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Theory of the Riemann Zeta and Allied Functions | EMS Press
Martin N. Huxley
Nihon University of Science and Technology, Tokyo, Japan
This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a hike in the mountains and piano recitals by Peter Elliott and Valentin Blomer. \smallskip Since the times of Dirichlet and Riemann, zeta functions and Dirichlet series have played a central role in analytic number theory, and in recent times connections have been found with other areas of mathematics and its applications, including theoretical physics. The talks represented the various aspects of the theory of zeta functions. In particular, the following topics were discussed, among others: {\small \begin{list}{(\hspkl\roman{anum}\hspkl)}{ \leftmargin4mm\labelsep2mm\listparindent0mm \labelwidth3mm\itemsep1mm\topsep1mm \rightmargin0mm\usecounter{anum}\parsep0mm\partopsep0mm} \item[
\bullet
] Connections of classical zeta functions with automorphic functions and spectral theory. \item[
\bullet
] Estimates of the size of zeta and
L\!
-functions, both at individual points and in mean value. \item[
\bullet
] Problems concerning the zeros of zeta functions (Riemann's hypothesis and other questions such as the Siegel zero and the distribution of zeros of Epstein's zeta functions). \item[
\bullet
] Applications of zeta and
L\!
-functions to arithmetic functions, and the duality between arithmetic and analysis. \item[
\bullet
] Random Matrix Theory, which shed new light light on mean value estimates and their consequences. \item[
\bullet
] Numerical calculations related to the zeros of Riemann's zeta function and other computational projects. \elist }
Martin N. Huxley, Matti Jutila, Yoichi Motohashi, Samuel J. Patterson, Theory of the Riemann Zeta and Allied Functions. Oberwolfach Rep. 1 (2004), no. 4, pp. 2419–2490
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A Coincidence Best Proximity Point Problem in G-Metric Spaces
2015 A Coincidence Best Proximity Point Problem in
G
M. Abbas, A. Hussain, P. Kumam
The aim of this paper is to initiate the study of coincidence best proximity point problem in the setup of generalized metric spaces. Some results dealing with existence and uniqueness of a coincidence best proximity point of mappings satisfying certain contractive conditions in such spaces are obtained. An example is provided to support the result proved herein. Our results generalize, extend, and unify various results in the existing literature.
M. Abbas. A. Hussain. P. Kumam. "A Coincidence Best Proximity Point Problem in
G
-Metric Spaces." Abstr. Appl. Anal. 2015 1 - 12, 2015. https://doi.org/10.1155/2015/243753
M. Abbas, A. Hussain, P. Kumam "A Coincidence Best Proximity Point Problem in
G
-Metric Spaces," Abstract and Applied Analysis, Abstr. Appl. Anal. 2015(none), 1-12, (2015)
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Mathematics - Aloe Protocol
A summary of the math involved in building Blend
Portion of Funds in Uniswap
Aloe Blend is designed to mimic the payoff curve of Uniswap V2, which means matching its liquidity density from
0 \rightarrow \infty
. But what exactly is that density?
To keep things simple, let's look at the no-fee scenario for a Uniswap V2 pool containing assets X and Y. At price
p_l=\frac{y_l}{x_l}
we have the constant product formula
x_l y_l = k
. Imagine someone trades
\Delta y
units of Y for
\Delta x
units of X. We now have
x_u y_u = k
and a new price
p_u=\frac{y_u}{x_u}
x_u=x_l-\Delta x
y_u=y_l+\Delta y
. Isolating
\Delta x
\Delta y
, we get the following:
\Delta x=\sqrt{k} \left(\frac{1}{\sqrt{p_l}}-\frac{1}{\sqrt{p_u}}\right)
\Delta y=\sqrt{k} \left(\sqrt{p_u}-\sqrt{p_l}\right)
Key Insight: In this context, the liquidity between
p_l
p_u
\Delta x
\Delta y
, depending on which price is considered "current." If
p \ge p_u
, then liquidity is
\Delta y
units of Y. If
p \le p_l
\Delta x
These results can be massaged to match equations 6.29 and 6.30 in the Uniswap V3 Whitepaper with the slight modification that
\Delta L=\sqrt{k}
. That's all it takes to mimic V2 on V3! This is all the background we need to implement the basics of Aloe Blend.
Each Blend Vault has some inventory,
\left( r_x,r_y \right)
, but only part of that inventory is deposited to Uniswap V3:
\left( \Delta x, \Delta y \right)
. Since we want the vault to behave like Uniswap V2, we set
r_x r_y = k
p=\frac{r_y}{r_x}=1.0001^a
p_l
p_u
is arbitrary as long as
p_l \lt p \lt p_u
, but to make the math work out nicely we use
\left(p_l,p_u \right)=1.0001^{a \pm b}
. Combining these expressions with the previous equations yields the following:
\Delta x=r_x \left(1 - 1.0001^{-b/2} \right)
\Delta y=r_ y\left(1 - 1.0001^{-b/2} \right)
/// @dev Computes amounts that should be placed in primary Uniswap position
function _computeMagicAmounts(
uint256 _inventory0,
int24 _halfWidth
) internal pure returns (uint256 amount0, uint256 amount1) {
// the fraction of total inventory (X96) that should be put into primary Uniswap order to mimic Uniswap v2
uint96 magic = uint96(Q96 - TickMath.getSqrtRatioAtTick(-_halfWidth));
amount0 = FullMath.mulDiv(_inventory0, magic, Q96);
These equations allow Blend to compute the portion of vault inventory that should be deposited to Uniswap V3. The symmetry and 1.0001 exponentials are quite convenient here, as they reduce on-chain computation and allow existing TickMath library functions to be reused.
