algebra” might be daunting, even the youngest students can readily understand this kind of math. The development of
algebraic concepts has become a central focus of the NCTM math standards. Classroom teachers in elementary school must engage students in developmentally appropriate tasks that build the necessary
foundation for understanding algebra. This workshop emphasizes hands-on experience and includes ideas of patterns and relationships, equalities and inequalities, functions, and models. Leave with
ideas you can use in your classroom to address the Algebra standard tomorrow.
This workshop addresses what the Algebra standard looks like when working with students.
Participants will receive an extensive ready-to-use resource full of ideas to motivate all learners.
Hands on Geometry (K-6)
Did you know ... "Children who develop a strong sense of spatial relationships and master concepts of geometry are better prepared to learn number and measurement ideas ...." from Curriculum and
Evaluation Standards for Mathematics.
Participants will have the opportunity to explore, to model and solve problems, search for
This workshop addresses what the Geometry standard looks like when working with students. Best practices in mathematics will be modeled and shared throughout the workshop.
Participants will receive an extensive ready-to-use resource full of ideas to motivate all learners.
Effective Math Instructional Strategies for Tier 1, Tier 2, and Tier 3 (K-6)
Effective Instructional Practices to Support Tier I, II, and III in Math. Join Michelle and explore the effectiveness of mathematics instruction and enhance the learning, motivation, and confidence
of struggling math learners in the K-6 classroom. We will discuss strategic ways to enhance Tier I and Tier 2 math instruction and interventions for struggling learners, as well as learn about the
critical attributes of Tier 3 math intervention. Several research-based instructional strategies will be experienced including Signapore Math, Cognitively Guided Instruction, Guided Math framework,
and many more.
Developing Mathematical Ideas: Numbers and Operations(K-8)
Developing Mathematical Ideas - a case study, takes us to a whole new cognitive level for mathematics professional development.
Developing Mathematical Ideas (DMI) is a professional development curriculum designed to help teachers think through the major ideas of K-7 mathematics and examine how children develop those ideas.
At the heart of the materials are sets of classroom episodes (cases) illustrating student thinking as described by their teachers. The curriculum also offers teachers opportunities: to explore
mathematics in lessons led by a facilitator; to share and discuss the work of their own students; to view and discuss the videotapes of mathematics classrooms; to write their own classroom episodes;
to analyze lessons taken from innovative elementary mathematics curricula; and to read overviews of related research.
In order to develop the depth and breadth of knowledge required for teaching, teachers need to learn in and from practice. They need to learn to elicit students’ thinking and to make sense of it in
the context of classroom work. Teachers need to adopt a stance of inquiry in which they would “frame, guide, and revise tasks and pose questions, so as to learn more about students’ ideas and
understanding.” And teachers need to learn how to use this knowledge to make shifts in lessons based on what students do and do not understand.
Each participant will be provided with a Casebook.
Thinking Mathematically - Integrating Arithmetic and Algebra in Elementary Schools (K-8)
This two day study group is designed for elementary math teachers who wants to examine how children learn mathematics. It is the bases of this study group that as educators we need to reconsider how
arithmetic is taught and learned. We will examine children's conceptions and misconceptions that students bring to learning mathematics in the elementary grades.
Learning mathematics involves learning ways of thinking. It involves learning powerful mathematical ideas rather than a collection of disconnected procedures. But it also entails learning how to
generate those ideas, how to express them using words and symbols, and how to justify to oneself and to others that those ideas are true.
We will use the book Thinking Mathematically - written by Thomas Carpenter. Author of Children's Mathematics - Cognitively Guided Instruction. This book provides a rich portrait of arithmetic set in
a broader perspective on mathematics. The book is loaded with ideas to support the mathematical work of the teacher in pressing students, provoking, and supporting.
Developing Mathematical Ideas: Focus on Geometry (K-8)
This 4-week class is classified as Results-Oriented Staff Development
Developing Mathematical Ideas (DMI) helps teachers improve the way they teach the big ideas of K-6 mathematics and examines how children approach and understand mathematics. DMI is a program of
seminars to help teachers learn more mathematics and improve their mathematics instruction. Seminar participants focus on classroom episodes, or cases, that illustrate students’ mathematical
During this study group participants will examine aspects of two- and three-dimensional shapes, develop geometric vocabulary, and explore both definitions and properties of geometric objects. The
study group includes a study of angle, similarity, congruence, and the relationships between 3-D objects and their 2-D representations.
Using Children’s Literature to Teach Math (K-6)
I believe strongly that motivation in students is a necessary component in creating a love for learning, especially in the area of mathematics. I have found that children’s books are wonderful for
sparking students imaginations in ways that textbooks and worksheets often don’t. Students who find their strengths in reading, find the mathematics less intimidating through literature books.
This day will be packed with several literature books and mathematics activities. Some may be short, one-day activities, others will be full week integrated mathematics lessons. | {"url":"http://michellef.essdack.org/?q=node/5","timestamp":"2014-04-18T05:30:30Z","content_type":null,"content_length":"22581","record_id":"<urn:uuid:318ea760-35af-4201-8df7-3d2debb3c6ec>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00458-ip-10-147-4-33.ec2.internal.warc.gz"} |
Prisoner teamwork
Author Prisoner teamwork
Sheriff Ok, I'm telling you all right up front that I don't know the answer to this one, so no name calling later on... This puzzle has been haunting me for months now, so if you need for me to
have the answer just back away now... and there will be no hard feelings...
Oct 14, Some big number of prisoners (I think I remember it being 21 but I might be off by a few, my guess is any big number will do, so for instance if you can do it with 17 you would still get
2002 tons of accolades), are given this challenge...
Posts: The warden brings ~21 prisoners into a room. The room has two, two way switches. The warden explains the following:
8764 Every once in a while I will bring one of you into this room. When you arrive, you MUST flip exactly one of these two-way switches.
You will have no contact with each other once this meeting is over.
5 I will let you meet now, and devise a plan.
When any of you are brought into the room, and you can tell for sure that you have ALL visited the room at least once, tell me.
If you are correct, I will set you all free! If you are wrong, the challenge is over, and you will all serve your full terms.
What plan could the prisoners devise to communicate with each other, via the two switch settings, so that they could tell when they had ALL visited the room?
p.s. I heard this on Car Talk's weekly puzzle segment
[ August 15, 2003: Message edited by: Bert Bates ]
[ August 15, 2003: Message edited by: Bert Bates ]
Spot false dilemmas now, ask me how!
(If you're not on the edge, you're taking up too much room.)
damn you bert... I've got work to do and now I'm trying to think about prisoners and light switches....
Joined: My first thought was to count sequentially with the lights...
Oct 17, Before the first guy goes in -- both switches are OFF (00).
2001 01 - after guy 1 leaves.
Posts: 11 - after guy 2 leaves.
4313 10 - after guy 3 leaves.
00 - after guy 4 leaves.
I SO, up to this point all these guys can tell how many have been in before them. If we continue in the same pattern:
like... 01 - after guy 5 leaves.
11 - after guy 6 leaves.
10 - after guy 7 leaves.
00 - after guy 8 leaves.
But all guy # 8 knows for sure is that 3 people came in before him... he can't be sure that guys 1-4 were there... so -- the quesiton is, how do you count to 17 or 21 or something with only
2 bits?
What if... you can balance the switch in the middle position between on and off... then you can have base 3 #s
01 - after guy 1 leaves.
02 - after guy 2 leaves.
12 - after guy 3 leaves.
11 - after guy 4 leaves.
10 - after guy 5 leaves.
20 - after guy 6 leaves.
21 - after guy 7 leaves.
22 - after guy 8 leaves.
.... but then you can't get back to 00 without having one guy flip two switches or without jumping directly into the middle of the sequence. Either way you're still in the same situation
where the guys won't know if 8 people have been in before them or 1.
OK -- after a little searchng -- here's the actual problem according to CarTalk's website (doen't make it any easier though...):
Oct 17,
Posts: A warden meets with 23 new prisoners when they arrive. He tells them, "You may meet today and plan a strategy. But after today, you will be in isolated cells and will have no
4313 communication with one another.
"In the prison is a switch room, which contains two light switches labeled A and B, each of which can be in either the 'On' or the 'Off' position. I am not telling you their present
I positions. The switches are not connected to anything.
"After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches
and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none, either. Then he'll be led back to his cell.
"No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners at random. I may choose the
same guy three times in a row, or I may jump around and come back.
"But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time anyone of you may declare to me, 'We have all visited the switch
"If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators."
Here's the question:
What is the strategy the prisoners devise?
Oct 14,
2002 Jess -
Posts: Apart from the alligators, and 23 vs. 21 prisoners, does your version have any logical differences?
Jul 16, A little googling...and voila! This is no fun...but it's been haunting you. You don't have to peek
2001 the answer
Oct 14,
Posts: I didn't peek, and I'm happy to know that there IS an answer, now I can get back to being haunted...
Oct 17, Well, the fact that a prisoner can and probably will be taken into the room more than once is key. Also this line is important too: "But, given enough time, everyone will eventually visit
2001 the switch room as many times as everyone else."
Posts: And let me just say that my attempt at an answer was totally off -- I peaked at the answer from the page I cut& paster the problem from. I would never have come up with their answer... but
4313 at least now I can get back to this dang fake requirements document for school.
[ August 15, 2003: Message edited by: Jessica Sant ]
Joined: i think the fact that there are 23 rather than 21 prisoners is important for some reason because 23 is 1 less than a multiple of 4 (24). of course, 21 is 1 greater than 20...
Apr 12,
Sheriff I confess, I was weak of will, and followed Roy's link. Cool puzzle.
Apart from the alligators, and 23 vs. 21 prisoners, does your version have any logical differences?
Joined: The version Jess found does state explicitly that the warden's prisoner selections are random, and that the initial state of the light switches in unknown. Both are potentially useful bits
Jan 30, of info to have confirmed. To ensure eventual success, I would want assurance that the room visits will continue indefintely. Otherwise the warden can just stop when he feels like it, and
2000 screw everyone. Jess' version does guarantee "given enough time, everyone will eventually visit the switch room as many times as everyone else" - but that's a bit too vague for my taste.
Posts: For simplicity, just assume there will be one visit per day, unto eternity, and each visitor will be selected at random, and no prisoners will ever die. (Or if there's any death, the warden
18671 will inform the other prisoners of it.)
"I'm not back." - Bill Harding, Twister
Oct 17, Originally posted by Greg Harris:
2001 i think the fact that there are 23 rather than 21 prisoners is important for some reason because 23 is 1 less than a multiple of 4 (24). of course, 21 is 1 greater than 20...
you'd think that 'cause 23 is prime, that might mean something too -- but nope, the solution given can work for any number of prisoners (given enough time as Jim said)
Dec 04, When the last prisoner completes his term and is about to be set free, you can declare that everyone has visited the room, since by then, everone else would've had to or else the warden's
2000 claim of everyone visiting it would not be true.
Posts: --Mark
Some thoughts on a better solution... Now I'm assuming there's no trick (e.g. they all write their name of the wall when they come in there).
Joined: You start with 0 bits of information about the state of the room, i.e. there are two unknown, random bits. Each prisoner effectively ANDs one bit with one of the two random bits. This makes
Dec 04, it hard to send information through the switch states.
2000 Also since each person must flip a bit each time, and since any person may apprear N times (N >= 1) we cannot assume each person represents a bit in some master code.
Posts: --Mark
6037 [ August 15, 2003: Message edited by: Mark Herschberg ]
subject: Prisoner teamwork | {"url":"http://www.coderanch.com/t/35147/Programming/Prisoner-teamwork","timestamp":"2014-04-17T19:04:39Z","content_type":null,"content_length":"48221","record_id":"<urn:uuid:55991537-7317-4703-99b7-32d763c99ceb>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00200-ip-10-147-4-33.ec2.internal.warc.gz"} |
Jumbling Puzzle Challenge
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Post subject: Jumbling Puzzle Challenge
Oskar Posted:
Fri Apr 13, 2012 1:42 pm
[Admin: Split from Puzzles that Jumble, this deserves its own topic]
Joined: Mon Nov 30, 2009 1:03
pm Hi fans of Twisty Puzzles that Jumble,
Yesterday, I presented "Twisty puzzles that jumble, a challenge looking for a mathematician" at the Dutch Mathematics Congress in Eindhoven. Here is my presentation:
As one would expect with such an audience, there were more questions than answers. Here are some interesting questions that were raised during and after my talk.
-What constitutes a "state" of a jumbling puzzle?
-Is the number of "states" of a jumbling puzzle always a finite, integer number?
-Is it essential to keep the core fixed when evaluating the number of states?
-If so, how would that work for deep-cut jumbling puzzles?
Prof.dr. Hans Zantema suggested that we start with the underlying group structure of a jumbling puzzle (c.f.
Rubik's Cube group
) and then consider the "basic operations". Basic operations are what takes a twisty puzzle from one state to an adjacent state (c.f. Rubik: F,B,U,D,L,R).
Food for thought ...
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Last edited by Oskar on Sun Apr 22, 2012 12:12 pm, edited 2 times in total.
Post subject: Re: Puzzles that jumble
Andreas Nortmann Posted:
Sun Apr 15, 2012 12:22 pm
Oskar wrote:
Joined: Mon Aug 02, 2004 7:03
am Prof.dr. Hans Zantema suggested that we start with the underlying group structure of a jumbling puzzle (c.f.
Location: Koblenz, Germany
Rubik's Cube group
) and then consider the "basic operations". Basic operations are what takes a twisty puzzle from one state to an adjacent state (c.f. Rubik: F,B,U,D,L,R).
Here begins the problem for all jumbling puzzle like the traditionally bandaged puzzles as well:
On a Rubiks cube the six "basic operations" can be applied on every permutation. On a jumbling or bandaged puzzle the set of possible moves depends on the current
It is indeed time that some professionials deal with these puzzles!
Post subject: Re: Jumbling Puzzle Challenge
Allagem Posted:
Mon Apr 16, 2012 8:23 am
To quickly throw out an idea and make an obligatory post to show I still exist.......
Joined: Sun
Oct 08, 2006 As far as stable-cored (i.e. not puzzles like Mixup Cube) jumbling puzzles go, there are only a finite number of types of pieces that can only be configured in a finite number of
1:47 pm shapes. Viewing a puzzle like the Rubik's Cube from the mechanism out, every state is identical. It's not until you add sticker colors that states become distinguishable. Many of the
Location: properties that are true on twistypuzzles can be proven because the states form a mathematical group. A puzzle can be the realized form of a group ONLY if every move is available in
Houston/San every state, so there are some good reasons to try to cleanly handle the blocked moves in jumbling puzzles.
Texas Based on the above observations, perhaps where we have considered non-jumbling puzzles as a set of positions and a universal set of moves, we should consider jumbling puzzles as a set
of mechanical arrangements, each having its own set of configurations and states. Take the Helicopter Cube for example. When the puzzle is in a cube shape, there are 12 available
rotation vectors (or planes, however you prefer to call them) and 24!/(4!)^6*8!*3^7/24 = approx. 1.19 * 10^22 possible sticker states (sticker arrangements). Each of the rotation
vectors can stop at 1 of 5 angles, not including its current position, to allow another move to interact with the same pieces altered by the previous move. 1 of these angles (180
degrees) returns you to another state in the same mechanical arrangement. However, the other angles lead you to 1 of 2 different mechanical states per rotation vector. From these
different mechanical states, the set of available rotations changes, both in vectors and angles. There are the same number of sticker states (sticker arrangements) although permutations
and orientations will have to be redefined somewhat. I conjecture there is about 150 possible mechanical states each with 1.19 * 10^22 sticker states.
By careful defining/tracing out how piece permutations/orientations transfer from mechanical state to mechanical state, we can even prove what parities are possible and what are not.
For example, I have scrambled and solved Oskar's Meteor Madness 6 or 7 times and have observed that every "edge" (piece with two stickers) can end up in every edge location in either
orientation and every "corner" (piece with 3 stickers) can end up in every corner location in any of 3 orientations. Swapping two edges, swapping two corners, flipping a single edge,
and rotating a single corner all appear to be impossible. I think it would be interesting to try to prove that for this puzzle. In fact, it seems to me that any piece-type for which no
move changes the orientation but not the position of any single instance of this piece type is restricted by this orientation parity, on both non-jumbling and jumbling puzzles alike.
Does anyone know of a counter-example to this? Could there be some subtle law of geometry that forces this?
Anyway that's my idea. Define the set of moves based on the puzzle's mechanical state, allowing multiple definitions for moves throughout a single puzzle. This keeps the number of
overall puzzle states (mechanical state * sticker state) finite and integral. I am a huge fan of keeping the core of a puzzle fixed, but with non-jumbling puzzles, it is mathematically
sound to fix another piece of the puzzle, or even to fix no piece at all. I BELIEVE if a piece other than the core of a stable-core puzzle (this is important!, ask Carl
Good Questions!
Matt Galla
Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Mon Apr 16, 2012 10:04 am
Nice topic. I don't have too much time at the moment but I wanted to get a few quick answers out there. Great presentation Oskar. I wish I could have been there.
Joined: Thu
Dec 02, 2004 Oskar wrote:
12:09 pm
Location: -What constitutes a "state" of a jumbling puzzle?
Starting from the solved state a new state is reached anytime after a turn in made where an additional turn is allowed which moves some of the same pieces moved in the first turn plus
some additional pieces not moved in the first turn. Not sure of a better way to word that. Looking at the normal 3x3x3 you need to exclude the case where the top layer is turned by 45
degrees. Without such a definition one could call this a new state as one is still allowed to turn the bottom layer while the puzzle is in this position. Note the plus part is needed
to cover the case where the slice and bottom layer are turned together by 45 degrees and the following argument that this is a new state because I can now turn the bottom layer by
itself again.
You can use this rule from any state of the puzzle to find new states.
Oskar wrote:
-Is the number of "states" of a jumbling puzzle always a finite, integer number?
Yes. As long as the puzzle has a finite number of pieces.
Oskar wrote:
-Is it essential to keep the core fixed when evaluating the number of states?
It doesn't need to be the core but any "holding point" should work and produce the same number of states. Note a holding point can be virtual but not imaginary. These are terms I've
defined elsewhere. If you don't hold something fixed I think you can run into a case where you might find an infinite number of states but many would be just global rotations of the
entire puzzle from other states.
Oskar wrote:
-If so, how would that work for deep-cut jumbling puzzles?
Just pick any piece that is in the puzzle and consider that your holding point. Look at this
where I define the pieces in the Complex 3x3x3 by considering a corner of the 3x3x3 as your holding point. It doesn't need to be the core and by doing this I show how it can be
generalized to define the pieces of a Complex 2x2x2 or Complex 4x4x4 both of which would be deep cut puzzles.
Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Mon Apr 16, 2012 10:19 am
Allagem wrote:
Joined: Thu Dec
02, 2004 12:09 I am a huge fan of keeping the core of a puzzle fixed, but with non-jumbling puzzles, it is mathematically sound to fix another piece of the puzzle, or even to fix no piece at all.
Location: I believe you need to fix "something" but as I said it doesn't always need to be a piece in the puzzle. It could be a "virtual piece" which is defined based on the geometry of the
Missouri puzzle itself. Think of a Skewb. It has two virtual cores. One is made physical in the actual construction of the mech which holds the puzzle together but you can imagine you are
holding on to the other core by keeping the other corners not attached to the core fixed in position and just allowing them to rotate.
Allagem wrote:
I BELIEVE if a piece other than the core of a stable-core puzzle (this is important!, ask Carl
I need to think about this. I tend to think that if the fixed piece (aka holding point) hasn't moved how did it jumble? But when I imagine a Helicopter Cube being held by a corner I
think I see what you mean. It may very well be the case that some holding points are considered "better" then others.
I do agree that two states which only differ by a global rotation of the entire puzzle shouldn't be counted as seperate states.
Post subject: Re: Jumbling Puzzle Challenge
Oskar Posted:
Sun Apr 22, 2012 12:31 pm
Hi Jumble-specialists,
Joined: Mon Nov 30, 2009 1:03 pm
Prof. Zantema suggests that we first properly define jumbling in 2D before going to 3D. Doug Engel wrote this
excellent overview of 2D twisty puzzles
(some of which beg to be implemented in the physical realm).
I copied some graphs from Doug Engels work below. To you expert opinions, which of these 2D twisty puzzles jumble and which don't?
Line periodic circle puzzles.jpg [ 72.27 KiB | Viewed 6030 times ]
Non-periodic circle puzzles.jpg [ 47.09 KiB | Viewed 6030 times ]
Hybrid circle puzzles.jpg [ 83.04 KiB | Viewed 6030 times ]
Offset circle puzzles.jpg [ 53.27 KiB | Viewed 6030 times ]
Dual offset grid circle puzzles.jpg [ 60.69 KiB | Viewed 6030 times ]
Penrose circle puzzles.jpg [ 76.19 KiB | Viewed 6030 times ]
Partially symmetric circle puzzles.jpg [ 64.3 KiB | Viewed 6030 times ]
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Post subject: Re: Jumbling Puzzle Challenge
Andreas Nortmann Posted:
Mon Apr 23, 2012 11:27 am
I tried hard to give a useful answer but this all I could find:
Joined: Mon Aug 02, 2004 7:03 am
Location: Koblenz, Germany The puzzles in the first images do not jumble if they are restricted to 180°-turns.
Why don't you use the example you and Bram used for your article in CFF?
Maybe we can construct a jumbling 2D-puzzle from scratch?
It could be puzzle with 2 circles. Beside the usual (?) turns of 120° (or any other rational number) a move must be possible after a turn of a irrational angle.
EDIT: Corrected minor language mistakes.
Last edited by Andreas Nortmann on Tue Apr 24, 2012 12:14 pm, edited 1 time in total.
Post subject: Re: Jumbling Puzzle Challenge
Coaster1235 Posted:
Mon Apr 23, 2012 1:01 pm
Here are my guesses whether the puzzles jumble or not. I guess it's pretty useless as it's based on no deeper analysis than looking at the puzzles.
Joined: Wed Dec 14, 2011 12:25 pm
Location: Finland Line-periodic:
The two puzzles that contain many small circles jumble, except if turned by 180 degrees only as Andreas said. The other two don't jumble.
I'm very sure that thing jumbles. It would at least be a mess unbandaged anyway
Hybrid circle puzzles:
The three of these look like they're combinations of six-fold and four-fold cuts, and can be unbandaged to 12-fold.
Offset circle puzzles:
Again those jumble unless there's some specific angle used.
Dual offset grid circle puzzles:
I'm rather sure that it jumbles, unless some specific offset value is used. It's two four-fold cut patterns offset.
Penrose circle puzzles:
Since the prototiles have rational angles, the circles are rotated in rational angles, but that puzzle unbandaged would be very scary.
Partially symmetric circle puzzles:
The first one is identical to one of the hybrid circle puzzles, and the other doesn't jumble.
I don't really know if this is worth anything, but that's my view of them
My pen-and-paper puzzles
Post subject: Re: Jumbling Puzzle Challenge
Oskar Posted:
Mon Apr 30, 2012 1:34 pm
Hans Zantema implemented Doug Engel's Binary Bisect 5 puzzle (
Joined: Mon Nov 30, 2009 1:03 pm
Slide Rule Duel
) in software. You can see his webpage
. You can also directly download the
executable. Hans made this implementation to study the concept of jumbling.
Enjoy and respond!
Doug Engel's Binary Bisect 5 - programmed by Hans Zantema.jpg [ 38.51 KiB | Viewed 5874 times ]
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Post subject: Re: Jumbling Puzzle Challenge
Coaster1235 Posted:
Mon Apr 30, 2012 1:59 pm
I'm a bit confused with why the Slide Rule Duel is considered jumbling, it unbandages fine. Is it because one could slide the half-circle to some random position and
Joined: Wed Dec 14, 2011 expect it to turn from there aswell?
12:25 pm
Location: Finland Attachment:
unbandagedslideruleduel.png [ 33.42 KiB | Viewed 5859 times ]
unbandagedslideruledueljumble.png [ 37.29 KiB | Viewed 5859 times ]
My pen-and-paper puzzles
Post subject: Re: Jumbling Puzzle Challenge
Jared Posted:
Mon Apr 30, 2012 10:11 pm
Coaster, THANK YOU for that picture. I've been wondering the same thing since Oskar's first article.
Joined: Mon Aug 18, 2008 10:16 pm
Location: Somewhere Else
Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Tue May 01, 2012 10:58 am
Oskar wrote:
Joined: Thu
Dec 02, 2004 Hans made this implementation to study the concept of jumbling.
12:09 pm
Location: I don't expect this to help him very much. As Coaster1235 points out, this puzzle doesn't jumble. If 45 degree turns are allowed then its a bandaged puzzle.
Oskar wrote:
Prof. Zantema suggests that we first properly define jumbling in 2D before going to 3D.
I'm not sure this is the best approach. I tend to think of the 2D puzzle as being on the surface of a very large (r=infinite) sphere. So the circles you see are just very shallow cuts
into the sphere. In that sense the 2D puzzles are a subset of the 3D puzzle and any approach to define jumbling in 3D should be applicable to 2D as well. However if you start with 2D
you are looking at infinitely shallow cuts in an infinitely large sphere and you are looking at paterns which in principle could go on forever. If you start with a helicopter cube for
example everything is finite and there are a finite number of cuts.
So let's say a puzzle is bandaged if it requires only a finite number of cuts to totally unbandage it into a doctrinaire puzzle. And we can also say a puzzle jumbles if it requires an
infinite number of cuts to totally unbandage it into a doctrinaire puzzle.
Now this is exactly where you run into an issue with 2D. Let's look at Doug Engel's Binary Bisect 5 puzzle again. This is a 2D puzzle and as presented its bandaged. You can count the
cuts Coaster1235 had to add above. However let's imagine a version of Doug Engel's Binary Bisect 5 puzzle where the top of the patern is copied infinitely many times. The puzzle now
has infinitely many pieces. I'd argue that the puzzle can still be unbandaged and its easy to see how but it now requires infinitely many cuts so does it jumble?
I think it should be easier to start with 3D puzzles (say a sphere of radius=1) cut with a finite number of 2D surfaces with depth>0. This is a case where I think cataloging all
possibilites in 2D will actually be harder then dealing with the 3D problem.
Post subject: Re: Jumbling Puzzle Challenge
Coaster1235 Posted:
Tue May 01, 2012 11:52 am
Oskar wrote:
Joined: Wed Dec
14, 2011 12:25 pm Prof. Zantema suggests that we first properly define jumbling in 2D
Location: Finland
Haven't we done this already? A 2D circle-based puzzle jumbles if a circle has to be turned an irrational amount of degrees. Wasn't that how Bram defined jumbling?
Moving to the 3D twisties, I think the correspondent definition goes: if a puzzle's face (when visualized as faceturning) is turned less than it's rotational symmetry (sorry I
don't know a better wording for that) and subsequent turns can be made, the puzzle jumbles. Obviously from this we can see that if the faces of a puzzle have no rotational
symmetry, it purely jumbles. Can someone think of a counterexample?
My pen-and-paper puzzles
Post subject: Re: Jumbling Puzzle Challenge
Oskar Posted:
Tue May 01, 2012 12:59 pm
Coaster1235 wrote:
Joined: Mon Nov 30,
2009 1:03 pm I'm a bit confused with why the Slide Rule Duel is considered jumbling, it unbandages fine.
I obviously made a mistake. Your illustration proves to me that Slide Rule Duel does not jumble.
Coaster1235 wrote:
A 2D circle-based puzzle jumbles if a circle has to be turned an irrational amount of degrees.
I disagree. The
Offset Circle puzzles
do not jumble, despite their irrational angle. I call "stored cuts".
Hans Zantema posed an interesting conjecture in an email to me today.
Hans Zantema wrote:
If all states of the puzzle have exactly the same number of neighbouring states, then the puzzle does not jumble.
Of course, this criterion is insufficient to prove that a puzzle jumbles. And his conjecture may even be incorrect. Who knows a counterexample of a twisty puzzle that
definitely jumbles, but where each state still has the same number of neighbouring states?
Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club
Post subject: Re: Jumbling Puzzle Challenge
Coaster1235 Posted:
Tue May 01, 2012 1:21 pm
Oskar wrote:
Joined: Wed Dec
14, 2011 12:25 pm Coaster1235 wrote:
Location: Finland
A 2D circle-based puzzle jumbles if a circle has to be turned an irrational amount of degrees.
I disagree. The
Offset Circle puzzles
do not jumble, despite their irrational angle. I call "stored cuts".
Looking at it from your point of view makes it not jumbling, looking at it from my point of view makes it jumbling. To be able to turn all the circles you'd have to introduce new
cuts, and because of the irrational offset angle the puzzle jumbles. Of course solving the puzzle is like it had stored cuts if any unbandaging isn't done, kind of like the
Bermuda cubes (they also jumble).
offsetcircleunbandaging.png [ 55.05 KiB | Viewed 5656 times ]
My pen-and-paper puzzles
Last edited by Coaster1235 on Tue May 01, 2012 1:49 pm, edited 1 time in total.
Post subject: Re: Jumbling Puzzle Challenge
Oskar Posted:
Tue May 01, 2012 1:34 pm
Coaster1235 wrote:
Joined: Mon Nov 30, 2009
1:03 pm To be able to turn all the circles you'd have to introduce new cuts, and because of the irrational offset angle the puzzle jumbles. Of course solving wise it acts like the
puzzle has stored cuts ...
Interesting point. I argue that two puzzles should fall in the same category, if they are identical from a solving perspective. A mathematician would probably rephrase my
argument in terms of mapping state spaces onto each other.
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Post subject: Re: Jumbling Puzzle Challenge
Coaster1235 Posted:
Tue May 01, 2012 2:14 pm
Coaster1235 wrote:
Joined: Wed Dec
14, 2011 12:25 pm Moving to the 3D twisties, I think the correspondent definition goes: if a puzzle's face (when visualized as faceturning) is turned less than it's rotational symmetry (sorry I
Location: Finland don't know a better wording for that) and subsequent turns can be made, the puzzle jumbles. Obviously from this we can see that if the faces of a puzzle have no rotational
symmetry, it purely jumbles. Can someone think of a counterexample?
I just thought of one: cuboids. By my definition, a 3x4x5 jumbles but that is not true. What do you think of that?
My pen-and-paper puzzles
Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Tue May 01, 2012 4:40 pm
Coaster1235 wrote:
Joined: Thu
Dec 02, 2004 By my definition, a 3x4x5 jumbles but that is not true.
12:09 pm
Location: What is your definition again? I think I'm missing something.
Also let's go back and look at Oskar's statement here again:
Oskar wrote:
I disagree. The
Offset Circle puzzles
do not jumble, despite their irrational angle. I call "stored cuts".
When I think of "stored cuts" I think of puzzles like the
Fuzed Cube
. Its doctrinaire as after every turn it returns to the same state if stickers aren't considered. Another example would be my doctrinaire
Deep Uniaxial 3x3x3
. It also makes heavy use of stored cuts and is a doctrinaire puzzle.
However these two 2D puzzles (the Offset Circle Puzzles) are NOT doctrinaire. So you can either say they are bandaged or that they jumble. If you try to "unbandage" these puzzles you
end up cutting things to dust in the process of trying to turn these into doctrinaire puzzles, provided we are talking about irrational offset angles. So I'm not sure how one would say
these don't jumble.
But how does that imply a 3x4x5 jumbles? You can view the 3x4x5 as a bandaged 60x60x60 and if you go to unbandage that you can get to a doctrinaire 60x60x60 with a finite number of
cuts. Not sure that is the best way to prove it but I'd say the 3x4x5 is a bandaged puzzle and not one which jumbles.
Back to the Offset Circle Puzzles... again as drawn I'd say they jumble. However, does the "offset" really add anything to these puzzles? Could removing the "offset" be viewed as a form
of unbandaging? In which case the puzzle on the left moves the bottom circle to a position in line with one of the top 2 circles and the puzzle on the right becomes a strait line. If I
do that have I really removed anything from these puzzles? In this case these puzzles ARE doctrinaire and don't jumble and DO make use of stored cuts. If we are allowed to move these
elements around then I guess I would consider the puzzles as drawn shape modifications of doctrinaire puzzles. In 3D this would be equivalent to moving an axis of rotation which
generally isn't allowed if one is expected to produce the same puzzle but here in 2D it appears that is an option in some cases.
So now after saying all that I now think I agree with Oskar. These two Offset Circle Puzzles are shape modifications which make use of stored cuts. Actually don't ALL 2D puzzles make
use of stored cuts? If you didn't have any stored cuts I think you'd have an infinite puzzle.
Post subject: Re: Jumbling Puzzle Challenge
Coaster1235 Posted:
Wed May 02, 2012 12:07 am
wwwmwww wrote:
Joined: Wed
Dec 14, 2011 What is your definition again? I think I'm missing something.
12:25 pm
Location: Read this again:
Coaster1235 wrote:
Moving to the 3D twisties, I think the correspondent definition goes: if a puzzle's face (when visualized as faceturning) is turned less than it's rotational symmetry (sorry I don't
know a better wording for that) and subsequent turns can be made, the puzzle jumbles. Obviously from this we can see that if the faces of a puzzle have no rotational symmetry, it
purely jumbles.
On the 3x4x5 the 3x5 sides (which are rectangles, twofold symmetry) can make 90 degree turns, which is not two-fold. But I think the definition holds for many puzzles. I don't know if
my definition is that useful though. What's your opinion on the Bermuda cubes? They can be viewed as a heavily bandaged cube whose faces can turn 45 degrees (and would jumble), but it
could also be viewed as if it had stored cuts, in a way.
wwwmwww wrote:
Back to the Offset Circle Puzzles... again as drawn I'd say they jumble. However, does the "offset" really add anything to these puzzles? Could removing the "offset" be viewed as a
form of unbandaging? In which case the puzzle on the left moves the bottom circle to a position in line with one of the top 2 circles and the puzzle on the right becomes a strait
line. If I do that have I really removed anything from these puzzles? In this case these puzzles ARE doctrinaire and don't jumble and DO make use of stored cuts. If we are allowed to
move these elements around then I guess I would consider the puzzles as drawn shape modifications of doctrinaire puzzles.
To allow for this the definition would need some addendum, such as "...and if the puzzle can't be readjusted into a doctrinaire puzzle without changing the solving process." Though,
that is pretty cumbersome in my opinion, so I'd still want to view it as a jumbling puzzle. Solvingwise the jumbling doesn't add anything new though.
wwwmwww wrote:
So now after saying all that I now think I agree with Oskar. These two Offset Circle Puzzles are shape modifications which make use of stored cuts. Actually don't ALL 2D puzzles make
use of stored cuts? If you didn't have any stored cuts I think you'd have an infinite puzzle.
My pen-and-paper puzzles
Post subject: Re: Jumbling Puzzle Challenge
Oskar Posted:
Wed May 02, 2012 1:51 pm
wwwmwww wrote:
Joined: Mon Nov
30, 2009 1:03 pm You can view the 3x4x5 as a bandaged 60x60x60 and if you go to unbandage that you can get to a doctrinaire 60x60x60 with a finite number of cuts. Not sure that is the best way to
prove it but I'd say the 3x4x5 is a bandaged puzzle and not one which jumbles.
I agree. We should not get confused by the shapes of those beautiful cuboids. For the unbandaging process, the only thing that matters is the puzzle's equivalent
Jaap's Sphere
wwwmwww wrote:
However these two 2D puzzles (the Offset Circle Puzzles) are NOT doctrinaire. So you can either say they are bandaged or that they jumble. If you try to "unbandage" these puzzles
you end up cutting things to dust in the process of trying to turn these into doctrinaire puzzles, provided we are talking about irrational offset angles. So I'm not sure how one
would say these don't jumble.
I wonder how the state diagram of an Offset Circle Puzzle (and hence the solving experience) relates to the state diagram of the associated "Non-Offset Circle Puzzle". I suspect
that those state diagrams trivially map onto one another, in which case I would argue that both puzzles should be classified the same, i.e. both are doctrinaire.
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Post subject: Re: Jumbling Puzzle Challenge
Coaster1235 Posted:
Wed May 02, 2012 2:36 pm
Oskar wrote:
Joined: Wed Dec 14,
2011 12:25 pm I wonder how the state diagram of an Offset Circle Puzzle (and hence the solving experience) relates to the state diagram of the associated "Non-Offset Circle Puzzle". I suspect
Location: Finland that those state diagrams trivially map onto one another, in which case I would argue that both puzzles should be classified the same, i.e. both are doctrinaire.
Since all it takes is a little bit of extra rotation to align the center circle to either of the groups connected, they do map. That means our definition gets its first addendum!
A 2D circle based puzzle jumbles if a circle has to be rotated an irrational amount of degrees to line up cuts AND the puzzle can't be rearranged into a doctrinaire puzzle
without affecting the solving experience.
I find that to be a bit cumbersome.
My pen-and-paper puzzles
Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Wed May 02, 2012 2:59 pm
Oskar wrote:
Joined: Thu Dec 02,
2004 12:09 pm I wonder how the state diagram of an Offset Circle Puzzle (and hence the solving experience) relates to the state diagram of the associated "Non-Offset Circle Puzzle". I suspect
Location: Missouri that those state diagrams trivially map onto one another, in which case I would argue that both puzzles should be classified the same, i.e. both are doctrinaire.
Before I finished my above post I had changed my stance myself. I now MOSTLY agree with you. However I would still not call the Offset Circle Puzzles doctrinaire. To see why lets
go back and look at my favorite post to refer to on this topic.
Bram was the first to define doctrinaire
The Non-Offset Circle Puzzles clearly are doctrinaire BUT these Offset Circle Puzzles I'd call shape mods also defined by Bram in the same post. Note if you "remove all the
coloration then every single position would [NOT] look exactly the same" so these aren't doctrinaire.
These fall in the same boat as the Fisher Cube which Bram also mentions. All of its states can be trivially mapped to a Rubiks Cube yet here you can tell states apart even after
the coloration is removed.
Post subject: Re: Jumbling Puzzle Challenge
Oskar Posted:
Wed May 02, 2012 11:54 pm
wwwmwww wrote:
Joined: Mon Nov
30, 2009 1:03 pm I don't like it that shape-mods would be considered non-doctrinaire, as it confuses the discussion. How about "classic doctrinaire" versus "shape-modded doctrinaire"?
Coaster1235 wrote:
A 2D circle based puzzle jumbles if a circle has to be rotated an irrational amount of degrees to line up cuts AND ...
Maybe, geometry is not the essence. We have identified several jumbling puzzles, both in 2D and 3D, that have rational angles. Based on my discussions with Prof. Zantema, I wonder
whether we can have definitions that ignore geometry and only consider the state space of the puzzle.
First of all, note that a Rubik's Cube has an infinite number of states, if you include all mid-turn states. My conjecture is that there exists an unambiguous algorithm to reduce
such an infinite state space into the finite state space that we all consider the state space of the Rubik's Cube. Let's call this algorithm call "reduction of the state space", and
let's call the result the "reduced state space". Most likely, there exist proper mathematical terms for these.
With these terms defined, we could now define
1) "A puzzle is doctrinaire if all states of its reduced state space are equivalent"
This definition would make discussions about shape-modding irrelevant.
Also the process of bandaging could be defined in these terms
2) "Puzzle A is a bandaged version of puzzle B, if the reduced state space of puzzle A is equivalent to a sub-set of the reduced state space of puzzle B"
This definition would make discussions about stored cuts irrelevant. It also removes the concept of unbandaging from the physical realm.
Subsequently, the definition of jumbling would become
3) "A puzzle jumbles if its reduced state space cannot be extended into one where all states are equivalent"
I wonder how well these definitions hold for twisty puzzles in the physical realm. Science is about
Do we know doctrinaire puzzles that do not satisfy 1)?
Do we know bandaged puzzles that do not satisfy 2)?
Do we know jumbling puzzles that do not satisfy 3)?
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Post subject: Re: Jumbling Puzzle Challenge
Bram Posted:
Thu May 03, 2012 3:26 am
Oskar wrote:
Joined: Sat Mar 22, 2003 9:11 am
Location: Marin, CA Based on my discussions with Prof. Zantema, I wonder whether we can have definitions that ignore geometry and only consider the state space of the puzzle.
I doubt it. If you're willing to allow extraordinary amounts of fudging, getting into smushing and stretching, it's amazing what can be made doctrinaire.
Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Thu May 03, 2012 11:51 am
Oskar wrote:
Joined: Thu
Dec 02, 2004 wwwmwww wrote:
12:09 pm
Location: I don't like it that shape-mods would be considered non-doctrinaire, as it confuses the discussion. How about "classic doctrinaire" versus "shape-modded doctrinaire"?
Bram was the first to propose the definition and I think it has served the community very well for nearly 3 years so I'm not really wanting to see it changed but as Bram has joined this
discussion he's more then welcome to comment on this topic. What you appear to be after is a term which encompases doctrinaire and shape mods. I'm happy to have two such groups as I
believe all here would say the Fisher Cube is a different puzzle from a Rubik's cube even though at their core they are the same basic idea. It also allows us to talk about the
doctrinaire shapes for certain puzzles. The 3x3x3 can be a cube, or a sphere, etc. For the Mixup Cube you could put 18 face centers on the puzzle and see a possible doctrinaire shape it
could take and so on. How about if we come up with a new term for the union of the sets of doctrinaire and shape mod puzzles? Or maybe we could just say the equivalent Jaap's Sphere is
doctrinaire. I'm certainly open to ideas.
Oskar wrote:
Also the process of bandaging could be defined in these terms
2) "Puzzle A is a bandaged version of puzzle B, if the reduced state space of puzzle A is equivalent to a sub-set of the reduced state space of puzzle B".
This definition would make discussions about stored cuts irrelevant. It also removes the concept of unbandaging from the physical realm.
I see problems here. The Fuzed Cube is a sub-set of the 3x3x3 yet its not considered a bandaged puzzle. It is doctrinaire. However the terminology here certainly could be improved as
one could argue that its possible to make a Fuzed Cube by bandaging a 3x3x3. So the verb bandage and the adjective bandaged don't really mean the same thing. In fact I'm sure there are
many (just yesterday I was tempted to start a thread asking how many and I many yet) doctrinaire subsets of the 3x3x3. For another example see this puzzle.
Its four copies of the same doctrinaire subset of the 3x3x3 fuzed into one puzzle. Where Andreas calls it "non-bandaged" I would have said doctrinaire.
If you are proposing that we do call these puzzles bandaged and not doctrinaire then you run into another problem. You'd have to consider the 3x3x3 itself a bandaged puzzle as its
equivalent to a sub-set of the 5x5x5. And that leads to a never ending can of worms.
Post subject: Re: Jumbling Puzzle Challenge
Andreas Nortmann Posted:
Thu May 03, 2012 12:51 pm
Oskar wrote:
Joined: Mon Aug 02, 2004
7:03 am wwwmwww wrote:
Location: Koblenz,
Germany I don't like it that shape-mods would be considered non-doctrinaire, as it confuses the discussion. How about "classic doctrinaire" versus "shape-modded doctrinaire"?
Now I can contribute at least something small:
I do not see a problem with defining shape-mods as non-doctrinaire. Shape-mods are nothing more than shape-mods. They are not solved any different (at least it math is
considered) and can be completely thrown out of the discussion.
Post subject: Re: Jumbling Puzzle Challenge
Oskar Posted:
Thu May 03, 2012 1:21 pm
wwwmwww wrote:
Joined: Mon Nov 30,
2009 1:03 pm I see problems here. The Fuzed Cube is a sub-set of the 3x3x3 yet its not considered a bandaged puzzle. It is doctrinaire. However the terminology here certainly could be
improved as one could argue that its possible to make a Fuzed Cube by bandaging a 3x3x3.
What is the problem? I see no contradiction between the following statements.
1) Fuzed Cube is doctrinaire.
2) Fuzed Cube is a bandaged version of Rubiks Cube.
So we may need to distinguish "non-doctrinaire bandaged puzzles" from "doctrinaire bandaged puzzle".
Oskar wrote:
With these terms defined, we could now define
1) "A puzzle is doctrinaire if all states of its reduced state space are equivalent".
Prof Zantema informed me that the proper mathematical term is
vertex-transitive graph
Andreas Nortmann wrote:
I do not see a problem with defining shape-mods as non-doctrinaire.
wwwmwww wrote:
Bram was the first to propose the definition and I think it has served the community very well for nearly 3 years so I'm not really wanting to see it changed
The definition of a "planet" had been stable for centuries. Recently, scientific considerations and debate resulted in a sharper definition, and
Pluto losing its status as a planet
. I call that scientific progress.
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Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Thu May 03, 2012 3:46 pm
Oskar wrote:
Joined: Thu
Dec 02, 2004 The definition of a "planet" had been stable for centuries. Recently, scientific considerations and debate resulted in a sharper definition, and
12:09 pm
Location: Pluto losing its status as a planet
. I call that scientific progress.
I'm not against progress and I'm not against making the definitions more mathematically accurate. What I was hoping to avoid was what has already happened here in these forums with
words like "order". I'm aware of at least 3 different ways its has been defined and there are groups here which hold on to each. I can follow conversations as long as I know which
definition is being used and I understand the reasons behind the different mindsets, so I won't go so far as to say any of them are wrong. I can also follow your definitions. Its just
that the groups you are defining are different from the groups Bram defined with these terms. Here are a few Vinn diagrams which I hope show where you are making changes.
Venn.png [ 15.87 KiB | Viewed 5278 times ]
Bram has set up each definition such that all puzzles (maybe I should say all twisty puzzles) fall into one and only one of these bins. Your structure is much different and using your
definition of bandaging ALL puzzles are bandaged. I don't see how that buys us anything. Sure its a possible definition and its writen more mathematically but I'm not seeing the
Some examples:
Curvy Copter
is a jumbling puzzle. But its also a bandaged
Curvy Copter Plus
The Fisher Cube is a shape mod of a 3x3x3. Its also doctrinaire using your definition. And its a bandaged puzzle because all its states make up a subset of the 5x5x5.
I do agree that making a statement like the Fuzed Cube can be made from bandaging a 3x3x3 and at the same time saying the Fuzed Cube is NOT a bandaged puzzle can be confusing. No one
has yet offered a better word or definition that I'm aware of. If you want to say that the 3x3x3 isn't a bandaged puzzle how do you get around the statement that the 5x5x5 can be
bandaged into a 3x3x3?
Post subject: Re: Jumbling Puzzle Challenge
Coaster1235 Posted:
Thu May 03, 2012 4:19 pm
The way I understood Bram's definitions was that the two main categories are doctrinaire and jumbling (since they are mutually exclusive by definition), and bandaged and shapemod
Joined: Wed Dec would be subcategories found in both categories (jumbling+bandaged I imagined to be purposefully bandaged).
14, 2011 12:25 pm
Location: Finland In my opinion I see both my interpretation of Bram's view and Oskar's view to collapse to the same thing, just grouped differently (Oskar's system sees doctrinaire puzzles to be a
special case of bandaged instead of vice versa).
The Fused Cube can be put to many categories (either a bandaged 3x3x3 or a 2x2x2 with altered cut depths), as can many others as well, for example the Offset Skewb (bandaged
Master Skewb or Skewb with altered cut depths) and Carl's Uniaxial Cube (bandaged 5x5x5 or a 3x3x3 with altered cut depths). Is that a problem? Solving wise this category is
pretty diverse.
My pen-and-paper puzzles
Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Thu May 03, 2012 5:26 pm
Coaster1235 wrote:
Joined: Thu Dec
02, 2004 12:09 pm The way I understood Bram's definitions was that the two main categories are doctrinaire and jumbling (since they are mutually exclusive by definition), and bandaged and shapemod
Location: Missouri would be subcategories found in both categories (jumbling+bandaged I imagined to be purposefully bandaged).
Let me repost Bram's definitions here.
Bram wrote:
I think I've figured out how to explain what's jumbling in a well-defined manner. For the sake of simplicity, I'm going to be skipping a discussion of puzzles with gaps.
Let's define a 'doctrinaire' puzzle as one where if you were to remove all the coloration then every single position would look exactly the same. The Rubik's Cube is a doctrinaire
puzzle, as is the Skewb and Megaminx. Also the Sphere Xyz, Chromo Ball, Puck puzzles, and a bunch of other puzzles which don't have slices like a Rubik's Cube but still have
A shape mod is a non-doctrinaire puzzle which can be shape modded to a doctrinaire puzzle. The Fisher Cube is a shape mod, as is the Mixup Cube.
A bandage puzzle is a non-doctrinaire one where by cutting the pieces into smaller parts it's possible to transform it into a doctrinaire puzzle.
A jumble puzzle is one which is non-doctrinaire but where it isn't possible to shape mod or unbandage it into a doctrinaire puzzle. Examples include the Helicopter Cube, 24-cube,
Jumbleprism, Uncanny Cube, and Battle Gears.
The 24-cube is a somewhat confusing case because if one were to make an identical-looking puzzle which was physically blocked from doing anything but the 180 degree moves then it
would be a doctrinaire puzzle, but as it is it's a jumble puzzle, and it's surprisingly difficult to figure out a way of keeping it from jumbling.
So by definition a shape mod is not a doctrinaire puzzle. Neither is a jumbling puzzle.
Another issue is the two meanings of the word bandage. The definition of "bandage puzzle" is NOT a puzzle which has been bandaged, i.e pieces have been glued together to restrict
Coaster1235 wrote:
In my opinion I see both my interpretation of Bram's view and Oskar's view to collapse to the same thing, just grouped differently (Oskar's system sees doctrinaire puzzles to be a
special case of bandaged instead of vice versa).
That is not what I'm seeing.
Coaster1235 wrote:
The Fused Cube can be put to many categories (either a bandaged 3x3x3 or a 2x2x2 with altered cut depths), as can many others as well, for example the Offset Skewb (bandaged
Master Skewb or Skewb with altered cut depths) and Carl's Uniaxial Cube (bandaged 5x5x5 or a 3x3x3 with altered cut depths). Is that a problem? Solving wise this category is
pretty diverse.
All depends on one's set of definitions. Using Bram's definitions the Fused Cube and my Uniaxial Cube are NOT bandaged puzzles. Both can be made by bandaging other puzzles but
since they are doctrinaire, they are not bandaged by definition. I agree the group "bandaged puzzle" could have a better name to avoid this confusion.
Post subject: Re: Jumbling Puzzle Challenge
Andreas Nortmann Posted:
Fri May 04, 2012 9:48 am
Oskar wrote:
Joined: Mon Aug 02,
2004 7:03 am wwwmwww wrote:
Location: Koblenz,
Germany I see problems here. The Fuzed Cube is a sub-set of the 3x3x3 yet its not considered a bandaged puzzle. It is doctrinaire. However the terminology here certainly could be
improved as one could argue that its possible to make a Fuzed Cube by bandaging a 3x3x3.
What is the problem? I see no contradiction between the following statements.
1) Fuzed Cube is doctrinaire.
2) Fuzed Cube is a bandaged version of Rubiks Cube.
So we may need to distinguish "non-doctrinaire bandaged puzzles" from "doctrinaire bandaged puzzle".
I do not have a problem with considering the Fused Cube as doctrinaire and bandaged. I tried to coin the term "restricted" for all subgroups which are still doctrinaire this
but it didn't stick so far.
BTW: The Fused cube is a popular example of a bandaged doctrinaire puzzle. Another example is the Siamese Cube.
And to further irritate you: Gear Cube and Gear Cube Extreme are doctrinaire too. Both represent subgroups of the 3x3x3.
Post subject: Re: Jumbling Puzzle Challenge
wwwmwww Posted:
Fri May 04, 2012 12:09 pm
Let me propose this idea...
Joined: Thu Dec
02, 2004 12:09 Attachment:
Location: Venn2.png [ 12 KiB | Viewed 5186 times ]
The groups Bram described are the same. The Vertex-Transitive puzzles are the union of the doctrinaire and shape mod puzzles.
In other words let's define a Vertex-Transitive puzzle as one where if you were to remove all the coloration AND shape differences between mathematically-alike-pieces then every
single position would look exactly the same.
I'm also not opposed to calling a Fused Cube a restricted 3x3x3 but for this term to be useful you need a definition that seperates the 3x3x3 as an unrestricted puzzle and I don't
know an easy way to do that. You can say the Fused Cube is restricted because its a subset of the 3x3x3 but as the 3x3x3 is a subset of the 5x5x5 how to you keep from saying all
puzzles are restricted. You could try to define "order" again and somehow use that but then also note the 3x3x3 is a subset of the Complex-3x3x3.
To me these puzzles aren't bandaged in the same way as the group of bandaged puzzles which Bram defined. Let's look at the turns which are allowed on the Fused Cube. You can say
your independant layers of rotation are F,U,R and these 3 faces can ALWAYS be turned... they are never blocked at any time. So even the term restricted sounds odd to me when talking
about these puzzles.
So since every puzzle is a subset of some greater puzzle creating a new label/classification seems pointless to me. So going back to mathematical terms maybe instead of calling the
Fused Cube a restricted or bandaged 3x3x3 we should simply call the Fused Cube a subset of the 3x3x3 and acknowledge that all sets can be subsets of some greater set and not use
restricted/bandaged as adjectives to classify these puzzles. Though I'm still curious to see if restricted can be defined in a meaningful way.
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The changing phase of quantum materials
The latest news from academia, regulators research labs and other things of interest
Posted: May 10, 2013
The changing phase of quantum materials
(Nanowerk News) Matter is categorized as either conductive, semi-conductive or resistive
to the flow of electrons based on its bulk properties. However, physicists have now
predicted a new state of matter in which the bulk of the material is insulating—resisting
electron flow—but where electrons are free to move along its edges. The possibility of
such a material, known as a ‘topological insulator’, has caused a great deal of excitement
among physicists because its surface conducting states are unusually stable, making them a
promising resource for use in quantum computers. Bohm-Jung Yang and Naoto Nagaosa from the
RIKEN Center for Emergent Matter Science and their co-workers have now devised a general
theory for how an insulator changes into a topological insulator ("Theory of Topological
Quantum Phase Transitions in 3D Noncentrosymmetric Systems"), which should aid in the
practical search for such materials.
General energy band diagrams for topological insulators showing the change from insulator
(left) to semi-metal (center) and topological insulator (right) with increasing pressure.
A full understanding of a material requires knowledge of how its properties vary as the
external environment changes. Increasing the pressure of a gas, for example, can change it
to a liquid, and higher pressures cause its atoms to bond together to form a solid
crystal. Such changes of state at temperatures near absolute zero are known as quantum
phase transitions. The theoretical model developed by Nagaosa’s team describes quantum
phase transitions involving topological insulator states.
“We want to understand how two insulating phases with distinct topological properties can
be transformed from one to the other when external perturbations are applied,” explains
Nagaosa. “Our theory shows the importance of atomic symmetry in understanding this
topological phase transition.”
A topological quantum phase transition was recently experimentally observed in bismuth
thallium sulfide selenide, a compound with an ‘inversion symmetric’ atomic arrangement—a
structure that looks the same when reflected with respect to a point. The model put
forward by Nagaosa and his colleagues goes beyond such materials by understanding phase
transitions in ‘noncentrosymmetric’ materials, which do not exhibit this simplifying
The researchers' sophisticated model predicts that three-dimensional noncentrosymmetric
materials can either change directly from a conventional insulator to a topological
insulating state, or pass through an intermediate semi-metal state (Fig. 1). Their model
provides detailed estimates for the temperature dependence of many of the properties of
this semi-metallic phase, and the conditions required for a phase transition, known as
quantum critical points. “The unique physical properties of the semi-metallic state that
we have identified will provide a useful guideline for experimental proof of a topological
phase transition in three-dimensional noncentrosymmetric systems,” says Nagaosa. | {"url":"http://www.nanowerk.com/news2/newsid=30438.php","timestamp":"2014-04-20T03:15:32Z","content_type":null,"content_length":"36982","record_id":"<urn:uuid:801a3324-25de-4bb9-9d53-3710b25000be>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00400-ip-10-147-4-33.ec2.internal.warc.gz"} |
I am trying to write a program for GCD of 2 integers. I have compiled my program with no errors or warnings using Sunblast. The problem is I am not getting the output that I need. It is asking me for
another input after my first input and it is not recognizing the 3 in the y input. I also attached a copy of the output at the bottom of this code. Can someone Please Help me out since I cannot
figure out what I am doing wrong. Thank you very much for you help and your time...
#include <stdio.h>
int GCD (int x, int y)
int b;
if((x%y) == 0)
b = y;
else if((y%x) == 0)
b = x;
GCD(y, x%y);
return b;
int get_input()
int a, i=0;
scanf("%d\n", &a);
for (i = 0; a <= 0; i++)
printf("Please enter an integer greater than 0\n");
scanf("%d\n", &a);
return a;
int main()
int x, y, z;
printf("This program computes the greatest common divisor\n");
printf("of positive integers x and y, entered by the user\n");
printf("Inputs must be integers greater than zero\n\n");
printf("Enter integer x:");
x = get_input();
printf("Enter integer y:");
y = get_input();
z = GCD(x,y);
printf("The GCD of %d and %d is %d\n", x, y, z);
return 0;
----------------------------------------------OUTPUT-------------------------------------------------------------- with example numbers input highlighted. This is exactly how it appears on SSH
This program computes the greatest common divisor
of positive integers x and y, entered by the user
Inputs must be integers greater than zero
Enter integer x:
Enter integer y:
The GCD of 6 and 4 is 4
This post has been edited by Dabucs78: 17 February 2009 - 03:11 PM | {"url":"http://www.dreamincode.net/forums/topic/87290-program-for-gcd/","timestamp":"2014-04-19T01:54:23Z","content_type":null,"content_length":"95226","record_id":"<urn:uuid:a269d5b7-080a-43f6-a336-36c981f95d25>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00225-ip-10-147-4-33.ec2.internal.warc.gz"} |
Probability Relations Between Separated Systems
Results 1 - 10 of 59
, 1992
"... The quantum formalism is a "measurement" formalism--a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising
from Bohmian mechanics, which is what emerges from Schr6dinger's equation for a system of particles when ..."
Cited by 112 (47 self)
Add to MetaCart
The quantum formalism is a "measurement" formalism--a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from
Bohmian mechanics, which is what emerges from Schr6dinger's equation for a system of particles when we merely insist that "particles " means particles. While distinctly non-Newtonian, Bohmian
mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that an
appearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by "p = IV [ 2.,, A crucial ingredient in our analysis of the origin of this randomness is
the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising
in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.
- J. Math. Phys
"... We present an elementary proof of the quantum de Finetti representation theorem, a quantum analogue of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with
the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanc ..."
Cited by 44 (7 self)
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We present an elementary proof of the quantum de Finetti representation theorem, a quantum analogue of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the
original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti
theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states
of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an “unknown
quantum state ” in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be
states of knowledge rather than states of nature. We further demonstrate that the theorem fails for real Hilbert spaces and discuss the significance of this point. I.
- Foundations of Physics , 2003
"... We show that three fundamental information-theoretic constraints—the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them,
the impossibility of broadcasting the information contained in an unknown physical state, and the impossibil ..."
Cited by 28 (3 self)
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We show that three fundamental information-theoretic constraints—the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the
impossibility of broadcasting the information contained in an unknown physical state, and the impossibility of unconditionally secure bit commitment—suffice to entail that the observables and state
space of a physical theory are quantum-mechanical. We demonstrate the converse derivation in part, and consider the implications of alternative answers to a remaining open question about nonlocality
and bit commitment. KEY WORDS: quantum theory; information-theoretic constraints. Of John Wheeler’s ‘‘Really Big Questions,’ ’ the one on which most progress has been made is It from Bit?—does
information play a significant role at the foundations of physics? It is perhaps less ambitious than some of the other Questions, such as How Come Existence?, because it does not necessarily require
a metaphysical answer. And unlike, say, Why the Quantum?, it does not require the discovery of new laws of nature: there was room for hope that it might be answered through a better understanding of
the laws as we currently know them, particularly those of quantum physics. And this is what has happened: the better understanding is the quantum theory of information and computation. 1
- SIGMA 2, Paper 66 , 2006
"... The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts
about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 1 ..."
Cited by 19 (14 self)
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The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts
about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15×15 multiplication table of the associated four-dimensional matrices exhibits a
so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. All lines in each pencil carry mutually commuting operators; in one of the
pencils, which we call the kernel, the observables on two lines share a base of maximally entangled states. The three operators on any line in each pencil represent a row or column of some Mermin’s
“magic ” square, thus revealing an inherent geometrical nature of the latter. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a
cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct
product of n copies of the Galois field GF(2), with n = 2, 3 and 4.
, 1997
"... This paper is a commentary on the foundational significance of the Clifton-Bub-Halvorson theorem characterizing quantum theory in terms of three information-theoretic constraints. I argue that:
(1) a quantum theory is best understood as a theory about the possibilities and impossibilities of informa ..."
Cited by 16 (1 self)
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This paper is a commentary on the foundational significance of the Clifton-Bub-Halvorson theorem characterizing quantum theory in terms of three information-theoretic constraints. I argue that: (1) a
quantum theory is best understood as a theory about the possibilities and impossibilities of information transfer, as opposed to a theory about the mechanics of nonclassical waves or particles, (2)
given the information-theoretic constraints, any mechanical theory of quantum phenomena that includes an account of the measuring instruments that reveal these phenomena must be empirically
equivalent to a quantum theory, and (3) assuming the information-theoretic constraints are in fact satisfied in our world, no mechanical theory of quantum phenomena that includes an account of
measurement interactions can be acceptable, and the appropriate aim of physics at the fundamental level then becomes the representation and manipulation of information.
"... A framework for the mind-matter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean
descriptions as elements for a more comprehensive non-Boolean description of a world without an apriorigiven mind-mat ..."
Cited by 6 (0 self)
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A framework for the mind-matter problem in a holistic universe which has no parts is outlined. The conceptual structure of modern quantum theory suggests to use complementary Boolean descriptions as
elements for a more comprehensive non-Boolean description of a world without an apriorigiven mind-matter distinction. Such a description in terms of a locally Boolean but globally non-Boolean
structure makes allowance for the fact that Boolean descriptions play a privileged role in science. If we accept the insight that there are no ultimate building blocks, the existence of holistic
correlations between contextually chosen parts is a natural consequence. The main problem of a genuinely non-Boolean description is to find an appropriate partition of the universe of discourse. If
we adopt the idea that all fundamental laws of physics are invariant under time translations, then we can consider a partition of the world into a tenseless and a tensed domain. In the sense of a
regulative principle, the material domain is defined as the tenseless domain with its homogeneous time. The tensed domain contains the mental domain with a tensed time characterized by a privileged
position, the Now. Since this partition refers to two complementary descriptions which are not given apriori,wehavetoexpectcorrelations between these two domains. In physics it corresponds to
Newton’s separation of universal laws of nature and contingent initial conditions. Both descriptions have a non-Boolean structure and can be encompassed into a single non-Boolean description. Tensed
and tenseless time can be synchronized by holistic correlations. 1.
, 2000
"... I show how quantum mechanics, like the theory of relativity, can be understood as a 'principle theory' in Einstein's sense, and I use this notion to explore the approach to the problem of
interpretation developed in my book ..."
Cited by 5 (0 self)
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I show how quantum mechanics, like the theory of relativity, can be understood as a 'principle theory' in Einstein's sense, and I use this notion to explore the approach to the problem of
interpretation developed in my book
- in Foundations of Physics. quant-ph/0408020
"... I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles. The notion
of quantum information is to be understood as a new physical primitive—just as, following Einstein’s spec ..."
Cited by 5 (1 self)
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I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles. The notion of
quantum information is to be understood as a new physical primitive—just as, following Einstein’s special theory of relativity, a field is no longer regarded as the physical manifestation of
vibrations in a mechanical medium, but recognized as a new physical primitive in its own right. 1
- arXiv:quant-ph/0512125. Forthcoming in Butterfield and Earman (eds.) Handbook of Philosophy of Physics , 2005
"... This Chapter deals with theoretical developments in the subject of quantum information and quantum computation, and includes an overview of classical information and some relevant quantum
mechanics. The discussion covers topics in quantum communication, quantum cryptography, and quantum computation, ..."
Cited by 4 (0 self)
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This Chapter deals with theoretical developments in the subject of quantum information and quantum computation, and includes an overview of classical information and some relevant quantum mechanics.
