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'''Physisorption''', also called '''physical adsorption''', is a process in which the electronic structure of the atom or molecule is barely perturbed upon [[adsorption]].<ref>{{citation|author=K. Oura et al.|title=Surface Science, An Introduction|location= Berlin|publisher= Springer|year= 2003| isbn =978-3-540-00545-2}}</ref><ref name=ConceptsinSurfacePhysics>{{citation |author=M. C. Desjonqueres et al |title=Concepts in surface physics |edition=2nd |url=http://books.google.com/?id=XW_Wvjwt5nIC&printsec=frontcover&dq=Concepts+in+surface+physics |place=New York |publisher=Springer-Verlag |date= 1996. Corrected printing 1998 |isbn=3-540-58622-9 |accessdate=29 August 2012}}</ref><ref>{{Citation|author=Hans Luth et al.|title=Surfaces and interfaces of solids|publisher= Springer-Verlag|year= 1993| isbn =978-3-540-56840-7}}</ref>
{{about|Euler's theorem in number theory|other meanings|List of topics named after Leonhard Euler}}
In [[number theory]], '''Euler's theorem''' (also known as the '''Fermat–Euler theorem''' or '''Euler's totient theorem''') states that if ''n''  and ''a'' are [[coprime]] positive integers, then
:<math>a^{\varphi (n)} \equiv 1 \pmod{n}</math>
where φ(''n'') is [[Euler's totient function]]. (The notation is explained in the article [[Modular arithmetic]].) It was first stated and proved by [[Leonhard Euler]] in 1736.<ref>{{mathworld|EulersTotientTheorem|Euler's Totient Theorem}}</ref>


==Introduction==
There is a converse of Euler's theorem: if the above congruence is true then ''a'' and ''n'' must be coprime.


The fundamental interacting force of physisorption is caused by [[van der Waals force]]. Even though the interaction energy is very weak (~10&ndash;100&nbsp;meV), physisorption plays an important role in nature.  For instance, the van der Waals attraction between surfaces and foot-hairs of [[gecko]]s provides the remarkable ability to climb up vertical walls.<ref>{{Citation | author=K. Autumn ''et al.''|title= Adhesive force of a single gecko foot-hair| journal=Nature| volume=405 | pages= 681–5| year=2000 | doi=10.1038/35015073 | pmid=10864324 | issue=6787}}</ref>  Van der Waals forces originate from the interactions between induced, permanent or transient electric dipoles.
The theorem is a generalization of [[Fermat's little theorem]], and is further generalized by [[Carmichael function|Carmichael's theorem]].


In comparison with [[chemisorption]], in which the electronic structure of bonding atoms or molecules is changed and covalent or ionic bonds form, physisorption, generally speaking, can only be observed in the environment of low temperature (thermal energy at room temperature ~26&nbsp;meV) and the absence of the relatively strong chemisorptions. In practice, the categorisation of a particular adsorption as physisorption or chemisorption depends principally on the [[binding energy]] of the adsorbate to the substrate.
The theorem may be used to easily reduce large powers modulo ''n''. For example, consider finding the ones place decimal digit of 7<sup>222</sup>, i.e. 7<sup>222</sup> (mod 10). Note that 7 and 10 are coprime, and φ(10) = 4. So Euler's theorem yields 7<sup>4</sup> ≡ 1 (mod 10), and we get 7<sup>222</sup> ≡ 7<sup>4×55 + 2</sup> ≡ (7<sup>4</sup>)<sup>55</sup>×7<sup>2</sup> ≡ 1<sup>55</sup>×7<sup>2</sup> ≡ 49 ≡ 9 (mod 10).


==Modeling by image charge==
In general, when reducing a power of ''a'' modulo ''n'' (where ''a'' and ''n'' are coprime), one needs to work modulo φ(''n'') in the exponent of ''a'':
[[Image:physisorption 1.jpg|thumbnail|200px|Fig. 1. Schematic illustration of an adsorbed hydrogen atom near a perfect [[electrical conductor|conductor]] interacting with its [[image charge]]s.]]
:if ''x'' ≡ ''y'' (mod φ(''n'')), then ''a''<sup>''x''</sup> ≡ ''a''<sup>''y''</sup> (mod ''n'').


