Signal-to-quantization-noise ratio

There are close parallels between the mathematical expressions for the thermodynamic entropy, usually denoted by S, of a physical system in the statistical thermodynamics established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s, and the information-theoretic entropy, usually expressed as H, of Claude Shannon and Ralph Hartley developed in the 1940s. Shannon, although not initially aware of this similarity, commented on it upon publicizing information theory in A Mathematical Theory of Communication.

This article explores what links there are between the two concepts, and how far they can be regarded as connected.

Equivalence of form of the defining expressions

Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.

The defining expression for entropy in the theory of statistical mechanics established by Ludwig Boltzmann and J. Willard Gibbs in the 1870s, is of the form:

$S=-k_{B}\sum _{i}p_{i}\ln p_{i},\,$

where $p_{i}$ is the probability of the microstate i taken from an equilibrium ensemble.

The defining expression for entropy in the theory of information established by Claude E. Shannon in 1948 is of the form:

$H=-\sum _{i}p_{i}\log _{b}p_{i},\,$

where $p_{i}$ is the probability of the message $m_{i}$ taken from the message space M and b is the base of the logarithm used. Common values of b are 2, Euler's number Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "[http://www.blitzkappers.com/?option=com_k2&view=itemlist&task=user&id=64019 public listed property developers In singapore] agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

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Mathematically H may also be seen as an average information, taken over the message space, because when a certain message occurs with probability p_i, the information −log(p_i) will be obtained.

If all the microstates are equiprobable (a microcanonical ensemble), the statistical thermodynamic entropy reduces to the form, as given by Boltzmann,

$S=k_{B}\ln W\,$

where W is the number of microstates.

If all the messages are equiprobable, the information entropy reduces to the Hartley entropy

$H=\log _{b}|M|\,$

where $|M|$ is the cardinality of the message space M.

The logarithm in the thermodynamic definition is the natural logarithm. It can be shown that the Gibbs entropy formula, with the natural logarithm, reproduces all of the properties of the macroscopic classical thermodynamics of Clausius. (See article: Entropy (statistical views)).

The logarithm can also be taken to the natural base in the case of information entropy. This is equivalent to choosing to measure information in nats instead of the usual bits. In practice, information entropy is almost always calculated using base 2 logarithms, but this distinction amounts to nothing other than a change in units. One nat is about 1.44 bits.

The presence of Boltzmann's constant k in the thermodynamic definitions is a historical accident, reflecting the conventional units of temperature. It is there to make sure that the statistical definition of thermodynamic entropy matches the classical entropy of Clausius, thermodynamically conjugate to temperature. For a simple compressible system that can only perform volume work, the first law of thermodynamics becomes

$dE=-pdV+TdS\,$

But one can equally well write this equation in terms of what physicists and chemists sometimes call the 'reduced' or dimensionless entropy, σ = S/k, so that

$dE=-pdV+k_{B}Td\sigma \,$

Just as S is conjugate to T, so σ is conjugate to k_BT (the energy that is characteristic of T on a molecular scale).

Theoretical relationship

Despite all that, there is an important difference between the two quantities. The information entropy H can be calculated for any probability distribution (if the "message" is taken to be that the event i which had probability p_i occurred, out of the space of the events possible), while the thermodynamic entropy S refers to thermodynamic probabilities p_i specifically.

Furthermore, the thermodynamic entropy S is dominated by different arrangements of the system, and in particular its energy, that are possible on a molecular scale. In comparison, information entropy of any macroscopic event is so small as to be completely irrelevant.

However, a connection can be made between the two. If the probabilities in question are the thermodynamic probabilities p_i:, the (reduced) Gibbs entropy σ can then be seen as simply the amount of Shannon information needed to define the detailed microscopic state of the system, given its macroscopic description. Or, in the words of G. N. Lewis writing about chemical entropy in 1930, "Gain in entropy always means loss of information, and nothing more". To be more concrete, in the discrete case using base two logarithms, the reduced Gibbs entropy is equal to the minimum number of yes/no questions needed to be answered in order to fully specify the microstate, given that we know the macrostate.

Furthermore, the prescription to find the equilibrium distributions of statistical mechanics —such as the Boltzmann distribution— by maximising the Gibbs entropy subject to appropriate constraints (the Gibbs algorithm) can be seen as something not unique to thermodynamics, but as a principle of general relevance in all sorts of statistical inference, if it is desired to find a maximally uninformative probability distribution, subject to certain constraints on the behaviour of its averages. (These perspectives are explored further in the article Maximum entropy thermodynamics.)

Information is physical

Szilard's engine

N-atom engine schematic

A physical thought experiment demonstrating how just the possession of information might in principle have thermodynamic consequences was established in 1929 by Leó Szilárd, in a refinement of the famous Maxwell's demon scenario.

Consider Maxwell's set-up, but with only a single gas particle in a box. If the supernatural demon knows which half of the box the particle is in (equivalent to a single bit of information), it can close a shutter between the two halves of the box, close a piston unopposed into the empty half of the box, and then extract $k_{B}T\ln 2$ joules of useful work if the shutter is opened again. The particle can then be left to isothermally expand back to its original equilibrium occupied volume. In just the right circumstances therefore, the possession of a single bit of Shannon information (a single bit of negentropy in Brillouin's term) really does correspond to a reduction in the entropy of the physical system. The global entropy is not decreased, but information to energy conversion is possible.