The discussion covers topics in quantum communication, quantum cryptography, and quantum computation, and concludes by considering whether a perspective in terms of quantum information | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=234437","timestamp":"2014-04-20T01:38:43Z","content_type":null,"content_length":"37853","record_id":"<urn:uuid:ddbc6d36-7f72-4a8f-821a-38c52a1bc635>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00580-ip-10-147-4-33.ec2.internal.warc.gz"} |
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How to use node similarity to measure subgraph similarity
up vote 0 down vote favorite
For a semantic annotation task I am trying to calculate the semantic similarity between two sets of annotations: S1 and S2. Both sets consist out of multiple nodes from within one graph (in my case
an ontology). The sets do not necessarily contain the same amount of nodes.
I measure the similarity between two nodes using a path-based similarity metric (Leacock & Chodorow's). By measuring the similarity between each node in both subgraphs, I can generate a semantic
similarity graph, see example here: http://graus.nu/thesis/measure-and-visualize-semantic-similarity-between-subgraphs/ (in this case, the red and blue nodes are nodes from S1 and S2, the edges
represent similarity between nodes).
But now I want to know how to calculate the average, mean and standard deviation of the similarity between the two subgraphs as a whole. How would I go about that? I thought about taking the shortest
path for each node in S1 to each node in S2, and using these to get my avg/mean/std deviation, however different sized sets means I miss information in some cases (for example in the case where S1
has 3 nodes and S2 has 8 nodes; I'd get 3 paths). Couldn't I just use all possible edges between all nodes from S1 to all nodes from S2 to get these numbers?
Researching so far has brought me distance matrices, graph bijection, but I'm a bit lost here, any help in pointing me in the right direction will be greatly appreciated!
graph-theory similarity
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1 Answer
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I'm approaching it now by taking each shortest path from S1 to a node in S2, and then the other way around (shortest path from node in S2 to node in S1), omitting duplicate paths. I
up vote 1 imagine that as long as I'm consistent with this approach over different annotation sets, it can suffice to use as measure, right?
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Finding parameters in a ODE system
June 24th 2009, 07:52 AM #1
Jun 2009
Finding parameters in a ODE system
I have a problem: I have the Lotka-Volterra system
x'= x(A-By)
y'= y(Cx -E)
and sperimental data (I used some values of the parameters and I solved the system with procedure like "dsolve" of maple or similar).
I would like to find again my parameters starting only to the datas.
I find some book on this problem (Comincioli, Bard) but they say that I should find the explicit function:
x(t)=.... (es. x(t)=Acos(5t)+Be^t -Csin(3t) + D)
and then use the Least Mean Square to find parameters:
Err= 1/2 (explicit x & y depending of A,B,C,E - datas)^2
For this type of ODE, however, it is impossible.
I tried to use Taylor's serie, but it doesn't depend to the parameters.
Now I try to use the energy H (that is a constant in this particular case), but I am not sure that is a good idea: I should use a 5° parameter T:
Err= 1/2 (H(A,B,C,E) - T)^2 and... I don't know T!!!
Someone know this problem and can say me how find information (books, articles, web page...)?
Thank you very much,
I have a problem: I have the Lotka-Volterra system
x'= x(A-By)
y'= y(Cx -E)
and sperimental data (I used some values of the parameters and I solved the system with procedure like "dsolve" of maple or similar).
I would like to find again my parameters starting only to the datas.
I find some book on this problem (Comincioli, Bard) but they say that I should find the explicit function:
x(t)=.... (es. x(t)=Acos(5t)+Be^t -Csin(3t) + D)
and then use the Least Mean Square to find parameters:
Err= 1/2 (explicit x & y depending of A,B,C,E - datas)^2
For this type of ODE, however, it is impossible.
I tried to use Taylor's serie, but it doesn't depend to the parameters.
Now I try to use the energy H (that is a constant in this particular case), but I am not sure that is a good idea: I should use a 5° parameter T:
Err= 1/2 (H(A,B,C,E) - T)^2 and... I don't know T!!!
Someone know this problem and can say me how find information (books, articles, web page...)?
Thank you very much,
Are your experiment data points points only $(x,y)$ or $(x,y)$ for specific $t$ values?
Realize that you can integrate if you divide the two ODEs, i.e.
$<br /> \frac{dy}{dx} = \frac{y(cx - e)}{x(a-by)}<br />$
which separates and integrates to
$<br /> c x - e \ln x + b y - a \ln y - k = 0<br />$
With only $(x,y)$ data points, you could preform a linear regression on this linear equation for the constants $a, b, c, e$ and $k$.
My datas depend to time (how the 2 populations x and y growth in time t) and your formula is exactly the energy (my "T" is your "k").
My problem is: ok, I know that there is k... but I don't know its value! So if I minimize the error... it minimize also k! But k strictly depends on parameters... I don't want the best value of
k, but the best values of a,b,c,e. However, I don't know the true value of k (ok, in this case yes because I created the datas with some specific parameters, but in an hypotetic case the datas
are "real" (and probably with errors) and I don't know at all parameters).
This is the link for Wikipedia on the Lotka-Volterra equations
Lotka?Volterra equation - Wikipedia, the free encyclopedia
Thank you for your patience!
My datas depend to time (how the 2 populations x and y growth in time t) and your formula is exactly the energy (my "T" is your "k").
My problem is: ok, I know that there is k... but I don't know its value! So if I minimize the error... it minimize also k! But k strictly depends on parameters... I don't want the best value of
k, but the best values of a,b,c,e. However, I don't know the true value of k (ok, in this case yes because I created the datas with some specific parameters, but in an hypotetic case the datas
are "real" (and probably with errors) and I don't know at all parameters).
This is the link for Wikipedia on the Lotka-Volterra equations
Lotka?Volterra equation - Wikipedia, the free encyclopedia
Thank you for your patience!
Maybe you could first approximate k. For this approximation, you might try picking 5 points and numerical solve for a - k. Repeat with a different set of data points then take the average. Then
do a regression on the integrated equation for a, b, c, and e. Just an idea.
June 24th 2009, 02:05 PM #2
June 24th 2009, 02:14 PM #3
June 25th 2009, 12:08 AM #4
Jun 2009
June 25th 2009, 04:02 AM #5 | {"url":"http://mathhelpforum.com/differential-equations/93633-finding-parameters-ode-system.html","timestamp":"2014-04-20T07:23:33Z","content_type":null,"content_length":"48856","record_id":"<urn:uuid:4779120c-7300-4fbb-8dc1-267af888e770>","cc-path":"CC-MAIN-2014-15/segments/1397609538022.19/warc/CC-MAIN-20140416005218-00244-ip-10-147-4-33.ec2.internal.warc.gz"} |
Search Results for
Search Results for solution*
1. Finkel biography
□ One is his Mathematical Solution Book, and the other achievement, by far the most significant, is his founding of The American Mathematical Monthly.
□ Let us look first at his own description of how he came to publish the Mathematical Solution Book [Amer.
□ While teaching in a country school in Union County, Ohio, in 1887 I began the writing of my 'Mathematical Solution Book', designed to aid in improving the teaching of elementary mathematics
in the rural schools, high schools and academies, and got it ready for publication the following year.
□ The Mathematical Solution Book has a title which takes up a large part of the page, namely A mathematical solution book containing systematic solutions of many of the most difficult problems.
□ Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical
Journals of the United States, and Many Original Problems and Solutions with Notes and Explanations by B F Finkel.
□ The Preface to Finkel's Solution Book is at THIS LINK .
□ The author is the well-known Professor of Mathematics and Physics at Drury College, and his aim is to provide systematic solutions of difficult questions in the earlier subjects.
□ He is very averse from "Short Cuts" and "Lightning Methods"; and insists that solutions should be written out step by step in logical order and the chain of reasoning made complete in every
□ At the same time, at my leisure, I was contributing problems and solutions to [various journals].
□ Most of our existing Journals deal almost exclusively with subjects beyond the reach of the average student or teacher of Mathematics or at least with subjects with which they are not
familiar, and little, if any space, is devoted to the solution of problems.
□ Most of our existing Journals deal almost exclusively with subjects beyond the reach of the average student or teacher of Mathematics or at least with subjects with which they are not
familiar, and little, if any space, is devoted to the solution of problems.
□ B F Finkel's Mathematical Solution Book .
2. Kellogg Bruce biography
□ A major theme in Bruce's opus is the numerical solution of singular perturbation problems.
□ Another major theme of his research was the behaviour of solutions to partial differential equations near corners and interfaces.
□ An alternating direction iteration method is formulated, and convergence is proved, for the solution of certain systems of nonlinear equations.
□ The results also prove that the transport equation itself has a unique solution for the boundary conditions considered.
□ The author determines the behaviour of the solutions of second order elliptic differential equations in two independent variables at points where two interface curves cross, where an
interface curve meets the boundary, or where an interface or boundary has a discontinuous tangent.
□ Our object is to represent the solution to the problem as the sum of a finite number of special solutions, which may be considered as known, plus a remainder term whose regularity depends on
the regularity of the data of the problem.
□ Kellogg's own summary to Discontinuous solutions of the linearized, steady state, compressible, viscous, Navier-Stokes equations (1988) is as follows:- .
□ Jump conditions across a possible curve of discontinuities of a solution of the linearised system are derived.
□ In a particular case, a discontinuous solution of the linearised system is constructed.
□ The singular perturbation causes boundary layers and interior layers in the solution, and the corners of the polygon cause corner singularities in the solution.
□ The paper considers pointwise bounds for derivatives of the solution that show the influence of these layers and corner singularities.
□ Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present.
□ Sharpened pointwise bounds on the solution and its derivatives are derived.
□ We consider a one-dimensional convection-diffusion boundary value problem, whose solution contains a boundary layer at the outflow boundary, and construct a finite element method for its
□ It is shown that, measured in an e-weighted energy norm, the Galerkin finite element solution attains the same order of accuracy as the bilinear nodal interpolant.
3. Kerr Roy biography
□ In 1963, Roy Kerr, a New Zealander, found a set of solutions of the equations of general relativity that described rotating black holes.
□ If the rotation is zero, the black hole is perfectly round and the solution is identical to the Schwarzschild solution.
□ it was conjectured that any rotating body that collapsed to form a black hole would eventually settle down to a stationary state described by the Kerr solution.
□ Finally, in 1973, David Robinson at Kings College, London, used Carter's and my results to show that the conjecture had been correct: such a black hole had indeed to be the Kerr solution.
□ In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein's equations of general relativity,
discovered by the New Zealand mathematician, Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the universe.
□ I was trying to look at the whole structure - the Bianchi identities, the Einstein equations and these Tetrads - to see how they fitted together and it all seemed to be pretty nice and it
looked like lots of solutions were going to come out.
□ Teddy Newman and Roger Penrose were working on a similar set of methods, but Teddy had come out with this as-yet unpublished theorem that basically 'proved' that my solution couldn't exist!
Luckily, my neighbour, who was playing around with relativity, too, got hold of a preprint and I just scanned through it (I'm a lazy reader) and hit the crucial part which proved to me that
my solution could exist! After that, I kept working like mad and found the solution in a few weeks.
□ Kip Thorne, in [Black Holes and Time Warps: Einstein\'s Outrageous Legacy (W W Norton & Co, 1995).',3)">3], recalls when Kerr announced his solution in a 10-minute presentation at the Texas
Symposium on Relativistic Astrophysics in 1963:- .
□ He, Papapetrou, had been trying for 30 years to find such a solution to Einstein's equation and had failed, as had other relativists.
□ In 1965, in collaboration with Alfred Schild who was a colleague at the University of Texas, Kerr published Some algebraically degenerate solutions of Einstein's gravitational field equations
which introduced what are today known as Kerr-Schild spacetimes and the Kerr-Schild metric.
□ In the early 1960s Professor Kerr discovered a specific solution to Einstein's field equations which describes a structure now termed a Kerr black hole.
□ Not only was the solution especially complex, lacking symmetry of previous solutions, but it became apparent that any stationary black hole can be described by Kerr's solution.
4. Bernstein Sergi biography
□ Bernstein returned to Paris and submitted his doctoral dissertation Sur la nature analytique des solutions des equations aux derivees partielles du second ordre to the Sorbonne in the spring
of 1904.
□ This problem, posed by Hilbert at the International Congress of Mathematicians in Paris in 1900, was on analytic solutions of elliptic differential equations and asked for a proof that all
solutions of regular analytical variational problems are analytic.
□ In 1906 he passed his Master's examination at St Petersburg but only with difficulty since Aleksandr Nikolayevich Korkin, who examined him on differential equations, expected him to use
classical methods of solution (some sources say that Bernstein only passed the examination at the second attempt).
□ He moved to Kharkov in 1908 where he submitted a thesis Investigation and Solution of Elliptic Partial Differential Equations of Second Degree for yet another Master's degree.
□ As well as describing his approach to solving Hilbert's 19th Problem, it also solved Hilbert's 20th Problem on the analytic solution of Dirichlet's problem for a wide class of non-linear
elliptic equations.
□ Charles-Jean de La Vallee Poussin had asked in 1908: is it possible to approximate the ordinate of a polygonal line by means of a polynomial of degree n with error less than 1/n? Both de La
Vallee Poussin and Bernstein made some progress in the following years and then the Belgium Academy of Science offered a prize for a solution.
□ Bernstein gave a complete solution in 1911, introducing what are now called the Bernstein polynomials and giving a constructive proof of Weierstrass's theorem (1885) that a continuous
function on a finite subinterval of the real line can be uniformly approximated as closely as we wish by a polynomial.
□ Mathematicians for a long time have confined themselves to the finite or algebraic integration of differential equations, but after the solution of many interesting problems the equations
that can be solved by these methods have to all intents and purposes been exhausted, and one must either give up all further progress or abandon the formal point of view and start on a new
analytic path.
□ As constructive function theory we want to call the direction of function theory which follows the aim to give the simplest and most pleasant basis for the quantitative investigation and
calculation both of empirical and of all other functions occurring as solutions of naturally posed problems of mathematical analysis (for instance, as solutions of differential or functional
□ He proved a special case of his own problem in Solution of a mathematical problem related to the theory of inheritance (1924).
5. Khayyam biography
□ An approximate numerical solution was then found by interpolation in trigonometric tables.
□ Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a
result which would not be proved for another 750 years.
□ Khayyam also wrote that he hoped to give a full description of the solution of cubic equations in a later work [Scripta Math.
□ Indeed Khayyam did produce such a work, the Treatise on Demonstration of Problems of Algebra which contained a complete classification of cubic equations with geometric solutions found by
means of intersecting conic sections.
□ In the science of algebra one encounters problems dependent on certain types of extremely difficult preliminary theorems, whose solution was unsuccessful for most of those who attempted it.
□ As for the Ancients, no work from them dealing with the subject has come down to us; perhaps after having looked for solutions and having examined them, they were unable to fathom their
difficulties; or perhaps their investigations did not require such an examination; or finally, their works on this subject, if they existed, have not been translated into our language.
□ Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution.
□ He demonstrated the existence of equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions.
□ He did hope that "arithmetic solutions" might be found one day when he wrote (see for example [Dictionary of Scientific Biography (New York 1970-1990).
□ University of Georgia (Geometric solution of the cubic) .
6. Mazya biography
□ As Maz'ya did not make this a secret, his fellow students all decided not to submit their solutions.
□ Solution of Dirichlet's problem for an equation of elliptic type (Russian) was published in 1959 and Classes of domains and imbedding theorems for function spaces (Russian) in 1960.
□ He published the two papers Some estimates of solutions of second-order elliptic equations (Russian) and p-conductivity and theorems on imbedding certain functional spaces into a C-space
(Russian) in 1961, and then four further papers in 1962, the year in which he was awarded his Candidate degree (equivalent to a doctorate) from Moscow State University.
□ The book is in two parts, the first is on the higher-dimensional potential theory and the solution of the boundary problems for regions with irregular boundaries while the second part is on
the space of functions whose derivatives are measures.
□ In collaboration with S A Nazarov and B A Plamenevskii, Maz'ya published Asymptotic behavior of solutions of elliptic boundary value problems under singular perturbations of the domain in
□ The book deals with the construction of asymptotic expansions of solutions of elliptic boundary value problems under singular perturbations of the domains (i.e.
□ blunted angles, cones or edges, small holes, narrow slits, etc.) A general approach is suggested, its main feature being systematic application of solutions to the so-called 'limit' problems.
□ Singularities of these solutions do not increase from one step to another.
□ The main term of an asymptotic expansion for a solution of the Dirichlet problem for the Laplacian in a three-dimensional domain with a narrow slit is obtained in the third chapter.
□ The fourth chapter deals with asymptotic expansions of solutions to a quasilinear equation of the second order.
□ For example we list a few recent works without detailing the co-authors: Spectral problems associated with corner singularities of solutions to elliptic equations (2000); Asymptotic theory of
elliptic boundary value problems in singularly perturbed domains (2000); Spectral problems associated with corner singularities of solutions to elliptic equations (2001); and Linear water
waves (2002).
7. Bhaskara II biography
□ To give some examples before we examine his work in a little more detail we note that he knew that x2 = 9 had two solutions.
□ When p = 61 he found the solutions x = 226153980, y = 1776319049.
□ When p = 67 he found the solutions x = 5967, y = 48842.
□ Bhaskaracharya is finding integer solution to 195x = 221y + 65.
□ He obtains the solutions (x, y) = (6, 5) or (23, 20) or (40, 35) and so on.
□ Joy and happiness is indeed ever increasing in this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction of fractions, multiplication
and involution, pure and perfect as are the solutions, and tasteful as is the speech which is exemplified.
□ Equations leading to more than one solution are given by Bhaskaracharya:- .
□ The problem leads to a quadratic equation and Bhaskaracharya says that the two solutions, namely 16 and 48, are equally admissible.
□ The problem is to find integer solutions to an equation of the form ax + by + cz = d.
□ Of course such problems do not have a unique solution as Bhaskaracharya is fully aware.
□ He finds one solution, which is the minimum, namely horses 85, camels 76, mules 31 and oxen 4.
8. Diophantus biography
□ Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers.
□ The Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations.
□ The work considers the solution of many problems concerning linear and quadratic equations, but considers only positive rational solutions to these problems.
□ Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless.
□ In other words how could a problem lead to the solution -4 books? There is no evidence to suggest that Diophantus realised that a quadratic equation could have two solutions.
□ However, the fact that he was always satisfied with a rational solution and did not require a whole number is more sophisticated than we might realise today.
□ For example to find a square between 5/4 and 2 he multiplies both by 64, spots the square 100 between 80 and 128, so obtaining the solution 25/16 to the original problem.
□ In Book V he solves problems such as writing 13 as the sum of two square each greater than 6 (and he gives the solution 66049/10201 and 66564/10201).
9. Ferro biography
□ Today we write the solutions to ax2 + bx + c = 0 as .
□ In del Ferro's time, although such solutions were known, they were not known in this form.
□ There has been much conjecture as to whether del Ferro came to work on the solution to cubic equations as a result of a visit which Pacioli made to Bologna.
□ It is not known whether the two discussed the algebraic solution of cubic equations, but certainly Pacioli had included this topic in his famous treatise the Summa which he had published
seven years earlier.
□ The subsequent developments in the story of the solution of the cubic, namely the contest in 1535 between Antonio Maria Fior (a student of del Ferro) and Tartaglia, then the involvement of
Cardan, are told in detail in our biographies of Tartaglia and of Cardan.
□ As far as this biography of del Ferro is concerned we should stress that it was Cardan's discovery that del Ferro had been the first to solve the cubic and not Tartaglia which made him feel
that he could honour his oath to Tartaglia not to divulge his method and still publish the solution in Ars Magna for there Cardan considered he is giving del Ferro's method, not that of
□ Four years ago when Cardano was going to Florence and I accompanied him, we saw at Bologna Hannibal della Nave, a clever and humane man who showed us a little book in the hand of Scipione del
Ferro, his father-in-law, written a long time ago, in which that discovery [solution of cubic equations] was elegantly and learnedly presented.
□ Scipione Ferro of Bologna, almost thirty years ago, discovered the solution of the cube and things equal to a number [which in today's notation is the case x3+ mx = n], a really beautiful and
admirable accomplishment.
□ The story that Fior was the only person to whom del Ferro divulged his solution is common in most histories of mathematics, yet it is false.
□ As we have seen above the solution was written down by del Ferro and certainly was known to Nave.
□ Pompeo Bolognetti, who lectured at the University of Bologna on mathematics from 1554 to 1568, also had access to the original solution by del Ferro as well as the solution as given by Cardan
in Ars Magna which had been published by then.
□ Dal Ferro's rule for the solution of cubic equations.
□ The manuscript gives a method of solution which is applied to the equation 3x3 + 18x = 60.
10. Ljunggren biography
□ The journal presented a collection of problems and each year the Crown Prince Olav Prize was given to the pupil who gave the best solutions to these problems.
□ He proved that there is only a finite number of solutions and that it is possible to determine an upper limit for this number; in the special case D = 2, D1 = 3 he showed that the only
solution is x = 3, y = 2, z = 1.
□ where the left-hand side has no squared factor in x, has only a finite number of solutions.
□ However Mordell did not find the solution, nor was he able to find bounds on the finite number of solutions.
□ In the paper Ljunggren found bounds for the number of integer solutions for some special equations of this type.
□ In the first of these he proves that the equation in question has at most two positive integer solutions and gives an example of D = 1785 which does indeed have two solutions, namely x = 13,
y = 4 and x = 239, y = 1352.
□ In the second of the two papers he proves that, under certain conditions on D, there are again at most two positive integer solutions.
□ where D + 1 is not a square has at most two solutions if n ≠ 2, and if there are two solutions, these will be determined by the fundamental unit of the domain Z[√D] and its second or fourth
□ [Ljunggren] was also a very gifted problem solver, contributing many original solutions to problems posed by his peers.
11. Cherry biography
□ His first papers On the form of the solution of the equations of dynamics, On Poincare's theorem of 'the non-existence of uniform integrals of dynamical equations', and Note on the employment
of angular variables in celestial mechanics were all published in 1924 and Some examples of trajectories defined by differential equations of a generalised dynamical type in the following
□ In 1937 he published Topological Properties of the Solutions of Ordinary Differential Equations and in 1947 he published the first part of Flow of a compressible fluid about a cylinder.
□ The author states that he has solved the problem of finding the exact solution of a two-dimensional uniform flow of a compressible perfect fluid about a cylinder.
□ The solution contains an infinite number of parameters which theoretically can be fixed to determine the shape of the cylinder ..
□ Since the author uses the solutions of Chaplygin, in the form of an infinite series of hypergeometric functions, of the linear second order partial differential equation in the hodograph
variables of the potential function, this series diverges for values of the velocity whose speeds exceed the speed at infinity.
□ The essential part of the paper is to overcome this difficulty by successfully continuing "analytically" the solutions into the region of higher speeds.
□ Also in 1949 he published Numerical solutions for transonic flow which:- .
□ presents the flow patterns past a cylinder, produced by superposition of a cosine-term solution and a sine-term solution to that generated from an incompressible flow past a cylinder without
12. Choquet-Bruhat biography
□ Of very great significance is that Yvonne Bruhat's analysis enabled her to prove rigorously for the first time local-in-time existence and uniqueness of solutions of the Einstein equations.
□ Her research covers a very wide range of knowledge from the first mathematical proof for the existence of solutions of Einstein's relativistic theory of gravitation to the study of the
conversion of electromagnetic waves into gravitational waves (or the reverse) in the vicinity of a black hole.
□ Ondes Asymptotiques et Approchees pour des Systemes d'Equations aux Derivees Partielles non Lineaires, published in 1969, gives a method for constructing asymptotic and approximate wave
solutions about a given solution for nonlinear systems of partial differential equations.
□ Global Solutions of the Problem of Constraints on a Closed Manifold, published in 1973, shows that the existence of global solutions of the constraint equations of general relativity on a
closed manifold depend on subtle properties of the manifold.
□ In 1963 she published Recueil de problemes de mathematiques a l'usage des physiciens which was translated into English as Problems and solutions in mathematical physics (1967).
□ It consists of problems and their solutions required for certificate examinations in the mathematical methods of physics.
□ for their separate as well as joint work in proving the existence and uniqueness of solutions to Einstein's gravitational field equations so as to improve numerical solution procedures with
relevance to realistic physical solutions.
13. Lewy biography
□ In this paper criteria are given for determining conditions which guarantee the stability of numerical solutions of certain classes of differential equations.
□ On the basis of this, and using the daring idea of converting an elliptic equation into a hyperbolic one by penetrating into the complex domain, he developed a new proof of the analyticity of
solutions of analytic elliptic equations in two independent variables, one which far exceeded the known proof in its elegance and simplicity.
□ Nirenberg [D Kinderlehrer (ed.), Hans Lewy Selecta (Boston, MA, 2002).',6)">6] lists Lewy's mathematical papers under the following topics: (i) partial differential equations involving
existence and regularity theory for elliptic and hyperbolic equations, geometric applications, approximation of solutions; (ii) existence and regularity of variational problems, free boundary
problems, theory of minimal surfaces; (iii) partial differential equations connected with several complex variables; (iv) partial differential equations connected with water waves and fluid
dynamics; (v) offbeat properties of solutions of partial differential equations.
□ Among the first papers he published after emigrating to the United States were A priori limitations for solutions of Monge-Ampere equations (two papers, the first in 1935, the second two
years later), and On differential geometry in the large : Minkowski's problem (1938).
□ The dock problem, written jointly with Friedrichs two years later, gives an explicit solution for the dock problem over a fluid of infinite depth.
□ The solution is given by the sum of two integrals of Laplace type taken over a complex path of integration.
□ His paper An example of a smooth linear partial differential equation without solution (1957) gave a simple partial differential equation which has no solution, a result which had a
substantial impact on the area.
14. Bergman biography
□ This led him further to a general theory of integral operators that map arbitrary analytic functions into solutions of various partial differential equations.
□ It has been known for a century that the problem of finding the two-dimensional potential flow of an incompressible fluid can be solved by means of complex variables: To each analytic
function of a complex variable corresponds a particular solution of the potential problem and vice versa.
□ Bergman gave explicit formulae which allow a solution of a given differential equation to derive from an arbitrarily chosen analytic function (in some instances from a pair of real functions)
and proved that all solutions can be derived in this way.
□ They consider a special type of differential equation, yet more general than the potential equation, and build up a system of solutions in close analogy to the procedure followed in the
theory of analytic functions.
□ Though all solutions obtained by Bers and Gelbart can be derived by Bergman's methods also, it must be expected that the new approach will prove very useful.
□ The second part lays more stress on rigor, and treats fundamental solutions, reduction of boundary value problems to integral equations, orthonormal systems and kernel functions, eigenvalue
problems associated with the kernels, variational theory of domain functions, comparison domains, basic existence theorems, and dependence of solutions on the boundary data or on the
coefficients of the differential equation.
□ This treatise gives a summary of the author's numerous contributions from 1926 to 1961 to the theory of solutions of linear partial differential equations in two and three real variables by
means of integral operators which usually involve analytic functions of one, or sometimes two, complex variables.
□ Results in the theory of one complex variable on such topics as analytic continuation, the residue theorem, Hadamard's theorems on the connection between the coefficients of the power series
development of an analytic function and the character and location of the singularities and on Abelian integrals are used to give information concerning domains of regularity, series
expansion, singularities and integral relations for the solutions.
15. Baker Alan biography
□ where m is an integer and f is an irreducible homogeneous binary form of degree at least three, with integer coefficients, have at most finitely many solutions in integers.
□ Turan goes on to say that Carl Siegel and Klaus Roth generalised the classes of Diophantine equations for which these conclusions would hold and even bounded the number of solutions.
□ Baker however went further and produced results which, at least in principle, could lead to a complete solution of this type of problem.
□ for any solution (x0, y0) of f (x, y) = m.
□ Of course this means that only a finite number of possibilities need to be considered so, at least in principle, one could determine the complete list of solutions by checking each of the
finite number of possible solutions.
□ Hilbert himself remarked that he expected this problem to be harder than the solution of the Riemann conjecture.
□ Secondly, it shows that a direct solution of a deep problem develops itself quite naturally into a healthy theory and gets into early and fruitful contact with significant problems of
16. Lions biography
□ 70 (2) (1996), 125-135.',3)">3] is his work on "viscosity solutions" for nonlinear partial differential equations.
□ such nonlinear partial differential equation simply do not have smooth or even C1 solutions existing after short times.
□ The only option is therefore to search for some kind of "weak" solution.
□ Lions and Crandall at last broke open the problem by focusing attention on viscosity solutions, which are defined in terms of certain inequalities holding wherever the graph of the solution
is touched on one side or the other by a smooth test function.
□ In 1989 Lions, in joint work with DiPerma, was the first to give a rigorous solution with arbitrary initial data.
□ In spite of what Lions himself liked to call the 'truly diabolical' complexity of the set of partial differential equations, boundary conditions, transmission conditions, nonlinearities,
physical hypotheses, etc., that appeared in those models, Lions, in collaboration with Roger Temam and Shou Hong Wang, was able to study the questions of the existence and uniqueness of
solutions, to establish the existence of attractors, and to do a numerical analysis of these models.
□ And, amazingly enough, he was the first person to establish (in 2000) a result of existence and uniqueness of the solution of this type of problem.
17. Arino biography
□ Solutions periodiques d'equations differentielles a argument retarde.
□ Oscillations autour d'un point stationnaire, conditions suffisantes de non-existence (1980); "Following a note by P Seguier the authors give some results on the non-existence of a nontrivial
periodic solution to differential equations with delay, using mainly properties of monotonicity.
□ We also give sufficient conditions for a solution to stay in a weakly closed set." .
□ Solutions oscillantes d'equations differentielles autonomes a retard (1978); "We show some results proving the existence, and specifying the behaviour, of solutions oscillating near a
stationary point for some equations of the type x '(t) = L(xt) + N(xt) which have certain monotone and continuity properties.
□ Comportement des solutions d'equations differentielles a retard dans un espace ordonne (1980); "Using vectorial Ljapunov functionals, we give here some results related to the behaviour at
infinity of solutions of a differential equation with delay in an ordered Banach space." .
□ Arino studied for a doctorate supervised by Maurice Gaultier and was awarded the degree in 1980 from the University of Bordeaux for his thesis Contributions a l'etude des comportements des
solutions d'equations differentielles a retard par des methodes de monotonie et bifurcation.
□ Some of the problems dealt with from a mathematical point of view involved obtaining asymptotic properties of the solutions, in the framework of semigroup theory of positive operators as well
as the application of aggregation of variables methods to models formulated with two time scales.
18. Al-Haytham biography
□ Huygens found a good solution which Vincenzo Riccati and then Saladini simplified and improved.
□ Ibn al-Haytham wrote a treatise Solution of doubts in which he gives his answers to these questions.
□ In Opuscula ibn al-Haytham considers the solution of a system of congruences.
□ Ibn al-Haytham gives two methods of solution:- .
□ The problem is indeterminate, that is it admits of many solutions.
□ Here ibn al-Haytham gives a general method of solution which, in the special case, gives the solution (7 - 1)! + 1.
□ Ibn al-Haytham's second method gives all the solutions to systems of congruences of the type stated (which of course is a special case of the Chinese Remainder Theorem).
□ Article: The Telegraph (The solution of Alhazen's problem) .
19. Lax Peter biography
□ In 1957 he published an extremely important paper Asymptotic solutions of oscillating initial value problems where the beginnings of the theory of Fourier integral operators appears.
□ for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.
□ The equations that arise in such fields as aerodynamics, meteorology and elasticity are nonlinear and much more complex: their solutions can develop singularities.
□ He constructed explicit solutions, identified classes of especially well-behaved systems, introduced an important notion of entropy, and, with Glimm, made a penetrating study of how solutions
behave over a long period of time.
□ In addition, he introduced the widely used Lax-Friedrichs and Lax-Wendroff numerical schemes for computing solutions.
□ Inspired by Richtmyer, Lax established with this theorem the conditions under which a numerical implementation gives a valid approximation to the solution of a differential equation.
□ In the late 1960s a revolution occurred when Kruskal and co-workers discovered a new family of examples, which have "soliton" solutions: single-crested waves that maintain their shape as they
□ Lax became fascinated by these mysterious solutions and found a unifying concept for understanding them, rewriting the equations in terms of what are now called "Lax pairs".
□ Together with Phillips, Lax developed a broad theory of scattering and described the long-term behaviour of solutions (specifically, the decay of energy).
□ In 1970 Lax and Glimm published Decay of solutions of systems of nonlinear hyperbolic conservation laws, a difficult work which requires familiarity with earlier work of both authors.
20. Harriot biography
□ he produced a practical numerical solution of the Mercator problem, most probably by the addition of secants ..
□ He gave a solution to Alhazen's problem which involved considering an equivalent problem, namely the problem of the maximum intercept formed between a circle and a diameter of a chord
rotating about a point on a circle.
□ He came very close to a vector analysis solution of the problem of finding the velocity of the projectile and, certainly by 1607, he came to the conclusion that the path of the projectile was
a tilted parabola.
□ As an example of his abilities to solve equations, even when the roots are negative or imaginary, we reproduce his solution of an equation of degree 4.
□ As we have seen from the example above, Harriot did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look like a
present day solution.
□ For example, it does not discuss negative solutions.
21. Besicovitch biography
□ The solutions submitted were carefully read and annotated by Besicovitch and the announcement "Perfect solutions of Problem 12 were sent in by M and N" spurred several young mathematicians on
to develop their analytic powers.
□ When solving a problem most mathematicians need to make a commitment as to the nature of the solution long before the solution has been found, and this commitment interposes a psychological
barrier to the consideration of other possibilities.
□ One of the achievements, with which he will always be associated, was his solution of the Kakeya problem on minimising areas.
□ Often he showed that the "obvious solution" to certain problems is false.
□ The solution generally accepted for this problem by around 1950 was that however the man moved, the lion first aimed to get onto the line joining the man to the centre of the arena (which it
could always achieve) and then keeping on this radius however the man moved, it would end up catching the man.
□ He was more likely than anyone else to solve a problem which had seemed intractable, commonly the solution needed, by way of proof or counter-example, an ingenious and intricate construction.
22. Menaechmus biography
□ Menaechmus's solution is described by Eutocius in his commentary to Archimedes' On the sphere and cylinder.
□ Of course we must emphasis that this in no way indicates the way that Menaechmus solved the problem but it does show in modern terms how the parabola and hyperbola enter into the solution to
the problem.
□ Immediately following this solution, Eutocius gives a second solution.
□ ',1)">1], [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3] and [Greek Geometry from Thales to Euclid (Dublin-London, 1889), 153-179.',4)">4] all consider a problem associated
with these solutions.
□ Plutarch says that Plato disapproved of Menaechmus's solution using mechanical devices which, he believed, debased the study of geometry which he regarded as the highest achievement of the
human mind.
□ However, the solution described above which follows Eutocius does not seem to involve mechanical devices.
□ The solution proposed to this question in [Dictionary of Scientific Biography (New York 1970-1990).
□ What has come to be known as Plato's solution to the problem of duplicating the cube is widely recognised as not due to Plato since it involves a mechanical instrument.
□ it seems probable that someone who had Menaechmus's second solution before him worked to show how the same representation of the four straight lines could be got by a mechanical construction
as an alternative to the use of conics.
23. Mytropolsky biography
□ He further developed asymptotic methods and applied them to the solution of practical problems.
□ Using a method of successive substitutes, he constructed a general solution for a system of nonlinear equations and studied its behaviour in the neighbourhood of the quasi-periodic solution.
□ Asymptotic solutions of differential equations are worked out in great detail, the author always being willing to go the second mile with the reader in obtaining the inherently complicated
formulas that arise.
□ We give various algorithms, schemes and rules for constructing approximate solutions of equations with small and large parameters, and obtain examples which in many cases graphically
illustrate the effectiveness of the method of averaging and the breadth of its application to various problems which are, at first glance, very disparate.
□ Among the many co-authored works we mention Lectures on the application of asymptotic methods to the solution of partial differential equations (1968) co-authored with his former student
Boris Illich Moseenkov, Lectures on the methods of integral manifolds (1968) co-authored with his former student Olga Borisovna Lykova, Lectures on the theory of oscillation of systems with
lag (1969) co-authored with his former student Dmitrii Ivanovich Martynyuk, Asymptotic solutions of partial differential equations (1976) co-authored with his former student Boris Illich
Moseenkov, Periodic and quasiperiodic oscillations of systems with lag (1979) also co-authored with D I Martynyuk, Mathematical justification of asymptotic methods of nonlinear mechanics
(1983) co-authored with his former student Grigorii Petrovich Khoma, Group-theoretic approach in asymptotic methods of nonlinear mechanics (1988) co-authored with his former student Aleksey
Konstantinovich Lopatin, and Asymptotic methods for investigating quasiwave equations of hyperbolic type (1991) co-authored with his former students G P Khoma and Miron Ivanovich Gromyak.
24. Kantorovich biography
□ Kantorovich gave two lectures, "On conformal mappings of domains" and "On some methods of approximate solution of partial differential equations".
□ The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these
approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind.
□ The trust's laboratory seemed unable to arrive at a satisfactory solution that could not be further improved upon.
□ The mathematical formulation of production problems of optimal planning was presented here for the first time and the effective methods of their solution and economic analysis were proposed.
□ Kantorovich introduced many new concepts into the study of mathematical programming such as giving necessary and sufficient optimality conditions on the base of supporting hyperplanes at the
solution point in the production space, the concept of primal-dual methods, the interpretation in economics of multipliers, and the column-generation method used in linear programming.
□ The authors are particularly concerned with applications of functional analysis to the theory of approximation and the theory of existence and uniqueness of solutions of differential and
integral equations (both linear and non-linear).
□ These other areas include functional analysis and numerical analysis and within these topics he published papers on the theory of functions, the theory of complex variables, approximation
theory in which he was particularly interested in using Bernstein polynomials, the calculus of variations, methods of finding approximate solutions to partial differential equations, and
descriptive set theory.
25. Rudolff biography
□ This is evident from old books on algebra, written many years ago, in which quantities are represented, not by characters, but by words written out in full, 'drachm', 'thing', 'substance',
etc., and in the solution of each special example the statement was put, 'one thing', in such words as ponatur, una res, etc.
□ From this quote we see that he must have read the Latin Regensburg algebra of 1461 for in that work the solution to all problems begin with the words 'Pono quod lucrum sit una res'.
□ In looking at the case of a quadratic of the form ax2 + b = cx he believed at first that there was only one solution to this equation which will solve the original problem, but he later
recognised his error and realised that such equations have two solutions.
□ For several such problems concerned with "splitting the bill" (Zechenaufgaben) Rudolff supplied all the possible solutions.
□ Rudolff does not work out their solutions because, as he stated, he wanted to stimulate further algebraic research.
□ Rudolff was aware of the double root of the equation ax2 + b = cx and gave all the solutions to indeterminate first-degree equations.
26. Faddeev biography
□ He was able to extend significantly the class of equations of the third and fourth degree that admit a complete solution.
□ When he was studying, for example, the equation x3+ y3= A, Faddeev found estimates of the rank of the group of solutions that enabled him to solve the equation completely for all A ≤ 50.
□ Until then it had been possible to prove only that there were non-trivial solutions for some A.
□ For the equation x4+ Ay4= ±1 he proved that there is at most one non-trivial solution; this corresponds to the basic unit of a certain purely imaginary field of algebraic numbers of the
fourth degree and exists only when the basic unit is trinomial.
□ Two features are very characteristic of the mode of presentation: on the one hand the extensive use of geometrical considerations as a background for the true understanding of complicated
situations which otherwise would remain obscure, and on the other hand, the care shown by the authors in inventing effective methods of solution, illustrated by actual application to
numerical examples and to the construction of valuable tables.
□ Much of this work was done in collaboration with his wife Vera Nikolaevna Faddeeva but his first few papers on this topic are single authored: On certain sequences of polynomials which are
useful for the construction of iteration methods for solving of systems of linear algebraic equations (1958), On over-relaxation in the solution of a system of linear equations (1958), and On
the conditionality of matrices (1959).
□ This book consists of 983 problems (209 pages); hints to selected problems (37 pages), mostly the briefest possible; and 250 pages of solutions, ranging from mere answers to numerical
problems to complete proofs ..
27. Fermat biography
□ I have also found many sorts of analyses for diverse problems, numerical as well as geometrical, for the solution of which Viete's analysis could not have sufficed.
□ has no non-zero integer solutions for x, y and z when n > 2.
□ The second of the two problems, namely to find all solutions of Nx2 + 1 = y2 for N not a square, was however solved by Wallis and Brouncker and they developed continued fractions in their
□ Brouncker produced rational solutions which led to arguments.
□ Fermat posed further problems, namely that the sum of two cubes cannot be a cube (a special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof
of the general result was incorrect), that there are exactly two integer solutions of x2 + 4 = y3 and that the equation x2 + 2 = y3 has only one integer solution.
28. Bogolyubov biography
□ He wrote his first scientific paper On the behavior of solutions of linear differential equations at infinity (Russian) in 1924.
□ The works of his first period, some of which were carried out by him jointly with his teacher N M Krylov, deal with direct methods of the calculus of variations, to the theory of
nearly-periodic functions and approximate solutions of boundary-value differential equations.
□ By applying the Poincare-Lyapunov theory and the Poincare-Denjoy theory on trajectories on a torus, he examined the nature of the exact stationary solution in the vicinity of an approximate
solution for a sufficiently small value of the parameter and proved theorems on the existence and stability of quasi-periodic solutions.
□ Moreover, general statements of methods for solving equations are illustrated by the explicit solution of examples.
□ In solving problems of this type it is exceedingly important to provide the degree of rigor required for the correct solution of these problems.
29. Kublanovskaya biography
□ the method is applied to particular problems such as, for instance, solution of systems of linear equations, determination of eigenvalues and eigenvectors of a matrix, integration of
differential equations by series, solution of Dirichlet problem by finite differences, solution of integral equations, etc.
□ For all these cases, well-chosen numerical examples are analyzed and the solutions are tabulated.
□ Vera Kublanovskaya submitted her first QR summary 'Certain algorithms for the solution of the complete eigenvalue problem' on 5 July 1960, followed by two subsequent papers 'Some algorithms
for the solution of the complete eigenvalue problem' (1961) and 'The solution of the eigenvalue problem for an arbitrary matrix' (1962) with details.
□ Kublanovskaya continued publishing significant papers on related topics (all written in Russian) such as: (with Vera Faddeeva) Computational methods for the solution of the general eigenvalue
problem (1962); On a method of orthogonalizing a system of vectors (1964); Reduction of an arbitrary matrix to tridiagonal form (1964); A numerical scheme for the Jacobi process (1964); Some
bounds for the eigenvalues of a positive definite matrix (1965); An algorithm for the calculation of eigenvalues of positive definite matrices (1965); On a certain process of supplementary
orthogonalisation of vectors (1965); and A method for solving the complete problem of eigenvalues of a degenerate matrix (1966).
□ The paper is concerned with finding, without the use of the Gaussian transformation, the normal generalized (in the sense of the least-squares method) solution for a system of linear
algebraic equations with a rectangular matrix.
30. Robinson Julia biography
□ Along with Martin Davis and Hilary Putman she gave a fundamental result which contributed to the solution to Hilbert's Tenth Problem, making what became known as the Robinson hypothesis.
□ She also did important work on that problem with Matijasevic after he gave the complete solution in 1970.
□ .,w) = 0 such that the sets of all values of x in all solutions of P = 0 is too complicated a set to be calculated by any method whatever.
□ .,w) = 0 has a solution for a given value of a, then we would have a method of calculating whether a belongs to the set S, and this is impossible.
□ As a result of her work at RAND she published An iterative method of solving a game in the Annals of Mathematics in 1951 in which she proved the convergence of an iterative process for
approximating solutions for each player in a finite two-person zero-sum game.
□ In 1971 at a conference in Bucharest Robinson gave a lecture Solving diophantine equations in which she set the agenda for continuing to study Diophantine equations following the negative
solution to Hilbert's Tenth Problem problem.
□ Instead of asking whether a given Diophantine equation has a solution, ask "for what equations do known methods yield the answer?" .
31. Matiyasevich biography
□ Does there exist an algorithm to determine whether a Diophantine equation has a solution in natural numbers? .
□ The fact that the second statement asks for a solution in natural numbers while the first asks for a solution in integers is not significant.
□ On the morning of 3 January 1970, I believed I had a solution of Hilbert's tenth problem, but by the end of that day I had discovered a flaw in my work.
□ This paper shows that every recursively enumerable relation is diophantine and so completes the solution of Hilbert's tenth problem in the negative sense.
□ Matiyasevich also published Solution of the tenth problem of Hilbert in Hungarian in 1970.
□ In this thesis, as well as giving a simplified proof that no algorithm exists to determine whether Diophantine equations have integer solutions, he gave a Diophantine representation of a wide
class of natural number sequences produced by linear recurrence relations.
□ The first five lead to the negative solution of Hilbert's Tenth Problem; the remaining chapters are devoted to various applications of the method used by the author, which is, in a sense,
more important than the solution itself: it has applications to Hilbert's eighth problem, decision problems in number theory, Diophantine complexity, decision problems in calculus, and
Diophantine games.
32. Marchenko biography
□ The solution of an operator equation in the form of a travelling wave (a one-soliton solution) is elementary.
□ The solutions of the original equation are obtained from the one-soliton operator solutions by bordering them with special finite-dimensional projectors.
□ Arbitrariness in the choice of the operator algebra and the bordering projectors allows us to find broad classes of solutions of the Korteweg-de Vries, Kadomtsev-Petviashvili, nonlinear
Schrodinger, sine-Gordon, Toda lattice, Langmuir and other equations.
□ In these classes solutions are contained that can be obtained by the inverse problem method and by the methods of algebraic geometry, and also solutions that do not reduce to these methods.
33. Serrin biography
□ A set of conditions is given for the solution of flow problems involving curved boundaries.
□ In addition to his work on hydrodynamics, he also began publishing very significant results on elliptic differential equations with papers such as On the Phragmen-Lindelof theorem for
elliptic partial differential equations (1954), On the Harnack inequality for linear elliptic equations (1956) and (with David Gilbarg) On isolated singularities of solutions of second order
linear elliptic equations (1956).
□ One of the fundamental questions which should be answered concerning any problem of applied mathematics is whether it is well set, that is, whether solutions actually exist and whether they
are unique.
□ We shall be concerned here with the initial value problem for compressible fluid flow, and we shall study in particular the uniqueness of its solutions.
□ It is during that period that his two articles ['Local behavior of solutions of quasi-linear equations' (1964) and 'Isolated singularities of solutions of quasi-linear equations' (1965)] on
isolated singularities were published in 'Acta Mathematica'.
□ The maximum principle enables us to obtain information about solutions of differential equations and inequalities without any explicit knowledge of the solutions themselves, and thus can be a
valuable tool in scientific research.
34. Word problems
□ Here there are above all three fundamental problems whose solution is very difficult and which will not be possible without a penetrating study of the subject.
□ Each knotted space curve, in order to be completely understood, demands the solution of the three above problems in a special case.
□ He used his solution to show that right and left trefoils are distinct.
□ He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group.
□ The solution to the word problem for these groups began with Dehn who stated the Freiheitssatz: .
□ In the following year Magnus published a paper containing a special case of the word problem for 1-relator groups, then in 1932 he published a complete proof of the solution of the word
problem for this class of groups.
□ It required computability theory and developments in mathematical logic to even make the questions precise, but these areas were to not only provide explicit questions, they also provided
solutions to the questions.
□ While sitting in the dentist's chair waiting for this unpleasant experience, inspiration struck and suddenly he saw the route to the solution.
35. Cartan biography
□ This enabled Cartan to define what the general solution of an arbitrary differential system really is but he was not only interested in the general solution for he also studied singular
□ He did this by moving from a given system to a new associated system whose general solution gave the singular solutions to the original system.
□ He failed to show that all singular solutions were given by his technique, however, and this was not achieved until four years after his death.
36. De Giorgi biography
□ as a child I had a special taste for puzzling out solutions to little problems, but I also had a certain passion for experimenting with little gadgets - experiments, if not of physics, of
□ In 1955 De Giorgi gave an important example which showed nonuniqueness for solutions of the Cauchy problem for partial differential equations of parabolic type whose coefficents satisfy
certain regularity conditions.
□ In the following year he proved what has become known as "De Giorgi's Theorem" concerning the Holder continuity of solutions of elliptic partial differential equations of second order.
□ The authors of this paper are all students of De Giorgi and they describe his contributions to geometric measure theory, the solution of Hilbert's XIXth problem in any dimension, the solution
of the n-dimensional Plateau problem, the solution of the n-dimensional Bernstein problem, some results on partial differential equations in Gevrey spaces, convergence problems for
functionals and operators, free boundary problems, semicontinuity and relaxation problems, minimum problems with free discontinuity set, and motion by mean curvature.
37. Word problems
□ Here there are above all three fundamental problems whose solution is very difficult and which will not be possible without a penetrating study of the subject.
□ Each knotted space curve, in order to be completely understood, demands the solution of the three above problems in a special case.
□ He used his solution to show that right and left trefoils are distinct.
□ He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group.
□ The solution to the word problem for these groups began with Dehn who stated the Freiheitssatz: .
□ In the following year Magnus published a paper containing a special case of the word problem for 1-relator groups, then in 1932 he published a complete proof of the solution of the word
problem for this class of groups.
□ It required computability theory and developments in mathematical logic to even make the questions precise, but these areas were to not only provide explicit questions, they also provided
solutions to the questions.
□ While sitting in the dentist's chair waiting for this unpleasant experience, inspiration struck and suddenly he saw the route to the solution.
38. Lemaitre biography
□ Einstein was at the conference and he spoke to Lemaitre in Brussels telling him that the ideas in his 1927 paper had been presented by Friedmann in 1922, but he also said that although he
thought Lemaitre's solutions of the equations of general relativity were mathematically correct, they presented a solution which was not feasible physically.
□ Lemaitre then applied these ideas to accelerate the orthodox process of iteration, taking the Picard iterative solution of first order differential equations as an example.
□ In Sur un cas limite du probleme de Stormer (1945) he studied trajectories of an electron in the neighborhood of lines of force of a magnetic dipole field, then returned to his study of
numerical solutions to first order differential equations in Interpolation dans la methode de Runge-Kutta (1947).
□ It is shown how these equations can be applied toward the solution of the well-known problem of uniform distribution in a homogeneous, expanding universe.
□ The paper opens with a rapid expository review of the general relativity theory of gravitation, including discussion of kinematics, conservation laws, spherical symmetry, and the solutions of
Schwarzschild and de Sitter in terms of comoving coordinates.
39. Krylov Nikolai biography
□ He worked mainly on interpolation and numerical solutions to differential equations, where he obtained very effective formulas for the errors.
□ For example he published On the approximate solution of the integro-differential equations of mathematical physics (1926), and Approximation of periodic solutions of differential equations in
French in 1929.
□ With his collaborator and former student N N Bogolyubov, he published On Rayleigh's principle in the theory of differential equations of mathematical physics and on Euler's method in calculus
of variations (1927-8) and On the quasiperiodic solutions of the equations of the nonlinear mechanics.
□ Moreover, general statements of methods for solving equations are illustrated by the explicit solution of examples.
□ Before publishing this book with Bogolyubov, in 1931 Krylov had published the important monograph Les methodes de solution approchee des problemes de la physique mathematique.
40. Schubert biography
□ Schubert is famed for his work on enumerative geometry which considers those parts of algebraic geometry that involves a finite number of solutions.
□ Algebraically, the solution of the problems of enumerative geometry amounts to finding the number of solutions for certain systems of algebraic equations with finitely many solutions.
□ Since the direct algebraic solution of the problems is possible only in the simplest cases, mathematicians sought to transform the system of equations, by continuous variation of the
constants involved, into a system for which the number of solutions could be determined more easily.
41. Kruskal Martin biography
□ An important paper on astronomy was Maximal extension of Schwarzschild's metric (1960) which showed that, using what are now called Kruskal coordinates, certain solutions of the equations of
general relativity which are singular at the origin are not singular away from the origin, so allowing the study of black holes.
□ He was led to asymptotic analysis in his plasma physics studies and from there to solutions of Hamiltonian equations as in Asymptotic theory of Hamiltonian and other systems with all
solutions nearly periodic (1962).
□ Methods for exact solution published in 1974 was fundamental, and the ideas developed in it were later extended to dynamical systems, inverse scattering, and symplectic geometry.
□ Before it, there was no general theory for the exact solution of any important class of nonlinear differential equations.
□ For his influence as a leader in nonlinear science for more than two decades as the principal architect of the theory of soliton solutions of nonlinear equations of evolution.
42. Halphen biography
□ The first result which brought him to the attention of mathematicians world-wide was his solution in 1873 of a problem of Chasles [Dictionary of Scientific Biography (New York 1970-1990).
□ Halphen's solution was ingenious ..
□ He defined the concepts of proper and improper solutions to an enumerative problem involving conics.
□ Then a particular number associated with a problem about conics has enumerative significance when it counts the number of proper solutions.
□ Halphen and Schubert engaged in a heated debate on whether an enumerative formula should be allowed to count degenerate solutions along with the nondegenerate solutions.
43. Magiros biography
□ Magiros wrote other papers on stability such as The stability in the sense of Lyapunov, Poincare and Lagrange of some precessional phenomena (1970) and Remarks on stability concepts of
solutions of dynamical systems (1974).
□ Linearization by approximate methods in which he points out that "approximate" linearizations may lose the whole qualitative behaviour of the original nonlinear equation; and Characteristic
properties of linear and nonlinear systems in which he gives many examples, recalls the importance of identifying characteristic properties of solutions, such as the superposition property
for linear systems, and the possibility of limit cycles and self-excited oscillations in nonlinear systems.
□ Two papers which Magiros published in 1977 are: Nonlinear differential equations with several general solutions in which he gives specific devices for finding solutions of some nonlinear
ordinary differential equations; and The general solutions of nonlinear differential equations as functions of their arbitrary constants presenting some nonlinear differential equations for
which, surprisingly, some superposition does occur, that is, there are families of solutions depending linearly on arbitrary constants.
□ covers a variety of topics from special functions and transforms to numerical methods for the solution of nonlinear differential equations and optimal control problems.
44. Stampioen biography
□ In 1633 he challenged Descartes to a public competition by giving him a geometric problem whose solution involved the solution of a quartic equation.
□ Descartes presented a solution but it was rejected by Stampioen as not being complete.
□ He then posed two further public challenges under the alias of John Baptista of Antwerp involving the solution of cubics.
□ Having posed the problems as if by John Baptista of Antwerp, he then proceeded to give solutions to the problems under his own name, using his methods for finding the cube root of a + √b.
□ Stampioen rejected Waessenaer's solution which prompted Waessenaer to reply with a broadly based attack on the mathematics contained in Stampioen's Algebra or the New Method.
□ He published a topographical map in 1650 and, were it not for the fact that he is mentioned as being a member of a panel set up to adjudicate a proposed solution to the longitude problem in
1689, we might have wrongly supposed that he died shortly after 1650.
45. Abel biography
□ While in his final year at school, however, Abel had begun working on the solution of quintic equations by radicals.
□ In Abel's third paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation.
□ Geometers have occupied themselves a great deal with the general solution of algebraic equations and several among them have sought to prove the impossibility.
□ The second of these explanations does seem the more likely, especially since Gauss had written in his thesis of 1801 that the algebraic solution of an equation was no better than devising a
symbol for the root of the equation and then saying that the equation had a root equal to the symbol.
□ He had been working again on the algebraic solution of equations, with the aim of solving the problem of which equations were soluble by radicals (the problem which Galois solved a few years
□ Also after Abel's death unpublished work on the algebraic solution of equations was found.
46. Al-Quhi biography
□ Al-Quhi's solution to the problem is given in [Centaurus 38 (2-3) (1996), 140-207.',5)">5].
□ It is a classical style of solution using results from Euclid's Elements, Apollonius's Conics and Archimedes' On the sphere and cylinder.
□ If a solution exists, al-Quhi showed that it will have coordinates which lie on a particular rectangular hyperbola that he has constructed.
□ Next al-Quhi introduces the "cone of the surface" which, after many deductions, leads to showing that the solution has coordinates lying on a parabola.
□ One, which requires the solution of a quadratic equation, had been found by Abu Kamil in the ninth century.
□ The other, which requires the solution of a quartic equation, is the one presented by al-Quhi.
□ Topics covered are quite varied, ranging from a discussion of what "known" means to solutions of specific problems such as the following Suppose we are given a circle and two intersecting
straight lines l and m.
47. Ferrari biography
□ They worked on problems set by Zuanne da Coi and eventually were able to extend solutions discovered in these special cases.
□ Ferrari discovered the solution of the quartic equation in 1540 with a quite beautiful argument but it relied on the solution of cubic equations so could not be published before the solution
of the cubic had been published.
□ Cardan and Ferrari satisfied della Nave that they could solve the ubiquitous cosa and cube problem, and della Nave showed them in return the papers of the late del Ferro, proving that
Tartaglia was not the first to discover the solution of the cubic.
□ Cardan published both the solution to the cubic and Ferrari's solution to the quartic in Ars Magna (1545) convinced that he could break his oath since Tartaglia was not the first to solve the
48. Norlund biography
□ He studied the factorial series and interpolation series entering in their solutions, determining their region of convergence and by analytic prolongation and different summation methods he
extended them in the complex plane, determining their singularities and their behaviour at infinity, also by use of their relations to continued fractions and asymptotic series.
□ This is the first book to develop the theory of the difference calculus from the function-theoretic point of view and to include a significant part of the recent researches having to do with
the analytic and asymptotic character of the solutions of linear difference equations.
□ This is The logarithmic solutions of the hypergeometric equation (1963) which was reviewed by L J Slater:- .
□ In this important paper the author discusses in a clear and detailed way the complete logarithmic solutions of the hypergeometric differential equation satisfied by the Gauss function ..
□ Complete tables are given of the linear and quadratic relations which hold between the various solutions in every possible special case.
□ Tables are also given for the continuation formulae which hold between the logarithmic and other cases of Riemann's P-function, and the paper concludes with a very clear statement of the
logarithmic solutions of the confluent hypergeometric equation satisfied by Kummer's function ..
□ This coolness could be felt as an aloofness; maybe it was due to some sort of shyness, but on the other hand his words thus gained more importance and one could feel how he exerted himself to
find the right solution to problems.
49. Fantappie biography
□ The geometrical significance in abstract spaces of such notions as characteristic strips and singular solutions is given.
□ Suddenly I saw the possibility of interpreting a wide range of solutions (the anticipated potentials) of the wave equation which can be considered the fundamental law of the Universe.
□ These solutions had been always rejected as "impossible", but suddenly they appeared "possible", and they explained a new category of phenomena which I later named "syntropic", totally
different from the entropic ones, of the mechanical, physical and chemical laws, which obey only the principle of classical causation and the law of entropy.
□ Syntropic phenomena, which are instead represented by those strange solutions of the "anticipated potentials", should obey two opposite principles of finality (moved by a final cause placed
in the future, and not by a cause which is placed in the past): differentiation and non-causable in a laboratory.
□ Finally let us look briefly at some of the papers which Fantappie published in the last seven years of his life: Costruzione effettiva di prodotti funzionali relativisticamente invarianti
(1949) constructs functional scalar products of two functions, as required in quantum mechanics, which are relativistically invariant; Caratterizzazione analitica delle grandezze della
meccanica quantica (1952) gives conditions on an hermitian operator that he claims are necessary and sufficient for it to satisfy to represent a physically real observable; Determinazione di
tutte le grandezze fisiche possibili in un universo quantico (1952) discusses aspects of group invariance of wave equations; Gli operatori funzionali vettoriali e tensoriali, covarianti
rispetto a un gruppo qualunque (1953) discusses the role of operators and Lie groups in a quantum-mechanical universe; Deduzione della legge di gravitazione di Newton dalle proprieta del
gruppo di Galilei (1955) shows that the inverse square law is a necessary consequence if certain specific assumptions are made; Les nouvelles methodes d'integration, en termes finis, des
equations aux derivees partielles (1955) applies analytic functionals to find explicit solutions of partial differential equations; and Sur les methodes nouvelles d'integration des equations
aux derivees partielles au moyen des fonctionnelles analytiques (1956) gives a new method for the solution of Cauchy's problem.
50. Sluze biography
□ This work had been inspired by Mersenne who had informed them of Torricelli's solution of the problem of calculating the volume of the solid generated by revolving a hyperbola about the axis.
□ De Sluze had gained the experience necessary to solve such problems and sent his solution to Pascal who praised it highly.
□ This work was on geometrical construction in which he discussed the cubature of various solids and the solutions to third and fourth degree equations which he obtained geometrically using the
intersection of any conic section with a circle.
□ They had both been stimulated to work on it by reading Isaac Barrow's Lectiones Opticae (1669) where he gave only a partial solution to the problem.
□ Huygens sent his results to Oldenburg in which he stated that he had found an elegant solution.
□ Some time later de Sluze told Oldenburg that he had found a good method of solution and was then sent details of Huygens' method by Oldenburg.
51. Caccioppoli biography
□ For the linear case he considered a linear transformation acting on the vectors of a linear space (in which the solution is to be found).
□ If the image set entirely covers the second linear space then solutions exist independently on the given data.
□ the image set is a linear, closed subspace in the second linear space) then necessary and sufficient conditions are placed on the data set so that the problem has solutions.
□ In the period between 1933 and 1938 Caccioppoli applied his method to elliptic equations, providing the a priori upper bound for their solutions, in a more general way than Bernstein did for
the two-dimensional case.
□ In 1935 he dealt with the question introduced in 1900 by Hilbert during the International Congress of Mathematicians, namely whether or not the solutions of analytical elliptic equations are
□ Caccioppoli proved the analyticity of C2-class solutions.
52. Lorenz Edward biography
□ The first few terms of a particular series solution are obtained explicitly.
□ Solutions of these equations can be identified with trajectories in phase space.
□ For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can
evolve into considerably different states.
□ Systems with bounded solutions are shown to possess bounded numerical solutions.
53. Mahavira biography
□ He also discussed integer solutions of first degree indeterminate equation by a method called kuttaka.
□ The kuttaka (or the "pulveriser") method is based on the use of the Euclidean algorithm but the method of solution also resembles the continued fraction process of Euler given in 1764.
□ There is no unique solution but the smallest solution in positive integers is p = 15, x = 1, y = 3, z = 5.
□ Any solution in positive integers is a multiple of this solution as Mahavira claims.
54. Pogorelov biography
□ It contains a number of new results on the setting of boundary-value problems, and on questions of uniqueness and regularity of generalized solutions.
□ The booklet contains a solution of Hilbert's well-known fourth problem concerning the determination of all realizations up to isomorphism of the system of axioms of classical geometries
(Euclidean and non-Euclidean) supposing that the axioms of congruency are replaced by the axiom "triangle inequality".
□ Pogorelov's solution to Hilbert's Fourth Problem, which he presented to a meeting of the Kharkov Mathematical Society held at the Kharkov University, was described by I Kra as a "mathematical
jewel" [Ukrainian Math.
□ In my article published in 1973 I have admitted some immodesty when I entitled it as "The Complete Solution of the Fourth Hilbert Problem".
□ In fact, it did not contain a complete solution of the fourth problem, because only the two-dimensional case was examined.
□ He was peerless in his skill in overcoming difficulties in the solution of hard mathematical problems.
55. Fox Leslie biography
□ For example he published Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations (1947), A short account of relaxation methods
(1948), and The solution by relaxation methods of ordinary differential equations (1949).
□ This book was The numerical solution of two-point boundary problems in ordinary differential equations and it is a great tribute to his expository skills that it was reprinted by Dover
Publications in 1990.
□ The book summarises at an elementary level the methods for numerical construction of the solutions of boundary-value problems which can be expressed in terms of ordinary differential
equations of orders one to four.
□ Another collaboration between Fox and Mayers led to Numerical solution of ordinary differential equations published in 1987, four years after Fox retired.
□ It considers the success achieved in the production of new techniques, machine-oriented techniques, error analysis, mathematical theorems and the solution of practical problems, and contrasts
this with corresponding work in the field of linear algebra.
56. Sokolov biography
□ Other applications include On the determination of dynamic pull in shaft-lifting cables (Ukrainian) (1955) and On approximate solution of the basic equation of the dynamics of a hoisting
cable (Ukrainian) (1955).
□ One of the topics which will always be associated with Sokolov's name is his method for finding approximate solutions to differential and integral equations.