To give a simple illustration of physisorption, we can first consider an adsorbed hydrogen atom in front of a perfect conductor, as shown in Fig. 1.  A nucleus with positive charge is located at '''R'''&nbsp;=&nbsp;(0,&nbsp;0,&nbsp;''Z''), and the position coordinate of its electron, '''r'''&nbsp;=&nbsp;(''x'',&nbsp;''y'',&nbsp;''z'') is given with respect to the nucleus.  The adsorption process can be viewed as the interaction between this hydrogen atom and its image charges of both the nucleus and electron in the conductor. As a result, the total electrostatic energy is the sum of attraction and repulsion terms:
Euler's theorem also forms the basis of the [[RSA (algorithm)|RSA]] encryption system: encryption and decryption in this system together amount to exponentiating the original text by ''k''φ(''n'')+1 for some positive integer ''k'', so Euler's theorem shows that the decrypted result is the same as the original.


:<math>V = {e^2\over 4\pi\varepsilon_0}\left(\frac{-1}{|2\mathbf R|}+\frac{-1}{|2\mathbf R+\mathbf r-\mathbf r'|}+\frac{1}{|2\mathbf R-\mathbf r'|}+\frac{1}{|2\mathbf R+\mathbf r|}\right).</math>
== Proofs ==


The first term is the attractive interaction of nucleus and its image charge, and the second term is due to the interaction of the electron and its image charge. The repulsive interaction is shown in the third and fourth terms arising from the interaction of nucleus-image electron and electron-image nucleus, respectively.
1. Euler's theorem can be proven using concepts from the [[group (mathematics)|theory of groups]]:<ref>Ireland & Rosen, corr. 1 to prop 3.3.2</ref>
The residue classes (mod ''n'') that are coprime to ''n'' form a group under multiplication (see the article [[Multiplicative group of integers modulo n]] for details.) [[Lagrange's theorem (group theory)|Lagrange's theorem]] states that the order of any subgroup of a finite group divides the order of the entire group, in this case φ(''n''). If ''a'' is any number coprime to ''n'' then ''a'' is in one of these residue classes, and its powers ''a'', ''a''<sup>2</sup>, ..., ''a''<sup>''k''</sup> &equiv; 1 (mod ''n'') are a subgroup. Lagrange's theorem says ''k'' must divide φ(''n''), i.e. there is an integer ''M'' such that ''kM'' = φ(''n''). But then,
:<math>
a^{\varphi(n)} =
a^{kM} =
(a^{k})^M \equiv
1^M =
1 \pmod{n}.
</math>


By [[Taylor expansion]] in powers of |'''r'''|&nbsp;/&nbsp;|'''R'''|, this interaction energy can be further expressed as:
2. There is also a direct proof:<ref>Hardy & Wright, thm. 72</ref><ref>Landau, thm. 75</ref> Let ''R'' = {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>φ(''n'')</sub>} be a [[reduced residue system]] (mod  ''n'') and let ''a'' be any integer coprime to ''n''. The proof hinges on the fundamental fact that multiplication by ''a'' permutes the ''x''<sub>''i''</sub>: in other words if ''ax''<sub>''j''</sub>  &equiv; ''ax''<sub>''k''</sub> (mod ''n'') then ''j'' = ''k''. (This law of cancellation is proved in the article [[Multiplicative_group_of_integers_modulo_n#Group_axioms|Multiplicative group of integers modulo n]].<ref>See [[Bézout's lemma]]</ref>) That is, the sets ''R'' and ''aR'' = {''ax''<sub>1</sub>, ''ax''<sub>2</sub>, ..., ''ax''<sub>φ(''n'')</sub>}, considered as sets of congruence classes (mod ''n''), are identical (as sets - they may be listed in different orders), so the product of all the numbers in ''R'' is congruent (mod ''n'') to the product of all the numbers in ''aR'':
 