To use the equations above and figure out what percentage of funds to deposit to Uniswap, Blend must first choose how wide its position will be. This is tough because it has to optimize over the following criteria:
Maximizing silo deposits to earn more yield (smaller Uniswap position is better)
Keeping the Uniswap position in-range (larger Uniswap position is better)
Keeping rebalances as infrequent as possible so that maintenance fees paid to bots are kept to a minimum (larger Uniswap position is better)
These trade-offs are made even more complicated by the fact that Blend should work for all sorts of trading pairs, and stable-stable pairs behave very differently from others. Blend addresses this by measuring implied volatility on-chain and scaling position width accordingly.
Implied volatility (IV) differs from historical volatility (HV) in that it's forward-looking rather than backward-looking. In general, high IV values mean that the market thinks a given price is going to swing around a lot. This is perfect information for Blend, because if it observes high IV, it can preemptively increase its Uniswap position width to keep it in-range.
Surprisingly, computing IV on-chain is not only possible, but also more precise and gas-efficient than computing HV. Guillaume Lambert was the first to discover that you can estimate IV from Uniswap V3 data. What follows is a summary of Guillaume's results, plus some modifications to make everything run in the EVM.
IV = \sigma = 2 \gamma \sqrt{\frac{Daily Volume}{Tick Liquidity}}
This equation operates on data from a single Uniswap pool and determines the implied volatility between the two pool assets.
\gamma
is the fee tier, daily volume is self-explanatory, and tick liquidity is the amount of liquidity available at the current tick. Since IV is a dimensionless quantity, daily volume and tick liquidity must be denominated in the same asset.
\gamma
isn't as simple as it seems. Uniswap governance can excise a protocol fee on one or both assets independently.
There's no direct way to get daily volume on-chain. It must be estimated from daily LP fee revenue.
With this in mind, let's rework the equation such that it's actually implementable:
IV=\sigma=2 \sqrt{\gamma_0 \gamma_1} \sqrt{\frac{{fees}_0 \frac{p_{mean}}{\gamma_0} + {fees}_1 \frac{1}{\gamma_1}}{TickLiquidity}}=2\sqrt{\frac{\gamma_1 fees_0 p_{mean} + \gamma_0 fees_1}{TickLiquidity}}
There are a few approximations here. First, we've replaced
\gamma
with the geometric mean of
\gamma_0
\gamma_1
. This may or may not be the "correct" mean to use, but it helps the math work out cleanly. We're also assuming that
volume_0=\frac{p_{mean}}{\gamma_0} \sum fees_{0, i}
\frac{1}{\gamma_0} \cdot \vec{p} \cdot \overrightarrow{fees_0}
p
fees_0
are vectors containing one element for each trade.
The only other tricky thing is making sure that these calculations don't overflow. You can find most of the code here and the oracle documentation here.
The Aloe Labs team did a brief analysis of the accuracy of this volume approximation and found that it was good enough for Blend. Prices were modeled with GBM, and simulations were run for both uniformly-distributed and normally-distributed trade sizes.
The approximation error seems to be proportional to
\mu^2 \propto \left( log \frac{p_n}{p_0} \right) ^ 2
with negligible dependence on GBM's
\sigma
. For reasonable daily price movements (
\pm 30
%) the error is just 1%.
Mapping IV to Ticks
The last step is to map IV values to position widths, measured in ticks. To encompass 95% of trading activity, Blend's Uniswap position should cover 2 standard deviations of price movement:
p (1\pm2\sigma)
. Unfortunately, this would require an uneven number of ticks above the current price (
\log_{1.0001}(1+2\sigma)
) and below the current price (
\log_{1.0001}(\frac{1}{1-2\sigma})
). To make other math simpler, the larger value was chosen.
/// @dev Computes position width based on volatility
function _computeNextPositionWidth(uint256 _sigma) internal pure returns (int24) {
if (_sigma <= 9.9491783619e15) return MIN_WIDTH; // \frac{1e18}{B} (1 - \frac{1}{1.0001^(MIN_WIDTH / 2)})
if (_sigma >= 3.7500454036e17) return MAX_WIDTH; // \frac{1e18}{B} (1 - \frac{1}{1.0001^(MAX_WIDTH / 2)})
_sigma *= B; // scale by a constant factor to increase confidence
uint160 ratio = uint160((Q96 * 1e18) / (1e18 - _sigma));
return TickMath.getTickAtSqrtRatio(ratio);
Since 95% of trading activity should stay in-range of Blend's Uniswap positions, most rebalances can be done by just shuffling liquidity between Uniswap and the silos. No swaps or limit orders necessary.
If/when a primary Uniswap position does slide out of range (or when interest income piles up in one silo faster than the other), Blend is able to place limit orders to get back to a 50/50 inventory ratio. These limit orders are carefully placed to avoid locking in impermanent loss.