□ Examples of his papers on this topic are On a method of approximate solution of linear integral and differential equations (Ukrainian) (1955), Sur la methode du moyennage des corrections
fonctionnelles (Russian) (1957), Sur l'application de la methode des corrections fonctionnelles moyennes aux equations integrales non lineaires (Russian) (1957), On a method of approximate
solution of systems of linear integral equations (Russian) (1961), On a method of approximate solution of systems of nonlinear integral equations with constant limits (Russian) (1963), and On
sufficient tests for the convergence of the method of averaging of functional corrections (Russian) (1965).
□ This basic approach is developed by the author and applied to the approximate solution of Fredholm and Volterra-type integral equations of the second kind, to their nonlinear counterparts, to
integral equations of mixed type, to linear and nonlinear one-dimensional boundary value problems, to initial-value problems in ordinary differential equations and to certain elliptic,
hyperbolic and parabolic equations.
57. Ollerenshaw biography
□ Without fail, on waking in the morning, the details, the logical argument required or the facts that I needed to recall were clearly imprinted in my mind and, because of clarity, any required
solution would often be clearly 'written' on the partition.
□ It was moreover a matter of geometry -- pure mathematics -- a nice problem that had a neat and successful solution.
□ Critical lattices relate to whole numbers in two or more dimensions and lead, by geometrical methods, to solutions concerned with 'close packing', for example, how best to stack tins in a
cupboard or oranges in a box.
□ This became a good outlet for her papers which included the first general solution Rubik's cube, a solution to the twelve penny problem, and a solution to the nine prisoners problem.
58. Tikhonov biography
□ The thesis applied an extension of Emile Picard's method of approximating the solution of a differential equation and gave applications to heat conduction, in particular cooling which obeys
the law given by Josef Stefan and Boltzmann.
□ However, his mathematical investigations are never confined to the solution of a given concrete problem, but serve as the starting point for stating a general mathematical problem that is a
broad generalisation of the first problem.
□ However, in 1948 he began to study a new type of problem when he considered the behaviour of the solutions of systems of equations with a small parameter in the term with the highest
□ Under his guidance many algorithms for the solution of various problems of electrodynamics, geophysics, plasma physics, gas dynamics, ..
□ He defined a class of regularisable ill-posed problems and introduced the concept of a regularising operator which was used in the solution of these problems.
□ Combining his computing skills with solving problems of this type, Tikhonov gave computer implementations of algorithms to compute the operators which he used in the solution of these
59. Morton biography
□ One may indeed expect a rapid expansion of this activity in the next few years as the solution of physically interesting two- and three-dimensional flows becomes a practical and economic
□ This is that from the inception of the subject scientists attempting the solution of fluid flow problems have continuously made outstanding contributions to the subject's development.
□ We should mention two important books he published: (with David F Mayers) Numerical Solution of Partial Differential Equations: an introduction (1994, 2nd edition 2005); and Numerical
Solution of Convection-Diffusion Problems (1996).
□ The book includes parabolic, hyperbolic, and elliptic equations, each section starting with an analysis of the behaviour of solutions of the partial differential equations.
□ The level of presentation is ideal for anyone with some knowledge of numerical analysis who wishes to learn about the solution of convection-diffusion problems.
60. Cercignani biography
□ The papers Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem and Solutions of linearized gas-dynamics Boltzmann equation
and application to slip-flow problem appeared in print in 1962 while, in the following year, he published further papers, including two written jointly with his physics advisor Sergio
Albertoni, namely Numerical Evaluation of the Slip Coefficient and Slip-coefficient expression is derived using an exact analytical solution of the slip flow problem.
□ During the 1980's, he studied the evaporation-condensation interface between a gas and a liquid, stating an important conjecture for the long-time behaviour of the Boltzmann equation
□ A few years ago, a collaboration with Sasha Bobylev on self-similar solutions of the Boltzmann equation even led him to a new pretty formula for the inversion of the Laplace transform.
□ He established important theoretical results, including properties of existence and uniqueness of solutions of initial value problems, which are the basis of recent developments in methods
for numerical simulation of gas in networks.
61. Yang Hui biography
□ Firstly he explains the logic behind the problem, secondly he gives a numerical solution to the problem, and thirdly he shows how the method he has presented can be modified to solve similar
□ For example, if the problem reduced to the solution of a quadratic equation, then Yang would solve it numerically, then show how to solve a general quadratic equation numerically.
□ Although Yang has presented a problem straight from the Nine Chapters his method of solution is quite different.
□ the additive method of multiplication and the subtractive method of division [relative to the] ten problems and their solutions.
□ Yang's solution is quoted in [The nine chapters on the mathematical art : Companion and commentary (Beijing, 1999).',8)">8]:- .
□ Then a modern solution would set up equations .
62. Loyd biography
□ Click Solution White plays Ng4 as first move.
□ The subtle solutions are when Black plays Kf3 or Kh3 as first move.
□ Now if Black plays Kh3 or g2 then Qh7 for White mates, while if Black plays f3 then White mates with Rh8.')">HERE for the solution.
□ A prize of $1000, offered for the first correct solution to the problem, has never been claimed, although there are thousands of persons who say they have performed the required feat.
□ At first sight one would not expect there to be a unique solution to this problem, but Loyd was well aware that a logical argument would find the one and only one solution.
63. Viete biography
□ (If I asked for a solution to ax = b nobody asks: "For which quantity do I solve the equation ?") .
□ The problem is that if we ask for a solution of x3 + x = 1 then we ask for the solution to a problem which does not make sense geometrically.
□ Viete therefore looked for solutions of equations such as A3 + B2A = B2Z where, using his convention, A was unknown and B and Z were knowns.
□ He gave geometrical solutions to doubling a cube and trisecting an angle in this book.
64. Deligne biography
□ Weil's work related questions about integer solutions to polynomial equations to questions in algebraic geometry.
□ He conjectured results about the number of solutions to polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation.
□ A solution of the three Weil conjectures was given by Deligne in 1974.
□ A solution to these problems required the development of a new kind of algebraic topology.
□ These conjectures were both exceptionally hard to settle (the best specialists, including A Grothendieck, had worked on them) and most interesting in view of the far-reaching consequences of
their solution.
65. Spencer Tony biography
□ As an example of his continued research activity, let us quote his own summary of his paper Exact solutions for a thick elastic plate with a thin elastic surface layer published in 2005:- .
□ A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly
elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate.
□ The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions,
a solution of the three-dimensional elasticity equations for the inhomogeneous material.
□ It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity
66. Gale biography
□ in 1949 for his thesis Solutions of Finite Two-Person Games.
□ Related to his thesis were the papers (with S Sherman) Solutions of finite two-person games, and two papers written jointly with H W Kuhn and A W Tucker, On symmetric games, and Reductions of
game matrices.
□ The paper on symmetric games shows that if a zero-sum two-person game has an m × n pay-off matrix, then an optimal solution is immediately derivable from an optimal solution of a symmetric
zero-sum two-person game with a square pay-off matrix of order m+n+1.
□ His contributions range from optimal assignment problems in a general setting to major contributions to mathematical economics such as his solution of the n-dimensional 'Ramsey Problem' and
his important theory of optimal economic growth.
67. Fowler biography
□ Early in his career, after receiving his degree, Fowler took to examining the behavior of the solutions to certain second-order differential equations.
□ He rightly deduced Emden's equation must have other solutions.
□ When Milne divulged his thoughts to Fowler, Ralph immediately developed a new solution for different values of n and all types of boundary solutions.
□ These ions are closely packed leaving the free electrons to form a degenerate gas which Fowler described as "like a gigantic molecule in its lowest state." The equilibrium of the white dwarfs
was later found to be described by a solution to Emden's equation as generalized by Fowler in the above equation with n = 3/2.
68. Carmichael biography
□ He taught for three years at the Presbyterian College in Anniston and by 1909 he had around 170 publications in the American Mathematical Monthly, mostly problems and solutions to problems,
as well as 13 papers in the Annals of Mathematics and the Bulletin of the American Mathematical Society.
□ in 1911 for his thesis Linear Difference Equations and their Analytic Solutions Linear Difference Equations and their Analytic Solutions.
□ Show that if the equation φ(x) = n has one solution it always has a second solution, n being given and x being the unknown.
69. Douglas biography
□ It was during this period that he worked out a complete solution to the Plateau problem which had been posed by Lagrange in 1760 and then had been studied by leading mathematicians such as
Riemann, Weierstrass and Schwarz.
□ Before Douglas's solution only special cases of the problem had been solved.
□ In a series of papers from 1927 onwards Douglas worked towards the complete solution: Extremals and transversality of the general calculus of variations problem of the first order in space
(1927), The general geometry of paths (1927-28), and A method of numerical solution of the problem of Plateau (1927-28).
□ Douglas presented full details of his solution in Solution of the problem of Plateau in the Transactions of the American Mathematical Society in 1931.
□ After giving a complete solution to the Plateau Problem, Douglas went on to study generalisations of it.
□ In particular the award was for three papers all published in 1939: Green's function and the problem of Plateau and The most general form of the problem of Plateau published in the American
Journal of Mathematics and Solution of the inverse problem of the calculus of variations published in the Proceedings of the National Academy of Sciences.
□ The third paper does not give the compete proof for the solution of the inverse problem of the calculus of variations but is an announcement of the result.
70. Rado biography
□ In 1930 Rado published the work for which he is most famous, namely his solution to the Plateau Problem.
□ Plateau was a physicist who experimented with dipping thin wire frames into a soap solution and examining the soap film which was then stretched across the wire.
□ Garnier made a major breakthrough in 1928 followed soon after by independent solutions to the general problem by Douglas and by Rado.
□ His solution appeared 1930 in The problem of least area and the problem of Plateau published in Mathematische Zeitschrift.
□ Let us remark that the solution to the Plateau problem by both Douglas and by Rado did not exclude the possibility that the minimal surface contained a singularity.
71. Keller Joseph biography
□ By appropriate approximations, the solution of the integral equation is reduced to the evaluation of a surface integral.
□ Keller's first two single-author papers appeared in 1948: On the solution of the Boltzmann equation for rarefied gases; and The solitary wave and periodic waves in shallow water.
□ Two problems are inverses if the formulation of each involves all or part of the solution of the other.
□ The direct problem of the pair has been extensively studied or has a solution readily obtained by standard methods.
□ They can have no or only partly determined solutions even though the corresponding direct problems are well posed.
72. Wu biography
□ Although many approximate treatments, such as linear theory and shallow-water theory as well as numerical computations, have been used to explain many important phenomena, it is certainly of
importance to study the solutions of the equations which include the effects neglected by approximate models.
□ By applying tools from harmonic analysis (singular integrals and Clifford algebra), she proves that the Taylor sign condition always holds and that there exists a unique solution to the water
wave equations for a finite time interval when the initial wave profile is a Jordan surface.
□ We discuss results on the existence and uniqueness of solutions for given data, the regularity of solutions, singularity formation and the nature of the solutions after the singularity
formation time.
73. Margulis biography
□ Margulis's most spectacular achievement has been the complete solution of that problem and, in particular, the proof of the conjecture in question.
□ Margulis proved the full conjecture in 1986 and gives a beautiful survey of the work leading to this solution in [Fields Medallists Lectures (Singapore, 1997), 272-327.',3)">3].
□ One was the solution to a problem posed by Rusiewicz, about finitely additive measures on spheres and Euclidean spaces.
□ Though his work addresses deep unsolved problems, his solutions are housed in new conceptual and methodological frameworks of broad and enduring application.
□ Besides his celebrated results on super-rigidity and arithmeticity of irreducible lattices of higher rank semisimple Lie groups, and the solution of the Oppenheim conjecture on values of
irrational indefinite quadratic forms at integral points, he has also initiated many other directions of research and solved a variety of famous open problems.
74. Black Fischer biography
□ The solution was to demand a new level of mathematical technique.
□ The solution worked using a number of steps :- .
□ The famous Black-Scholes formula for the solution is:- .
□ He was in the forefront of recognising the importance of computer technology and good trading/engineering systems and thought that trying to model reality was more important than closed form
analytical solutions.
□ ',98)">98] and by Feynman [Review of Modern Physics, 20, 367-387.',89)">89], where it was shown that the solution of Fourier's equation could be expressed as the distribution function of a
random variable arising of a large number of random walks each with n steps (and with each step size proportional to √(t/n)) and by letting n become very large (i.e.
75. Catalan biography
□ Liouville began publication of Journal de Mathematiques Pures et Appliquees in 1836 and a paper by Catalan, Solution d'un probleme de Probabilite relatif au jeu de rencontre, was published in
the second volume in 1837.
□ The second of these contains the 'Catalan numbers' which appears in the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals.
□ Four papers by Catalan are published in Volume 4 in 1839: Note sur la Theorie des Nombres; Solution nouvelle de cette question: Un polygone etant donne, de combien de manieres peut-on le
partager en triangles au moyen de diagonales?; Addition a la Note sur une Equation aux differences finies; and Memoire sur la reduction d'une classe d'integrales multiples.
□ Two consecutive whole numbers, other than 8 and 9, cannot be consecutive powers; otherwise said, the equation xm - yn = 1 in which the unknowns are positive integers only admits a single
□ He wrote: Elements de geometrie (1843); the two volume work Traite elementaire de geometrie descriptive (1850-52) which ran to 5 editions with the last appearing in 1881; Theoremes et
problemes de geometrie elementaire (1852) which ran to 6 editions with the last appearing in 1879; Nouveau manuel des aspirants au baccalaureat es sciences (1852) which ran to 12 editions;
Solutions des problemes de mathematique et de physique donnes a la Sorbonne dans les compositions du baccalaureat es sciences (1855-56); two volumes of Manuel des candidats a l'Ecole
Polytechnique (1857-58); Notions d'astronomie (1860) which ran to 6 editions; Traite elementaire des series (1860); Histoire d'un concours (1865) with a second edition published in 1867; and
Cours d'analyse de l'universite de Liege (1870) with a second edition published in 1880.
76. Krasnosel'skii biography
□ For example Positive solutions of operator equations (1962) which studied the existence, uniqueness, and properties of positive solutions of linear and non-linear equations in a partially
ordered Banach space, Vector fields in the plane (1963) which the angular variation of a plane vector field relative to a curve, and Displacement operators along trajectories of differential
equations (1966) which is described by C Olech as follows:- .
□ The second subject covered in the book concerns the existence and uniqueness of positive periodic solutions for systems satisfying certain monotonicity assumptions.
□ The third part is concerned with the study of the connection between convexity and concavity of the displacement operator and the stability or instability of periodic solutions.
□ For example Approximate solution of operator equations (1969):- .
77. Gromov biography
□ He did, however, contribute the text of his lecture A topological technique for the construction of solutions of differential equations and inequalities which was published in the Conference
Proceedings in 1971.
□ Some of those problems were long standing, and their unexpected solutions caused wonder and surprise due to the originality and elegance of the method conceived by Gromov: famous instances
are his proof of the old conjecture according to which a finitely generated group of polynomial growth has a nilpotent subgroup of finite index, or the beautiful construction (together with I
Pyatetski-Shapiro) of non-arithmetic discrete groups of hyperbolic transformations in arbitrary dimension.
□ To summarise, Gromov has brought about not only solutions to famous and time-old problems, but also the bases of new fields of study for many scholars.
□ for his work in Riemannian geometry, which revolutionized the subject; his theory of pseudoholomorphic curves in symplectic manifolds; his solution of the problem of groups of polynomial
growth; and his construction of the theory of hyperbolic groups.
□ The Abel committee says: "Mikhail Gromov is always in pursuit of new questions and is constantly thinking of new ideas for solutions to old problems.
78. Lavrentev biography
□ Lavrent'ev gave excellent examples of practical application of theoretical solutions.
□ For example, he applied variational properties of conformal mappings and reduced the important problem of flow around a wing to the solution of a singular integral equation of the first kind.
□ In his works written together with Mstislav Vsevolodovich Keldysh, important results obtained in the theory of conformal mappings were used for the solution of numerous applied problems.
□ The scientific work carried out at the Institute of Mathematics of the Ukrainian Academy of Sciences under the guidance of Lavrent'ev was aimed at the solution of not only fundamental
theoretical problems but also numerous important applied problems.
□ A great merit of Lavrent'ev lies in the fact that he resolutely directed the theoretical investigations at the Institute of Mathematics of the Ukrainian Academy of Sciences towards the
solution of the urgent problems of national economy.
79. Kirkman biography
□ After Steiner asked his question, a solution was given by M Reiss in 1859.
□ The solution to the Fifteen Schoolgirls Problem is not particularly hard.
□ Cayley published a solution first, then Kirkman published his own solution, which of course he knew before asking the question.
□ Kirkman continued to study mathematics until his 89th year sending questions and solutions to the Educational Times up to a few months before his death.
80. Bernoulli Johann biography
□ Five solutions were obtained, Jacob Bernoulli and Leibniz both solving the problem in addition to Johann Bernoulli.
□ The solution of the cycloid had not been found by Galileo who had earlier given an incorrect solution.
□ Johann's solution to this problem was less satisfactory than that of Jacob but, when Johann returned to the problem in 1718 having read a work by Taylor, he produced an elegant solution which
was to form a foundation for the calculus of variations.
81. Sundman biography
□ His methods were applicable more generally, however, and he went on to make a major breakthrough in the solution of the three-body problem.
□ The most famous contribution of Sundman was his solution of the three-body problem which he accomplished using analytic methods to prove the existence of an infinite series solution.
□ In fact the Academy was so impressed by his solution that, after receiving a report on his work from a committee headed by emile Picard, they decided to double the usual value of the prize in
recognition of the brilliance of the work.
□ One of the reasons why the Academy was so impressed by Sundman's solution was that it was a problem to which Henri Poincare had devoted much effort.
□ The rigorous solution of the three-body problem is no further advanced today than it was in Lagrange's time, and one could say that it was clearly impossible.
□ In particular, their divergences reflect different views about the mathematical model of the three-body problem; and, finally, different conceptions about the idea of 'solution' of a
physico-mathematical problem.
□ Sundman will always be known for his remarkable solution to the extremely difficult three-body problem, but he did other important work.
82. Petryshyn biography
□ In 1962, Direct and iterative methods for the solution of linear operator equations in Hilbert space was published which does much toward developing a unified point of view toward a number of
important methods of solving linear equations.
□ In the same year, The generalized overrelaxation method for the approximate solution of operator equations in Hilbert space appeared and in the following year the two papers On a general
iterative method for the approximate solution of linear operator equations and On the generalized overrelaxation method for operation equations.
□ His major results include the development of the theory of iterative and projective methods for the constructive solution of linear and nonlinear abstract and differential equations.
□ He has shown that the theory of A-proper type maps not only extends and unifies the classical theory of compact maps with some recent theories of condensing and monotone-accretive maps, but
also provides a new approach to the constructive solution of nonlinear abstract and differential equations.
□ This outstanding reference/text develops an essentially constructive theory of solvability on linear and nonlinear abstract and differential equations involving A-proper operator equations in
separable Banach spaces, treats the problem of existence of a solution for equations involving pseudo-A-proper and weakly-A-proper mappings, and illustrates their applications.
□ In this monograph we develop the generalised degree theory for densely defined A-proper mappings, and then use it to study the solubility (sometimes constructive) and the structure of the
solution set of [an] important class of semilinear abstract and differential equations ..
□ A-proper mappings arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation..
83. Qin Jiushao biography
□ Again equations of high degree appear, one problem involving the solution of the equation of degree 10.
□ [Solution: x = 3, so diameter of city is x2= 9 li] .
□ Throughout the text, in addition to the tenth degree equation above, Qin also reduces the solution of certain problems to a cubic or quartic equation which he solves by the standard Chinese
method (namely that which today is called the Ruffini-Horner method).
□ Note the solution x = 1008 is not given] .
□ His next move, therefore, is to make various passes through the mk making replacements in such a way that eventually the new moduli are pairwise coprime but the solution to the original
problem remains unchanged by the replacements.
□ He then solves each congruence and finally reassembles the answers to give the solution to the system of simultaneous congruences.
□ Shen discusses Qin's method of solution fully in [J.
□ How impressive is this work? Well suffice to say that Euler failed to provide a satisfactory solution to these problems and it was left to Gauss, Lebesgue and Stieltjes to rediscovered this
method of solving systems of congruences.
84. Stein biography
□ This theory led to important connections between harmonic analysis and probability theory, and facilitated the solution of numerous problems.
□ His explicit approximate solutions for the ∂-problems made it possible to prove sharp regularity results for solutions in strongly pseudoconvex domains.
□ Before Stein tells you his solution, the problems involved look utterly hopeless.
85. Eutocius biography
□ the account of the solutions of the problem of the duplication of the cube, or the finding of two mean proportionals, by Plato, Heron, Philon, Apollonius, Diocles, Pappus, Sporus, Menaechmus,
Archytas, Eratosthenes, Nicomedes; .
□ the fragment discovered by Eutocius himself containing the missing solution, promised by Archimedes in On the Sphere and Cylinder Book II.
□ 4, of the auxiliary problem amounting to the solution by means of conics of the cubic equation (a - x) x2 = b c2.
□ the solutions (a) by Diocles of the original problem of II.4 without bringing in the cubic, (b) by Dionysodorus of the auxiliary cubic equation.
86. Blanch biography
□ For the Mathieu equation y" + (a - 2 q cos 2x)y = 0, it is well known that certain values of a, described as characteristic values, lead to periodic solutions.
□ The author remarks, "there does not seem to appear in the literature any method for improving the accuracy of the characteristic values, except by cumbersome iteration." She then develops a
method which corrects not only an approximate characteristic value, but also the coefficients in the series for the periodic solutions.
□ Among other papers that Blanch wrote before moving to Wright Patterson Air Force Base were: (with Roselyn Siegel) Table of modified Bernoulli polynomials (1950), On the numerical solution of
equations involving differential operators with constant coefficients (1952), On the numerical solution of parabolic partial differential equations (1953) and (with Henry E Fettis) Subsonic
oscillatory aerodynamic coefficients computed by the method of Reissner and Haskind (1953).
87. Viviani biography
□ I was able to benefit from our intelligent conversations and his precious teachings and he was content that in the study of mathematics, which I had only recently begun, I could turn to his
own voice for the solution of those doubts and difficulties that I often found through the natural weakness of my intellect.
□ The first of these contained solutions to twelve geometrical problems which had been published as challenges by Cristoforo Sadler.
□ It appears that Viviani found these easy and did not consider it worthwhile to publish his solutions but Prince Leopoldo, brother of the Grand Duke, had strongly encouraged him to do so.
□ His solution involved using the intersection of four right cylinders, the bases of which are tangent to the base of the hemisphere.
88. Yamabe biography
□ Not only did Yamabe put the final touches to his solution to Hilbert's Fifth Problem in Princeton but while he was there his first child Kimiko was born and he submitted his doctoral
dissertation to Osaka University and was awarded his doctorate.
□ In the following year Yamabe published A unique continuation theorem for solutions of a parabolic differential equation written jointly with Seizo Ito.
□ It was proved by P J Heawood that the four-colouring of a given "normal" map (having 2n vertices, 3n edges, and n + 2 faces or countries) is equivalent to the solution of a system of n + 2
congruences modulo 3 for 2n unknowns (each equal to ±1, and each occurring in three of the congruences).
□ The authors have coded this problem on the UNIVAC Scientific 1103 computer, which took 25 minutes to obtain the 146 solutions in a typical case with n = 18.
89. Adian biography
□ The teacher asked everyone to solve only a couple of problems from each section, and he was immensely surprised when one of the students, Sergei Adian, handed him a thick notebook with
complete solutions, drawings included, of all the problems from Rybkin's book! It is not surprising that the Education Department of Kirovabad submitted to Baku, the capital of the Azerbaijan
Republic, a petition to send Sergei Adian to Moscow State University (MSU) to continue his education after completing his secondary school studies.
□ (Clearly, all continuous solutions of the equation are linear functions.) This result was not published at the time.
□ thesis, this new problem was more interesting, was mentioned in Kurosh's monograph, and was a difficult problem that had resisted solution by Novikov's methods.
□ Completing the project took intensive efforts from both collaborators in the course of eight years, and in 1968 their famous paper Infinite periodic groups appeared, containing a negative
solution of the problem for all odd periods n > 4381, and hence for all multiples of those odd integers as well.
90. Brahmagupta biography
□ Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate equation of the type ax + c = by.
□ For example he solves 8x2 + 1 = y2 obtaining the solutions (x, y) = (1, 3), (6, 17), (35, 99), (204, 577), (1189, 3363), ..
□ For the equation 11x2 + 1 = y2 Brahmagupta obtained the solutions (x, y) = (3, 10), (161/5, 534/5), ..
□ He also solves 61x2 + 1 = y2 which is particularly elegant having x = 226153980, y = 1766319049 as its smallest solution.
91. Bateman biography
□ Two further papers appeared in print in 1904, namely The solution of partial differential equations by means of definite integrals, and Certain definite integrals and expansions connected
with the Legendre and Bessel functions.
□ One of these 1908 papers is his first publication on transformations of partial differential equations and their general solutions.
□ In 1904 he extended Whittaker's solution of the potential and wave equation by definite integrals to more general partial differential equations.
□ The finest contribution Bateman made to mathematics, however, was his work on transformations of partial differential equations, in particular his general solutions containing arbitrary
92. Remez biography
□ Remez generalised Chebyshev-Markov characterisation theory and used it to obtain approximate solutions of differential equations.
□ He also worked on approximate solutions of differential equations and the history of mathematics.
□ The book has two parts: Part I - Properties of the solution of the general Chebyshev problem; Part II - Finite systems of inconsistent equations and the method of nets in Chebyshev
□ All of Remez's work is characterised by great skill in applying the subtlest theoretical studies to finding a numerical solution to concrete problems.
93. Aubin biography
□ The analytic problem requires one to prove the existence of a solution of a highly nonlinear (complex Monge-Ampere) differential equation.
□ Professor Aubin is widely known for his contribution to the solutions of the Calabi conjecture as well as the Yamabe problem.
□ This became the basic tool of the compactness argument for a lot of subsequent work on this equation and later lead to the solution of prescribing curvature problem with no assumption on the
symmetry of the curvature.
□ The inclusion of a large number of interesting exercises (some of them with complete solutions) enhances the educational value of this book.
94. MacDuffee biography
□ After this bit of number theory it is easy to attack the problem of finding the integral solutions of an equation having integral coefficients, and the rational solutions of an equation
having rational coefficients.
□ Let us consider for a moment the theorem that if an equation with integral coefficients has a rational solution, when this solution is expressed in lowest terms the numerator is a divisor of
the constant term of the equation.
95. Bombelli biography
□ It is unclear exactly how Bombelli learnt of the leading mathematical works of the day, but of course he lived in the right part of Italy to be involved in the major events surrounding the
solution of cubic and quartic equations.
□ He has no reservations about doing this, even though in the problems he subsequently treats he neglects possible negative solutions.
□ He then showed that, using his calculus of complex numbers, correct real solutions could be obtained from the Cardan-Tartaglia formula for the solution to a cubic even when the formula gave
an expression involving the square roots of negative numbers.
96. Taylor biography
□ For example Taylor wrote to Machin in 1712 providing a solution to a problem concerning Kepler's second law of planetary motion.
□ The paper gives a solution to the problem of the centre of oscillation of a body, and it resulted in a priority dispute with Johann Bernoulli.
□ The second version occurs as Corollary 2 to Proposition 7 and was thought of as a method of expanding solutions of fluxional equations in infinite series.
□ These include singular solutions to differential equations, a change of variables formula, and a way of relating the derivative of a function to the derivative of the inverse function.
97. Riccati Vincenzo biography
□ It probably came too late, at the end of the period of construction of the curves, when geometry has given way to algebra, and when series became the tool of choice to represent the solutions
of differential equations.
□ Vincenzo studied hyperbolic functions and used them to obtain solutions of cubic equations.
□ Al-Haytham had a less than satisfactory solution to his own problem, Christiaan Huygens found a good solution which Vincenzo Riccati and Saladini simplified and improved.
98. Tartaglia biography
□ In fact Tartaglia had also discovered how to solve one type of cubic equation since his friend Zuanne da Coi had set two problems which had led Tartaglia to a general solution of a different
type from that which Fior could solve, namely 'squares and cubes equal to numbers' or (in modern notation) x3 + ax2 = b.
□ As public lecturer of mathematics at the Piatti Foundation in Milan, he was aware of the problem of solving cubic equations, but, until the contest, he had taken Pacioli at his word and
assumed that, as Pacioli stated in the Suma published in 1494, solutions were impossible.
□ In 1545 Cardan published Artis magnae sive de regulis algebraicis liber unus, or Ars magna as it is more commonly known, which contained solutions to both the cubic and quartic equations and
all of the additional work he had completed on Tartaglia's formula.
□ For all the brilliance of his discovery of the solution to the cubic equation problem, Tartaglia was still a relatively poor mathematics teacher in Venice.
99. Forsythe biography
□ The books he wrote were: Bibliography of Russian Mathematics Books (1956); (with Wolfgang Wasow) Finite-Difference Methods for Partial Differential Equations (1967); and (with Cleve B Moler)
Computer Solution of Linear Algebraic Systems (1967).
□ The solution of partial differential equations by finite-difference methods constitutes one of the key areas in numerical analysis which have undergone rapid progress during the last decade.
□ As a result, the numerical solution of many types of partial differential equations has been made feasible.
□ The authors of this book have made an important contribution in this area, by assembling and presenting in one volume some of the best known techniques currently being used in the solution of
partial differential equations by finite-difference methods.
□ Next we present two extracts from reviews of Computer Solution of Linear Algebraic Systems.
□ The aim of this monograph is to present, at the senior-graduate level, an up-to-date account of the methods presently in use for the solution of systems of linear equations.
□ The student should have cultivated and practiced the solution of mathematical problems new to him.
100. De L'Hopital biography
□ L'Hopital had published a few brief mathematical notes, but in 1692, while Bernoulli was giving him lessons at Ouques, l'Hopital sent a solution of de Beaune's problem to Huygens.
□ Florimond de Beaune had asked for a curve for which the subtangent had a fixed length and Bernoulli had included the solution in the course he had given l'Hopital.
□ L'Hopital did not claim that the solution he sent Huygens was his own but Huygens made the reasonable assumption that it was.
□ Shortly after this l'Hopital published the solution under a pseudonym.
□ By the time Bernoulli saw the published solution he was back in Basel and, naturally enough, he was highly displeased.
□ It does appear, however, that he made few if any mathematical discoveries of his own and his solution of the brachystochrone problem was probably not his own.
□ The fact that this problem was solved independently by Newton, Leibniz and Jacob Bernoulli would put l'Hopital in very good company indeed if the solution was indeed due to him.
101. Floer biography
□ Floer developed a new method for "counting" the solutions of maximum-minimum problems arising in geometry.
□ A certain quantity called the "index" traditionally used to classify solutions was infinite, and therefore unhelpful, in many important but apparently intractable problems.
□ Andreas realized that the difference between the indices of any two solutions could still be defined and could be used where the index itself was useless.
□ Combining this observation with detailed, careful analysis, and using work of many other mathematicians as well as his own, Andreas developed a theory that led to the solution of a number of
outstanding problems.
102. Krylov Aleksei biography
□ He studied the acceleration of convergence of Fourier series in a paper in 1912, and studied the approximate solutions to differential equations in a paper published in 1917.
□ This paper On the numerical solution of the equation by which, in technical matters, frequencies of small oscillations of material systems are determined deals with eigenvalue problems.
□ is to present simple methods of composition of the secular equation in the developed form, after which, its solution, i.e.
□ Krylov's practical interests were combined with a deep understanding of the ideas and methods of classical mathematics and mechanics of the seventeenth, eighteenth, and nineteenth centuries;
and in the world of Newton, Euler, and Gauss, he found forgotten methods that were applicable to the solution of contemporary problems.
103. Roomen biography
□ This should result in tables of sines, tangents and secants, and in a solution of the circle squaring problem, which for him meant the calculation of the proportion between the circumference
and the diameter of a circle.
□ Section 11 would examine the many faulty or simply wrong solutions to the problem of squaring of the circle.
□ Viete's solution was published in 1595 and, at the end of his booklet, he proposed the Apollonian Problem of drawing a circle to touch three given circles.
□ Viete published a ruler and compass solution to the Apollonian Problem in 1600 which greatly impressed van Roomen [Descartes\'s Mathematical Thought (Springer, New York, 2003).',2)">2]:- .
104. Haselgrove biography
□ Some of the applications of computers were immediately obvious; many well-formulated problems in science and engineering which required numerical solutions could benefit directly from faster
□ This work involves the elucidation of classes of mathematical problems which are suitable for solution by a standardised approach.
□ The solution methods have to be studied analytically and tested on an extensive range of problems, to determine their applicability and limitations.
□ In The solution of non-linear equations and of differential equations with two-point boundary conditions (1961) Haselgrove suggests general iterative techniques, based on an n-dimensional
extension of the Newton-Raphson process.
105. Olech biography
□ He was not curious about the details, but he asked me to write out my solution, which of course I did.
□ Wazewski had there presented his ideas of applying the topological notion of a retract to the study of the solutions of differential equations and Lefschetz had seen the idea as being one of
the most significant advances in the study of differential equations.
□ In particular he worked with Philip Hartman and they published the joint paper On global asymptotic stability of solutions of differential equations (1962).
□ He also solved very important problems concerning autonomous systems on the plane with stable Jacobian matrix at each point of the plane and applied the Wazewski topological method in
studying the asymptotic behaviour of solutions of differential equations.
106. Kloosterman biography
□ Kloosterman was examining the number of solutions in integers xn, to the equation .
□ He had managed to find, provided s ≥ 5 and the an satisfy suitable congruence conditions, an asymptotic formula for the number of solution to the equation (*).
□ Under these conditions (1) always has a solution for large values of m.
□ His solution of this case appeared in his paper On the representation of numbers in the form ax2 + by2 + cz2 + dt2 which was published in Acta Mathematica in 1926.
107. Martin biography
□ Artemas Martin, a self-taught mathematician whose activity covered almost six decades, has been described as 'a unique example of what an inherent love of the solution of mathematical
problems can do to a man even if he has not the advantages of advanced schooling'.
□ With his library there is a number of notebooks which contain his solutions to mathematical problems and well as lists of books that he was trying to purchase.
□ Martin published a very large number of problems and solutions to problems in a wide range of publications [American National Biography 14 (Oxford, 1999), 587.',3)">3]:- .
□ has that rare and happy faculty of presenting his solutions in the simplest mathematical language, so that those who have mastered the elements of the various branches of mathematics, are
able to understand his reasoning.
108. Colson biography
□ Notice that (a - b)3 + 3ab(a - b) = a3 - b3 so if a and b satisfy ab = m and a3 - b3 = 2n then a - b is a solution of x3 + 3mx = 2n.
□ Colson tested each of these 9 possible solutions to see if it satisfies the original equation, and was able to identify the three actual solutions.
□ This paper by Colson was the first to give all three solutions to a cubic equation.
109. Ladyzhenskaya biography
□ As in the previous decade, during the 1960s she continued obtaining results about existence and uniqueness of solutions of linear and quasilinear elliptic, parabolic, and hyperbolic partial
differential equations.
□ At the start of the last century Sergei Bernstein proposed an approach to the study of the classical solvability of boundary-value problems for equations based on a priori estimates for
solutions as well as describing conditions that are necessary for such solvability.
□ They developed a complete theory for the solvability of boundary-value problems for uniformly parabolic and uniformly elliptic quasilinear second-order equations and of the smoothness of
generalized solutions.
□ One result gave the solution of Hilbert's 19th problem for one second-order equation.
110. Sobolev biography
□ published a number of profound papers in which he put forward a new method for the solution of an important class of partial differential equations.
□ Working with Smirnov, Sobolev studied functionally invariant solutions of the wave equation.
□ These methods allowed them to find closed form solutions to the wave equation describing the oscillations of an elastic medium.
□ These ideas in the main concern generalised solutions of non-classical boundary value problems.
111. Mikhlin biography
□ Dealing with the plane elasticity problem, he proposed two methods for its solution in multiply connected domains.
□ The second method is a certain generalisation of the classical Schwarz algorithm for the solution of the Dirichlet problem in a given domain by reducing it to simpler problems in smaller
domains whose union is the original one.
□ He studied the error of the approximate solution for shells, similar to plane plates, and found out that this error is small for the so-called purely rotational state of stress.
□ When applied to the variational method, this notion enabled him to state necessary and sufficient conditions in order to minimise errors in the solution of the given problem when the error
arising in the numerical construction of the algebraic system resulting from the application of the method itself is sufficiently small, no matter how large is the system's order.
□ Mikhlin also studied the finite element approximation in weighted Sobolev spaces related to the numerical solution of degenerate elliptic equations.
□ He found the optimal order of approximation for some methods of solution of variational inequalities.
□ Algorithm error: is the intrinsic error of the algorithm used for the solution of the approximating problem.
112. Gregory biography
□ However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood:
Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor;
he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general
cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a
definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is
impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an
algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could
not be solved by radicals.
□ James Gregory's manuscripts on algebraic solutions of equations .
113. Babuska biography
□ His first papers, all written in Czech, were Welding stresses and deformations (1952), Plane elasticity problem (1952), A contribution to the theoretical solution of welding stresses and some
experimental results (1953), A contribution to one method of solution of the biharmonic problem (1954), Solution of the elastic problem of a half-plane loaded by a sequence of singular forces
(1954), (with L Mejzlik) The stresses in a gravity dam on a soft bottom (1954), On plane biharmonic problems in regions with corners (1955), (with L Mejzlik) The method of finite differences
for solving of problems of partial differential equations (1955), and Numerical solution of complete regular systems of linear algebraic equations and some applications in the theory of
frameworks (1955).
□ Basically, the mathematical problem Babuska's group had to solve was to find a numerical solution to a nonlinear partial differential equation.
□ An original contribution is the axiomatic construction of the fundamentals of plane elasticity, the accuracy and generality of the mathematical procedures and some new numerical methods of
□ His next important book, published in collaboration with Milan Prager and Emil Vitasek in 1964, was Numerical Solution of Differential Equations (Czech).
114. Schoen biography
□ For the former, an outline is given of the recent solution of Yamabe's conjecture (that every metric on a compact manifold is pointwise conformally equivalent to one with constant scalar
curvature), including the use of the positive mass theorem and a discussion of regularity of weak solutions of Yamabe's equation.
□ The solution of the Yamabe problem on compact manifolds, which Schoen discussed in this lecture, is one of his greatest achievements.
□ for his work on the application of partial differential equations to differential geometry, in particular his completion of the solution to the Yamabe Problem in "Conformal deformation of a
Riemannian metric to constant scalar curvature".
115. Al-Samawal biography
□ In Book 2 of al-Bahir al-Samawal describes the theory of quadratic equations but, rather surprisingly, he gave geometric solutions to these equations despite algebraic methods having been
fully described by al-Khwarizmi, al-Karaji, and others.
□ Al-Samawal also described the solution of indeterminate equations such as finding x so that a xn is a square, and finding x so that axn + bxn-1 is a square.
□ In Book 4 al-Samawal also classifies problems into necessary problems, namely ones which can be solved; possible problems, namely ones where it is not known whether a solution can be found or
not, and impossible problems which [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',3)">3]:- .
□ if one could assume the existence of their solution, this existence would lead to an absurdity.
116. Nirenberg biography
□ Some other highlights are his research on the regularity of free boundary problems with [David] Kinderlehrer and [Joel] Spuck, existence of smooth solutions of equations of Monge-Ampere type
with [Luis] Caffarelli and Spuck, and singular sets for the Navier-Stokes equations with Caffarelli and [Robert] Kohn.
□ His study of symmetric solutions of non-linear elliptic equations using moving plane methods with [Basilis] Gidas and [Wei Ming] Ni and later with [Henri] Berestycki, is an ingenious
application of the maximum principle.
□ This is used in Chapters III and IV in the discussion of bifurcation theory (the highlight being a complete proof of Rabinowitz' global bifurcation theorem) and the solution of nonlinear
partial differential equations (the highlight being the global theorem of Landesman and Lazer).
□ Caffarelli mentions Nirenberg's areas of interest in partial differential equations: Regularity and solvability of elliptic equations of order 2n; the Minkowski problem and fully nonlinear
equations; the theory of higher regularity for free boundary problems; and symmetry properties of solutions to invariant equations.
117. Kumano-Go biography
□ During these years Kumano-Go published a series of papers which studied the local and global uniqueness of the solutions of the Cauchy problem for partial differential equations.
□ In two papers Kumano-Go also studied non-uniqueness of solutions of the Cauchy problem.
□ the construction of the fundamental solution of a first order hyperbolic system and the study of the wave front sets of solutions.
118. Fichera biography
□ In 1949 he published the important paper Analisi esistenziale per le soluzioni dei problemi al contorno misti, relativi all'equazione e ai sistemi di equazioni del secondo ordine di tipo
ellittico, autoaggiunti on uniqueness and existence of solutions of certain mixed boundary value problems.
□ In pure mathematics Gaetano Fichera achieved considerable results in the following fields: mixed boundary value problems of elliptic equations; generalized potential of a simple layer; second
order elliptic-parabolic equations; well posed problems; weak solutions; semicontinuity of quasi-regular integrals of the calculus of variations; two-sided approximation of the eigenvalues of
a certain type of positive operators and computation of their multiplicity; uniform approximation of a complex function f(z); extension and generalization of the theory for potentials of
simple and double layer; specification of the necessary and sufficient conditions for the passage to the limit under integral sign for an arbitrary set; analytic functions of several complex
variables; solution of the Dirichlet problem for a holomorphic function in a bounded domain with a connected boundary, without the strong conditions assumed by Francesco Severi in a former
study; construction of a general abstract axiomatic theory of differential forms; convergence proof of an approximating method in numerical analysis and explicit bounds for the error.
□ concern the existence, uniqueness and regularity of solutions.
119. Gleason biography
□ In his talk he sketched a possible approach to the solution of Hilbert's fifth problem, emphasizing the importance of one-parameter (local) subgroups in a locally Euclidean group G.
□ Then, in 1952, Gleason's paper Groups without small subgroups taken together with the results of Montgomery and Zippin, and Yamabe, gave a complete solution to Hilbert's problem.
□ Gleason won the Newcomb Cleveland Prize from the American Association for the Advancement of Science for his contribution to the solution of the problem.
□ In 1980 Gleason, together with R E Greenwood and L M Kelly, published The William Lowell Putnam Mathematical Competition which gave all the problems and their solutions from the beginning of
the competition in 1938 up to 1964.
120. Varadhan biography
□ To Daniel Stroock and Srinivasa Varadhan for their four papers 'Diffusion processes with continuous coefficients I and II' (1969), 'On the support of diffusion processes with applications to
the strong maximum principle (1970), Multidimensional diffusion processes (1979), in which they introduced the new concept of a martingale solution to a stochastic differential equation,
enabling them to prove existence, uniqueness, and other important properties of solutions to equations which could not be treated before by purely analytic methods; their formulation has been
widely used to prove convergence of various processes to diffusions.
□ In his landmark paper 'Asymptotic probabilities and differential equations' in 1966 and his surprising solution of the polaron problem of Euclidean quantum field theory in 1969, Varadhan
began to shape a general theory of large deviations that was much more than a quantitative improvement of convergence rates.
□ A striking application is their solution of a conjecture of Mark Kac concerning large time asymptotics of a tubular neighbourhood of the Brownian motion path, the so-called 'Wiener sausage'.
121. Dudeney biography
□ Click Solution Player one will fail if K tries to capture WP and H tries to capture BP.
□ It is a parity problem.')">HERE for the solution.
□ Other puzzles simply reduced to systems of linear equations if a mathematical solution was sought.
□ How could he have done it? There is no necessity to give measurements, for if the smaller piece (which is half a square) be made a little too large or small, it will not effect the method of
□ Click Solution The problem can be solved with only two cuts, creating five pieces as shown.')">HERE for the solution.
122. Al-Tusi Sharaf biography
□ He then looks at eight types of cubic equation which always have a positive solution, then five types which may have no positive solution.
□ We illustrate the method by showing how al-Tusi examined one of the five types of equation which under certain conditions has a solution, namely the equation x3 + a = bx, where a, b are
□ We use, of course, modern notation to make the solution easy to understand, while al-Tusi would express all his mathematics in words.
□ Al-Tusi's first comment is that if t is a solution to this equation then t3 + a = bt and, since a > 0, t3 < bt so t < √b.
□ Thus the equation bx - x3 = a has a solution if a ≤ 2(b/3)3/2.
123. Trahtman biography
□ My first publications were [in semigroup theory], in particular, the solution of the Tarski problem ..
□ We present a solution of the road colouring problem.
□ Quite often it just requires a cunning new way to think about the problem and a solution drops out.
□ Unlike Fermat's Last Theorem which required lots of technical and sophisticated mathematical techniques to crack, the maths behind Trakhtman's solution is not complicated, it just required a
clever new way to look for the solution.
□ Now he's working on a real algorithm to implement his solution.
124. Carcavi biography
□ (2) The equation x3 + y3 = z3 has no solutions in integers.
□ (3) The equation y2 + 2 = x3 admits no solutions in integers except x = 3, y = 5.
□ (4) The equation y2 + 4 = x3 admits no solutions in integers except x = 2, y = 2 and x = 5, y = 11.
□ Pascal published a challenge under the name of Dettonville offering two prizes for solutions to these problems, and he lodged the prizes together with his own solutions with Carcavi.
□ He asked Carcavi and Roberval to judge the solutions submitted showing his respect for Carcavi's mathematical abilities.
125. Keldysh Mstislav biography
□ The paper gives a very clear and concise exposition of various recent results concerning the solvability of Dirichlet's problem and also the stability of the solution when the boundary of the
domain varies.
□ In Chapter I the author gives an exposition of the generalized solution of Dirichlet's problem in the sense of Wiener, and discusses the notions of regular and irregular points.
□ Chapter III is devoted to Wiener's criterion for regularity of a point, and to discussion of the behaviour of the solution at an irregular point.
□ Chapter IV contains an exposition of the harmonic measure and integral representation of the generalized solution.
□ Here he was able to use his experience in solving the flutter problem and his solution to shimmy, together with detailed instructions to engineers on how to overcome the problem, was
described in Shimmy of the front gear of the three wheels undercart (1945).
□ He was named Hero of the Socialist Labour in 1956 for his solution to defence problems and received the Lenin Prize in the following year.
126. Li Zhi biography
□ To solve the above equation Li Zhi would bring the leading coefficient to -1 and then give the solution; in this case 20.
□ The type of problem which worried mathematicians in Islamic countries, and in Europe, concerning the solution of cubic, quartic, and higher order equations did not seem to arise in China.
□ If we examine Li Zhi's solution closely we see a remarkable depth of understanding of equations.
□ Knowing that the solution cannot be a negative number (x = -16), Li Zhi works with the cubic factor and solves that to find the solution.
□ He gives the solution 20 pu which is the diameter of the pond.
127. Crank biography
□ His main work was on the numerical solution of partial differential equations and, in particular, the solution of heat-conduction problems.
□ John Crank is best known for his joint work with Phyllis Nicolson on the heat equation, where a continuous solution u(x, t) is required which satisfies the second order partial differential
equation .
□ They considered numerical methods which find an approximate solution on a grid of values of x and t, replacing ut(x, t) and uxx(x, t) by finite difference approximations.
□ Richardson's method yielded a numerical solution which was very easy to compute, but alas was numerically unstable and thus useless.
□ Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.
128. Samoilenko biography
□ His most original contribution was the numeric-analytic method for the study of periodic solutions of differential equations with periodic right hand side.
□ They wrote Numerical-analytic methods for the study of periodic solutions (Russian) in 1976 and followed this with a new work on similar topics entitled Numerical-analytic methods for
investigating the solutions of boundary value problems (Russian) in 1986.
□ In this monograph we present new promising directions in the development of numerical-analytic methods for studying the solutions of nonlinear boundary value problems in the case of a general
form of boundary conditions, problems with controlling parameters, and also boundary value problems for impulse systems.
□ We give the solutions of typical problems in a course on ordinary differential equations.
129. Dantzig George biography
□ Using hand-operated desk calculators, approximately 120 man-days were required to obtain a solution.
□ The particular problem solved was one which had been studied earlier by George Stigler (who later became a Nobel Laureate) who proposed a solution based on the substitution of certain foods
by others which gave more nutrition per dollar.
□ He did not claim the solution to be the cheapest but gave his reasons for believing that the cost per annum could not be reduced by more than a few dollars.
□ Indeed, it turned out that Stigler's solution (expressed in 1945 dollars) was only 24 cents higher than the true minimum per year $39.69.
□ His work permits the solution of many previously intractable problems and has made linear programming into one of the most frequently used techniques of modern applied mathematics.
130. Leray biography
□ This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.
□ He studied solutions of the initial value problem for three-dimensional Navier-Stokes equations.
□ He examined not only the existence and uniqueness of solutions but he showed that the solutions remained smooth for only a finite time after which turbulent solutions arise.
131. Orszag biography
□ This book contains a wealth of solved problems and of techniques for approximating the exact solutions.
□ The title of this volume is somewhat misleading in that the subjects discussed are approximate analytic solutions of ordinary differential and difference equations, and no other topics are
□ There are many examples and exercises, and a welcome feature is the large number of diagrams which compare the approximate solutions with the known solutions of some of the problems
□ It is not only the authors who hope for a similar book on approximate solutions of partial differential equations.
132. Boruvka biography
□ To many people Boruvka is best known for his solution of the Minimal Spanning Tree problem which he published in 1926 in two papers On a certain minimal problem (Czech) and Contribution to
the solution of a problem of economical construction of electrical networks (Czech).
□ He spent 1926-27 in Paris, where he lectured on his solution to the Minimal Spanning Tree problem, then returned to Masaryk University in Brno where he habilitated and was made a dozent in
□ And that is why I came with the idea that the solution of that problem was possible only in the following way that in the first period one would acquire some experience in the simplest cases
and only in the second period, on the basis of the concepts introduced and experience obtained, one would go to the extension of those results to the most general case.
□ I would like to remember facts in my life that were essential not only for me personally, but chiefly for mathematics and for the future mathematical generation: Before every serious task I
try to find carefully and dutifully how to fulfil it in the best way, and when I find a solution, I carry it out as best as I can according to my best sense and conscience and with all my
133. Hall Marshall biography
□ This mathematical problem had been studied since about 1893, but the solution to the 92 by 92 matrix was unproven until 1961 because it required extensive computation.
□ Perhaps his best known result in group theory is his solution of the Burnside problem for groups of exponent 6.
□ He outlined his proof in Solution of the Burnside problem of exponent 6 (1957) and gave full details in the 22-page paper Solution of the Burnside problem of exponent six (1958).
□ John Thompson, then a graduate student at the University of Chicago, persuaded Saunders Mac Lane to invite me to talk on this subject [his solution of the Burnside problem for exponent 6].
134. Fuchs biography
□ He discussed problems of the following kind: What conditions must be placed on the coefficients of a differential equation so that all solutions have prescribed proberties (e.g.
□ He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane.
□ In a series of papers (1880-81) Fuchs studied functions obtained by inverting the integrals of solutions to a second-order linear differential equation in a manner generalising Jacobi's
inversion problem.
□ Fuchs also investigated how to find the matrix connecting two systems of solutions of differential equations near two different points.
□ In this interesting paper Gray also discusses the relationships between Fuchs' ideas and his mathematical tools, and illustrates how solutions of some problems led Fuchs to the study of
further problems.
135. Golub biography
□ in 1959 for his thesis The Use of Chebyshev Matrix Polynomials in the Iterative Solution of Linear Equations Compared to the Method of Successive Overrelaxation which developed ideas in a
paper by von Neumann.
□ In 1980 Golub lectured on the numerical solution of large linear systems at a summer school in France.
□ A large part of scientific computing is concerned with the solution of differential equations, and thus differential equations are an appropriate focus for an introduction to scientific
□ The need to solve differential equations was one of the original and primary motivations for the development of both analog and digital computers, and the numerical solution of such problems
still requires a substantial fraction of all available computing time.
□ Although there are many existing packages for such problems, or at least for the main subproblems such as the solution of linear systems of equations, we believe that it is important for
users of such packages to understand the underlying principles of the numerical methods.
136. Mazur biography
□ We discussed problems proposed right there, often with no solution evident even after several hours of thinking.
□ We found a solution to a problem involving infinite dimensional vector spaces.
□ As with many of the problems in the Scottish Book the proposer would offer a prize for their solution.
□ Per Enflo showed in 1972 that the problem had a negative solution and, while in Warsaw lecturing on his solution, Mazur presented him with his prize, the live goose! .
137. Friedmann biography
□ In his last year at the University he was working on an essay on the subject I assigned: 'Find all orthogonal substitutions such that the Laplace equation, transformed for the new variables,
admits particular solutions in the form of a product of two functions, one of which depends only on one, and the other on the other two variables'.
□ In January of this year, Mr Friedmann submitted to me an extensive study of about 130 pages, in which he gave a quite satisfactory solution of the problem.
□ In reality it turns out that the solution given in it does not satisfy the field equations.
138. Fatou biography
□ Although not giving a complete solution, Fatou's work also made a major contribution to finding a solution to the related question of whether conformal mapping of Jordan regions onto the open
disc can be extended continuously to the boundary.
□ Using existance theorems for the solutions to differential equations, Fatou was able to prove rigorously certian results on planetary orbits which Gauss had suggested by only verified with an
intuitive argument.
139. McCowan biography
□ A regular attendee at meetings of the Edinburgh Mathematical Society, he presented the papers: On a representation of elliptic integrals by curvilinear arcs (12 June 1891); On the solution of
non-linear partial differential equations of the second order (13 May 1892); and Note on the solution of partial differential equations by the method of reciprocation (11 November 1892).
□ J McCowan of University College at Dundee discussed this topic [waves] more fully and arrived at exact and complete solutions for certain cases.
140. Appell biography
□ In 1878 he noted the physical significance of the imaginary period of elliptic functions in the solution of the pendulum which had been though to be purely a mathematical curiosity.
□ Appell submitted a solution which won second place.
□ his scientific work consists of a series of brilliant solutions of particular problems, some of the greatest difficulty.
141. Perelman biography
□ He returned to St Petersburg at the end of April 2002 and, in July, put Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, the third instalment of his
work, on the web.
□ In March 2010 the Clay Mathematics Institute announced that Perelman had met the conditions for the award of one million US dollars which they had offered for the solution of the Poincare
□ I consider that the American mathematician Hamilton's contribution to the solution of the problem is no less than mine.
142. Dupre biography
□ Athanase Dupre has determined the surface tensions of solutions of soap by different methods.
□ A statistical method gives for one part of common soap in 5000 of water a surface tension about one-half as great as for pure water, but if the tension be measured on a jet close to the
orifice, the value (for the same solution) is sensibly identical with that of pure water.
□ He explains these different values of the surface tension of the same solution as well as the great effect on the surface tension which a very small quantity of soap or other trifling
impurity may produce, by the tendency of the soap or other substance to form a film on the surface of the liquid.
143. Woods biography
□ This work led to two papers in 1950: Improvements to the accuracy of arithmetical solutions to certain two-dimensional field problems and The two-dimensional subsonic flow of an inviscid
fluid about on aerofoil of arbitrary shape.
□ He now published a whole series of papers - the next two were: A new relaxation treatment of flow with axial symmetry (1951), and The numerical solution of two-dimensional fluid motion in the
neighbourhood of stagnation points and sharp corners (1952).
□ If F is harmonic or is a solution to Poisson's equation, it may have singular points in the field or on the boundary at which it (a) has finite values, but has infinite derivatives, (b) has
logarithmic infinities, or (c) has simple discontinuities.
144. Sintsov biography
□ Of course through his many years of research his interests varied but the main areas on which he worked were the theory of conics and applications of this geometrical theory to the solution
of differential equations and, perhaps most important of all, the theory of nonholonomic differential geometry.
□ Sintsov gave in 1903 an elegant proof of its general real solution, which has the form f (x, y) = q(x) - q(y), where q is an arbitrary function in one variable.
□ in 1903) elementary simple proofs of its general real solutions.
145. Sneddon biography
□ The book discusses applications of Fourier, Mellin, Laplace and Hankel transforms to the solution of problems in physics and engineering.
□ It is a major text containing around 550 pages and is mainly concerned with applications which involve the solution of ordinary differential equations, and boundary value and initial value
problems for partial differential equations.
□ The aim of this book is to present the elements of the theory of partial differential equations in a form suitable for the use of students and research workers whose main interest in the
subject lies in finding solutions of particular equations rather than in the general theory.
146. Horner biography
□ This discussion is somewhat moot because the method was anticipated in 19th century Europe by Paolo Ruffini (it won him the gold medal offered by the Italian Mathematical Society for Science
who sought improved methods for numerical solutions to equations), but had, in any case, been considered by Zhu Shijie in China in the thirteenth century.
□ It is also worth noting that he gave a solution to what has come to be known as the "butterfly problem" which appeared in The Gentleman's Diary for 1815 [Math.
□ The butterfly problem, whose name becomes clear on looking at the figure, has led to a wide range of interesting solutions.
147. Pascal biography
□ Pascal published a challenge offering two prizes for solutions to these problems to Wren, Laloubere, Leibniz, Huygens, Wallis, Fermat and several other mathematicians.
□ Wallis and Laloubere entered the competition but Laloubere's solution was wrong and Wallis was also not successful.
□ Pascal published his own solutions to his challenge problems in the Letters to Carcavi.
148. Fejer biography
□ Encouraged by Maksay, Fejer began submitting his solutions to the problems to Budapest [The Mathematical Intelligencer 15 (2) (1993), 13-26.',6)">6]:- .
□ Laszlo Racz, a secondary school teacher who led a problem study group in Budapest, often opened his session by saying, "Lipot Weiss has again sent in a beautiful solution." .
□ Poisson's integral provides a valid solution for Dirichlet's problem for the circle.
149. Wald biography
□ Wald reported to the seminar on his work in econometrics, in particular he wrote a paper for the seminar on the existence of a solution to the competitive economic model.
□ He proved important results, perhaps the most significant being the existence of a solution to the competitive economic model which, as we noted above was written for Menger's seminar.
□ seasonal corrections to time series, approximate formulas for economic index numbers, indifference surfaces, the existence and uniqueness of solutions of extended forms of the Walrasian
system of equations of production, the Cournot duopoly problem, and finally, in his much used work written with Mann (1943), stochastic difference equations.
150. Prthudakasvami biography
□ The solution of a first-degree indeterminate equation by a method called kuttaka (or "pulveriser") was given by Aryabhata I.
□ This method of finding integer solutions resembles the continued fraction process and can also be seen as a use of the Euclidean algorithm.
□ Brahmagupta seems to have used a method involving continued fractions to find integer solutions of an indeterminate equation of the type ax + c = by.
151. Roach biography
□ His early papers were: On the approximate solution of elliptic, self adjoint boundary value problems (1967); Fundamental solutions and surface distributions (1968); Approximate Green's
functions and the solution of related integral equations (1970); and (jointly with Robert A Adams) An intrinsic approach to radiation conditions (1972).
152. Kochin biography
□ The seminar participants also reviewed works containing solutions to specific problems related to the frontal model.
□ Soon Kochin's article was published, specifying Defant's solution.
□ He gave the solution to the problem of small amplitude waves on the surface of an uncompressed liquid in the paper Towards a Theory of Cauchy-Poisson Waves (Russian) in 1935.
153. Peano biography
□ In 1886 Peano proved that if f (x, y) is continuous then the first order differential equation dy/dx = f (x, y) has a solution.
□ The existence of solutions with stronger hypothesis on f had been given earlier by Cauchy and then Lipschitz.
□ Four years later Peano showed that the solutions were not unique, giving as an example the differential equation dy/dx = 3y2/3 , with y(0) = 0.
154. Abu Kamil biography
□ The Book on algebra by Abu Kamil is in three parts: (i) On the solution of quadratic equations, (ii) On applications of algebra to the regular pentagon and decagon, and (iii) On Diophantine
equations and problems of recreational mathematics.
□ Rather it presents a number of rules, some of which are far from easy, each given for the numerical solution of a geometric problem.
□ The Book of rare things in the art of calculation is concerned with solutions to indeterminate equations.
155. Jeffery biography
□ He did one years teacher training in 1911 but he was already undertaking research and his first paper On a form of the solution of Laplace's equation suitable for problems relating to two
spheres was read to the Royal Society in 1912.
□ He made effective use of Whittaker's general solution to Laplace's equation which Whittaker found in 1903.
□ Jeffery also worked on general relativity and produced exact solutions to Einstein's field equations in certain special cases.
156. Bondi biography
□ The correct number of solutions to the problem is 332.
□ Bondi defines when two solutions are considered equivalent, then finds 123 distinct solution.
157. Penrose biography
□ In this paper Penrose defined a generalized inverse X of a complex rectangular (or possibly square and singular) matrix A to be the unique solution to the equations AXA = A, XAX = X, (AX)T =
AX, (XA)T = XA.
□ In the following year Penrose published On best approximation solutions of linear matrix equations which used the generalized inverse of a matrix to find the best approximate solution X to AX
= B where A is rectangular and non-square or square and singular.
158. Euler biography
□ Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem.
□ He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of
variation of constants, integrating factors, a method of approximating solutions, and many others.
159. Gershgorin biography
□ In 1929 Gershgorin published On electrical nets for approximate solution of the differential equation of Laplace (Russian) in which he gave a method for finding approximate solutions to
partial differential equations by constructing a model based on networks of electrical components.
□ L Lichtenstein [in 'Zur Theorie der konformen Abbildung: Konforme Abbildung nicht-analytischer, singularitatenfreier Flachenstucke auf ebene Gebiete' (1916)] had reduced that important
problem to the solution of a Fredholm integral equation.
160. Gray Marion biography
□ Marion C Gray, A modification of Hallen's solution of the antenna problem, J.
□ Marion C Gray and S A Schelkunoff, The approximate solution of linear differential equations.
□ Various papers by Gray were read to the Society: The equation of telegraphy (which appeared in volume 42 of the Proceedings and she read to the meeting of the Society in November 1923), The
equation of conduction of heat (which also appeared in volume 42 of the Proceedings), and On the equation of heat (which appeared as Particular solutions of the equation of conduction of heat
in one dimension in volume 43 of the Proceedings).
161. Ferrers biography
□ His first book was "Solutions of the Cambridge Senate House Problems, 1848 - 51".
□ Not at all - Ferrers contribution was the method of solution.
□ We see that there is a 1-1 correspondence between a partition and its conjugate and this 1-1 correspondence provides the solution to the problem stated by Sylvester (and stated by Adams in
the Tripos paper of 1847).
162. Runge biography
□ Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients.
□ There were three standard methods for the numerical solution of such equations, namely by Newton, Bernoulli and Graffe, and the method found by Runge had all three of the standard methods as
special cases.
□ He worked out many numerical and graphical methods, gave numerical solutions of differential equations, etc.
163. Macaulay biography
□ Such problems have no complete solution, but Macaulay looks for structural properties of the set of solutions.
□ He also contributed a number of articles: Bolyai's science of absolute space (1900), On continued fractions (1900), Projective geometry (1906), On the axioms and postulates employed in the
elementary plane constructions (1906), On a problem in mechanics and the number of its solutions (1906), and Some inequalities connected with a method of representing positive integers
164. Cimmino biography
□ Cimmino was only nineteen years old when he graduated with his thesis on approximate methods of solution for the heat equation in 2-dimensions, but he was appointed as an assistant to Picone
who held the chair of analytical geometry at the University of Naples.
□ Towards the end of that period, Professor Cimmino devised a numerical method for the approximate solution of systems of linear equations that he reminded me of in these days, following the
recent publication by Dr Cesari ..
□ We have seen that Cimmino made contributions to partial differential equations of elliptic type and to computing approximate solutions to systems of linear equations.
165. Fibonacci biography
□ Three of these problems were solved by Fibonacci and he gives solutions in Flos which he sent to Frederick II.
□ And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.
□ Without explaining his methods, Fibonacci then gives the approximate solution in sexagesimal notation as 1.22.7.42.33.4.40 (this is written to base 60, so it is 1 + 22/60 + 7/602 + 42/603 +
166. Schelp biography
□ Interaction with Paul Erdős started in 1972 as a result of a solution of an Erdős-Bondy problem on Ramsey numbers for cycles.
□ Enthusiasm in talking about it, in trying to solve a maths problem, in appreciating others' solutions, in taking part of the whole process of ups and downs culminating in a solution.
