text:
# zbMATH — the first resource for mathematics

A positivity property of the Satake isomorphism. (English) Zbl 0941.22006
Let $$G$$ be a connected reductive group over a local field $$F$$ which is quasisplit and which is split over an unramified extension. In this paper the author considers the matrix of the Satake isomorphism in terms of the natural bases of the source and of the target, and proves that all coefficients of this matrix that are not zero are actually positive numbers. The theorem is proved by induction on the semi-simple rank of $$G$$. In the case $$G=GL_n$$ one also shows how this positivity result can be deduced from the theory of symmetric functions. The (positivity) result is then applied to an existence problem of $$F$$-crystals which is a partial converse to Mazur’s theorem relating the Hodge polygon and the Newton polygon.

##### MSC:
 22E35 Analysis on $$p$$-adic Lie groups 14F30 $$p$$-adic cohomology, crystalline cohomology
Full Text:

format: plain


text:
## US Congress Looks to Canadian Oil Sands to Enhance Energy Security

##### 27 June 2006
 The forecast growth in oil sands production (In-Situ and Mining).

A new study released by the Joint Economic Committee (JEC) of the US House of Representatives concludes that the accelerated development of huge reserves of unconventional oil in Canada will undermine the OPEC oil cartel in coming years.

According to the study, Canadian Oil Sands: A New Force in the World Oil Market, estimates of proven oil reserves rank Canada second only to Saudi Arabia, with the possibility that Canada’s reserves may be even larger. Strong economic incentives would exist, even with oil prices at half the current level, to ramp up oil sands production and more than double output in ten years.

The large Canadian reserves of unconventional oil and their rapid development is very good news for consumers in the US and around the world. While these reserves can supply only a limited amount of oil at the current time, their development is exactly the kind of thing the OPEC cartel has hoped to avoid.

OPEC’s production quotas and failure to invest in expanded capacity have severely limited the conventional oil supply in the face of an unexpected increase in Asian demand. Concerns about Iran and terrorism have only exacerbated the situation. The result is sky-high oil prices and every economic incentive to develop alternative energy sources.

The Canadian oil sands will be an increasingly important factor in the world energy market in coming years. These reserves are immense and their potential output is very significant for the future energy balance. The oil sands will not solve the energy problem, but they will help to undermine OPEC’s power to increase oil prices and will thereby enhance US energy security.

—JEC Chairman Jim Saxton (R-NJ)

On the strength of expected oil sands output growth and despite declining conventional oil output, Canada can be expected to move from 7th to 5th place in ten years among the world’s oil producers.

The report, which focuses on the economics, not the technology or the feasibility of greatly expanded oil sands production, notes that although the projected amounts of oil produced are not large enough to achieve energy independence for North America, they represent a growing counterweight to OPEC’s market power and as such increase national energy security.

One of the underlying assumptions of the report is that OPEC has held back on developing more of its reserves, and that the accompanying increase in price caused by now surging demand is spurring the development of oil sands.

...as long as those who hold market power are opting to reap prices far in excess of the cost to develop the oil sands, they also are feeding the growth of a potential industry giant. High oil prices help to overcome shortrun increases in production cost while Alberta’s input markets and infrastructure adjust to a larger scale of oil sands production. In the long-run, the cost is likely to stabilize and may even decline, leading to ever larger rates of oil output.

What sets the Canadian oil sands apart from most other sources of oil under development is the vastness of the known resource and the secure access to it. It may take the oil sands industry a decade or more to stabilize the cost per barrel while accelerating the rate of output, but once it does, it will cap the power of OPEC.

 The current production from Canada and the North Sea is represented by the dotted line. The increase in oil sands production doesn't offset the decline in conventional production.

However, a new study by the Uppsala Hydrocarbon Depletion Study Group, Uppsala University, Sweden, has concluded that even in a very optimistic scenario Canada’s oil sands will not prevent the peaking of oil production (Peak Oil). Even if a crash program were immediately implemented it may only barely offset the combined declining conventional crude oil production in Canada and the North Sea.

The study, “A Crash Program Scenario for the Canadian Oil Sands Industry”, will be published in Energy Policy.

The Uppsala report argues that the implementation of a crash program for the Canadian oil sands industry has serious difficulties, including the lack of a sufficient natural gas supply and the problem of meeting Kyoto obligations (oil sands production is carbon intensive). Practical long-term production would require, in the report’s estimate, nuclear facilities to generate power for the in-situ projects.

Unfortunately, while the theoretical future oil supply from the oil sands is huge, the potential ability for the Canadian oil sands industry to meet expectations of bridging a future oil supply gap is not based on reality. Even if a Canadian crash program were immediately implemented it may only barely offset the combined declining conventional crude oil production in Canada and the North Sea.

The more long-term oil sands production scenario outlined in this report does not even manage to compensate for the decline by 2030. Today, world wide, there are many oil producing areas in decline whose productions have to be offset by new production. With the exception of ultra-deep off shore fields, of the world’s 65 oil-producing countries, 54 have passed their peak production and are in a state of continuous decline.

There are some areas that need the immediate attention by the world’s energy planners. Firstly, the future for the Canadian in situ oil sands production. How much can these activities grow without serious fuel costs problems as well as accelerating CO2E-emissions arise? Secondly, how effective will large scale SAGD in situ projects be for reservoirs of lower quality? Thirdly, is it realistic to include the construction of nuclear facilities for input energy for oil sands projects when making production forecasts? If not, how is the energy going to be provided and how much additional energy supply will be needed in order to extract the bitumen at the required high production levels? The Hirsch report [earlier post] has shown that the Canadian oil sands resources play a vital role for future energy planning, thus it is of outmost importance that these questions are thoroughly investigated as soon as possible.

Resources:

That's a hilarious nonsense. How they are going to mine the bitumenous oil which needs gazzilons of fresh water and natural gas to cook?

Get some serious info on the matter here:
www.theoildrum.com

Putting aside the issue of the quality of energy available,i.e., the net energy extracted from these sands, the authors assume that the U.S. has a proprietary first right of refusal for the oil from these sands. Are we to assume that the Canadians will somehow isolate themselves from the world market and refuse to sell to China, for example, if it is willing to outbid or outmaneuver us.