:<math>
:<math>V = {-e^2\over 16\pi\varepsilon_0 Z^3}\left(\frac{x^2+y^2}{2}+z^2\right)+ {3e^2\over 32\pi\varepsilon_0 Z^4}\left(\frac{x^2+y^2}{2}{z}+z^3\right)+O\left(\frac{1}{Z^5}\right).</math>
\prod_{i=1}^{\varphi(n)} x_i \equiv
 
\prod_{i=1}^{\varphi(n)} ax_i \equiv
One can find from the first non-vanishing term that the physisorption potential depends on the distance ''Z'' between adsorbed atom and surface as ''Z''<sup>&minus;3</sup>, in contrast with the ''r''<sup>&minus;6</sup> dependence of the molecular [[van der Waals]] potential, where ''r'' is the distance between two [[dipoles]].
a^{\varphi(n)}\prod_{i=1}^{\varphi(n)} x_i \pmod{n},
 
</math> and using the cancellation law to cancel the ''x''<sub>''i''</sub>s gives Euler's theorem:
==Modeling by quantum-mechanical oscillator==
 
The [[van der Waals force|van der Waals]] binding energy can be analyzed by another simple physical picture: modeling the motion of an electron around its nucleus by a three-dimensional simple [[harmonic oscillator]] with a potential energy ''V<sub>a</sub>'':
 
:<math>V_a = \frac{m_e}{2}{\omega^2}(x^2+y^2+z^2),</math>
 
where ''m<sub>e</sub>'' and ''&omega;'' are the mass and vibrational frequency of the electron, respectively.
 
As this atom approaches the surface of a metal and forms adsorption, this potential energy ''V<sub>a</sub>'' will be modified due to the image charges by additional potential terms which are quadratic in the displacements:
 
:<math>V_a = \frac{m_e}{2}{\omega^2}(x^2+y^2+z^2)-{e^2\over 16\pi\varepsilon_0 Z^3}\left(\frac{x^2+y^2}{2}+z^2\right)+\ldots</math>  (from the Taylor expansion above.)
 
Assuming
 
:<math> m_e \omega^2>>{e^2\over 16\pi\varepsilon_0 Z^3},</math>
 
the potential is well approximated as
 
:<math>V_a \sim \frac{m_e}{2}{\omega_1^2}(x^2+y^2)+\frac{m_e}{2}{\omega_2^2}z^2</math>,
 
where


:<math>
:<math>
\begin{align}
a^{\varphi(n)}\equiv 1 \pmod{n}.
\omega_1 &= \omega - {e^2\over 32\pi\varepsilon_0 m_e\omega Z^3},\\
\omega_2 &= \omega - {e^2\over 16\pi\varepsilon_0 m_e\omega Z^3}.
\end{align}
</math>
</math>


If one assumes that the electron is in the ground state, then the van der Waals binding energy is essentially the change of the zero-point energy:
==See also==
* [[Carmichael function]]
* [[Euler's criterion]]
* [[Fermat's little theorem]]
* [[Wilson's theorem]]


:<math>V_v = \frac{\hbar}{2}(2\omega_1+\omega_2-3\omega)= - {\hbar e^2\over 16\pi\varepsilon_0 m_e\omega Z^3}.</math>
==Notes==
{{reflist}}


This expression also shows the nature of the ''Z''<sup>&minus;3</sup> dependence of the van der Waals interaction.
==References==
The ''[[Disquisitiones Arithmeticae]]'' has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.


