Particle-size distribution

In mathematics, the Menger curvature of a triple of points in n-dimensional Euclidean space Rⁿ is the reciprocal of the radius of the circle that passes through the three points. It is named after the Austrian-American mathematician Karl Menger.

Definition

Let x, y and z be three points in Rⁿ; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let Π ⊆ Rⁿ be the Euclidean plane spanned by x, y and z and let C ⊆ Π be the unique Euclidean circle in Π that passes through x, y and z (the circumcircle of x, y and z). Let R be the radius of C. Then the Menger curvature c(x, y, z) of x, y and z is defined by

$c(x,y,z)={\frac {1}{R}}.$

If the three points are collinear, R can be informally considered to be +∞, and it makes rigorous sense to define c(x, y, z) = 0. If any of the points x, y and z are coincident, again define c(x, y, z) = 0.

Using the well-known formula relating the side lengths of a triangle to its area, it follows that

$c(x,y,z)={\frac {1}{R}}={\frac {4A}{|x-y||y-z||z-x|}},$

where A denotes the area of the triangle spanned by x, y and z.

Another way of computing Menger curvature is the identity

$c(x,y,z)={\frac {2\sin \angle xyz}{|x-z|}}$

where $\angle xyz$ is the angle made at the y-corner of the triangle spanned by x,y,z.

Menger curvature may also be defined on a general metric space. If X is a metric space and x,y, and z are distinct points, let f be an isometry from $\{x,y,z\}$ into $\mathbb {R} ^{2}$. Define the Menger curvature of these points to be

$c_{X}(x,y,z)=c(f(x),f(y),f(z)).$

Note that f need not be defined on all of X, just on {x,y,z}, and the value c_X (x,y,z) is independent of the choice of f.

Integral Curvature Rectifiability

Menger curvature can be used to give quantitative conditions for when sets in $\mathbb {R} ^{n}$ may be rectifiable. For a Borel measure $\mu$ on a Euclidean space $\mathbb {R} ^{n}$ define

$c^{p}(\mu )=\int \int \int c(x,y,z)^{p}d\mu (x)d\mu (y)d\mu (z).$

• A Borel set $E\subseteq \mathbb {R} ^{n}$ is rectifiable if $c^{2}(H^{1}|_{E})<\infty$, where $H^{1}|_{E}$ denotes one-dimensional Hausdorff measure restricted to the set $E$.

The basic intuition behind the result is that Menger curvature measures how straight a given triple of points are (the smaller $c(x,y,z)\max\{|x-y|,|y-z|,|z-y|\}$ is, the closer x,y, and z are to being collinear), and this integral quantity being finite is saying that the set E is flat on most small scales. In particular, if the power in the integral is larger, our set is smoother than just being rectifiable

• Let $p>3$, $f:S^{1}\rightarrow \mathbb {R} ^{n}$ be a homeomorphism and $\Gamma =f(S^{1})$. Then $f\in C^{1,1-{\frac {3}{p}}}(S^{1})$ if $c^{p}(H^{1}|_{\Gamma })<\infty$.

• If $0<H^{s}(E)<\infty$ where $0<s\leq {\frac {1}{2}}$, and $c^{2s}(H^{s}|_{E})<\infty$, then $E$ is rectifiable in the sense that there are countably many $C^{1}$ curves $\Gamma _{i}$ such that $H^{s}(E\backslash \bigcup \Gamma _{i})=0$. The result is not true for ${\frac {1}{2}}<s<1$, and $c^{2s}(H^{s}|_{E})=\infty$ for $1<s\leq n$.:

In the opposite direction, there is a result of Peter Jones:

• If $E\subseteq \Gamma \subseteq \mathbb {R} ^{2}$, $H^{1}(E)>0$, and $\Gamma$ is rectifiable. Then there is a positive Radon measure $\mu$ supported on $E$ satisfying $\mu B(x,r)\leq r$ for all $x\in E$ and $r>0$ such that $c^{2}(\mu )<\infty$ (in particular, this measure is the Frostman measure associated to E). Moreover, if $H^{1}(B(x,r)\cap \Gamma )\leq Cr$ for some constant C and all $x\in \Gamma$ and r>0, then $c^{2}(H^{1}|_{E})<\infty$. This last result follows from the Analyst's Traveling Salesman Theorem.

Analogous results hold in general metric spaces:

See also

• Menger-Melnikov curvature of a measure

External links

• Template:Cite web

References

• 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