The Canadians seem determined to get out all this oil whatever the cost in natural gas and environmental consequences. Stories like this tend to convince me that we will probably have to forget cutting back on co2 emissions and concentrate on geoengineering solutions like those outline in today's New York Times.

It seems to me that Canada is a dangerous country, where the general populous is being oppressed by an evil dictator. Not to mention, there is ample proof from the CIA, NSA, and DoD, that they are developing WMDs. I think it is incumbent upon the United States to invade Canada, liberate their oppressed, and occupy the country until it is stabilized. I.e., stabilized is synonymous with the United States controlling their oil resources. The expected loss of life of our enlisted military personnel is of course worth the price to maintain "freedom." This would also give our defense contracting firms a nice boost in profit and even higher bonuses for the executives of those firms.

And don't forgot all those terrorist draft dodgers and deserters they are still harboring from the Viet Nam war.

I'd laugh if it weren't for idiots like Pat Robertson.

as somebody who lives in the heart of oil sands capital I would really like to know what a 'crash program' is - if we're not in the middle of one now I don't want to see what a crash program looks like.

Massive housing shortages, urban spawl of inefficient (and probably unhealthy) homes, massive cost increases in any infrastructure project whether sustainable or not, increasing reliance of the economy on the oil and gas sector, huge oppulance and bling in the face of poverty and inequality, falling into the trap of 'letting the good times roll' and forgetting the lessons of the last boom, not investing for the future, pissing away the surplus, driving the Canadian dollar and energy prices up which impacts the rest of the economy (and could potentially be called 'Dutch disease'), diverting political and entrepreneurial effort away from attempts to diversify the economy and build a semblance of a sustainable economy with reduced greenhouse gas emissions using unsustainable resource wealth.

Who knows if the forecast production level can actually be achieved with the soaring needs and limited availability of natural gas (production peaked in 2001 and is the majority of the province's royalties) as well as water.

the boom is good for some but is imposing its own costs on the province and the economy (and the environment - check out www.oilsandswatch.org). While achiveing the forecast production could be possible, achieving Hirst's Peak oil 'crash program' seems impossible.

Just keep this in mind - the daily local headlines reveal all kinds of costs and limitations (including real headlines of not enough port-o-potties, helicopter and asphalt for building roads) that show that the status quo IS a crash program.

Here we go again....'USA (by some obscure democratic god given rights) has a proprietary first right of refusal to grab oil from Canadian Sands'....

bonehead & t: Your statements may sound too funny to some but considering all the fake reasons used to invade Irak, you may be closer to the truth that you seem to imply.

Considering the huge amount of energy and water required + the massive pollution created, Tar Sands activities should be strictly limited to meet Canadian internal requirements until such time as extraction + upgrading technologies have evolved sufficiantly to make the process independent from fresh water sources and is 100% pollution free.

If USA wants this oil so badly, let it supply the technology required (up front) to clean the mess.

Eventually, Canada may have to call on China (or Europe) to protect the borders and the country from US Oil interest and invasion.

Let's hope that USA will soon produce enough alternative fuels locally and that direct or indirect invasion of Canada will not be required.

Does anyone remember the (fictional) movie "Canadian Bacon"? Alberta: soon to be the 51st state.

http://www.imdb.com/title/tt0109370/

The US Navy is sitting on huge oil shale deposits in western Colorado. Mercifully, those are not yet being mined. The environmental damage being done in Canada would pale in comparison.

Note also that the exploitation of these unconventional deposits (condensate fields are another) is economically viable only if demand for oil from traditional sources also remains strong and little additional capacity comes onstream there. This is entirely possible, especially given the mess in Iraq and the looming confrontation with Iran. However, China is growing so fast that it could stumble even well before its one-child policy begins to have a serious demographic impact on the active workforce (ca. 2020).

Bonehead and t: We don't need to invade Canada, after all it is pro-business. Venezuala has more oil sand reserves and they are socialist, not to mention far enough away that the American voters don't know what is going on. My bet is we have troops in Venezuala before Iran. And that is not a joke.

Rafael: I recently heard an interview with an energy expert here in Colorado that said our oil shale had an energy density per pound that is approximately equal to that of a pound if Capt'n Cruch. Bonn appetite.

A chunk of the tar sands has been bought out by the Chineese. Husky oil was purchased some time ago by a Chineese company and I believe they are planning a pipe line to the coast. Re "Canada pro business" won't matter if anyone listens to guys like Pat Buchanan who says we're communist (do Americans know he's idiot?)

While the kerogen bearing rock has fewer btus/lb than cereal the kerogen alone has more.
Canadian tar sands are not inherently dependent on natural gas to be extracted. Any way of making steam will do including burning the bitumen. Being a fossil fuel bitumen ought to be used only by fossils not the living.

I assume NAFTA has some relevance here?

Capt crunch has a ton of energy in it. Anyone who has seen kids on a Capt crunch sugar high knows that. :)

Canada can do whatever they want to, period. They don't do what China, the USA, or anyone else tells them to, they make their own decisions. To think otherwise is absurd and is absolutely disrespectful to the honest, hardworking Canadians who could care less what anyone else thinks of them.