Furthermore by introducing the atomic [[polarizability]],
*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | last2 = Clarke | first2 = Arthur A. (translator into English) 
  | title = Disquisitiones Arithemeticae (Second, corrected edition)
  | publisher = [[Springer Publishing|Springer]]
  | location = New York
  | date = 1986
  | isbn = 0-387-96254-9}}
*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | last2 = Maser | first2 = H. (translator into German) 
  | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
  | publisher = Chelsea
  | location = New York
  | date = 1965
  | isbn = 0-8284-0191-8}}
*{{citation
  | last1 = Hardy  | first1 = G. H.
  | last2 = Wright | first2 = E. M.
  | title = An Introduction to the Theory of Numbers (Fifth edition)
  | publisher = [[Oxford University Press]]
  | location = Oxford
  | date = 1980
  | isbn = 978-0-19-853171-5}}
*{{citation
  | last1 = Ireland  | first1 = Kenneth
  | last2 = Rosen  | first2 = Michael
  | title = A Classical Introduction to Modern Number Theory (Second edition)
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = New York
  | date = 1990
  | isbn = 0-387-97329-X}}
*{{citation
  | last1 = Landau | first1 = Edmund
  | title = Elementary Number Theory
  | publisher = Chelsea
  | location = New York
  | date = 1966}}


:<math> \alpha= \frac {e^2} {m_e\omega^2},</math>
== External links ==
* {{mathworld|EulersTotientTheorem|Euler's Totient Theorem}}
* [http://planetmath.org/encyclopedia/EulersTheorem.html Euler's Theorem] at [http://planetmath.org PlanetMath]


the van der Waals potential can be further simplified:
[[Category:Modular arithmetic]]
 
[[Category:Theorems in number theory]]
:<math>V_v = - {\hbar \alpha \omega\over 16\pi\varepsilon_0 Z^3}= -\frac{C_v}{Z^3},</math>
[[Category:Articles containing proofs]]
 
{{Link GA|es}}
where
 
:<math>C_v = {\hbar \alpha \omega\over 16\pi\varepsilon_0},</math>
 
is the van der Waals constant which is related to the atomic polarizability.
 
Also, by expressing the fourth-order correction in the Taylor expansion above as (''aC<sub>v</sub>Z''<sub>0</sub>)&nbsp;/&nbsp;(Z<sup>4</sup>), where ''a'' is some constant, we can define ''Z''<sub>0</sub> as the position of the ''dynamical image plane'' and obtain
[[Image:physisorption table.jpg|thumbnail|400px|Table 1. The van der Waals constant ''C<sub>v</sub>'' and the position of the dynamical image plane ''Z''<sub>0</sub> for various rare gases atoms adsorbed on noble metal surfaces obtained by the jellium model. Note that ''C<sub>v</sub>'' is in eV/Å<sup>3</sup> and ''Z''<sub>0</sub> in Å.]]
 
:<math>V_v = - \frac{C_v}{(Z-Z_0)^3}+O\left(\frac{1}{Z^5}\right).</math>
 
The origin of ''Z''<sub>0</sub> comes from the spilling of the electron wavefunction out of the surface.  As a result, the position of image plane representing the reference for the space coordinate is different from the substrate surface itself and modified by ''Z''<sub>0</sub>.
 
Table 1 shows the [[jellium]] model calculation for van der Waals constant ''C<sub>v</sub>'' and dynamical image plane ''Z''<sub>0</sub> of rare gas atoms on various metal surfaces.  The increasing of ''C<sub>v</sub>'' from He to Xe for all metal substrates is caused by the larger atomic [[polarizability]] of the heavier rare gas atoms.  For the position of the dynamical image plane, it decreases with increasing dielectric function and is typically on the order of 0.2&nbsp;Å.
 
==Physisorption potential==
[[Image:physisorption 2.jpg|thumbnail|400px|Fig. 2. Calculated physisorption potential energy for He adsorbed on various [[jellium]] metal surfaces. Note that the weak van der Waals attraction forms shallow wells with energy about few meV.<ref name="Kohn"/>]]
 
Even though the [[van der Waals interaction]] is attractive, as the adsorbed atom moves closer to the surface the wavefunction of electron starts to overlap with that of the surface atoms.  Further the energy of the system will increase due to the orthogonality of wavefunctions of the approaching atom and surface atoms.
 