167. Voevodsky biography
□ He continued to work on ideas coming from Grothendieck and in 1991 published Galois representations connected with hyperbolic curves (Russian) which gave partial solutions to conjectures of
Grothendieck, made on nonabelian algebraic geometry, contained in his 1983 letter to Faltings and also in his unpublished 'Esquisse d'un programme' mentioned above.
□ In the present paper, the authors offer a very different solution to the problem of providing an algebraic formulation of singular cohomology with finite coefficients.
□ One consequence of Voevodsky's work, and one of his most celebrated achievements, is the solution of the Milnor Conjecture, which for three decades was the main outstanding problem in
algebraic K-theory.
168. Malgrange biography
□ The idea of finding an elementary solution for all differential operators with constant coefficients might seem a bit far fetched.
□ However, I had made the suggestion of the existence of such a solution using the theory of distributions.
□ Malgrange was awarded his doctorate in 1955 from the Universite Henri Poincare at Nancy for his thesis Existence et approximation des solutions des equations aux derivees partielles et des
equations de convolution.
169. Romberg biography
□ I submitted the solution and received the following response: "The assignment was completely solved by the sender.
□ His only solution was to fly from Prague and so avoid the German checks.
□ It is also indicated how the method may reduce the labour for obtaining solutions of the analytical eigenvalue problem.
170. Osgood biography
□ Osgood's main work was on the convergence of sequences of continuous functions, solutions of differential equations, the calculus of variations and space filling curves.
□ In 1898 Osgood published an important paper on the solutions of the differential equation dy/dx = f(x, y) satisfying the prescribed initial conditions y(a) = b.
□ Osgood showed that if f(x, y) is merely continuous there exists at least one solution ..
171. Box biography
□ One feature of particular interest is practical discussion of genuinely nonlinear fitting problems and their solution with the help of tact and a special, publicly available, IBM-704 program.
□ The authors - all statistical practitioners themselves - take a fresh approach to statistics oriented toward the solution of problems in the physical, engineering, biological and social
□ The authors typically start with the statement of a problem faced by an experimenter, and then present one or more possible solutions, stating clearly the assumptions required for the
validity of each.
172. Cramer Harald biography
□ One interesting paper by Cramer over this period which we should note is one he published in 1920 discussing prime number solutions x, y to the equation ax + by = c, where a, b, c are fixed
□ Note that if a = b = 1 then the question of whether this equation has a solution for all c is Goldbach's conjecture, while if a = 1, b = -1, c = 2, then the question about prime solutions to
x = y + 2 is the twin prime conjecture.
173. Lopatynsky biography
□ His research interests then moved towards differential equations with his first paper on this topic Solution of the equation y ' = f (x, y) published in 1939, proving a general existence
□ We consider the basic methods of solving differential equations and methods of qualitative investigation of these solutions.
□ We illustrate the theoretical material with an analysis of the solution of many examples.
174. Warschawski biography
□ He published The convergence of expansions resulting from a self-adjoint boundary problem in the Duke Mathematical Journal in 1940, jointly with A S Galbraith, which studied a problem of the
Riesz-Fischer type concerning the expansion of a function with n derivatives in terms of the characteristic solutions of a self-adjoint boundary value problem of the second order.
□ The first was a single author paper On the solution of the Lichtenstein-Gershgorin integral equation in conformal mapping.
□ Theory while the second, On the solution of the Lichtenstein-Gershgorin integral equation in conformal mapping.
175. Picard Emile biography
□ He used methods of successive approximation to show the existence of solutions of ordinary differential equations solving the Cauchy problem for these differential equations.
□ Picard's solution was represented in the form of a convergent series.
□ He studied the transmission of electrical pulses along wires finding a beautiful solution to the problem.
176. Seitz biography
□ Finkel writes [A mathematical solution book (1893), 440-441.',6)">6]:- .
□ The more difficult the question, the more determined was he to master it, and from [1872 to 1879], I never knew him to fail in the solution of any problem he undertook.
□ Seitz, although he died at the age of 37, contributed over 500 published problems and solutions in the Analyst, the Mathematical Visitor, the Mathematical Magazine, the School Visitor, and
the Educational Times of London.
177. Biot biography
□ Having discovered these laws he used them in analysis of saccharine solutions using an instrument called a polarimeter which he invented.
□ For this work on the polarisation of light passing through chemical solutions he was awarded the Rumford Medal of the Royal Society of London in 1840.
□ Arago supported the Daguerre photographic process with silver plates while Biot championed an approach with paper soaked in a silver solution as developed by Henry Fox Talbot.
178. Chaplygin biography
□ He published a famous paper On gas streams in 1902 giving exact solutions to many cases of noncontinuous flow of a compressible gas.
□ Three decades later, however, Chaplygin's dissertation served as a starting point for many studies by aerodynamics specialists and provided the basis for the solution of problems of subsonic
□ This postulate - the so-called Chaplygin-Zhukovsky postulate - gives a complete solution to the problem of the forces exerted by a stream on a body passing through it.
179. Zelmanov biography
□ Let me explain the background to the restricted Burnside problem, the solution of which was the main reason for the award of the Medal, and also explain how Zelmanov, not a group theorist by
training, came to solve one of the most fundamental questions in group theory.
□ This is equivalent to saying that a positive solution to the Restricted Burnside problem would show that there are only finitely many finite factor groups of B(d, n).
□ The General Burnside problem was shown to have a negative solution by Golod in 1964.
□ The greatest early contribution to the Restricted Burnside problem was by Hall and Higman in 1956 where they showed that, if the Schreier conjecture holds, then the Restricted Burnside
problem has a positive solution if it could be proved for all prime powers n.
180. Thompson John biography
□ The solution of Frobenius's conjecture was not done by simply pushing the existing techniques further than others had done; rather it was achieved by introducing many highly original ideas
which were to lead to many developments in group theory.
□ To classify finite groups therefore reduces to two problems, namely the classification of finite simple groups and the solution of the extension problem, that is the problem of how to fit the
building blocks together.
□ I like to say that I would like to see the solution of the problem of the finite simple groups and the part I expect Thompson's work to play in it.
□ His work on coding theory was to lay the foundation for the solution of a long standing problem, namely the fact that there is no finite plane of order 10.
181. Yau biography
□ The analytic problem is that of proving the existence of a solution of a highly nonlinear (complex Monge-Ampere ) differential equation.
□ Yau's solution is classical in spirit, via a priori estimates.
□ However, there were still questions relating to whether Douglas's solution, which was known to be a smooth immersed surface, is actually embedded.
□ for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.
182. Zhang Qiujian biography
□ However, no reasons are given for the method of solution.
□ There are problems on extracting square and cube roots, problems on finding the solution to quadratic equations, problems on finding the sum of an arithmetic progression, and on solving
systems of linear equations.
□ Zhang gives the solution by solving a quadratic equation, but his formulae are not particularly accurate.
□ A modern solution sets A to have x coins, B to have y and C to have z.
183. Whittaker biography
□ He studied these special functions as arising from the solution of differential equations derived from the hypergeometric equation.
□ His results in partial differential equations (described as 'most sensational' by Watson) included a general solution of the Laplace equation in three dimensions in a particular form and the
solution of the wave equation.
□ He also worked on electromagnetic theory giving a general solution of Maxwell's equation, and it was through this topic that his interest in relativity arose.
184. Steinitz biography
□ He submitted a solution to a prize problem announced by the university and this won him not only 200 marks but also the right to submit his doctoral dissertation without payment of the usual
□ Steinitz's solution of Konig's theorem twenty years before Konig is discussed in detail in [Acta.
□ If the reader is unaware of the solution, he will in places hardly be able to guess what is meant.
□ This does not indicate that M Levy has not also found a solution for the general case and it indicates even less that he was not able to find it.
185. Mordell biography
□ For his Smith's Prize essay Mordell studied solutions of y2 = x3 + k, an equation which had been considered by Fermat.
□ Thue had already proved a result which, combined with Mordell's work showed that this equation had only finitely many solutions but Mordell only learned about Thue's work at a later date.
□ At the time he wrote the essay Mordell believed that for some k there may be infinitely many solutions.
□ However he solved the equation for many values of k, giving complete solutions for some values.
186. Piatetski-Shapiro biography
□ He had given a solution to a problem which had been posed by Raphael Salem.
□ Among his main achievements are: the solution of Salem's problem about the uniqueness of the expansion of a function into a trigonometric series; the example of a non symmetric homogeneous
domain in dimension 4 answering Cartan's question, and the complete classification (with E Vinberg and G Gindikin) of all bounded homogeneous domains; the solution of Torelli's problem for K3
surfaces (with I Shafarevich); a solution of a special case of Selberg's conjecture on unipotent elements, which paved the way for important advances in the theory of discrete groups, and
many important results in the theory of automorphic functions, e.g., the extension of the theory to the general context of semi-simple Lie groups (with I Gelfand), the general theory of
arithmetic groups operating on bounded symmetric domains, the first 'converse theorem' for GL(3), the construction of L-functions for automorphic representations for all the classical groups
(with S Rallis) and the proof of the existence of non arithmetic lattices in hyperbolic spaces of arbitrary large dimension (with M Gromov).
187. Eudoxus biography
□ Eutocius wrote about Eudoxus's solution but it appears that he had in front of him a document which, although claiming to give Eudoxus's solution, must have been written by someone who had
failed to understand it.
□ Tannery's ingenious suggestion was that Eudoxus had used the kampyle curve in his solution and, as a consequence, the curve is now known as the kampyle of Eudoxus.
□ Eudoxus was, I think, too original a mathematician to content himself with a mere adaptation of Archytas's method of solution.
188. Erlang biography
□ During this time he kept up his interest in mathematics, and he received an award in 1904 for an essay on Huygens' solution of infinitesimal problems which he submitted to the University of
□ Jensen persuaded Erlang to apply his skills to the solution of problems which arose from a study of waiting times for telephone calls.
□ In this paper he showed that if telephone calls were made at random they followed the Poisson distribution, and he gave a partial solution to the delay problem.
□ In 1917 he published Solution of some problems in the theory of probability of significance in automatic telephone exchanges in which he gave a formula for loss and waiting time which was
soon used by telephone companies in many countries including the British Post Office.
189. Nicolson biography
□ Phyllis Nicolson is best known for her joint work with John Crank on the heat equation, where a continuous solution u(x, t) is required which satisfies the second order partial differential
equation .
□ They considered numerical methods which find an approximate solution on a grid of values of x and t, replacing ut(x, t) and uxx(x, t) by finite difference approximations.
□ Richardson's method yielded a numerical solution which was very easy to compute, but alas was numerically unstable and thus useless.
□ Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.
190. Steiner biography
□ In his paper Several laws governing the division of planes and space, which also appeared in the first volume of Crelle's Journal, he considers the problem: What is the maximum number of
parts into which a space can be divided by n planes? It is a beautiful problem and has the solution (n3 + 5n + 6)/6.
□ See [One Hundred Great Problems of Elementary Mathematics, Their History and Solution (Dover, 1965).',3)">3] for a solution.
□ The proof, essentially as given by Steiner, is reproduced in [One Hundred Great Problems of Elementary Mathematics, Their History and Solution (Dover, 1965).',3)">3].
191. Picone biography
□ Resulting from this were Picone's results on a priori bounds for the solutions of ordinary differential equations, as well as for those of linear partial differential equations of elliptic
type and parabolic type for which the bound is obtained by means of the boundary data and the known terms; these results are contained in his well-known 'Notes on higher analysis' (Italian) a
volume published in 1940 and which was, for its time, "truly avant-garde".
□ Gaetano Fichera highlights Picone's 1936 memoir which contains a characterization of a large class of linear partial differential equations whose solutions enjoy mean-value properties termed
"integral properties" by Picone; using this theory Picone reconstructed M Nicolescu's theory of polyharmonic functions.
□ However, the works which led to the broadest and most important research are those based on the translation of boundary value problems for linear partial differential equations into systems
of Fischer-Riesz integral equations; this method, whose object is the numerical calculation of the solutions, is similar to that of subsequent authors, who considered weak solutions of the
same problems.
192. Coulomb biography
□ A reason, perhaps, for the relative neglect of this portion of Coulomb's work was that he sought to demonstrate the use of variational calculus in formulating methods of approach to
fundamental problems in structural mechanics rather than to give numerical solutions to specific problems.
□ his simple, elegant solution to the problem of torsion in cylinders and his use of the torsion balance in physical applications were important to numerous physicists in succeeding years.
193. Mathieu Emile biography
□ From his late twenties his main efforts were devoted to the then unfashionable continuation of the great French tradition of mathematical physics, and he extended in sophistication the
formation and solution of partial differential equations for a wide range of physical problems.
□ The Mathieu functions are solutions of the Mathieu equation which is .
194. Tschirnhaus biography
□ In it he discussed several mathematical questions including the solution of higher equations.
□ In his letter Leibniz also criticises Tschirnhaus's solution of algebraic equations.
□ Tschirnhaus worked on the solution of equations and the study of curves.
195. Moulton biography
□ He published On a class of particular solutions of the problem of four bodies in 1900 and A class of periodic solutions of the problem of three bodies with application to the lunar theory in
1906 both in the Transactions of the American Mathematical Society.
□ Other papers, this time in the Annals of Mathematics, include The straight line solutions of the problem of n bodies (1910) and The deviations of falling bodies (1913).
196. Drinfeld biography
□ Although he only proved a special case of the Langlands conjecture, Drinfeld has introduced important new ideas in his solution and made a real breakthrough.
□ The interactions between mathematics and mathematical physics studied by Atiyah led to the introduction of instantons - solutions, that is, of a certain nonlinear system of partial
differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
197. Benedetti biography
□ In this work he discussed the general solution of all problems in Euclid's Elements, and other geometric problems, using only a compass of fixed opening.
□ [Benedetti] opens the section titled "De rationibus operationum perspectivae" with an impatient review of a perspective error committed by both Durer and Setlio, and he proceeds at once to
prove the correct solution.
□ He must have been aware that Tartaglia considered that Cardan had stolen his solution of the cubic, although in this case Cardan fully acknowledged Tartaglia's contributions.
198. Carleson biography
□ The citation emphasizes not only Carleson's fundamental scientific contributions, the best known of which perhaps are the proof of Luzin's conjecture on the convergence of Fourier series, the
solutions of the corona problem and the interpolation problem for bounded analytic functions, the solution of the extension problem for quasiconformal mappings in higher dimensions, and the
proof of the existence of 'strange attractors' in the Henon family of planar maps, but also his outstanding role as scientific leader and advisor.
199. Chernikov biography
□ The practical importance of convenient algorithms for the solution of systems of linear inequalities and their connection with the theory of linear programming is well known.
□ the basis of this theory lies in the principle of boundary solutions; all its results are deduced from it by means of only a few finite methods..
200. Machin biography
□ While Newton was planning for a third edition, he received two independent solutions of the problem of the motion of the nodes of the moon's orbit, one by John Machin, the other by Henry
□ One other publication by Machin is worth noting, namely The solution of Kepler's problem which was published in the Philosophical Proceedings of the Royal Society in 1738.
201. Gelbart biography
□ In 1935 a solution to this problem by Abe Gelbart, Student, Central High School, Paterson, New Jersey was published.
□ The basic idea was to construct a theory similar to complex function theory for the solutions of a system of generalized Cauchy-Riemann equations arising in the mechanics of continua.
202. Manfredi biography
□ The difficulties in finding a solution were great for there were economic issues, technical issues, political issues and legal issues to overcome.
□ Pope Clement XI called the experts to a meeting in April 1718 but the proposed solution ran into technical objections.
□ This particular effort to find a solution came to an end in March 1721 when Clement XI died.
203. Van Ceulen biography
□ Goudaan had posed a geometric problem which Van Ceulen solved but his solution was not accepted by Goudaan.
□ When Goudaan published his own solution to the problem, Van Ceulen realised that it was incorrect.
□ In 1595 the two men competed in the solution of a forty-fifth degree equation proposed by van Roomen in his 'Ideae mathematicae' (1593) and recognised its relation to the expression of sin
45A in terms of sin A.
204. Janovskaja biography
□ There then follows general comments concerning the theory of algorithms, and the mathematical concepts of proof, construction and solution.
□ An analysis is given for the problem of finding geometric solutions for algebraic equations of degree higher than two by locating points of intersection of conic sections with other curves.
205. Hamming biography
□ In particular, Hamming investigated the Green's function and also the characteristic solutions for which he obtained asymptotic expressions.
□ Work in codes is related to packing problems and the error-correcting codes discovered by Hamming led to the solution of a packing problem for matrices over finite fields.
206. Hensel biography
□ He showed, at least for quadratic forms, that an equation has a rational solution if and only if it has a solution in the p-adic numbers for each prime p and a solution in the reals.
207. Fenyo biography
□ The structure of the solutions is then examined, including singular points and limit cycles, and the book concludes with an account of the elementary theory of non-linear oscillations.
□ The soundness of his basic culture, coupled with his innate curiosity, led Istvan Fenyo to seek the solution of problems in various areas of mathematics.
208. Ulugh Beg biography
□ This excellent book records the main achievements which include the following: methods for giving accurate approximate solutions of cubic equations; work with the binomial theorem; Ulugh
Beg's accurate tables of sines and tangents correct to eight decimal places; formulae of spherical trigonometry; and of particular importance, Ulugh Beg's Catalogue of the stars, the first
comprehensive stellar catalogue since that of Ptolemy.
□ The calculation is built on an accurate determination of sin 1° which Ulugh Beg solved by showing it to be the solution of a cubic equation which he then solved by numerical methods.
209. Cholesky biography
□ After his death one of his fellow officers, Commandant Benoit, published Cholesky's method of computing solutions to the normal equations for some least squares data fitting problems in Note
sur une methode de resolution des equations normales provenant de l'application de la methode des moindres carres a un systeme d'equations lineaires en nombre inferieure a celui des
□ To solve Ax = b one now needs to solve LL'x = b so put y = L'x which gives Ly = b which is solved for y, then y = L'x is solved for x to obtain the solution.
210. Kadets biography
□ The first steps towards a solution had been taken by Stanislaw Mazur, a student of Banach's, in 1929 and then by Stefan Kaczmarz, a colleague of Banach's, in 1932.
□ To the solution of the problem he applied arguments of approximation theory suggested by Bernstein's theorem on the recovery of a continuous function from its least deviations from
□ He continued to work towards a complete solution of the Frechet-Banach problem and achieved this in 1966.
211. Dynkin biography
□ a class of measure-valued Markov processes [which] can be used to give probabilistic solutions to certain nonlinear PDE's in a way which is analogous to the classical solution of the
Dirichlet problem by means of Brownian motion.
212. Walsh biography
□ Memoir on the Invention of Partial Equations; The Theory of Partial Functions; Irish Manufactures: A New Method of Tangents; An Introduction to the Geometry of the Sphere, Pyramid and Solid
Angles; General Principles of the Theory of Sound; The Normal Diameter in Curves; The Problem of Double Tangency; The Geometric Base; The Theoretic Solution of Algebraic Equations of the
Higher Orders.
□ Thus, in a page headed Cubic Equations, he writes the name of Cardan opposite to a well-known algebraic solution, that of Walsh opposite to the same result put under another and less
convenient form, and below these he gives a formula headed For a Complete Cubic by Walsh only.
□ Discovered the general solution of numerical equations of the fifth degree at 114 Evergreen Street, at the Cross of Evergreen, Cork, at nine o'clock in the forenoon of July 7th, 1844; exactly
twenty-two years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science.
213. Motzkin biography
□ It was written as a partial solution to a problem which had been posed by Ostrowski and it gave Motzkin particular pleasure when he returned to the problem many years later and was able to
give a complete solution.
□ The proof is very typical of Motzkin in that the Euclidean algorithm is given a new formulation, which at first seems to be leading away from the problem at hand, but is suddenly seen to be
the decisive key to its solution.
214. Girard Albert biography
□ He was the first who understood the use of negative roots in the solution of geometrical problems.
□ For those having only one root he indicated, beside Cardano's rules, an elegant method of numerical solution by means of trigonometric tables and iteration.
□ The negative solution is explained in geometry by moving backward, and the minus sign moves back when the + advances.
215. Ajima biography
□ The first, the Gion shrine problem, he solved in an unpublished manuscript of 1774 entitled Kyoto Gion Dai Toujyutsu (The Solution to the Gion Shrine Problem).
□ Although his solution was unpublished, nevertheless Ajima became famous for his work on this problem.
□ Malfatti assumed that the solution would involve three circles, each of which is tangent to the other two.
216. Truesdell biography
□ In its use of formal methods, its reliance on special kinematic hypotheses (membranes of revolution only) and its presentation of series solutions of a great many special problems, this work
can be considered properly isolated from his other work.
□ Most important of all, he taught me that careful scholarship and the persistent search for insight and understanding are far more important than facile skill in the use of contemporary
techniques for the solution of currently popular problems.
217. Borok biography
□ Her papers published in 1954-1959 contain a range of "inverse" theorems that allow partial differential equations to be characterized as parabolic or hyperbolic, by certain properties of
their solutions.
□ In the early 1960s Valentina worked on fundamental solutions and stability for partial differential equations well-posed in the sense of Petrovskii.
□ Her central results include the construction of maximal classes of uniqueness and well-posedness, Phragmen-Lindelof type theorems, and the study of asymptotic properties and stability of
solutions of boundary-value problems in infinite layers.
218. Simpson biography
□ It was his obvious mathematical skills demonstrated in these solutions which first brought his to the attention of other mathematicians of the day.
□ Simpson's attempt at an analytical solution is interpreted.
219. Riccati biography
□ He considered many general classes of differential equations and found methods of solution which were widely adopted.
□ He is chiefly known for the Riccati differential equation of which he made elaborate study and gave solutions for certain special cases.
220. Manfredi Gabriele biography
□ He first studied equations with algebraic solutions, then those that lead to transcendental curves, then moved on to equations that are solved by means of substitution of variables.
□ The difficulties in finding a solution were great for there were economic issues, technical issues, political issues and legal issues to overcome.
221. Delone biography
□ Two features are very characteristic of the mode of presentation: on the one hand the extensive use of geometrical considerations as a background for the true understanding of complicated
situations which otherwise would remain obscure, and on the other hand, the care shown by the authors in inventing effective methods of solution, illustrated by actual application to
numerical examples and to the construction of valuable tables.
□ He also published a number of texts aimed at school pupils including (with O K Zhitomirski) Problems with solutions for a revision course in elementary mathematics (1928), (with O K
Zhitomirski) Problems in geometry (1935), Analytical geometry I (1948), and (with D A Raikov) Analytical geometry II (1949).
222. Moser Jurgen biography
□ First we mention Lectures on Hamiltonian systems (1968) which examines problems of the stability of solutions, the convergence of power series expansions, and integrals for Hamiltonian
systems near a critical point.
□ The missing final two chapters would have been on KAM theory and unstable hyperbolic solutions.
□ for his contributions to the theory of Hamiltonian dynamical systems, especially his proof of the stability of periodic solutions of Hamiltonian systems having two degrees of freedom and his
specific applications of the ideas in connection with this work.
223. Laguerre biography
□ Laguerre studied approximation methods and is best remembered for the special functions the Laguerre polynomials which are solutions of the Laguerre differential equations.
□ When Darboux proved the orthogonality of systems of homofocal ovals, he also showed that ovals provide a geometrical interpretation of the addition theorem and that they constitute the
algebraic form of the integral solution.
224. Zhu Shijie biography
□ Therefore Zhu does not necessarily give the simplest solution to a problem, but rather often introduces complications explicitly designed to illustrate how to handle more complicated
□ The following problem in the Siyuan yujian is reduced by Zhu to a polynomial equation of degree 5 (see [First Australian Conference on the History of Mathematics (Clayton, 1980) (Clayton,
1981), 103-134.',7)">7] for a detailed solution as given by Zhu):- .
□ The Siyuan yujian also contains a transformation method for the numerical solution of equations which is applied to equations up to degree 14.
225. Cesari biography
□ The paper also contains applications of his general methods to more specialised problems, solutions to which had been found by Carl Jacobi and Richard von Mises.
□ Particularly noted for his study of the existence theorems for optimal solutions for both single- and multi-dimensional systems, he also contributed to the theory of necessary conditions and
the analysis of Pareto problems.
□ He also investigated the existence of solutions to certain quasi-linear hyperbolic systems.
226. Al-Karaji biography
□ Often he explicitely says that he is giving a solution in the style of Diophantus.
□ The solutions of quadratics are based explicitly on the Euclidean theorems ..
227. Weise biography
□ Also mentioned are existence theorems as well as solutions by iteration, power series, and numerical methods.
□ Weise acted as supervisor of PhD students from a wide range of mathematical fields, a dozen of them went on to become professors, among them Wolfgang Gaschutz (finite groups), Wolfgang Haken
(knot theory and the solution of the four-colour-problem), Wilhelm Klingenberg (differential geometry) and Jens Mennicke (topology).
228. Schwarz biography
□ Weierstrass had shown that Dirichlet's solution to this was not rigorous, see [Rend.
□ An idea from this work, in which he constructed a function using successive approximations, led Emile Picard to his existence proof for solutions of differential equations.
229. Kolosov biography
□ He passed his Master's examinations in 1893 and his Master's dissertation On certain modifications of Hamilton's principle and its application to the solution of problems of mechanics of
solid bodies (Russian) (1903) contained his first really significant result.
□ Kolosov's thesis contained a formal solution of the plane problems of the theory of elasticity.
□ In addition to the important results we have mention above, we note that in 1907 Kolosov derived the solution for stresses around an elliptical hole.
230. Schiffer biography
□ The 'Calculus of Variations' - formulating and solving problems in terms of a quantity to be maximized or minimized and analysing the properties of such extremal solutions - had already been
and remains an established, highly developed, and highly effective area of mathematical analysis and its applications.
□ The second part lays more stress on rigour, and treats fundamental solutions, reduction of boundary value problems to integral equations, orthonormal systems and kernel functions, eigenvalue
problems associated with the kernels, variational theory of domain functions, comparison domains, basic existence theorems, and dependence of solutions on the boundary data or on the
coefficients of the differential equation.
231. Zolotarev biography
□ Zolotarev was able to give a much more effective solution.
□ They were able to give complete solutions in the case of four variables and of five variables.
232. Archytas biography
□ Archytas solved the problem with a remarkable geometric solution (not of course a ruler and compass construction).
□ One interesting innovation which Archytas brought into his solution of finding two mean proportionals between two line segments was to introduce movement into geometry.
□ We know of Archytas's solution to the problem of duplicating the cube through the writings of Eutocius of Ascalon.
233. Frenicle de Bessy biography
□ We know that Frenicle found four solutions to the first of these problems on the day that he was given the problem, and found another six solutions the next day.
□ He gave solutions to both problems in Solutio duorm problematum ..
234. Schmidt F-K biography
□ Jean Dieudonne, writing in [The Mathematical Intelligencer 10 (1975), 7-21.',3)">3] about the solution of the Weil Conjectures, states that F-K Schmidt was one of the main contributors of
essential ideas to the ultimate solution.
□ At the annual meeting of the Deutsche Mathematiker Vereinigung in September 1930 in Konigsberg, F K Schmidt gave a talk about [this] question and his solution.
235. Petersen biography
□ The interest he had shown in ruler and compass constructions when he was at school had continued to influence his research topic and his doctoral thesis was entitled On equations which can be
solved by square roots, with application to the solution of problems by ruler and compass.
□ First published three years later, his Methods and theories for the solution of problems of geometrical construction appeared in various editions and languages.
□ Petersen never returned to cryptography; this seems to be another instance of a problem that must have occupied him intensely for a period, until he found a satisfactory solution and moved on
to something else.
236. Narayana biography
□ He then finds the solutions x = 6, y = 19 which give the approximation 19/6 = 3.1666666666666666667, which is correct to 2 decimal places.
□ Narayana then gives the solutions x = 228, y = 721 which give the approximation 721/228 = 3.1622807017543859649, correct to four places.
□ Finally Narayana gives the pair of solutions x = 8658, y = 227379 which give the approximation 227379/8658 = 3.1622776622776622777, correct to eight decimal places.
237. Arnold biography
□ I spent a whole day thinking on this oldies, and the solution ..
□ The examining committee for the theis, which contained a solution to Hilbert's 13th problem, consisted of A G Vitushkin and L V Keldysh.
□ thesis contained a solution to Hilbert's 13th problem.
238. Al-Kashi biography
□ In the richness of its contents and in the application of arithmetical and algebraic methods to the solution of various problems, including several geometric ones, and in the clarity and
elegance of exposition, this voluminous textbook is one of the best in the whole of medieval literature; it attests to both the author's erudition and his pedagogical ability.
□ He was not the first to look at approximate solutions to this equation since al-Biruni had worked on it earlier.
239. Niven biography
□ In the present state of our knowledge of the resistance of the air to shot, the problem of integrating the equations of motion of the shot and of plotting-out a representation of the curve
described by it is peculiar, because, according to the best experiments we possess, the law of the retardation cannot be expressed by a single exact formula which is available for the
□ We are therefore compelled to give a solution adapted to Tables, the magnitudes of the retardation being set down in those Tables for velocities which are common in practice.
□ The amount of labour, however, in calculating all the quantities for a single component arc, even with the aid of copious tables, is so great that I was led to examine whether any thing could
be done towards simplifying the solution and reducing the amount of calculation.
240. Bruns biography
□ He worked on the three-body problem showing that the series solutions of the Lagrange equations can change between convergent to divergent for small perturbations of the constants on which
the coefficients of the time depend.
□ A few years later Poincare extended Bruns' work to show that no solution to the three-body problem was possible given by algebraic expressions and integrals.
241. Magenes biography
□ The paper considers the problem of the existence of solutions of the differential equation in the title which pass through a given point and are tangent to a given curve.
□ The first of these papers examines the values of λ for which the equation in the title, subject to certain boundary conditions, has a solution.
242. Fontaine des Bertins biography
□ In 1732 Fontaine gave a solution to the brachistochrone problem, in 1734 he gave a solution of the tautochrone problem which was more general than that given by Huygens, Newton, Euler or
Jacob Bernoulli, and in 1737 he gave a solution to an orthogonal trajectories problem.
243. Montgomery biography
□ Iwasawa, himself a contributor to the solution of Hilbert's Fifth Problem, wrote in a review:- .
□ In the meantime, the theory of topological groups has made outstanding progress, culminating in the solution of Hilbert's fifth problem by Gleason and by the authors of the present book.
□ The next two chapters are devoted to the study of the structure of locally compact groups which leads to a solution of Hilbert's problem.
244. Bers biography
□ The nonparametric differential equation of minimal surfaces may be considered the most accessible significant example revealing typical qualities of solutions of non-linear partial
differential equations.
□ The author sets as his goal the development of a function theory for solutions of linear, elliptic, second order partial differential equations in two independent variables (or systems of two
first-order equations).
□ One of the chief stumbling blocks in such a task is the fact that the notion of derivative is a hereditary property for analytic functions while this is clearly not the case for solutions of
general second order elliptic equations.
245. Clairaut biography
□ The following year Clairaut studied the differential equations now known as 'Clairaut's differential equations' and gave a singular solution in addition to the general integral of the
□ When d'Alembert attacked Clairaut's solution of the three-body problem as being too much based on observation and not, like his own work, based on theoretical results, Clairaut strongly
attacked d'Alembert in the most bitter dispute of their lives.
□ The algebra book was an even more scholarly work and took the subject up to the solution of equations of degree four.
246. Graffe biography
□ Graffe is best remembered for his "root-squaring" method of numerical solution of algebraic equations, developed to answer a prize question posed by the Berlin Academy of Sciences.
□ Perhaps it is also the fact that for the general solution of equations that exceed the 4th degree, insuperable obstacles seem to stand in the way, which gives a peculiar charm to these
investigations, which almost every mathematician is trying to use his powers to consider.
□ The fact that he lost out because of errors in the way he submitted his solution for the prize was a major disappointment to Graffe.
247. Ghetaldi biography
□ Also in 1607 Ghetaldi produced a pamphlet Variorum problematum collectio with 42 problems with solutions.
□ It is reasonable to ask: what is the most impressive ideas contained in Ghetaldi's work? Without doubt, it is his application of algebraic methods to the solution of problems in geometry.
248. Vladimirov biography
□ The physicists passed mathematical assignments to the team in which Vladimirov was working and, prompted by these problems, he developed a new technique for the numerical solution of boundary
value problems specifically designed for the type of problems which were encountered.
□ In this thesis he presented his theoretical investigation of the numerical solution, using the method of characteristics, of the single-velocity transport equation for a multilayered sphere.
□ Thus, he first proved the theorem on the uniqueness, existence, and smoothness of the solution of the single-velocity transport equation, established properties of the eigenvalues and
eigenfunctions, and gave a new variational principle (the Vladimirov principle).
249. Richardson biography
□ Having developed these methods by which he was able to obtain highly accurate solutions, it was a natural step to apply the same methods to solve the problems of the dynamics of the
atmosphere which he encountered in his work for the Meteorological Office.
□ It was a remarkable piece of work but in a sense it was ahead of its time since the time taken for the necessary hand calculations in a pre-computer age took so long that, even with many
people working to solve the equations, the solution would be found far too late to be useful to predict the weather.
250. Vallee Poussin biography
□ Vallee Poussin's first mathematical research was on analysis, in particular concentrating on integrals and solutions of differential equations.
□ In fact the solution of this major open problem was one of the major motivations for the development of complex analysis during the period from 1851 to 1896.
251. Ford biography
□ Following his contributions to the war effort, Ford joined the faculty at the Rice Institution, Houston, Texas and while there he published papers such as On the closeness of approach of
complex rational fractions to a complex irrational number (1925), The Solution of Equations by the Method of Successive Approximations (1925), On motions which satisfy Kepler's first and
second laws (1927/28), and The limit points of a group (1929).
□ He had gained a reputation as an excellent expositor and he wrote outstanding articles as well as contributing many mathematical problems and solutions.
252. Zippin biography
□ He slowly returned to his joint research with Deane Montgomery, and together they made progress towards a solution to Hilbert's fifth problem [The honors class: Hilbert\'s problems and their
solvers (A K Peters, 2002).',1)">1]:- .
□ It then follows immediately that every locally euclidean group is a Lie group, namely, the solution of Hilbert's fifth problem.
□ In the meantime, the theory of topological groups has made outstanding progress, culminating in the solution of Hilbert's fifth problem by Gleason and by the authors of the present book.
253. La Faille biography
□ He determined the longitude by studying the phases of the moon and put his solution forward for the prize offered by Spain for a solution to the longitude problem.
□ De la Faille supported van Langren's solution to the longitude problem but no decision was reached about awarding the prize [Jesuit Science and the Republic of Letters (MIT Press, 2003),
339-340.',2)">2]:- .
254. Schwarzschild biography
□ Schwarzschild's relativity papers give the first exact solution of Einstein's general gravitational equations, giving an understanding of the geometry of space near a point mass.
□ I had not expected that one could formulate the exact solution of the problem in such a simple way.
□ However, Schwarzschild himself makes clear that he believes that the theoretical solution is physically meaningless, so making it very clear that he did not believe in the physical reality of
black holes.
255. Robins biography
□ This was shown to Dr Pemberton, who, thence, conceiving a good opinion of the writer, for a further trial of his proficiency sent him some problems, of which the Doctor required elegant
solutions, not those founded on algebraical calculations; adding an example of such a solution that the young geometer might the more readily comprehend his meaning.
256. Fiedler biography
□ He published his thesis in three parts (1954, 1955, 1956) but these were not his first publications, having already published Solution of a problem of Professor E Čech (1952), On certain
matrices and the equation for the parameters of singular points of a rational curve (1952), and (with L Granat) Rational curve with the maximum number of real nodal points (1954).
□ Examples of papers he published on these topics are: Numerical solution of algebraic equations which have roots with almost the same modulus (1956); Numerical solution of algebraic equations
by the Bernoulli-Whittaker method (1957); On some properties of Hermitian matrices (1957); (with Jiri Sedlacek) On W-bases of directed graphs (1958); and (with Josej Bily and Frantisek
Nozieka) Die Graphentheorie in Anwendung auf das Transportproblem (1958).
257. Blichfeldt biography
□ This was not the immediate solution to all of Hans's problems for once in the United States he spent four years as a labourer working on farms and in sawmills.
□ Some of the many topics that he covered were diophantine approximations, orders of linear homogeneous groups, theory of geometry of numbers, approximate solutions of the integers of a set of
linear equations, low-velocity fire angle, finite collineation groups, and characteristic roots.
258. Varga biography
□ 'Matrix Iterative Analysis' belongs in the personal library of every numerical analyst interested in either the practical or theoretical aspects of the numerical solution of partial
differential equations.
□ However, the title should prepare the reader for a modern treatment (thoroughly imbedded in functional analysis), for the subject not to be regarded merely as an end in itself (the numerical
solution of elliptic and parabolic boundary value problems is the author's eventual target) and for theoretical depth (there are no details of explicit numerical procedures).
□ At the same time it shows that there are still a lot of unsolved mathematical problems, the solution of which may require deep mathematics, but probably also fast computers and
highly-accurate numerical software.
259. Atiyah biography
□ His first major contribution (in collaboration with F Hirzebruch) was the development of a new and powerful technique in topology (K-theory) which led to the solution of many outstanding
difficult problems.
□ Subsequently (in collaboration with I M Singer) he established an important theorem dealing with the number of solutions of elliptic differential equations.
260. Todd John biography
□ Solution of differential equations by recurrence relations (1950); Experiments on the inversion of a 16 × 16 matrix (1953); Experiments in the solution of differential equations by Monte
Carlo methods (1954); The condition of the finite segments of the Hilbert matrix (1954); Motivation for working in numerical analysis (1954); and A direct approach to the problem of stability
in the numerical solution of partial differential equations (1956).
261. Kato biography
□ II in 1950, and Note on Schwinger's variational method, On the existence of solutions of the helium wave equation, Upper and lower bounds of scattering phases and Fundamental properties of
Hamiltonian operators of Schrodinger type in 1951.
□ Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line ..
262. Ree biography
□ Ree managed to solve the problem and sent his solution to Max Zorn.
□ When Zorn received Ree's solution he was impressed and sent it to the Bulletin of the American Mathematical Society.
□ One might imagine that Ree would be overjoyed to have his first paper published in a prestigious journal but he did not realise that the paper had been published for over five years after he
sent his solution to Zorn.
263. Hellins biography
□ Hellins published many papers; the following were all in the Philosophical Transactions of the Royal Society: A new method of finding the equal roots of an equation by division (1782); Dr
Halley's method of computing the quadrature of the circle improved; being a transformation of his series for that purpose, to others which converge by the powers of 60 (1794); Mr Jones'
computation of the hyperbolic logarithm of 10 compared (1796); A method of computing the value of a slowly converging series, of which all the terms are affirmative (1798); An improved
solution of a problem in physical astronomy, by which swiftly converging series are obtained, which are useful in computing the perturbations of the motions of the Earth, Mars, and Venus, by
their mutual attraction (1798); A second appendix to the improved solution of a problem in physical astronomy (1800); and On the rectification of the conic sections (1802).
□ The utility of hyperbolic and elliptic arches, in the solution of various problems, and particularly in the business of computing fluents, has been shown by those eminent mathematicians,
Maclaurin, Simpson and Landen; the last of whom has written a very ingenious paper on hyperbolic and elliptic arches, which was published in the first volume of his 'Mathematical Memoirs', in
the year 1780.
264. Delannoy biography
□ In all he seems to have published eleven articles but he also published many problems and solutions to problems set by others.
□ His first publication was Emploi de l'echiquier pour la solution de problemes arithmetiques (1886) followed by Sur la duree du jeu (1888).
265. Ramsey biography
□ He accepted Russell's solution to remove the logical paradoxes of set theory arising from, for example, "the set of all sets which are not members of themselves".
□ Unsolved problems abound, and additional interesting open questions arise faster than solutions to the existing problems.
266. Rolle biography
□ He made his solution known through publishing it in the Journal des scavans.
□ Rolle published another important work on solutions of indeterminate equations in 1699, Methode pour resoudre les equations indeterminees de l'algebre.
267. Tarski biography
□ He had already made clear how pleased mathematicians should be that there is no solution to the general decision problem.
□ the solution of the decision problem in its most general form is negative.
□ Perhaps sometimes in their sleepless nights they thought with horror of the moment when some wicked metamathematician would find a positive solution, and design a machine which would enable
us to solve any mathematical problem in a purely mechanical way..
268. Ruffini biography
□ Certainly no mathematician has published such a claim and even Lagrange in his famous paper Reflections on the resolution of algebraic equations says he will return to the question of the
solution of the quintic and, clearly, he still hoped to solve it by radicals.
□ In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown
that the algebraic solution of the general equation of degree greater than four is impossible.
□ The algebraic solution of general equations of degree greater than four is always impossible.
269. Bianchi biography
□ As preparation for the general solution, the author applies the Lie-Killing methods for finding all three-dimensional spaces in which the motions of figures with given degrees of freedom are
possible - this is enough to outline the goal and the general train of thought of Bianchi's work.
□ The importance of his results is known to every reader who is familiar with the awards of the Royal Jablonowski Society for 1901; their citation states that the strength of the methods and
the elegance of the solutions need not be pointed out when we are talking about a paper whose author is Bianchi.
270. Mason biography
□ The first problem which Hilbert suggested to him for a thesis topic was rapidly solved and he wrote up an elegant solution in two pages.
□ He published seven papers in the Transactions of the American Mathematical Society between 1904 and 1910: Green's theorem and Green's functions for certain systems of differential equations
(1904), The doubly periodic solutions of Poisson's equation in two independent variables (1905), A problem of the calculus of variations in which the integrand is discontinuous (1906), On the
boundary value problems of linear ordinary differential equations of second order (1906), The expansion of a function in terms of normal functions (1907); The properties of curves in space
which minimize a definite integral (1908) and Fields of extremals in space (1910).
271. Anosov biography
□ His work was supervised by Pontryagin and during this period Anosov published a number of papers including: On stability of equilibrium states of relay systems (Russian) (1959); Averaging in
systems of ordinary differential equations with rapidly oscillating solutions (Russian) (1960); and Limit cycles of systems of differential equations with small parameters in the highest
derivatives (Russian) (1960).
□ The contemporary version of Hilbert's 21st problem is to find conditions implying a positive or negative solution to the Riemann-Hilbert problem.
272. Sitter biography
□ He found solutions to Einstein's field equations in the absence of matter.
□ This is a particularly simple solution of the field equations of general relativity for an expanding universe.
273. Al-Khwarizmi biography
□ Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations.
□ He uses both algebraic methods of solution and geometric methods.
□ The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded.
274. Abraham biography
□ It contains the complete solution of the general quadratic and is the first text in Europe to give such a solution.
□ Rather strangely, however, 1145 was also the year that al-Khwarizmi's algebra book was translated by Robert of Chester so Abraham bar Hiyya's work was rapidly joined by a second text giving
the complete solution to the general quadratic equation.
275. Efimov biography
□ The theorem which P S Aleksandrov refers to in this quote was one that Efimov became interested in while working for his doctorate but it was not until around 1950 that he began to
concentrate all his efforts on obtaining a solution.
□ Only a few mathematicians have the determination shown by Efimov to spend over twelve years of their lives trying to solve a single problem but he did so and published his solution in 1963 in
The impossibility in three-dimensional Euclidean space of a complete regular surface with a negative upper bound of the Gaussian curvature.
□ Thus, although applications (e.g., to rigid body dynamics and elasticity theory) are mentioned and the usual matrix theory is covered (including, e.g., reduction to the Jordan canonical
form), there is none of the standard material on the solution of systems of linear equations.
276. Watson biography
□ Although Watson was not interested in how best to model the situation, he was, however, very interested in using his expertise to determine mathematical solutions to the given model which
others might then check against observations.
□ He obtained solutions to the problem in 1918 which showed conclusively that the model was not a satisfactory one.
□ Watson showed that if the layer was about 100 km above the Earth's surface and it had a certain conductivity, then indeed the solutions obtained closely matched observations.
277. Saurin biography
□ His two papers on this topic both appeared in 1709, the first being Solutions et analyses de quelques probleme appartenants aux nouvelles methodes, and the second Solution generale du
probleme ..
278. Bendixson biography
□ Bendixson also made interesting contributions to algebra when he investigated the classical problem of the algebraic solution of equations.
□ In examining periodic solutions of differential equations Bendixson used methods based on continued fractions.
279. Collatz biography
□ The book Aufgaben aus der Angewandten Mathematik (1972) (with J Albrecht) provides a collection of problems (with their solutions) on the solution of equations and systems of equations,
interpolation, quadrature, approximation, and harmonic analysis.
280. Rellich biography
□ In this dissertation he generalised the Riemann's integration method, namely the explicit representation of the solution of the initial value problem of a linear hyperbolic differential
equation of second order, to the case of such equations any order.
□ He is also known for Rellich's theorem on entire solutions of differential equations which he proved in 1940.
281. Poisson biography
□ His approach to these problems was to use series expansions to derive approximate solutions.
□ Poisson submitted the first part of his solution to the Academy on 9 March entitled Sur la distribution de l'electricite a la surface des corps conducteurs.
282. Jerrard biography
□ Jerrard wrote a further two volume work on the algebraic solution of equations An essay on the resolution of equations (1858).
□ Jerrard did not accept that the algebraic solution of the quintic equation was impossible.
□ James Cockle was another mathematician who could not accept that Abel had proved the solution to be impossible, but Hamilton supported Abel and pointed out errors in Jerrard's work.
283. Duarte biography
□ He published papers on the general solution of a diophantine equation of the third degree x3 + y3 + z3 - 3xyz = v3, simplified Kummer's criterion and gave a simple proof of the impossibility
of solving the Fermat equation x3 + y3 + z3 = 0 in nonzero integers.
□ Duarte also contributed to that part of mathematics and proposed problems and solutions to the American Mathematical Monthly for several years, and also to the journal Ciencia y Ingenieria
(Science and Engineering) published in Merida.
284. Tao biography
□ Imposing cylindrical symmetry on the equations leads to the "wave maps" problem where, although it has yet to be solved, Tao's contributions have led to a great resurgence of interest since
his ideas seem to have made a solution possible.
□ Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving mathematical problems includes numerous exercises and model solutions throughout.
285. Kalman biography
□ The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
□ Analytic solutions are available in some cases.
286. Steggall biography
□ It was while he was at Owens College, Manchester, that Steggall published London University Pure Mathematics Questions and Solutions which gave the University of London examination questions
from 1877 to 1881 together with Steggall's solutions.
□ The generation of British mathematics to which Steggall belonged delighted in proposing and working out problems whose solutions might require the aid of any branch of pure or applied
287. Praeger biography
□ Praeger had studied the functional equation x(n+1) - x(n) = x2(n), where x2(n) = x(x(n)) and x is an integer-valued function of the integer variable n, and found a three-parameter family of
□ In Enumeration of rooted trees with a height distribution (1985) written jointly with P Schultz and N C Wormald, the authors used generating functions to find a new solution to the problem of
determining the number of rooted trees whose vertices have a given height distribution.
288. Dilworth biography
□ Indeed, it was the hope of many of the early researchers that lattice-theoretic methods would lead to the solution of some of the important problems in group theory.
□ Furthermore, these and other current problems are sufficiently difficult that imaginative and ingenious methods will be required in their solution.
289. Kingman biography
□ This paper gave, in some sense, a complete solution to a problem which Kingman had been studying since his paper Markov transition probabilities.
□ Between this first paper and the complete solution in 1971, Kingman had published several further contributions building up to the final elegant result: Markov transition probabilities.
290. Belanger biography
□ The revised paper was published as Essai sur la Solution Numerique de quelques Problemes Relatifs au Mouvement Permanent des Eaux Courantes (Essay on the numerical solution of some problems
relating to the steady flow of water).
291. Schlafli biography
□ are then values of the unknowns belonging to a single solution.
□ As in that theory, a 'group' of values of coordinates determines a point, so in this one a 'group' of given values of the n variables will determine a solution.
292. Reinhardt biography
□ This was a step towards providing a solution to Hilbert's Eighteenth Problem for it solved the first part, showing that there are only finitely many essentially different space groups in
n-dimensional Euclidean space.
□ It was in the last of these papers that Reinhardt completed the solution of Hilbert's Eighteenth Problem by finding a polyhedron which, although it is not the fundamental region of any space
group, tiles 3-dimensional Euclidean space.
293. Polya biography
□ What was the great novelty which made Polya and Szego's book of analysis problems so different? It was Polya's idea to classify the problems not by their subject, but rather by their method
of solution.
□ its purpose is to discover the solution of the present problem.
294. Artin biography
□ Artin himself proved that when O is the field of algebraic numbers, the subfield K of real algebraic numbers solves the problem and, moreover, it is the unique solution up to automorphisms of
the field O.
□ Artin gave a complete solution in the paper Uber die Zerlegung definiter Funcktionen in Quadrate also published in 1927.
295. Linfoot biography
□ In fact despite still being an undergraduate, Linfoot was already undertaking research and published his first paper The domains of convergence of Kummer's solutions to the Riemann P-equation
in 1926.
□ During World War II, he remained at Bristol but did important work for the Ministry of Aircraft Production on optical systems for air reconnaissance and also for Mott's research group which
was trying to provide rapid solutions to problems of a technical nature.
296. Bolibrukh biography
□ Hilbert believed that the question had a positive solution and the problem appeared settled in 1908 when Josip Plemelj proved this by giving a reduction of the problem to a known result.
□ It was still believed that what was required was a correction in Plemelj's method to give the expected positive answer but Bolibrukh produced a major surprise when he proved in 1989 that
certain prescribed conditions on the singularities led to a negative solution.
297. Brocard biography
□ In 1876, Brocard asked if the only solutions to the equation n! + 1 = m2, in positive integers (n, m), are (4, 5), (5, 11), (7, 71).
□ It is not even known whether there are only finitely many solutions.
298. Haar biography
□ Each issue of the Kozepiskolai Matematikai Lapok contained a number of selected exercises from mathematics and shortly thereafter from physics, as well as solutions to the past months'
problems and a list of those pupils who had sent in correct solutions.
299. MacColl biography
□ He had published 'a pamphlet on ratios' in 1861 before moving to France, but once in Boulogne he became much more active in publishing, particularly in the 'Questions, Problems' section and
in the 'Solutions' section of the Educational Times.
□ As to his books, his first was Algebraical Exercises and Problems with Elliptical Solutions (1870).
300. Fergola biography
□ At this point (after the geometrical analysis, which is "an ontological principle of reduction"), we must proceed to the "geometrical composition" of the problem, that is, the construction of
the solution, where the order of the analysis is reversed.
□ This last step is crucial: "the construction is the essential condition for the proper solution of a geometrical problem." Both geometrical analysis and composition must he accomplished
according to the ancient criteria of elegance ..
301. Al-Mahani biography
□ Therefore, this solution was declared impossible until the appearance of Ja'far al-Khazin who solved the equation with the help of conic sections.
□ It would be too easy to say that since al-Mahani has proposed a method of solution which he could not carry through then his work was of little value.
302. Kalton biography
□ Invariably, in a few days he would have a solution for them.
□ Nigel solved it in forty-eight hours, and later this solution was used to answer a problem of Ramanujan.
303. Poretsky biography
□ On 25 May 1886 Poretsky defended his thesis for a master's degree in astronomy, entitled "On the solution of some of the normal systems occurring in spherical astronomy, with an application
to identify errors in the division of the Kazan Observatory meridian circle" which he had submitted to the Physical-Mathematical Faculty of Kazan University [Dictionary of Scientific
Biography (New York 1970-1990).',1)">1]:- .
□ He published major works on methods of solution of logical equations, and on the reverse mode of mathematical logic.
304. Lions Jacques-Louis biography
□ By this we mean that approximate solutions are constructed by a reduction of the problem to a finite-dimensional one, and these are then shown to form a relatively compact family in a
suitable topology, by means of a priori estimates and other evaluations.
□ These comprise methods in which the approximate solutions are regularized, or smoothed, before the passage to the limit is performed .
305. Cardan biography
□ I have sent to enquire after the solution to various problems for which you have given me no answer, one of which concerns the cube equal to an unknown plus a number.
□ In it he gave the methods of solution of the cubic and quartic equation.
306. Davis biography
□ Does there exist an algorithm to determine whether a Diophantine equation has a solution in natural numbers? .
□ is a completely self-contained exposition of the proof that there is no algorithm for determining whether an arbitrary Diophantine polynomial equation with integer coefficients has an integer
307. Sankara biography
□ It is a text which covers the standard mathematical methods of Aryabhata I such as the solution of the indeterminate equation by = ax ± c (a, b, c integers) in integers which is then applied
to astronomical problems.
□ ',2)">2] then it refers to the optional number in a guessed solution and it is a feature which differs from the original method as presented by Bhaskara I.
308. Heaviside biography
□ The solution of the algebraic equation could be transformed back using conversion tables to give the solution of the original differential equation.
309. Menshov biography
□ He showed Luzin his solution to the problem that Luzin had just posed and before the end of 1914 the two had begun a firm mathematical friendship.
□ Menshov does not belong among the ranks of those mathematicians who undertake the solution of comparatively easy problems, or who continue the research of other authors on a course that has
already been indicated.
310. Kruskal Joseph biography
□ Roughly a year later, I had put a lot of work into this problem, but was still not close to a solution.
□ His work combined with mine finally led to a solution.
311. Iyanaga biography
□ He studied this topic in T Yosiye's course and became interested in conditions under which dy/dx = f (x, y) has a unique solution.
□ conditions assure the uniqueness of solution.
312. Linnik biography
□ In 1948-49 Linnik obtained results which contained, in principle, a complete solution to two central problems in the theory of the summation of variables forming a Markov chain.
□ Linnik substantially improved and developed the methods of his predecessors and gave an almost definitive solution of the problem for an inhomogeneous chain with an arbitrary finite number of
313. Dezin biography
□ Already in his diploma work he developed a technique involving operators of averaging with variable radius, which even at present remains an effective tool in the theory of extension of
functions and in the theory of boundary-value problems, in investigations of the problem of when weak and strong solutions coincide.
□ These papers include Existence and uniqueness theorems for solutions of boundary problems for partial differential equations in function spaces (1959), Boundary value problems for invariant
elliptic systems (1960), and Invariant elliptic systems of equations (1960).
314. Maddison biography
□ When she first reached Bryn Mawr College, Maddison continued to work on this topic but later, advised by Scott, she began to work on singular solutions of differential equations.
□ in 1896 for her thesis On Singular Solutions of Differential Equations of the First Order in Two Variables and the Geometrical Properties of Certain Invariants and Covariants of Their
Complete Primitives and in the same year appointed as Reader in Mathematics at Bryn Mawr.
315. Lagrange biography
□ He solved the resulting system of n+1 differential equations, then let n tend to infinity to obtain the same functional solution as Euler had done.
□ His different route to the solution, however, shows that he was looking for different methods than those of Euler, for whom Lagrange had the greatest respect.
316. Graham biography
□ By the end of the semester, I finished the book and had a solution.
□ Every one of his collaborators knows that Ron will somehow find whatever hours or days are needed to come up with some substantial suggestion, and frequently with the crucial step towards the
317. Faddeeva biography
□ The second chapter deals with numerical methods for the solution of systems of linear equations and the inversion of matrices, and the third with methods for computing characteristic roots
and vectors of a matrix.
□ We mentioned above the 20 joint papers by Faddeeva and her husband, noting that some of the last few of these were: Natural norms in algebraic processes (1970), On the question of the
solution of linear algebraic systems (1974), Parallel calculations in linear algebra (Part 1 in 1977, Part 2 in 1982), and A view of the development of numerical methods of linear algebra
318. Shtokalo biography
□ After 1945 he became particularly interested in the qualitative and stability theory of solutions of systems of linear ordinary differential equations in the Lyapunov sense and in the 1940s
and 1950 published a series of articles and three monographs in these areas.
□ Shtokalo's work had a particular impact on linear ordinary differential equations with almost periodic and quasi-periodic solutions.
319. Cafiero biography
□ He had begun published papers during this period, the first of these being Sull'approssimazione mediante poligonali degli integrali del sistema differenziale: y' = F(x, y), y(x0) = y0 (1947),
which, under the assumption that F(x, y) is continuous in a rectangle, develops a method of polygonal approximation which produces every solution of the system specified in the title.
□ His next paper was Un'osservazione sulla continuita rispetto ai valori iniziali degli integrali dell'equazione: y' = f (x, y) (1947), which proves that any group of conditions sufficient to
assure the existence and uniqueness, with respect to the initial values, of the integral of the equation y' = f (x, y) is also sufficient to assure the continuous dependence of the solution
on the initial values.
320. Spence David biography
□ By similarity considerations, the displacements are expressed in terms of the solution of a pair of nonlinear ordinary differential equations satisfying two-point boundary conditions.
□ For such 'noncanonical' data, coefficients in the eigenfunction expansion can be found only from the solution of infinite sets of linear equations, for which a variety of methods of
formulation have been proposed.
321. Ringrose biography
□ The third and fourth volumes contain the solutions to the exercises.
□ R S Doran says the authors' solutions:- .
322. Ibrahim biography
□ This is in contrast to The selected problems in which 41 difficult geometrical problems are solved, usually by analysis only, without a discussion of the number of solutions or conditions
which make the solutions possible.
323. Carleman biography
□ Names such as Carleman inequality, Carleman theorems (Denjoy-Carleman theorem on quasi-analytic classes of functions, Carleman theorem on conditions of well-definedness of moment problems,
Carleman theorem on uniform approximation by entire functions, Carleman theorem on approximation of analytic functions by polynomials in the mean), Carleman singularity of orthogonal system,
integral equation of Carleman type, Carleman operator, Carleman kernel, Carleman method of reducing an integral equation to a boundary value problem in the theory of analytic functions,
Jensen-Carleman formula in complex analysis, Carleman continuum, Carleman linearization or Carleman embedding technique, Carleman polynomials, Carleman estimate in the unique continuation
problem for solutions of partial differential equations and Carleman system in the kinetic theory of gas are well-known in mathematics (see [Encyclopaedia of Mathematics 2 (Kluwer 1988),
□ Results on unique continuation for solutions to partial differential equations are important in many areas of applied mathematics, in particular in control theory and inverse problems.
324. Stewartson biography
□ Keith Stewartson's abiding passion in mathematical research lay in the solution of the equations governing the motion of liquids and gases, and in the comparison of his theoretical
predictions with experiment and observation.
□ The methods used are a combination of experimental observation, computation often on a very large scale, and analysis of the structure of the asymptotic form of the solution as the friction
tends to zero.
325. Bevan-Baker biography
□ Huygens' geometrical construction, with its restriction that only one sheet of the envelope of the spherical wavelets is to be considered, is first justified in Chapter I by Poisson's
analytical solution of the equation of wave-motions.
□ The formula is interpreted physically and the question of the uniqueness of the solution discussed.
326. Titchmarsh biography
□ At Russell's first lecture the room was packed to the doors, and Russell said: "Ah, there's my clever pupil Mr Titchmarsh - he knows it all, he can go away." Russell dictated his lectures
word for word and examples were handed out - and then, if necessary, solutions to examples.
□ Some of Titchmarsh's solutions replaced the official ones.
327. Chuquet biography
□ In this work negative numbers, used as coefficients, exponents and solutions, appear for the first time.
□ The sections on equations cover quadratic equations where he discusses two solutions.
328. Adamson biography
□ No solutions of the exercises, no proofs of the theorems are included in the first part of the book - this is a 'Workbook' and readers are invited to try their hand at solving the problems
and proving the theorems for themselves.
□ The second part of the book contains complete solutions to all but the most utterly trivial exercises and complete proofs of the theorems.
329. Lipschitz biography
□ Lipschitz is remembered for the 'Lipschitz condition', an inequality that guarantees a unique solution to the differential equation y' = f (x, y).
□ Peano gave an existence theorem for this differential equation, giving conditions which guarantee at least one solution.
330. Kempe biography
□ Hence many problems - such as, for example, the trisection of an angle - which can readily be effected by employing other simple means, are said to have no geometrical solution, since they
cannot be achieved by straight lines and circles only.
□ The first solution was found by a French army officer called Peaucellier and was brought to England by Professor Sylvester in a lecture at the Royal Institution in January 1874.
331. Luzin biography
□ The study of effective sets that he embarked upon was pursued intensively for more than two decades and led to the solution of many important problems of set theory ..
□ The solution of the large problems that he undertook is distinguished by their subtlety, elegance, and simplicity of presentation.
332. Bellman biography
□ His doctoral dissertation on the stability of differential equations was concerned with the behaviour of the solutions of real differential equations as the independent variable t tends to
□ These include, in addition to those already mentioned: A Survey of the Theory of the Boundedness, Stability, and Asymptotic Behavior of Solutions of Linear and Nonlinear Differential and
Difference Equations (1949); A survey of the mathematical theory of time-lag, retarded control, and hereditary processes (1954); Dynamic programming of continuous processes (1954); Dynamic
programming (1957); Some aspects of the mathematical theory of control processes (1958); Introduction to matrix analysis (1960); A brief introduction to theta functions (1961); An
introduction to inequalities (1961); Adaptive control processes: A guided tour (1961); Inequalities (1961); Applied dynamic programming (1962); Differential-difference equations (1963);
Perturbation techniques in mathematics, physics, and engineering (1964); and Dynamic programming and modern control theory (1965).
333. Steinhaus biography
□ Steinhaus found a proportional but not envy free solution for n = 3.
□ An envy free solution to Steinhaus's problem for n = 3 was found in 1962 by John H Conway and, independently, by John Selfridge.
334. Novikov Sergi biography
□ These include a systematic study of finite-gap solutions of two-dimensional integrable systems, formulation of the equivalence of the classification of algebraic-geometric solutions of the KP
equation with the conformal classification of Riemann surfaces, and work (with Krichever) on "almost commuting" operators that appear in string theory and matrix models ("Krichever-Novikov
algebras", now widely used in physics).
335. Day biography
□ One of the first was in the paper A simple solution to the word problem for lattices (1970) where he gave a simple solution to the word problem in free lattices.
336. Hua biography
□ Hua wrote several papers with H S Vandiver on the solution of equations in finite fields and with I Reiner on automorphisms of classical groups.
□ The newfound interest in applicable mathematics took him in the 1960s, accompanied by a team of assistants, all over China to show workers of all kinds how to apply their reasoning faculty to
the solution of shop-floor and everyday problems.
337. D'Alembert biography
□ d'Alembert has tried to undermine [my solution to the vibrating strings problem] by various cavils, and that for the sole reason that he did not get it himself.
□ He wished to publish in our journal not a proof, but a bare statement that my solution is defective.
338. Sturm biography
□ The first to give a complete solution was Cauchy but his method was cumbersome and impractical.
□ Sturm achieved fame with his paper which, using ideas of Fourier, gave a simple solution.
339. Galois biography
□ On 25 May and 1 June he submitted articles on the algebraic solution of equations to the Academie des Sciences.
□ However the papers reached Liouville who, in September 1843, announced to the Academy that he had found in Galois' papers a concise solution .
340. Rogosinski biography
□ Rogosinski's solution greatly pleased Landau and it was from this time that their close friendship developed.
□ Nevertheless, it is a very powerful tool for the solution of many questions in the theory of functions.
341. Andrews biography
□ Andrews had published three papers by the time he had completed his thesis work: An asymptotic expression for the number of solutions of a general class of Diophantine equations (1961); A
lower bound for the volume of strictly convex bodies with many boundary lattice points (1963); and On estimates in number theory (1963).
□ This last paper, in the American Mathematical Monthly, gave a method for finding an upper bound for the number of solutions of a Diophantine equation of the form y = f (x).
342. Minkowski biography
□ In 1881 the Academy of Sciences (Paris) announced that the Grand Prix for mathematical science to be awarded in 1883 would be for a solution to the problem of the number of representations of
an integer as the sum of five squares.
□ Minkowski, although only eighteen years old at the time, reconstructed Eisenstein's theory of quadratic forms and produced a beautiful solution to the Grand Prix problem.
343. Remak biography
□ [In order to answer questions about] (i) identification of an economic optimum, (ii) identification of an approximate economic solution, ..
□ There is, however, work in progress concerning the numerical solution of linear equations with several unknowns using electrical circuits.
344. Kuczma biography
□ For example in the student years he published papers such as: (with Stanislaw Golab and Z Opial) La courbure d'une courbe plane et l'existence d'une asymptote (1958), On convex solutions of
the functional equation g[a(x)] - g(x) = j(x) (1959), On the functional equation j(x) + j [f (x)] = F(x) (1959), On linear differential geometric objects of the first class with one component
(1959), Bemerkung zur vorhergehenden Arbeit von M Kucharzewski (1959), Note on convex functions (1959), and (with Jerzy Kordylewski) On some functional equations (1959).