One question that arises is how large a limit order should be such that, when it's pushed through, the vault's inventory ratio will be exactly 50/50. As long as one really can create a limit order (a sufficiently thin range order on Uniswap V3), the answer is simple:
error=|inventory_0 \cdot p - inventory_1|
value_{1,limit order}=\frac{error}{2}
This approximation is good enough for Blend, but we've provided more exact formulas here.
|
In philosophy, events are objects in time or instantiations of properties in objects. On some views, only changes in the form of acquiring or losing a property can constitute events, like the lawn's becoming dry.[1] According to others, there are also events that involve nothing but the retaining of a property, e.g. the lawn's staying wet.[1][2] Events are usually defined as particulars that, unlike universals, cannot repeat at different times.[2] Processes are complex events constituted by a sequence of events.[3] But even simple events can be conceived as complex entities involving an object, a time and the property exemplified by the object at this time.[4][5] Traditionally, metaphysicians tended to emphasize static being over dynamic events. This tendency has been opposed by so-called process philosophy or process ontology, which ascribes ontological primacy to events and processes.[6][7]
Jaegwon Kim theorized that events are structured.
They are composed of three things:
Object(s)
{\displaystyle [x]}
a property
{\displaystyle [P]}
time or a temporal interval
{\displaystyle [t]}
Events are defined using the operation
{\displaystyle [x,P,t]}
a) the existence condition and
b) the identity condition.
The existence condition states “
{\displaystyle [x,P,t]}
exists if and only if object
{\displaystyle x}
exemplifies the
{\displaystyle n}
{\displaystyle P}
{\displaystyle t}
.” This means a unique event exists if the above is met. The identity condition states “
{\displaystyle [x,P,t]}
{\displaystyle [y,Q,t']}
{\displaystyle x=y}
{\displaystyle P=Q}
{\displaystyle t=t'}
Kim uses these to define events under five conditions:
One, they are unrepeatable, unchangeable particulars that include changes and the states and conditions of that event.
Two, they have a semi-temporal location.
Three, only their constructive property creates distinct events.
Four, holding a constructive property as a generic event creates a type-token relationship between events, and events are not limited to their three requirements (i.e.
{\displaystyle [x,P,t]}
). Critics of this theory such as Myles Brand have suggested that the theory be modified so that an event had a spatiotemporal region; consider the event of a flash of lightning. The idea is that an event must include both the span of time of the flash of lightning and the area in which it occurred.
Other problems exist within Kim's theory, as he never specified what properties were (e.g. universals, tropes, natural classes, etc.). In addition, it is not specified if properties are few or abundant. The following is Kim's response to the above.
. . . [T]he basic generic events may be best picked out relative to a scientific theory, whether the theory is a common-sense theory of the behavior of middle-sized objects or a highly sophisticated physical theory. They are among the important properties, relative to the theory, in terms of which lawful regularities can be discovered, described, and explained. The basic parameters in terms of which the laws of the theory are formulated would, on this view, give us our basic generic events, and the usual logical, mathematical, and perhaps other types of operations on them would yield complex, defined generic events. We commonly recognize such properties as motion, colors, temperatures, weights, pushing, and breaking, as generic events and states, but we must view this against the background of our common-sense explanatory and predictive scheme of the world around us. I think it highly likely that we cannot pick out generic events completely a priori.[8]
There is also a major debate about the essentiality of a constitutive object. There are two major questions involved in this: If one event occurs, could it have occurred in the same manner if it were another person, and could it occur in the same manner if it would have occurred at a different time? Kim holds that neither are true and that different conditions (i.e. a different person or time) would lead to a separate event. However, some consider it natural to assume the opposite.
Donald Davidson and John Lemmon proposed a theory of events that had two major conditions, respectively: a causal criterion and a spatiotemporal criterion.
The causal criterion defines an event as two events being the same if and only if they have the same cause and effect.
The spatiotemporal criterion defines an event as two events being the same if and only if they occur in the same space at the same time. Davidson however provided this scenario; if a metal ball becomes warmer during a certain minute, and during the same minute rotates through 35 degrees, must we say that these are the same event? However, one can argue that the warming of the ball and the rotation are possibly temporally separated and are therefore separate events.
David Lewis theorized that events are merely spatiotemporal regions and properties (i.e. membership of a class). He defines an event as “e is an event only if it is a class of spatiotemporal regions, both thisworldly (assuming it occurs in the actual world) and otherworldly.” The only problem with this definition is it only tells us what an event could be, but does not define a unique event. This theory entails modal realism, which assumes possible worlds exist; worlds are defined as sets containing all objects that exist as a part of that set. However, this theory is controversial. Some philosophers have attempted to remove possible worlds, and reduce them to other entities. They hold that the world we exist in is the only world that actually exists, and that possible worlds are only possibilities.
Lewis’ theory is composed of four key points. Firstly, the non-duplication principle; it states that x and y are separate events if and only if there is one member of x that is not a member of y (or vice versa). Secondly, there exist regions that are subsets of possible worlds and thirdly, events are not structured by an essential time.
In Being and Event, Alain Badiou writes that the event (événement) is a multiple which basically does not make sense according to the rules of the "situation," in other words existence. Hence, the event "is not," and therefore, in order for there to be an event, there must be an "intervention" which changes the rules of the situation in order to allow that particular event to be ("to be" meaning to be a multiple which belongs to the multiple of the situation — these terms are drawn from or defined in reference to set theory). In his view, there is no "one," and everything that is is a "multiple." "One" happens when the situation "counts," or accounts for, acknowledges, or defines something: it "counts it as one." For the event to be counted as one by the situation, or counted in the one of the situation, an intervention needs to decide its belonging to the situation. This is because his definition of the event violates the prohibition against self-belonging (in other words, it is a set-theoretical definition which violates set theory's rules of consistency), thus does not count as extant on its own.[9]
Gilles Deleuze lectured on the concept of event on March 10, 1987. A sense of the lecture is described by James Williams.[10] Williams also wrote, "From the point of view of the difference between two possible worlds, the event is all important".[11] He also stated, "Every event is revolutionary due to an integration of signs, acts and structures through the whole event. Events are distinguished by the intensity of this revolution, rather than the types of freedom or chance."[12] In 1988 Deleuze published a magazine article "Signes et événements"[13]
In his book Nietszche and Philosophy, he addresses the question "Which one is beautiful?" In the preface to the English translation he wrote:
The one that ... does not refer to an individual, to a person, but rather to an event, that is, to the forces in their various relationships to a proposition or phenomenon, and the genetic relationship that determines these forces (power).[14]
The Danish philosopher Ole Fogh Kirkeby deserves mentioning, as he has written a comprehensive trilogy about the event, or in Danish "begivenheden". In the first work of the trilogy "Eventum tantum – begivenhedens ethos"[15] (Eventum tantum - the ethos of the event) he distinguishes between three levels of the event, inspired from Nicholas of Cusa: Eventum tantum as non aliud, the alma-event and the proto-event.