This [[Pauli exclusion]] and repulsion are particularly strong for atoms with closed valence shells that dominate the surface interaction.  As a result, the minimum energy of physisorption must be found by the balance between the long-range van der Waals attraction and short-range [[Pauli repulsion]].  For instance, by separating the total interaction of physisorption into two contributions- a short-range term depicted by [[Hartree&ndash;Fock]] theory and a long-range van der Waals attraction, the equilibrium position of physisorption for rare gases adsorbed on [[jellium]] substrate can be determined.<ref name="Kohn">{{Citation | author=E. Zaremba and W. Kohn|title= Theory of helium adsorption on simple and noble-metal surfaces| journal= Phys. Rev. B| volume=15 | issue=4 | pages= 1769| year=1977 | doi=10.1103/PhysRevB.15.1769|bibcode = 1977PhRvB..15.1769Z }}</ref>  Fig. 2 shows the physisorption potential energy of He adsorbed on Ag, Cu, and Au substrates which are described by the [[jellium]] model with different densities of smear-out background positive charges.  It can be found that the weak van der Waals interaction leads to shallow attractive energy wells (<10&nbsp;meV).  One of the experimental methods for exploring physisorption potential energy is the scattering process, for instance, inert gas atoms scattered from metal surfaces.  Certain specific features of the interaction potential between scattered atoms and surface can be extracted by analyzing the experimentally determined angular distribution and cross sections of the scattered particles.
 
==Comparison with chemisorption==
 
* Physisorption is a general phenomenon and occurs in any solid/fluid or solid/gas system. [[Chemisorption]] is characterized by chemical specificity.
* In physisorption, perturbation of the electronic states of adsorbent and adsorbate is minimal.  For chemisorption, changes in the electronic states may be detectable by suitable physical means.
* Typical binding energy of physisorption is about 10&ndash;100 meV.  Chemisorption usually forms bonding with energy of 1&ndash;10&nbsp;eV.
* The elementary step in physisorption from a gas phase does not involve an activation energy.  Chemisorption often involves an activation energy.
* For physisorption, under appropriate conditions, gas phase molecules can form multilayer adsorption.  In chemisorption, molecules are adsorbed on the surface by valence bonds and only form monolayer adsorption.
 
==See also==
*[[Adsorption]]
*[[Chemisorption]]
*[[van der Waals force]]
 
==References==
{{reflist}}


[[Category:Surface chemistry]]
[[ar:مبرهنة أويلر]]
[[bg:Теорема на Ойлер]]
[[ca:Teorema d'Euler]]
[[cs:Eulerova věta (teorie čísel)]]
[[da:Eulers sætning]]
[[de:Satz von Euler]]
[[el:Θεώρημα του Όιλερ]]
[[es:Teorema de Euler]]
[[fa:قضیه اویلر]]
[[fr:Théorème d'Euler (nombres)]]
[[ko:오일러의 정리]]
[[id:Teorema Euler]]
[[is:Eulersregla]]
[[it:Teorema di Eulero (aritmetica modulare)]]
[[he:משפט אוילר]]
[[kk:Эйлер теоремасы (сандар теориясы)]]
[[hu:Euler–Fermat-tétel]]
[[nl:Stelling van Euler]]
[[ja:オイラーの定理 (数論)]]
[[pl:Twierdzenie Eulera (teoria liczb)]]
[[pt:Teorema de Euler]]
[[ro:Teorema lui Euler]]
[[ru:Теорема Эйлера (теория чисел)]]
[[fi:Eulerin lause (lukuteoria)]]
[[sv:Eulers sats]]
[[tr:Euler teoremi]]
[[uk:Теорема Ейлера (теорія чисел)]]
[[vi:Định lý Euler]]
[[zh:欧拉定理 (数论)]]

Revision as of 20:38, 9 August 2014

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 number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that if n and a are coprime positive integers, then

aφ(n)≡1(modn)

where φ(n) is Euler's totient function. (The notation is explained in the article Modular arithmetic.) It was first stated and proved by Leonhard Euler in 1736.[1]

There is a converse of Euler's theorem: if the above congruence is true then a and n must be coprime.