□ Fundamental notions such as existence and uniqueness of solutions of equations under consideration are treated throughout the book as well as a surprisingly wide scale of examples showing
applications of the theory in dynamical systems, ergodic theory, functional analysis, functional equations in several variables, functional inequalities, geometry, iteration theory, ordinary
differential equations, partial differential equations, probability theory and stochastic processes.
345. Pisier biography
□ describes the development centred around the six problems formulated and discussed at the end of the Resume, and presents various results which led to their solutions.
□ For each of them, the solutions that were known (at the time of the writing of the book) are given with complete proofs and the necessary background.
346. Brunelleschi biography
□ Huge engineering problems faced the placing of a dome on the octagonal Baptistry, and much argument had taken place on how to solve this and Brunelleschi set to work on finding an innovative
□ He now combined his artistic skills, his mathematical skills, and his understanding of mechanical devices when he made a proposal to the wardens of works of the cathedral when they set up a
competition in 1418 to find the best solution to the problem of designing and constructing the dome.
347. Coble biography
□ His early papers, written while he was at Johns Hopkins University, include: On the relation between the three-parameter groups of a cubic space curve and a quadric surface (1906); An
application of the form-problems associated with certain Cremona groups to the solution of equations of higher degree (1908); An application of Moore's cross-ratio group to the solution of
the sextic equation (1911); An application of finite geometry to the characteristic theory of the odd and even theta functions (1913); and Point sets and allied Cremona groups (1915).
348. Dahlquist biography
□ Awarded to a young scientist (normally under 45) for original contributions to fields associated with Germund Dahlquist, especially the numerical solution of differential equations and
numerical methods for scientific computing.
□ He has created the fundamental concepts of stability, A-stability and the nonlinear G-stability for the numerical solution of ordinary differential equations.
349. Cheng Dawei biography
□ Here is a modern solution.
□ How does Cheng Da Wei solve the problem? Basically he uses proportion supposing that the solution to the problem is that A has 10 sheep.
350. Kline biography
□ His research publications during his first years as director of the Division of Electromagnetic Research, now in applied areas, included: Some Bessel equations and their application to guide
and cavity theory (1948); A Bessel function expansion (1950); An asymptotic solution of Maxwell's equations (1950); and An asymptotic solution of linear second-order hyperbolic differential
equations (1952).
351. Federer biography
□ Federer and Morse wrote up the solution to the problem as the joint paper Some properties of measurable functions which was published in 1943.
□ It became dominant partly because of its successful application in the solution of the classical Plateau problem.
352. Pacioli biography
□ XII (Wiesbaden, 1985), 237-246.',9)">9], although the solution he gave is incorrect.
□ During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations.
353. Trudinger biography
□ In another 1967 paper On Harnack type inequalities and their application to quasilinear elliptic equations Trudinger examines weak solutions, subsolutions and supersolutions of certain
quasilinear second order differential equations.
□ It had two new chapters one of which examined strong solutions of linear elliptic equations, and the other was on fully nonlinear elliptic equations.
354. Ostrowski biography
□ One consequence of this association was his monograph Solution of equations and systems of equations which was published in 1960 and was the result of a series of lectures he had given at the
National Bureau of Standards.
□ By 1973 the third edition of this monograph appeared, this time with a new title: Solution of equations in Euclidean and Banach spaces.
355. Gray Andrew biography
□ that not only had he written out the solutions of those questions which had been answered incorrectly, but he had also written out complete solutions of all those which he had not attempted.
356. Tinseau biography
□ Two papers were published in 1772 on infinitesimal geometry Solution de quelques problemes relatifs a la theorie des surfaces courbes et des lignes a double courbure and Sur quelques
proptietes des solides renfermes par des surfaces composees des lignes droites.
□ He also wrote Solution de quelques questions d'astronomie on astronomy but it was never published.
357. Malfatti biography
□ In 1802 he gave the first solution to the problem of describing in a triangle three circles that are mutually tangent, each of which touches two sides of the triangle, the so-called Malfatti
□ His solution was published in a paper of 1803 on un problema stereotomica.
358. Rennie biography
□ The JCMN is mainly concerned with mathematical problems, and their solution.
□ he offers a solution to the tomography problem (that is, to find out about the inside of a system through measurements on the outside) for electrical circuits with two-terminal linear
components such as resistors.
359. Szafraniec biography
□ The papers he wrote while he was undertaking research included: On a certain sequence of ordinary differential equations (1963); (with Andrzej Lasota) Sur les solutions periodiques d'une
equation differentielle ordinaire d'ordre n (1966) and (with Andrzej Lasota) Application of the differential equations with distributional coefficients to the optimal control theory (1968).
□ Out of his many noticeable results, we mention a few: simplified forms (including the diagonal one) of the boundedness condition in the famous Szokefalvi-Nagy general dilation theory together
with related integral representations of exponentially bounded operator-valued functions on abelian *-semigroups (unfortunately often attributed exclusively to a paper by Berg and Maserick,
which appeared later), foundations of the theory of unbounded subnormal operators (together with Jan Stochel), new solutions to multidimensional real and complex moment problems (together
with Jan Stochel), fresh look on interpolation theory, three term recurrence relations for orthogonal polynomials of several variables (together with Dariusz Cichon and Jan Stochel), and
advances in the theory of quantum harmonic oscillators and canonical commutation relations.
360. Schrodinger biography
□ The solution of the natural boundary value problem of this differential equation in wave mechanics is completely equivalent to the solution of Heisenberg's algebraic problem.
361. Foulis biography
□ Hyla wrote the solutions manual to Dave's first Calculus book, providing solutions to approximately 5000 problems, an impressive feat before the availability of graphing calculators.
362. Straus biography
□ An approximate solution of the field equations for empty space is obtained and the gravitational potentials thus determined are required to piece together continuously with the known
gravitational potentials for a pressure free, spatially constant density of matter.
□ Under these conditions it is possible to show that the Schwarzschild field can be transformed into a solution of the problem.
363. Petrovsky biography
□ the 1937 paper on the Cauchy problem for hyperbolic systems, the 1939 paper on the analyticity of solutions of elliptic systems, and the 1945 paper on lacunae for solutions of hyperbolic
364. Wilkinson biography
□ He began to put his greatest efforts into the numerical solution of hyperbolic partial differential equations, using finite difference methods and the method of characteristics.
□ In my opinion Wilkinson is single-handedly responsible for the creation of almost all of the current body of scientific knowledge about the computer solution of the problems of linear
365. Butzer biography
□ Moreover, it has already become indispensable in classical approximation theory, in the study of the initial and boundary behaviour of solutions of partial differential equations and in the
theory of singular integrals, because of the new results obtained by the authors in these areas.
□ Special as this approach may seem, it not only embraces many of the topics of the classical theory but also leads to significant new results, e.g., on summation processes of Fourier series,
conjugate functions, fractional integration and differentiation, limiting behaviour of solutions of partial differential equations, and saturation theory.
366. Siegel biography
□ the Lagrangian solutions for the three-body problem.
□ He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series.
367. Gelfond biography
□ Gelfond developed basic techniques in the study of transcendental numbers, that is numbers that are not the solution of an algebraic equation with rational coefficients.
□ This book is very much in the spirit of the modern Russian school concerned with the so-called constructive theory of functions, approximative methods for the solution of differential
equations, and so forth.
368. Wiener Christian biography
□ He also sought to simplify individual problems as much as possible and to find the easiest graphical solutions for them.
□ He was also interested in the problems and their solutions (such as shadow construction and brightness distribution), as well as in the development of the necessary geometric aids.
369. Mansur biography
□ Abu Nasr Mansur's main achievements are his commentry on the Spherics of Menelaus, his role in the development of trigonometry from Ptolemy's calculation with chords towards the trigonometric
functions used today, and his development of a set of tables which give easy numerical solutions to typical problems of spherical astronomy.
□ Menelaus's work formed the basis for Ptolemy's numerical solutions of spherical astronomy problems in the Almagest.
370. Adleman biography
□ Altogether only about one fiftieth of a teaspoon of solution was used.
□ 542-545.',10)">10]) a DNA solution to another famous 'NP-complete' problem - the so-called "satisfaction" problem (SAT).
371. Jungius biography
□ Empiricus remained verifiable through experience, Epistemonicus is grounded in principles and rules - as are the axioms of Euclid's geometry - and Heureticus reveals new methods for the
solution of problems previously insoluble.
□ A similar replacement occurs when copper itself is introduced into a solution of silver in aqua fortis.
372. Simon biography
□ The 1994 Bocher Prize is awarded to Leon Simon for his profound contributions towards understanding the structure of singular sets for solutions of variational problems.
□ These results left open basic questions about the structure of the set of singularities exhibited by the solutions of such variational problems.
373. Hironaka biography
□ Hironaka gave a general solution of this problem in any dimension in 1964 in Resolution of singularities of an algebraic variety over a field of characteristic zero.
□ Hironaka talked about his solution in his lecture On resolution of singularities (characteristic zero) to the International Congress of Mathematicians in Stockholm in 1962.
374. Von Neumann biography
□ If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper.
□ Haar's construction of measure in groups provided the inspiration for his wonderful partial solution of Hilbert's fifth problem, in which he proved the possibility of introducing analytical
parameters in compact groups.
375. Garnir biography
□ In particular, he studied Green's functions as solutions to boundary value problems for the wave and diffusion equations.
□ In the later part of his career, Garnir became interested in the propagation of singularities of solutions of boundary value problems for evolution partial differential equations.
376. Wronski biography
□ A piece of work which he had undertaken during this period resulted in a publication Resolution generale des equations de tous degres in 1812 claiming to show that every equation had an
algebraic solution.
□ For good measure, it contains a summary of the "general solution of the fifth degree equation".
377. Somov biography
□ In the theory of elliptical functions and their application to mechanics, he completed the solution of the problem concerning the rotation of a solid body around an immobile point in the
Euler-Poinsot and Lagrange-Poisson examples.
□ The first in Russia to deal with the solution of kinematic problems, Somov included a chapter on this topic in his textbook on theoretical mechanics.
378. Bartholin biography
□ The problem is the first example of an inverse tangent problem which in modern notation results in requiring the solution to the differential equation .
□ This has solution y = x + a(e-x/a - 1).
379. Levin biography
□ Levin solved the problem and, a few years later, published his solution in the paper Generalization of a theorem of Holder (Russian) (1934).
□ Of his results on the spectral theory of differential operators we shall mention only the construction, dating from the 50's, of the operator "attached to infinity" of the transformation for
the Schrodinger equation, which played an important part in the solution of the inverse problem in the theory of scattering.
380. Mendelsohn biography
□ He wrote papers on a wide variety of combinatorial problems, for example: Symbolic solution of card matching problems (1946), Applications of combinatorial formulae to generalizations of
Wilson's theorem (1949), Representations of positive real numbers by infinite sequences of integers (1952), A problem in combinatorial analysis (1953), The asymptotic series for a certain
class of permutation problems (1956), and Some elementary properties of ill conditioned matrices and linear equations (1956).
□ Two of Mendelsohn's papers An algorithmic solution for a word problem in group theory (1964) and (with Clark T Benson) A calculus for a certain class of word problems in groups (1966) were
particularly important in launching this strand of my own research career - thank you Nathan! .
381. Al-Khalili biography
□ The calculation of the direction of Mecca, as a function of terrestrial latitude and longitude, was one of the hardest of all problems of spherical trigonometry for which Islam required a
□ One possible solution is that al-Khalili had computed more accurate auxiliary tables before calculating his tables for the direction of Mecca but these are now lost.
382. Milne-Thomson biography
□ In 1948 he published Applications of elliptic functions to wind tunnel interference while in 1957 he wrote a review paper A general solution of the equations of hydrodynamics which M G
Scherberg reviews as follows:- .
□ Again in the case of torsion the presence of a couple about the axis suffices to give the distribution which leads to the solution.
383. Sridhara biography
□ Often after stating a rule Sridhara gives one or more numerical examples, but he does not give solutions to these example nor does he even give answers in this work.
□ In [Ganita 1 (1950), 1-12.',7)">7] Shukla examines Sridhara's method for finding rational solutions of Nx2 ± 1 = y2, 1 - Nx2 = y2, Nx2 ± C = y2, and C - Nx2 = y2 which Sridhara gives in the
384. Mascheroni biography
□ This encouraged him and he continued his work with two purposes in mind: to give a theoretical solution to the problem of constructions with compasses alone and to offer practical
constructions that might be of help in making precision instruments.
□ From the solution to these problems he is able to prove theoretically that any construction which can be made using a ruler and compasses can be made with compasses alone.
385. Banachiewicz biography
□ This, however, had been burnt down by the Germans on 15 September 1944, so after he took up his duties again after the war, Banachiewicz began to look for another solution.
□ For example: An outline of the Cracovian algorithms of the method of least squares (1942); On the accuracy of least squares solution (1945); Sur la resolution des equations normales de la
methode des moindres carres (1948); Sur l'interpolation dans le cas des intervalles inegaux (1949); A general least squares interpolation formula (1949); Les cracoviens et quelques-unes de
leurs applications en geodesie (1949); On the general least squares interpolation formula (1950); and Resolution d'un systeme d'equations lineaires algebriques par division (written much
earlier by only published in 1951 due to World War II) [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
386. Fields biography
□ After the award of the degree in 1887 for his thesis Symbolic Finite Solutions, and Solutions by Definite Integrals of the Equation (dn/dxn)y - (xm)y = 0, he remained teaching at Johns
Hopkins for a further two years.
387. Rosenhain biography
□ Adolph Gopel independently solved the same problem but he did not submit his solution for the Paris Academy prize so basically they only received one solution to the problem.
388. Bayes biography
□ The Essay, then, mainly, and perhaps justly, remembered for the solution of the problem posed by Bayes, should also be remembered for its contribution to pure mathematics.
□ This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.
389. Borelli biography
□ Daniele Spinola and Pietro Emmanuele both gave solutions to the problem and Borelli was asked to judge.
□ Although Borelli was critical of both solutions, he preferred that of Spinola and this led to an argument which became very heated.
390. Meshchersky biography
□ This paper generalised the solution of the problem of the flow of a jet around a symmetric wedge obtained by D K Bobylev in 1881 to a nonsymmetric wedge.
□ Meshchersky obtained a complete solution for this more complex case of flow around a nonsymmetric wedge and the paper [Studies in the history of physics and mechanics (Moscow, 1988),
201-217.',3)">3] considers in detail the mathematical methods which he used, in particular comparing his methods to analogous ones of Western authors.
391. Ince biography
□ Ince was the first to prove the proving the uniqueness of the Mathieu functions as periodic solutions.
□ It takes the existence of solutions for granted ..
392. Kummer biography
□ In fact the prize of 3000 francs was offered for a solution to Fermat's Last Theorem but when no solution was forthcoming, even after extending the date, the Prize was given to Kummer even
though he had not submitted an entry for the Prize.
393. La Roche biography
□ Here is an example of one of his problems with solution which we have put into modern notation:- .
□ Solution: Let the number be x.
394. Xiahou Yang biography
□ We have comments by Zhang Qiujian which criticise the accuracy of one of the solutions given in the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual).
□ The treatise contains three chapters in the usual style of problems and solutions.
395. Novikov biography
□ Jointly with Adian he showed that the problem of the finiteness of periodic groups proposed by Burnside in 1902 had a negative solution.
396. Oppenheim biography
□ The conjecture concerns Diophantine approximation and solutions of real quadratic forms which are not multiples of a rational form.
397. Woodward biography
□ This problem was one requiring for its solution mathematical work of the highest order and, in addition, the experience of the engineer, so to shape his formulas that they could be applied
directly by the computer.
398. Freedman biography
□ The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e.
399. Mohr Ernst biography
□ One of these 1951 papers looks at the numerical solution of the differential equation dy/dx = f (x, y).
400. Al-Biruni biography
□ The contents of the work include the Arabic nomenclature of shade and shadows, strange phenomena involving shadows, gnomonics, the history of the tangent and secant functions, applications of
the shadow functions to the astrolabe and to other instruments, shadow observations for the solution of various astronomical problems, and the shadow-determined times of Muslim prayers.
401. Stokes biography
□ He pursued the usual school studies, and attracted the attention of the mathematical master by his solution of geometrical problems.
402. Nagata biography
□ In fact Nagata announced his negative solution to Hilbert's 14th problem in his invited lecture On the fourteenth problem of Hilbert at the International Congress of Mathematicians held in
Edinburgh, Scotland, in August 1958.
403. Nash-Williams biography
□ This situation has, however, not deterred graph-theorists from studying the problem and obtaining some results which, although far from constituting a complete solution, are nevertheless
404. Francais Jacques biography
□ He applied his methods to the famous problem of finding a sphere tangent to four given spheres, publishing a number of notes on the topic between 1808 and 1812, and giving a complete solution
in the 1812 paper which appeared in Gergonne's Journal.
405. Feigenbaum biography
□ The modernisation of cartography done to archival standards poses many problems, the solutions for which are strongly illuminated by the ideas and methods of nonlinear systems.
406. Birnbaum biography
□ He showed me over and over again how mathematics could be used to look at a specific actuarial transaction, give it a general formulation, obtain a general solution, and make it part of an
inventory of techniques to be used in similar situations in the future.
407. Zeeman biography
□ You have to invent maths to get a solution to a problem but, in the process, I often discover a whole lot more which I didn't expect.
408. Henrici Peter biography
□ His next contribution Bergmans Integraloperator erster Art und Riemannsche Funktion (1952) is an elegant study of the representation of solutions of an elliptic partial differential equation
in terms of analytic functions.
409. Delamain biography
□ After having largely explained the use of the 'Grammelogia' in the solution of a variety of questions in proportion, in interest, in annuities, in the extraction of roots, etc., the author
concludes thus: "If there be composed three circles of equal thickness, A, B, and C, so that the inner edge of B, and the outer edge of A, be answerably graduated with logarithmic sines; and
the outer edge of B, and the inner edge of C, with logarithms, and then, on the backside, be graduated the logarithmic tangents, and again the logarithmic sines opposite to the former
graduations, it shall be fitted for the resolution of plain and spherical triangles.
410. Feldman biography
□ Then follows a treatment of the methods of Gelfond and Schneider which led to the solution of Hilbert's seventh problem.
411. Mazur Barry biography
□ Mazur had much earlier received the Cole prize for work which would prove important in the solution of Fermat's last Theorem.
412. Mydorge biography
□ Mydorge left an unpublished manuscript Traite de geometrie of over 1000 geometric problems and their solutions.
413. Lichtenstein biography
□ The solutions as functions of the boundary values and the parameter).
414. Schauder biography
□ This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.
415. Girard Pierre biography
□ I thank you for the solution you sent me to the geometry problem..
416. Mises biography
□ His Institute rapidly became a centre for research into areas such as probability, statistics, numerical solutions of differential equations, elasticity and aerodynamics.
417. Wilder biography
□ He suggested Wilder write up the solution to the problem for his doctorate which indeed he did, becoming Moore's first Texas doctorate in 1923 with his dissertation Concerning Continuous
418. Kneser Hellmuth biography
□ For example he produced a beautiful solution to the functional equation f ( f (x) ) = ex which he published in 1950, and the deep understanding he achieved of the strange properties of
manifolds without a countable basis of neighbourhoods between 1958 and 1964.
419. Gopel biography
□ finally, after ingenious calculations, obtained the result that the quotients of two theta functions are solutions of the Jacobian problem for p = 2.
420. Weierstrass biography
□ ., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I
made this decision in my seventh semester ..
421. Pascal Etienne biography
□ At the beginning of 1637 Fermat wrote his "Solution d'un probleme propose par M de Pascal".
422. Broglie biography
□ He wrote at least twenty-five books including Ondes et mouvements (Waves and motions) (1926), La mecanique ondulatoire (Wave mechanics) (1928), Une tentative d'interpretation causale et non
lineaire de la mecanique ondulatoire: la theorie de la double solution (1956), Introduction a la nouvelle theorie des particules de M Jean-Pierre Vigier et de ses collaborateurs (1961), Etude
critique des bases de l'interpretation actuelle de la mecanique ondulatoire (1963).
423. Vijayanandi biography
□ This system led to much work on integer solutions of equations and their application to astronomy.
424. Plemelj biography
□ Riemann's problem, concerning the existence of a linear differential equation of the Fuchsian class with prescribed regular singular points and monodromy group, had been reduced to the
solution of an integral equation by Hilbert in 1905.
425. Bliss Nathaniel biography
□ This was a period when clocks were proving to be the solution to the longitude problem and Harrison's clock H4 was being tested during his time in Greenwich.
426. Picard Jean biography
□ Prodigious engineering efforts went into the solutions to this problem, and Picard's new levelling instruments with telescopic sights helped determine routes and avoided costly errors.
427. Jensen biography
□ The theorem is important, but does not lead to a solution of the Riemann Hypothesis as Jensen had hoped.
428. Mellin biography
□ He also extended his transform to several variables and applied it to the solution of partial differential equations.
429. Anthemius biography
□ Heath [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2] gives Anthemius's solution:- .
430. Vandiver biography
□ When he was eighteen years old he began to solve many of the number theory problems which were posed in the American Mathematical Monthly, regularly submitting solutions.
431. Van der Waerden biography
□ These are solutions of the Burnside problem.
432. Schmidt Harry biography
□ A rigorous solution of the Navier-Stokes equations as an example.
433. Fefferman biography
□ Professor Charles Fefferman's contributions and ideas have had an impact on the development of modern analysis, differential equations, mathematical physics and geometry, with his most recent
work including his sharp (computable) solution of the Whitney extension problem.
434. Dijkstra biography
□ she had a great agility in manipulating formulae and a wonderful gift for finding very elegant solutions.
435. Epstein biography
□ Emigration would probably have been impossible in any case with the outbreak of war only a few weeks away, and Hitler's so-called final solution to the Jewish problem following shortly
436. Mitchell biography
□ He was instrumental in bringing the subject of spurious solutions to the fore.
437. Gateaux biography
□ He recalled that Volterra introduced this notion to study problems including an hereditary phenomenon, but also that it was used by others (Jacques Hadamard and Paul Levy) to study some
problems of mathematical physics - such as the equilibrium problem of fitted elastic plates - finding a solution to equations with functional derivatives, or, in other words, by calculating a
relation between this functional and its derivative.
438. Andreotti biography
□ Classical theorems on removable singularities and existence and uniqueness of solutions to the Cauchy problem are extended to some systems of partial differential operators through this
439. Lorentz George biography
□ Chapters nine, ten, and eleven concern entropy and Kolmogorov's solution of Hilbert's thirteenth problem.
440. Diocles biography
□ The solution of this problem would, of course, have interesting consequences for the construction of a sundial.
441. Thompson Abigail biography
□ Her paper "Thin position and the recognition problem for S3 " (1994), used the idea of thin position to reinterpret Rubenstein's solution to the recognition problem of the 3-sphere in a
startling way.
442. Mobius biography
□ His intuition, the problems he set himself, and the solutions that he found, all exhibit something extraordinarily ingenious, something original in an uncontrived way.
443. Peierls biography
□ Surprises in theoretical physics are either theoretical results in disagreement with naive physical intuition, or simple solutions to apparently unmanageable problems.
444. Jacobsthal biography
□ He also showed that it is possible to find a solution p = x2 + y2 where x and y can be expressed with simple sums over Legendre symbols.
445. Wessel biography
□ He possesses a lot of theoretical knowledge of algebra, trigonometry and mathematical geometry, and as far as the last point is concerned, he has come up with some new and beautiful solutions
to the most difficult problems in geographical surveying.
446. Pringsheim biography
□ Before he left Germany to go to Zurich, Pringsheim gave to his friend Caratheodory a present of a very rare text from Jacob Bernoulli to his brother Johann Bernoulli containing the solution
to the isoperimetric problem.
447. Riesz Marcel biography
□ In Problems related to characteristic surfaces Riesz extended these ideas to obtain the solution of the wave equation for a very general class of characteristic boundaries.
448. Emerson biography
□ He returned to fluxions, publishing The Method of Increments herein the principles are demonstrated and the Practice thereof shown in the Solution of Problems in 1763.
449. Ward Seth biography
□ Arithmetic and geometry are sincerely and profoundly taught, analytical algebra, the solution and application of equations, containing the whole mystery of both those sciences, being
faithfully expounded in the Schools by the Professor of Geometry, and in several Colleges by particular tutors.
450. Coulson biography
□ In almost every case the fundamental problem is the same, since it consists in solving the standard equation of wave motion; the various applications differ chiefly in the conditions imposed
upon these solutions.
451. Hopf Eberhard biography
□ He held this post until 1947 by which time he had returned to the United States, where he presented the definitive solution of Hurewicz's problem.
452. Pitiscus biography
□ The word 'trigonometry' is due to Pitiscus and first occurs in the title of his work Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus first published in Heidelberg
in 1595 as the final section of A Scultetus's Sphaericorum libri tres methodice conscripti et utilibus scholiis expositi.
453. Shannon biography
□ At the Massachusetts Institute of Technology he also worked on the differential analyser, an early type of mechanical computer developed by Vannevar Bush for obtaining numerical solutions to
ordinary differential equations.
454. Cooper William biography
□ In the first part, Cooper and Henderson present the complete solution of a numerical example by the "simplex method" of G B Dantzig.
455. Lehmer Derrick N biography
□ "Even a worm will turn," and now electricity and light, which have in the past gone to mathematics for solutions of their intricate problem, turn about and solve problems in mathematics which
would require scores of years to complete.
456. Smirnov biography
□ With Sobolev he devised a method for obtaining solutions on the propagation of waves in elastic media with plane boundaries.
457. Schmidt biography
□ He showed that in this case the integral equation had real eigenvalues, Hilbert's word, and the solutions corresponding to these eigenvalues he called eigenfunctions.
458. Fuss biography
□ Most of Fuss's papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations.
459. Tarry biography
□ He published a solution to the problem of finding the way out of a maze in 1895, a problem which had been of interest from classical times.
460. Artin Michael biography
□ The point of the extension is that Artin's theorem on approximating formal power series solutions allows one to show that many moduli spaces are actually algebraic spaces and so can be
studied by the methods of algebraic geometry.
461. Le Cam biography
□ He made many contributions to the solution of practical problems such as studying stochastic models for rainfall, for the effects of radiation on living cells, sodium channel modelling and
for cancer metastasis.
462. Hedrick biography
□ He was awarded a doctorate by Gottingen in February 1901 for a dissertation, supervised by Hilbert, Uber den analytischen Charakter der Losungen von Differentialgleichungen (On the analytic
character of solutions of differential equations).
463. Bieberbach biography
□ There he worked out the details of his solution to the first part of Hilbert's eighteenth problem publishing them in two papers Uber die Bewegungsgruppen der Euklidischen Raume (1911, 1912).
464. Morera biography
□ his results consisted of] solutions to complicated and difficult problems.
465. Koch biography
□ Yet this work can be said to be the first step on the long road which eventually led to functional analysis, since it provided Fredholm with the key for the solution of his integral equation.
466. Thomae biography
□ He also discovered methods of solving difference equations giving solutions in the form of definite integrals.
467. Birman biography
□ We had a course in Euclidean geometry, and every single night we would have telephone conversations and argue over the solutions to the geometry problems.
468. Bezout biography
□ His first paper on the theory of equations Sur plusieurs classes d'equations de tous les degres qui admettent une solution algebrique examined how a single equation in a single unknown could
be attacked by writing it as two equations in two unknowns.
469. Hippocrates biography
□ Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.
470. Skopin biography
□ At that time, a positive solution of the restricted Burnside problem was known only for a prime exponent.
471. Bell John biography
□ The solution to the EPR problem that Einstein would have liked, rejecting (1) but retaining (2) was illegitimate.
472. Bass biography
□ The solution of the congruence subgroup problem is presented as a pivotal event.
473. Pfaff biography
□ In 1810 he contributed to the solution of a problem due to Gauss concerning the ellipse of greatest area which could be drawn inside a given quadrilateral.
474. Lax Anneli biography
□ this book includes problems, solutions, historical notes, and bibliographical references that go beyond undergraduate enrichment.
475. Schneider biography
□ The work on transcendence which had led Schneider to his solution of Hilbert's Seventh Problem led him to extend to a more general programme studying the transcendence of elliptic functions,
modular functions and abelian functions.
476. Severi biography
□ His most impressive work came before he went to Rome but, despite spending less time on mathematics, after this he still managed to produce work of the greatest importance like the solution
of the Dirichlet problem and his development of the theory of rational equivalence.
477. Lusztig biography
□ He was able to answer the question but they had asked F P Peterson, who was at the Massachusetts Institute of Technology, the same question a couple of months earlier and it was soon
discovered that he had found a similar solution.
478. Weingarten biography
□ In this work he reduced the problem of finding all surfaces isometric to a given surface to the problem of determining all solutions to a partial differential equation of the Monge-Ampere
479. Lafforgue biography
□ This law allows one to describe, for any positive integer d, the primes p for which the congruence x2 = d mod p has a solution.
480. Hill biography
□ His new idea on how to approach the solution to the three body problem involved solving the restricted problem.
481. Lovasz biography
□ This book presents a nice survey of some recent developments towards the efficient solution of computational problems in areas like graph theory, number theory, and combinatorial
482. Adams Edwin biography
□ The solution of two-dimensional electrical and hydrodynamical problems connected with a grating of rounded bars was obtained by H W Richmond beginning with a grating of bars of rectangular
483. Dougall biography
□ Examples of papers he read at meeting of the Society are Elementary Proof of the Collinearity of the Mid Points of the Diagonals of a Complete Quadrilateral on Friday 12 February 1897;
Methods of Solution of the Equations of Elasticity on 10 December 1897; and Notes on Spherical Harmonics on 12 December 1913.
484. Perron biography
□ One of the things he is best-known for is the Perron paradox which highlights the danger of assuming that a solution to a problem exists.
485. Turing biography
□ He was criticised for his handwriting, struggled at English, and even in mathematics he was too interested with his own ideas to produce solutions to problems using the methods taught by his
486. Kahler biography
□ His thesis advisor was Leon Lichtenstein but Kahler chose himself the topic for his doctoral dissertation Uber die Existenz von Gleichgewichtsfiguren, die sich aus gewissen Losungen des
n-Korperproblems ableiten (On the existence of equilibrium figures that are derived from certain solutions of the n-body problem).
487. Boscovich biography
□ His solution to this minimising problem took a geometric form.
488. Szasz Domokos biography
□ The paper presents a partial solution of a classical open problem in mathematical physics, which is to prove rigorously the ergodicity of a system consisting of any number of identical hard
balls in a box with periodic boundary conditions (i.e., on a torus).
489. Cartwright biography
□ For something to do we went on and on at the thing with no earthly prospect of "results"; suddenly the whole vista of the dramatic fine structure of solutions stared us in the face.
490. Tukey biography
□ the usefulness and limitation of mathematical statistics; the importance of having methods of statistical analysis that are robust to violations of the assumptions underlying their use; the
need to amass experience of the behaviour of specific methods of analysis in order to provide guidance on their use; the importance of allowing the possibility of data's influencing the
choice of method by which they are analysed; the need for statisticians to reject the role of 'guardian of proven truth', and to resist attempts to provide once-for-all solutions and tidy
over-unifications of the subject; the iterative nature of data analysis; implications of the increasing power, availability and cheapness of computing facilities; the training of
491. Kodaira biography
□ Profusely illustrated and with plenty of examples, and problems (solutions to many of which are included), this book should be a stimulating text for advanced courses in complex analysis.
492. Pairman biography
□ Her thesis advisor was George Birkhoff and after submitting her thesis Expansion Theorems for Solution of a Fredholm's Linear Homogeneous Integral Equation of the Second Kind with Kernel of
Special Non-Symmetric Type she was awarded a Ph.D.
493. Al-Farisi biography
□ He noted the impossibility of giving an integer solution to the equation .
494. Fine Nathan biography
□ Fine was also interested in problem solving and contributed both problems and solutions to problems to several different journals.
495. Bessel-Hagen biography
□ Caratheodory thought Bessel-Hagen's disertation the first important advance in the theory of discontinuous solutions for problems in the calculus of variations since his own work in 1905.
496. Skolem biography
□ It was entitled Einige Satze uber ganzzahlige Losungen gewisser Gleichungen und Ungleichungen, and was on integral solutions of certain algebraic equations and inequalities.
497. Lagny biography
□ In about 1690 he developed a method of giving approximate solutions of algebraic equations and, in 1694, Halley published a twelve page paper in the Philosophical Transactions of the Royal
Society giving his method of solving polynomial equations by successive approximation which is essentially the same as that given by Lagny a few years earlier.
498. Chernoff biography
□ He took courses by Bers, Feller, Loewner, Tamarkin, and others, and wrote a Master's dissertation Complex Solutions of Partial Differential Equations under Bergman's supervision.
499. Brisson biography
□ The main idea in these reports was the application of the functional calculus, through symbols, to the solution of certain kinds of linear differential equations and of linear equations with
finite differences.
500. Clifford biography
□ Without any diagram or symbolic aid he described the geometrical conditions on which the solution depended, and they seemed to stand out visibly in space.
501. Hilbert biography
□ we hear within ourselves the constant cry: There is the problem, seek the solution.
502. Baer biography
□ All of this will require careful planning, and there are some problems for which I cannot adequately envision solutions, and for this reason I am turning to you for advice.
503. Bellavitis biography
□ In algebra he continued Ruffini's work on the numerical solution of algebraic equations and he also worked on number theory.
504. Hall biography
□ He introduced the idea of a normal form which he used in the solution of the word problem for Lie rings and also for nilpotent groups.
505. Piaggio biography
□ The author effectively remarks that this is not a necessary condition for the existence of a solution.
506. Laurent Hermann biography
□ The last three volumes are devoted entirely to the solution and application of ordinary and partial differential equations.
507. Donaldson biography
□ Using instantons, solutions to the equations of Yang-Mills gauge theory, he gained important insight into the structure of closed four-manifolds.
508. Barclay biography
□ Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussion of new problems or new solutions, and
comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.
509. Gillman biography
□ This paper is concerned with the solution of an example of a zero-sum two-person game with essentially bounded measurable functions on (0, ) as pure strategies.
510. Miranda biography
□ Examples of his work around this time are: Su un problema di Minkowski (1939) which considers the problem of determining a convex surface of given Gaussian curvature; Su alcuni sviluppi in
serie procedenti per funzioni non necessariamente ortogonali (1939) which examines expansion theorems in terms of the characteristic solutions of an integral equation whose kernel, although
symmetric, involves the characteristic parameter; Nuovi contributi alla teoria delle equazioni integrali lineari con nucleo dipendente dal parametro (1940) which examines the development of
the Hilbert-Schmidt theory for a particular type of linear integral equation; and Observations on a theorem of Brouwer (1940) which gave an elementary proof of the equivalence of Brouwer's
fixed point theorem and a special case of Kronecker's index theorem.
511. Tapia biography
□ In it Tapia considered the solution of the equation P(x) = 0, where P is a nonlinear mapping between Banach spaces.
512. Nikodym biography
□ the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, the Nikodym-Grothendieck boundedness theorem), in functional analysis (the Radon-Nikodym property of a Banach space,
the Frechet-Nikodym metric space, a Nikodym set), projections onto convex sets with applications to Dirichlet problem, generalized solutions of differential equations, descriptive set theory
and the foundations of quantum mechanics.
513. Sato biography
□ Sato explained the new theory of microlocal analysis in his lecture Regularity of hyperfunctions solutions of partial differential equations at the International Congress of Mathematicians at
Nice in 1970, but the details appear in the 165 page paper by Sato, Kawai and Kashiwara Microfunctions and pseudo-differential equations in the proceedings of the Katata Conference held in
514. Cooper biography
□ Other papers in which deal with applications include The uniqueness of the solution of the equation of heat conduction (1950).
515. Sporus biography
□ His solution of the problem of duplicating the cube is similar to that of Diocles and in fact Pappus also followed a similar construction.
516. Drach biography
□ of classifying the transcendental quantities satisfying the rational system verified by the solutions.
517. Smullyan biography
□ Problems which had a unique solution, yet looked quite impossible.
518. Friedrichs biography
□ This second paper gives an exact solution to the problem of the behaviour of a thin circular plate after buckling under a uniform edge force applied in the plane of the plate.
519. Thue biography
□ If f (x, y) is a homogeneous polynomial with integer coefficients, irreducible in the rationals and of degree > 2 and c is a non-zero integer then f (x, y) = c has only a finite number of
integer solutions.
520. Scorza biography
□ His first publication, which appear in print when he was only eighteen years old, gave the solution to two mathematical problems posed in the journal for secondary mathematics teaching.
521. Koopmans biography
□ He showed that the desired result is obtainable by the straightforward solution of a system of equations involving the costs of the materials at their sources and the costs of shipping them
by alternative routes.
522. Ozanam biography
□ He also wrote many works on mathematics, for example Methode generale pour tracer les cadrans (1673), La geometrie pratique du sr Boulenger (1684), Traite de la construction des equations
pour la solution des problemes indeterminez (1687), Traite des lieux geometriques (1687), Traite des lignes du premier genre (1687), De l'usage du compas de proportion (1688).
523. Dase biography
□ With small numbers, everybody that possesses any readiness in reckoning, sees the answer to such a question [the divisibility of a number] at once directly, for greater numbers with more or
less trouble; this trouble grows in an increasing relation as the numbers grow, till even a practiced reckoner requires hours, yes days, for a single number; for still greater numbers, the
solution by special calculation is entirely impractical.
524. Valerio biography
□ (The authors mean this claim in the sense that Valerio was the first to systematize this ancient device, in the first three theorems of 'De centro'.) In applying these general theorems to the
solution of a wide class of problems, Valerio thus advanced beyond his ancient Renaissance predecessors.
525. Hobbes biography
□ it is clear that he hoped to assert preeminence in the learned world largely on the basis of the solution of the problem of squaring the circle.
526. Kirchhoff biography
□ Kirchhoff considered an electrical network consisting of circuits joined at nodes of the network and gave laws which reduce the calculation of the currents in each loop to the solution of
algebraic equations.
527. Moser William biography
□ He edited (some with Ed Barbeau) the booklets containing the problems, solutions and results of the Canadian mathematical Olympiads from 1969 to 1978.
528. Uhlenbeck Karen biography
□ Among other things, she sketched some aspects of Simon Donaldson's work on the geometry of four-dimensional manifolds, instantons - solutions, that is, of a certain nonlinear system of
partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
529. Baudhayana biography
□ The Sulbasutra of Baudhayana contains geometric solutions (but not algebraic ones) of a linear equation in a single unknown.
530. Vajda biography
□ Of course, when these problems are cast in a Linear Programming form, the optimal solutions are integral, which results in the relevance of Linear Programming.
531. Young Andrew biography
□ He read the paper On the quasi-periodic solutions of Mathieu's differential equation to the Society at its meeting on Friday 13 February 1914.
532. Bassi biography
□ In the end Benedict XIV went for a compromise solution by appointing Bassi as the twenty-fifth Benedettini but not giving her the same voting rights as the others.
533. Sullivan biography
□ Beyond the solution of difficult outstanding problems, his work has generated important and active areas of research pursued by many mathematicians.
534. Weldon biography
□ Biometrika will include (a) memoirs on variation, inheritance, and selection in animals and plants, based upon the examination of statistically large numbers of specimens (this will of course
include statistical investigations of anthropometry); (b) those developments of statistical theory which are applicable to biological problems; (c) numerical tables and graphical solutions
tending to reduce the labour of statistical arithmetic; (d) abstracts of memoirs, dealing with these subjects, which are published elsewhere; and (e) notes on current biometric work and
unsolved problems.
535. Hayes David biography
□ If a seemingly insuperable problem appeared, he would keep attempting a solution until all roadblocks were cleared and the way to further progress was open.
536. Williams biography
□ In 1925 Cox was awarded his doctorate by Cornell University for his thesis Polynomial solutions of difference equations.
537. Possel biography
□ announces the solution of several variants of the game of Nim with the following rules: Two players alternate in removing matches from p piles T1,..
538. Christoffel biography
□ The procedure Christoffel employed in his solution of the equivalence problem is what Gregorio Ricci-Curbastro later called covariant differentiation, Christoffel also used the latter concept
to define the basic Riemann-Christoffel curvature tensor.
539. Jia Xian biography
□ Jia Xian is known to have written two mathematics books: Huangdi Jiuzhang Suanjing Xicao (The Yellow Emperor's detailed solutions to the Nine Chapters on the Mathematical Art), and Suanfa
Xuegu Ji (A collection of ancient mathematical rules).
540. Schubert Hans biography
□ Under certain conditions, he obtains solutions in terms of Bessel and Hankel functions and computes the resulting formulas for downwash.
541. Saint-Venant biography
□ In the 1850s Saint-Venant derived solutions for the torsion of non-circular cylinders.
542. Crelle biography
□ The solution was simple, even if it required a change in policy, and that was to have a second journal for more practical mathematics and this he moved to a second journal which he started in
1829, the Journal fur die Baukunst.
543. Blades biography
□ For example, he communicated On Spheroidal Harmonics and Allied Functions, by Mr G B Jeffery to the meeting on Friday 11 June 1915 and Transformations of Axes for Whittaker's Solution of
Laplace's Equation, by Dr G B Jeffery to the meeting on Friday 9 March 1917.
544. Bukreev biography
□ The geodesics as solutions of the Euler equation, Gauss curvature, geodesic curvature; II.
545. Julia biography
□ The present book contains a small number of carefully chosen problems, each problem followed by one or more complete solutions.
546. Miller Kelly biography
□ But he realized early that the Negro college student of that period and in the years immediately ahead needed to be awakened to a realization of the problems of the race and an interest in
their solution.
547. Routledge biography
□ Hall gave his students old examination papers to solve and then criticised the solutions they produced.
548. Bombieri biography
□ He has repeatedly demonstrated an ability to quickly master essentials of a complicated new field, to select important problems which are accessible, and to apply intense energy and insight
to their solution, making liberal use of deep results of other mathematicians in widely differing areas.
549. Airy biography
□ He had earlier published a paper On a peculiar Defect in the Eye on this problem for which he was the first to provide a practical solution.
550. Butters biography
□ He also contributed to the mathematical work of the Society, For example at the meeting of the Society on Friday 11 January 1889, J Watt Butters discussed the solution of an algebraic
551. Beaugrand biography
□ In 1634 he was appointed to the committee which was set up by Cardinal Richelieu to evaluate Jean-Baptiste Morin's solution to the longitude problem by measuring absolute time from the
position of the Moon relative to the stars.
552. Post biography
□ Post showed that the word problem for semigroups was recursively insoluble in 1947, giving the solution to a problem which had been posed by Thue in 1914.
553. Rademacher biography
□ A large part of the recent developments were initiated by the author's solution of the problem of unrestricted partitions and many of the results are due either to him or to his direct
554. Lexell biography
□ It involves the solution of polygons given certain sides and angles between them, their mensuration, division by diagonals, circumscribing polygons around circles and inscribing polygons in
555. Wren biography
□ Of course Charles II was not having an observatory built to push forward scientific research, rather he wanted a solution to the longitude problem which would give England a huge advantage
over its competitors as a sea-faring nation.
556. Fine Henry biography
□ Two further paper On the functions defined by differential equations with an extension of the Puiseux polygon construction to these equations, and Singular solutions of ordinary differential
equations appeared in 1889 and 1890 respectively.
557. Farkas biography
□ In 1881 Gyula Farkas published a paper on Farkas Bolyai's iterative solution to the trinomial equation, making a careful study of the convergence of the algorithm.
558. Lerch biography
□ He is remembered today for his solution of integral equations in operator calculus and for the 'Lerch formula' for the derivative of Kummer's trigonometric expansion for log G(v).
559. Slater biography
□ He wrote papers on this topic such as: (with R H Fowler) Collision numbers in solutions (1938), The rates of unimolecular reactions in gases (1939), Aspects of a theory of unimolecular
reaction rates (1948), and Gaseous unimolecular reactions: theory of the effects of pressure and of vibrational degeneracy (1953).
560. Stifel biography
□ One of the advances in Stifel's notes is an early attempt to use negative numbers to reduce the solution of a quadratic equation to a single case.
561. Plancherel biography
□ He also contributed to the solutions to variational problems via Ritz' method and to ergodic theory.
562. Gegenbauer biography
□ The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.
563. Strassen biography
□ Using this new matrix multiplication routine, Strassen was able to show that Gaussian elimination (an efficient algorithm for solving systems of linear equations) is not an optimal solution.
564. Watson Henry biography
□ The only solution received did not please Galton ..
565. Ehrenfest biography
□ He corresponded with Klein who told him that what was required was a survey, not a complete solution of all the problems of the subject by Ehrenfest himself.
566. Amsler biography
□ That was the problem of the attraction of an ellipsoid, which was first studied in depth by Ivory whose solution was later generalised by Poisson.
567. Malcev biography
□ mathematical work is distinguished by the abundance of new ideas and the creation of new mathematical trends on the one hand, and the solution of a number of classical problems on the other
568. Mahler biography
□ I almost immediately posed him the following problem: An integer is called powerful if p | m implies p2 | m; are there infinitely many consecutive powerful numbers? Mahler immediately
answered: Trivially, yes! x2 - 8 y2 = 1 has infinitely many solutions.
569. Aryabhata I biography
□ This work is the first we are aware of which examines integer solutions to equations of the form by = ax + c and by = ax - c, where a, b, c are integers.
570. Morawetz biography
□ During the 1970s she extended this work to examine other solutions to the wave equation.
571. Hermite biography
□ He had found general solutions to the equations in terms of theta-functions.
572. Beurling biography
□ Quite possibly the finest feat of cryptoanalysis performed during the Second World War was Arne Beurling's solution of the secret of the Geheimschreiber.
573. Rouche biography
□ This is the well-known criterion which says that a system of linear equations has a solution if and only if the rank of the matrix of the associated homogeneous system is equal to the rank of
the augumented matrix of the system.
574. De Beaune biography
□ According to Beaugrand, the first of these problems - which in the present state of textual study appears to concern itself only with the determination of the tangent to an analytically
defined curve - interested Debeaune "in a design touching on dioptrics." As to the second of these problems, the one that has been particularly identified with Debeaune and that ushered in
what was called at the end of the seventeenth century the "inverse of tangents" i.e., the determination of a curve from a property of its tangent - Debeaune told Mersenne on 5 March 1639 that
he sought a solution with only one precise aim: to prove that the isochronism of string vibrations and of pendulum oscillations was independent of the amplitude.
575. Euclid biography
□ Euclid's geometric solution of a quadratic equation .
576. Hopf biography
□ The boldness of the questions deserves as much admiration as the surprising results of the solutions.
577. Jyesthadeva biography
□ Other mathematical results presented by Jyesthadeva include topics studied by earlier Indian mathematicians such as integer solutions of systems of first degree equation solved by the kuttaka
method, and rules of finding the sines and the cosines of the sum and difference of two angles.
578. Stallings biography
□ This paper is an ingenious solution of the conjecture for dimension n ≥ 7.
579. Tacquet biography
□ Tacquet rejected all notions that solids are composed of planes, planes of lines, and so on, except as heuristic devices for finding solutions.
580. Peterson biography
□ by means of a uniform general method, he deduced nearly all the devices known at that time for finding general solutions of different classes of equations.
581. Wilkins biography
□ He looked at the problem of different languages in different parts of the world and looked for a solution to this curse which hindered learning.
582. Browder Felix biography
□ In nonlinear functional analysis the introduction of monotone and, later, accretive operator theory led to the solution of problems that had heretofore been out of reach.
583. Plato biography
□ He remained in Syracuse for part of 360 BC but did not achieve a political solution to the rivalry.
584. Lie biography
□ He examined his contact transformations considering how they affected a process due to Jacobi of generating further solutions of differential equations from a given one.
585. De Moivre biography
□ Dupont looks at this problem, and Todhunter's solution, in [Atti Accad.
586. Bott biography
□ Synge then came up with the perfect solution - why not move straight to a doctorate? Richard Duffin became Bott's supervisor and the first problem they solved was one which Bott suggested
587. Adler biography
□ His theoretical solution involved giving specific constructions, such as bisecting a circular arc, using only a compass.
588. Kirillov biography
□ This text, based on courses and seminars at Moscow University, consists of three main parts: expository text, problems and hints for solution.
589. Bronowski biography
□ In 1933 he published a solution of the classical functional Waring problem, to determine the minimal n such that a general degree d polynomial f can be expressed as a sum of dth powers of n
linear forms, but his argument was incomplete.
590. Prodi biography
□ This book was in the line of the work of Renato Caccioppoli and like the pioneering monograph 'Problemi di esistenza in analisi funzionale' of Carlo Miranda [1949], it put the emphasis upon
the application of global implicit function theorems to the existence and multiplicity of solutions of nonlinear elliptic partial differential equations.
591. Kneser biography
□ But above all, the decisive advances towards the solution of the so-called Mayer Problem, recently introduced to the calculus of variations, are due to Kneser.
592. Halley biography
□ Halley's other activities included studying archaeology, geophysics, the history of astronomy, and the solution of polynomial equations.
593. Bobillier biography
□ The second and the third Books, deal with the solution of problems, and the equations which derive from them; the latter, with certain algebraic methods which enable numerical calculations to
be shortened.
594. Eisenstein biography
□ When he was about ten years old his parents tried to find a solution to his continual health problems by sending him to Cauer Academy in Charlottenburg, a district of Berlin which was not
incorporated into the city until 1920.
595. Green biography
□ The formula connecting surface and volume integrals, now known as Green's theorem, was introduced in the work, as was "Green's function" the concept now extensively used in the solution of
partial differential equations.
596. Kostrikin biography
□ The meaning of an algebraic concept can be of a number-theoretic or geometric nature, and frequently its roots lie in computational aspects of mathematics and in the solution of equations.
597. Lobachevsky biography
□ This method of numerical solution of algebraic equations, developed independently by Graffe to answer a prize question of the Berlin Academy, is today a particularly suitable method for using
computers to solve such problems.
598. Mauchly biography
□ For his pioneering contributions to automatic computing by participating in the design and construction of the ENIAC, the world's first all-electronic computer, and of the BINAC and the
UNIVAC, and for his pioneering efforts in the application of electronic computers to the solution of scientific and business problems.
599. Wigner biography
□ epoch-making work on how symmetry is implemented in quantum mechanics, the determination of all the irreducible unitary representations of the Poincare group, and his work with Bargmann on
realizing those irreducible unitary representations as the Hilbert spaces of solutions of relativistic wave equations, ..
600. Jordanus biography
□ The solution illustrates the use of letters by Jordanus:- .
601. Ehrenfest-Afanassjewa biography
□ They corresponded with Klein who told them that what was required was a survey, not a complete solution of all the problems of the subject by the Ehrenfests themselves.
602. Netto biography
□ Despite this, Netto's "proof" was widely accepted as providing a solution to the dimension problem until Jurgens' criticism in 1899 of Netto's proof.
603. Salem biography
□ Much of this progress is due to [Salem] and people influenced by his ideas, and acquaintance with his work seems to be a prerequisite for those who would like to contribute to the solution of
the problem.
604. Wang Xiaotong biography
□ Try to set up the necessary equations in these two cases in a similar way to our solution to Problem 15 above.
605. Van der Pol biography
□ We explain the history of the development of the equation carrying his name, and also the origins of the method of finding the first approximation to the solution of this equation (the method
of slowly varying coefficients).
606. Haret biography
□ Taking also into account commensurabilities, and using generalized Fourier series (which generate quasiperiodic solutions), Poincare proved the divergence of these series, which means
instability, confirming in this way Haret's result.
607. Sripati biography
□ Other mathematics included in Sripati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by
Brahmagupta .
608. Macdonald biography
□ He corrected his 1903 solution to the problem of a perfectly conducting sphere embedded in an infinite homogeneous dielectric in 1904 after a subtle error was pointed out by Poincare.
609. Maurolico biography
□ Demonstratio algebrae, which is an elementary text looking at quadratic equations and problems whose solution reduces to solving a quadratic; .
610. Somerville biography
□ In this correspondence they discussed the mathematical problems set in the Mathematical Repository and in 1811 Mary received a silver medal for her solution to one of these problems.
611. Dodgson biography
□ As early as 1894 Dodgson used truth tables for the solution of specific logic problems.
612. Helly biography
□ He taught in a Gymnasium, gave private tuition, and wrote solution manuals for a series of standard textbooks.
613. Kaczmarz biography
□ There is Kaczmarz's algorithm for the approximate solution for systems of linear equations which appears in his paper Angenaherte Auflosung von Systemen linearer Gleichungen published in the
Bulletin International de l'Academie Polonaise des Sciences et des Lettres in 1937.
614. Morin Jean-Baptiste biography
□ His solution, proposed in 1634, was based on measuring absolute time by the position of the Moon relative to the stars.
615. Saunderson biography
□ The final book presents the solution of cubic and quartic equations.
616. Gowers biography
□ Dr Gowers' achievements include the following: a solution to the notorious Banach hyperplane problem (to find a Banach space which is not isomorphic to any hyperplane), a counterexample to
the Banach space Schroder-Bernstein theorem, a proof that if all closed infinite-dimensional subspaces of a Banach space are isomorphic then it is a Hilbert space, and an example of a Banach
space such that every bounded operator is a Fredholm operator.
617. Cohen Wim biography
□ It concerned fundamental research in direct relation to practical engineering problems: his elegant and deep mathematical solution of the problem of stresses and displacements in helicoidal
shells and ship propeller blades has proved of great importance in ship engineering building.
618. Alberti biography
□ It was again Alberti who found the solution that remained influential up to our own days.
619. Levy Hyman biography
□ Levy's main work was in numerical methods, numerical solution of differential equations, finite difference equations and statistics.
620. Hipparchus biography
□ it seems highly probable that Hipparchus was the first to construct a table of chords and thus provide a general solution for trigonometrical problems.
621. Gemma Frisius biography
□ It is worth noting that although there were many methods of finding longitude proposed in the 250 years following Gemma Frisius's work, ultimately the methods he proposed were to become the
solution to finding the longitude at sea.
622. Ricci Giovanni biography
□ Hilbert himself remarked that he expected this Seventh Problem to be harder than the solution of the Riemann conjecture.
623. Flugge-Lotz biography
□ Her contributions have spanned a lifetime during which she demonstrate, in a field dominated by men, the value and quality of a woman's intuitive approach in searching for and discovering
solutions to complex engineering problems.
624. Goursat biography
□ Volume 3 surveys variations of solutions and partial differential equations of the second order and integral equations and calculus of variations.
625. Caramuel biography
□ He published the philosophy work Rationalis et realis philosophia in 1642 and, in the following year, published his theological work Theologia moralis ad prima atque clarissima principia
reducta which sought solutions to theological problems through applying mathematical rules.
626. Fresnel biography
□ The mathematical difficulties were formidable, and a solution was to require many months of effort.
627. Faber biography
□ Only in the 1980s was Faber's idea seen to be an important ingredient for the efficient solution of partial differential equations.
628. Ghizzetti biography
□ In the next three parts, the author applies the basic material to the solution of some of the ordinary and partial differential equations in electrotechnics.
629. Hadamard biography
□ The topic proposed for the prize had been one on geodesics and Hadamard's work in studying the trajectories of point masses on a surface led to certain non-linear differential equations whose
solution also gave properties of geodesics.
630. Begle biography
□ Throughout, Ed dictated no solutions but strove to harmonise the opinions of all.
631. Voronoy biography
□ In our exposition the resolution of these questions is based on a detailed study of the solutions of third-degree equations relative to a prime and a composite modulus.
632. Eckert Wallace biography
□ The first is the development of the theory or the solution of the differential equations of motion expressing the coordinates of the moon as explicit functions of time.
633. Iacob biography
□ The author's more than twenty years' research on boundary-value problems for plane harmonic functions is reflected in a thorough account of the potential-theoretical background for the
solution of plane incompressible-flow and linearized compressible-flow problems.
634. Levi-Civita biography
□ Their results include the conception of the localized induction approximation for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of motion, the
asymptotic potential theory applied to vortex tubes, the derivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations, and the stability
analysis of circular vortex filaments.
635. Aiken biography
□ These plans were made for a very specific purpose, for Aiken's research had led to a system of differential equations which had no exact solution and which could only be solved using
numerical techniques.
636. Lissajous biography
□ This apparatus which is particularly remarkable because it represents the solution to a difficult problem, has, unfortunately, no chance of being sold in quantity, since the number of
pictures need is very considerable: not less than 32 pictures are required.
637. Kaplansky biography
□ He completed the solution of Kurosh's problem on algebraic algebras of bounded degree, where Jacobson had made a decisive reduction, and considered numerous questions in the area of Banach
algebras, always from the algebraist's viewpoint.
638. Franklin biography
□ In A Step-Polygon of a Denumerable Infinity of Sides which Bounds No Finite Area (1933), written in collaboration with Jesse Douglas, the authors gave an explicit construction for functions
which had been shown to exist by Jesse Douglas in his solution of Plateau's problem in his paper published in 1931 (for which he received a Fields Medal).
639. Torricelli biography
□ Around 1640, Torricelli devised a geometrical solution to a problem, allegedly first formulated in the early 1600s by Fermat: 'given three points in a plane, find a fourth point such that the
sum of its distances to the three given points is as small as possible'.
640. Hoyle biography
□ Fred believed that, as a general rule, solutions to major unsolved problems had to be sought by exploring radical hypotheses, whilst at the same time not deviating from well-attested
scientific tools and methods.
641. Whiston biography
□ Having been one of the main enthusiasts for the Longitude Act, Whiston now proposed a number of methods of finding the longitude at sea; there was a lot of money for a good solution but he
never succeeded.
642. Nash biography
□ After this Nash worked on ideas that would appear in his paper Continuity of solutions of parabolic and elliptic equations which was published in the American Journal of Mathematics in 1958.
643. Goldstein biography
□ He studied numerical solutions to steady-flow laminar boundary-layer equations in 1930.
644. Ortega biography
□ In the second part of the book, devoted mostly to geometry, Ortega gives a method of extracting square roots very accurately using Pell's equation, which is surprising since a general
solution to Pell's equation does not appear to have been found before Fermat over 100 years later.
645. Goldberg biography
□ He gave the lecture On the growth of entire solutions of algebraic differential equations which was published in 2005.
646. Geocze biography
□ No difficulties at home, no noise from his children, no horrors of the trenches or the thunder of guns were able to distract Zoard Geocze from concentrating on the solution of his favourite
problem and making efforts to widen and deepen our knowledge of the subject.
647. Robertson biography
□ Around this time he built on de Sitter's solution of the equations of general relativity in an empty universe and developed what are now called Robertson-Walker spaces [Biographical Memoirs
National Academy of Sciences 51 (1980), 343-361.',2)">2]:- .
648. Vinogradov biography
□ His methods reached their height in Some theorems concerning the theory of prime numbers written in 1937 which provides a partial solution to the Goldbach conjecture.
649. Budan de Boislaurent biography
□ Fashions in mathematics change and solving a problem which Lagrange had deemed important does not guarantee your solution will achieve fame.
650. Cramer biography
□ This leads to 5 linear equations in 5 unknowns and he refers the reader to an appendix containing Cramer's rule for their solution.
651. Smith biography
□ Hermite asked Smith if he would cooperate in trying not to make the Academy look foolish, and simply submit a solution to the Grand Prix question.
652. Chazy biography
□ In the same year, Chazy published a paper on the three-body problem, Sur les solutions isosceles du Probleme des Trois Corps, an area he had begun to study in 1919 and for which he has become
653. Juel biography
□ Many readers must have felt that if all that projective geometry could tell us of a problem involving a cubic equation was that it has at least one solution, and not more than three, then
projective geometry had not by any means justified its claims to replace the ordinary algebraic kind.
654. De Vries biography
□ They found explicit, closed-form, travelling-wave solutions to the Korteweg - de Vries equation that decay rapidly.
655. Schoenberg biography
□ Schoenberg is noted worldwide for his realisation of the importance of spline functions for general mathematical analysis and in approximation theory, their key relevance in numerical
procedures for solving differential equations with initial and/or boundary conditions, and their role in the solution of a whole host of variational problems.
656. Ahmes biography
□ The Verso has 87 problems on the four operations, solution of equations, progressions, volumes of granaries, the two-thirds rule etc.
657. Boussinesq biography
□ In his first derivation of the solitary wave, published in 1871 in the 'Comptes rendus', Boussinesq sought an approximate solution of Euler's equations that propagated at the constant speed c
without deformation in a rectangular channel.
658. Orlicz biography
□ Working in Lvov Orlicz participated in the famous meetings at the Scottish Cafe (Kawiarnia Szkocka) where Stefan Banach, Hugo Steinhaus, Stanislaw Ulam, Stanislaw Mazur, Marek Kac, Juliusz
Schauder, Stefan Kaczmarz and many others talked about mathematical problems and looked for their solutions.
659. Wallace biography
□ He published two books after he retired, A Geometrical Treatise on the Conic Sections with an Appendix Containing Formulae for their Quadrature (1838) and Geometrical Theorems and Analytical
Formulae with their application to the Solution of Certain Geodetical Problems and an Appendix (1839).
660. Hawking biography
□ In 2005 Hawking published Information loss in black holes in which he proposed a solution to the information loss paradox.
661. Mercator Nicolaus biography
□ A known theoretical solution was to devise a clock which would keep accurate time at sea.
662. Lansberge biography
□ In the solution of spherical triangles Van Lansberge employs a device similar to that of Maurice Bressieu in his 'Metrices astronomicae' (Paris, 1581), the marking of the given parts of a
triangle by two strokes.
663. Oleinik biography
□ Much of the book is devoted to the study of the asymptotic behaviour of solutions to nonlinear elliptic second-order equations.
664. Menabrea biography
□ He had always worked for a compromise solution to the Italian unification question and up to 1859 believed that a compromise between the Vatican and the state was possible.
665. Huygens biography
□ At this time Huygens patented his design of pendulum clock with the solution of the longitude problem in mind.
666. Lindelof biography
□ Lindelof's first work in 1890 was on the existence of solutions for differential equations.
667. Boutroux biography
□ There he lectured at the College de France on functions which are the solutions of first order differential equations.
668. Bjerknes Vilhelm biography
□ Vilhelm Bjerknes and his associates at Bergen succeeded in devising equations relating the measurable components of weather, but their complexity precluded the rapid solutions needed for
669. Mineur biography
□ He was awarded his doctorate in 1924 for his thesis Discontinuous solutions of a class of functional equations in which he established an addition theorem for Fuchsian functions.
670. Taussky-Todd biography
□ who provided significant contributions to solutions of problems associated with applications of computers.
671. Hollerith biography
□ Hollerith realised that cards would provide a better solution.
672. Kubilius biography
□ He published three papers (all in Russian) during his time as a research student: On the application of I M Vinogradov's method to the solution of a problem of the metric theory of numbers
(1949); The distribution of Gaussian primes in the sectors and contours (1950); and On the decomposition of prime numbers as the sum of two squares (1951).
673. Camus biography
□ treatment of toothed wheels and their use in clocks, studies of the raising of water from wells by buckets and pumps, an evaluation of an alleged solution to the problem of perpetual motion,
and works on devices and standards of measurement.
674. Turan biography
□ Turan mentioned these problems and told me that they were not only interesting in themselves but their positive solution would have many applications.
675. Korteweg biography
□ They found explicit, closed-form, travelling-wave solutions to the Korteweg - de Vries equation that decay rapidly.
676. Boole biography
□ Boole had begun to correspond with De Morgan in 1842 and when in the following year he wrote a paper On a general method of analysis applying algebraic methods to the solution of differential
equations he sent it to De Morgan for comments.
677. Kolchin biography
□ Algebraic groups can be viewed as groups in the category of algebraic varieties, where the latter are taken to be locally given as sets of simultaneous solutions of algebraic equations.
678. Povzner biography
□ He published The representation of smallest degree isomorphic to a given Abelian group (1937), written to give his partial solution to a problem stated by Otto Yulevich Schmidt in his book
The abstract theory of groups, namely given an abstract group, find a permutation representation of least degree.
679. Egorov biography
□ In this paper, in addition to the independent, very elegant and simple solution of the problem proposed, the originality and logical rigour of the exposition of the basic general geometrical
principles deserve special mention, as does also the very successful working out of many details.
680. Antonelli biography
□ However the war had ended before the machine came into service but it was still used for the numerical solution of differential equations as intended.