Free play (Derrida)
^ a b Honderich, Ted (2005). "events". The Oxford Companion to Philosophy. Oxford University Press.
^ a b Kim, Jaegwon; Sosa, Ernest; Rosenkrantz, Gary S. "event theory". A Companion to Metaphysics. Wiley-Blackwell.
^ Craig, Edward (1996). "processes". Routledge Encyclopedia of Philosophy. Routledge.
^ Audi, Robert. "event". The Cambridge Dictionary of Philosophy. Cambridge University Press.
^ Schneider, Susan. "Events". Internet Encyclopedia of Philosophy.
^ Seibt, Johanna (2020). "Process Philosophy". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 9 January 2021.
^ Hustwit, J. R. "Process Philosophy". Internet Encyclopedia of Philosophy. Retrieved 9 January 2021.
^ Jaegwon Kim (1993) Supervenience and Mind, page 37, Cambridge University Press
^ Alain Badiou (1988) L'Être et l'Événement
^ Charles J. Stivale (editor) (2011) Gilles Deleuze: Key Concepts, 2nd edition, chapter 6: Event, pp 80–90
^ James Williams (2003) Gilles Deleuze’s Difference and Repetition: A Critical Introduction and Guide, page 78, Edinburgh University Press
^ Williams 2003 p xi
^ Gilles Deleuze (1988) "Signes et événements", Magazine Littéraire, #257, pages 16 to 25
^ Michael Hart (1993) Gilles Deleuze: An apprenticeship in philosophy, page 31, University of Minnesota Press ISBN 0-8166-2160-8
^ Ole Fogh Kirkeby (2005) Eventum tantum : Begivenhedens ethos. København: Samfundslitteratur
Roberto Casati & Achille Varzi, Events, from Stanford Encyclopedia of Philosophy.
Susan Schneider, Events, from The Internet Encyclopedia of Philosophy.
Byron Kaldis, Events, from Encyclopedia of Philosophy and the Social Sciences.
Gilles Deleuze – Félix Guattari
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Deterministic 3D Ground‐Motion Simulations (0–5 Hz) and Surface Topography Effects of the 30 October 2016 Mw 6.5 Norcia, Italy, Earthquake | Bulletin of the Seismological Society of America | GeoScienceWorld
Deterministic 3D Ground‐Motion Simulations (0–5 Hz) and Surface Topography Effects of the 30 October 2016 Mw 6.5 Norcia, Italy, Earthquake
Arben Pitarka;
Arben Pitarka *
Corresponding author: pitarka1@llnl.gov
Aybige Akinci;
National Institute of Geophysics and Volcanology, Rome, Italy
Pasquale De Gori;
Arben Pitarka, Aybige Akinci, Pasquale De Gori, Mauro Buttinelli; Deterministic 3D Ground‐Motion Simulations (0–5 Hz) and Surface Topography Effects of the 30 October 2016
Mw
6.5 Norcia, Italy, Earthquake. Bulletin of the Seismological Society of America 2021;; 112 (1): 262–286. doi: https://doi.org/10.1785/0120210133
Mw
6.5 Norcia, Italy, earthquake occurred on 30 October 2016 and caused extensive damage to buildings in the epicentral area. The earthquake was recorded by a network of strong‐motion stations, including 14 stations located within a 5 km distance from the two causative faults. We used a numerical approach for generating seismic waves from two hybrid deterministic and stochastic kinematic fault rupture models propagating through a 3D Earth model derived from seismic tomography and local geology. The broadband simulations were performed in the 0–5 Hz frequency range using a physics‐based deterministic approach modeling the earthquake rupture and elastic wave propagation. We used SW4, a finite‐difference code that uses a conforming curvilinear mesh, designed to model surface topography with high numerical accuracy.
The simulations reproduce the amplitude and duration of observed near‐fault ground motions. Our results also suggest that due to the local fault‐slip pattern and upward rupture directivity, the spatial pattern of the horizontal near‐fault ground motion generated during the earthquake was complex and characterized by several local minima and maxima. Some of these local ground‐motion maxima in the near‐fault region were not observed because of the sparse station coverage. The simulated peak ground velocity (PGV) is higher than both the recorded PGV and predicted PGV based on empirical models for several areas located above the fault planes. Ground motions calculated with and without surface topography indicate that, on average, the local topography amplifies the ground‐motion velocity by 30%. There is correlation between the PGV and local topography, with the PGV being higher at hilltops. In contrast, spatial variations of simulated PGA do not correlate with the surface topography. Simulated ground motions are important for seismic hazard and engineering assessments for areas that lack seismic station coverage and historical recordings from large damaging earthquakes.
