Q-function

The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by Template:HarvsPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digitsPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. The deadline for the contest was May 20, 2002. In the end, 20 teams solved all of the problems perfectly within the required precision, and an anonymous donor aided in producing the required prize moniesPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. The challenge and its solutions were described in detail in the book Template:Harvs.

The problems

From Template:Harv:

1. $\underset{\epsilon \to 0}{\lim }\underset{\epsilon }{\overset{1}{\int }}{x}^{-1}\cos (x^{-1}\log x)dx$
2. A photon moving at speed 1 in the xy-plane starts at t = 0 at (x, y) = (0.5, 0.1) heading due east. Around every integer lattice point (i, j) in the plane, a circular mirror of radius 1/3 has been erected. How far from the origin is the photon at t = 10?
3. The infinite matrix A with entries $a_{11}=1,a_{12}=1/2,a_{21}=1/3,a_{13}=1/4,a_{22}=1/5,a_{31}=1/6,\dots$ is a bounded operator on ${\ell }^2$. What is $\left|\right|A\left|\right|$?
4. What is the global minimum of the function $\exp (\sin \left(50x\right))+\sin \left(60e^y\right)+\sin (70\sin x)+\sin (\sin \left(80y\right))-\sin \left(10(x+y)\right)+1/4(x^2+y^2)$
5. Let $f(z)=1/\mathrm{\Gamma }(z)$, where $\mathrm{\Gamma }(z)$ is the gamma function, and let $p(z)$ be the cubic polynomial that best approximates $f(z)$ on the unit disk in the supremum norm $\left|\right|.|{|}_{\mathrm{\infty }}$. What is $\left|\right|f-p|{|}_{\mathrm{\infty }}$?
6. A flea starts at $(0,0)$ on the infinite 2D integer lattice and executes a biased random walk: At each step it hops north or south with probability $1/4$, east with probability $1/4+\epsilon$, and west with probability $1/4-\epsilon$. The probability that the flea returns to (0, 0) sometime during its wanderings is $1/2$. What is $\epsilon$?
7. Let A be the 20000×20000 matrix whose entries are zero everywhere except for the primes 2, 3, 5, 7, ..., 224737 along the main diagonal and the number 1 in all the positions $a_{ij}$ with $|i-j|=1,2,4,8,\dots ,16384$. What is the (1, 1) entry of $A^{-1}$?
8. A square plate $[-1,1]\times [-1,1]$ is at temperature $u=0$. At time $t=0$, the temperature is increased to $u=5$ along one of the four sides while being held at $u=0$ along the other three sides, and heat then flows into the plate according to $u_t=\mathrm{\Delta }u$. When does the temperature reach $u=1$ at the center of the plate?
9. The integral $I(\alpha )=\underset{0}{\overset{2}{\int }}[2+\sin \left(10\alpha \right)]x^{\alpha }\sin (\alpha /(2-x))dx$ depends on the parameter α. What is the value of α in [0, 5] at which I(α) achieves its maximum?
10. A particle at the center of a 10×1 rectangle undergoes Brownian motion (i.e., 2D random walk with infinitesimal step lengths) till it hits the boundary. What is the probability that it hits at one of the ends rather than at one of the sides?

Solutions

1. 0.3233674316
2. 0.9952629194
3. 1.274224152
4. −3.306868647
5. 0.2143352345
6. 0.06191395447
7. 0.7250783462
8. 0.4240113870
9. 0.7859336743
10. 3.837587979 × 10⁻⁷

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References

• Template:Cite web
• Template:Cite web
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  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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