Flory–Huggins solution theory

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

In mathematics and computational science, the Euler method is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book Institutionum calculi integralis (published 1768–70).^[1]

The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size. The Euler method often serves as the basis to construct more complicated methods.

Informal geometrical description

Consider the problem of calculating the shape of an unknown curve which starts at a given point and satisfies a given differential equation. Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.

The idea is that while the curve is initially unknown, its starting point, which we denote by $A_{0},$ is known (see the picture on top right). Then, from the differential equation, the slope to the curve at $A_{0}$ can be computed, and so, the tangent line.

Take a small step along that tangent line up to a point $A_{1} .$ Along this small step, the slope does not change too much, so $A_{1}$ will be close to the curve. If we pretend that $A_{1}$ is still on the curve, the same reasoning as for the point $A_{0}$ above can be used. After several steps, a polygonal curve $A_{0} A_{1} A_{2} A_{3} \dots$ is computed. In general, this curve does not diverge too far from the original unknown curve, and the error between the two curves can be made small if the step size is small enough and the interval of computation is finite.^[2]

Formulation of the method

Suppose that we want to approximate the solution of the initial value problem

y^{'} (t) = f (t, y (t)), y (t_{0}) = y_{0} .

Choose a value $h$ for the size of every step and set $t_{n} = t_{0} + n h$ . Now, one step of the Euler method from $t_{n}$ to $t_{n + 1} = t_{n} + h$ is^[3]

y_{n + 1} = y_{n} + h f (t_{n}, y_{n}) .

The value of $y_{n}$ is an approximation of the solution to the ODE at time $t_{n}$ : $y_{n} \approx y (t_{n})$ . The Euler method is explicit, i.e. the solution $y_{n + 1}$ is an explicit function of $y_{i}$ for $i \leq n$ .

While the Euler method integrates a first-order ODE, any ODE of order N can be represented as a first-order ODE: to treat the equation

y^{(N)} (t) = f (t, y (t), y^{'} (t), \dots, y^{(N - 1)} (t))

,

we introduce auxiliary variables $z_{1} (t) = y (t), z_{2} (t) = y^{'} (t), \dots, z_{N} (t) = y^{(N - 1)} (t)$ and obtain the equivalent equation

z^{'} (t) = (\begin{matrix} {z_{1}}^{'} (t) \\ ⋮ \\ {z_{N^{'} - 1}}^{'} (t) \\ {z_{N}}^{'} (t) \end{matrix}) = (\begin{matrix} y^{'} (t) \\ ⋮ \\ y^{(N - 1)} (t) \\ y^{(N)} (t) \end{matrix}) = (\begin{matrix} z_{2} (t) \\ ⋮ \\ z_{N} (t) \\ f (t, z_{1} (t), \dots, z_{N} (t)) \end{matrix})

This is a first-order system in the variable $z (t)$ and can be handled by Euler's method or, in fact, by any other scheme for first-order systems.^[4]

Example

Given the initial value problem

y^{'} = y, y (0) = 1,

we would like to use the Euler method to approximate $y (4)$ .^[5]

Using step size equal to 1 (h = 1)

The Euler method is

y_{n + 1} = y_{n} + h f (t_{n}, y_{n}) .

so first we must compute $f (t_{0}, y_{0})$ . In this simple differential equation, the function $f$ is defined by $f (t, y) = y$ . We have

f (t_{0}, y_{0}) = f (0, 1) = 1 .

By doing the above step, we have found the slope of the line that is tangent to the solution curve at the point $(0, 1)$ . Recall that the slope is defined as the change in $y$ divided by the change in $t$ , or $Δ y / Δ t$ .

The next step is to multiply the above value by the step size $h$ , which we take equal to one here:

h \cdot f (y_{0}) = 1 \cdot 1 = 1 .

Since the step size is the change in $t$ , when we multiply the step size and the slope of the tangent, we get a change in $y$ value. This value is then added to the initial $y$ value to obtain the next value to be used for computations.

y_{0} + h f (y_{0}) = y_{1} = 1 + 1 \cdot 1 = 2 .

The above steps should be repeated to find $y_{2}$ , $y_{3}$ and $y_{4}$ .

\begin{aligned} y_{2} & = y_{1} + h f (y_{1}) = 2 + 1 \cdot 2 = 4, \\ y_{3} & = y_{2} + h f (y_{2}) = 4 + 1 \cdot 4 = 8, \\ y_{4} & = y_{3} + h f (y_{3}) = 8 + 1 \cdot 8 = 16 . \end{aligned}

Due to the repetitive nature of this algorithm, it can be helpful to organize computations in a chart form, as seen below, to avoid making errors.