681. Redei biography
□ 32-33 (1990), 199-211.',1)">1] examines the work which led up to the solution of the problem by Redei in 1953:- .
682. Banach biography
□ We discussed problems proposed right there, often with no solution evident even after several hours of thinking.
683. Banneker biography
□ The surviving manuscript journal contains mathematical puzzles and their solutions.
684. Razmadze biography
□ He also did important work on discontinuous solutions.
685. Adrain biography
□ Adrain's first papers in the Mathematical Correspondent concerned the steering of a ship and Diophantine algebra (the study of rational solutions to polynomial equations).
686. Schwerdtfeger biography
□ Admirable features of the book include a list of examples for solution, footnotes with historical notes, references to the original literature and to extensions of the topics treated, a
geometric framework for the algebraic results, and a summary of the principal problems considered ..
687. Nicomedes biography
□ As indicated in this quote Pappus also wrote about Nicomedes, in particular he wrote about his solution to the problem of trisecting an angle (see for example [A History of Greek Mathematics
(2 Vols.) (Oxford, 1921).',2)">2]):- .
688. Stevin biography
□ In the latter Stevin presented a unified treatment for solving quadratic equations and a method for finding approximate solutions to algebraic equations of all degrees.
689. Castelli biography
□ He remarks with wry amusement on the gay times had by the many knights and gentlefolk in the cardinal's entourage, while he devoted himself instead to the solution of hundreds of equations.
690. Wilton biography
□ Papers Wilton published during this period include: On plane waves of sound (1913); On the highest wave in deep water (1913); On deep water waves (1914); Figures of equilibrium of rotating
fluid under the restriction that the figure is to be a surface of revolution (1914); On the potential and force function of an electrified spherical bowl (1914-15); On ripples (1915); On the
solution of certain problems of two-dimensional physics (1915); A pseudo-sphere whose equation is expressible in terms of elliptic functions (1915); and A formula in zonal harmonics
691. Al-Jayyani biography
□ The work, which is published together with a Spanish translation and a commentary in [La trigonometria europea en el siglo XI : Estudio de la obra de Ibn Mu\'ad, \'el Kitab mayhulat\'
(Barcelona, 1979).',3)">3], contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle.
692. Philon biography
□ The solution is effectively produced by the intersection of a circle and a rectangular hyperbola.
693. Gergonne biography
□ Gergonne provided an elegant solution to the Problem of Apollonius in 1816.
694. Lifshitz biography
□ This 1974 prize was awarded jointly to Lifshitz, V A Belinskii and I M Khalatnikov for their work on the singularities of cosmological solutions of the gravitational equations which was
presented in sixteen papers between 1961 and 1985.
695. Chrystal biography
□ The result is that algebra, as we teach it, is neither an art nor a science, but an ill-farrago of rules; whose object is the solution of examination problems.
696. Cohen biography
□ These results present the long-awaited solutions of the most outstanding open problems of axiomatic set theory and should be rated as the most important advance in the study of axiomatic set
theory since the publication of Godel's 1940 monograph 'The consistency of the continuum hypothesis' (1940).
697. Bari biography
□ Already this first piece of work by Nina Bari testified to her great mathematical talent, since it included the solution of several very difficult problems in the theory of trigonometric
series that had lately been engaging the attention of many outstanding mathematicians.
698. Richmond biography
□ It is true that the scope of these methods is restricted, but there is compensation in the fact that when geometry is successful in solving a problem the solution is almost invariably both
simple and beautiful.
699. Kluvanek biography
□ Indeed, to express the required solutions in integral form one may have to integrate with respect to a vector-valued measure of infinite total variation.
700. Zeno of Elea biography
□ Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution.
701. Seki biography
□ For example, in 1683, he considered integer solutions of ax - by = 1 where a, b are integers.
702. Wazewski biography
□ he succeeded in applying with amazing effect the topological notion of retract (introduced by K Borsuk) to the study of the solutions of differential equations.
703. Plateau biography
□ He later used a solution of soapy water and glycerine and dipped wire contours into it, noting that the surfaces formed were minimal surfaces.
704. Cox Elbert biography
□ In 1925 Cox was awarded his doctorate for his thesis Polynomial solutions of difference equations.
705. Leonardo biography
□ Leonardo studied Euclid and Pacioli's Suma and began his own geometry research, sometimes giving mechanical solutions.
706. Poincare biography
□ His results applied only to restricted classes of functions and Poincare wanted to generalise these results but, as a route towards this, he looked for a class functions where solutions did
not exist.
707. Capelli biography
□ Capelli had proved the theorem, known today as the Rouche-Capelli theorem, which gives conditions for the existence of the solution of a system of linear equations.
708. Bhaskara I biography
□ ',12)">12], [Ganita 23 (1) (1972), 57-79',13)">13] and [Ganita 23 (2) (1972), 41-50.',14)">14] Shukla discusses some features of Bhaskara's mathematics such as: numbers and symbolism, the
classification of mathematics, the names and solution methods of equations of the first degree, quadratic equations, cubic equations and equations with more than one unknown, symbolic
algebra, unusual and special terms in Bhaskara's work, weights and measures, the Euclidean algorithm method of solving linear indeterminate equations, examples given by Bhaskara I
illustrating Aryabhata I's rules, certain tables for solving an equation occurring in astronomy, and reference made by Bhaskara I to the works of earlier Indian mathematicians.
709. Wilks biography
□ In 1947 he was awarded the Presidential Certificate of Merit for his contributions to antisubmarine warfare and the solution of convoy problems.
710. Pincherle biography
□ the 1888 paper (in Italian) of S Pincherle on the 'Generalized Hypergeometric Functions' led him to introduce the afterwards named Mellin-Barnes integral to represent the solution of a
generalized hypergeometric differential equation investigated by Goursat in 1883.
711. Fricke biography
□ Fricke's long experience with the latter subject made it easy for him to give a simple authoritative exposition of those portions of it which suffice for the transcendental solutions of
equations of low degrees.
712. Guo Shoujing biography
□ The equation has two real roots, the smaller being the solution to the problem while the other, being numerically larger than the length of the arc, was rightly discarded by Guo.
713. Christiansen biography
□ Yet his great integrity led him to be intolerant of injustice, of those who were rude, self-seeking, inefficient and not disposed to think, and of those who peddled simple solutions to
complex problems.
714. Wrinch biography
□ After discussions they decided (correctly it turned out) that computers would not give an immediate solution because of the difficult problem of determining phases.
715. Wilkins Ernest biography
□ In 1944 four of his papers appeared: On the growth of solutions of linear differential equations; Definitely self-conjugate adjoint integral equations; Multiple integral problems in
parametric form in the calculus of variations; and A note on skewness and kurtosis.
716. Polozii biography
□ Original results in the theory of functions of a complex variable were obtained in the 1950s and 1960s by G Polozii of Kiev, who introduced a new notion of p-analytic functions, defined the
notion of derivative and integral for these functions, developed their calculus, obtained a generalised Cauchy formula, and devised a new approximation method for solution of problems in
elasticity and filtration.
717. Brioschi biography
□ One of his most important results was his application of elliptical modular functions to the solution of equations of the fifth degree in 1858.
718. Ostrovskii biography
□ The authors describe the applications to the interpolation by entire functions, to entire and meromorphic solutions of ordinary differential equations, to the Riemann boundary problem with an
infinite index and to the arithmetic of the convolution semigroup of probability distributions.
719. Warga biography
□ This is a scholarly, clear presentation of that aspect of optimal control theory which deals with the existence of usual, generalized (relaxed) and approximate optimal solutions, and
necessary conditions for optimality.
720. Menelaus biography
□ Another Arab reference to Menelaus suggests that his Elements of Geometry contained Archytas's solution of the problem of duplicating the cube.
721. Boggio biography
□ In the first of these, Boggio obtained a solution for the problem of an elastic membrane, displaced in its own plane with known displacements on the boundary.
722. Adams Frank biography
□ in recognition of his solution of several outstanding problems of algebraic topology and of the methods he invented for this purpose which have proved of prime importance in the theory of
that subject.
723. Third biography
□ In the Mathematical Questions and Solutions section of the Educational Times of 1902 he published The perpendicular from the isogonal conjugate of any point on the Euler line of a triangle to
the trilinear polar of the point passes through the orthocentre.
724. Zhukovsky biography
□ Those Joukowski aerofoils were actually used on some aircraft, and today these techniques provide a mathematically rigorous reference solution to which modern approaches to aerofoil design
can be compared for validation.
725. Simplicius biography
□ Damascius had written Problems and Solutions about the First Principles which develops the Neoplatonist philosophy as expounded by Proclus.
726. Lehmer Emma biography
□ To perform the operation with pencil and paper one must start with the million or so numbers among which the solution is known to lie.
727. Chudakov biography
□ Chudakov made a substantial contribution to the solution when he proved that all, except possibly a finite number, of even integers greater than 2 can be represented as the sum of two primes.
728. Nygaard biography
□ In 1952 he published On the solution of integral equations by Monte-Carlo methods as a Norwegian Defence Research Establishment Report.
729. Lehmer Derrick biography
□ He was a pioneer in the application of mechanical methods, including digital computers, to the solution of problems in number theory and he talked about some of the methods used to factorise
numbers including: factor tables, trial division, Legendre's method, factor stencils, the continued fraction method, Fermat's method, methods based on quadratic forms, and Shanks' method.
730. Kovacs biography
□ It is an open-book competition and the competitors have ten days in which to produce solutions.
731. Darboux biography
□ brilliant are his reductions of various geometrical problems to a common analytic basis, and their solution and development from a common point of view.
732. Gromoll biography
□ provided one of the cornerstones of the Poincare Conjecture solution.
733. Higman biography
□ This result plays a vital part in Zelmanov's positive solution to the restricted Burnside problem in the early 1990s.
734. Thomason biography
□ For example in a 1983 paper he found a partial solution of Grothendieck's absolute cohomological purity conjecture.
735. Bonferroni biography
□ More than anything else, however, I was struck by his personal style and the simplifying solutions to the very complex procedures which he proposed.
736. Kolmogorov biography
□ The time of their graduate studies remains for all of Kolmogorov's students an unforgettable period in their lives, full of high scientific and cultural strivings, outbursts of scientific
progress and a dedication of all one's powers to the solutions of the problems of science.
737. Wielandt biography
□ I am indebted to that time for valuable discoveries: on the one hand the applicability of abstract tools to the solution of concrete problems, on the other hand, the - for a pure
mathematician - unexpected difficulty and unaccustomed responsibility of numerical evaluation.
738. Chatelet Albert biography
□ Clairin, who applied group theory to the solution of differential equations, had published Cours de mathematiques generales (1910).
739. Scheffe biography
□ Scheffe's doctoral dissertation The Asymptotic Solutions of Certain Linear Differential Equations in Which the Coefficient of the Parameter May Have a Zero was supervised by Rudolph E Langer.
740. Lob biography
□ Examples of papers by Lob in the Journal of Symbolic Logic in the 1950s are Concatenation as basis for a complete system of arithmetic (1953), Solution of a problem of Leon Henkin (1955), and
Formal systems of constructive mathematics (1956).
741. Vessiot biography
□ In 1892 he submitted his doctoral dissertation on groups of linear transformations, in particular studying the action of these groups on the independent solutions of a differential equation.
742. Cauchy biography
□ Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial
differential equations, the Cauchy-Riemann equations and Cauchy sequences.
743. Rudolph biography
□ Although we all tried very hard to solve it; Dan beat us all with a ingenious solution.
744. Householder biography
□ For his impact and influence on computer science in general and particularly for his contributions to the methods and techniques for obtaining numerical solutions to very large problems
through the use of digital computers, and for his many publications, including books, which have provided guidance and help to workers in the field of numerical analysis, and for his
contributions to professional activities and societies as committee member, paper referee, conference organiser, and society President.
745. Frenet biography
□ It contains problems with full solutions and often historical remarks.
746. Lehto biography
□ It led in due time to a simple solution of the geometric problem of moduli, and there are encouraging signs of a fruitful theory in several dimensions.
747. Rahn biography
□ in the solutions, and in the arithmetic too, I make use of a completely new method, which has not been used by any writer on algebra in a published work, that I first learned from an eminent
and very learned person to whom I should very gladly acknowledge indebtedness and humble respect, had he permitted.
748. Simson biography
□ a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions.
749. Bartel biography
□ Throughout the book care is taken to present the theoretical foundations on which are based the solutions of the problems considered.
750. Bachmann biography
□ Bachmann surveyed the attempts that had been made over nearly 300 years attempting to give a positive or a negative solution to Fermat's Last Theorem in Das Fermatproblem in Seiner Bisherigen
Entwicklung (1919).
751. Scholtz biography
□ He also submitted solutions to problems that had been posed in this journal.
752. Copson biography
□ by Poisson's analytical solution of the equation of wave-motions.
753. Karlin biography
□ Karlin published papers such as Solutions of discrete, two-person games; Polynomial games; and Games with continuous, convex pay-off all in 1950.
754. Humbert Georges biography
□ He thus enriched analysis and gave the complete solution of the two great questions of the transformation of hyperelliptic functions and of their complex multiplication.
755. Frobenius biography
□ On the algebraic solution of equations, whose coefficients are rational functions of one variable.
756. Church biography
□ This generalization of the Laplace transform is the topic of study of the second paper, again using the method to obtain solutions of second-order partial differential equations.
757. Carre biography
□ Between 1701 and 1705, Carre published over a dozen papers on a variety of mathematical and physical subjects: Methode pour la rectification des lignes courbes par les tangentes (1701);
Solution du probleme propose aux Geometres dans les memoires de Trevoux, des mois de Septembre et d'Octobre (1701); Reflexions ajoutees par M Carre a la Table des Equations (1701);
Observation sur la cause de la refraction de la lumiere (1702); Pourquoi les marees vont toujours en augmentant depuis Brest jusqu'a Saint-Malo, et en diminuant le long des cotes de Normandie
(1702); Nombre et noms des instruments de musique (1702); Observations sur la vinaigre qui fait rouler de petites pierres sur un plan incline (1703); Observation sur la rectification des
caustiques par reflexions formees par le cercle, la cycloide ordinaire, et la parabole, et de leurs developpees, avec la mesure des espaces qu'elle renferment (1703); Methode pour la
rectification des courbes (1704); Observation sur ce qui produit le son (1704); Examen d'une courbe formee par le moyen du cercle (1705); Experiences physiques sur la refraction des balles de
mousquet dans l'eau, et sur la resistance de ce fluide (1705); and Probleme d'hydrodynamique sur la proportion des tuyaux pour avoir une quantite d'eau determinee (1705).
758. Ulam biography
□ While Ulam was at Los Alamos, he developed the 'Monte-Carlo method' which searched for solutions to mathematical problems using a statistical sampling method with random numbers.
759. Recorde biography
□ In his study of quadratic equations, Recorde does not allow solutions which are negative, but he does allow negative coefficients.
760. Shelah biography
□ The solution appears in the second edition of his book 'Classification Theory and the Number of Non-isomorphic Models' (1990).
761. Ceva Giovanni biography
□ The first mathematical problem he attacked was the classic problem of squaring the circle and he produced several incorrect solutions to it.
762. Bremermann biography
□ During this time he continued to produce high quality results in complex analysis continuing to push the results of his doctoral dissertation towards a general solution to the Levi problem.
763. Thymaridas biography
□ then Thymaridas gives the solution .
764. Durell biography
□ All chapters conclude with a series of exercises, with solutions at the end of the book.
765. Hudde biography
□ Hudde proposed a "mechanical" solution, which was not mathematically exact.
766. Gronwall biography
□ Only a mathematician with Gronwall's gift for analysis and most uncommon grasp of the literature of chemistry and physics could have contributed the elegant solution which he gave.
767. Bollobas biography
□ All come with solutions, many with hints, and most with illustrations.
768. Eckmann biography
□ Peter Hilton, who had been a personal friend of Eckmann's for many years spoke in detail of Eckmann's research in topology: continuous solutions of systems of linear equations, a
group-theoretical proof of the Hurwitz-Radon theorem, complexes with operators, spaces with means, simple homotopy type.
769. De Bruijn biography
□ Also in 1943, in addition to his doctoral thesis, he published On the absolute convergence of Dirichlet series, On the number of solutions of the system ..
770. Kochina biography
□ For example in 1948 she studied numerical solutions of a partial differential equation in On a nonlinear partial differential equation arising in the theory of filtration.
771. Descartes biography
□ Descartes' geometric solution of a quadratic equation .
772. Magnitsky biography
□ Geometry and trigonometry were not abstract entities, but solution methods for navigational problems, just as contemporary English and American texts relied on examples of commercial
transactions to induce students to do their calculations.
773. McClintock biography
□ He published A simplified solution of the cubic in 1900 in the Annals of Mathematics.
774. Gaschutz biography
□ This is an example that shows how minor variations of the initial conditions can influence the solutions of an equation considerably.
775. Fraser biography
□ Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussion of new problems or new solutions, and
comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.
776. Courant biography
□ The first application as a numerical method, however, was given by Courant in 1943 in his solution of a torsion problem.
777. Roth Klaus biography
□ Speaking of Roth's solution to this problem of approximating algebraic numbers Davenport said [2]:- .
778. Schramm biography
□ This work led to the solution of many problems by him, many together with his collaborators Greg Lawler now at Cornell University, and Wendelin Werner in Strasbourg, France, as well as by
many other mathematical researchers.
779. Ostrogradski biography
□ They include a special case of Green's theorem, a general development (the first such, according to Yushkevich) of the method of separation of variables, and the first solution of the problem
of heat diffusion in a triangular prism.
□ His important work on ordinary differential equations considered methods of solution of non-linear equations which involved power series expansions in a parameter alpha.
780. Fock biography
□ The reviewer feels that the author has made a major contribution to the understanding of gravitation theory, especially by his insistence on studying the solutions of the field equations and
not merely the formal properties of the equations.
781. Neumann Carl biography
□ In 1890 Emile Picard used Neumann's results to develop his method of successive approximation which he used to give existence proofs for the solutions of partial differential equations.
782. Sun Zi biography
□ In fact the solution given, although in a special case, gives exactly the modern method.
783. Rudin biography
□ Her 1952 paper A primitive dispersion set of the plane provided a positive solution to an unsolved problem contained in R L Wilder's book Topology of manifolds (1949).
784. Burgess biography
□ And Concurrency of lines joining vertices of a triangle to opposite vertices of triangles on its sides; determinants connected with the periodic solutions of Mathieu's equation.
785. Conway biography
□ To really understand and prove everything in the book, not to mention to attempt solutions of the many questions inspired on every page of the book, will engage many people for many years.
786. Akhiezer biography
□ His most outstanding work consisted of deep approximation results in the constructive function theory, including the solution of the problem of Zolotarev.
787. Verhulst biography
□ He named the solution to the equation he had proposed in his 1838 paper the 'logistic function'.
788. Fredholm biography
□ Two years later in Stockholm a lecture about the 'principal solutions' of Roux and their connections with Volterra's equation led to a vivid discussion Finally, after a long silence Fredholm
spoke and remarked in his usual slow drawl: in potential theory there is also such an equation.
789. D'Ocagne biography
□ Nomography consists in the construction of graduated graphic tables, nomograms, or charts, representing formulas or equations to be solved, the solutions of which were provided by inspection
of the tables.
790. Eratosthenes biography
□ Eratosthenes erected a column at Alexandria with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [A History of Greek Mathematics (2
vols.) (Oxford, 1921).',4)">4]:- .
791. Freitag biography
□ She has also contributed prodigiously to the Elementary Problems and Solutions Section of the Fibonacci Quarterly and published many papers in that journal.
792. Ricci-Curbastro biography
□ In the paper, applications are given by Ricci-Curbastro and Levi-Civita to the classification of the quadratic forms of differentials and there are other analytic applications; they give
applications to geometry including the theory of surfaces and groups of motions; and mechanical applications including dynamics and solutions to Lagrange's equations.
793. Bring biography
□ This work describes Bring's contribution to the algebraic solution of equations.
794. Vilant biography
□ Those parts for which some originality may be claimed are: (a) a method for finding the cube root of binomials of form R ± √S, where S may be positive or negative, and (b) a method for
finding rational and whole-number solutions of indeterminate problems involving linear, quadratic and cubic equations.
795. Tamarkin biography
□ I proposed to Tamarkin that he think about the asymptotic solution of differential equations (i.e.
796. Barsotti biography
□ Thus it is not surprising that Chevalley's solution of the problem has no evident link with the methods that, according to the suggestions of the classical geometers, should have been used in
order to define the intersection multiplicity; rather, it is linked to the analytical approach, and it is therefore a strictly "local" theory, thus having the advantage of providing an
intersection multiplicity also for algebroid varieties.
797. Hertz Heinrich biography
□ A prize had been announced by the Philosophy Faculty for the solution of an experimental problem concerning electrical inertia and Hertz was very keen to enter.
798. La Hire biography
□ He began with their focal definitions and applied Cartesian analytic geometry t the study of equations and the solution of indeterminate problems; he also displayed the Cartesian method for
solving certain types of equations by intersections of curves.
799. Roberval biography
□ He therefore solved this problem before Torricelli who found a solution after 1644.
800. Greenhill biography
□ The chief characteristics of Greenhill's work were a desire for concrete realisation of abstract theories and the direction of investigation to the solution of definite problems.
801. Taylor Geoffrey biography
□ Taylor continued his research after the end of the War, taking the opportunity to complete some more thorough investigations into problems where previously the pressure of finding solutions
had prevented him from taking his study further.
802. Von Staudt biography
□ Von Staudt also gave a nice geometric solution to quadratic equations.
803. Rees David biography
□ [He] was never happier than when sitting in front of the television scribbling down algebraic equations to find a solution to some mathematical challenge he had set himself.
804. Lhuilier biography
□ Lhuilier also corrected Euler's solution of the Konigsberg bridge problem.
805. Sidler biography
□ In his paper Die Schale Vivianis (1901) he showed an elementary solution to the problem and new results relating to Viviani's curve.
History Topics
1. Pell's equation
□ where n is a given integer and we are looking for integer solutions (x, y).
□ In other words, if (a, b) and (c, d) are solutions to Pell's equation then so are .
□ This fundamentally important fact generalises easily to give Brahmagupta's lemma, namely that if (a, b) and (c, d) are integer solutions of 'Pell type equations' of the form .
□ are both integer solutions of the 'Pell type equation' .
□ Now of course the method of composition can be applied again to (a, b) and (2ab, b2+ na2) to get another solution and Brahmagupta immediately saw that from one solution of Pell's equation he
could generate many solutions.
□ He also noted that, using a similar argument to what we have just given, if x = a, y = b is a solution of nx2 + k = y2 then applying the method of composition to (a, b) and (a, b) gave (2ab,
b2 + na2) as a solution of nx2 + k2 = y2 and so, dividing through by k2, gives .
□ as a solution of Pell's equation nx2 + 1 = y2.
□ Well if k = 2 then, since (a, b) is a solution of nx2 + k = y2 we have na2 = b2 - 2.
□ and this is an integer solution to Pell's equation.
□ If k = -2 then essentially the same argument works while if k = 4 or k = -4 then a more complicated method, still based on the method of composition, shows that integer solutions to Pell's
equation can be found.
□ So Brahmagupta was able to show that if he could find (a, b) which "nearly" satisfied Pell's equation in the sense that na2 + k = b2 where k = 1, -1, 2, -2, 4, or -4 then he could find one,
and therefore many, integer solutions to Pell's equation.
□ Often he could find trial solutions which worked for one of these values of k and so in many cases he was able to give solutions.
□ as another solution.
□ Among the examples Brahmagupta gives himself is a solution of Pell's equation .
□ and applies his method to find the solution .
□ We can now generate a sequence of solutions (x,y): .
□ He discovered the cyclic method, called chakravala by the Indians, which was an algorithm to produce a solution to Pell's equation nx2 + 1 = y2 starting off from any "close" pair (a, b) with
na2 + k = b2.
□ We can assume that a and b are coprime, for otherwise we could divide each by their gcd and get a "closer" solution with smaller k.
□ is a solution to .
□ With such a choice of m he therefore has integer solutions .
□ If (m2 - n)/k is one of 1, -1, 2, -2, 4, -4 then we can apply Brahmagupta's method to find a solution to Pell's equation nx2 + 1 = y2.
□ If (m2 - n)/k is not one of these values then repeat the process starting this time with the solution x = (am + b)/k, y = (bm + na)/k to the 'Pell type equation' nx2 + (m2 - n)/k = y2 in
exactly the same way as we applied the process to na2 + k = b2.
□ as a solution to the 'Pell type equation' nx2 - 4 = y2.
□ as the smallest solution to 61x2 + 1 = y2.
□ Secondly the algorithm always reaches a solution of Pell's equation after a finite number of steps without stopping when an equation of the type nx2 + k = y2 where k = -1, 2, -2, 4, or -4 is
reached and then applying Brahmagupta's method.
□ However, when one writes down a proof it should become clear that the algorithm switching to Brahmagupta's method is never necessary (although can reach the solution more quickly).
□ Now Narayana applies Brahmagupta's method, in the form we gave above for equations with k = 2, to obtain the solutions .
□ His next example is a solution of Pell's equation .
□ Finally Narayana applies Brahmagupta's method to this last equation to get the solution .
□ We await these solutions, which, if England or Belgic or Celtic Gaul do not produce, then Narbonese Gaul will.
□ Narbonese Gaul, of course, was the area around Toulouse where Fermat lived! One of Fermat's challenge problems was the same example of Pell's equation which had been studied by Bhaskara II
500 years earlier, namely to find solutions to .
□ Brouncker discovered a method of solution which is essentially the same as the method of continued fractions which was later developed rigorously by Lagrange.
□ Frenicle de Bessy tabulated the solutions of Pell's equation for all n up to 150, although this was never published and his efforts have been lost.
□ Brouncker found the smallest solutions, using his method, which is .
□ We should note that by this time several mathematicians had claimed that Pell's equation nx2 + 1 = y2 had solutions for any n.
□ Wallis, describing Brouncker's method, had made that claim, as had Fermat when commenting on the solutions proposed to his challenge.
□ In fact Fermat claimed, correctly of course, that for any n Pell's equation had infinitely many solutions.
□ This established rigorously the fact that for every n Pell's equation had infinitely many solutions.
□ The solution depends on the continued fraction expansion of √n.
□ will be the smallest solution to Pell's equation .
□ To find the infinite series of solutions take the powers of 170 + 39√19.
□ will give a second solution to the equation.
□ as the next solution.
□ Here are the first few powers of (170 + 39√19), starting with its square, which gives the first few solutions to the equation 19x2 + 1 = y2 .
2. Braids arithmetic
□ May 12th 1868." In addition to the collection of arithmetic problems with solutions written by William Braid, he has written a Scots Proverb at the end of most of the problems.
□ The Voluntary Exercises in Arithmetic are mostly elementary and the solutions for those near the beginning are not worth recording.
□ However, we give an occasional example of a solution to illustrate William Braid's methods.
□ The examples dealing with repeating decimals fall into this category and we give more details of Braid's solutions (including his errors) when giving these.
□ Solution.
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3. Alcuin's book
□ Alcuin gives solutions in the book but often these give an answer with a verification that it is correct rather than a method of proof.
□ We shall usually give both a modern approach to solving the problem as well as a comment on Alcuin's solution.
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□ It would appear from the surviving manuscripts that Alcuin didn't give a solution to this puzzle.
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□ Again it would appear that Alcuin didn't give a solution.
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□ This is Alcuin's solution, although of course he writes it out in words rather than the symbols we have used.
□ See if you can find a solution with only 9 crossings.
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□ Alcuin's solution appears simply wrong unless the statement of the puzzle has been corrupted..
□ Solution.
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□ Of course, as well as the incorrect formula for the area of the field, this solution has another problem, namely that you can't fit that many houses into the field.
□ Solution.
□ This problem is similar to the previous one and Alcuin's solution has all the same errors in it.
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□ These are the only 5 possible solutions with x, y, z all positive.
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□ These are the only 6 possible solutions with x, y, z all positive.
□ Solution.
□ The original problem that Alcuin is basing this puzzle on might be based on Islamic law or Alcuin may be assuming the reader will base the solution on Roman law.
□ However, his solution assumes that the each will receive the average of the two possibilities, namely: .
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□ Alcuin gives a good solution to this puzzle.
□ Solution.
□ Of course the problem has no solution since the sum of three odd numbers can never be even.
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□ This is precisely Alcuin's solution.
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□ Alcuin gives a solution in which the camel makes three trips to a point 20 leagues from the start and moves the grain to here at a cost of 60 measures.
□ Solution.
4. Doubling the cube
□ We next consider the solution proposed by Archytas.
□ This is a quite beautiful solution showing quite outstanding innovation by Archytas.
□ The solution by Archytas is the most remarkable of all, especially when his date is considered (first half of the fourth century BC), because it is not a plane construction but a bold
construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution..
□ This is the solution of Archytas, reported by Eudemus: .
□ Through the writings of Eutocius, we know that Eudoxus also gave a solution to the problem of doubling the cube.
□ His solution is lost, however, since the version which Eutocius had in front of him was rather trivially incorrect and he therefore did not reproduce it.
□ Nobody believes that Eudoxus had an elementary error in his solution (he was far too good a mathematician for that) so the error must have been an error introduced when his solution was
copied by someone who did not understand it properly.
□ Paul Tannery suggested that Eudoxus's solution was a two-dimensional version of the one given by Archytas which we have just described, in effect the solution obtained by projecting
Archytas's construction onto a plane.
□ too original a mathematician to content himself with a mere adaptation of Archytas's method of solution.
□ Menaechmus's solution to finding two mean proportionals is described by Eutocius in his commentary to Archimedes' On the sphere and cylinder.
□ Menaechmus gave two solutions.
□ Of course we must emphasis again that this in no way indicates the way that Menaechmus solved the problem but it does show in modern terms how the parabola and hyperbola enter into the
solution to the problem.
□ For his second solution Menaechmus uses the intersection of the two parabolas y2 = bx and x2 = ay which are the second and third equations in our list.
□ One of the great puzzles concerning the solution of the problem of doubling the cube is that there is a mechanical solution known as Plato's machine.
□ Now it seems highly unlikely that Plato would give a mechanical solution, particularly given his views on such solutions.
□ One theory is that Plato invented the mechanical solution to show how easy it is to devise such solutions, but the more widely held theory is that Plato's machine was invented by one of his
followers at the Academy.
□ He erected a column at Alexandria dedicated to King Ptolemy with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [A history of Greek
mathematics I (Oxford, 1931).',2)">2]:- .
□ Other solutions to the problem were by Philon and Heron who both gave similar methods.
□ Their solution is effectively produced by the intersection of a circle and a rectangular hyperbola.
□ Nicomedes, who was highly critical of Eratosthenes mechanical solution, gave a construction which used the conchoid curve which he also used to solve the problem of trisection of an angle.
□ Although these many different methods were invented to double the cube and remarkable mathematical discoveries were made in the attempts, the ancient Greeks were never going to find the
solution that they really sought, namely one which could be made with a ruler and compass construction.
5. Brachistochrone problem
□ Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument.
□ If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.
□ Perhaps we are reading too much into Johann Bernoulli's references to Pascal and Fermat, but it interesting to note that Pascal's most famous challenge concerned the cycloid, which Johann
Bernoulli knew at this stage to be the solution to the brachistochrone problem, and his method of solving the problem used ideas due to Fermat.
□ Leibniz persuaded Johann Bernoulli to allow a longer time for solutions to be produced than the six months he had originally intended so that foreign mathematicians would also have a chance
to solve the problem.
□ Five solutions were obtained, Newton, Jacob Bernoulli, Leibniz and de L'Hopital solving the problem in addition to Johann Bernoulli.
□ Newton sent his solution to Charles Montague, the Earl of Halifax, who was an innovative finance minister and the founder of the Bank of England.
□ He was President of the Royal Society during the years 1695 to 1698 so it was natural that Newton send him his solution to the brachistochrone problem.
□ The Royal Society published Newton's solution anonymously in the Philosophical Transactions of the Royal Society in January 1697.
□ His solution was explained to Montague as follows:- .
□ Solution.
□ The May 1697 publication of Acta Eruditorum contained Leibniz's solution to the brachistochrone problem on page 205, Johann Bernoulli's solution on pages 206 to 211, Jacob Bernoulli's
solution on pages 211 to 214, and a Latin translation of Newton's solution on page 223.
□ The solution by de L'Hopital was not published until 1988 when, nearly 300 years later, Jeanne Peiffer presented it as Appendix 1 in [Die Gesammelten Werke der Mathematiker und Physiker der
Familie Bernoulli (Basel, 1988).',1)" onmouseover="window.status='Click to see reference';return true">1].
□ Johann Bernoulli's solution divides the plane into strips and he assumes that the particle follows a straight line in each strip.
□ Huygens had shown in 1659, prompted by Pascal's challenge about the cycloid, that the cycloid is the solution to the tautochrone problem, namely that of finding the curve for which the time
taken by a particle sliding down the curve under uniform gravity to its lowest point is independent of its starting point.
□ Johann Bernoulli ended his solution of the brachistochrone problem with these words:- .
□ Despite the friendly words with which Johann Bernoulli described his brother Jacob Bernoulli's solution to the brachistochrone problem (see above), a serious argument erupted between the
brothers after the May 1697 publication of Acta Eruditorum.
□ Euler, however, commented that his geometrical approach to these problems was not ideal and it only gave necessary conditions that a solution has to satisfy.
□ The question of the existence of a solution was not solved by Euler's contribution.
□ The solution was found by considering special cases, and it was only some time later, in research isoperimetric curves, that the great mathematician of whom we speak and his famous brother
Jacob Bernoulli gave some general rules for solving several other problems of the same type.
6. Bakhshali manuscript
□ The Bakhshali manuscript is a handbook of rules and illustrative examples together with their solutions.
□ The solution to the example is then given and finally a proof is set out.
□ To illustrate we give the following indeterminate problem which, of course, does not have a unique solution:- .
□ The solution, translated into modern notation, proceeds as follows.
□ We seek integer solutions x1, x2, x3 and k (where x1 is the price of an asava, x2 is the price of a haya, and x3 is the price of a horse) satisfying .
□ For integer solutions k - (x1 + x2 + x3) must be a multiple of the lcm of 4, 6 and 7.
□ This is the indeterminate nature of the problem and taking different multiples of the lcm will lead to different solutions.
□ This is not the minimum integer solution which would be k = 131.
□ so we obtain integer solutions by taking k = 131 which is the smallest solution.
□ This solution is not given in the Bakhshali manuscript but the author of the manuscript would have obtained this had he taken k - (x1 + x2 + x3) = lcm(4, 6, 7) = 84.
□ Here is another equalisation problem taken from the manuscript which has a unique solution:- .
7. Trisecting an angle
□ Later, however, they trisected the angle by means of the conics, using in the solution the verging described below ..
□ Now one of the reasons why the problem of trisecting an angle seems to have attracted less in the way of reported solutions by the best ancient Greek mathematicians is that the construction
above, although not possible with an unmarked straight edge and compass, is nevertheless easy to carry out in practice.
□ A mechanical type of solution is easily found.
□ So as a practical problem there was little left to do although the Greeks still were not satisfied in general with mechanical solutions from a purely mathematical point of view they did not
find them.
□ There is another mechanical solution given by Archimedes.
□ Pappus gives two solutions which both involve the drawing of a hyperbola.
□ The passage from Pappus from which this solution is taken is remarkable as being one of three passages in Greek mathematical works still extant ..
□ These constructions described by Pappus show how the Greeks 'improved' their solutions to the problem of trisecting an angle.
□ From a mechanical solution they had progressed to a solution involving conic sections.
□ They could never progress to plane solutions since we know that such are impossible.
8. Weather forecasting
□ Basically, this method allows finding approximate solutions to differential equations.
□ All the different methods need to be stable, in other words, it has to be guaranteed that the numerical solution does not diverge from the true solution as the time span for which the
forecast is made increases.
□ The residual function is zero when the solution of the equation above is exact, therefore the series coefficients an should be chosen such that the residual function is minimised, i.e.
□ Spherical harmonics Ynm(λ, φ) are the angular part of the solution to Laplace's equation.
□ At the poles, the solutions to differential equations become infinitely differentiable; therefore the poles are usually excluded from the spectral space, which actually simplifies the method
[Chebyshev and Fourier Spectral Methods, second edition (Mineola NY, 2000) ',5)">5, p.
□ As a result, the solutions of many problems are very accurate.
□ A third technique for finding approximate solutions to partial differential equations and hence to the primitive equations is the finite element method.
□ The domain for which the partial differential equations have to be solved is divided into a number of subdomains, and a different polynomial is used to approximate the solution for each
9. Tartaglia versus Cardan
□ Tartaglia: I cannot do that, because as soon as he shall have one of the said cases with its solution, his excellency will at once understand the rule discovered by me, with which many other
rules may perhaps be found, based on the same material.
□ I know them by the two last, because a similar one to the seventh he sent me two years ago, and I made him confess that he did not understand the same, and a similar one to that last (which
induces an operation of the square and cube equal to a number) I gave him out of courtesy solved, not a year ago, and for that solution I found a rule specially bearing upon such problems.
□ I send you two questions with their solutions, but the solutions shall be separate from the questions, and the messenger will take them with him; and if you cannot solve the questions he will
place the solutions in your hand.
□ In addition to this, be pleased to send me the propositions offered by you to Master Antonio Maria Fior, and if you will not send me the solutions, keep them by you, they are not so very
□ And if it should please you, in receiving the solutions of my said questions - should you yourself be unable to solve them, after you have satisfied yourself that my first six questions are
different in kind - to send me the solution of any one of them, rather for friendship's sake, a for a test of your great skill, than for any other purpose, you will do me a very singular
□ Tartaglia to Cardano (August 1539): Master Girolamo, I have received a letter of yours, in which you write that you understand the rule; but that when the cube of one-third of the coefficient
of the unknown is greater in value than the square of one-half of the number you cannot resolve the equation by following the rule, and therefore you request me to give you the solution of
this equation "One cube equal to nine unknowns plus ten".
10. Squaring the circle
□ This is really asking whether squaring the circle is a 'plane' problem in the terminology of Pappus given above (we shall often refer to a 'plane solution' rather than use the more cumbersome
'solutions using ruler and compass").
□ The ancient Greeks, however, did not restrict themselves to attempting to find a plane solution (which we now know to be impossible), but rather developed a great variety of methods using
various curves invented specially for the purpose, or devised constructions based on some mechanical method.
□ Oenopides is thought by Heath to be the person who required a plane solution to geometry problems.
□ It only led to a greater flood of amateur solutions to the problem of squaring the circle and in 1775 the Paris Academie des Sciences passed a resolution which meant that no further attempted
solutions submitted to them would be examined.
□ A few years later the Royal Society in London also banned consideration of any further 'proofs' of squaring the circle as large numbers of amateur mathematicians tried to achieve fame by
presenting the Society with a solution.
□ The final solution to the problem of whether the circle could be squared using ruler and compass methods came in 1880 when Lindemann proved that π was transcendental, that is it is not the
root of any polynomial equation with rational coefficients.
11. Fair book
□ Although no 'common rule' is specified, the solution tells us that the rule gives the tonnage as x .
□ Although no words appear in the solution, we see that the tonnage, found from log tables, is 651.5.
□ Here Walker produces a strange method of solution.
□ The answer is correct but an extremely lengthy and difficult method to obtain the solution.
□ It is worth noting that some (but not all) of the solutions have a tick to the left of the question.
□ The solution is carried out with two applications of Heron's formula to the triangles ABD and BCD.
□ No solution given.
□ No solution is given.
□ The cask is drawn but no calculations or solution are given.
□ No solution is worked out.
□ At this point the solution terminates and no answer is given.
□ The solution to this question again terminates and no answer is given.
12. Quadratic etc equations
□ Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber embadorum published in 1145 which is the first book published in Europe to give the complete
solution of the quadratic equation.
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□ However, without the Hindu's knowledge of negative numbers, dal Ferro would not have been able to use his solution of the one case to solve all cubic equations.
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□ Cardan invited Tartaglia to visit him and, after much persuasion, made him divulge the secret of his solution of the cubic equation.
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□ Here, in modern notation, is Cardan's solution of x3 + mx = n.
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□ so if a and b satisfy 3ab = m and a3 - b3 = n then a - b is a solution of x3 + mx = n.
□ Then x = a - b is the solution to the cubic.
□ Cardan knew that you could not take the square root of a negative number yet he also knew that x = 4 was a solution to the equation.
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□ Here, again in modern notation, is Ferrari's solution of the case: x4 + px2 + qx + r = 0.
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□ Solve this quadratic and we have the required solution to the quartic equation.
□ In the years after Cardan's Ars Magna many mathematicians contributed to the solution of cubic and quartic equations.
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□ One of the most elementary to us, yet showing a marked improvement in understanding, was the observation that if x = b, x = c, x = d are solutions of a cubic then the cubic is .
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13. Fair book insert
□ The problems on this page all give solutions in square links.
□ The first solution given to this problem is incorrect but then the correct solution is given using the rule area = (a + b)h/2 where a, b are the lengths of the parallel sides and h is the
perpendicular distance between them.
□ [We have corrected this problem which is incorrectly gives the diagonal as AB.] The solution just computes the areas of the two triangles ABC and ADC.
□ The method of solution here is to solve triangles, one after the other, using b = a sin B/sin A, all done with six-figure logs.
□ It is the first time that a method of solution is given, all the previous problems being solved by use of a rule (there is no evidence that the rule is thought of in terms of a formula,
rather it seems to be thought of as a procedure which has been learnt by heart).
□ The x here is my invention - it is not used in the solution where he has calculated that 36 goes in the space where I have the x.
□ Various errors are made in the solution.
□ There then follows a problem which is not stated but the solution clearly indicates that the problem was to solve a right angled triangle where the two shorter sides are 500 and 480.
□ Solution given using methods as above is correct hypotenuse is 693.1 and the angles are 43°50 and 46°10.
14. Mathematical games
□ In fact Lucas (the inventor of the Towers of Hanoi) gives a pretty solution to Cardan's Ring Puzzle using binary arithmetic.
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□ Tartaglia, who with Cardan jointly discovered the algebraic solution of the cubic, was another famous inventor of mathematical recreations.
Go directly to this paragraph
□ He invented many arithmetical problems, and contributed to problems with weighing masses with the smallest number of weights and Ferry Boat type problems which now have solutions using graph
Go directly to this paragraph
□ Ozanam and Montucla quote the solutions of both De Moivre and Montmort.
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□ There is a unique solution (up to symmetry) to the 6 × 6 problem and the puzzle, in the form of a wooden board with 36 holes into which pins were placed, was sold on the streets of London for
one penny.
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□ Solutions for n = 9, 15, 27 were given in 1850 and much work was done on the problem thereafter.
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□ This problem was shown by computer to have exactly 65 solutions in 1958.
15. Word problems
□ Here there are above all three fundamental problems whose solution is very difficult and which will not be possible without a penetrating study of the subject.
□ Each knotted space curve, in order to be completely understood, demands the solution of the three above problems in a special case.
□ He used his solution to show that right and left trefoils are distinct.
□ He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group.
□ The solution to the word problem for these groups began with Dehn who stated the Freiheitssatz: .
□ In the following year Magnus published a paper containing a special case of the word problem for 1-relator groups, then in 1932 he published a complete proof of the solution of the word
problem for this class of groups.
□ It required computability theory and developments in mathematical logic to even make the questions precise, but these areas were to not only provide explicit questions, they also provided
solutions to the questions.
□ While sitting in the dentist's chair waiting for this unpleasant experience, inspiration struck and suddenly he saw the route to the solution.
16. Longitude2
□ Hooke, therefore, like almost all scientists of that time was a biased judge of longitude solutions since he hoped to solve the problem himself.
□ Jonas Moore, although keen to see a solution of the longitude problem, seems to have seen his role as making it possible for others to solve it rather than himself.
□ More and more pressure was mounting for a solution to the longitude problem as the continuing failure to solve it was costing England vast sums of money.
□ Everyone believed that mathematicians and astronomers would provide the solution but it is not to be.
□ every hour, on the dot, immerse the bandage in a solution of the powder of sympathy and the dog on shipboard would yelp the hour.
□ James Bradley, who had succeeded Halley as Astronomer Royal in 1742, and Tobias Mayer were convinced that the lunar distance method would lead to the solution of the longitude problem.
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17. Debating topics
□ Does the equation ax = b always have a solution? Do quadratic, cubic and quartic equations always have solutions? .
□ The equation x2 + 1 = 0 has no real number solution.
□ Let i be a symbol representing its solution.
□ Do we need to introduce negative numbers to get solutions of such equations? .
18. function concept
□ It was a concept whose introduction was particularly well timed as far as Johann Bernoulli was concerned for he was looking at problems in the calculus of variations where functions occur as
□ In 1746 d'Alembert published a solution to the problem of a vibrating stretched string.
□ The solution, of course, depended on the initial form of the string and d'Alembert insisted in his solution that the function which described the initial velocities of the each point of the
string had to be E-continuous, that is expressed by a single analytic expression.
□ In this work Condorcet distinguished three types of functions: explicit functions, implicit functions given only by unsolved equations, and functions which are defined from physical
considerations such as being the solution to a diffferential equation.
19. Kepler's Planetary Laws
□ Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
□ And because it was expressed geometrically, the solution would potentially be exact - the closed orbit of a single planet in a plane round the fixed Sun.
□ This is the process that was described (in Section 4) as idealization because it ensured an exact solution (of the one-body problem) which was uniquely simple.
□ The solutions reached in each case are in some senses provisional, but they are certainly vital steps on the way to the presentday solution.
20. Quadratic etc equations references
□ B Hughes, The earliest correct algebraic solutions of cubic equations, Vita mathematica (Washington, DC, 1996), 107-112.
□ C Romo Santos, Cardano's 'Ars magna' and the solutions of cubic and quartic equations (Spanish), Rev.
□ G S Smirnova, Geometric solution of the cubic equations in Raffaele Bombelli's 'Algebra' (Russian), Istor.
□ P D Yardley, Graphical solution of the cubic equation developed from the work of Omar Khayyam, Bull.
21. The Scottish Book
□ From items at the end of this collection, it will be seen that some Russian mathematicians must have visited the town; they left several problems (and prizes for their solutions).
□ Many of the problems have since found their solution, some in the form of published papers.
□ (I know of some of my own problems, solutions to which were published in periodicals, among them, e.g.
□ I should be grateful if the recipients of this collection were willing to point out errors, supply information about solution to problems, or indicate developments contained in recent
literature in topics connected with the subjects discussed in the problems.
22. Quadratic etc equations references
□ B Hughes, The earliest correct algebraic solutions of cubic equations, Vita mathematica (Washington, DC, 1996), 107-112.
□ C Romo Santos, Cardano's 'Ars magna' and the solutions of cubic and quartic equations (Spanish), Rev.
□ G S Smirnova, Geometric solution of the cubic equations in Raffaele Bombelli's 'Algebra' (Russian), Istor.
□ P D Yardley, Graphical solution of the cubic equation developed from the work of Omar Khayyam, Bull.
23. Egyptian Papyri
□ The method of solution was to "get rid of" the fractions by multiplying through.
□ Now the answer to the red auxiliary equation is 4 so the original equation had solution twice × (twice × 1/15).
□ Doubling this gives 1/5 + 1/15 which is the required solution to Problem 21.
□ Finally Ahmes checks his solution, or proves his answer is correct.
□ As a final look at the Rhind papyrus let us give the solution to Problem 50.
□ What is its area? Here is the solution as given by Ahmes.
□ Notice that the solution is equivalent to taking π = 4(8/9)2 = 3.1605.
24. Fund theorem of algebra
□ The formula when applied to the equation x3 = 15x + 4 gave an answer involving √-121 yet Cardan knew that the equation had x = 4 as a solution.
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□ Viete gave equations of degree n with n roots but the first claim that there are always n solutions was made by a Flemish mathematician Albert Girard in 1629 in L'invention en algebre .
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□ However he does not assert that solutions are of the form a + bi, a, b real, so allows the possibility that solutions come from a larger number field than C.
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25. Longitude1
□ A great solution if one were able to determine where land was relative to the line .
□ The method is theoretically correct but Werner had not solved the longitude problem since the cross-staff could not make accurate enough measurements, and more seriously there was no
mathematical theory of the Moons orbit (and even when Newton gave his theory of gravitation 150 years later the Moon's motion, a three body problem, was beyond solution).
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□ The position of the Spice Islands was in dispute and Spain sought a solution to these costly problems.
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□ The fact that no solution to this problem had been found was costing countries vast sums of money.
□ A solution had to be found, so countries began to adopt the standard method, namely to offer money, prizes, pensions, wealth beyond belief to mathematicians and astronomers who could give a
method to find the longitude at sea.
□ The Academie Royale was desperate to examine every chance for a solution and money was no problem.
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□ The members of the Academie Royale des Sciences made observations of the Moon over the years 1667 to 1669 which convinced them that the mathematics of the position of the Moon was too
difficult to make it useful as a solution to the longitude problem.
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26. Babylonian mathematics
□ In this article we now examine some algebra which the Babylonians developed, particularly problems which led to equations and their solution.
□ When a solution was found for y then x was found by x = by/a.
□ The solution given by the scribe is to compute 0; 40 times 0; 40 to get 0; 26, 40.
□ The form that their solutions took was, respectively .
27. Planetary motion
□ Nowadays astronomers accept that planetary motion has to be treated dynamically, as a many-body problem, for which there is bound to be no exact solution.
□ The astronomical solution to the one-body problem consists of the two laws: .
□ This composite solution represents what is in fact the earliest instance of a planetary orbit: it will be succinctly referred to in what follows as 'the Sun-focused ellipse'.
□ We shall now prove that, subject to its obvious external limitations, this unique solution is of universal applicability as a self-contained piece of mathematics.
□ Moreover, the kinematical solution is qualitatively different from any later, dynamical one in that it possesses exact geometrical representation - while the adoption of the auxiliary angle
as variable ensures that the treatment turns out to be the simplest possible.
28. Greek sources II
□ Although Archimedes promises a solution later in his text, it does not appear.
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□ Eutocius quotes from a solution by Diocles of this problem.
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□ A number of different solutions have been proposed but this is leading us away from the question of dating which we are discussing in this article.
29. Neptune and Pluto
□ Strangely Airy, who now knew that both Adams and Le Verrier had come to almost identical solutions to the same problem, did not tell either of them about the other, nor did he tell Le Verrier
of his plans to begin a search.
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□ His first solution had depended on assuming a distance for the "new planet" of twice that of Uranus from the Sun.
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□ He was unhappy with this arbitrary part of his solution and he had redone the mathematical analysis finding a better estimate of the distance of the "new planet" by testing different
distances against the observed perturbations of Uranus.
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30. Topology history
□ In 1736 Euler published a paper on the solution of the Konigsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of
a problem relating to the geometry of position.
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□ involved looking at the totality of all solutions rather than at particular trajectories as had been the case earlier.
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31. General relativity
□ Many possible solutions were proposed, Venus was 10% heavier than was thought, there was another planet inside Mercury's orbit, the sun was more oblate than observed, Mercury had a moon and,
really the only one not ruled out by experiment, that Newton's inverse square law was incorrect.
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□ Immediately after Einstein's 1915 paper giving the correct field equations, Karl Schwarzschild found in 1916 a mathematical solution to the equations which corresponds to the gravitational
field of a massive compact object.
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□ At the time this was purely theoretical work but, of course, work on neutron stars, pulsars and black holes relied entirely on Schwarzschild's solutions and has made this part of the most
important work going on in astronomy today.
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32. Arabic mathematics
□ This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric
construction of equations arose.
□ Omar Khayyam (born 1048) gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.
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□ Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work [Arch.
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33. Classical time
□ During the 16th century the solution of problems relating to time became of utmost importance because of its relation to finding the longitude.
□ In an age of exploration on a world scale, determining position became a crucial problem and much effort was put into its solution.
□ Theoretically this provided a solution to the longitude problem, but in practice observing the eclipses of Jupiter's moons from the deck of a ship was essentially impossible.
□ Several large prizes were offered for a solution to the problem of determining longitude and Galileo tried the persuade the Spanish Court in 1616 that he could determine absolute time using
Jupiter's moons and, after failing to convince them, tried to persuade Holland of his method when they offered a large prize in 1636.
34. The four colour theorem
□ De Morgan kept asking if anyone could find a solution to Guthrie's problem and several mathematicians worked on it.
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□ However the final ideas necessary for the solution of the Four Colour Conjecture had been introduced before these last two results.
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□ The year 1976 saw a complete solution to the Four Colour Conjecture when it was to become the Four Colour Theorem for the second, and last, time.
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35. Water-clocks
□ The solution to the first problem was to have the klepsydra supplied from a large reservoir of water kept at a constant level.
□ Vitruvius gives several solutions, the simplest being as follows [Vitruvius : ten books on architecture / translation [from the Latin] by Ingrid D Rowland ; commentary and illustrations by
Thomas Noble Howe ; with additional commentary by Ingrid D Rowland and Michael J Dewar, ed.
36. Matrices and determinants
□ from which the solution can be found for the third type of corn, then for the second, then the first by back substitution.
□ had a solution because .
□ In 1772 Laplace claimed that the methods introduced by Cramer and Bezout were impractical and, in a paper where he studied the orbits of the inner planets, he discussed the solution of
systems of linear equations without actually calculating it, by using determinants.
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37. History overview
□ Major progress in mathematics in Europe began again at the beginning of the 16th Century with Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic
Go directly to this paragraph
□ There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0.
38. Chinese overview
□ Perhaps it is most famous for presenting the 'Hundred fowls problem' which is an indeterminate problem with three non-trivial solutions.
□ He improved methods for finding square and cube roots, and extended the method to the numerical solution of polynomial equations computing powers of sums using binomial coefficients
constructed with Pascal's triangle.
39. Fermat's last theorem
□ has no non-zero integer solutions for x, y and z when n > 2.
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□ Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved.
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40. Indian mathematics
□ He introduced trigonometry in order to make his astronomical calculations, based on the Greek epicycle theory, and he solved with integer solutions indeterminate equations which arose in
astronomical theories.
□ Far from it for he made other major contributions in to the understanding of integer solutions to indeterminate equations and to interpolation formulas invented to aid the computation of sine
□ Citrabhanu was a sixteenth century mathematicians from Kerala who gave integer solutions to twenty-one types of systems of two algebraic equations.
41. EMS History
□ Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussion of new problems or new solutions, and
comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.
□ (Professor of Mathematics in the University of Edinburgh), on The Solution of Algebraic and Transcendental Equations in the Mathematical Laboratory.
42. Forgery 1
□ If this is so then he had not lost the ability to produce mathematics of the highest quality since his solution of the problem to determine the number of conics tangent to five given conics
which he found in 1864 was remarkable, particularly as it corrected a previous incorrect solution by the outstanding mathematician Steiner.
43. Pell's equation references
□ A A Antropov, Two methods for the solution of Pell's equation in the work of J Wallis (Russian), Istor.
□ A A Antropov, Wallis' method of "approximations" as applied to the solution of the equation xÛ - nyÛ = 1 in integers (Russian), Istor.-Mat.
44. Burnside problem references
□ I N Sanov, Solution of Burnside's problem for n = 4, Leningrad State University Annals (Uchenyi Zapiski) Math.
□ M Hall Jr., Solution of the Burnside Problem for Exponent Six, Illinois J.
45. Pell's equation references
□ A A Antropov, Two methods for the solution of Pell's equation in the work of J Wallis (Russian), Istor.
□ A A Antropov, Wallis' method of "approximations" as applied to the solution of the equation xÛ - nyÛ = 1 in integers (Russian), Istor.-Mat.
46. Harriot's manuscripts
□ He confessed that he had been listening to his master and a guest arguing about the solution to a mathematical problem.
□ Both, he politely pointed out, were wrong in their opinions and he proceeded to explain the correct solution.
47. Tait's scrapbook
□ The Scrapbook contains information about Tait's solution of the 15 puzzle.
□ Tait had solved the puzzle and submitted a paper on the solution when he saw two articles in the American Journal of Mathematics in 1879, one by W W Johnson and one by W E Story.
48. Burnside problem
□ If the Restricted Burnside Problem has a positive solution for some m, n then we may factor B(m, n) by the intersection of all subgroups of finite index to obtain B0(m,n), the universal
finite m-generator group of exponent n having all other finite m-generator groups of exponent n as homomorphic images.
□ 1994Zelmanov was awarded a Fields medal for his positive solution of the Restricted Burnside Problem.
49. Ring Theory
□ The equation xn+ yn= zn has no solution for positive integers x, y, z when n > 2.
□ In 1847 Lame announced that he had a solution of Fermat's Last Theorem and sketched out a proof.
50. Nine chapters
□ Essentially linear equations are solved by making two guesses at the solution, then computing the correct answer from the two errors.
□ Then the correct solution is .
51. Babylonian Pythagoras
□ ',17)">17], claims that the tablet is connected with the solution of quadratic equations and has nothing to do with Pythagorean triples:- .
□ This however is not the method of solution given by the Babylonians and really that is not surprising since it rests heavily on our algebraic understanding of equations.
52. Burnside problem references
□ I N Sanov, Solution of Burnside's problem for n = 4, Leningrad State University Annals (Uchenyi Zapiski) Math.
□ M Hall Jr., Solution of the Burnside Problem for Exponent Six, Illinois J.
53. Ten classics
□ In fact the solution given, although in a special case, gives exactly the modern method.
54. Abstract groups
□ The first version of Galois' important paper on the algebraic solution of equations was submitted to the Paris Academie des Sciences in 1829.
55. Indian Sulbasutras
□ This has the solution x = 1/(3 × 4 × 34) which is approximately 0.002450980392.
56. Golden ratio
□ multiplied the number of propositions concerning the section which had their origin in Plato, employing the method of analysis for their solution.
57. Gravitation
□ There was no simple solution to the problems that the different theories posed.
58. Real numbers 1
□ He then went on to compute an approximate solution.
59. Cartography
□ Al-Biruni wrote a textbook on the general solution of spherical triangles around 1000 then, some time after 1010, he applied these methods on spherical triangles to geographical problems.
60. Set theory
□ The first person to explicitly note that he was using such an axiom seems to have been Peano in 1890 in dealing with an existence proof for solutions to a system of differential equations.
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61. Pi history
□ Shortly after Shanks' calculation it was shown by Lindemann that π is transcendental, that is, π is not the solution of any polynomial equation with integer coefficients.
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62. Knots and physics
□ Its solution will be the proposed new kinetic theory of gases.
63. Egyptian mathematics
□ Some problems ask for the solution of an equation.
64. 20th century time
□ My solution was really for the very concept of time, that is, that time is not absolutely defined but there is an inseparable connection between time and the velocity of light.
65. Ledermann interview
□ He went over the problems from the previous week and very seldom, when somebody had given a really good solution, he was called on to go to the blackboard and show it to the other people.
66. Measurement
□ Diderot and d'Alembert in their Encyclopedie greatly regretted the diversity, but saw no possible acceptable solution to the problem.
67. Infinity
□ Fermat used his method to prove that there were no positive integer solutions to .
68. Group theory
□ Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group le groupe of permutations related to the equation.
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69. Real numbers 2
□ Clearly √2 is the root of a polynomial equation with rational coefficients, namely x2 = 2, and it is easy to see that all roots of rational numbers arise as solutions of such equations.
70. Squaring the circle references
□ T Albertini, La quadrature du cercle d'ibn al-Haytham : solution philosophique ou mathematique?, J.
71. Chrystal and the RSE
□ Scientifically speaking, uneducated themselves, they seem to think that they will catch the echo of the fact or the solution of an arithmetical problem by putting their ears to the
sounding-shell of uneducated public opinion.
72. Indian mathematics references
□ R Lal and R Prasad, Integral solutions of the equation Nx^2+1 = y^2 in ancient Indian mathematics (cakravala or the cyclic method), Ganita Bharati 15 (1-4) (1993), 41-54.
73. Brachistochrone problem references
□ H Erlichson, Johann Bernoulli's brachistochrone solution using Fermat's principle of least time, European J.
74. Brachistochrone problem references
□ H Erlichson, Johann Bernoulli's brachistochrone solution using Fermat's principle of least time, European J.
75. Cosmology
□ This solution implied that the Universe had been born at one moment, about ten thousand million years ago in the past and the galaxies were still travelling away from us after that initial
Go directly to this paragraph
76. Indian Sulbasutras references
□ A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulbasutra of Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin,
Studies in the history of mathematics 19 "Nauka" (Moscow, 1974), 220-222; 302.
77. Squaring the circle references
□ T Albertini, La quadrature du cercle d'ibn al-Haytham : solution philosophique ou mathematique?, J.
78. Real numbers 3
□ Solutions proposed by some mathematicians would only allow mathematics to treat objects which could be constructed.
79. Brachistochrone problem references
□ H Erlichson, Johann Bernoulli's brachistochrone solution using Fermat's principle of least time, European J.
80. Indian Sulbasutras references
□ A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulbasutra of Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin,
Studies in the history of mathematics 19 "Nauka" (Moscow, 1974), 220-222; 302.
81. Indian mathematics references
□ R Lal and R Prasad, Integral solutions of the equation Nx^2+1 = y^2 in ancient Indian mathematics (cakravala or the cyclic method), Ganita Bharati 15 (1-4) (1993), 41-54.
82. Babylonian numerals
□ How do we know this? Well if they had really found that the system presented them with real ambiguities they would have solved the problem - there is little doubt that they had the skills to
come up with a solution had the system been unworkable.
83. Newton's bucket
□ In 1918 Joseph Lense and Hans Thirring obtained approximate solutions of the equations of general relativity for rotating bodies.
84. Poincaré - Inspector of mines
□ The Davy safety lamp, invented about 1815, provided the solution at the time Poincare worked.
85. Mathematical classics
□ In fact the solution given, although in a special case, gives exactly the modern method.
Famous Curves
1. Cycloid
□ Pascal published a challenge (not under his own name but under the name of Dettonville) offering two prizes for solutions to these problems.
□ Wallis and Lalouere entered but Lalouere's solution was wrong and Wallis was also not successful.
□ Pascal published his own solutions to his challenge problems together with an extension of Wren's result which he was able to find.
□ He already knew the brachistochrone property of the cycloid and published his solution in 1697.
2. Involute
□ Finding a clock which would keep accurate time at sea was a major problem and many years were spent looking for a solution.
3. Trisectrix
□ Like so many curves it was studied to provide a solution to one of the ancient Greek problems, this one is in relation to the problem of trisecting an angle.
Societies etc
1. European Mathematical Society Prize
□ This problem deals with a variational problem with a singular boundary set, and proposes a finite representation of the optimum solution.
□ He succeeded in computing explicit solutions for "Asian options" where the pay-off is given by a time-average of geometric Brownian motion.
□ Thereby he gave a solution to a long-standing problem, open for more than 30 years, that has resisted the efforts of the greatest specialists of elliptic curves.
□ This last problem attracted the attention and efforts of many geometers for more than 20 years, and the method developed by Perelman yielded an astonishingly short solution.
□ Group invariant valuations were studied since Dehn's solution of Hilbert's third problem, with later contributions by Blaschke and others, and culminating in Hadwiger's celebrated
characterization theory for the intrinsic volumes.
□ Further contributions include discovery of a functional form of isoperimetric inequalities and a recent solution (with Artstein, Ball and Naor) of a long-standing Shannon's problem on entropy
production in random systems.
□ has introduced an entirely new perspective to the theory of discontinuous solutions of one-dimensional hyperbolic conservation laws, representing solutions as local superposition of
travelling waves and introducing innovative Glimm functionals.
□ His ideas have led to the solution of the long standing problem of stability and convergence of vanishing viscosity approximations.
□ In the technically demanding proof the travelling waves are constructed as solutions of a functional equation, applying centre manifold theory in an infinite dimensional space.
□ One of the earlier contributions is his surprising solution of the symplectic packing problem, completing work of Gromov, McDuff and Polterovich, showing that compact symplectic manifolds can
be packed by symplectic images of equally sized Euclidean balls without wasting volume if the number of balls is not too small.
□ Answering affirmatively Melnikov's conjecture, Tolsa provides a solution of the Painleve problem in terms of the Menger curvature.
2. Young Mathematician prize
□ for a study of short-wave asymptotics of the solution of the diffraction problem on a convex cylinder.
□ for work on the stability of solutions of operator Hamiltonian equations with periodic coefficients.
