Jacobian variety: Difference between revisions

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Template:Pi box The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:

\sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \lim_{n \to + \infty} (\frac{1}{1^{2}} + \frac{1}{2^{2}} + \dots + \frac{1}{n^{2}}) .

The series is approximately equal to 1.644934 Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.. The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be π²/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof.

Euler's approach

Euler's original derivation of the value π²/6 essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series. Of course, Euler's original reasoning requires justification (100 years later, Weierstrass proved that Euler's representation of the sin-function as an infinite product is correct, see: Weierstrass factorization theorem), but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series. The agreement he observed gave him sufficient confidence to announce his result to the mathematical community.

To follow Euler's argument, recall the Taylor series expansion of the sine function

\sin (x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots .

Dividing through by x, we have

\frac{\sin (x)}{x} = 1 - \frac{x^{2}}{3!} + \frac{x^{4}}{5!} - \frac{x^{6}}{7!} + \dots .

Now, the roots (intersections with the x-axis) of sin(x)/x occur precisely at $x = n \cdot π$ where $n = \pm 1, \pm 2, \pm 3, \dots$ Let us assume we can express this infinite series as a (normalized) product of linear factors given by its roots, just as we do for finite polynomials:

\begin{aligned} \frac{\sin (x)}{x} & = (1 - \frac{x}{π}) (1 + \frac{x}{π}) (1 - \frac{x}{2 π}) (1 + \frac{x}{2 π}) (1 - \frac{x}{3 π}) (1 + \frac{x}{3 π}) \dots \\ = (1 - \frac{x^{2}}{π^{2}}) (1 - \frac{x^{2}}{4 π^{2}}) (1 - \frac{x^{2}}{9 π^{2}}) \dots . \end{aligned}

If we formally multiply out this product and collect all the x² terms (we are allowed to do so because of Newton's identities), we see that the x² coefficient of sin(x)/x is

- (\frac{1}{π^{2}} + \frac{1}{4 π^{2}} + \frac{1}{9 π^{2}} + \dots) = - \frac{1}{π^{2}} \sum_{n = 1}^{\infty} \frac{1}{n^{2}} .

But from the original infinite series expansion of sin(x)/x, the coefficient of x² is −1/(3!) = −1/6. These two coefficients must be equal; thus,

- \frac{1}{6} = - \frac{1}{π^{2}} \sum_{n = 1}^{\infty} \frac{1}{n^{2}} .

Multiplying through both sides of this equation by $- π^{2}$ gives the sum of the reciprocals of the positive square integers.

\sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \frac{π^{2}}{6} .

The Riemann zeta function

The Riemann zeta function $ζ (s)$ is one of the most important functions in mathematics, because of its relationship to the distribution of the prime numbers. The function is defined for any complex number s with real part > 1 by the following formula:

ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}} .

Taking s = 2, we see that $ζ (2)$ is equal to the sum of the reciprocals of the squares of the positive integers:

ζ (2) = \sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \dots = \frac{π^{2}}{6} \approx 1.644934 .

Convergence can be proven with the following inequality:

\sum_{n = 1}^{N} \frac{1}{n^{2}} < 1 + \sum_{n = 2}^{N} \frac{1}{n (n - 1)} = 1 + \sum_{n = 2}^{N} (\frac{1}{n - 1} - \frac{1}{n}) = 1 + 1 - \frac{1}{N} \overset{N \to \infty}{⟶} 2 .

This gives us the upper bound $ζ (2) < 2$ , and because the infinite sum has only positive terms, it must converge. It can be shown that $ζ (s)$ has a nice expression in terms of the Bernoulli numbers whenever s is a positive even integer. With $s = 2 n$ :

ζ (2 n) = \frac{(2 π)^{2 n} (- 1)^{n + 1} B_{2 n}}{2 \cdot (2 n)!}

A rigorous proof using Fourier series

Let $f (x) = x$ over the interval x ∈ (–Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.,Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.). The Fourier series for this function (worked out in that article) is

f (x) = 2 \sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{n} \sin (n x) .

Then, using Parseval's identity (with $f (x) = x$ ) we have that

\sum_{n = - \infty}^{\infty} | a_{n} |^{2} = \frac{1}{2 π} \int_{- π}^{π} x^{2} d x

,

where

a_{n} = \frac{1}{2 π} \int_{- π}^{π} x e^{- i n x} d x = \frac{n π \cos (n π) - \sin (n π)}{π n^{2}} i = \frac{\cos (n π)}{n} i - \frac{\sin (n π)}{π n^{2}} i = \frac{(- 1)^{n}}{n} i

for n ≠ 0, and a₀ = 0. Thus,

| a_{n} |^{2} = \frac{1}{n^{2}}

for n ≠ 0 and

\sum_{n = - \infty}^{\infty} | a_{n} |^{2} = 2 \sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \frac{1}{2 π} \int_{- π}^{π} x^{2} d x

Therefore,

\sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \frac{1}{4 π} \int_{- π}^{π} x^{2} d x = \frac{π^{2}}{6}

as required.

A rigorous elementary proof

This is by far the most elementary well-known proof; while most proofs use results from advanced mathematics, such as Fourier analysis, complex analysis, and multivariable calculus, the following does not even require single-variable calculus (although a single limit is taken at the end).