Norcia Italy
Norcia earthquake 2016
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Reduced Groebner basis of an ideal I in Weyl algebra
{\displaystyle A_{n}}
Weyl.WRGB(L:LIST):LIST
This function converts Groebner basis GB computed by implementation in CoCoALib into reduced Groebner Basis. If GB is not a Groebner basis then the output will not be reduced Groebner basis. In fact, this function reduces a list GB of Weyl polynomals using WNR(F,L) into a new list L such that Ideal(L) = Ideal(GB).
This function is used inside the function WGB(I) to get a list of minimal Groebner basis elements for the ideal I.
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For other uses, see Titanium (disambiguation).
2.5 Anticancer therapy studies
4.1.1 Forming and forging
{\displaystyle {\ce {2FeTiO3 + 7Cl2 + 6C ->[900^oC] 2FeCl3 + 2TiCl4 + 6CO}}}
{\displaystyle {\ce {TiCl4 + 2Mg ->[1100^oC] Ti + 2MgCl2}}}
^ AWS G2.4/G2.4M:2007 Guide for the Fusion Welding of Titanium and Titanium Alloys. Miami: American Welding Society. 2006. Archived from the original on 10 December 2010. {{cite book}}: CS1 maint: bot: original URL status unknown (link)
^ Titanium Metals Corporation (1997). Titanium design and fabrication handbook for industrial applications. Dallas: Titanium Metals Corporation. Archived from the original on 9 February 2009. {{cite book}}: CS1 maint: bot: original URL status unknown (link)
^ Gafner, G. (1989). "The development of 990 Gold-Titanium: its Production, use and Properties" (PDF). Gold Bulletin. 22 (4): 112–122. doi:10.1007/BF03214709. S2CID 114336550. Archived from the original on 29 November 2010. {{cite journal}}: CS1 maint: unfit URL (link)
^ "Fine Art and Functional Works in Titanium and Other Earth Elements". Archived from the original on 13 May 2008. Retrieved 8 August 2009. {{cite web}}: CS1 maint: bot: original URL status unknown (link)
^ Alwitt, Robert S. (2002). "Electrochemistry Encyclopedia". Chemical Engineering Department, Case Western Reserve University, U.S. Archived from the original on 2 July 2008. Retrieved 30 December 2006. {{cite web}}: CS1 maint: unfit URL (link)
^ Turgeon, Luke (20 September 2007). "Titanium Titan: Broughton immortalised". The Gold Coast Bulletin. Archived from the original on 28 September 2013. {{cite news}}: CS1 maint: unfit URL (link)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Titanium&oldid=1088532857"
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§ Ordinals and cardinals
This a rough sketch of a part of set theory I know very little about, which I'm encountering as I solve the "supplementary exercises" in Munkres, chapter 1.
§ Ordinals
Two totally ordered sets have the same order type if there is a monotone isomorphism between them. That is, there's a function
f
which is monotone, and has an inverse. The inverse is guaranteed to be motone (1), so we do not need to stipulate a monotone inverse.
Definition of well ordered set : totally ordered set where every subset has a least element.
Theorem: The set of well ordered sets is itself well ordered.
Definition ordinals : Consider equivalence classes of well ordered sets under order type. of well ordered sets with the same order type. The equivalence classes are ordinals .
§ (1) Inverse of a Monotone function is monotone.
f: A \rightarrow B
be monotone:
a < a'
f(a) < f(a')
. Furthermore, there is a function
g: B \rightarrow A
g(f(a)) = a
f(g(b)) = b
Claim: if
b < b'
g(b) < g(b')
b < b'
. We must have (a)
g(b) < g(b')
, or (b)
g(b) = g(b')
, or (c)
g(b) > g(b')
g(b) < g(b')
we are done.
Suppose for contradiction
g(b) \geq g(b')
f(g(b)) \geq f(g(b'))
f
f, g
are inverses we get
b \geq b'
. This contradicts the assumption
b < b'
This doesn't work for partial orders because we may get
b
b'
as incomparable .
§ Von Neumann Ordinals
Von neumann ordinals: Representatives of equivalence classes of ordinals. Formally, each Von-Neumann ordinal is the well-ordered set of all smaller ordinals.
Formal defn of Von-Neumann ordinal
o
: (1) every element
x \in o
will be a subset of
o
x
is itself a set { ordinal < x }, which is a subset of { ordinal < o }. (2) the set
o
is well ordered by set membership, since two such ordinals will always be comparable, and one must contain the other.
For example of Von Neumann ordinals, consider 0 = {}, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}. We can order 3 based on membership: 0 ∈ 1, 2 so 0 < 1, 2. 1 ∈ 2 hence 1 < 2. This totally orders 3based on set membership. Next, also see that a member of 3, such as 2, is in face 2 = {0, 1}, which is a subset of 3. So every member of 3 is a subset of 3. (Not vice versa: not every subset is a member! The subset {1, 2} is not a member of 3).
§ Limit ordinals
A limit ordinal is an ordinal that cannot be written as the successor of some other ordinal.
Theorem : An ordinal must be either zero, or the successor of some other ordinal, or a limit ordinal (2)
References on ordinals
§ Cardinality and cardinals
We can define cardinality as equivalence classes of sets that are equinumerous : ie, sets with bijections between them. This does not strictly speaking work due to set-theoretic issues, but let's go with it.
In each such equivalence class of sets which are equinumerous, there will be many well ordered sets. The smallest such well ordered set (recall that the set of well ordered sets is itself totally ordered). This is called as the cardinal for that cardinality.