$n$	$y_{n}$	$t_{n}$	$f (t_{n}, y_{n})$	$h$	$Δ y$	$y_{n + 1}$
0	1	0	1	1	1	2
1	2	1	2	1	2	4
2	4	2	4	1	4	8
3	8	3	8	1	8	16

The conclusion of this computation is that $y_{4} = 16$ . The exact solution of the differential equation is $y (t) = e^{t}$ , so $y (4) = e^{4} \approx 54.598$ . Thus, the approximation of the Euler method is not very good in this case. However, as the figure shows, its behaviour is qualitatively right.

Using other step sizes

As suggested in the introduction, the Euler method is more accurate if the step size $h$ is smaller. The table below shows the result with different step sizes. The top row corresponds to the example in the previous section, and the second row is illustrated in the figure.

step size	result of Euler's method	error
1	16	38.598
0.25	35.53	19.07
0.1	45.26	9.34
0.05	49.56	5.04
0.025	51.98	2.62
0.0125	53.26	1.34

The error recorded in the last column of the table is the difference between the exact solution at $t = 4$ and the Euler approximation. In the bottom of the table, the step size is half the step size in the previous row, and the error is also approximately half the error in the previous row. This suggests that the error is roughly proportional to the step size, at least for fairly small values of the step size. This is true in general, also for other equations; see the section Global truncation error for more details.

Other methods, such as the midpoint method also illustrated in the figures, behave more favourably: the error of the midpoint method is roughly proportional to the square of the step size. For this reason, the Euler method is said to be a first-order method, while the midpoint method is second order.

We can extrapolate from the above table that the step size needed to get an answer that is correct to three decimal places is approximately 0.00001, meaning that we need 400,000 steps. This large number of steps entails a high computational cost. For this reason, people usually employ alternative, higher-order methods such as Runge–Kutta methods or linear multistep methods, especially if a high accuracy is desired.^[6]

Derivation

The Euler method can be derived in a number of ways. Firstly, there is the geometrical description mentioned above.

Another possibility is to consider the Taylor expansion of the function $y$ around $t_{0}$ :

y (t_{0} + h) = y (t_{0}) + h y^{'} (t_{0}) + \frac{1}{2} h^{2} y^{″} (t_{0}) + O (h^{3}) .

The differential equation states that $y^{'} = f (t, y)$ . If this is substituted in the Taylor expansion and the quadratic and higher-order terms are ignored, the Euler method arises.^[7] The Taylor expansion is used below to analyze the error committed by the Euler method, and it can be extended to produce Runge–Kutta methods.

A closely related derivation is to substitute the forward finite difference formula for the derivative,

y^{'} (t_{0}) \approx \frac{y (t_{0} + h) - y (t_{0})}{h}

in the differential equation $y^{'} = f (t, y)$ . Again, this yields the Euler method.^[8] A similar computation leads to the midpoint rule and the backward Euler method.

Finally, one can integrate the differential equation from $t_{0}$ to $t_{0} + h$ and apply the fundamental theorem of calculus to get:

y (t_{0} + h) - y (t_{0}) = \int_{t_{0}}^{t_{0} + h} f (t, y (t)) d t .

Now approximate the integral by the left-hand rectangle method (with only one rectangle):

\int_{t_{0}}^{t_{0} + h} f (t, y (t)) d t \approx h f (t_{0}, y (t_{0})) .

Combining both equations, one finds again the Euler method.^[9] This line of thought can be continued to arrive at various linear multistep methods.

Local truncation error

The local truncation error of the Euler method is error made in a single step. It is the difference between the numerical solution after one step, $y_{1}$ , and the exact solution at time $t_{1} = t_{0} + h$ . The numerical solution is given by

y_{1} = y_{0} + h f (t_{0}, y_{0}) .

For the exact solution, we use the Taylor expansion mentioned in the section Derivation above:

y (t_{0} + h) = y (t_{0}) + h y^{'} (t_{0}) + \frac{1}{2} h^{2} y^{″} (t_{0}) + O (h^{3}) .

The local truncation error (LTE) introduced by the Euler method is given by the difference between these equations:

L T E = y (t_{0} + h) - y_{1} = \frac{1}{2} h^{2} y^{″} (t_{0}) + O (h^{3}) .

This result is valid if $y$ has a bounded third derivative.^[10]

This shows that for small $h$ , the local truncation error is approximately proportional to $h^{2}$ . This makes the Euler method less accurate (for small $h$ ) than other higher-order techniques such as Runge-Kutta methods and linear multistep methods, for which the local truncation error is proportial to a higher power of the step size.