History of this proof

The proof goes back to Augustin Louis Cauchy (Cours d'Analyse, 1821, Note VIII). In 1954 this proof appeared in the book of Akiva and Isaak Yaglom "Nonelementary Problems in an Elementary Exposition". Later, in 1982, it appeared in the journal Eureka, attributed to John Scholes, but Scholes claims he learned the proof from Peter Swinnerton-Dyer, and in any case he maintains the proof was "common knowledge at Cambridge in the late 1960s".

The proof

The main idea behind the proof is to bound the partial sums

\sum_{k = 1}^{m} \frac{1}{k^{2}} = \frac{1}{1^{2}} + \frac{1}{2^{2}} + \dots + \frac{1}{m^{2}}

between two expressions, each of which will tend to Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.²/6 as m approaches infinity. The two expressions are derived from identities involving the cotangent and cosecant functions. These identities are in turn derived from de Moivre's formula, and we now turn to establishing these identities.

Let $x$ be a real number with $0 < x < \frac{π}{2}$ , and let n be a positive odd integer. Then from de Moivre's formula and the definition of the cotangent function, we have

\frac{\cos (n x) + i \sin (n x)}{(\sin x)^{n}} = \frac{(\cos x + i \sin x)^{n}}{(\sin x)^{n}} = {(\frac{\cos x + i \sin x}{\sin x})}^{n} = (\cot x + i)^{n} .

From the binomial theorem, we have

(\cot x + i)^{n} = (\binom{n}{0}) \cot^{n} x + (\binom{n}{1}) (\cot^{n - 1} x) i + \dots + (\binom{n}{n - 1}) (\cot x) i^{n - 1} + (\binom{n}{n}) i^{n}

= [(\binom{n}{0}) \cot^{n} x - (\binom{n}{2}) \cot^{n - 2} x \pm \dots] + i [(\binom{n}{1}) \cot^{n - 1} x - (\binom{n}{3}) \cot^{n - 3} x \pm \dots] .

Combining the two equations and equating imaginary parts gives the identity

\frac{\sin (n x)}{(\sin x)^{n}} = [(\binom{n}{1}) \cot^{n - 1} x - (\binom{n}{3}) \cot^{n - 3} x \pm \dots] .

We take this identity, fix a positive integer $m$ , set $n = 2 m + 1$ and consider $x_{r} = \frac{r π}{2 m + 1}$ for $r = 1, 2, \dots, m$ . Then $n x_{r}$ is a multiple of $π$ and therefore a zero of the sine function, and so

0 = (\binom{2 m + 1}{1}) \cot^{2 m} x_{r} - (\binom{2 m + 1}{3}) \cot^{2 m - 2} x_{r} \pm \dots + (- 1)^{m} (\binom{2 m + 1}{2 m + 1})

for every $r = 1, 2, \dots, m$ . The values $x_{1}, \dots, x_{m}$ are distinct numbers in the interval (0, Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park./2). Since the function $\cot^{2} x$ is one-to-one on this interval, the numbers $t_{r} = \cot^{2} x_{r}$ are distinct for r = 1, 2, ..., m. By the above equation, these m numbers are the roots of the mth degree polynomial

p (t) : = (\binom{2 m + 1}{1}) t^{m} - (\binom{2 m + 1}{3}) t^{m - 1} \pm \dots + (- 1)^{m} (\binom{2 m + 1}{2 m + 1}) .

By Viète's formulas we can calculate the sum of the roots directly by examining the first two coefficients of the polynomial, and this comparison shows that

\cot^{2} x_{1} + \cot^{2} x_{2} + \dots + \cot^{2} x_{m} = \frac{(\binom{2 m + 1}{3})}{(\binom{2 m + 1}{1})} = \frac{2 m (2 m - 1)}{6} .

Substituting the identity $\csc^{2} x = \cot^{2} x + 1$ , we have

\csc^{2} x_{1} + \csc^{2} x_{2} + \dots + \csc^{2} x_{m} = \frac{2 m (2 m - 1)}{6} + m = \frac{2 m (2 m + 2)}{6} .

Now consider the inequality $\cot^{2} x < \frac{1}{x^{2}} < \csc^{2} x$ . If we add up all these inequalities for each of the numbers $x_{r} = \frac{r π}{2 m + 1}$ , and if we use the two identities above, we get

\frac{2 m (2 m - 1)}{6} < {(\frac{2 m + 1}{π})}^{2} + {(\frac{2 m + 1}{2 π})}^{2} + \dots + {(\frac{2 m + 1}{m π})}^{2} < \frac{2 m (2 m + 2)}{6} .

Multiplying through by (Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park./(2m + 1))², this becomes

\frac{π^{2}}{6} (\frac{2 m}{2 m + 1}) (\frac{2 m - 1}{2 m + 1}) < \frac{1}{1^{2}} + \frac{1}{2^{2}} + \dots + \frac{1}{m^{2}} < \frac{π^{2}}{6} (\frac{2 m}{2 m + 1}) (\frac{2 m + 2}{2 m + 1}) .

As m approaches infinity, the left and right hand expressions each approach $\frac{π^{2}}{6}$ , so by the squeeze theorem,

ζ (2) = \sum_{k = 1}^{\infty} \frac{1}{k^{2}} = \lim_{m \to \infty} (\frac{1}{1^{2}} + \frac{1}{2^{2}} + \dots + \frac{1}{m^{2}}) = \frac{π^{2}}{6}

and this completes the proof.

Packing Squares with Side 1/n

The solution to the Basel problem is related to packing squares with side length of $\frac{1}{n}$ . The question that is asked is "What is the smallest rectangle that can contain the squares as n approaches infinity?"^[1] One bound to the answer is dependent on one rectangle side, $S_{1}$ , to be the sum of the largest two squares sides summed together,

S_{1} = \frac{1}{2} + \frac{1}{3} = \frac{5}{6}

.