Using a phase-contrast microscope equipped with a high speed camera connected to a computer, as demon, the principle has been actually demonstrated. In this experiment, information to energy conversion is performed on a Brownian particle by means of feedback control; that is, synchronizing the work given to the particle with the information obtained on its position. Computing energy balances for different feedback protocols, has confirmed that the Jarzynski equality requires a generalization that accounts for the amount of information involved in the feedback.

Landauer's principle

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In fact one can generalise: any information that has a physical representation must somehow be embedded in the statistical mechanical degrees of freedom of a physical system.

Thus, Rolf Landauer argued in 1961, if one were to imagine starting with those degrees of freedom in a thermalised state, there would be a real reduction in thermodynamic entropy if they were then re-set to a known state. This can only be achieved under information-preserving microscopically deterministic dynamics if the uncertainty is somehow dumped somewhere else – i.e. if the entropy of the environment (or the non information-bearing degrees of freedom) is increased by at least an equivalent amount, as required by the Second Law, by gaining an appropriate quantity of heat: specifically kT ln 2 of heat for every 1 bit of randomness erased.

On the other hand, Landauer argued, there is no thermodynamic objection to a logically reversible operation potentially being achieved in a physically reversible way in the system. It is only logically irreversible operations – for example, the erasing of a bit to a known state, or the merging of two computation paths – which must be accompanied by a corresponding entropy increase. When information is physical, all processing of its representations, i.e. generation, encoding, transmission, decoding and interpretation, are natural processes where entropy increases by consumption of free energy.

Applied to the Maxwell's demon/Szilard engine scenario, this suggests that it might be possible to "read" the state of the particle into a computing apparatus with no entropy cost; but only if the apparatus has already been SET into a known state, rather than being in a thermalised state of uncertainty. To SET (or RESET) the apparatus into this state will cost all the entropy that can be saved by knowing the state of Szilard's particle.

Negentropy

Shannon entropy has been related by physicist Léon Brillouin to a concept sometimes called negentropy. In his 1962 book Science and Information Theory, Brillouin described the Negentropy Principle of Information or NPI, the gist of which is that acquiring information about a system’s microstates is associated with a decrease in entropy (work is needed to extract information, erasure leads to increase in thermodynamic entropy). There is no violation of the second law of thermodynamics, according to Brillouin, since a reduction in any local system’s thermodynamic entropy results in an increase in thermodynamic entropy elsewhere. Negentropy was considered as controversial because its earlier understanding can yield Carnot efficiency higher than one.

In 2009, Mahulikar & Herwig redefined thermodynamic negentropy as the specific entropy deficit of the dynamically ordered sub-system relative to its surroundings. This definition enabled the formulation of the Negentropy Principle, which is mathematically shown to follow from the 2nd Law of Thermodynamics, during order existence.

Black holes

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Quantum theory

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One could speak of the "joint entropy" of the position and momentum distributions in this quantity by considering them independent, but since they are not jointly observable, they cannot be considered as a joint distribution. Note that this entropy is not the accepted entropy of a quantum system, the Von Neumann entropy, −Tr ρ lnρ = −⟨lnρ⟩. In phase-space, the Von Neumann entropy can nevertheless be represented equivalently to Hilbert space, even though positions and momenta are quantum conjugate variables; and thus leads to a properly bounded entropy distinctly different (more detailed) than Hirschman's; this one accounts for the full information content of a mixture of quantum states.

(Dissatisfaction with the Von Neumann entropy from quantum information points of view has been expressed by Stotland, Pomeransky, Bachmat and Cohen, who have introduced a yet different definition of entropy that reflects the inherent uncertainty of quantum mechanical states. This definition allows distinction between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.)

The fluctuation theorem

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. The fluctuation theorem provides a mathematical justification of the second law of thermodynamics under these principles, and precisely defines the limitations of the applicability of that law for systems away from thermodynamic equilibrium.

Topics of recent research

Is information quantized?

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• The supposition that there is such a thing as an observable state (for instance the upper face of a dice or a coin) before the observation begins
• The fact that knowing this state does not depend on the order in which observations are made (commutativity)

Anton Zeilinger's and Caslav Brukner's article synthesized and developed these remarks. The so-called Zeilinger's principle suggests that the quantization observed in QM could be bound to information quantization (one cannot observe less than one bit, and what is not observed is by definition "random"). Nevertheless, these claims remain quite controversial. Detailed discussions of the applicability of the Shannon information in quantum mechanics and an argument that Zeilinger's principle cannot explain quantization have been published, that show that Brukner and Zeilinger change, in the middle of the calculation in their article, the numerical values of the probabilities needed to compute the Shannon entropy, so that the calculation makes little sense.

Extracting work from quantum information in a Szilárd engine

In 2013, a description was published of a two atom version of a Szilárd engine using Quantum discord to generate work from purely quantum information. Refinements in the lower temperature limit were suggested.

References

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Additional references

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• 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
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (A highly technical collection of writings giving an overview of the concept of entropy as it appears in various disciplines.)
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 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
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 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 (as PDF)

External links

• Information Processing and Thermodynamic Entropy Stanford Encyclopedia of Philosophy.
• An Intuitive Guide to the Concept of Entropy Arising in Various Sectors of Science — a wikibook on the interpretation of the concept of entropy.