sudarshan wrote:Amber G. wrote:
The standard term used (in text books etc) for calculus is "kalan" (कलन ) (Some times, additional words (Rashi-kalan or chal rashi karan - राशी कलन, or चल राशी कलन ) are added.
Thanks, Amber, much appreciate the detailed response.
Regards,
Sudarshan
skumar wrote:Amber G. wrote:<snip>
For the Yugoslav problem, the answer is 2, (That is rabbit escapes if it can run 2x (or more) times faster). As gakkakad mentioned, the answer is by no means trivial.
<snip>
AmberG,
From the diagram above, r1 is the initial position of the rabbit, f1 is the initial position of the fox. The base of the triangle represents the X axis.....
The fox adopts the following strategy -...
<snip>
What remains is to trivially determine this minimum speed at which the x coords remain equal. If the rabbit keeps going to the left towards e, the fox will catch up with it if it travel at 1/sqrt(3) times the speed of the rabbit. Motions in the opposite direction are similar i.e. the fox always has to travel at 1/sqrt(3) times the speed of the rabbit so that the x coords match while it comes closer along the y-axis until it finally catches up with it.
So, if the aim is for the fox to catch the rabbit, the rabbit does not require to travel @ less than 2x but only at less than sqrt(3)x. To put it another way, the fox is guaranteed to catch the rabbit if it is faster than 1/sqrt(3) times the rabbit. Indeed, it is trivially possible to calculate the angle at which the fox should travel for a given ratio of speeds.
.
I think this problem as given is simpler than the duck-fox lake problem but it might be tougher to prove the minimum ratio at which the duck can never be caught, especially for the range between sqrt(3) and 2
Gakakkad, my (older) sister went into medicine, and she still enjoys math. In those days to get MBBS/MS/MD one need not even know calculus but she kept her interest. (I wish I learned more bio-science from her *smile* ). My soon to become DIL is finishing residency in neurology but is extremely good in, and always enjoyed math/physics (majored, and has taken graduate level math course)...So please keep up with physics/maths .. we need good doctors with strong math physics background.
.. He now teaches basic biology at MIT ... In fact it was neuroscience that got him interested in Mathematics...There is a terrorist boat (max speed v) chased by a coast-guard boat (max speed 2v ) and suddenly the first boat enters in a dense fog and no one can see anything. The coast-guard captain knows the terrorist's thinking, it will change direction once and then will flee with max speed in a straight direction. Can coast-guard boat catch the terrorist's boat? How?
Amber G. wrote:<snip>
(This way the minimum speed (for fox) is "improved" from (1/(sqrt(3)) to 0.5. )
Added later: For clarity, see my next post for few more comments.
.Amber G. wrote:Let me end this with a less known but a beautiful theorem in Geometry.
Take any triangle, trisect each angle. The inner triangle thus formed (try to draw it and you will know what I mean) will always be a equilateral triangle. (If outer triangle is ABC, and inner PQR while angles
CBP = PBR = RBA , and angles BAR=RAQ=QAC and angles ACQ=QCP=PCB – triangle PQR will always be a equilateral triangle no matter what shape is ABC)
Neshant wrote:I'd like to raise my knowledge of applied math in general. i.e. learn techniques of problem solving primarily for engineering problems. I know the field of math is vast but what book can you recommend that starts at a basic level (and i emphasize basic) yet leaves me with a reasonably good knowledge of math by the end of it.
Due to time constraints, I'd prefer doing an online course watching a video or watching a DVD of some guy explaining the concepts. Anything like that out there that goes topic by topic?
Modus Indorum In the Liber Abaci, Fibonacci says the following introducing the so-called "Modus Indorum" or the method of the Indians, today known as Arabic numerals.
After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days.
There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business.
I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. (Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters.
Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this
knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.
The nine Indian figures are:
9 8 7 6 5 4 3 2 1
With these nine figures, and with the sign 0 ... any number may be written. (Sigler 2003 and Grimm 1973)
ArmenT wrote:Amber G. wrote:Let me end this with a less known but a beautiful theorem in Geometry.
Take any triangle, trisect each angle. The inner triangle thus formed (try to draw it and you will know what I mean) will always be a equilateral triangle. (If outer triangle is ABC, and inner PQR while angles
CBP = PBR = RBA , and angles BAR=RAQ=QAC and angles ACQ=QCP=PCB – triangle PQR will always be a equilateral triangle no matter what shape is ABC)
Don't you mean "bisect" instead of "trisect"? It would seem that I'd get a six-sided figure if I divided each angle into 3 parts? (I'll admit I haven't drawn an actual figure yet, so I may be wrong here). Bliss to clarify onlee?
amit_s wrote:Here is one just using law of sines and some trigonometry.
...