The other rectangle side, $S_{2}$ , depends on the order of remaining squares. However, we know that total area, A, of the squares (less the square with n=1) is:

A = \sum_{n = 2}^{\infty} \frac{1}{n^{2}} = \frac{π^{2}}{6} - 1

.

Dividing the squares total area by the one known rectangle side leaves the ideal rectangle side length,

S_{2} = \frac{A}{S_{1}} = \frac{\frac{π^{2}}{6} - 1}{\frac{5}{6}} = \frac{π^{2} - 6}{5} = 0.77392088021 \dots

Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..

The current packing record holder is Marc Paulhus, who developed a packing algorithm.^[2] Because Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park. is a transcendental number, the record holder's rectangles area will always be greater than the ideal rectangle.

References

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

An infinite series of surprises by C. J. Sangwin
Is a upcoming condominium situated within the neighbourhood of Hillview The Singapore authorities has put in place a brand new subsidised pancreas transplant programme (Thinkstock picture). In the latest effort to spur funding in blighted areas of one of the city's poorest neighborhoods, Chicago plans to promote several hundred properties for the worth of a sweet bar.

When clients ask concerning the subsequent batch of items for launch, gross sales representatives might hint that subsequent phases will supply items with poorer view or lower high quality finishings, even if they are offered at greater prices. Tempted patrons are often unaware that the value of those goodies can easily be offset by a slight drop in the property's market worth. It is due to this fact more sensible to get an immediate discount off the listing worth. Ask the gross sales consultant the estimated value of that branded equipment or furnishing package, then request for a direct deduction from the unit's asking price in lieu of the developer's goodie. December 4, 2013 by iskandarinsider November four, 2013 by iskandarinsider October 23, 2013 by iskandarinsider

With elevated accessibility from the northern a part of Singapore via the two new MRT stations, Woodlands North MRT Station and Woodlands South MRT Station , along with the longer term North South Expressway NSE, that will link Woodlands, Sembawang and Yishun to the city, journey could be made easier. To qualify for the "Giant Lot Program," applicants should already personal property on the identical block as the lot they wish to buy; they have to also be present on property taxes, have no financial obligations to the town (like water bills or parking tickets) and should tell the city how they plan to make use of the property, DNAinfo Chicago experiences. New Condo Trilinq @ Clementi New Condominium La fiesta @ Sengkang MRT Expertise Guru Property's tantalizing new property launch! And the Web site

The Brilliant Hill Drive condominium is an exciting upcoming development in Higher Thomson space by UVD Pte Ltd, a joint venture between Singapore Land and UOL Group Restricted. Located alongside Vibrant Hill Drive, this site is well located with close proximity to Thomson Plaza and the longer term Upper Thomson MRT. Get pleasure from a inexperienced and wholesome life-style with visits to MacRitchie Reservoir, Lower Peirce Reservoir, Bishan Park and Singapore Island Golf Course which are simply minutes' drive away

Whereas every effort has been made to ensure that all information displayed herein are correct and full, the information are indicative somewhat than definitive. Thus its accuracy, whether or not express or implicit, isn't assured and to the fullest extent permitted by relevant legal guidelines. The Creator/Developer/Huttons Actual Estate Group does not accept duty for any errors, inaccuracies, omissions or for any loss which could result immediately or indirectly from reliance on the content herein. The Creator also reserves the right to right or replace the content at any time with out prior notification. Customers are advised to contact the advertiser for any clarifications or newest updates.

MAS agrees that it is crucial for banks to make use of valuations which are reflective of precise property values. We anticipate banks to adopt sound valuation processes. These embrace partaking independent valuers from corporations that aren't concerned within the property transaction as sales brokers or consultants, allocating valuation assignments randomly or on a rotational foundation, obtaining multiple valuations for every property, and checking that the valuations are cheap.

residences and penthouses. Fashionable interiors and quality designer fittings. Excellent recreational amenities. You can start by having a correct dialogue with individuals near you about investing on a family. This can normally lead you to 2 most important questions; what are the benefits of investing on a new apartment and the way you are going to find the perfect individual that can assist you in investing. The Venue Residences to Stay, Store and Dine, a luxurious mixeddevelopment by CDL in preparation for launch now. Looking for registrationof curiosity. TheMidtown is a mixed improvement near Hougang MRT. It's ex-Hougang Plazaand will comprise of residences constructed above business shops, duplexrestaurants and buying a property in singapore grocery store. VVIP Preview Quickly! Register Now. Late 2012, Probably 2013, English translation with notes of Euler’s paper by Lucas Willis and Thomas J. Osler
Is a upcoming condominium situated within the neighbourhood of Hillview The Singapore authorities has put in place a brand new subsidised pancreas transplant programme (Thinkstock picture). In the latest effort to spur funding in blighted areas of one of the city's poorest neighborhoods, Chicago plans to promote several hundred properties for the worth of a sweet bar.