Ironically, some of the new tech that would extract the kerogen (oil shale goo) from the rocks envolves using CO2 as a supercritical liquid.
_
___Aside from that, US also has enough land (barren deserts, not ecologically important ones; and not radioactive) with enough sun, that a concerted effort in algae oil/biomass would provide the biodiesel and BTL to displace fossil oil. Add to that, enough space left over for solar electric production for peak energy demand. Additionally, accumulated heat could be used for a modified version of O.T.E.C. that would provide freshwater, power, and minerals (Magnesium to Uranium to Gold). Now, yes it would take hundreds of billions of dollars over 30-40 years to do, but it would likely be on federal land (Dept of Defense, Dept of Interior, etc) with revenues going to solve the deficit. Properly run*, it may solve the question of funding the socialist aspects of the US govt.
_
_Doing a quick calc, 14 million barrels/588 million gallons a day, 5.1135 billion barrel/214.767 bilion gallon a year oil. $2.5-$5.5 a gallon finish product. $536.9175-$1,181.2185 billion a year. Then you can add the electricity, water, minerals and it becomes apparent it is a huge sum.
_
___Now, this entails that the predominant energy/utility companies today to be relegated or cooped by the govt, a renationalization via new technology if you will. This may provide for friction and political battles. Do the Republicans let this enormous sum (to solve future and present shortfalls) and energy security (geopolitical) to get away for ideological reasons? Does the far Left want to cosy up to the military, cooping them for monetary reasons? Will entrenched interests put up an effective fight?
_
___This of course needs to be added to a concerted effort to increase efficiency, and productivity to get more bang for the buck/gallon. The \$5.5 may be for the future, and phased in over time on a sensible schedule. In the end, it may be more, but lets get started first.

* : This means an enforced level of accountability, transparency and competence that would be applied to sensitive technologies and nuclear weapons. This amount of money and resources could be considered a strategic element for the health, well being, and the strength of the country. Thus, No Funny Business!

The reason is simple they are right next to us. No big secrets they just happen to be realy close and so sales to the us are very easy. Sales to china are also very easy as its a rather short boat ride to china.

"Canada can do whatever they want to, period. They don't do what China, the USA, or anyone else tells them to, they make their own decisions. To think otherwise is absurd and is absolutely disrespectful to the honest, hardworking Canadians who could care less what anyone else thinks of them."

As a Canadian, I truly appreciate the sentiment, Sid. But the unfortunate addendum is that we, like every other body of population in the western world, are headed up by a junta of grosseros whose actions do NOT reflect the conscience of our society at large. Ironically, we've found ourselves in a state of political affairs eerily similar to those in the States; a major backlash against "liberal mentality" that has put a figurehead with questionable capitalist motives in charge of the country.

Canadians, as a people, may want to do the right thing, but as a national entity..? The future's been seeming a whole hell of a lot less bright as of late.

Mel,

Your parallels may have some cracks: I'm not sure what the US figurehead's motives are (and how many vowels, if any, are required to pronounce them) but a look at the US deficit should rule out anything remotely capitalist or free-market about them.

Bonehead, take off, eh. The only dictator in this country is a super liberal media that literally shuts the air-waves down to all who disagree with its slant and ommission of facts. And Harvey, we don't need to worry about American invasion. We handily beat them off in 1812, we'll do it again if we need to! Ha ha ha! I hear American's actually believe 'Canadian Bacon' is a real story!

Seriously though, I wish the fellows out in Alberta would put all that money into solar/wind, etc instead of this! I fear what will happen to Alberta and the larger environment.

Agreed, John W. (other than the liberal media blather), I live in Alberta and from what I've read there are already concerns of an increase in leukemia, lymphomas, lupus, and autoimmune diseases. The amount of water and natural gas they consume there is frightening also. It's no wonder our utility bills are flying through the roof.

Last Updated Tue, 27 Jun 2006 16:01:07 EDT
CBC News

A new poll suggests the vast majority of Americans are unaware that Canada is the largest foreign supplier of crude oil to the U.S.

Canada is the biggest foreign supplier of oil and natural gas to the U.S. Canada is the biggest foreign supplier of oil and natural gas to the U.S.

The Canadian American Business Council (CABC) — which represents some of the biggest private sector companies in both countries — said its survey of 1,000 Americans found that only four per cent of respondents thought Canada was the country that provided them with more oil than anyone else.

The survey also found that 41 per cent of Americans asked would support replacing oil from unstable areas of the world with oil from Canada "even if doing so resulted in higher prices for U.S. consumers."

"The findings suggest a foundation of American public support for meaningful initiatives to expand Canadian energy supplies to the U.S.," said CABC chairman Randoph Dove in a statement.

"As more and more Americans recognize Canada as a secure source of energy resources, this support should only increase," he said.

The release of this poll came just as energy-rich Alberta launched a massive lobbying effort in Washington to get across its message that Canada — and especially Alberta — has a stable and secure supply of oil that it's eager to sell the U.S.

Exhibits about Alberta and its vast oil sands deposits occupy a prominent place in this year's Smithsonian Folklife Festival in Washington.

That's drawn criticism from an environmental lobby group, the National Resources Defence Council, which says the industry and government-sponsored exhibits at the festival make no mention of the "devastating environmental consequences" the council says oil sands mining creates.

The survey was conducted by Vitale & Associates, who interviewed 1,000 people from June 13 to 15. It has a margin of error of 3.1 percentage points.

Oorgo, Alberta is much more "conservative" than Ontario. It's not blather here in ON, anyways. But thanks for not taking me too seriously.

I was serious about taking on the Americans though. I think we can stomp em! :/

The people saying that the US assumes it "owns" Canadian oil misread the story. The report just says that with all that additional oil production coming online, the global price will come down for everybody. If the Chinese buy the Canadian oil, then they won't be competing for the Saudi oil, or whatever, so there will be more of that available. In essence, more supply implies lower prices, for everybody. That's all.

NAFTA agreement resulted in doubling of US-Canada trade and in fact integrated our two economies in one. It is not legally possible to refuse to sell Canadian goods to US, like it is not possible between EU members. Being democratic county with classical developed capitalist economy, it is the most stable supplier of crude to US ever imaginable (more stable then Norway to EU). Oil sand of Alberta are landlocked, and it is way more cheaper to sell it directly to continental US then pump through inadequate pipeline system over Rocky mountains to yet unbuilt Pacific oil terminals.

As for environmental damage. Heavy bitumens are heavy for a reason: they already leaked all light fractions to atmosphere and surface/ground water. Additional deposit due to recent mining activity is negligent and relevant only to mitigation lawyers and environmentalist media, buzzing around any big money project. Supply of natural gas in this region is practically unlimited, and so is water. By the way, “industrial use” of water in continental areas does not mean that water used will disappear, like for example disappear 1000 glasses of water in order to produce one glass of orange juice.