□ for works on the properties of a solution of non-homogeneous Cauchy-Riemann system.
□ for giving a solution of J-P Serre's problem.
□ for giving periodic solutions in control systems.
□ for finite-gap and isodromic solutions of equations of nonlinear Schrodinger type.
□ for giving fast algorithms for polynomial expansions and solutions of algebraic systems.
□ for work in representation theory for solutions of the Young-Baxter equation.
□ for work on regularity of solutions of some problems in mechanics.
3. AMS Steele Prize
□ for three fundamental papers: "On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two
complex variables", "An example of a smooth linear partial differential equation without solution", and "On hulls of holomorphy".
□ He is one of the founders of the modern theory of transformation groups and is particularly known for his contributions to the solution of Hilbert's fifth problem.
□ for his paper "Solution in the large for nonlinear hyperboic systems of conservation laws".
□ for two seminal papers "Viscosity solutions of Hamilton-Jacobi equations" (joint with P-L Lions), and "Generation of semi-groups of nonlinear transformations on general Banach spaces" (joint
with T M Liggett).
□ for the "Evans-Krylov theorem" as first established in the papers Lawrence C Evans "Classical solutions of fully nonlinear convex, second order elliptic equations", and N V Krylov "Boundedly
inhomogeneous elliptic and parabolic equations".
□ Methods for exact solution".
4. Clay Award
□ for his ground-breaking work in analysis, notably his optimal restriction theorems in Fourier analysis, his work on the wave map equation (the hyperbolic analogue of the harmonic map
equation), his global existence theorems for KdV type equations, as well as significant work in quite distant areas of mathematics, such as his solution with Allen Knutson of Horn's
conjecture, a fundamental problem about hermitian matrices that goes back to questions posed by Hermann Weyl in 1912 .
□ This conjecture posits an essentially geometric necessary and suffcient condition, "Psi", for a pseudo-differential operator of principal type to be locally solvable, i.e., for the equation
Pu = f to have local solutions given a finite number of conditions on F Dencker's work provides a full mathematical understanding of the surprising discovery by Hans Lewy in 1957 that there
exist a linear partial differential operator - a one-term, third-order perturbation of the Cauchy-Riemann operator - which is not local solvable in this sense.
□ for their work on local and global Galois representations, partly in collaboration with Clozel and Shepherd-Barron, culminating in the solution of the Sato-Tate conjecture for elliptic curves
with non-integral j-invariants.
□ Taubes' affirmative solution of the Weinstein conjecture for any 3-dimensional contact manifold is based on a novel application of the Seiberg-Witten equations to the problem.
□ for their solutions of the Marden Tameness Conjecture, and, by implication through the work of Thurston and Canary, of the Ahlfors Measure Conjecture.
5. AMS Bôcher Prize
□ for his memoirs "Green's function and the problem of Plateau", "The most general form of the problem of Plateau", and "Solution of the inverse problem of the calculus of variations".
□ for his solution of several outstanding problems in diffraction theory and scattering theory and for developing the analytical tools needed for their resolution.
□ for his work on the application of partial differential equations to differential geometry, in particular his completion of the solution to the Yamabe Problem in "Conformal deformation of a
Riemannian metric to constant scalar curvature".
□ for his profound contributions toward understanding the structure of singular sets for solutions of variational problems.
6. Paris Academy of Sciences
□ The 1857 prize was offered for a solution to Fermat's Last Theorem and, not surprisingly, no solutions were submitted even when the deadline was extended.
7. Norwegian Mathematical Society
□ That problem found a temporary solution when Heegaard succeeded in obtaining funds for a series of pamphlets, Norsk matematisk forenings skrifter ..
□ For many years starting in 1922, Crown Prince Olav awarded a prize for the best solutions to a series of problems posed in the Journal.
8. Sylvester Medal
□ for his solution of several outstanding problems of algebraic topology and of the methods he invented for this purpose which have proved of prime importance in the theory of the subject.
□ for his many contributions to number theory and in particular his solution of the famous problem concerning approximating algebraic numbers by rationals.
9. AMS Fulkerson Prize
□ for 'Informational complexity and effective methods of solution for convex extremal problems', Ekonomika i Matematicheskie Metody 12 (1976), 357-369.
□ for 'The solution of van der Waerden's problem for permanents', Akademiia Nauk SSSR.
10. NAS Award in Applied Mathematics
□ for his innovative and imaginative use of mathematics in the solution of a wide variety of challenging and significant scientific and engineering problems.
□ for his profound and penetrating solution of outstanding problems of statistical mechanics.
11. Rolf Schock Prize
□ .for his outstanding work in mathematical physics, particularly for his contribution to the mathematical understanding of the quantum-mechanical many-body theory and for his work on exact
solutions of models in statistical mechanics and quantum mechanics.
12. Abel Prize
□ for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.
13. AMS/SIAM Birkhoff Prize
□ for his contributions to the theory of Hamiltonian dynamical systems, especially his proof of the stability of periodic solutions of Hamiltonian systems having two degrees of freedom and his
specific applications of the ideas in connection with this work.
14. RSS Guy Medal in Gold
□ intended to encourage the cultivation of statistics in their scientific aspects and promote the application of numbers to the solution of important problems in all the relations of life in
which the numerical method can be employed, with a view to determining the laws which regulate them.
15. New York Mathematical Society
□ It is believed that the meetings of the Society may be rendered interesting by the discussion of mathematical subjects, the criticism of current mathematical literature, and solutions to
problems proposed by its members and correspondents.
16. Lagrange Prize
□ To give only a few examples one should mention first the systematic use of functional analysis and weak solutions for solving elliptic and parabolic differential equations, both theoretically
and numerically, further the various methods he developed for solving nonlinear problems and his profound studies on control problems for systems governed by partial differential equations,
optimal control first and controllability later with the introduction of the now standard Hilbert Uniqueness Method.
17. AMS Satter Prize
□ for her deep contributions to algebraic geometry, and in particular for her recent solutions to two long-standing open problems: the Kodaira problem ("On the homotopy types of compact Kahler
and complex projective manifolds") and Green's Conjecture ("Green's canonical syzygy conjecture for generic curves of odd genus"; and "Green's generic syzygy conjecture for curves of even
genus lying on a K3 surface".
18. AMS Wiener Prize
□ for his contributions to applied mathematics in the areas of supersonic aerodynamics, plasma physics and hydromagnetics, and especially for his contributions to the truly remarkable
development of inverse scattering theory for the solution of nonlinear partial differential equations.
19. AMS Cole Prize in Algebra
□ for their solution of Abhyankar's conjecture.
20. Dahlquist Prize
□ The prize, established in 1995, is awarded to a young scientist (normally under 45) for original contributions to fields associated with Germund Dahlquist, especially the numerical solution
of differential equations and numerical methods for scientific computing.
21. Wilks Award of the ASS
□ His originality and ability to provide practical solutions to real-world statistical problems illuminate his extensive writings; a notable example is his classic text Survey Sampling, which
is widely consulted and referenced by practitioners of statistics everywhere.
22. History of the EMS
□ Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussions of new problems or new solutions, an
comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.
23. Serbian Academy of Sciences
□ The Minister of Education saw that the only solution was to merge the two, which he did in 1892.
24. AMS Veblen Prize
□ for his work in differential geometry and, in particular, the solution of the four-dimensional Poincare conjecture.
25. International Congress Speaker
□ Carlos Kenig, The Global Behavior of Solutions to Critical Non-linear Dispersive Equations .
26. RSS Guy Medal in Bronze
□ intended to encourage the cultivation of statistics in their scientific aspects and promote the application of numbers to the solution of important problems in all the relations of life in
which the numerical method can be employed, with a view to determining the laws which regulate them.
27. Pioneer Prize
□ for his work on the applications of theoretical work in inverse problems to the solution of a wide range of industrial problems; for his promotion worldwide of industrial/applied mathematics
problem solving; for his initiative to include very active applied mathematics components in the Austrian Mathematical Community; and for the founding of the Austrian Academy of Sciences
sponsored RICAM, the Radon Institute for Computational and Applied Mathematics.
28. RSS Guy Medal in Silver
□ intended to encourage the cultivation of statistics in their scientific aspects and promote the application of numbers to the solution of important problems in all the relations of life in
which the numerical method can be employed, with a view to determining the laws which regulate them.
29. Edinburgh Mathematical Society
□ Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussions of new problems or new solutions, a
comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.
30. BMC 1956
□ de Rham, G WElementary solutions of certain differential equations .
31. BMC 2001
□ Bournaveas, N Low regularity solutions of the Klein-Gordon-Dirac equations .
32. Scientific Committee 2007
□ Using hotels, or moving the BMC to September are possible solutions.
33. BMC 1984
□ Read, C JA solution to the invariant subspace problem .
34. Minutes for 2006
□ Edmund Robertson offered to come to York to find a reasonable financial solution of this question.
35. BMC 2007
□ Qian, Z On strong solutions of the 3D-Navier-Stokes equations .
36. BMC 1964
□ Lighthill, M JAsymptotic properties of Fourier integrals and of solutions of partial differential equations .
37. BMC 2005
□ Maz'ya, V Unsolved mysteries of solutions to PDEs near the boundary .
38. Mathematical Association of America
□ Most of our existing journals deal almost exclusively with subjects beyond the reach of the average student or teacher of mathematics or at least with subjects with which they are familiar,
and little, if any, space, is devoted to the solution of problems É No pains will be spared on the part of the Editors to make this the most interesting and most popular journal published in
1. References for Newton
□ E J Aiton, The solution of the inverse-problem of central forces in Newton's 'Principia', Arch.
□ J B Brackenridge, Newton's mature dynamics and the 'Principia' : a simplified solution to the Kepler problem, Historia Math.
□ H Erlichson, Newton's first inverse solutions, Centaurus 34 (4) (1991), 345-366.
□ H Erlichson, Newton's solution to the equiangular spiral problem and a new solution using only the equiangular property, Historia Math.
□ H Erlichson, Newton's 1679/80 solution of the constant gravity problem, Amer.
□ B Pourciau, Newton's solution of the one-body problem, Arch.
2. References for Adleman
□ L Adleman, 'Molecular Computation of Solutions to Combinatorial Problems', Science, 266, November 11, 1994, pp.
□ R J Lipton, 'DNA Solution of Hard Computational Problems', Science, 268, April 28 1994, pp.
3. References for Cardan
□ E Kenney, Cardano : 'Arithmetic subtlety' and impossible solutions, Philosophia Mathematica II (1989), 195-216.
□ E Kenney, Cardano : 'arithmetic subtlety' and impossible solutions, Philos.
□ C Romo Santos, Cardano's 'Ars magna' and the solutions of cubic and quartic equations (Spanish), Rev.
4. References for Euler
□ H H Frisinger, The solution of a famous two-centuries-old problem : The Leonhard Euler Latin square conjecture, Historia Math.
□ E Knobloch, Leibniz and Euler : problems and solutions concerning infinitesimal geometry and calculus, Conference on the History of Mathematics (Rende, 1991), 293-313.
5. References for Huygens
□ P Dupont and C S Roero, The treatise 'De ratiociniis in ludo aleae' of Christiaan Huygens, with the 'Annotationes' of Jakob Bernoulli ('Ars conjectandi', Part I), presented in an Italian
translation, with historical and critical commentary and modern solutions (Italian), Mem.
□ E Shoesmith, Huygens' solution to the gambler's ruin problem, Historia Math.
6. References for De Moivre
□ P Dupont, Critical elaboration of de Moivre's solutions of the 'jeu de rencontre' (Italian), Atti Accad.
□ P Dupont, On the 'gamblers' ruin' problem : critical review of the solutions of De Moivre and Todhunter of a classical example (Italian), Atti Accad.
□ A Hald, On de Moivre's solutions of the problem of duration of play, 1708-1718, Arch.
7. References for Chazy
□ K G Atrokhov and V I Gromak, Solution of the Chazy system (Russian), Differ.
□ K G Atrokhov and V I Gromak, Solution of the Chazy system, Differ.
8. References for Harriot
□ M Kalmar, Thomas Hariot's 'De reflexione corporum rotundorum' : an early solution to the problem of impact, Arch.
□ J V Pepper, Some clarifications of Harriot's solution of Mercator's problem, Hist.
9. References for Schauder
□ W Forster, J Schauder : Fragments of a portrait, Numerical Solution of Highly Nonlinear Problems (Amsterdam, 1980), 417-425.
□ J Leray, My friend Julius Schauder, Numerical Solution of Highly Nonlinear Problems (Amsterdam, 1980), 427-439.
10. References for Al-Haytham
□ T Albertini, La quadrature du cercle d'ibn al-Haytham : solution philosophique ou mathematique?, J.
□ A S Dallal, ibn al-Haytham's universal solution for finding the direction of the qibla by calculation, Arabic Sci.
11. References for Al-Haytham
□ T Albertini, La quadrature du cercle d'ibn al-Haytham : solution philosophique ou mathematique?, J.
□ A S Dallal, ibn al-Haytham's universal solution for finding the direction of the qibla by calculation, Arabic Sci.
12. References for Delamain
□ J Dawplucker, Critical retrospect of works on the sliding-rule, and various mathematical problems and solutions in the 'Mechanics' magazine', Mechanics' Magazine (Saturday, 4 September 1830).
□ J Dawplucker, Critical retrospect of works on the sliding-rule, and various mathematical problems and solutions in the 'Mechanics' magazine', Iron: An illustrated weekly journal for iron and
steel manufacturers, metallurgists, mine proprietors, engineers, shipbuilders, scientists, capitalists 14 (1931), 5-6.
13. References for Wallis
□ A A Antropov, Two methods for the solution of Pell's equation in the work of J Wallis (Russian), Istor.
□ A A Antropov, Wallis' method of 'approximations' as applied to the solution of the equation xÛ - nyÛ = 1 in integers (Russian), Istor.-Mat.
14. References for Forsythe
□ A S Householder, Review: Computer Solution of Linear Algebraic Systems by George E Forsythe and Cleve B Moler, Mathematics of Computation 24 (110) (1970), 482.
□ R L Johnson, Review: Computer Solution of Linear Algebraic Systems by George E Forsythe and Cleve B Moler, SIAM Review 10 (3) (1968), 384-385.
15. References for Gini
□ G Favero, A Totalitarian Solution: Corrado Gini and Italian Economic Statistics.
16. References for Kublanovskaya
□ N K Jain and K Singhal, On Kublanovskaya's approach to the solution of the generalized latent value problem for functional-matrices, SIAM J.
17. References for Maschke
□ G Zappa, History of the solution of fifth- and sixth-degree equations, with an emphasis on the contributions of Francesco Brioschi (Italian), Rend.
18. References for Todhunter
□ P Dupont, On the 'gamblers' ruin' problem: critical review of the solutions of De Moivre and Todhunter of a classical example (Italian), Atti Accad.
19. References for Apastamba
□ A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulba Sutra Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin,
Studies in the history of mathematics 19 'Nauka' (Moscow, 1974), 220-222; 302.
20. References for Galileo
□ S Quan, Galileo and the problem of concentric circles : A refutation, and the solution, Ann.
21. References for Plancherel
□ Sur la convergence en moyenne des suites de solutions d'une equation aux derivees partielles du second ordre lineaire et de type elliptique.
22. References for Heun
□ A Ishkhanyan and K-A Suominen, New solutions of Heun's general equation, J.
23. References for De Beaune
□ A B Stykan and O A Petrova, Solution of a problem that was presented to Descartes by de Beaune (Russian), Istor.-Mat.
24. References for Al-Khwarizmi
□ Y Dold-Samplonius, Developments in the solution to the equation cxÛ + bx = a from al-Khwarizmi to Fibonacci, in From deferent to equant (New York, 1987), 71-87.
25. References for Al-Samawal
□ Y Dold-Samplonius, The solution of quadratic equations according to al-Samaw'al, in Mathemata, Boethius : Texte Abh.
26. References for Finkel
□ R F Davis, Review: A Mathematical Solution Book, by B F Finkel, The Mathematical Gazette 2 (41) (1903), 342-343.
27. References for Sripati
□ R S Lal and R Prasad, Contributions of Sripati (1039 A.D.) in the solution of first degree simultaneous indeterminate equations, Math.
28. References for Horner
□ M H Bektasova, From the history of numerical methods for the solution of equations (Russian), in Collection of questions on mathematics and mechanics No.
29. References for Torricelli
□ J Krarup and S Vajda, On Torricelli's geometrical solution to a problem of Fermat, Duality in practice, IMA J.
30. References for Banneker
□ B Lumpkin, From Egypt to Benjamin Banneker : African origins of false position solutions, in R Calinger (ed.), Vita mathematica (Washington, DC, 1996), 279-289.
31. References for Leonardo
□ E B Thro, Leonardo da Vinci's solution to the problem of the pinhole camera, Arch.
32. References for Qin Jiushao
□ Pai Shang-shu, An inquiry into the solution techniques of nine surveying problems raised by Chin Chiu-shao, in Discourses on the history of mathematics of the Sung and Yuan period (Peking
33. References for Steiner
□ H Dorrie, One Hundred Great Problems of Elementary Mathematics, Their History and Solution (Dover, 1965).
34. References for Leibniz
□ E Knobloch, Leibniz and Euler : problems and solutions concerning infinitesimal geometry and calculus, in Conference on the History of Mathematics (Rende, 1991), 293-313.
35. References for Brioschi
□ Zappa, History of the solution of fifth- and sixth-degree equations, with an emphasis on the contributions of Francesco Brioschi (Italian), Rend.
36. References for Bombelli
□ G S Smirnova, Geometric solution of the cubic equations in Raffaele Bombelli's 'Algebra' (Russian), Istor.
37. References for Ceva Giovanni
□ A Brigaglia and P Nastasi, The solutions of Girolamo Saccheri and Giovanni Ceva to Ruggero Ventimiglia's 'Geometram quaero' : Italian projective geometry in the late seventeenth century
(Italian), Arch.
38. References for Faedo
□ M Benzi and E Toscano, Mauro Picone, Sandro Faedo, and the Numerical Solution of Partial Differential Equations in Italy (1928-1953), Department of Mathematics and Computer Science, Emory
University, Atlanta, Georgia.
39. References for Democritus
□ D E Hahm, Chrysippus' solution to the Democritean dilemma of the cone, Isis 63 (217) (1972), 205-220.
40. References for Crank
□ A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc.
41. References for Courant
□ C A Felippa, 50 year classic reprint : an appreciation of R Courant's 'Variational methods for the solution of problems of equilibrium and vibrations', Internat.
42. References for Mollweide
□ C N Mills, Discussions: On Checking the Solution of a Triangle, Amer.
43. References for Rejewski
□ M Rejewski, Mathematical solution of the Enigma cipher, Cryptologia 6 (1) (1982), 1-18.
44. References for Chrysippus
□ D E Hahm, Chrysippus' solution to the Democritean dilemma of the cone, Isis 63 (217) (1972), 205-220.
45. References for Khayyam
□ P D Yardley, Graphical solution of the cubic equation developed from the work of Omar Khayyam, Bull.
46. References for D'Adhemar
□ J Hadamard, Recherches sur les solutions fondamentales et l'integration des equations lineaires aux derivees partielles (deuxieme memoire), Annales Scientifiques de l'ENS, troisieme serie 22,
(1905), 101-141.
47. References for Cholesky
□ L Fox, H D Huskey and J H Wilkinson, Notes on the solutions of algebraic linear simultaneous equations, Quart.
48. References for Al-Kashi
□ E M Bruins, Numerical solution of equations before and after al-Kashi, in Mathemata, Boethius : Texte Abh.
49. References for Gauss
□ W Benham, The Gauss anagram : an alternative solution, Ann.
50. References for Mikusinski
□ R Hilfer, Y Luchko and Z Tomovski, Operational method for the solution of fractional differential equations with generalised Riemann-Liouville fractional derivatives, Fractional Calculus and
Applied Analysis 12 (3) (2009), 299-318.
51. References for Mersenne
□ J MacLachlan, Mersenne's solution for Galileo's problem of the rotating earth, Historia Math.
52. References for Nicolson
□ A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc.
53. References for Seitz
□ B F Finkel, Biography: E B Seitz, in B F Finkel, A mathematical solution book (1893), 440-441.
54. References for Dudeney
□ M Kobayashi, Kiyasu-Zen'iti and G Nakamura, A solution of Dudeney's round table problem for an even number of people, J.
55. References for Banachiewicz
□ T Angelitch, The solution of systems of algebraic linear equations by the method of Banachiewicz (Serbo-Croat), Srpska Akad.
56. References for Mahavira
□ B Datta, On Mahavira's solution of rational triangles and quadrilaterals, Bull.
57. References for Von Staudt
□ E John Hornsby, Geometrical and Graphical Solutions of Quadratic Equations, The College Mathematics Journal 21 (5) (1990), 362-369.
58. References for Saccheri
□ A Brigaglia and P Nastasi, The solutions of Girolamo Saccheri and Giovanni Ceva to Ruggero Ventimiglia's 'Geometram quaero' : Italian projective geometry in the late seventeenth century
(Italian), Arch.
59. References for Sridhara
□ K Shankar Shukla, On Sridhara's rational solution of Nx^2+1=y^2, Ganita 1 (1950), 1-12.
60. References for Delaunay
□ Ja O Matviisin, Charles Eugene Delaunay (1816-1872) : an outline of his life and scientific activity (Ukrainian), in Projective-iterative methods for the solution of differential and integral
equations (Ukrainian) (Kiev, 1974), 116-130.
61. References for Roomen
□ P P A Henry, La solution de Francois Viete au probleme d'Adriaan van Roomen, Ecole Polytechnique Federale de Lausanne.
Additional material
1. Finkel's Solution Book
□ B F Finkel's Mathematical Solution Book .
□ A mathematical solution book containing systematic solutions of many of the most difficult problems.
□ Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical
Journals of the United States, and Many Original Problems and Solutions.
□ This work is the outgrowth of eight years' experience in teaching in the Public Schools, during which time I have observed that a work presenting a systematic treatment of solutions of
problems would be serviceable to both teachers and pupils.
□ Anyone who can write out systematic solutions of problems can resort to "Short Cuts" at pleasure; but, on the other hand, let a student who has done all his work in mathematics by formulae,
"Short Cuts," and "Lightning Methods" attempt to write out a systematic solution - one in which the work explains itself - and he will soon convince one of his inability to express his
thoughts in a logical manner.
□ One solution, thoroughly analysed and criticised by a class, is worth more than a dozen solutions the difficulties of which are seen through a cloud of obscurities.
□ It has been the aim to give a solution of every problem presenting anything peculiar, and those which go the rounds of the country.
□ In this edition I have added a chapter on Longitude and Time, the biographies of a few more mathematicians, several hundred more problems for solution, an introduction to the study of
Geometry, and an introduction to the study of Algebra.
□ http://www-history.mcs.st-andrews.ac.uk/Extras/Finkel_solution.html .
2. David Hilbert: 'Mathematical Problems
□ If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems
which the science of today sets and whose solution we expect from the future.
□ It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.
□ It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.
□ The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal.
□ The fruitful methods and the far-reaching principles which Poincare has brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the
circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.
□ The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the
theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory
of potential - to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics.
□ But, in the further development of a branch of mathematics, the human mind, encouraged by the success of its solutions, becomes conscious of its independence.
□ It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem.
□ I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are
implied in the statement of the problem and which must always be exactly formulated.
□ While insisting on rigour in the proof as a requirement for a perfect solution of a problem, I should like, on the other hand, to oppose the opinion that only the concepts of analysis, or
even those of arithmetic alone, are susceptible of a fully rigorous treatment.
□ Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed.
□ The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated.
□ In later mathematics, the question as to the impossibility of certain solutions plays a pre-eminent part, and we perceive in this way that old and difficult problems, such as the proof of the
axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another
sense than that originally intended.
□ It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a
proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the
impossibility of its solution and therewith the necessary failure of all attempts.
□ However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number
of purely logical processes.
□ Seek its solution.
3. Euler Elogium.html.html
□ These discoveries led Euler to an important discovery by observing the unique characteristics of exponential and logarithmic quantities born within the circle and following methods by which
the solutions make the problems disappear, the terms of the imaginaries which would then be present and which would have complicated the calculation, even though the are known to collapse,
reduced the formulas to simpler and more convenient expressions.
□ They are quickly enriched by the solutions to a great number of problems that the Mathematicians dared not deal with because of the difficulty and the physical impossibility to conduct their
calculations to a satisfactory conclusion.
□ His novel research into the series of indefinite products provided the necessary resources into solutions to a great many useful and curious questions.
□ He is responsible for the general solution of linear equations which are so varied and useful as well as the first of all formulas for approximations.
□ Often he preferred not to reveal the process of his thinking rather than to be exposed to the suspicion of a slight of hand and that he arrived at the solution only after the fact.
□ Euler arrived at the conclusion that differential equations are susceptible to particular solutions which are not included in general solutions in which Mr.
□ Euler who has shown why these particular integrals are excluded from the general solution and he is the first to have devoted some time to this theory, which has since been perfected by other
celebrated geometers and the memoire in which Mr.
□ de la Grange has left nothing unknown concerning the nature of these integrals and their use in the solution of problems.
□ The solutions to the problems of the least resistant solid, and the curve of quickest descent and the problem of largest isoperimetric areas were much celebrated in Europe.
□ The general method for solving the problem was hidden in these solutions, especially in those of Jakob Bernoulli who had found the answer to the isoperimetric question which provided him with
an advantage over his brother since so many masterpieces subsequently fathered by Johann Bernoulli.
□ The solution to the problem that is sought for the motion of an object which is launched into space and is attracted towards two points has become famous by Euler's ability of make the
necessary substitutions thought a reduction to quadratic equations so that their complexity and form might have made them appear to be insoluble.
□ In such a way that in the course of his work there sometimes appeared a unique method to integrate a differential equation or sometimes a remark concerning a question in Analysis or Mechanics
lead him to a solution to a very complicated differential equation which did not lend itself to direct methods.
□ At other times it would be a problem that appeared insurmountable that he resolved in an instant by a very simple method or an elementary problem with a very difficult solution that could
only be overcome with the greatest efforts.
4. ELOGIUM OF EULER
□ These discoveries led Euler to an important discovery by observing the unique characteristics of exponential and logarithmic quantities born within the circle and following methods by which
the solutions make the problems disappear, the terms of the imaginaries which would then be present and which would have complicated the calculation, even though the are known to collapse,
reduced the formulas to simpler and more convenient expressions.
□ They are quickly enriched by the solutions to a great number of problems that the Mathematicians dared not deal with because of the difficulty and the physical impossibility to conduct their
calculations to a satisfactory conclusion.
□ His novel research into the series of indefinite products provided the necessary resources into solutions to a great many useful and curious questions.
□ He is responsible for the general solution of linear equations which are so varied and useful as well as the first of all formulas for approximations.
□ Often he preferred not to reveal the process of his thinking rather than to be exposed to the suspicion of a slight of hand and that he arrived at the solution only after the fact.
□ Euler arrived at the conclusion that differential equations are susceptible to particular solutions which are not included in general solutions in which Mr.
□ Euler who has shown why these particular integrals are excluded from the general solution and he is the first to have devoted some time to this theory, which has since been perfected by other
celebrated geometers and the memoire in which Mr.
□ de la Grange has left nothing unknown concerning the nature of these integrals and their use in the solution of problems.
□ The solutions to the problems of the least resistant solid, and the curve of quickest descent and the problem of largest isoperimetric areas were much celebrated in Europe.
□ The general method for solving the problem was hidden in these solutions, especially in those of Jakob Bernoulli who had found the answer to the isoperimetric question which provided him with
an advantage over his brother since so many masterpieces subsequently fathered by Johann Bernoulli.
□ The solution to the problem that is sought for the motion of an object which is launched into space and is attracted towards two points has become famous by Euler's ability of make the
necessary substitutions thought a reduction to quadratic equations so that their complexity and form might have made them appear to be insoluble.
□ In such a way that in the course of his work there sometimes appeared a unique method to integrate a differential equation or sometimes a remark concerning a question in Analysis or Mechanics
lead him to a solution to a very complicated differential equation which did not lend itself to direct methods.
□ At other times it would be a problem that appeared insurmountable that he resolved in an instant by a very simple method or an elementary problem with a very difficult solution that could
only be overcome with the greatest efforts.
5. James Gregory's manuscripts
□ James Gregory's manuscripts on algebraic solutions of equations .
□ Ever since the famous discoveries of solutions for cubic and biquadratic equations by Scipio Ferro, Tartaglia and Cardan, of the Italian school, mathematicians of all countries had attempted
a generalization, and particularly addressed themselves to find an algebraic solution of the quintic equation.
□ It was not, indeed, until the beginning of the nineteenth century that the matter was settled, when Abel demonstrated the impossibility of such a solution, in general, for the quintic and
higher equations.
□ A comparison between these notes and the allusions contained in the letters of Gregory and Collins, from May to October, 1675, make it evident that Gregory believed he had hit upon a general
algebraic solution for all orders.
□ "I have now abundantly satisfied myself in these things I was searching after in the analytics, which are all about reduction and solution of equations.
□ A week or two later Tschirnhausen, then a young man of twenty-four years, arrived in London, to spend the summer in a search for the solution of all equations up to the eighth dimension.
□ Unfortunately the elimination processes would have yielded at least one irreducible equation of degree 6, had the attempted work been completed: and thus the hoped-for solution - by lowering
the degree from 5 to 4 - would have had to be abandoned.
6. L R Ford - Differential Equations
□ The first three chapters lead up to the later chapters by their discussions of direction fields, of solutions in series, of the Wronskian and linear dependence.
□ The method of successive approximations leads naturally into the Chapters VI and VII on interpolation and numerical integration and solutions.
□ Attention should be called to the treatments of finite differences and of the symbolic operators, also to the emphasis placed on whole families of solutions.
□ In Chapter X on partial differential equations of the first order the distinction between complete and general solutions is well brought out, also the geometrical interpretations of solutions
are emphasized.
□ What has been attempted here has been the presentation of a compact, connected, and (it is believed) teachable body of material which exhibits those elementary methods of solution which are
of commonest use." Here again the author has succeeded.
□ It is unusual to find Clairaut's equation and simple examples of solution in series in the first chapter of a text-book on differential equations, but the idea is a good one.
□ From the beginning the student learns that successive approximation to a solution may be the best we can do and that singular solutions may exist.
□ Subsequent chapters cover special methods for equations of first order, linear equations of any order with a brief account of the use of the Laplace transform, solution in series of the
hypergeometric, Legendre's and Bessel's equations, approximate numerical solutions, and two chapters on partial differential equations.
□ Numerical solutions are preceded by a good chapter on finite differences, including approximate differentiation and integration and the algebra of operators i and E.
□ General solutions of simple types of partial differential equations are obtained before separation of variables is used to solve problems of vibration and the Laplace equation in two
7. De Montmort: 'Essai d'Analyse
□ Having set down the rules, he solves simple cases in a method somewhat reminiscent of Huygens, and then takes a plunge into a general solution which appears to be correct but is not always
demonstrably so.
□ The preceding solution furnishes a singular use of the figurate numbers (of which I shall speak later), for I find in examining the formula, that Pierre's chance is expressible by an infinite
series of terms which have alternate + and - signs, and such that the numerator is the series of numbers which are found in the Table (i.e.
□ He doesn't bother with the rules (they must have been entirely established by this time) and he calculates several simple chances but remarks that in the majority of situations the solution
cannot be found.
□ I think I should add that this problem was posed by me to a Lady, who gave me almost immediately the correct solution using the Arithmetic Triangle.
□ He had a wide circle of correspondents among mathematicians of all countries, including Newton and Leibniz, exchanging with them news about mathematical problems and discussing solutions to
the problems of the day (and fray).
□ who, upon occasion of a French tract, called l'Analyse des Jeux de Hasard, which had lately been published, was pleased to propose to me some Problems of much greater difficulty than any he
had found in that Book; which having solved to his satisfaction, he engaged me to methodise these Problems, and to lay down the Rules which had led me to their Solution ..
□ Huygens, first, as I know, set down rules for the solution of the same kind of problem as those which the new French author illustrates freely with diverse examples.
□ Montmort has come to maturity and discusses, wherever he is able, the general solution to the problem he sets himself, rather than beginning with special cases and then plunging into a
general statement which is often unsupported by mathematical argument.
□ The expansion involved in the solution of his Proposition XVI:- .
□ De Moivre probably took the problem from the first edition, generalised it, and gave the solution with no indication of method of proof.
□ Montmort reached the solution by himself also, since a letter to Johann Bernoulli in 1710 shows that he had already obtained it.
□ The matching distribution is presented with a proof of the general case; this proof was not given in the first edition, implying that Montmort either guessed the original solution or was
dissatisfied with his first method of proof.
□ Both Johann and Nicolaus take as their focus of comment the first edition of the Essai d'Analyse, but it is Nicolaus who comes forward with helpful and sometimes new solutions.
□ Nicolaus is soothing to Montmort but fair to de Moivre, pointing out in several places in his commentary that de Moivre had shown to him his general solutions to various problems when he was
in London, and trying to explain that de Moivre had not intended to slight Montmort by his introduction.
8. Kantorovich books.html
□ The object of the present work, according to the author, is to show that the ideas and methods of functional analysis can be used for the development of effective, practical algorithms for
the explicit solutions of practical problems with just as much success as that with which they have been used for the theoretical study of these problems.
□ this book by Kantorovich and Krylov, originally titled Methods for the Approximate Solution of Partial Differential Equations, ..
□ This book is concerned with approximate methods used in the solution of partial differential equations, conformal mapping and the approximate solution of integral equations.
□ The book itself is concerned mainly with the numerical solution of partial differential equations, as the title to the first edition (1936) indicated.
□ Next come methods of solution of Fredholm integral equations with applications to the Dirichlet problem.
□ The work is concerned with what Lanczos calls 'parexic analysis', a study of processes which lead to approximate solutions of the problems of mathematical physics by rigorous methods.
□ He lays particular emphasis on the use of efficient prices, derived from the solution of a linear program, to bring about marginal improvements in resource allocation, without the need to
resort to a recomputation of the entire program.
□ Tables for the numerical solution of boundary value problems of the theory of harmonic functions (1963), by L V Kantorovich, V I Krylov and K Ye Chernin.
□ An approximate solution of the Dirichlet problem for a fixed region with arbitrarily prescribed boundary values [is] be obtained from tabulated values of [certain] coefficient functions ..
□ The authors are particularly concerned with applications of functional analysis to the theory of approximation and the theory of existence and uniqueness of solutions of differential and
integral equations (both linear and non-linear).
□ The first of these contained fundamental advances and determined the content and further development of this discipline: it examined the mathematically new type of "extremal" problems; it
evolved a universal method for their solution (method of solution multipliers) as well as various efficient numerical algorithms derived from it; it indicated the more important fields of
technical-economic problems where these methods could be most usefully applied; and it brought out the economic significance of indicators resulting from an analysis of problems by this
method which is particularly essential in problems of a socialist economy.
□ The discussion in academic circles in the Soviet Union around the proposals of Kantorovich has shown that his solution to some is quite unthinkable and unmarxist, while others with a knack
for higher mathematics and less Marx tend to support some of his main propositions.
9. Valiant Turing Award
□ This difficulty was characterized by complexity classes, such as P (tractable problems) and NP (problems for which a solution can easily be checked once it has been produced, but perhaps not
easily found in the first place).
□ For example, he and Vijay Vazirani wrote the influential paper "NP is as easy as detecting single solutions" (Theoretical Computer Science, 1986).
□ Before this paper, many researchers believed that the hardest search problems were those with many solutions (sparsely embedded among far more non-solutions), because the multiplicity of
solutions could confuse search algorithms and keep them from being able to narrow in on a single solution.
□ Valiant and Vazirani gave a dramatic and beautiful demonstration that this idea was completely wrong, by showing how a good algorithm for finding unique solutions could be used to solve any
search problem.
□ Valiant discovered a brilliant and simple randomized solution to the problem.
□ Following this articulation of the challenge, Valiant proposed the BSP model as a candidate solution.
10. Charles Bossut on Leibniz and Newton Part 2
□ It produced challenges of very difficult problems, the solution of which gave rise to new theories, and considerably extended the domain of geometry.
□ The English already triumphed: but Johann Bernoulli, taking up the cause of Leibniz who had just died, laughed at this scheme of a solution.
□ However, they all agreed in considering Newton's solution as insufficient and of no use.
□ Without stopping to develop Newton's solution, he gave one of his own in the Philosophical Transactions for 1717 which answered the question as proposed by Leibniz in it's full extent.
□ Had he contented himself with this he would have merited only praise: but, urged on by his resentment against Johann Bernoulli who had spoken a little slightingly of him on another occasion,
he prefixed to his solution some insulting reflections on the partisans of Leibniz, having principally in view Johann Bernoulli their leader.
□ Among other things, he said that if they did not perceive how Newton's solution led to the equations of the problem, it must be attributed to their ignorance: illorum imperitiae tribuendum.
□ In a dissertation on orthogonal trajectories in the Leipzig Transactions for 1718, composed jointly by Johann Bernoulli and his son Nicholas, it was agreed that Dr Taylor's solution was
accurate, and even evinced some sagacity; but then it was shown that it was far from being sufficiently general and that there existed a great number of resolvable cases to which it could not
be applied.
□ Of this he complained with acrimony; and at the same time retorted the accusation by showing that Johann Bernoulli, in his last solution of the isoperimetrical problem, had travestied the
solution of his brother, and that all the simplifications he had made in it had not changed it's nature.
□ We are again obliged to say that Johann Bernoulli retained his superiority here by the simplicity and elegance of his solutions.
□ This case Keill proposed to Johann Bernoulli, who not only resolved it in a very short time, but extended the solution to the general hypothesis in which the resistance of the medium should
be as any power of the velocity of the projectile.
□ When he had discovered this theory, he offered repeatedly to send it to a confidential person in London on condition that Keill would give up his solution likewise; but Keill, though strongly
urged, maintained a profound silence.
□ In this conjecture he was cruelly mistaken: and his challenge, which was something more than indiscreet, drew on him a reprimand from the Swiss geometrician that was so much the more poignant
as the only mode of answering it satisfactorily was by a solution of the problem which he could neither effect by his own skill nor by the assistance of his friends.
11. Eulogy to Euler by Fuss
□ His superiority in analysis provided the necessary recognition, however what truly made his glory was the solution to the isoperimetric problem, so famous by its controversy between the two
Bernoulli brothers, Johann and Jakob each of whom pretended to have found the solution but neither knew of it in its entirety.
□ Bernoulli possessed the greater ability in physical principals combined with a patience to help in the solution of the problems which calculations brought about by experiments conducted with
the utmost focus and manipulation.
□ While reading through the fifty-four letters that the King wrote to him between 1741 and 1777, amongst which there are letters in the King's own handwriting and I have noted that more often
that not his particularly brilliant solutions were used.
□ He had provided real and immediate solutions to problems concerning the salt works at Schonebeck, the fountain pumps at Sans-Souci and various financial projects.
□ Bernoulli's solution taken from the Taylorian troichoides is not general and certainly not sufficient to explain it.
□ The solution to the important problem regarding the precession of the equinox and the nutation of the earth's axis that Mr.
□ In his last memoire he had found a solution as to how to understand the number of eccentricities in lunar motion, which he had not been able to determine in his first theory due to the
complicated calculations and the incomplete method that was available to him at that time.
□ One finds the most felicitous integrations and a multitude of contrivances and refinements of the most sublime analysis, truly deep research on the nature and the properties of numbers, the
ingenious proofs of a numbers of Fermat's theorems, the solution to a number of very difficult problems concerning equilibrium and the motion of solid bodies both flexible and elastic and the
unraveling of a number of apparent paradoxes.
12. Eulogy to Euler by Fuss
□ His superiority in analysis provided the necessary recognition, however what truly made his glory was the solution to the isoperimetric problem, so famous by its controversy between the two
Bernoulli brothers, Johann and Jakob each of whom pretended to have found the solution but neither knew of it in its entirety.
□ Bernoulli possessed the greater ability in physical principals combined with a patience to help in the solution of the problems which calculations brought about by experiments conducted with
the utmost focus and manipulation.
□ While reading through the fifty-four letters that the King wrote to him between 1741 and 1777, amongst which there are letters in the King's own handwriting and I have noted that more often
that not his particularly brilliant solutions were used.
□ He had provided real and immediate solutions to problems concerning the salt works at Schonebeck, the fountain pumps at Sans-Souci and various financial projects.
□ Bernoulli's solution taken from the Taylorian troichoides is not general and certainly not sufficient to explain it.
□ The solution to the important problem regarding the precession of the equinox and the nutation of the earth's axis that Mr.
□ In his last memoire he had found a solution as to how to understand the number of eccentricities in lunar motion, which he had not been able to determine in his first theory due to the
complicated calculations and the incomplete method that was available to him at that time.
□ One finds the most felicitous integrations and a multitude of contrivances and refinements of the most sublime analysis, truly deep research on the nature and the properties of numbers, the
ingenious proofs of a numbers of Fermat's theorems, the solution to a number of very difficult problems concerning equilibrium and the motion of solid bodies both flexible and elastic and the
unraveling of a number of apparent paradoxes.
13. Finkel and The American Mathematical Monthly
□ Most of our existing Journals deal almost exclusively with subjects beyond the reach of the average student or teacher of Mathematics or at least with subjects with which they are not
familiar, and little, if any space, is devoted to the solution of problems.
□ While realizing that the solution of problems is one of the lowest forms of Mathematical research, and that, in general, it has no scientific value, yet its educational value can not be over
□ 'The American Mathematical Monthly' will, therefore, devote a due portion of its space to the solution of problems, whether they be the easy problems in Arithmetic, or the difficult problems
in the Calculus, Mechanics, Probability, or Modern Higher Mathematics.
□ Teachers, students and all lovers of mathematics are, therefore, cordially invited to contribute problems, solutions and papers on interesting and important subjects in mathematics.
□ All problems, solutions, and articles intended for publication in the February Number, should be received on or before February 1st, 1894.
□ Solutions to problems in this Number will appear in March Number, but should be mailed to Editors before February 15th.
□ Professor Finkel remained one of the editors of the department of Problems and Solutions through 1933, and was still a member of the board of editors at the time of his death.
14. Percy MacMahon addresses the British Association in 1901, Part 2
□ Their solutions did not further the general progress, but were merely valuable in connection with the special problems.
□ Starting with this notion, Euler developed a theory of generating functions on the expansion of which depended the formal solutions of many problems.
□ I propose to give some account of these problems, and to add a short history of the way in which a method of solution has been reached.
□ One of the most important questions awaiting solution in connection with the theory of finite discontinuous groups is the enumeration of the types of groups of given order, or of Latin
Squares which satisfy additional conditions.
□ For a general investigation, however, it is more scientific to start by designing functions and operations, and then to ascertain the problems of which the solution is furnished.
□ a further condition being that one solution only is given by a group of numbers satisfying the equation; that in fact permutations amongst the quantities a, b, g ..
□ A generating function can be formed which involves in its construction the Diophantine equation and inequalities, and leads after treatment to a representative, as well as enumerative,
solution of the problem.
15. John Williamson papers
□ P S Dwyer,E P Starke, J Williamson and J H M Wedderburn, Problems and Solutions: Advanced Problems: Solutions: 3645, Amer.
□ R Robinson and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 3667, Amer.
□ J W Cell and J Williamson, Problems and Solutions: Elementary Problems: Solutions: E280, Amer.
□ J H M Wedderburn and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 3856, Amer.
□ J A Greenwood and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 4023, Amer.
□ C E Springer and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 3994, Amer.
□ F J Duarte and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 4159, Amer.
□ D H Lehmer, D M Smiley, M F Smiley and J Williamson, Problems and Solutions: Elementary Problems: Solutions: E710, Amer.
16. Poincaré on the future of mathematics
□ Many times already men have thought that they had solved all the problems, or at least that they had made an inventory of all that admit of solution.
□ And then the meaning of the word solution has been extended; the insoluble problems have become the most interesting of all, and other problems hitherto undreamed of have presented
□ For the Greeks a good solution was one that employed only rule and compass; later it became one obtained by the extraction of radicals, then one in which algebraical functions and radicals
alone figured.
□ An algebraical formula which gives us the solution of a type of numerical problems, if we finally replace the letters by numbers, is the simple example which occurs to one's mind at once.
□ What is it that gives us the feeling of elegance in a solution or a demonstration? It is the harmony of the different parts, their symmetry, and their happy adjustment; it is, in a word, all
that introduces order, all that gives them unity, that enables us to obtain a clear comprehension of the whole as well as of the parts.
□ Briefly stated, the sentiment of mathematical elegance is nothing but the satisfaction due to some conformity between the solution we wish to discover and the necessities of our mind, and it
is on account of this very conformity that the solution can be an instrument for us.
□ And since it enables us to foresee whether the solution of these problems will be simple, it shows us at least whether the calculation is worth undertaking.
□ Formerly an equation was not considered to have been solved until the solution had been expressed by means of a finite number of known functions.
□ It then remains to find the exact solution of the problem.
□ Can this be regarded as a true solution? The story goes that Newton once communicated to Leibnitz an anagram somewhat like the following: aaaaabbbeeeeii, etc.
□ To-day a similar solution would no longer satisfy us, for two reasons - because the convergence is too slow, and because the terms succeed one another without obeying any law.
□ Nevertheless an imperfect solution may happen to lead us towards a better one.
□ When the problems relating to congruents with several variables have been solved, we shall have made the first step towards the solution of many questions of indeterminate analysis.
□ Or rather I am wrong, for they would certainly have presented themselves, since their solution is necessary for a host of questions of analysis, but they would have presented themselves
isolated, one after the other, and without our being able to perceive their common link.
17. Taylor versus Continental mathematicians
□ Jacob Bernoulli had published a correct solution in 1701 but Johann Bernoulli's solution, obtained at the same time, was not satisfactory.
□ The author has confined himself solely to his subject; he has avoided express mention of what has been done by others because that would necessarily have forced him to note several mistakes
and imperfections found in their solutions: for that reason he has not judged it apropos to speak of the solut)rowed from these solutions the analysis he uses to resolve these problems, these
solutions having the defect of being restricted to particular cases, although the problems are proposed in general terms.
□ would be regarded as a favour done to the author of this solution, because, if he had mentioned it, he would not have been able to dispense with censuring three or four very considerable
mistakes which one encounters there.
18. Pólya: 'How to solve it' Preface
□ A great discovery solves a great problem but there is a grain of discovery in the solution of any problem.
□ He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: "Yes, the solution seems to work, it
appears to be correct; but how is it possible to invent such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such facts? And how could I
invent or discover such things by myself?" Today the author is teaching mathematics in a university; he thinks or hopes that some of his more eager students ask similar questions and he tries
to satisfy their curiosity.
□ Trying to understand not only the solution of this or that problem but also the motives and procedures of the solution, and trying to explain these motives and procedures to others, he was
finally led to write the present book.
□ Behind the desire to solve this or that problem that confers no material advantage, there may be a deeper curiosity, a desire to understand the ways and means, the motives and procedures, of
□ The following pages are written somewhat concisely, but as simply as possible, and are based on a long and serious study of methods of solution.
19. Vajda books
□ In fact, this book would better be described as a digest of articles which deal with the solution of problems by linear programming, with emphasis on the use of the transportation method.
□ The book is rich in exercises and illustrative examples, and it covers a wide variety of theorems ranging from existence of solutions to their construction.
□ The only significant change is the addition of seven new chapters devoted to brief introductions to and examples of Discrete Linear Programming, Dynamic Programming, Stochastic Programming
and Quadratic Programming, and also of a section on the solutions to the problems posed at the ends of some chapters.
□ There is no increase in the size of the book; this has been achieved partly by the regrettable removal of problems for the reader and their solutions.
□ However, unless the reader already has a good working knowledge of the subjects, he will find the descriptions of the techniques and the solutions to the problems rather brief.
□ This is a collection of 236 problems, with answers or full solutions.
□ Some hypothetical situations are presented as exercises for each of the four chapters, and for the serious student their solutions are presented.
20. Reviews Landau Lifshitz.html
□ For example, while a basic treatment of the Dirac equation will discuss its solution for spherically symmetric potentials, ending with the bound-state solutions to the Coulomb problem, this
book goes on to give the scattering states in a Coulomb potential, a treatment of ultra-relativistic scattering, the solution to the Dirac equation in an external electromagnetic plane wave,
and more.
□ Quite a number of examples, in the form of problems with solutions, appear throughout the text.
□ No problems for solution by the student are given.
21. Halmos: creative art
□ The modern angle trisector either doesn't know those rules, or he knows them but thinks that the idea is to get a close approximation, or he knows the rules and knows that an exact solution
is required but lets wish be father to the deed and simply makes a mistake.
□ Let me, therefore, conclude this particular tack by mentioning a tiny and trivial mathematical problem and describing its solution-possibly you'll then get (if you don't already have) a
little feeling for what attracts and amuses mathematicians and what is the nature of the inspiration I have been talking about.
□ All cognoscenti know, therefore, that the presence in the statement of a problem of a number like 1 + 210 is bound to be a strong hint to its solution; the chances are, and this can be
guessed even before the statement of the problem is complete, that the solution will depend on doubling - or halving - something ten times.
□ That's a simple job that pencil and paper can accomplish in a few seconds; the answer (and hence the solution of the problem) is 1024.
□ The trouble with this solution is that it's much too special.
□ His solution works, but it is as free of inspiration as the layman's.
□ The problem has also an inspired solution, that requires no computation, no formulas, no numbers - just pure thought.
□ In mathophysics the question always comes from outside, from the 'real world', and the satisfaction the scientist gets from the solution comes, to a large extent, from the light it throws on
22. Kepler's Planetary Laws
□ Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
□ And because it was expressed geometrically, the solution would potentially be exact - the closed orbit of a single planet in a plane round the fixed Sun.
□ This is the process that was described (in Section 4) as idealization because it ensured an exact solution (of the one-body problem) which was uniquely simple.
□ The solutions reached in each case are in some senses provisional, but they are certainly vital steps on the way to the presentday solution.
23. Kepler's Planetary Laws
□ Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
□ And because it was expressed geometrically, the solution would potentially be exact - the closed orbit of a single planet in a plane round the fixed Sun.
□ This is the process that was described previously as idealization because it ensured an exact solution (of the one-body problem) which was uniquely simple.
□ The solutions reached in each case are in some senses provisional, but they are certainly vital steps on the way to a modern solution.
24. Al-Khwarizmi and quadratic equations
□ In this book, which has given us the word 'algebra', al-Khwarizmi gives a complete solution to all possible types of quadratic equation.
□ Here is al-Khwarizmi's solution of the equation .
□ What is most remarkable is that in this case he knows that the quadratic has two solutions:- .
□ Now it is clear that al-Khwarizmi is intending to teach his readers general methods of solution and not just how to solve specific examples.
□ When you meet an instance which refers you to this case, try its solution by addition, and if that does not work subtraction will.
25. Carathéodory: 'Conformal representation
□ This work of Gauss appeared to give the whole inquiry its final solution; actually it left unanswered the much more difficult question whether and in what way a given finite portion of the
surface can be represented on a portion of the plane.
□ In the proof of this theorem, which forms the foundation of the whole theory, he assumes as obvious that a certain problem in the calculus of variations possesses a solution, and this
assumption, as Weierstrass (1815-1897) first pointed out, invalidates his proof Quite simple, analytic, and in every way regular problems in the calculus of variations axe now known which do
not always possess solutions.
□ Nevertheless, about fifty years after Riemann, Hilbert was able to prove rigorously that the particular problem which arose in Riemann's work does possess a solution; this theorem is known as
Dirichlet's Principle.
26. Henry Baker addresses the British Association in 1913, Part 2
□ Then I mention the historical fact that the problem of ascertaining when that well-known linear differential equation called the hypergeometric equation has all its solutions expressible in
finite terms as algebraic functions, was first solved in connection with a group of similar kind.
□ For any linear differential equation it is of primary importance to consider the group of interchanges of its solutions when the independent variable, starting from an arbitrary point, makes
all possible excursions, returning to its initial value.
□ But our whole physical outlook is based on the belief that the problems of Nature are expressible by differential equations; and our knowledge of even the possibilities of the solutions of
differential equations consists largely, save for some special types, of that kind of ignorance which, in the nature of the case, can form no idea of its own extent.
□ There are subjects whose whole content is an excuse for a desired solution of a differential equation; there are infinitely laborious methods of arithmetical computation held in high repute
of which the same must be said.
27. N S Krylov's monograph - Introduction
□ Closely connected with the first type of difficulties are the problem of the mechanical interpretation of irreversibility and, among other things, all the well-known objections to Boltzmann's
treatment of the H-theorem, and all the attempts still being made at achieving a quantum-mechanical solution of this problem.
□ The purpose of this work is, in considering together the two above-mentioned groups of difficulties, to give a solution to the problem that would explicitly introduce the basic concept of
statistical physics - the concept of relaxation - and would make possible the quantitative evaluation of relaxation time.
□ Finally, it should form a logically well built structure devoid, at any rate, of those contradictions that are characteristic, as will be shown further, of all the solutions that have ever
been proposed to the problem taken in all its generality (that is, comprising the problem of introducing irreversibility, ergodicity, and finite relaxation time into the theory).
□ Although the solution of the said basic problem must be brought about by the work as a whole, each of the above-mentioned parts may be regarded as constituting a more or less self-contained
28. The Dundee Numerical Analysis Conferences
□ This included an Honours course in numerical analysis, taught mainly by Jim Fulton, although Mike Osborne contributed some lectures on the numerical solution of differential equations.
□ If there were any records of the meeting, they have not survived, but I still possess a folder (unfortunately the original contents are long gone) with "Symposium on the solution of
differential equations, St Andrews, June 1965" written on it.
□ Mike talked about the recently described Nordsieck's method being equivalent to a multi-step method, Ron gave a survey talk on ADI methods, and Jack Lambert spoke on some aspect of the
numerical solution of ordinary differential equations.
□ This was held from 26-30 June, and called "Colloquium on the numerical solution of differential equations".
□ It was called "Conference on the numerical solution of differential equations", attracted 148 participants, and there were eight invited speakers, all from outside the UK: J Albrecht, E G
D'Jakonov, B Noble, K Nickel, G Strang, M Urabe, E Vitasek, O Widlund.
□ There was a Seminar on Ritz-Galerkin and the Finite Element Method from 8 - 9 July, 1971, and finally a Conference on the Numerical Solution of Ordinary Differential Equations from 5 - 6
August, 1971.
□ In 1973 a Conference on the Numerical Solution of Differential Equations was held from July 3 - 6.
29. Dickson: 'Theory of Equations
□ The exercises are so placed that a reasonably elegant and brief solution may be expected, without resort to tedious multiplications and similar manual labour.
□ An easy introduction to determinants and their application to the solution of systems of linear equations is afforded by Chapter XI, which is independent of the earlier chapters.
□ In the first chapter we are given also the graphical solution of a quadratic.
□ These puzzle students and often teachers, partly because the problem is not clearly understood, and partly because there is so obviously a solution; and yet their impossibility may readily be
made plausible to a student familiar with coordinate geometry and is here rigorously proved in an elementary way.
□ The use of elementary calculus allows a clear treatment and a complete solution of the problem, "given an equation to locate its real roots," while the methods of Chap.
□ Besides Horner's well-known method for the numerical computation of roots, Newton's is given and emphasized as one that is effective for non-algebraic as well as for algebraic equations; and
Graffe's little known but very ingenious scheme of solution by forming equations whose roots are powers of the roots of the given equation, and Lagrange's solution by continued fractions are
also explained.
30. Truesdell's books
□ The reader must constantly go back and forth between the actors on the stage, who, of course, do not yet know the solution of the knots, and Truesdell, who does.
□ It contains many recently determined solutions to the equations of motion for nonlinear fluids and uses these solutions to evaluate the fluid models.
□ The book is written very clearly and contains a large number of exercises and their solutions.
31. Whittaker EMS Obituary.html
□ It became particularly useful when wave mechanics was being developed in the years 1925 and 1926 and solutions of Schrodinger's equation were being ardently sought after.
□ Then in 1902 and 1903 he published two papers on the partial differential equations of mathematical physics in which he obtained the solution of Laplace's equation with which his name is
□ The advertisement intimating the opening of the laboratory in October 1913 specified the subjects to be taught (some of which were interpolation, method of least squares, solution of systems
of linear equations, evaluation of determinants, determination of roots of transcendental equations, practical Fourier analysis, evaluation of definite integrals, numerical solution of
differential equations, construction of tables of functions not previously tabulated such as parabolic cylinder functions) and also indicated that facilities were available for original
research and that the University would grant recognition, under certain conditions, to research students who would be permitted to offer themselves for the degree of D.Sc.
32. Todd: 'Basic Numerical Mathematics
□ Throughout both volumes we emphasize the idea of "controlled computational experiments": we try to check our programs and get some idea of errors by using them on problems of which we already
know the solution such experiments can in some way replace the error analyses which are not appropriate in beginning courses.
□ Then the direct solution of the inversion problem is taken up, first in the context of theoretical arithmetic (i.e., when round-off is disregarded) and then in the context of practical
□ Next, several iterative methods for the solution of systems of linear equations are examined.
□ It is then feasible to discuss two applications: the first, the solution of a two-point boundary value problem, and the second, that of least squares curve fitting.
□ We have not considered it necessary to give the machine programs required in the solution of the problems: the programs are almost always trivial and when they are not, the use of library
subroutines is intended.
□ A typical problem later in Volume 2 will require, e.g., the generation of a special matrix, a call to the library for a subroutine to operate on the matrix and then a program to evaluate the
error in the alleged solution provided by the machine.
33. Carl Runge: 'Graphical Methods
□ We may distinguish different stages in the solution of a problem.
□ Or to give another instance take Fermat's problem, for the solution of which the late Mr Wlolfskehl, of Darmstadt, has left $25,000 in his will.
□ So the solution of the problem may or may not end in its first stage.
□ In many other cases the first stage of the solution may be so easy, that we immediately pass on to the second stage of finding methods to calculate the unknown quantities sought for.
□ Or even if the first stage of the solution is not so easy, it may be expedient to pass on to the second stage.
□ So there arises a third stage of the solution of a mathematical problem in which the object is to develop methods for finding the result with as little trouble as possible.
34. Julia Robinson: Hilbert's 10th Problem
□ Julia Robinson and Martin Davis spent a large part of their lives trying to solve Hilbert's Tenth Problem: Does there exist an algorithm to determine whether a given Diophantine equation had
a solution in rational numbers? .
□ I even worked in the opposite direction, trying to show that there was a positive solution to Hilbert's problem, but I never published any of that work.
□ It followed that the solution to Hilbert's tenth problem is negative - a general method for determining whether a given diophantine equation has a solution in integers does not exist.
□ I have been told that some people think that I was blind not to see the solution myself when I was so close to it.
□ At that time, in connection with the solution of Hilbert's problem and the role played in it by the Robinson hypothesis, Linnik told me that I was the second most famous Robinson in the
Soviet Union, the first being Robinson Crusoe.
35. Julia Robinson: Hilbert's 10th Problem
□ Julia Robinson and Martin Davis spent a large part of their lives trying to solve Hilbert's Tenth Problem: Does there exist an algorithm to determine whether a given Diophantine equation had
a solution in rational numbers? .
□ I even worked in the opposite direction, trying to show that there was a positive solution to Hilbert's problem, but I never published any of that work.
□ It followed that the solution to Hilbert's tenth problem is negative - a general method for determining whether a given diophantine equation has a solution in integers does not exist.
□ I have been told that some people think that I was blind not to see the solution myself when I was so close to it.
□ At that time, in connection with the solution of Hilbert's problem and the role played in it by the Robinson hypothesis, Linnik told me that I was the second most famous Robinson in the
Soviet Union, the first being Robinson Crusoe.
36. Julia Robinson: Hilbert's 10th Problem
□ Julia Robinson and Martin Davis spent a large part of their lives trying to solve Hilbert's Tenth Problem: Does there exist an algorithm to determine whether a given Diophantine equation had
a solution in rational numbers? .
□ I even worked in the opposite direction, trying to show that there was a positive solution to Hilbert's problem, but I never published any of that work.
□ It followed that the solution to Hilbert's tenth problem is negative - a general method for determining whether a given diophantine equation has a solution in integers does not exist.
□ I have been told that some people think that I was blind not to see the solution myself when I was so close to it.
□ At that time, in connection with the solution of Hilbert's problem and the role played in it by the Robinson hypothesis, Linnik told me that I was the second most famous Robinson in the
Soviet Union, the first being Robinson Crusoe.
37. Mathematicians and Music 3
□ Of this equation he found the solution .
□ Euler immediately raised the question of the generality of the solution and set forth his interpretation.
□ He started with Taylor's particular solution and found, in effect, that the function for determining the position of the string after starting from rest could naturally be expressed in a form
later called a Fourier series.
□ Bernoulli remarked that since his solution was perfectly general it should include those of Euler and D'Alembert.
□ No mathematician would admit even the possibility of its solution till this was thoroughly demonstrated, in connection with certain problems in the flow of heat, by Fourier who gives due
credit to the suggestiveness of the work of those in the previous century to whom I have referred.
38. Planetary motion tackled kinematically
□ Nowadays astronomers accept that planetary motion has to be treated dynamically, as a many-body problem, for which there is bound to be no exact solution.
□ The astronomical solution to the one-body problem consists of the two laws: .
□ This composite solution represents what is in fact the earliest instance of a planetary orbit: it will be succinctly referred to in what follows as 'the Sun-focused ellipse'.
□ We shall now prove that, subject to its obvious external limitations, this unique solution is of universal applicability as a self-contained piece of mathematics.
□ As a consequence, our kinematical solution is qualitatively different from any later, dynamical one in that it is exact on the basis of geometry alone, as we shall demonstrate.
39. L'Hôpital: 'Analyse des infiniment petits' Preface
□ But since he was mainly concerned with the solution of equations he was interested in curves only as a way to finding roots.
□ M Descartes' Geometry made it fashionable to solve geometrical problems by means of equations, and opened up many possibilities of obtaining such solutions.
□ The ninth consists of solutions to various problems arising out of the earlier work.
40. John Couch Adams' account of the discovery of Neptune
□ I find among my papers the following memorandum, dated July 3, 1841: "Formed a design, in the beginning of this week, of investigating, as soon as possible after taking my degree, the
irregularities in the motion of Uranus, which are yet unaccounted for, in order to find whether they may be attributed to the action of an undiscovered planet beyond it, and, if possible,
thence to determine approximately the elements of its orbit, etc., which would probably lead to its discovery." Accordingly, in 1843, I attempted a first solution of the problem, assuming the
orbit to be a circle, with a radius equal to twice the mean distance of Uranus from the sun.
□ Meanwhile the Royal Academy of Sciences of Gottingen had proposed the theory of Uranus as the subject of their mathematical prize, and although the little time which I could spare from
important duties in my college prevented me from attempting the complete examination of the theory which a competition for the prize would have required, yet this fact, together with the
possession of such a valuable series of observations, induced me to undertake a new solution of the problem.
□ After obtaining several solutions differing little from each other, by gradually taking into account more and more terms of the series expressing the perturbations, I communicated to
Professor Challis, in September 1845, the final values which I had obtained for the mass, heliocentric longitude, and elements of the orbit of the assumed planet.
41. H S Ruse papers
□ W Feller writes: The authors study Riemannian Vn in connection with the fundamental solution of the corresponding equation Δ2u = 0, where Δ2 stands for the second differential parameter.
□ General solutions of Laplace's equation in a simply harmonic manifold (1963).
□ T J Willmore writes: Explicit formulae are obtained for general solutions of Laplace's equation in a real n-cell equipped with a simply harmonic riemannian metric.
42. R A Fisher: 'History of Statistics
□ We do know that the reason for his hesitation to publish was his dissatisfaction with the postulate required for the celebrated "Bayes' Theorem." While we must reject this postulate, we
should also recognise Bayes' greatness in perceiving the problem to be solved, in making an ingenious attempt at its solution, and finally in realising more clearly than many subsequent
writers the underlying weakness of his attempt.
□ Once the true nature of the problem was indicated, a large number of sampling problems were within reach of mathematical solution.
□ "Student" himself gave in this and a subsequent paper the correct solutions for three such problems - the distribution of the estimate of the variance, that of the mean divided by its
estimated standard deviation, and that of the estimated correlation coefficient between independent variates.
43. G A Miller - A letter to the editor
□ For instance, recent discoveries relating to the finding of at least one root by the ancient Babylonians of certain numerical quadratic and cubic equations throws new light on the history of
algebra and on the contributions made by the Greeks and the Arabians towards the solution of algebraic equations.