So we redefine cardinality as the smallet ordinal
\alpha
such that there is a bijection between
X
\alpha
. This is motivated from the "equivalence class of all equinumerous sets", but sidesteps set theoretic issues. For this to work, we need well ordering. Otherwise, there could a set with no ordering that is in bijection with it.
§ Rank
The rank of the empty set is zero. The rank of a set is recursively the smallest ordinal greater than the ranks of all the members of the set. Every ordinal has a rank equal to itself.
V_0
V_{n+1} \equiv 2^{V_n}
. This defines
V
for successor ordinals.
V_\lambda \equiv \cup_{\beta < \lambda} V_\beta
V
for limit ordinals.
V_\alpha
are also callled stages or ranks. We can define the rank of a set
S
\alpha
S \subseteq V_\alpha
§ Inaccessible cardinal
A cardinal that cannot be created by adding cardinals, taking unions of cardinals, taking power sets of cardinals. So the set of cardinals smaller than an inacessible cardinal give a model for ZFC. if
\kappa
is an inaccessible cardinal, then
V_\kappa
, collection of all sets of rank less than
\kappa
acts as a place to do mathematics safely, while still having access to the "set of all sets"
V_\kappa
(Grothendeick universes, apparently).
§ Alternative definition of cardinality using rank
Recall that we wanted to define cardinality as the equivalence class of of equinumerous sets, but this ran into set theoretic issues.
A fix (by Dana Scott) is for a set
A
, consider the least rank
\kappa
where some set in bijection with
A
appears. Then we define the cardinality of
A
to be the equivalence classes of sets in
V_\kappa
that are in bijection with
A
. This gives us the cardinals without needing us to consider all sets. This works even without well ordering.
I don't actually understand why this works. In my mind, the set {0} and {{0}} both have the same size, but {0} lives in V1 while {{0}}lives in V2, so they won't have the same cardinality? Actually, I think I do understand: for the set {{0}}, the set {0} which is in bijection with {{0}} occurs at rank 1, so the cardinality of {{0}} is given by the equivalence class in V1: [{0}].
The key part seems to be "find the smallest rank". I have no idea how one would formalize this.
§ Weak and strong limits
S
is a strong limit if it cannot be obtained by taking powersets of sets smaller than it.
In set theory, we as a rule of thumb replace powerset with successor to get some weaker statement.
S
is a weak limit if it cannot be obtained by taking successor of sets smaller than it.
Large Sets 3
|
Bijection — Wikipedia Republished // WIKI 2
{\displaystyle X}
{\displaystyle \mathbb {B} }
{\displaystyle \mathbb {B} }
{\displaystyle X}
{\displaystyle \mathbb {B} ^{n}}
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Z} }
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {R} }
{\displaystyle X}
{\displaystyle \mathbb {R} ^{n}}
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {C} }
{\displaystyle X}
{\displaystyle \mathbb {C} ^{n}}
{\displaystyle X}
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.[1] The term one-to-one correspondence must not be confused with one-to-one function (an injective function; see figures).
A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets.
A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms the symmetric group.
Bijective functions are essential to many areas of mathematics including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group, and projective map.
INJECTIVE, SURJECTIVE, and BIJECTIVE FUNCTIONS - DISCRETE MATHEMATICS
Examples On Bijection / Maths Algebra
Bijection Proof (a taste of math proof)
Bijective Functions and Why They're Important | Bijections, Bijective Proof, Functions and Relations
How to Prove a Function is a Bijection and Find the Inverse
2.1 Batting line-up of a baseball or cricket team
2.2 Seats and students of a classroom
3 More mathematical examples and some non-examples
4 Inverses
6 Cardinality
8 Category theory
9 Generalization to partial functions
Further information on notation: Function (mathematics) § Notation
Satisfying properties (1) and (2) means that a pairing is a function with domain X. It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y. Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions). Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions).[2] With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto".[3]
Batting line-up of a baseball or cricket team
Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball) and the set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list.
Seats and students of a classroom
For any set X, the identity function 1X: X → X, 1X(x) = x is bijective.
The function f: R → R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x) = y. More generally, any linear function over the reals, f: R → R, f(x) = ax + b (where a is non-zero) is a bijection. Each real number y is obtained from (or paired with) the real number x = (y − b)/a.
The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers
{\displaystyle \mathbb {R} ^{+}\equiv \left(0,\infty \right)}
, then g would be bijective; its inverse (see below) is the natural logarithm function ln.
{\displaystyle \mathbb {R} _{0}^{+}\equiv \left[0,\infty \right)}
By Cantor-Bernstein-Schroder theorem, given any two sets X and Y, and two injective functions f: X → Y and g: Y → X, there exists a bijective function h: X → Y.
A bijection f with domain X (indicated by f: X → Y in functional notation) also defines a converse relation starting in Y and going to X (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, in general, yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y. Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection.
{\displaystyle g\,\circ \,f}
{\displaystyle g\,\circ \,f}
{\displaystyle (g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})}
{\displaystyle g\,\circ \,f}
If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (∘), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).
If X and Y are finite sets with the same cardinality, and f: X → Y, then the following are equivalent:
For a finite set S, there is a bijection between the set of possible total orderings of the elements and the set of bijections from S to S. That is to say, the number of permutations of elements of S is the same as the number of total orderings of that set—namely, n!.