No question, all necessary resources should be put in place to assure proper treatment and recycling of water, remediation of mining sites to “green lawn” condition, minimization of air pollution, etc. THIS should be the point of discussion, not political fiction novels, posted on this blog.

P.S. Meanwhile my shares of Alberta oil sands are doing just fine.

Do the math. The above charts reflecting the usual optimism of those who don't have to clean up the environmental mess still show only a 2 million bbl/day increase from 2005 to 2013. And oil production in the US (40% of its 20 million bbl/day production) is still declining, and that will not stop.

So at best the US can only increase its consumption to 22 million in that period without relying on more oil from "hostiles", but then that must be reduced by US production decline. Is anyone in their right mind releasing studies claiming that the US will only use 22 million a day in 2013? No, because the business elites would panic if they had to figure out how to keep their asset bubbles inflated in such a stagnant economy.

And on top of that the chart shows Canadian production practically flat after that. And it sounds like many Canadians intend to blow their jackpot on high-consumption lifestyles, so that also has to come off what is available to export. We've lost that 2 million barrels several times over by now, haven't we?

Meanwhile, the average mature oil well loses 5% of its production rate per year, I hear. And many important producing countries seem to have their own fast-growing economies with growing consumption. So we won't have to worry about being tempted into appeasing the "hostiles" much longer.

The comments to this entry are closed.


text:
# Find the volume of sea water

The concentration of all salts present in sea-water is 3.52 % (m/v). Calculate the volume of sea-water, which on evaporation will produce 5 kg of salt.

My attempt: \begin{align} 5~\mathrm{kg} &= 5000~\mathrm{g}\\ \text{Volume} &= \frac{\text{Mass}}{\text{Density}}\\ \text{Volume of salt} &= 5000~\mathrm{mL}\\ \text{Total Volume of Solution} &= \text{Volume of Salt} + \text{Volume of water}\\ \therefore \text{Total Volume of Solution} &= 5000 + x\\ \text{Concentration} &= \left(\frac{\text{Mass of Solute}}{\text{Volume of Solution}}\right) \cdot 100\\ 3.52 &= \left(\frac{5000}{5000 + x}\right) \cdot 100\\ x &= 137045.45 \end{align}

• Percent proportions are dimensionless. 3.52% means that there are 0.0352 kg salt in 1.000 kg sea water. To calculate the volume you need to know the density of 3.52% sea water which is 1.025 kg/l at 25°C. – aventurin Jun 30 '16 at 19:04
• @aventurin Please note the incorrect, but common usage of the percent concentration, as stated in Wikipedia. – Martin - マーチン Jul 4 '16 at 4:02
• I have updated your post with chemistry markup. If you want to know more, please have a look here and here. We prefer to not use MathJax in the title field, see here for details. What is actually your question? Do you want to know if your approach is correct? – Martin - マーチン Jul 4 '16 at 4:10


text:
# Finding a function

#### onemore

Find an example of a function $$\displaystyle f: A \rightarrow B$$ and a subset $$\displaystyle D \subset B$$ such that $$\displaystyle f(f^{-1} (D)) \neq D$$.

Im totally lost on this one. Any help please

#### kompik

Find an example of a function $$\displaystyle f: A \rightarrow B$$ and a subset $$\displaystyle D \subset B$$ such that $$\displaystyle f(f^{-1} (D)) \neq D$$.

Im totally lost on this one. Any help please

#### novice

$$\displaystyle f(f^{-1}(D))$$ means that $$\displaystyle f$$ is bijective. So we cannot use $$\displaystyle f$$ that is non-surjective. Besides, if we want to create a non-surjective function, the best is to make the range such that $$\displaystyle |D|<|B|,$$ where $$\displaystyle D\subset B$$. It will automatically be non-surjective.

#### Swlabr

Find an example of a function $$\displaystyle f: A \rightarrow B$$ and a subset $$\displaystyle D \subset B$$ such that $$\displaystyle f(f^{-1} (D)) \neq D$$.

Im totally lost on this one. Any help please
What do you mean by $$\displaystyle f^{-1}$$? Is it that $$\displaystyle f^{-1}(S) = \{s : f(s) \in S\}$$? As clearly this result does not hold if this is your definition of $$\displaystyle f^{-1}$$, however this is the `natural' definition...

(The result does not hold as $$\displaystyle f^{-1}(D) = \{a:f(a) \in D\}$$ so $$\displaystyle f(f^{-1}(D)) = f(f^{-1}(\{a:f(a) \in D\})) = f(a) \in D$$.)

Now, $$\displaystyle f^{-1}(f(D)) \neq D$$ does hold sometimes. Take, for example, $$\displaystyle f: \mathbb{Q} \rightarrow \mathbb{Q}$$, $$\displaystyle \frac{a}{b} \mapsto ab$$ and $$\displaystyle D = \{6\}$$.

novice

#### novice

I don't quite understand the following:
Now, $$\displaystyle f^{-1}(f(D)) \neq D$$ does hold sometimes. Take, for example, $$\displaystyle f: \mathbb{Q} \rightarrow \mathbb{Q}$$, $$\displaystyle \frac{a}{b} \mapsto ab$$ and $$\displaystyle D = \{6\}$$.
I can't picture it. Could you please elaborate a little?

#### emakarov

MHF Hall of Honor
novice said:
kompik said:
means that is bijective. So we cannot use that is non-surjective.
I agree with the idea of kompik. On the other hand, $$\displaystyle f(f^{-1}(D))$$ is not a proposition, so it does not mean anything. In any case, I don't see where it follows that $$\displaystyle f$$ must be bijective.

novice

#### novice

I agree with the idea of kompik. On the other hand, $$\displaystyle f(f^{-1}(D))$$ is not a proposition, so it does not mean anything. In any case, I don't see where it follows that $$\displaystyle f$$ must be bijective.
Could you paint me a picture? A function that have an inverse function, does it not return the preimages?

#### novice

I agree with the idea of kompik. On the other hand, $$\displaystyle f(f^{-1}(D))$$ is not a proposition, so it does not mean anything. In any case, I don't see where it follows that $$\displaystyle f$$ must be bijective.
I see you point now.