□ It is to be emphasized that the complete solution of general quadratic and of general cubic equations could not be attained until our ordinary complex numbers began to be understood at about
the beginning of the nineteenth century, although complete formal solutions were used earlier in Europe.
44. John Couch Adams' account of the discovery of Neptune
□ I find among my papers the following memorandum, dated July 3, 1841: "Formed a design, in the beginning of this week, of investigating, as soon as possible after taking my degree, the
irregularities in the motion of Uranus, which are yet unaccounted for, in order to find whether they may be attributed to the action of an undiscovered planet beyond it, and, if possible,
thence to determine approximately the elements of its orbit, etc., which would probably lead to its discovery." Accordingly, in 1843, I attempted a first solution of the problem, assuming the
orbit to be a circle, with a radius equal to twice the mean distance of Uranus from the sun.
□ Meanwhile the Royal Academy of Sciences of Gottingen had proposed the theory of Uranus as the subject of their mathematical prize, and although the little time which I could spare from
important duties in my college prevented me from attempting the complete examination of the theory which a competition for the prize would have required, yet this fact, together with the
possession of such a valuable series of observations, induced me to undertake a new solution of the problem.
□ After obtaining several solutions differing little from each other, by gradually taking into account more and more terms of the series expressing the perturbations, I communicated to
Professor Challis, in September 1845, the final values which I had obtained for the mass, heliocentric longitude, and elements of the orbit of the assumed planet.
45. G H Hardy addresses the British Association in 1922
□ It is impossible, for me to give you, in the time at my command, any general account of the problems of the theory of numbers, or of the progress that has been made towards their solution
even during the last twenty years.
□ But there is no similar solution for our actual problem, nor, I need hardly say, for the analogous problems for fourth, fifth, or higher powers.
□ There is no case, except the simple case of squares, in which the solution is in any sense complete.
□ The first step towards a solution was made by Dirichlet, who proved for the first time, in 1837, that any such arithmetical progression contains an infinity of primes.
46. James Clerk Maxwell on the nature of Saturn's rings
□ There are some questions in Astronomy, to which we are attracted rather on account of their peculiarity, as the possible illustration of some unknown principle, than from any direct advantage
which their solution would afford to mankind.
□ There is a very general and very important problem in Dynamics, the solution of which would contain all the results of this Essay and a great deal more.
□ It is this: "Having found a particular solution of the equations of motion of any material system, to determine whether a slight disturbance of the motion indicated by the solution would
cause a small periodic variation, or a total derangement of the motion." .
47. Pólya on Fejér
□ Yet he could perceive the significance, the beauty, and the promise of a rather concrete not too large problem, foresee the possibility of a solution and work at it with intensity.
□ And, when he had found the solution, he kept on working at it with loving care, till each detail became fully transparent.
□ It is due to such care spent on the elaboration of the solution that Fejer's papers are very clearly written, and easy to read and most of his proofs appear very clear and simple.
□ Yet only the very naive may think that it is easy to write a paper that is easy to read, or that it is a simple thing to point out a significant problem that is capable of a simple solution.
48. Horace Lamb addresses the British Association in 1904
□ The practical character of the mathematical work of Stokes and his followers is shown especially in the constant effort to reduce the solution of a physical problem to a quantitative form.
□ It is now generally accepted that an analytical solution of a physical question, however elegant it may be made to appear by means of a judicious notation, is not complete so long as the
results are given merely in terms of functions defined by infinite series or definite integrals, and cannot be exhibited in a numerical or graphical form.
□ Moreover, the interest of the subject, whether mathematical or physical, is not yet exhausted; many important problems in Optics and Acoustics, for example, still await solution.
□ It would appear that there is an opening here for the mathematician; at all events, the numerical or graphical solution of any one of the numerous problems that could be suggested would be of
the highest interest.
49. James Clerk Maxwell on the nature of Saturn's rings
□ There are some questions in Astronomy, to which we are attracted rather on account of their peculiarity, as the possible illustration of some unknown principle, than from any direct advantage
which their solution would afford to mankind.
□ There is a very general and very important problem in Dynamics, the solution of which would contain all the results of this Essay and a great deal more.
□ It is this: "Having found a particular solution of the equations of motion of any material system, to determine whether a slight disturbance of the motion indicated by the solution would
cause a small periodic variation, or a total derangement of the motion." .
50. Max Planck: 'The Nature of Light
□ I shall doubtless mention much that is familiar to each of you, but I shall also deal with newer problems still awaiting solution.
□ When and how the last step will be made, the linking up of mechanics and electro-dynamics, cannot be said, and though many clever physicists are at present occupied with this question, the
time does not yet seem ripe for the solution.
□ It would have been possible to seek a solution by supposing that the theory would have been better had it abstained, in general, from making special hypotheses, based on immediate
observations, and to limit oneself to the pure facts, i.e.
□ It is not possible today to predict with certainty when any definite solution to this problem will be obtained.
51. Warga abstract
□ The first one is in well known problems of classical optimal control in which one approaches optimal solutions with rapidly oscillating control functions.
□ A limit solution is then obtained by imbedding the controls in the class of relaxed controls which are functions with values that are probability measures.
52. Von Neumann Silliman lectures
□ Then, during the late thirties, he became interested in questions of theoretical hydrodynamics, particularly in the great difficulties encountered in obtaining solutions to partial
differential equations by known analytical methods.
□ This was the period during which he became completely convinced, and tried to convince others in many varied fields, that numerical calculations done on fast electronic computing devices
would substantially facilitate the solution of many difficult, unsolved, scientific problems.
53. Edinburgh's tribute to A C Aitken
□ He found the solution of a sixth order difference equation which arose in Whittaker's Theory of Graduation, a problem which suited his special gifts in manipulative algebra and numerical
□ They contribute to a practical approach to mathematics which has enabled him the more readily to see the solutions of mathematical problems as well as to suggest many new practical methods to
his colleagues.
54. Konrad Knopp: Texts
□ Solution hints are provided, plus complete solutions to all problems.
55. Edward Sang on his tables
□ The quinquesection of these parts was effected by help of the method of the solution of equations of all orders, published by me in 1829; and the computation of the multiples of those parts
was effected by the use of the usual formula for second differences.
□ Kepler's celebrated problem has ever since his time exercised mathematicians, and, sharing the ambition of many others, I also sought often, and in vain, for an easy solution of it.
□ Accident brought it again before me, and this time, considering not the relations of the lines connected with it, but the relations of the areas concerned, an exceedingly simple solution was
56. Pappus on the trisection of an angle
□ Now it is considered a serious type of error for geometers to seek a solution to a plane problem by conics or linear curves and, in general, to seek a solution by a curve of the wrong type.
□ But later with the help of the conics they trisected the angle using the following 'vergings' for the solution.
57. Mathematical and Physical Journal for Secondary Schools
□ Each issue of the Kozepiskolai Matematikai Lapok contained a number of selected exercises from mathematics and shortly thereafter from physics, as well as solutions to past months' problems
and a list of those pupils who sent in correct solutions.
□ It regularly prints the best solutions of the students and the names of the best problems solvers.
58. What do mathematicians do?
□ In the 1920's, for example, the discovery of quantum mechanics went a very long way toward reducing chemistry to the solution of well-defined mathematical problems.
□ Much more difficult to establish is the beautiful result that solutions exist for the prime 2 and for precisely those odd primes which leave a remainder of 1 when divided by 4.
59. Ince obituary.html
□ The war years - a physical disability precluded active service - he spent in Edinburgh and Trinity College, Cambridge, as a research student; in various forms of national service (including a
period at University College, London, where he had contacts with that enthusiast for singular solutions of differential equations, M J M Hill); and as a temporary lecturer at the University
of Leeds.
□ The work on Mathieu's equation included a proof of the impossibility of the equation having more than one periodic solution, and culminated in a series of papers published in the Proceedings
of the Royal Society of Edinburgh in which he developed methods for the computation of the functions and carried these methods to fruition by the actual formation of the tables themselves.
60. André Weil: 'Algebraic Geometry
□ To take only one instance, a personal one, this book has arisen from the necessity of giving a firm basis to Severi's theory of correspondences on algebraic curves, especially in the case of
characteristic p ≠ 0 (in which there is no transcendental method to guarantee the correctness of the results obtained by algebraic means), this being required for the solution of a long
outstanding problem, the proof of the Riemann hypothesis in function-fields.
□ Our results include all that is required for a rigorous treatment of so-called "enumerative geometry", thus providing a complete solution of Hilbert's fifteenth problem.
□ IX; it contains such general comments as could not appropriately be made before, formulates problems, some of them of considerable importance, and, in some cases, makes tentative suggestions
about what seems to be at present the best approach to their solution; it is hoped that these may be helpful to the reader, to whom the author, having acted as his pilot until this point,
heartily wishes Godspeed on his sailing away from the axiomatic shore, further and further into open sea.
61. A A Albert: 'Structure of Algebras
□ The theory of linear associative algebras probably reached its zenith when the solution was found for the problem of determining all rational division algebras.
□ Since that time it has been my hope that I might develop a reasonably self-contained exposition of that solution as well as of the theory of algebras upon which it depends and which contains
the major portion of my own discoveries.
□ The results are also applied in the determination of the structure of the multiplication algebras of all generalized Riemann matrices, a result which is seen in Chapter XI to imply a complete
solution of the principal problem on Riemann matrices.
62. Heath: Everyman's Library 'Euclid' Introduction
□ 28 and 29: "These two problems, to the first of which the 27th proposition is necessary, are the most general and useful of all in the Elements, and are most frequently made use of by the
ancient geometers in the solution of other problems; and, therefore, are very ignorantly left out by Tacquet and Dechales in their editions of the Elements, who pretend that they are scarce
of any use." The important words here are those referring to the ancient geometers.
□ The propositions embody, in fact, the general method known as the "application of areas," which was of vital consequence to the Greek geometers, being the geometrical equivalent of the
solution of the general quadratic equations ax ∓ bx2/c = S so far as they have real roots.
□ The simplest case of "application of areas," which is equivalent to the solution for x of the simple equation ax = S, can be read in this volume (Eucl.
63. R A Fisher: 'Statistical Methods' Introduction
□ The solutions of problems of distribution (which may be regarded as purely deductive problems in the theory of probability) not only enable us to make critical tests of the significance of
statistical results, and of the adequacy of the hypothetical distributions upon which our methods of numerical inference are based, but afford real guidance in the choice of appropriate
statistics for purposes of estimation.
□ In view of the mathematical difficulty of some of the problems which arise it is also useful to know that approximations to the maximum likelihood solution are also in most cases efficient
64. Berge books
□ Of the several appendices, one lists fourteen questions still awaiting solution.
□ 2-person 0-sum games are introduced and solved by linear programming and by successive approximations (without mentioning George W Brown, Iterative solution of games by fictitious play,
Activity Analysis of Production and Allocation, T C Koopmans [Tjalling Charles Koopmans (1910-1985)], ed., John Wiley, New York, 1951, 374-376; Julia Robinson, An iterative method for solving
a game, Ann.
□ Their contents are respectively: separation theorems for convex sets, the Farkas-Minkowski [named after Gyula Farkas and Hermann Minkowski] and von Neumann minimax theorems, with divers
extensions and corollaries; the various forms of the minimization problem (with equivalence theorems); algorithms of "simplex" type for the solution of convex and quadratic programming
65. EMS 1914 Colloquium
□ (Professor of Mathematics in the University of Edinburgh), on THE SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS IN THE MATHEMATICAL LABORATORY.
□ This method bids fair to displace in practical application the older method of graphical solution equations, as it can very readily be applied to equations in any number of variables.
□ Professor Whittaker delivered the first of his two lectures on the solution of equations.
66. G H Hardy addresses the British Association in 1922, Part 1
□ It is impossible, for me to give you, in the time at my command, any general account of the problems of the theory of numbers, or of the progress that has been made towards their solution
even during the last twenty years.
□ But there is no similar solution for our actual problem, nor, I need hardly say, for the analogous problems for fourth, fifth, or higher powers.
□ There is no case, except the simple case of squares, in which the solution is in any sense complete.
67. Gibson History 7 - Robert Simson
□ In the higher geometry he prelected from his own Conics, and he gave a small specimen of the linear problems of the ancients by explaining the properties sometimes of the conchoid, sometimes
of the cissoid, with their application to the solution of such problems.
□ Till he produced his article in the Philosophical Transactions in 1723 there had been no elucidation of the mystery that had baffled every inquirer, and even then there was only an approach
to a solution, not the solution itself.
68. Phillip Griffiths Looks at 'Two Cultures' Today
□ Here at home, Congressman George Brown of California, a physicist and a former chair of the House Committee on Space, Science, and Technology, has been strongly influenced by Havel and now
sees little evidence that "objective scientific knowledge leads to subjective benefits for humanity." Congressman Brown has written that he wants to "make strong attempts to involve ordinary
citizens in processes of discussion and decision-making, including citizens who have not previously demonstrated expertise about such matters at all." To some, this recalls the tone of
Germany's "volkische" solution, and even of Mao Tse Tung's cultural revolution.
□ Such scientists, he said, seldom see that many problems of application and engineering are as intellectually exacting as so-called "pure" problems, and that many of the solutions are as
satisfying and beautiful.
69. Miller graduation address
□ There can certainly be no interest more fundamental or of greater concern to the human family than the solution of the problem - how men may dwell together in peace and prosperity, under a
stable social, civil, and political policy.
□ The Hebrews and Greeks have solved man's relation to the eternal mystery - the one in its religious, the other in its philosophical aspect; each has come as near the perfect solution as,
perhaps, it is possible for the human mind to reach.
□ Where this adjustment is complicated by diverse physical peculiarities and by different inherited or acquired characteristics, the problem becomes one of the greatest intricacy that has ever
taxed human wisdom and patience for solution.
70. Durell and Robson: 'Advanced Trigonometry
□ A Key is published, for the convenience of teachers, in which solutions are given in considerable detail, and in some cases alternative methods of solution are supplied, so that to some
extent the Key forms a supplementary teaching manual.
71. Mathematicians and Music 2.2
□ Among those of the sixteenth century achieving a reputation in mathematics and medicine none was better known than he, whose greatest mathematical work, Ars Magna (1545), contains the first
solution of the general cubic equation in print.
□ In them he suggests another solution of the problem of how suitably to arrive at a tempered scale.
72. R L Wilder: 'Cultural Basis of Mathematics I
□ I don't mean that it can solve these problems, but that it can point the way to solutions as well as show the kinds of solutions that may be expected.
73. Kline's books
□ This monograph was inspired by unpublished lecture notes of the late Rudolf Luneberg on the foundations of geometrical optics, based on solutions of the electromagnetic wave equation.
□ In his "proper direction for reform" he does not offer novel solutions.
74. Malcev: 'Foundations of Linear Algebra' Introduction
□ For example, the fundamental idea behind the solution of a system of linear equations in several unknowns is that of replacing such a system by a chain of these simple equations.
□ The search in the 18th century for the general solution of n linear equations in n unknowns led Leibniz and Cramer to the notion of the determinant.
75. John Walsh's delusions
□ Solution of Equations of the higher orders, 1845.
□ In his diary there is an entry: "Discovered the general solution of numerical equations of the fifth degree at 114 Evergreen Street, at the Cross of Evergreen, Cork, at nine o'clock in the
forenoon of 7 July 1844; exactly 22 years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science.] .
76. Edmund Whittaker: 'Physics and Philosophy
□ If the transitions from mathematics to moral values are not firmly established, Whittaker's attempt does not succeed in remedying the defects of Descartes' solution.
□ At the end of the lecture Whittaker just mentions the problem of how God's foreknowledge can be reconciled with man's freewill, and perhaps it may not be out of place to give briefly the
scholastics' solution of the problem.
77. James Jeans addresses the British Association in 1934, Part 2
□ It may seem strange, and almost too good to be true, that nature should in the last resort consist of something we can really understand; but there is always the simple solution available
that the external world is essentially of the same nature as mental ideas.
□ It is only a step from this to a solution of the problem which would have commended itself to many philosophers, from Plato to Berkeley, and is, I think, directly in line with the new
world-picture of modern physics.
78. Science at St Andrews
□ Despite its pugnacity the booklet is a tour de force, with a bewildering variety of novel ideas concealed in a verbiage of medieval geometry, yet containing a forthright solution of a problem
on the rhumb line that had mystified men for a century.
□ This period of discovery culminated for Gregory in the central expansion theorems of interpolation and the differential calculus, the former of which he announced in a letter to Collins,
November 1670, and the latter of which he exemplified in the following February by half a dozen examples and again a year later by the solution of Kepler's problem - on determining the
theoretical position in its orbit of a planet at a given time - which Gregory solved by invoking the properties of the cycloid and repeated differentiation.
79. EMS 1913 Colloquium
□ It was then shown that the electrodynamical equations were transformed in equations of the same type, and examples were given to show how from the solution of a problem in electrostatics the
solution of a problem dealing with moving distributions of electricity could be obtained.
80. Tietze: 'Famous Problems of Mathematics
□ In the discussion of the disputes over the priority of discovery of the solution of third-degree equations, a reference to the difference between objective and brachial methods of conflict
had a certain timeliness in view of the battles-royal that were then frequently "organized" at political meetings.
□ The lectures regularly contain a few biographical data on individual mathematicians who have a special relationship with the problem under discussion, or who discovered its solution.
81. The Works of Sir John Leslie
□ This is a sort of inverted form of solution.
□ He remarks quaintly: "The superior elegance and perspicuity with which the geometrical process unfolds the properties of these higher curves, may show that the fluxionary calculus should be
more sparingly employed, if not reserved for the solution of problems of a more arduous nature." After that it comes quite as a shock to meet mere differential equations masquerading in such
elegant geometrical company, but these are seen to be rank outsiders, members of the nouveaux riches.
82. Gregory-Collins correspondence
□ In December 1670 Collins began discussing the solution of algebraic equations with Gregory.
□ I have now abundantly satisfied myself in these things I was searching after in the analytics, which are all about reduction and solution of equations.
83. Muir on research in Scotland
□ Is it too Quixotic to suppose that had all mutual recriminations been laid aside, and a National Conference on the Higher Education been held several years ago, - a Conference including
professors, secondary schoolmasters, and the many influential laymen who, from their connection with school boards and other public bodies, take an interest in the subject, -a basis of action
could have been harmoniously arrived at, which would ere this have made a solution an immediate possibility? The evil is more clamant than ever now, when the far more difficult problem of
providing a national system of elementary education has received so satisfactory a solution; and may we not reasonably expect that the intellects and wills which solved the one are capable of
solving the other? .
84. Charles Bossut on Leibniz and Newton
□ He contents himself with saying that he has deduced them from the solution of a general problem which he expresses enigmatically by transposing the letters and the sense of which, as
explained after the business was known, is 'an equation containing flowing quantities being given, to find fluxions and inversely.' What light could Leibniz derive from such an anagram? All
we can conclude from such a letter is that at the time it was written Newton was in possession of the method of fluxions; by which, however, is to be understood simply the method of tangents
and quadratures; for the method of resolving differential equations was then out of the question, this not being invented till long after as has been said above.
□ Here then we have the clear and positive solution of the problem, the possession of which Newton so carefully endeavoured to reserve to himself.
85. EMS 1913 Colloquium 3.html.html
□ It was then shown that the electrodynamical equations were transformed in equations of the same type, and examples were given to show how from the solution of a problem in electrostatics the
solution of a problem dealing with moving distributions of electricity could be obtained.
86. George Chrystal's First Promoter's Address
□ It would be merest affection to say that I have not learned much from all this experience; but it is quite within the truth to say that there are many practical questions of high importance,
on the solution of which my experience throws no light whatever.
□ How one of them is to live by lecturing, say on definite integrals, is a mystery to the solution of which the advocates of pure and unadulterated extra-muralism have not addressed themselves.
87. Pappus on analysis and synthesis in geometry
□ Analysis, then, takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in
analysis we admit that which is sought as if it were already done and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by
so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards.
□ There are in all thirty-three Books, the contents of which up to the Conics of Apollonius I have set out for your consideration, including not only the number of the propositions, the
diorismi [a statement in advance as to when, how, and in how many ways the problem will be capable of solution] and the cases dealt with in each Book, but also the lemmas which are required;
indeed I have not, to the best of my belief, omitted any question arising in the study of the Books in question.
88. Goursat: 'Cours d'analyse mathématique
□ - Solutions periodiques et asymptotiques.
□ Solutions discontinues.
89. Max Planck: 'Quantum Theory
□ For me, such an object has, for a long time, been the solution of the problem of the distribution of energy in the normal spectrum of radiant heat.
□ Until this goal is attained the problem of the quantum of action will not cease to stimulate research and to yield results, and the greater the difficulties opposed to its solution, the
greater will be its significance for the extension and deepening of all our knowledge of physics.
90. NAS Award in Applied Mathematics and Numerical Analysis
□ for his innovative and imaginative use of mathematics in the solution of a wide variety of challenging and significant scientific and engineering problems.
□ for his profound and penetrating solution of outstanding problems of statistical mechanics.
91. Muir on research in Scotland
□ Is it too Quixotic to suppose that had all mutual recriminations been laid aside, and a National Conference on the Higher Education been held several years ago, - a Conference including
professors, secondary schoolmasters, and the many influential laymen who, from their connection with school boards and other public bodies, take an interest in the subject, -a basis of action
could have been harmoniously arrived at, which would ere this have made a solution an immediate possibility? The evil is more clamant than ever now, when the far more difficult problem of
providing a national system of elementary education has received so satisfactory a solution; and may we not reasonably expect that the intellects and wills which solved the one are capable of
solving the other? .
92. Kuratowski: 'Introduction to Topology
□ As a consequence, topology is often suitable for the solution of problems to which analysis cannot give the answer.
□ theorems on the existence of a solution of certain types of differential equations can be expressed as theorems on the existence of invariant points of a function space (the space of
continuous functions) under continuous transformations; these theorems can be proved by topological methods in a more general form and in a simpler way than was formerly done without the aid
of topology.
93. W H Young addresses ICM 1928
□ The solution of many even of the problems of today lies far beyond the Mathematics of our time, or of any definite epoch.
□ Verily in the clear statement of the problem lies more than half its solution.
94. Gowers laureation
□ Nevertheless, in 2009 he ventured to the other extreme, by posting a question on his blog 'Is massively collaborative mathematics possible?' He proposed a 'Polymath' project, where anyone who
wished could contribute their ideas on a blog page aiming to reach the solution of a difficult research problem collectively.
□ The solution was written up and published in a top journal under the name of Polymath.
95. Gauss: 'Disquisitiones Arithmeticae
□ The Analysis which is called indeterminate or Diophantine and which discusses the manner of selecting from an infinite set of solutions for an indeterminate problem those that are integral or
at least rational (and especially with the added condition that they be positive) is not the discipline to which I refer but rather a special part of it, just as the art of reducing and
solving equations (Algebra) is a special part of universal Analysis.
□ My greatest hope is that it pleases those who have at heart the development of science and that it proposes solutions that they have been looking for or at least opens the way for new
96. H L F Helmholtz: 'Theory of Music' Introduction
□ The horizons of physics, philosophy, and art have of late been too widely separated, and, as a consequence, the language, the methods, and the aims of any one of these studies present a
certain amount of difficulty for the student of any other of them; and possibly this is the principal cause why the problem here undertaken has not been long ago more thoroughly considered
and advanced towards its solution.
□ Meanwhile musical aesthetics has made unmistakable advances in those points which depend for their solution rather on psychological feeling than on the action of the senses, by introducing
the conception of movement in the examination of musical works of art.
97. Born Inaugural
□ The justification for considering this special branch of science as a philosophical doctrine is not so much its immense object, the universe from the atom to the cosmic spheres, as the fact
that the study of this object in its totality is confronted at every step by logical and epistemological difficulties; and although the material of the physical sciences is only a restricted
section of knowledge, neglecting the phenomena of life and consciousness, the solution of these logical and epistemological problems is an urgent need of reason.
□ In spite of this difficulty, I shall try to outline the problem and its solution, called quantum mechanics.
98. H Weyl: 'Theory of groups and quantum mechanics'Preface to Second Edition
□ At present no solution of the problem seems in sight; I fear that the clouds hanging over this part of the subject will roll together to form a new crisis in quantum physics.
99. Gheorghe Mihoc's books
□ The authors give an exposition, at a not too advanced mathematical level, of the main types of mathematical programming problems and of the more effective methods for their solution.
100. Ferrar: 'Textbook of Convergence
□ The majority are reasonably straightforward; hints for their solution are occasionally given.
101. Ernest Hobson addresses the British Association in 1910
□ Ample opportunities for the full discussion of all the detailed problems, the solution of which forms a great and necessary part of the work of those who are advancing science in its various
branches, are afforded by the special Societies which make those branches their exclusive concern.
102. EMS 1914 Colloquium 3.html
□ Professor Whittaker delivered the first of his two lectures on the solution of equations.
103. Rédei: Algebra
□ Galois theory, quadratic reciprocity, cyclic fields, solvability, the general equation, solution of cubic and quartic equations, geometric constructions, the normal basis theorem.
104. Cajori: 'A history of mathematics' Introduction
□ The interest which pupils take in their studies may be greatly increased if the solution of problems and the cold logic of geometrical demonstrations are interspersed with historical remarks
and anecdotes.
105. EMS obituary
□ This he put right by bringing such forms into line with the signature test that is essential for real quadratic forms: but he generously gave the problem over to his pupil, the late G Richard
Trott, who published the solution in his dissertation for a doctorate.
106. EMS 1914 Colloquium 1.html.html
□ This method bids fair to displace in practical application the older method of graphical solution equations, as it can very readily be applied to equations in any number of variables.
107. EMS obituary
□ Other titles were "On the uniqueness of the solution of the linear differential equation of the second order" (1903) and "The condition for the reality of the roots of an n-ic" (1906).
108. Harold Jeffreys on Logic and Scientific Inference
□ We do not say that it is the solution of the present difficulty, but a priori knowledge exists, and we shall have occasion later to consider instances of it at length.
109. Bartlett's reviews
□ Now there are a number of textbooks at all levels about the analytical solution of stochastic models, i.e.
110. A I Khinchin: 'Statistical Mechanics' Introduction
□ This problem, originated by Boltzmann, apparently is far from its complete solution even at the present time.
111. EMS obituary
□ Tweedie's brilliant solutions of the examples first brought him into notice.
112. James Jeans addresses the British Association in 1934
□ It seemed likely that Heisenberg had unravelled the secret of the structure of matter, and yet his solution was so far removed from the concepts of ordinary life that another parable had to
be invented to make it comprehensible.
113. Max Planck and the quanta of energy
□ For many years, such an aim for me was to find the solution to the problem of the distribution of energy in the normal spectrum of radiating heat.
114. Franklin's textbooks
□ It is designed to show them how to operate with complex quantities and how to solve problems for their solution on the use of Fourier series and integrals, and Laplace transforms.
115. Mitchell Feigenbaum: the interviewer
□ Briefly, he discovered a universal quantitative solution characterized by specific measurable constants that describes the crossover from simple to chaotic behaviors in many complex systems.
116. Gábor Szegó's student years
□ The names of the others who were in the same business were quickly known to me, and frequently I read with considerable envy how they had succeeded to solve some problems which I could not
handle with complete success, or how they had found a better solution (simpler, more elegant, or wittier) than the one I had sent in.
117. Oskar Bolza: 'Calculus of Variations
□ Weierstrass's discovery of the fourth necessary condition and his sufficiency proof for a so-called "strong" extremum, which gave for the first time a complete solution, at least for the
simplest type of problems, by means of an entirely new method based upon what is now known as Weierstrass's construction." .
118. Gibson History 10 - Matthew Stewart, John Stewart, William Trail
□ He undoubtedly obtained many important successes in this way; his solution of Kepler's problem being one of the most remarkable.
119. Archimedes: 'Quadrature of the parabola
□ But I am not aware that any one of my predecessors has attempted to square the segment bounded by a straight line and a section of a right-angled cone [a parabola], of which problem I have
now discovered the solution.
120. Mathematics in Aberdeen
□ Geometry, Algebra, Trigonometry (Plane and Spherical), Conic Sections, Theory of Equations, Analytical Geometry of Two and Three Dimensions, and Differential and Integral Calculus, including
the Solution of Differential Equations.
121. EMS obituary
□ Among longer papers may be mentioned "On the number and nature of the solutions of the Apollonian contact problem" in Vol.
122. Van der Waerden (print-only)
□ These are solutions of the Burnside problem.
123. Woodward (print-only)
□ This problem was one requiring for its solution mathematical work of the highest order and, in addition, the experience of the engineer, so to shape his formulas that they could be applied
directly by the computer.
124. Van der Waerden biography
□ These are solutions of the Burnside problem.
125. Woodward biography
□ This problem was one requiring for its solution mathematical work of the highest order and, in addition, the experience of the engineer, so to shape his formulas that they could be applied
directly by the computer.
126. Finlay Freundlich's Inaugural Address, Part 2
□ Here again, astronomy holds the key position to a final solution.
127. Turnbull lectures on Colin Maclaurin, Part 2
□ in such a manner as may suggest a synthetic demonstration that may serve to verify the solution.' .
128. Menger on the Calculus of Variations
□ So Newton's solution had no great effect on the development of mathematics.
129. Perelman's Fields Medal
□ In three preprints posted on the arXiv in 2002-2003 [The entropy formula for the Ricci flow and its geometric applications; Ricci flow with surgery on three-manifolds; Finite extinction time
for the solutions to the Ricci flow on certain three-manifolds], Perelman presented proofs of the Poincare conjecture and the geometrization conjecture.
130. Chrystal: 'Algebra' Preface
□ I suppose that the student has gone in this way the length of, say, the solution of problems by means of simple or perhaps even quadratic equations, and that he is more or less familiar with
the construction of literal formulae, such, for example, as that for the amount of a sum of money during a given term at simple interest.
131. Ernest Hobson addresses the British Association in 1910, Part 3
□ Only a firm grasp of the principles will give the necessary freedom in handling the methods of Mathematics required for the various practical problems in the solution of which they are
132. Bronowski and retrodigitisation
□ The topic was capped the following year by the then Editor of the Gazette, R L Goodstein (1912--1985), who noted [11] that Littlewood's approach gave a more succinct solution than Primrose's
133. W Burnside: 'Theory of Groups of Finite Order
□ The student of the theory of groups will find here a rich storehouse of material, and the investigator will find numerous suggestions in regard to problems which await solution and methods of
attacking them.
134. Peter Lax's student years
□ The problems can be very difficult, they can look impossible to solve, but if a solution is found, it is unbelievably worthwhile.
135. Paul Halmos: the Moore method
□ Often a student who hadn't yet found the proof of Theorem 11 would leave the room while someone else was presenting the proof of it - each student wanted to be able to give Moore his private
solution, found without any help.
136. G H Hardy addresses the British Association in 1922, Part 2
□ The first step towards a solution was made by Dirichlet, who proved for the first time, in 1837, that any such arithmetical progression contains an infinity of primes.
137. K Ollerenshaw: 'The Girls' School
□ The book as a whole, though it leaves no doubt as to the author's own convictions, is a fair-minded presentation of the problem confronting the girls' schools at the present time and of the
lines which their solution might be sought.
138. P G Tait's obituary of Listing
□ The solution of all such questions depends at once on the enumeration of the points of the complex figure at which an odd number of lines meet.
139. Anna Carlotta Leffler on Sonya Kovalevskaya
□ He did not believe her, but asked her to sit down beside him, after which he began to examine her solutions one by one.
140. Sheppard Papers
□ (Solutions in terms of differences and sums.)" Ibid., pp.
141. Somerville's Booklist
□ Mary submitted solutions to some of these and started a mathematical correspondence with him.
142. Vajda citation
□ That book was a key reference work since it not only introduced the families of optimization problems and the algorithms for their solution, but also set out the scope and limitations of
mathematical programming as normative models for managerial decisions.
143. Eddington: 'Mathematical Theory of Relativity' Introduction
□ We adopt what seems to be the commonsense solution of the difficulty.
144. Gaschutz' path
□ This is an example that shows how minor variations of the initial conditions can influence the solutions of an equation considerably.
145. Collins by Wood
□ On the 13th of October 1667 he was elected fellow of the Royal Society upon the publication in the 'Philosophical Transactions' of his 'Solution of a Problem concerning Time, that is, about
the Julian Period, with several different Perpetual Almanacks in single Verses; a Chronological Problem', and other things afterwards in the said 'Transactions' concerning 'Merchants
Accounts, Compound Interest', and 'Annuities', etc.
146. EMS obituary
□ The generation of British mathematicians to which Steggall belonged delighted in proposing and working out problems whose solution might require the aid of any branch of pure or applied
147. Archimedes' 'Quadrature of the parabola
□ But I am not aware that any one of my predecessors has attempted to square the segment bounded by a straight line and a section of a right-angled cone [a parabola], of which problem I have
now discovered the solution.
148. Marie-Louise Dubreil-Jacotin
□ Thanks to new methods, she was able to find the exact solution to certain problems in rotational movement or the study of waves in homogenous and heterogeneous liquids.
149. Pál Erdös's student years
□ They come from number theory, graph theory, geometry, set theory, and they range in difficulty from ingenious high school competition problems to the most difficult research problems- that
defy, and will continue to defy for many years to come, all attempts at solution; their common feature is that they are all fascinatingly interesting.
150. A D Aleksandrov's view of Mathematics
□ The same theory is useful, for example, in the solution of problems concerning the oozing of water under a dam, problems whose importance is obvious during the present period of construction
of huge hydroelectric stations.
151. Mathematicians and Music 2.1
□ Another of Paul Tannery's suggestions involves finding solutions of a Diophantine equation in three variables.
152. De Thou on François Viète
□ Adrian Romanus proposed a problem to all the mathematicians of Europe and Viete, who was the first to solve it, sent his solution to Romanus with corrections and a proof, together with
Apollonius Gallus.
153. University of Edinburgh Examinations
□ The solutions, when not integral, to be carried to two places of decimals.
154. Airy's work in engineering
□ The latter set includes a subset, those who admire Airy's sensible arrival at a solution to a problem which puzzled him for, perhaps, as long as a milli-second or two.
155. de Montessus publications
□ Solution du probleme fondamental de la statistique, Annales Societe sc.
156. Heinrich Tietze on Numbers
□ The solution to this problem is clearly reflected in the nomenclature of numbers.
157. Percy MacMahon addresses the British Association in 1901
□ The custom of offering prizes for the solutions of definite problems which are necessary to the general advance obtains more in Germany and in France than here, where, I believe, the Adams
Prize stands alone.
158. W H Young addresses ICM 1928 Part 2
□ The assumption of the wave- length as the characteristic of a monochromatic light, does not seem to have availed much towards the solution of the problem.
159. Isaac Todhunter: 'Euclid' Introduction
□ The construction then usually follows, which states the necessary straight lines and circles which must be drawn in order to constitute the solution of the problem, or to furnish assistance
in the demonstration of the theorem.
160. R A Fisher: the life of a scientist' Preface
□ It had the vitality of his immense pleasure in the process of thinking, the play of ideas, the solution of puzzles.
161. Sommerville obituary.html
□ The written solutions and comments went far beyond what was necessary for mere elucidation.
162. Planck's quanta.html
□ For many years, such an aim for me was to find the solution to the problem of the distribution of energy in the normal spectrum of radiating heat.
163. EMS session 5
□ Thereafter Mr J S Mackay read a paper on the solutions of Euclid's problems with one fixed aperture of the compasses by the Italian geometers of the 16th century; and communicated a note from
Mr R Tucker giving some novel properties connected with the triangle.
164. Mathematics at Aberdeen 4
□ When it was first awarded in 1795, for solutions of questions in geometry, the dies were made larger than intended, but, according to William Knight (a later Professor of Natural Philosophy),
this was 'better, as there is less temptation in future time to give away a large than a small medal'.
165. O Veblen's Opening Address to ICM 1950
□ The solution will not be to give up international mathematical meetings and organizations altogether, for there is a deep human instinct that brings them about.
166. Napier Tercentenary 4.html.html
□ He hoped by such comparisons to lead to a satisfactory solution of tile problem how best to arrange tabular matter, on what colour of paper, and with what kind of type.
167. James Jeans addresses the British Association in 1934, Part 3
□ It is only a step from this to a solution of the problem which would have commended itself to many philosophers, from Plato to Berkeley, and is, I think, directly in line with the new
world-picture of modern physics.
168. Schrödinger: 'Statistical Thermodynamics
□ This is the mathematical problem - always the same; we shall soon present its general solution, from which in the case of every particular kind of system every particular classification that
may be desirable can be found as a special case.
169. Coolidge: 'Origin of Polar Coordinates
□ Pascal used the same transformation to calculate the length of a parabolic are, a problem previously solved by Roberval, but his solution was not universally accepted as valid.
170. G H Hardy: 'Integration of functions
□ My object has been to do what I can to show that this impression is mistaken, by showing that the solution of any elementary problem of integration may be sought in a perfectly definite and
systematic way.
171. Archimedes on mechanical and geometric methods
□ [All these propositions have already been] proved [the solution to the problem of the centre of gravity of a cone must be in a work by Archimedes which has not survived].
172. EMS 1914 Colloquium 0.html.html
□ (Professor of Mathematics in the University of Edinburgh), on THE SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS IN THE MATHEMATICAL LABORATORY.
173. EMS obituary
□ In 1 Baker showed that the imaginary part of the same function was also, expressible as a multiple integral, as therefore was the complete function, and, instigated by another paper of
Poincare's published in 1902, gave in 2, using his result of 1, a simpler method of obtaining Poincare's solution of Weierstrass' problem.
174. G C McVittie papers
□ Cahill and McVittie (the authors) write: A solution of Einstein's field equations for the motion of a spherically symmetric distribution of perfect fluid is investigated in an isotropic
comoving coordinate system.
175. Veblen's Opening Address to ICM 1950
□ The solution will not be to give up international mathematical meetings and organizations altogether, for there is a deep human instinct that brings them about.
176. A N Whitehead addresses the British Association in 1916
□ The solution I am asking for is not a phrase however brilliant, but a solid branch of science, constructed with slow patience, showing in detail how the correspondence is effected.
177. Collins and Gregory discuss Tschirnhaus
□ I received lately two of your letters, whereby I perceive you have fallen in acquaintance with a very learned gentleman [Walter von Tschirnhaus] and a great admirer of Descartes, whom I also
admire so much that he or any other shall help him as to his solution of biquadratic and cubic equations.
178. EMS obituary
□ Among the best known of these is the theorem of analytical dynamics called after him, namely that, to any set of m invariant relations of a Hamiltonian system, which are in involution, there
corresponds a family of 8m particular solutions of the Hamiltonian system, whose determination depends upon the integration of a system of order (m - 1).
179. Prefaces Landau Lifshitz.html
□ We have not included in this book the various theories of ordinary liquids and of strong solutions, which to us appear neither convincing nor useful.
180. Laplace: 'Méchanique Céleste
□ The solution of this problem depends, at the same time, upon the accuracy of the observations, and upon the perfection of the analysis.
181. Dubreil-Jacotin on Maria Gaetana Agnesi
□ She left behind, under the name of Instituzioni analitiche, a "remarkable account of ordinary algebra, with the solution of several solved and unsolved geometric problems"; a second volume,
entirely devoted to infinitesimal analysis, a science then quite new, was declared "the most complete and the best done in this field" by the commissioners of the Academy of Sciences of
Paris, who were assigned to examine this work at their meeting of 6 December 1749.
182. Proclus and the history of geometry to Euclid
□ Leon also first discussed the diorismi (distinctions), that is the determination of the conditions under which the problem posed is capable of solution, and the conditions under which it is
183. Collected Papers of Paul Ehrenfest' Preface
□ Lorentz did not like to speak about a problem before he had arrived at a solution, and he reproved - always in an extremely polite and mild way - those who made remarks without due
184. Andrew Forsyth addresses the British Association in 1905, Part 2
□ This accumulation of facts is only one process in the solution of the universe: when the compelling genius is not at hand to transform knowledge into wisdom, useful work can still be done
upon them by the construction of organised accounts which shall give a systematic exposition of the results, and shall place them as far as may be in relative significance.
185. Levi-Civita.html
□ Among the best known of these is the theorem of analytical dynamics called after him, namely that, to any set of m invariant relations of a Hamiltonian system, which are in involution, there
corresponds a family of 8m particular solutions of the Hamiltonian system, whose determination depends upon the integration of a system of order (m - 1).
186. The Edinburgh Mathematical Society: the first hundred years
□ Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussion of new problems or new solutions, and
comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.
187. Wolfgang Pauli and the Exclusion Principle
□ It is therefore not surprising that I could not find a satisfactory solution of the problem at that time.
188. Henry Baker addresses the British Association in 1913
□ And, alas! to deal only with one of the earliest problems of the subject, though the finally sufficient conditions for a minimum of a simple integral seemed settled long ago, and could be
applied, for example, to Newton's celebrated problem of the solid of least resistance, it has since been shown to be a general fact that such a problem cannot have any definite solution at
189. Steggal obituary.html
□ The generation of British mathematicians to which Steggall belonged delighted in proposing and working out problems whose solution might require the aid of any branch of pure or applied
190. Charles Babbage and deciphering codes
□ I offered to give a few hours to the subject; and if I could see my way to a solution, to continue my researches; but if not on the road to success, to tell him I had given up the task.
191. A N Whitehead: 'Autobiographical Notes
□ The only point on which I feel certain is that there is no widespread, simple solution.
192. William Lowell Putnam Mathematical Competition
□ A full list of the questions from the Putnam competition (with solutions!) is available HERE .
193. Proclus and the history of geometry as far as Euclid
□ Leon also first discussed the diorismi (distinctions), that is the determination of the conditions under which the problem posed is capable of solution, and the conditions under which it is
194. Gibson History 4 - John Napier
□ The Descriptio defines a logarithm, lays down the rules for working with logarithms, illustrating their use particularly by applying them to the solution of triangles, and contains a Table of
the logarithms.
195. Library of Mathematics
□ Solutions of Laplace's equationD R Bland .
196. Thomas Bromwich: 'Infinite Series
□ The arrangement of the first seven chapters, as well as of Chapter IX, has undergone very little alteration: to the eighth chapter a discussion of the solution of linear differential
equations of the second order has been added.
197. J L Synge and Hamilton
□ For that reason it is not necessary to defend the application of the method to problems which would admit shorter special solutions.
198. Heinrich Tietze on Numbers, Part 2
□ The extension of the number system was required not only for geometric measurements, but also for the solution of algebraic equations.
199. J A Schouten's Opening Address to ICM 1954
□ Also the so-called "applied mathematics" came to new life and asked for more men well trained in mathematics and physics, because modern computing machines had made it possible to make use of
solutions that formerly only had theoretical value on account of the impossibility of doing the computing work in a reasonable time.
200. Cochran: 'Sampling Techniques' Introduction
□ On the other hand, if information is wanted for many subdivisions or segments of the population, it may be found that a complete enumeration offers the best solution.
201. Conforto Gottingen.html
□ Certainly the topic is not easy, but it is very likely that combining the ideas we have, for instance the theory of functionals, with the notions that are in common use here, we could find an
easy solution.
202. Napier Tercentenary
□ He hoped by such comparisons to lead to a satisfactory solution of tile problem how best to arrange tabular matter, on what colour of paper, and with what kind of type.
203. Letters from Galileo' Preface
□ Only the third question, 'Why did it happen', still divides scholars and others to such an extent that no one can pretend that even an approximate solution has been reached.
204. Whittaker RSE Prize
□ An early and brilliant example was his general solution of Laplace's equation, which might be considered the fundamental partial differential equation of the older physics.
205. The Edinburgh Mathematical Society: the first hundred years (1883-1983) Part 2
□ His experience in teaching mathematics in the University convinced him that 'algebra, as we teach it, is neither an art nor a science, but an ill-digested farrago of rules whose object is the
solution of examination problems.' This led him to write his monumental treatise of nearly 1200 pages on Algebra, which appeared in two parts in 1886 and 1889, and had a powerful effect on
the teaching of algebra in Great Britain and abroad.
206. M Bôcher: 'Integral equations
□ The theory of integral equations may be regarded as dating back at least as far as the discovery by Fourier of the theorem concerning integrals which bears his name; for, though this was not
the point of view of Fourier, this theorem may be regarded as a statement of the solution of a certain integral equation of the first kind.
207. Fermat's Journal des Sçavans obituary
□ He gave an introduction to loci, plane and three-dimensional, which is an analytical treatise on the solution of problems in two and three dimensions.
208. EMS summer 1937.html
□ "What is sauce for the goose," he said, "is another man's poison." He proceeded to show how a calculating machine could be used for the rapid numerical solution of complicated problems of
interpolation and of algebraic and differential equations.
209. Coulson: 'Electricity
□ We therefore break off, in Chapters IX and X, to discuss in detail a selected number of such problems, and to illustrate the technique required in their solution.
210. de Montessus publications
□ Solution du probleme fondamental de la statistique, Annales Societe sc.
211. Studies presented to Richard von Mises' Introduction
□ "If this goes on", writes von Mises, "the predictions of those who believe that the next step toward the solution of the basic sociological problems must come from physical annihilation of
one of the two groups of people will be borne out".
212. Huygens: 'Traité de la lumière
□ Since, however, the opinions offered, although ingenious, are not such that more intelligent people would need no further explanations of a more satisfying nature, I wish here to present my
thoughts on the subject so that, to the best of my ability, I might contribute to a solution of that part of science which, not without reason, is considered to be one of the most difficult.
213. Edinburgh Mathematics Examinations
□ The solutions, when not integral, to be carried to two places of decimals.
214. George Gibson: 'Calculus
□ them; and with the object of encouraging the student to put himself through the drill that is absolutely necessary for the acquisition of facility and confidence in applying the Calculus, I
have freely given hints towards the solution of the more important examples.
215. Three Sadleirian Professors
□ Among Professor Forsyth's own researches Cajori's History of Mathematics specially mentions Differential Invariants and Reciprocants, and Singular Solutions.
216. Footnote 12
□ He would become seriously upset with the attitude of indifference that my modest temperament made to assume when I told him the solution to a problem or a proof of a theorem that I was able
to find.
1. Quotations by Polya
□ "In order to solve this differential equation you look at it till a solution occurs to you." .
□ Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else.
□ We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
□ A GREAT discovery solves a great problem, but there is a grain of discovery in the solution of any problem.
2. Quotations by Gauss
□ There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our
destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.
3. Quotations by Bernoulli Johann
□ [After reading an anonymous solution to a problem that he realized was Newton's solution.] .
4. Quotations by Abel
□ The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it.
□ Opening of Memoir on algebraic equations, proving the impossibility of a solution of the general equation of the fifth degree (1824) .
5. Quotations by Poincare
□ What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all
that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.
□ When one tries to depict the figure formed by these two curves and their infinity of intersections, each of which corresponds to a doubly asymptotic solution, these intersections form a kind
of net, web or infinitely tight mesh .
6. A quotation by Peano
□ Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution.
7. Quotations by Klein
□ The answer to such arguments, however, is that the mathematician, even when he is himself operating with numbers and formulas, is by no means an inferior counterpart of the errorless machine,
"thoughtless thinker" of Thomas; but rather, he sets for himself his problems with definite, interesting, and valuable ends in view, and he carries them to solution in appropriate and
original manner.
8. Quotations by Heath
□ One feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the deliberation with which Archimedes
approaches the solution of any one of his main problems.
9. A quotation by Pacioli
□ And therefore when in your equations you find terms with different intervals without proportion, you shall say that the art, until now, has not given the solutions to this case ..
10. A quotation by Dudeney
□ A good puzzle should demand the exercise of our best wit and ingenuity, and although a knowledge of mathematics and of logic are often of great service in the solution of these things, yet it
sometimes happens that a kind of natural cunning and sagacity is of considerable value.
11. Quotations by De Morgan
□ The imaginary expression √(-a) and the negative expression -b, have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity.
12. Quotations by Dirac
□ I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.
13. Quotations by Eddington
□ The solution goes on famously; but just as we have got rid of all the other unknowns, behold! V disappears as well, and we are left with the indisputable but irritating conclusion: .
14. Quotations by Fuller
□ But when I have finished, if the solution is not beautiful, I know that it is wrong.
15. Quotations by Weyl
□ The question of the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in what direction it will find its final solution or even whether a final
objective answer can be expected at all.
16. Quotations by Descartes
□ The second, to divide each problem I examined into as many parts as was feasible, and as was requisite for its better solution.
17. Quotations by Alfven
□ The technologists claim that if everything works [in a nuclear fission reactor] according to their blueprints, fission energy will be a safe and very attractive solution to the energy needs
of the world.
1. Mathematical Chronology
□ Diophantus of Alexandria writes Arithmetica, a study of number theory problems in which only rational numbers are allowed as solutions.
□ Al-Khayyami (usually known as Omar Khayyam) writes Treatise on Demonstration of Problems of Algebra which contains a complete classification of cubic equations with geometric solutions found
by means of intersecting conic sections.
□ It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones and is one of the best textbooks in the whole of medieval literature.
□ Later known as Fermat's last theorem, it states that the equation xn + yn = zn has no non-zero solutions for x, y and z when n > 2.
□ which gives solutions to some of Fermat's number theory challenges.
□ He considers integer solutions of ax - by = 1 where a, b are integers.
□ The book discusses singular solutions to differential equations, a change of variables formula, and a way of relating the derivative of a function to the derivative of the inverse function.
□ He gives solutions for certain special cases to the equation which was first studied by Jacob Bernoulli.
□ The formula connecting surface and volume integrals, now known as "Green's theorem", appears for the first time in the work, as does the "Green's function" which would be extensively used in
the solution of partial differential equations.
□ Galois submits his first work on the algebraic solution of equations to the Academie des Sciences in Paris.
□ Liouville publishes Galois' papers on the solution of algebraic equations in Liouville's Journal.
□ Peano proves that if f(x, y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution.
□ Leray shows the existence of weak solutions to the Navier-Stokes equations.
□ This is a major contribution to the solution of the Goldbach conjecture.
□ Douglas gives a complete solution to the Plateau problem, proving the existence of a surface of minimal area bounded by a contour.
□ Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.
□ Quidong Wang finds infinite series solutions to the n-body problem (with minor exceptions).
□ A large prize is offered by banker Andrew Beal for a solution to the Beal Conjecture: the equation xp + yq = zr has no solutions for p, q, r > 2 and coprime integers x, y, z.
2. Chronology for 1990 to 2000
□ Quidong Wang finds infinite series solutions to the n-body problem (with minor exceptions).
□ A large prize is offered by banker Andrew Beal for a solution to the Beal Conjecture: the equation xp + yq = zr has no solutions for p, q, r > 2 and coprime integers x, y, z.
□ A prize of seven million dollars is put up for the solution of seven famous mathematical problems.
3. Chronology for 1930 to 1940
□ Leray shows the existence of weak solutions to the Navier-Stokes equations.
□ This is a major contribution to the solution of the Goldbach conjecture.
□ Douglas gives a complete solution to the Plateau problem, proving the existence of a surface of minimal area bounded by a contour.
4. Chronology for 1820 to 1830
□ The formula connecting surface and volume integrals, now known as "Green's theorem", appears for the first time in the work, as does the "Green's function" which would be extensively used in
the solution of partial differential equations.
□ Galois submits his first work on the algebraic solution of equations to the Academie des Sciences in Paris.
5. Chronology for 1970 to 1980
□ Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.
6. Chronology for 1720 to 1740
□ He gives solutions for certain special cases to the equation which was first studied by Jacob Bernoulli.
7. Chronology for 1880 to 1890
□ Peano proves that if f(x, y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution.
8. Chronology for 1840 to 1850
□ Liouville publishes Galois' papers on the solution of algebraic equations in Liouville's Journal.
9. Chronology for 1650 to 1675
□ which gives solutions to some of Fermat's number theory challenges.
10. Chronology for 900 to 1100
□ Al-Khayyami (usually known as Omar Khayyam) writes Treatise on Demonstration of Problems of Algebra which contains a complete classification of cubic equations with geometric solutions found
by means of intersecting conic sections.
11. Chronology for 1700 to 1720
□ The book discusses singular solutions to differential equations, a change of variables formula, and a way of relating the derivative of a function to the derivative of the inverse function.
12. Chronology for 1675 to 1700
□ He considers integer solutions of ax - by = 1 where a, b are integers.
13. Chronology for 1960 to 1970
□ Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.
14. Chronology for 1AD to 500
□ Diophantus of Alexandria writes Arithmetica, a study of number theory problems in which only rational numbers are allowed as solutions.
15. Chronology for 1300 to 1500
□ It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones and is one of the best textbooks in the whole of medieval literature.
16. Chronology for 1625 to 1650
□ Later known as Fermat's last theorem, it states that the equation xn + yn = zn has no non-zero solutions for x, y and z when n > 2.
This search was performed by Kevin Hughes' SWISH and Ben Soares' HistorySearch Perl script
Another Search Search suggestions
Main Index Biographies Index
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OP-SF WEB
Extract from OP-SF NET
Topic #10 ------------- OP-SF NET 5.2 ------------ March 15, 1998
From: Charles Dunkl
Subject: Review of book on Combinatorics
(This item appeared in our Activity Group's Newsletter, vol 8, no 2,
February 1998, pp. 15-16)
Introduction to Combinatorics
By Martin J. Erickson
Wiley, Chichester-New York-Brisbane-Toronto-Singapore, xii + 195 pp.,
ISBN 0-471-15408-3
This is a book of eleven chapters introducing itself as a textbook. This
review is written mainly from the point of view of evaluating it as such.
This reviewer admits to preferring textbooks to be systematic and
comprehensive. Some chapters of the present work tend to be sketchy rather
than systematic. We begin with a description of the contents.
Chapter 1 is a collection of needed information about sets, group theory,
linear algebra, and algebraic number theory. Sometimes the text can not
quite decide between just stating facts with reference to well-known
textbooks and actually giving definitions and some proofs. Some
definitions should have been avoided: what does it gain to define a finite
set as one having finitely many elements without defining "finitely" (or
discussing bijections involving the basic sets {1,2,3,...,n})? In the
rather important paragraph introducing the symmetric group the example of
the cycle decomposition has the element "3" appearing in two cycles (one
assumes that the second occurrence should have been "5", but this makes it
difficult for a student to understand the concept). The standard binomial
coefficient identities, like the Vandermonde sum, appear in this chapter,
but the proofs tend to be too swift for a beginner and uninteresting for
an expert.
Chapter 2 deals with the pigeonhole principle, has lots of detail and
presents interesting examples, like good approximation to irrational
numbers by rational ones. There is a brief introduction to graph theory
(but it refers to planar graphs without giving a definition, except later
in one of the problems).
Sequences and partial orders are taken up in Chapter 3. The author
displays more enthusiasm for the material here, which includes the
Erdos-Szekeres and Sperner theorems. The proofs are complete but place
some demands on the reader, like making a sketch or supplying more detail.
This style holds throughout the book. Chapter 4 continues with the theme
of existence theorems and takes up the topic of Ramsey numbers, graph
colorings, probabilistic methods and van der Waerden's theorem on
arithmetic progressions. These three chapters form the "existence" part.
The part on "enumeration" comprises Chapters 5 through 8. Counting
problems are introduced mostly using function concepts like injections and
surjections. The standard topics of Stirling numbers, Bell numbers,
derangements and their generating functions are covered here. The
unpleasant notation [x]^n is used for the Pochhammer symbol (x)_n. In six
pages, Chapter 7 discusses tableaux, hook-length formulas, and the
Robinson-Schensted correspondence. This seems rather sketchy and may not
be useful. On the other hand, Chapter 8 is a very good presentation of
Polya's theory of counting, with examples from graphs and cycle indices of
The third part of the book on "construction" comprises Chapters 9, 10 and
11, and presents linear codes, Hamming and Golay codes, t-designs, Latin
squares, Hadamard matrices and the Leech lattice.
There is a bibliography of some sixty monographs and a six-page index.
Also there are statements of open problems (most of them have been open
for some time!) and selections of problems from past Putnam Competition
As a textbook this work might be appropriate for an undergraduate seminar
for upper level mathematics students or as a supplement in a graduate
course. The coverage of Ramsey theory, Polya's counting methods, codes and
designs may be the strongest part of the book. Generally the proofs demand
careful perusal by the reader and have more words than formulas;
nevertheless the words are used very concisely! Every chapter has a
section of problems, which tend to be difficult. Somehow the reader is
left with the feeling of having been on a hasty tour of combinatorics
without a satisfactory amount of detail in the reader's favorite part of
the subject (whatever that might be). Indeed, similarly styled books have
been designated as "essays", indicating a collection of personal insights
and commentaries. This may be the better way of appreciating the present
Charles Dunkl
Back to Home Page of
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R plot function using ggplot2. Earthquake map ... | {"url":"http://www.r-bloggers.com/search/ggplot/page/112/","timestamp":"2014-04-17T19:01:24Z","content_type":null,"content_length":"38706","record_id":"<urn:uuid:ef7fc349-3c06-4695-bbb2-68a615fe20cc>","cc-path":"CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00462-ip-10-147-4-33.ec2.internal.warc.gz"} |
[FOM] Comparing the power of logics
Arnon Avron aa at tau.ac.il
Sat Oct 22 13:58:27 EDT 2005
There is one claim that has explicitly or implicitly been made several times
during the discussion concerning intuitionistic logic: that
intuitionistic logic is actually stronger than classical logic
(or that it is an extension of classical logic). It was put forward
in very clear terms by Andrej Bauer (in a private message):
> Perhaps we should do precisely what you suggest and have two sets of
> connectives: "i-or", "i-not", ... and "c-or", "c-not", ... We could even
> put them all together in one big happy logic. Except it would turn out
> that this logic is intuitionistic and that the c-connectives are
> definable in terms of the i-connectives
> This should
> be taken as evidence that intuitionistic logic is _more general_ than
> classical logic. Mathematicians like generality. Clearly, we need to
> educate them on this point.
Two questions concerning this claim.
First, I am not acquainted with any definition of c-not (say) in terms of the
i-connectives (like the way c-or is definable in classical logic in terms
of c-not and c-and). Does anybody know one which preserves the
*consequence relation* of classical logic? Is there even a definable unary
connective @ in intuitionistic logic such that for every A, B we have
that A and @@A intuitionistically follow from each other,
and B intuitionistically follows from the set {A, at A}?
(What I do know are some "translations" of classical logic into
intuitionistic logic, but they either only preserve theoremhood (not
the consequence relation!), or employ transformations of atomic formulas
to something different (like their double negations). None of these
translations provides a definition of c-not in terms of the i-connectives).
Second, let us examine the thesis about the greater generality of
intuitionistic logic in the light of an example of a logic
which is *really* more general than classical logic according to the
definability criterion. Assume that we use a language with four connectives:
the usual disjunction, conjunction and implication connectives,
together with a unary connective ~ ("negation"). Denote
by CL~ the logic which is obtained by deleting (=>~) from the standard
Gentzen-type system for classical logic, where ~ is taken as the "negation"
connective (Note that the positive fragment of CL~ is identical
to that of classical logic, but the "horror" called excluded middle
is not derivable in it. Isn't it great?). Now it was observed by
J. Y. Beziau that classical negation is definable in CL~ by:
-A = A->~A
Indeed, it is not difficult to prove that if we define a function * by
p* = p, (A->B)* = A*->B*, (A\/B)* = A*\/B*, (A&B)* = A*&B*, (\not A)* = -(A*)
we get a precise translation of CL into CL~, such that \gamma=>\delta
is derivable in CL iff \gamma*=>\delta* is derivable in CL~. On the
other hand it is easy to prove that no translation of this sort in
the other direction exits. Hence CL~ is strictly more general than CL!
But what is the *meaning* of ~, one may ask. Well, there is a thesis
according to which the meaning of a connective is completely determined by
the logical laws that govern it (I might discuss this thesis some other time).
But the main goal of this message is
to raise a difficult question about comparison of logics which
should be interesting also to people who think that the meaning
should determine the logical laws, not the other way around. Well, CL~ does
have a very simple semantics of the general type I have called
"non-deterministic matrices" ("Nmatrices", in short). In particular it
is strongly sound and complete w.r.t to the 2-valued Nmatrix in which
the truth-values are t and f, t is designated (and f is not), the
truth-tables of the positive connectives are the usual ones, while the
truth-table for ~ is given by ~t=f, ~f={t,f}. This means that any
valuation v which respects the truth-tables for the positive connectives,
and for which v(~A)=f in case v(A)=t, is legal (in terms of circuits,
one may think of ~ as a gate which outputs 0 if the input is 1, but
whose output is unpredictable and not uniform if the input is 0).
It is very easy to check that the truth-tale one obtains for the
definable connective - is indeed the classical truth-table for negation
(and it is hardly surprising that one cannot define a proper
nondeterministic two-valued connectives using the strictly deterministic
classical ones). So CL~ is strictly more general than classical logic
not only from the proof-theoretical point of view, but also
from the semantic point of view.
So should we start educating mathematicians
to use the more general CL~ rather than the dull CL? I am not sure, especially
that the job would not end at this point. Thus by moving to 3-valued
Namtrices, and then to 4-valued Nmatrices etc, we get logics which are provably
more and more general (in particular: more general then CL~), and we
shall need to educate mathematicians to learn and use
them instead of their poor Classical logic! Well, the sad truth is that
although` I have found in a series of papers many applications of Nmatrices
(in proof theory, uncertainty reasoning, and other areas), for the
time being it seems to me that mathematicians can go on doing
usual mathematics without them (and without intuitionistic logic too...).
Even in mathematics not always more generality is better.
But this discussion raises one difficult question, to which I admit
not to have a satisfactory answer: to what extent the possibility
of translating one Logic L1 into another L2 makes L2 stronger than L1?
It somehow seems paradoxical to say that by deleting a rule from
classical logic we get a stronger system! Yet we all think that the
classical logic of negation and disjunction (say) is stronger
than the pure implicational fragment of classical logic, because
classical implication is definable in terms of classical
negation and disjunction. What is the difference, if any?
Arnon Avron
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Is there a tool for finding probability distributions given some samples?
up vote 0 down vote favorite
I'm looking for a tool that does "probability distribution fitting" given a set of data points. Sort of like curve fitting, but tries to fit to standard density distributions.
For example if I input
(0, 0.0497871), (1, 0.149361), (2, 0.224042), (3, 0.224042), (4,0.168031), (5, 0.100819), (6, 0.0504094)
I would hope that it would tell me these data points fit a Poisson distribution.
1 This needs clarification: is your input supposed to be independent samples from the distribution, or points sampled from a graph of the density, or something else? Basically, what do you mean by
"sample"? – Darsh Ranjan Oct 21 '09 at 2:13
The sample data is the probability distribution function of a Poisson variable with mean 3. So it seems that the OP wants to give a pdf and find out what distribution it is. – Michael Lugo Oct 21
'09 at 2:22
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5 Answers
active oldest votes
It appears to me that you want to perform a goodness of fit test. What this test allows you to do is compare your sample data to the poisson distribution with a certain parameter via a
up vote 2 statistical hypothesis test. Check out the link for more information wikipedia.
down vote
This person might want something more, like a test that tells you whether a Poisson distribution, or Gaussian, or binomial, or something else is better. Given some explicit class of
1 cases to check (and a guarantee that the chosen distribution actually is one of them) this sort of statistical tool will probably work. But the general problem is probably too
open-ended to have a mathematical solution. – Kenny Easwaran Oct 21 '09 at 5:45
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Google suggests there are tools.
up vote 1 down vote
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You need to have some candidate distribution before you do a goodness of fit test.
up vote 1 down If you suspect your data follow a Poisson distribution, I'd start by computing the sample mean and variance. If these are equal, maybe you do have Poisson data. If your sample variance
vote is appreciably larger than your sample mean, you might try negative binomial next.
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The following document might be of some use:
Depending on whether you use R, it might be of a variable level of usefulness. The general approach still stands though.
up vote 0
down vote As a previous commenter has said, this isn't really a well-posed problem and does not have some closed mathematical solution. However, you can harness some computational tools to aide you
in this. A good starting point is always plotting the data to see what they look like, you can then hypothesize a distribution you think could fit and then finally test that hypothesis via
a goodness of fit test (as suggested by another commenter.)
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there is a software called @risk ( www.palisade.com) which can fit all the possible distributions and then order the distributions based on goodness of fit
up vote 0 down vote Hope this helps
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