When clients ask concerning the subsequent batch of items for launch, gross sales representatives might hint that subsequent phases will supply items with poorer view or lower high quality finishings, even if they are offered at greater prices. Tempted patrons are often unaware that the value of those goodies can easily be offset by a slight drop in the property's market worth. It is due to this fact more sensible to get an immediate discount off the listing worth. Ask the gross sales consultant the estimated value of that branded equipment or furnishing package, then request for a direct deduction from the unit's asking price in lieu of the developer's goodie. December 4, 2013 by iskandarinsider November four, 2013 by iskandarinsider October 23, 2013 by iskandarinsider

With elevated accessibility from the northern a part of Singapore via the two new MRT stations, Woodlands North MRT Station and Woodlands South MRT Station , along with the longer term North South Expressway NSE, that will link Woodlands, Sembawang and Yishun to the city, journey could be made easier. To qualify for the "Giant Lot Program," applicants should already personal property on the identical block as the lot they wish to buy; they have to also be present on property taxes, have no financial obligations to the town (like water bills or parking tickets) and should tell the city how they plan to make use of the property, DNAinfo Chicago experiences. New Condo Trilinq @ Clementi New Condominium La fiesta @ Sengkang MRT Expertise Guru Property's tantalizing new property launch! And the Web site

The Brilliant Hill Drive condominium is an exciting upcoming development in Higher Thomson space by UVD Pte Ltd, a joint venture between Singapore Land and UOL Group Restricted. Located alongside Vibrant Hill Drive, this site is well located with close proximity to Thomson Plaza and the longer term Upper Thomson MRT. Get pleasure from a inexperienced and wholesome life-style with visits to MacRitchie Reservoir, Lower Peirce Reservoir, Bishan Park and Singapore Island Golf Course which are simply minutes' drive away

Whereas every effort has been made to ensure that all information displayed herein are correct and full, the information are indicative somewhat than definitive. Thus its accuracy, whether or not express or implicit, isn't assured and to the fullest extent permitted by relevant legal guidelines. The Creator/Developer/Huttons Actual Estate Group does not accept duty for any errors, inaccuracies, omissions or for any loss which could result immediately or indirectly from reliance on the content herein. The Creator also reserves the right to right or replace the content at any time with out prior notification. Customers are advised to contact the advertiser for any clarifications or newest updates.

MAS agrees that it is crucial for banks to make use of valuations which are reflective of precise property values. We anticipate banks to adopt sound valuation processes. These embrace partaking independent valuers from corporations that aren't concerned within the property transaction as sales brokers or consultants, allocating valuation assignments randomly or on a rotational foundation, obtaining multiple valuations for every property, and checking that the valuations are cheap.

residences and penthouses. Fashionable interiors and quality designer fittings. Excellent recreational amenities. You can start by having a correct dialogue with individuals near you about investing on a family. This can normally lead you to 2 most important questions; what are the benefits of investing on a new apartment and the way you are going to find the perfect individual that can assist you in investing. The Venue Residences to Stay, Store and Dine, a luxurious mixeddevelopment by CDL in preparation for launch now. Looking for registrationof curiosity. TheMidtown is a mixed improvement near Hougang MRT. It's ex-Hougang Plazaand will comprise of residences constructed above business shops, duplexrestaurants and buying a property in singapore grocery store. VVIP Preview Quickly! Register Now. Late 2012, Probably 2013
Is a upcoming condominium situated within the neighbourhood of Hillview The Singapore authorities has put in place a brand new subsidised pancreas transplant programme (Thinkstock picture). In the latest effort to spur funding in blighted areas of one of the city's poorest neighborhoods, Chicago plans to promote several hundred properties for the worth of a sweet bar.

When clients ask concerning the subsequent batch of items for launch, gross sales representatives might hint that subsequent phases will supply items with poorer view or lower high quality finishings, even if they are offered at greater prices. Tempted patrons are often unaware that the value of those goodies can easily be offset by a slight drop in the property's market worth. It is due to this fact more sensible to get an immediate discount off the listing worth. Ask the gross sales consultant the estimated value of that branded equipment or furnishing package, then request for a direct deduction from the unit's asking price in lieu of the developer's goodie. December 4, 2013 by iskandarinsider November four, 2013 by iskandarinsider October 23, 2013 by iskandarinsider

With elevated accessibility from the northern a part of Singapore via the two new MRT stations, Woodlands North MRT Station and Woodlands South MRT Station , along with the longer term North South Expressway NSE, that will link Woodlands, Sembawang and Yishun to the city, journey could be made easier. To qualify for the "Giant Lot Program," applicants should already personal property on the identical block as the lot they wish to buy; they have to also be present on property taxes, have no financial obligations to the town (like water bills or parking tickets) and should tell the city how they plan to make use of the property, DNAinfo Chicago experiences. New Condo Trilinq @ Clementi New Condominium La fiesta @ Sengkang MRT Expertise Guru Property's tantalizing new property launch! And the Web site

The Brilliant Hill Drive condominium is an exciting upcoming development in Higher Thomson space by UVD Pte Ltd, a joint venture between Singapore Land and UOL Group Restricted. Located alongside Vibrant Hill Drive, this site is well located with close proximity to Thomson Plaza and the longer term Upper Thomson MRT. Get pleasure from a inexperienced and wholesome life-style with visits to MacRitchie Reservoir, Lower Peirce Reservoir, Bishan Park and Singapore Island Golf Course which are simply minutes' drive away

Whereas every effort has been made to ensure that all information displayed herein are correct and full, the information are indicative somewhat than definitive. Thus its accuracy, whether or not express or implicit, isn't assured and to the fullest extent permitted by relevant legal guidelines. The Creator/Developer/Huttons Actual Estate Group does not accept duty for any errors, inaccuracies, omissions or for any loss which could result immediately or indirectly from reliance on the content herein. The Creator also reserves the right to right or replace the content at any time with out prior notification. Customers are advised to contact the advertiser for any clarifications or newest updates.