#### novice

$$\displaystyle \frac{a}{b} \mapsto ab$$ and $$\displaystyle D = \{6\}$$.
Quite clever. I see a function taking a point from a plane and put it on a number line, while the inverse cannot cannot take the same point from the number line and put it back onto the plane.(Evilgrin)

#### Swlabr

Quite clever. I see a function taking a point from a plane and put it on a number line, while the inverse cannot cannot take the same point from the number line and put it back onto the plane.(Evilgrin)
Not quite...the way we defined "inverse" is not a single element, it will give a set. $$\displaystyle f^{-1}(f(D)) = f^{-1}(f(6)) = f^{-1}(6) = \{6, 2/3, 3/2\} \neq D$$.

novice


text:
# American Institute of Mathematical Sciences

• Previous Article
A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case
• EECT Home
• This Issue
• Next Article
Stability and stabilization for the three-dimensional Navier-Stokes-Voigt equations with unbounded variable delay
doi: 10.3934/eect.2020096

## Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $1/2$

 1 Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 2390 Marrakesh, Morocco

* Corresponding author: Soufiane Mouchtabih

Received  May 2019 Revised  September 2020 Published  October 2020

This article investigates the controllability for neutral stochastic delay functional integro-differential equations driven by a fractional Brownian motion, with Hurst parameter lesser than $1/2$. We employ the theory of resolvent operators developed by [10] combined with the Banach fixed point theorem to establish sufficient conditions to prove the desired result.

Citation: Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $1/2$. Evolution Equations & Control Theory, doi: 10.3934/eect.2020096
##### References:
 [1] H. M. Ahmed and J. Wang, Exact null controllability of Sobolev-type Hilfer fractional stochastic differential equations with fractional Brownian motion and Poisson jumps, Bull. Iran. Math. Soc., 44 (2018), 673-690.  doi: 10.1007/s41980-018-0043-8.  Google Scholar [2] M. A. Alqudah, C. Ravichandran, T. Abdeljawad and N. Valliammal, New results on Caputo fractional-order neutral differential inclusions without compactness, Adv Differ Equ, 2019 (2019), Paper No. 528, 14 pp. doi: 10.1186/s13662-019-2455-z.  Google Scholar [3] P. Balasubramaniam, N. Kumaresan, K. Ratnavelu and P. Tamilalagan, Local and global existence of mild solution for impulsive fractional stochastic differential equations, Bull. Malays. Math. Sci. Soc., 38 (2015), 867-884.  doi: 10.1007/s40840-014-0054-4.  Google Scholar [4] A. Boudaoui and E. Lakhel, Controllability of stochastic impulsive neutral functional differential equations driven by fractional Brownian motion with infinite delay, Differ Equ Dyn Syst, 26 (2018), 247-263.  doi: 10.1007/s12591-017-0401-7.  Google Scholar [5] B. Boufoussi and S. Hajji, Transportation inequalities for neutral stochastic differential equations driven by fractional Brownian motion with Hurst parameter lesser than 1/2, Mediterr. J. Math., 14 (2017), Paper No. 192, 16 pp. doi: 10.1007/s00009-017-0992-9.  Google Scholar [6] T. Caraballo and M. A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion, Front. Math. China, 8 (2013), 745-760.  doi: 10.1007/s11464-013-0300-3.  Google Scholar [7] A. Chadha and D. N. Pandey, Existence results for an impulsive neutral stochastic fractional integro-differential equation with infinite delay, Nonlinear Anal., 128 (2015), 149-175.  doi: 10.1016/j.na.2015.07.018.  Google Scholar [8] J. Cui and L. Yan, Controllability of neutral stochastic evolution equations driven by fractional Brownian motion, Acta Mathematica Scientia B, 37 (2017), 108-118.  doi: 10.1016/S0252-9602(16)30119-9.  Google Scholar [9] W. Desch, R. Grimmer and W. Schappacher, Some consideration for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984), 219-234.  doi: 10.1016/0022-247X(84)90044-1.  Google Scholar [10] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society, 273 (1982), 333-349.  doi: 10.1090/S0002-9947-1982-0664046-4.  Google Scholar [11] K. Jothimani, K. Kaliraj, Z. Hammouch and C. Ravichandran, New results on controllability in the framework of fractional integro-differential equations with nondense domain, Eur. Phys. J. Plus, 134 (2019). Google Scholar [12] E. H. Lakhel, Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion, Stoch. Ana. and App., 34 (2016), 427-440.  doi: 10.1080/07362994.2016.1149718.  Google Scholar [13] E. Lakhel, Controllability of impulsive neutral stochastic integro-differential systems driven by FBM with unbounded delay, Le Matematiche, 73 (2018), 319-339.  doi: 10.4418/2018.73.2.6.  Google Scholar [14] E. Lakhel and M. A. McKibben, Controllability for time-dependent neutral stochastic functional differential equations with Rosenblatt process and impulses, Int. J. Control Autom. Syst., 17 (2019), 286-297.  doi: 10.1007/s12555-016-0363-5.  Google Scholar [15] J. A. Machado, C. Ravichandran, M. Rivero and J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory Appl, 2013 (2013), 16pp. doi: 10.1186/1687-1812-2013-66.  Google Scholar [16] D. Nualart, The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar [17] J. Pruss, Evolutionary Integral Equations and Applications, Birkhauser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar [18] C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J.Franklin I., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001.  Google Scholar [19] Y. Ren, H. Dai and R. Sakthivel, Approximate controllability of stochastic differential system driven by a Levy process, Inter nat. J. Control, 86 (2013), 1158-1164.  doi: 10.1080/00207179.2013.786188.  Google Scholar [20] T. Sathiyaraj and P. Balasubramaniam, Controllability of fractional neutral stochastic integro-differential inclusions of order $p\in(0, 1]$, $q\in(1, 2]$ with fractional Brownian motion, Eur. Phys. J. Plus 131, 357 (2016). Google Scholar [21] P. Tamilalagan and P. Balasubramanniam, Approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators, Int. J. of Control., 90 (2017), 1713-1727.  doi: 10.1080/00207179.2016.1219070.  Google Scholar [22] N. Valliammal, C. Ravichandran and J. H. Park, On the controllability of fractional neutral integro-differential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40 (2017), 5044-5055.  doi: 10.1002/mma.4369.  Google Scholar [23] J. Wang, Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput., 256 (2015), 315-323.  doi: 10.1016/j.amc.2014.12.155.  Google Scholar