The notion of one-to-one correspondence generalizes to partial functions, where they are called partial bijections, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup.[4]
Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turns out to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of A and B′ is a subset of B.[5]
When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[6] An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.[7]
Ax–Grothendieck theorem
Bijection, injection and surjection
Bijective numeration
Bijective proof
Multivalued function
^ "Injective, Surjective and Bijective". www.mathsisfun.com. Retrieved 2019-12-07.
^ "Bijection, Injection, And Surjection | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2019-12-07.
^ Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.
^ Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge University Press. p. 289. ISBN 978-0-521-44179-7.
^ Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.
^ John Meakin (2007). "Groups and semigroups: connections and contrasts". In C.M. Campbell; M.R. Quick; E.F. Robertson; G.C. Smith (eds.). Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4. preprint citing Lawson, M. V. (1998). "The Möbius Inverse Monoid". Journal of Algebra. 200 (2): 428–438. doi:10.1006/jabr.1997.7242.
Cupillari (1989). The Nuts and Bolts of Proofs. Wadsworth. ISBN 9780534103200.
"Bijection", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Bijection". MathWorld.
Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.
|
Spud has a problem. He knows that the solutions for a quadratic equation are
x = 3 + 4i
x= 3 − 4i
, but in order to get credit for the problem he was supposed to have written down the original equation. Unfortunately, he lost the paper with the original equation on it. Luckily, his friends are full of advice.
Just remember when we made polynomials. If you wanted 7 and 4 to be the answers, you just used
(x - 7)(x - 4)
. So you just do
x
minus the first one times
x
minus the other.” Use
(x − (3 + 4i))(x − (3 − 4i))
to find the quadratic equation.
Hugo says, “No, no, no. You can do it that way, but that's too complicated. I think you just start with
x = 3 + 4i
and work backward. So
x - 3 = 4i
, then, hmmm... Yeah, that'll work.” Try Hugo's method.
Whose method do you think Spud should use? Explain your choice.
y = (x − (3 + 4i))(x − (3 − 4i))
2 by 2 rectangle labeled as follows: top edge, x, minus open parenthesis, 3, + 4, I, close parenthesis, left edge, x, minus open parenthesis, 3, minus 4, I, close parenthesis.
Labels added to the interior as follows: top left, x squared, top right, negative, x, times, open parenthesis, 3, + 4, I, close parenthesis, bottom left, negative, x, times, open parenthesis, 3, minus 4, I, close parenthesis, bottom right, open parenthesis, 3, minus 4, I, close parenthesis, times, open parenthesis, 3, + 4, I, close parenthesis.
Labels changed to the interior as follows: top right, negative 3, x, minus 4, x, I, bottom left, negative 3, x, + 4, x, I, bottom right, 25
y = x^2 - 6x + 25
(x - 3)^2 = (4i)^2
|
2020 Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations
Kandasamy Alagesan, Subaramaniyam Jaikumar, Govindasamy Ayyappan
In this paper, the authors studied oscillatory behavior of solutions of fourth-order delay difference equation
\Delta \left({c}_{3}\left(n\right)\Delta \left({c}_{2}\left(n\right)\Delta \left({c}_{1}\left(n\right)\Delta u\left(n\right)\right)\right)\right)+p\left(n\right)f\left(u\left(n-k\right)\right)=0
{\sum }_{n={n}_{0}}^{\infty }{c}_{i}\left(n\right)<\infty , i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3
. New oscillation criteria have been obtained which greatly reduce the number of conditions required for the studied equation. Some examples are presented to show the strength and applicability of the main results.
Kandasamy Alagesan. Subaramaniyam Jaikumar. Govindasamy Ayyappan. "Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations." Abstr. Appl. Anal. 2020 1 - 7, 2020. https://doi.org/10.1155/2020/1653081
Kandasamy Alagesan, Subaramaniyam Jaikumar, Govindasamy Ayyappan "Some New Oscillation Criteria for Fourth-Order Nonlinear Delay Difference Equations," Abstract and Applied Analysis, Abstr. Appl. Anal. 2020(none), 1-7, (2020)
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On arithmetic subgroups of a Q-rank 2 form of SU(2,2) and their automorphic cohomology
April, 2005 On arithmetic subgroups of a Q-rank 2 form of SU(2,2) and their automorphic cohomology
Takahiro HAYATA, Joachim SCHWERMER
{H}^{*}\left(\mathrm{\Gamma },E\right)
of an arithmetic subgroup
\mathrm{\Gamma }
of a connected reductive algebraic group
G
\mathbf{Q}
can be interpreted in terms of the automorphic spectrum of
\mathrm{\Gamma }
. In this frame there is a sum decomposition of the cohomology into the cuspidal cohomology ( i.e., classes represented by cuspidal automorphic forms for
G
) and the so called Eisenstein cohomology. The present paper deals with the case of a quasi split form