MAS agrees that it is crucial for banks to make use of valuations which are reflective of precise property values. We anticipate banks to adopt sound valuation processes. These embrace partaking independent valuers from corporations that aren't concerned within the property transaction as sales brokers or consultants, allocating valuation assignments randomly or on a rotational foundation, obtaining multiple valuations for every property, and checking that the valuations are cheap.

residences and penthouses. Fashionable interiors and quality designer fittings. Excellent recreational amenities. You can start by having a correct dialogue with individuals near you about investing on a family. This can normally lead you to 2 most important questions; what are the benefits of investing on a new apartment and the way you are going to find the perfect individual that can assist you in investing. The Venue Residences to Stay, Store and Dine, a luxurious mixeddevelopment by CDL in preparation for launch now. Looking for registrationof curiosity. TheMidtown is a mixed improvement near Hougang MRT. It's ex-Hougang Plazaand will comprise of residences constructed above business shops, duplexrestaurants and buying a property in singapore grocery store. VVIP Preview Quickly! Register Now. Late 2012, Probably 2013
Is a upcoming condominium situated within the neighbourhood of Hillview The Singapore authorities has put in place a brand new subsidised pancreas transplant programme (Thinkstock picture). In the latest effort to spur funding in blighted areas of one of the city's poorest neighborhoods, Chicago plans to promote several hundred properties for the worth of a sweet bar.

When clients ask concerning the subsequent batch of items for launch, gross sales representatives might hint that subsequent phases will supply items with poorer view or lower high quality finishings, even if they are offered at greater prices. Tempted patrons are often unaware that the value of those goodies can easily be offset by a slight drop in the property's market worth. It is due to this fact more sensible to get an immediate discount off the listing worth. Ask the gross sales consultant the estimated value of that branded equipment or furnishing package, then request for a direct deduction from the unit's asking price in lieu of the developer's goodie. December 4, 2013 by iskandarinsider November four, 2013 by iskandarinsider October 23, 2013 by iskandarinsider

With elevated accessibility from the northern a part of Singapore via the two new MRT stations, Woodlands North MRT Station and Woodlands South MRT Station , along with the longer term North South Expressway NSE, that will link Woodlands, Sembawang and Yishun to the city, journey could be made easier. To qualify for the "Giant Lot Program," applicants should already personal property on the identical block as the lot they wish to buy; they have to also be present on property taxes, have no financial obligations to the town (like water bills or parking tickets) and should tell the city how they plan to make use of the property, DNAinfo Chicago experiences. New Condo Trilinq @ Clementi New Condominium La fiesta @ Sengkang MRT Expertise Guru Property's tantalizing new property launch! And the Web site

The Brilliant Hill Drive condominium is an exciting upcoming development in Higher Thomson space by UVD Pte Ltd, a joint venture between Singapore Land and UOL Group Restricted. Located alongside Vibrant Hill Drive, this site is well located with close proximity to Thomson Plaza and the longer term Upper Thomson MRT. Get pleasure from a inexperienced and wholesome life-style with visits to MacRitchie Reservoir, Lower Peirce Reservoir, Bishan Park and Singapore Island Golf Course which are simply minutes' drive away

Whereas every effort has been made to ensure that all information displayed herein are correct and full, the information are indicative somewhat than definitive. Thus its accuracy, whether or not express or implicit, isn't assured and to the fullest extent permitted by relevant legal guidelines. The Creator/Developer/Huttons Actual Estate Group does not accept duty for any errors, inaccuracies, omissions or for any loss which could result immediately or indirectly from reliance on the content herein. The Creator also reserves the right to right or replace the content at any time with out prior notification. Customers are advised to contact the advertiser for any clarifications or newest updates.

MAS agrees that it is crucial for banks to make use of valuations which are reflective of precise property values. We anticipate banks to adopt sound valuation processes. These embrace partaking independent valuers from corporations that aren't concerned within the property transaction as sales brokers or consultants, allocating valuation assignments randomly or on a rotational foundation, obtaining multiple valuations for every property, and checking that the valuations are cheap.

residences and penthouses. Fashionable interiors and quality designer fittings. Excellent recreational amenities. You can start by having a correct dialogue with individuals near you about investing on a family. This can normally lead you to 2 most important questions; what are the benefits of investing on a new apartment and the way you are going to find the perfect individual that can assist you in investing. The Venue Residences to Stay, Store and Dine, a luxurious mixeddevelopment by CDL in preparation for launch now. Looking for registrationof curiosity. TheMidtown is a mixed improvement near Hougang MRT. It's ex-Hougang Plazaand will comprise of residences constructed above business shops, duplexrestaurants and buying a property in singapore grocery store. VVIP Preview Quickly! Register Now. Late 2012, Probably 2013, Fourteen proofs compiled by Robin Chapman

↑ Template:Cite web
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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

[1] Template:Cite web

[2] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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

[1]

[2]