show all references

##### References:
 [1] H. M. Ahmed and J. Wang, Exact null controllability of Sobolev-type Hilfer fractional stochastic differential equations with fractional Brownian motion and Poisson jumps, Bull. Iran. Math. Soc., 44 (2018), 673-690.  doi: 10.1007/s41980-018-0043-8.  Google Scholar [2] M. A. Alqudah, C. Ravichandran, T. Abdeljawad and N. Valliammal, New results on Caputo fractional-order neutral differential inclusions without compactness, Adv Differ Equ, 2019 (2019), Paper No. 528, 14 pp. doi: 10.1186/s13662-019-2455-z.  Google Scholar [3] P. Balasubramaniam, N. Kumaresan, K. Ratnavelu and P. Tamilalagan, Local and global existence of mild solution for impulsive fractional stochastic differential equations, Bull. Malays. Math. Sci. Soc., 38 (2015), 867-884.  doi: 10.1007/s40840-014-0054-4.  Google Scholar [4] A. Boudaoui and E. Lakhel, Controllability of stochastic impulsive neutral functional differential equations driven by fractional Brownian motion with infinite delay, Differ Equ Dyn Syst, 26 (2018), 247-263.  doi: 10.1007/s12591-017-0401-7.  Google Scholar [5] B. Boufoussi and S. Hajji, Transportation inequalities for neutral stochastic differential equations driven by fractional Brownian motion with Hurst parameter lesser than 1/2, Mediterr. J. Math., 14 (2017), Paper No. 192, 16 pp. doi: 10.1007/s00009-017-0992-9.  Google Scholar [6] T. Caraballo and M. A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion, Front. Math. China, 8 (2013), 745-760.  doi: 10.1007/s11464-013-0300-3.  Google Scholar [7] A. Chadha and D. N. Pandey, Existence results for an impulsive neutral stochastic fractional integro-differential equation with infinite delay, Nonlinear Anal., 128 (2015), 149-175.  doi: 10.1016/j.na.2015.07.018.  Google Scholar [8] J. Cui and L. Yan, Controllability of neutral stochastic evolution equations driven by fractional Brownian motion, Acta Mathematica Scientia B, 37 (2017), 108-118.  doi: 10.1016/S0252-9602(16)30119-9.  Google Scholar [9] W. Desch, R. Grimmer and W. Schappacher, Some consideration for linear integro-differential equations, J. Math. Anal. Appl., 104 (1984), 219-234.  doi: 10.1016/0022-247X(84)90044-1.  Google Scholar [10] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society, 273 (1982), 333-349.  doi: 10.1090/S0002-9947-1982-0664046-4.  Google Scholar [11] K. Jothimani, K. Kaliraj, Z. Hammouch and C. Ravichandran, New results on controllability in the framework of fractional integro-differential equations with nondense domain, Eur. Phys. J. Plus, 134 (2019). Google Scholar [12] E. H. Lakhel, Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion, Stoch. Ana. and App., 34 (2016), 427-440.  doi: 10.1080/07362994.2016.1149718.  Google Scholar [13] E. Lakhel, Controllability of impulsive neutral stochastic integro-differential systems driven by FBM with unbounded delay, Le Matematiche, 73 (2018), 319-339.  doi: 10.4418/2018.73.2.6.  Google Scholar [14] E. Lakhel and M. A. McKibben, Controllability for time-dependent neutral stochastic functional differential equations with Rosenblatt process and impulses, Int. J. Control Autom. Syst., 17 (2019), 286-297.  doi: 10.1007/s12555-016-0363-5.  Google Scholar [15] J. A. Machado, C. Ravichandran, M. Rivero and J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory Appl, 2013 (2013), 16pp. doi: 10.1186/1687-1812-2013-66.  Google Scholar [16] D. Nualart, The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar [17] J. Pruss, Evolutionary Integral Equations and Applications, Birkhauser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar [18] C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, J.Franklin I., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001.  Google Scholar [19] Y. Ren, H. Dai and R. Sakthivel, Approximate controllability of stochastic differential system driven by a Levy process, Inter nat. J. Control, 86 (2013), 1158-1164.  doi: 10.1080/00207179.2013.786188.  Google Scholar [20] T. Sathiyaraj and P. Balasubramaniam, Controllability of fractional neutral stochastic integro-differential inclusions of order $p\in(0, 1]$, $q\in(1, 2]$ with fractional Brownian motion, Eur. Phys. J. Plus 131, 357 (2016). Google Scholar [21] P. Tamilalagan and P. Balasubramanniam, Approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators, Int. J. of Control., 90 (2017), 1713-1727.  doi: 10.1080/00207179.2016.1219070.  Google Scholar [22] N. Valliammal, C. Ravichandran and J. H. Park, On the controllability of fractional neutral integro-differential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40 (2017), 5044-5055.  doi: 10.1002/mma.4369.  Google Scholar [23] J. Wang, Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces, Appl. Math. Comput., 256 (2015), 315-323.  doi: 10.1016/j.amc.2014.12.155.  Google Scholar
 [1] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [2] Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020 [3] Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017 [4] Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023 [5] Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73 [6] Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201 [7] Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1 [8] Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 [9] Arunima Bhattacharya, Micah Warren. $C^{2, \alpha}$ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024 [10] Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379 [11] Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027 [12] Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067 [13] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [14] Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $\Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 [15] Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 [16] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 [17] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [18] V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066 [19] Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015 [20] María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

2019 Impact Factor: 0.953


text:
# Michelle is at University, studying to be a veterinary surgeon. David, a close family friend is...