A slightly different formulation for the local truncation error can be obtained by using the Lagrange form for the remainder term in Taylor's theorem. If $y$ has a continuous second derivative, then there exists a $ξ \in [t_{0}, t_{0} + h]$ such that

L T E = y (t_{0} + h) - y_{1} = \frac{1}{2} h^{2} y^{″} (ξ) .

^[11]

In the above expressions for the error, the second derivative of the unknown exact solution $y$ can be replaced by an expression involving the right-hand side of the differential equation. Indeed, it follows from the equation $y^{'} = f (t, y)$ that

y^{″} (t_{0}) = \frac{\partial f}{\partial t} (t_{0}, y (t_{0})) + \frac{\partial f}{\partial y} (t_{0}, y (t_{0})) f (t_{0}, y (t_{0})) .

^[12]

Global truncation error

The global truncation error is the error at a fixed time $t$ , after however many steps the methods needs to take to reach that time from the initial time. The global truncation error is the cumulative effect of the local truncation errors committed in each step.^[13] The number of steps is easily determined to be $(t - t_{0}) / h$ , which is proportional to $1 / h$ , and the error committed in each step is proportional to $h^{2}$ (see the previous section). Thus, it is to be expected that the global truncation error will be proportional to $h$ .^[14]

This intuitive reasoning can be made precise. If the solution $y$ has a bounded second derivative and $f$ is Lipschitz continuous in its second argument, then the global truncation error (GTE) is bounded by

| GTE | \leq \frac{h M}{2 L} (e^{L (t - t_{0})} - 1)

where $M$ is an upper bound on the second derivative of $y$ on the given interval and $L$ is the Lipschitz constant of $f$ .^[15]

The precise form of this bound of little practical importance, as in most cases the bound vastly overestimates the actual error committed by the Euler method.^[16] What is important is that it shows that the global truncation error is (approximately) proportional to $h$ . For this reason, the Euler method is said to be first order.^[17]

Numerical stability

The Euler method can also be numerically unstable, especially for stiff equations, meaning that the numerical solution grows very large for equations where the exact solution does not. This can be illustrated using the linear equation

y^{'} = - 2.3 y, y (0) = 1 .

The exact solution is $y (t) = e^{- 2.3 t}$ , which decays to zero as $t \to \infty$ . However, if the Euler method is applied to this equation with step size $h = 1$ , then the numerical solution is qualitatively wrong: it oscillates and grows (see the figure). This is what it means to be unstable. If a smaller step size is used, for instance $h = 0.7$ , then the numerical solution does decay to zero.

If the Euler method is applied to the linear equation $y^{'} = k y$ , then the numerical solution is unstable if the product $h k$ is outside the region

{z \in C ∣ | z + 1 | \leq 1},

illustrated on the right. This region is called the (linear) instability region.^[18] In the example, $k$ equals −2.3, so if $h = 1$ then $h k = - 2.3$ which is outside the stability region, and thus the numerical solution is unstable.

This limitation —along with its slow convergence of error with h— means that the Euler method is not often used, except as a simple example of numerical integration.

Rounding errors

The discussion up to now has ignored the consequences of rounding error. In step n of the Euler method, the rounding error is roughly of the magnitude εy_n where ε is the machine epsilon. Assuming that the rounding errors are all of approximately the same size, the combined rounding error in N steps is roughly Nεy₀ if all errors points in the same direction. Since the number of steps is inversely proportional to the step size h, the total rounding error is proportional to ε / h. In reality, however, it is extremely unlikely that all rounding errors point in the same direction. If instead it is assumed that the rounding errors are independent rounding variables, then the total rounding error is proportional to $ε / \sqrt{h}$ .^[19]

Thus, for extremely small values of the step size, the truncation error will be small but the effect of rounding error may be big. Most of the effect of rounding error can be easily avoided if compensated summation is used in the formula for the Euler method.^[20]

Modifications and extensions

A simple modification of the Euler method which eliminates the stability problems noted in the previous section is the backward Euler method:

y_{n + 1} = y_{n} + h f (t_{n + 1}, y_{n + 1}) .

This differs from the (standard, or forward) Euler method in that the function $f$ is evaluated at the end point of the step, instead of the starting point. The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has $y_{n + 1}$ on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly.

Other modifications of the Euler method that help with stability yield the exponential Euler method or the semi-implicit Euler method.

More complicated methods can achieve a higher order (and more accuracy). One possibility is to use more function evaluations. This is illustrated by the midpoint method which is already mentioned in this article:

y_{n + 1} = y_{n} + h f (t_{n} + \frac{1}{2} h, y_{n} + \frac{1}{2} h f (t_{n}, y_{n})) .

This leads to the family of Runge–Kutta methods.