Now from 3 and 4
BP/BR = Sin(60+a)/Sin(60+c)
Hence Angle BRP = 60+c and angle BPR = 60+a in Triangle BPR.
...
Dan Mazer wrote:amit_s wrote:Here is one just using law of sines and some trigonometry.
...
Now from 3 and 4
BP/BR = Sin(60+a)/Sin(60+c)
Hence Angle BRP = 60+c and angle BPR = 60+a in Triangle BPR.
...
I didn't follow this step. Just because the ratio of the sides is expressible as the ratio of two sines, are they necessarily the angles of that triangle? Am I missing something obvious?
Dan Mazer wrote:amit_s wrote:Here is one just using law of sines and some trigonometry.
...
Now from 3 and 4
BP/BR = Sin(60+a)/Sin(60+c)
Hence Angle BRP = 60+c and angle BPR = 60+a in Triangle BPR.
...
I didn't follow this step. Just because the ratio of the sides is expressible as the ratio of two sines, are they necessarily the angles of that triangle? Am I missing something obvious?
....Due to time constraints, I'd prefer doing an online course watching a video or watching a DVD of some guy explaining the concepts. Anything like that out there that goes topic by topic?
C. K. Raju
Centre for Studies in Civilizations,
36, Tughlakabad Institutional Area, New Delhi 110 067
c k raju@vsnl.net
Racist history has been an instrument of inequity, and is still uncritically propagated in current Indian school texts in mathematics. As a first step towards equitable mathematics education, we need to do away with this. Critics have argued that the thrust for social justice in the mathematics classroom handicaps students. We argue to the contrary that the difficulties in teaching or learning mathematics arise because inequity and a brand of “theological correctness” are
already embedded into the history and philosophy of current formal (theorem-proving) mathematics. The philosophy of current formal mathematics derives from an analysis of the Elementsby Hilbert et al. That analysis proceeded from a historical narrative about Euclid and his method of proof. However, in the absence of any serious evidence for the historical “Euclid” this narrative must be rejected as a racist fantasy. The real philosophy of the Elements, and its religious significance for Greeks, is brought out by Proclus in his Commentary—virtually refuting point-by-point the inequitable post-Nicene (Augustinian) Christian theology with which he had to contend. This linkage of mathematics and religion persisted in Islamic rational theology (aql-¯ı-kal¯am) which too used the Elements to promote equity and justice. However, during the Crusades, history was Hellenized at Toledo. The Inquisition enforced theological correctness, and the Elements was reinterpreted to align it with the prevailing Christian theology. Current school texts use Hilbert’s synthetic reinterpretation, which substituted “equality” by “congruence”, and eliminated also the empirical, thus completing the process of making the Elements theologically correct. However, synthetic geometry (apart from being an invalid interpretation of the Elements) is harder to understand, and counter-intuitive, compared to metric or empirical or traditional geometry, and certainly does not add any practical value. This applies not only to geometry but to all formal mathematics: it is this “theologification” that has made mathematics difficult to learn or teach. The remedy is to “de-theologify” or secularize mathematics and teach it in the cultural and practical context in which it developed.
Amber G. wrote:His guru Paul Erdos is one of my hero.
shaardula wrote:might be trivial for folks here, but stumbled upon a coarse approximation to pythogoras theorem. c^2 = a^2 + b^2 ~ max(a,b) + min(a,b)/3. works better for slivers than it does for a=b, but still a good rough approximation.
matrimc wrote:Amber G. wrote:His guru Paul Erdos is one of my hero.
Amber G. ji
Hope you have read "The man who loved only numbers". If not, please do. It is a bio of Erdos.
Regards
)Amber G. wrote:shaardula wrote:might be trivial for folks here, but stumbled upon a coarse approximation to pythogoras theorem. c^2 = a^2 + b^2 ~ max(a,b) + min(a,b)/3. works better for slivers than it does for a=b, but still a good rough approximation.
(Assuming the above meant c ~ max(a,b) + min(a,b)/3)
Above is not that helpful or accurate unless b is zero (which is uninteresting) or the sides are (3,4,5) (or have the same ratio)
When a=b, the actual value is about 1.41a and approximation 1.33 a... besides sqrt(a^2+b^2) is simple enough...
except for the special case (of ratio 3:4:5 triangle), for thin slivers (that is if one side is relatively much smaller than the other).. just using, for example, max(a,b) alone will often give better approx than including the second( min(a,b)/3)) term.
(for example a=40, b=9 value of c (=41) is closer to 40 than 43)
umbar jee... mine is a simple geomtric proof. c >= max(a,b), simple projection tells you this. given that , the optimal fraction i have seen (through working matlab, dumb, i know) is slighlty >3. but/3 works for the road. from experience for on the road computation,this is good enough.
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