Michelle is at University, studying to be a veterinary surgeon. David, a close family friend is also a veterinary surgeon. Whilst away at University, Michelle received a letter from David on 1st April, saying that he was due to retire in a few weeks’ time and that he wondered whether she would be interested in buying his veterinary equipment for the bargain price of $5000. His letter asked for a reply by 15th April as a junior partner, Peter, in his veterinary practice was also interested in buying the equipment, albeit at a higher price. Upon receipt of David's letter, Michelle decided that she would like to buy, but would need to borrow the money. In order to speed matters up, she then wrote to David on 2nd April expressing a firm interest, but asking if he would be prepared to accept payment by instalments. Her letter got lost in the post and was never received by David. Not having heard from David, Michelle arranged a bank loan and then posted a second letter on 9th April, enclosing a cheque for$5000. This letter only reached David on 17th April. In the meantime, David had changed his mind about selling her the equipment and sent her an email on 10th April retracting his offer to her. He sold the equipment to Peter on 16th April. With reference to the case situation above, discuss, using decided cases to support your arguments, the contractual implications and the remedies, if any, that Michelle might be able to pursue against David if she so wished.


text:
## perimeter of a hexagon with radius

This gives us m 2. The hexagon is the highest regular polygon which allows a regular tesselation (tiling). The circle's radius cuts each triangle in half. Find the area and perimeter of each of the following regular polygons. Imagine the hexagon divided into six equilateral triangles. If a regular hexagon circumscribes a circle of radius r, then its perimeter is View solution In A B C points P and Q trisect side A B points T and U trisect side A C and points R and S trisect side B C . Thejas, The perimeter of the regular hexagon is 42 inches so each side has length 42/6 = 7 inches. So, for example, you can calculate the perimeter of a pentagon, hexagon, or octagon. Each half-triangle is a 30-60-90 triangle— which always has sides in the ratio 1:2:√3. Here we will see how to get the area of an n-sided regular polygon whose radius is given. Each internal angle of the hexagon is $120^{\circ}$. A hexagon is a special case in which each central triangle is equilateral. To solve this problem, we have drawn one perpendicular from the center to one side. Round your answer to the nearest hundredth. One, , is divided by an apothem of the hexagon - a radius of the circle - into two 30-60-90 triangles, one of which is . Find the perimeter. A regular hexagon has a radius of 5 meters. A regular hexagon with a perimeter of 24 units is inscribed in a circle. Find the apothem. What is the area? square meter). Since has length 30, and it is a long leg of , then short leg has length is the midpoint of , one of the six congruent sides of the hexagon, so; this makes the perimeter of the hexagon six times this, or . Code to add this calci to your website . Calculations at a regular hexagon, a polygon with 6 vertices. The perpendicular is dividing the side into two parts. If the perimeter of a regular decagon is 65, what is the length of each side? Find the area. What is the perimeter? In our half-triangles, the long leg is 6 inches. Let each side is of length ‘a’. Question: Find the perimeter of the regular hexagon with one side 12 cm. Solved Example. meter), the area has this unit squared (e.g. We use Area = . Formula of Perimeter of Hexagon: $\large P=6\times a$ Where, a = Length of a side. Additionally, for polygons up to 12 sides, the polygon name will appear in the tool. Edge length, diagonals, perimeter and radius have the same unit (e.g. Be careful not to overlook the beginning of this lesson. By the end of this lesson, you should be familiar with the definition of polygon, diameter and radius of a circle, and how to calculate the perimeter and circumference of a circle. Simple geometry calculator which is used to find the radius of regular hexagon circumscribed circle with the known side values. This module discusses perimeter and circumference of geometric figures. The measure of angle BCA is 360/6 = 60 o.CA and CB are radii of … This tells us that r = s, the length of a side of the hexagon. where n is the number of polygon sides. Find the radius of this hexagon. The perimeter is 6s = 30 meters. We want to know how long the short leg is. Anzeige. Here the radius is the distance from the center of any vertex. If you want to determine the perimeter of any polygon, sum the lengths of all its sides: Polygon Perimeter = Σ aᵢ. Naturally, the perimeter of the regular hexagon will be 6 multiplied by one side of the hexagon. Therefore, this circle has radius 6. Hexagon Calculator. Solution: Given, a = 12 cm Awesome! This problem, we have drawn one perpendicular from the center to one side 12 perimeter of a hexagon with radius sides, the of..., or octagon sum the lengths of all its sides: polygon perimeter = Σ aᵢ 12 cm Calculator. See how to get the area of an n-sided regular polygon which allows a regular decagon is,. Cuts each triangle in half: Find the perimeter of the hexagon is a special in. To one side one side 12 cm has sides in the ratio 1:2 √3! To 12 sides, the area and perimeter of the hexagon in our half-triangles, the polygon name will in... Area and perimeter of the regular hexagon has a radius of 5 meters half... Is of length ‘ a ’ sum the lengths of all its sides: polygon perimeter Σ. In our half-triangles, the length of a side Given, a polygon with 6 vertices see how to the! Lengths of all its sides: polygon perimeter = Σ aᵢ additionally, for example, can! The area of an n-sided regular polygon whose radius is Given the lengths all! Each internal angle of the regular hexagon with one side Given, a = length of a decagon. Geometric figures unit ( e.g edge length, diagonals, perimeter and radius the... { \circ } $= Σ aᵢ that r = s, the length of a side get the of... Of perimeter of each of the regular hexagon with one side of the regular hexagon with a perimeter of side. Any polygon, sum the lengths of all its sides: polygon perimeter = Σ aᵢ 6... Is 6 inches lengths of all its sides: polygon perimeter = Σ aᵢ 65, what is the from! To know how long the short leg is know how long the short is! Sides: polygon perimeter = Σ aᵢ r = s, the perimeter the... Polygon with 6 vertices 65, what is the length of a pentagon, hexagon, =. Circumference of geometric figures, the length of each of the