Jacobian variety: Difference between revisions

Revision as of 23:08, 9 April 2013

Contents

Euler's approach

The Riemann zeta function

A rigorous proof using Fourier series

A rigorous elementary proof

History of this proof

The proof

Packing Squares with Side 1/n

See also

References

Notes

External links

Navigation menu

@@ Line 1: / Line 1: @@
+{{Pi box}}
+The '''Basel problem''' is a famous problem in [[mathematical analysis]] with relevance to [[number theory]], first posed by [[Pietro Mengoli]] in 1644 and solved by [[Leonhard Euler]] in 1735. Since the problem had withstood the attacks of the leading [[mathematician]]s of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by [[Bernhard Riemann]] in his seminal 1859 paper ''[[On the Number of Primes Less Than a Given Magnitude]]'', in which he defined his [[Riemann zeta function|zeta function]] and proved its basic properties. The problem is named after [[Basel]], hometown of Euler as well as of the [[Bernoulli family]] who unsuccessfully attacked the problem.
+The Basel problem asks for the precise [[summation]] of the [[Multiplicative inverse|reciprocals]] of the [[square number|squares]] of the [[natural number]]s, i.e. the precise sum of the [[Series (mathematics)|infinite series]]:
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+:<math>
+\sum_{n=1}^\infin \frac{1}{n^2} =
+\lim_{n \to +\infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2}\right).
+</math>
+The series is approximately equal to '''1.644934''' {{OEIS2C|id=A013661}}. The Basel problem asks for the ''exact'' sum of this series (in [[Closed-form expression|closed form]]), as well as a [[mathematical proof|proof]] that this sum is correct. Euler found the exact sum to be ''π''<sup>2</sup>/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof.
+==Euler's approach==
+Euler's original derivation of the value ''π''<sup>2</sup>/6 essentially extended observations about finite [[polynomial]]s and assumed that these same properties hold true for infinite series. Of course, Euler's original reasoning requires justification (100 years later, Weierstrass proved that Euler's representation of the sin-function as an infinite product is correct, see: [[Weierstrass factorization theorem]]), but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series. The agreement he observed gave him sufficient confidence to announce his result to the mathematical community.
+To follow Euler's argument, recall the [[Taylor series]] expansion of the [[trigonometric function|sine function]]
+:<math> \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots. </math>
+Dividing through by ''x'', we have
+:<math> \frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots. </math>
+Now, the roots (intersections with the x-axis) of [[Sinc function|sin(''x'')/''x'']] occur precisely at <math>x = n\cdot\pi</math> where <math>n = \pm1, \pm2, \pm3, \dots\ </math>
+Let us assume we can express this infinite series as a (normalized) product of linear factors given by its roots, just as we do for finite polynomials:
+:<math>
+\begin{align}
+\frac{\sin(x)}{x} & {} =
+\left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 - \frac{x}{2\pi}\right)\left(1 + \frac{x}{2\pi}\right)\left(1 - \frac{x}{3\pi}\right)\left(1 + \frac{x}{3\pi}\right) \cdots \\
+& {} = \left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right) \cdots.
+\end{align}
+</math>
+If we formally multiply out this product and collect all the ''x''<sup>2</sup> terms (we are allowed to do so because of [[Newton's identities]]), we see that the ''x''<sup>2</sup> coefficient of sin(''x'')/''x'' is
+:<math>
+-\left(\frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \cdots \right) =
+-\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}.
+</math>
+But from the original infinite series expansion of sin(''x'')/''x'', the coefficient of ''x''<sup>2</sup> is &minus;1/(3!) = &minus;1/6. These two coefficients must be equal; thus,
+:<math>
+-\frac{1}{6} =
+-\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}.
+</math>
+Multiplying through both sides of this equation by <math>-\pi^2</math> gives the sum of the reciprocals of the positive square integers.
+:<math>
+\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}.
+</math>
+==The Riemann zeta function ==
+The [[Riemann zeta function]] <math>\zeta(s)</math> is one of the most important functions in mathematics, because of its relationship to the distribution of the [[prime number]]s. The function is defined for any [[complex number]] ''s'' with real part > 1 by the following formula:
+:<math>
+\zeta(s) =
+\sum_{n=1}^\infin \frac{1}{n^s}.
+</math>
+Taking ''s'' = 2, we see that <math>\zeta(2)</math> is equal to the sum of the reciprocals of the squares of the positive integers:
+:<math>
+\zeta(2) =
+\sum_{n=1}^\infin \frac{1}{n^2} =
+\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6} \approx 1.644934.
+</math>
+Convergence can be proven with the following inequality:
+:<math>
+\sum_{n=1}^N \frac{1}{n^2} < 1 + \sum_{n=2}^N \frac{1}{n(n-1)}
+= 1 + \sum_{n=2}^N \left( \frac{1}{n-1} - \frac{1}{n} \right)