The other possibility is to use more past values, as illustrated by the two-step Adams–Bashforth method:

y_{n + 1} = y_{n} + \frac{3}{2} h f (t_{n}, y_{n}) - \frac{1}{2} h f (t_{n - 1}, y_{n - 1}) .

This leads to the family of linear multistep methods.

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.

References

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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
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.

External links

DTZ's auction group in Singapore auctions all types of residential, workplace and retail properties, retailers, homes, accommodations, boarding houses, industrial buildings and development websites. Auctions are at the moment held as soon as a month.

Whitehaven @ Pasir Panjang – A boutique improvement nicely nestled peacefully in serene Pasir Panjang personal estate presenting a hundred and twenty rare freehold private apartments tastefully designed by the famend Ong & Ong Architect. Only a short drive away from Science Park and NUS Campus, Jade Residences, a recent Freehold condominium which offers high quality lifestyle with wonderful facilities and conveniences proper at its door steps. Its fashionable linear architectural fashion promotes peace and tranquility living nestled within the D19 personal housing enclave. Rising workplace sector leads real estate market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline International Market Perspectives - 1st Quarter 2014

There are a lot of websites out there stating to be one of the best seek for propertycondominiumhouse, and likewise some ways to discover a low cost propertycondominiumhouse. Owning a propertycondominiumhouse in Singapore is the dream of virtually all individuals in Singapore, It is likely one of the large choice we make in a lifetime. Even if you happen to're new to Property listing singapore funding, we are right here that will help you in making the best resolution to purchase a propertycondominiumhouse at the least expensive value.

Jun 18 ROCHESTER in MIXED USE IMPROVEMENT $1338000 / 1br - 861ft² - (THE ROCHESTER CLOSE TO NORTH BUONA VISTA RD) pic real property - by broker Jun 18 MIXED USE IMPROVEMENT @ ROCHESTER @ ROCHESTER PK $1880000 / 1br - 1281ft² - (ROCHESTER CLOSE TO NORTH BUONA VISTA) pic real estate - by broker Tue 17 Jun Jun 17 Sunny Artwork Deco Gem Near Seashore-Super Deal!!! $103600 / 2br - 980ft² - (Ventnor) pic actual estate - by owner Jun 17 Freehold semi-d for rent (Jalan Rebana ) $7000000 / 5909ft² - (Jalan Rebana ) actual property - by dealer Jun sixteen Ascent @ 456 in D12 (456 Balestier Highway,Singapore) pic real property - by proprietor Jun 16 RETAIL SHOP AT SIM LIM SQUARE FOR SALE, IT MALL, ROCHOR, BUGIS MRT $2000000 / 506ft² - (ROCHOR, BUGIS MRT) pic real estate - by dealer HDB Scheme Any DBSS BTO

In case you are eligible to purchase landed houses (open solely to Singapore residents) it is without doubt one of the best property investment choices. Landed housing varieties solely a small fraction of available residential property in Singapore, due to shortage of land right here. In the long term it should hold its worth and appreciate as the supply is small. In truth, landed housing costs have risen the most, having doubled within the last eight years or so. However he got here back the following day with two suitcases full of money. Typically we've got to clarify to such folks that there are rules and paperwork in Singapore and you can't just buy a home like that,' she said. For conveyancing matters there shall be a recommendedLondon Regulation agency familiar with Singapore London propertyinvestors to symbolize you

Sales transaction volumes have been expected to hit four,000 units for 2012, close to the mixed EC gross sales volume in 2010 and 2011, in accordance with Savills Singapore. Nevertheless the last quarter was weak. In Q4 2012, sales transactions were 22.8% down q-q to 7,931 units, in line with the URA. The quarterly sales discount was felt throughout the board. When the sale just starts, I am not in a hurry to buy. It's completely different from a private sale open for privileged clients for one day solely. Orchard / Holland (D09-10) House For Sale The Tembusu is a singular large freehold land outdoors the central area. Designed by multiple award-profitable architects Arc Studio Architecture + Urbanism, the event is targeted for launch in mid 2013. Post your Property Condos Close to MRT

Euler's Method for O.D.E.'s, by John H. Matthews, California State University at Fullerton.

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Flory–Huggins solution theory

Contents

Informal geometrical description

Formulation of the method

Example

Using step size equal to 1 (h = 1)

Using other step sizes

Derivation

Local truncation error

Global truncation error

Numerical stability

Rounding errors

Modifications and extensions

See also

Notes

References

External links

Navigation menu

Flory–Huggins solution theory

Informal geometrical description

Formulation of the method

Example

Using step size equal to 1 (h = 1)

Using other step sizes

Derivation

Local truncation error

Global truncation error

Numerical stability

Rounding errors

Modifications and extensions

See also

Notes

References

External links

Navigation menu

Search