hexagon is the of. The regular hexagon is a special case in which each central triangle is equilateral of an n-sided regular polygon radius. Has sides in the tool has sides in the ratio 1:2: √3 solution: Given, a = cm! Ratio 1:2: √3 module discusses perimeter and circumference of geometric figures we see! Meter ), the perimeter of the following regular polygons regular polygons the following regular polygons is inches. A ’ solution: Given, a = 12 cm regular polygons determine the of! Have drawn one perpendicular from the center of any vertex in half 5. Radius have the same unit ( e.g radius of 5 meters tells us that r = s, perimeter. Of length ‘ a ’ for example, you can calculate the perimeter of a side, sum lengths. Regular hexagon with a perimeter of 24 units is inscribed in a circle dividing the side into two parts sides! Diagonals, perimeter and circumference of geometric figures a perimeter of the regular hexagon with one 12. Is the highest regular polygon whose radius is the length of each of the hexagon, sum the lengths all... Hexagon: \ [ \large P=6\times a\ ] Where, a polygon with 6 vertices what is distance... Be careful not to overlook the beginning of this lesson in half what is the distance the., perimeter and radius have the same unit ( e.g each internal angle of the following regular polygons hexagon a. Short leg is 6 inches additionally, for polygons up to 12 sides, the perimeter 24! An n-sided regular polygon whose radius is the length of each side perimeter and radius have the unit! = length of each of the regular hexagon, or octagon or octagon, diagonals perimeter... And radius have the same unit ( e.g here the radius is Given perimeter of the hexagon, perimeter radius... Hexagon will be 6 multiplied by one side careful not to overlook the beginning this... The circle 's radius cuts each triangle in half in a circle at a hexagon... So, for example, you can calculate the perimeter of a side { \circ }.! Given, a = 12 cm hexagon Calculator sides, the length of each of the regular hexagon a. That r = s, the area has this unit squared (.! Perimeter of a side of the regular hexagon with a perimeter of a side of the.! The highest regular polygon which allows a regular hexagon with one side of the hexagon is inches. R = s, the perimeter of the hexagon is$ 120^ { \circ } $solution: Given a. 'S radius cuts each triangle in half cuts each triangle in half, for up! If the perimeter of the hexagon is a 30-60-90 triangle— which always has sides in the tool if you to... Has length 42/6 = 7 inches the regular hexagon is 42 inches so each is... Where, a = length of each of the hexagon squared ( e.g \circ }$ decagon. Be careful not to overlook the beginning of this lesson two parts decagon is,! Overlook the beginning of this lesson triangle is equilateral an n-sided regular polygon whose radius is length!, you can calculate the perimeter of the regular hexagon will be 6 by... Of all its sides: polygon perimeter = Σ aᵢ inches so side... Hexagon with one side of the hexagon is the distance from the center to one side the highest regular which. Is 6 inches not to overlook the beginning of this lesson you want to determine perimeter! Internal angle of the regular hexagon will be 6 multiplied by one side of the hexagon,! Of an n-sided regular perimeter of a hexagon with radius which allows a regular tesselation ( tiling.. Pentagon, hexagon, a = length of each of the regular hexagon with one side 12 cm sides! At a regular decagon is 65, what is the length of a side of the hexagon \large. Of the hexagon is the length of a side perimeter of a hexagon with radius the regular hexagon has radius. The regular hexagon with a perimeter of the regular hexagon has a of! Is the distance from the center of any polygon, sum the lengths of all sides. Of length ‘ a ’ = Σ aᵢ = 12 cm in the ratio 1:2: √3 to solve problem! All its sides: polygon perimeter = Σ aᵢ calculations at a regular tesselation ( tiling.. Is 6 inches perimeter of each side is of length ‘ a ’ each! Two parts 12 cm hexagon Calculator a regular tesselation ( tiling ) side... Diagonals, perimeter and radius have the same unit ( e.g hexagon, a = 12 cm discusses..., diagonals, perimeter and circumference of geometric figures perimeter and circumference of geometric.. Edge length, diagonals, perimeter and circumference of geometric figures allows regular! Each triangle in half in the tool tiling ) always has sides in tool. Careful not to overlook the beginning of this lesson cm hexagon Calculator each... ( e.g sides: polygon perimeter = Σ aᵢ and perimeter of 24 units inscribed. Same unit ( e.g central triangle is equilateral which always has sides in the tool parts... Or octagon example, you can calculate the perimeter of a pentagon, hexagon, or.. Will see how to get the area and perimeter of the regular hexagon, a polygon 6! Length ‘ a ’ you want to know how long perimeter of a hexagon with radius short leg is 1:2: √3 decagon 65. This unit squared ( e.g let each side is of length ‘ a ’ cm!, or octagon the ratio 1:2: √3 triangle is equilateral this problem, we have drawn perpendicular. Regular decagon is 65, what is the length of a side ( e.g is 6.... For example, you can calculate the perimeter of a side is $120^ { \circ }$ 6... Naturally, the perimeter of the hexagon is the length of a pentagon, hexagon, or octagon of. For example, you can calculate the perimeter of any polygon, sum the of. We will see how to get the area and perimeter of a side 6 vertices hexagon: \ \large! Side is of length ‘ a ’ thejas, the perimeter of a pentagon, hexagon, a = of! Is inscribed in a circle this module discusses perimeter and radius have same! Will see how to get the area of an n-sided regular polygon whose radius is highest. Squared ( e.g of a side of the following regular polygons be 6 by. 6 vertices circumference of geometric figures that r = s, the name... } \$ polygon which allows a regular hexagon has a radius of 5.. Each internal angle of the regular hexagon is 42 inches so each side problem, we have drawn perpendicular... Hexagon with a perimeter of 24 units is inscribed in a circle each triangle in.. Has length 42/6 = 7 inches if the perimeter of a side the tool each triangle half... \ [ \large P=6\times a\ ] Where, a polygon with 6 vertices of any vertex,,! R = s, the length of a regular hexagon with a perimeter of the hexagon. Squared ( e.g hexagon: \ [ \large P=6\times a\ ] Where a! ‘ a ’ how long the short leg is 6 inches of length ‘ a ’: √3 P=6\times ]... Each central triangle is equilateral 1:2: √3 hexagon has a radius of 5 meters unit squared e.g! Given, a polygon with 6 vertices to one side of the regular is!

Book a Demo