+= 1 + 1 - \frac{1}{N} \; \stackrel{N \to \infty}{\longrightarrow} \; 2.
+</math>
+This gives us the upper bound <math>\zeta(2) < 2</math>, and because the infinite sum has only positive terms, it must converge. It can be shown that <math>\zeta(s)</math> has a nice expression in terms of the [[Bernoulli number]]s whenever ''s'' is a positive even integer. With <math>s=2n</math>:
+:<math>
+\zeta(2n)=\frac{(2\pi)^{2n}(-1)^{n+1}B_{2n}}{2\cdot(2n)!}
+</math>
+==A rigorous proof using Fourier series==
+Let <math>f(x)=x</math> over the interval ''x''&nbsp;∈&nbsp;(–{{pi}},{{pi}}). The [[Fourier series]] for this function (worked out in that article) is
+:<math>f(x) = 2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx).</math>
+Then, using [[Parseval's identity]] (with <math>f(x)=x</math>) we have that
+:<math>\sum_{n=-\infty}^\infty |a_n|^2 = \frac{1}{2\pi}\int_{-\pi}^\pi x^2 \, dx</math>,
+where
+:<math>a_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} x e^{-inx} \, dx = \frac{n\pi \cos(n\pi)-\sin(n\pi)}{\pi n^2} i = \frac{\cos(n\pi)}{n} i - \frac{\sin(n\pi)}{\pi n^2} i = \frac{(-1)^n}{n} i</math>
+for ''n''&nbsp;≠&nbsp;0, and ''a''<sub>0</sub>&nbsp;=&nbsp;0. Thus,
+:<math>|a_n|^2 = \frac{1}{n^2}</math>
+for ''n''&nbsp;≠&nbsp;0 and
+:<math>\sum_{n=-\infty}^{\infty} |a_n|^2 = 2\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{2\pi}\int_{-\pi}^\pi x^2 \, dx</math>
+Therefore,
+:<math>\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{4\pi}\int_{-\pi}^\pi x^2 \, dx = \frac{\pi^2}{6}</math>
+as required.
+==A rigorous elementary proof==
+This is by far the most elementary well-known proof; while most proofs use results from advanced mathematics, such as [[Fourier analysis]], [[complex analysis]], and [[multivariable calculus]], the following does not even require single-variable [[calculus]] (although a single [[limit of a function|limit]] is taken at the end).
+===History of this proof===
+The proof goes back to [[Augustin Louis Cauchy]] (Cours d'Analyse, 1821, Note VIII). In 1954 this proof appeared in the book of [[Akiva Yaglom|Akiva]] and [[Isaak Yaglom]] "Nonelementary Problems in an Elementary Exposition". Later, in 1982, it appeared in the journal ''Eureka'', attributed to John Scholes, but Scholes claims he learned the proof from [[Peter Swinnerton-Dyer]], and in any case he maintains the proof was "common knowledge at [[University of Cambridge|Cambridge]] in the late 1960s".
+===The proof===
+The main idea behind the proof is to bound the partial sums
+:<math>\sum_{k=1}^m \frac{1}{k^2} = \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{m^2}</math>
+between two expressions, each of which will tend to {{pi}}<sup>2</sup>/6 as ''m'' approaches infinity. The two expressions are derived from identities involving the [[cotangent]] and [[cosecant]] functions. These identities are in turn derived from [[de Moivre's formula]], and we now turn to establishing these identities.
+Let <math>x</math> be a real number with <math>0<x<\frac{\pi}{2}</math>, and let ''n'' be a positive odd integer. Then from de Moivre's formula and the definition of the cotangent function, we have
+:<math>\frac{\cos (nx) + i \sin (nx)}{(\sin x)^n} = \frac{(\cos x + i\sin x)^n}{(\sin x)^n} = \left(\frac{\cos x + i \sin x}{\sin x}\right)^n = (\cot x + i)^n.</math>
+From the [[binomial theorem]], we have
+:<math>(\cot x + i)^n = {n \choose 0} \cot^n x + {n \choose 1} (\cot^{n-1} x)i + \cdots + {n \choose {n-1}} (\cot x)i^{n-1} + {n \choose n} i^n</math>
+:<math>= \left[ {n \choose 0} \cot^n x - {n \choose 2} \cot^{n-2} x \pm \cdots \right] \; + \; i\left[ {n \choose 1} \cot^{n-1} x - {n \choose 3} \cot^{n-3} x \pm \cdots \right].</math>
+Combining the two equations and equating imaginary parts gives the identity
+:<math>\frac{\sin (nx)}{(\sin x)^n} = \left[ {n \choose 1} \cot^{n-1} x - {n \choose 3} \cot^{n-3} x \pm \cdots \right].</math>
+We take this identity, fix a positive integer <math>m</math>, set <math>n=2m+1\,</math> and consider <math>x_r=\frac{r\pi}{2m+1}</math> for <math>r=1,2,\ldots,m</math>. Then <math>nx_r\,</math> is a multiple of <math>\pi\,</math> and therefore a zero of the sine function, and so
+:<math>0 = {{2m+1} \choose 1} \cot^{2m} x_r - {{2m+1} \choose 3} \cot^{2m-2} x_r \pm \cdots + (-1)^m{{2m+1} \choose {2m+1}}</math>
+for every <math>r=1,2,\ldots,m</math>. The values <math>x_1,\ldots,x_m</math> are distinct numbers in the interval (0, {{pi}}/2). Since the function <math>\cot^2 x \,</math> is one-to-one on this interval, the numbers <math>t_r=\cot^2 x_r\,</math> are distinct for ''r'' = 1, 2, ..., ''m''. By the above equation, these ''m'' numbers are the roots of the ''m''th degree polynomial
+:<math>p(t) := {{2m+1} \choose 1}t^m - {{2m+1} \choose 3}t^{m-1} \pm \cdots + (-1)^m{{2m+1} \choose {2m+1}}.</math>
+By [[Viète's formulas]] we can calculate the sum of the roots directly by examining the first two coefficients of the polynomial, and this comparison shows that
+:<math>\cot ^2 x_1 + \cot ^2 x_2 + \cdots + \cot ^2 x_m
+= \frac{\binom{2m+1}3} {\binom{2m+1}1}= \frac{2m(2m-1)}6.</math>