The theorem is a generalization of Fermat's little theorem, and is further generalized by Carmichael's theorem.

The theorem may be used to easily reduce large powers modulo n. For example, consider finding the ones place decimal digit of 7222, i.e. 7222 (mod 10). Note that 7 and 10 are coprime, and φ(10) = 4. So Euler's theorem yields 74 ≡ 1 (mod 10), and we get 7222 ≡ 74×55 + 2 ≡ (74)55×72 ≡ 155×72 ≡ 49 ≡ 9 (mod 10).

In general, when reducing a power of a modulo n (where a and n are coprime), one needs to work modulo φ(n) in the exponent of a:

if x ≡ y (mod φ(n)), then ax ≡ ay (mod n).

Euler's theorem also forms the basis of the RSA encryption system: encryption and decryption in this system together amount to exponentiating the original text by kφ(n)+1 for some positive integer k, so Euler's theorem shows that the decrypted result is the same as the original.

Proofs

1. Euler's theorem can be proven using concepts from the theory of groups:[2] The residue classes (mod n) that are coprime to n form a group under multiplication (see the article Multiplicative group of integers modulo n for details.) Lagrange's theorem states that the order of any subgroup of a finite group divides the order of the entire group, in this case φ(n). If a is any number coprime to n then a is in one of these residue classes, and its powers a, a2, ..., ak ≡ 1 (mod n) are a subgroup. Lagrange's theorem says k must divide φ(n), i.e. there is an integer M such that kM = φ(n). But then,

aφ(n)=akM=(ak)M≡1M=1(modn).

2. There is also a direct proof:[3][4] Let R = {x1, x2, ..., xφ(n)} be a reduced residue system (mod n) and let a be any integer coprime to n. The proof hinges on the fundamental fact that multiplication by a permutes the xi: in other words if axj ≡ axk (mod n) then j = k. (This law of cancellation is proved in the article Multiplicative group of integers modulo n.[5]) That is, the sets R and aR = {ax1, ax2, ..., axφ(n)}, considered as sets of congruence classes (mod n), are identical (as sets - they may be listed in different orders), so the product of all the numbers in R is congruent (mod n) to the product of all the numbers in aR:

∏i=1φ(n)xi≡∏i=1φ(n)axi≡aφ(n)∏i=1φ(n)xi(modn), and using the cancellation law to cancel the xis gives Euler's theorem:
aφ(n)≡1(modn).

See also

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

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

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    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

  • 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

    Here is my web site - cottagehillchurch.com
  • Euler's Theorem at PlanetMath

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ar:مبرهنة أويلر bg:Теорема на Ойлер ca:Teorema d'Euler cs:Eulerova věta (teorie čísel) da:Eulers sætning de:Satz von Euler el:Θεώρημα του Όιλερ es:Teorema de Euler fa:قضیه اویلر fr:Théorème d'Euler (nombres) ko:오일러의 정리 id:Teorema Euler is:Eulersregla it:Teorema di Eulero (aritmetica modulare) he:משפט אוילר kk:Эйлер теоремасы (сандар теориясы) hu:Euler–Fermat-tétel nl:Stelling van Euler ja:オイラーの定理 (数論) pl:Twierdzenie Eulera (teoria liczb) pt:Teorema de Euler ro:Teorema lui Euler ru:Теорема Эйлера (теория чисел) fi:Eulerin lause (lukuteoria) sv:Eulers sats tr:Euler teoremi uk:Теорема Ейлера (теорія чисел) vi:Định lý Euler zh:欧拉定理 (数论)

  1. ↑ 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

    Here is my web site - cottagehillchurch.com
  2. ↑ Ireland & Rosen, corr. 1 to prop 3.3.2
  3. ↑ Hardy & Wright, thm. 72
  4. ↑ Landau, thm. 75
  5. ↑ See Bézout's lemma
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