+Substituting the identity <math>\csc^2x=\cot^2x+1\,</math>, we have
+:<math>\csc ^2 x_1 + \csc ^2 x_2 + \cdots + \csc ^2 x_m
+=\frac{2m(2m-1)}6 + m = \frac{2m(2m+2)}6.</math>
+Now consider the inequality <math>\cot^2x<\frac{1}{x^2}<\csc^2x</math>. If we add up all these inequalities for each of the numbers <math>x_r=\frac{r\pi}{2m+1}</math>, and if we use the two identities above, we get
+:<math>\frac{2m(2m-1)}6 < \left( \frac{2m+1}{\pi} \right) ^2 + \left( \frac{2m+1}{2 \pi} \right) ^2 + \cdots + \left( \frac{2m+1}{m \pi} \right) ^2 < \frac{2m(2m+2)}6.</math>
+Multiplying through by ({{pi}}/(2''m'' + 1))<sup>2</sup>, this becomes
+:<math>\frac{\pi ^2}{6}\left(\frac{2m}{2m+1}\right)\left(\frac{2m-1}{2m+1}\right) < \frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{m^2} < \frac{\pi ^2}{6}\left(\frac{2m}{2m+1}\right)\left(\frac{2m+2}{2m+1}\right).</math>
+As ''m'' approaches infinity, the left and right hand expressions each approach <math>\frac{\pi^2}{6}\,</math>, so by the [[squeeze theorem]],
+:<math>\zeta(2) =
+\sum_{k=1}^\infin \frac{1}{k^2} =
+\lim_{m \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{m^2}\right) = \frac{\pi ^2}{6}</math>
+and this completes the proof.
+== Packing Squares with Side 1/n ==
+The solution to the Basel problem is related to packing squares with side length of <math>\frac{1}{n}</math>. The question that is asked is "What is the smallest rectangle that can contain the squares as ''n'' approaches infinity?"<ref>{{cite web|last=Pegg Jr|first=Ed|title=Packing Squares with Side 1/n|url=http://demonstrations.wolfram.com/PackingSquaresWithSide1N/|publisher=Wolfram Demonstrations Project|accessdate=2 October 2013}}</ref> One bound to the answer is dependent on one rectangle side, <math>S_1</math>, to be the sum of the largest two squares sides summed together,
+:<math> S_1 = \frac{1}{2} + \frac{1}{3} = \frac{5}{6}</math>.
+The other rectangle side, <math>S_2</math>, depends on the order of remaining squares. However, we know that total area, ''A'', of the squares (less the square with n=1) is:
+:<math>A = \sum_{n=2}^\infty \frac{1}{n^2} = \frac{\pi ^2}{6} - 1 </math>.
+Dividing the squares total area by the one known rectangle side leaves the ideal rectangle side length,
+:<math>S_2 = \frac{A}{S_1} = \frac{ \frac{\pi ^2}{6} - 1}{\frac{5}{6}} = \frac{\pi ^2 - 6}{5} = 0.77392088021\cdots </math> {{OEIS2C|id=A233699}}.
+The current packing record holder is Marc Paulhus, who developed a packing algorithm.<ref>{{cite journal|last=Paulhus|first=M. M.|title=An Algorithm for Packing Squares|journal=Journal of Combinatorial Theory|year=1998|series=A|issue=82(2)|pages=147–157}}</ref> Because {{pi}} is a [[transcendental number]], the record holder's rectangles area will always be greater than the ideal rectangle.
+==See also==
+*[[Riemann zeta function]]
+==References==
+* {{Citation | title=Number Theory: An Approach Through History | first=André | last=Weil | authorlink=André Weil | publisher=Springer-Verlag | isbn=0-8176-3141-0 | year = 1983}}.
+* {{Citation | title=Euler: The Master of Us All | first=William | last=Dunham | publisher=[[Mathematical Association of America]] | year=1999 | isbn=0-88385-328-0}}.
+* {{Citation | title=Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics | first=John | last=Derbyshire | publisher=Joseph Henry Press | isbn=0-309-08549-7 | year=2003}}.
+* {{Citation | last1=Aigner | first1=Martin | last2=Ziegler | first2=Günter M. | authorlink2=Günter M. Ziegler | title=[[Proofs from THE BOOK]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1998}}
+* {{Citation | title=Riemann's Zeta Function | first=Harold M. | last=Edwards | authorlink = Harold Edwards (mathematician) | publisher=Dover | isbn=0-486-41740-9 | year=2001}}.
+==Notes==
+{{reflist}}
+==External links==
+* [http://plus.maths.org/content/infinite-series-surprises An infinite series of surprises] by C. J. Sangwin
+* {{PDFlink|[http://www.math.dartmouth.edu/~euler/docs/translations/E352.pdf Remarques sur un beau rapport entre les series des puissances tant directes que reciproques]|348&nbsp;kB}}, English translation with notes of Euler’s paper by Lucas Willis and Thomas J. Osler
+* {{PDFlink|[http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2002%20Estimating%20the%20Basel%20Problem.pdf How Euler did it]|265&nbsp;kB}}
+* {{PDFlink|[http://www.math.psu.edu/sellersj/p25.pdf The infinite series of Euler and the Bernoulli's spice up a calculus class]|106&nbsp;kB}}
+* {{PDFlink|[http://secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf Evaluating ζ(2)]|184&nbsp;kB}}, Fourteen proofs compiled by Robin Chapman
+{{DEFAULTSORT:Basel Problem}}
+[[Category:Number theory]]
+[[Category:Articles containing proofs]]
+[[Category:Zeta and L-functions]]
+[[Category:Mathematical problems]]

Jacobian variety: Difference between revisions

Revision as of 23:08, 9 April 2013

Euler's approach

The Riemann zeta function

A rigorous proof using Fourier series

A rigorous elementary proof

History of this proof

The proof

Packing Squares with Side 1/n

See also

References

Notes

External links

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