G
\mathbf{Q}
-rank two of a unitary group of degree four. We describe in detail the Eisenstein series which give rise to non-trivial cohomology classes and the cuspidal automorphic forms for the Levi components of parabolic
\mathbf{Q}
-subgroups to which these classes are attached. Mainly the generic case will be treated, i.e., we essentially suppose that the coefficient system
E
Takahiro HAYATA. Joachim SCHWERMER. "On arithmetic subgroups of a Q-rank 2 form of SU(2,2) and their automorphic cohomology." J. Math. Soc. Japan 57 (2) 357 - 385, April, 2005. https://doi.org/10.2969/jmsj/1158242063
Keywords: associate parabolic subgroup , automorphic representation , cohomology of arithmetic subgroups , cuspidal cohomology , Eisenstein cohomology , minimal coset representatives
Takahiro HAYATA, Joachim SCHWERMER "On arithmetic subgroups of a Q-rank 2 form of SU(2,2) and their automorphic cohomology," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 57(2), 357-385, (April, 2005)
|
\stackrel{̄}{\epsilon }
\stackrel{̄}{\epsilon }
\stackrel{̄}{\epsilon }
\begin{array}{c}{h}_{s}=1\\ {h}_{s}=100\\ {h}_{s}=200\\ {h}_{s}=400\end{array}
\begin{array}{c}\hfill -0.62\hfill \\ \hfill -0.47\hfill \\ \hfill -0.27\hfill \\ \hfill -0.21\hfill \end{array}
\begin{array}{c}\hfill 0\hfill \\ \hfill 5\hfill \\ \hfill 23\hfill \\ \hfill 20\hfill \end{array}
\left\{\begin{array}{c}-0.47\hfill \\ -0.47\hfill \\ -0.45\hfill \\ -0.40\hfill \end{array}\right\}
\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 7\hfill \end{array}
\begin{array}{c}\hfill \left\{\begin{array}{c}-0.49\hfill \\ -0.46\hfill \end{array}\right\}\hfill \\ \hfill \left\{\begin{array}{c}-0.42\hfill \\ -0.42\hfill \end{array}\right\}\hfill \end{array}
\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}
\begin{array}{c}{k}_{q}=0.02\\ {k}_{q}=0.04\\ {k}_{q}=0.08\\ {k}_{q}=0.16\end{array}
\begin{array}{c}\hfill -0.37\hfill \\ \hfill -0.24\hfill \\ \hfill \left\{\begin{array}{c}-0.35\hfill \\ -0.33\hfill \end{array}\right\}\hfill \end{array}
\begin{array}{c}\hfill 5\hfill \\ \hfill 24\hfill \\ \hfill 14\hfill \\ \hfill 17\hfill \end{array}
\begin{array}{c}\hfill -0.60\hfill \\ \hfill -0.45\hfill \\ \hfill \left\{\begin{array}{c}-0.33\hfill \\ -0.29\hfill \end{array}\right\}\hfill \end{array}
\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 10\hfill \\ \hfill 12\hfill \end{array}
\begin{array}{c}\hfill -0.63\hfill \\ \hfill -0.44\hfill \\ \hfill \left\{\begin{array}{c}-0.34\hfill \\ -0.25\hfill \end{array}\right\}\hfill \end{array}
\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 13\hfill \\ \hfill 23\hfill \end{array}
\begin{array}{c}{k}_{s}=0.01\\ {k}_{s}=0.25\\ {k}_{s}=0.50\\ {k}_{s}=0.75\end{array}
\begin{array}{c}\hfill -0.48\hfill \\ \hfill \left\{\begin{array}{c}-0.37\hfill \\ -0.27\hfill \\ -0.24\hfill \end{array}\right\}\hfill \end{array}
\begin{array}{c}\hfill 0\hfill \\ \hfill 12\hfill \\ \hfill 23\hfill \\ \hfill 22\hfill \end{array}
\left\{\begin{array}{c}-0.50\hfill \\ -0.45\hfill \\ -0.45\hfill \\ -0.45\hfill \end{array}\right\}
\begin{array}{c}\hfill 0\hfill \\ \hfill 2\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}
\left\{\begin{array}{c}-0.48\hfill \\ -0.44\hfill \\ -0.42\hfill \\ -0.43\hfill \end{array}\right\}
\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}
\begin{array}{c}{h}_{p}\left({h}_{g}\right)=30;{k}_{rs}=1\cdot 1{0}^{-5}\\ {h}_{p}\left({h}_{g}\right)=60;{k}_{rs}=2\cdot 1{0}^{-5}\\ {h}_{p}\left({h}_{g}\right)=120;{k}_{rs}=4\cdot 1{0}^{-5}\\ {h}_{p}\left({h}_{g}\right)=240;{k}_{rs}=8\cdot 1{0}^{-5}\end{array}
\left\{\begin{array}{c}-0.28\hfill \\ -0.29\hfill \\ -0.35\hfill \\ -0.41\hfill \end{array}\right\}
\begin{array}{c}\hfill 22\hfill \\ \hfill 20\hfill \\ \hfill 13\hfill \\ \hfill 2\hfill \end{array}
\left\{\begin{array}{c}-0.46\hfill \\ -0.46\hfill \\ -0.44\hfill \\ -0.44\hfill \end{array}\right\}
\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}
\left\{\begin{array}{c}-0.45\hfill \\ -0.42\hfill \\ -0.39\hfill \\ -0.40\hfill \end{array}\right\}
\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 2\hfill \\ \hfill 5\hfill \end{array}
For each model and for each parameter value we present the average ε value (left column) and the number n of experiments with positive value of this coefficient (right column). The method for calculation of ε is described in section. Negative ε mean that TF noise is dampen in a loop. The extremal ε values achieved at intermediate parameter values are shown in bold. The Wilcoxon Rank-Sum Test was used to test for significance of difference between the ε values for adjacent parameter values. The differences which are statistically insignificant at the
\alpha =0.05
level are placed in parentheses.
|
Stochastic Analysis in Finance and Insurance | EMS Press
The workshop \emph{Stochastic Analysis in Finance and Insurance}, organised by Dmitry Kramkov (Pittsburgh), Martin Schweizer (Z\"{u}rich) and Nizar Touzi (Paris) was held January 27th -- February 2nd, 2008. The meeting had a total of 44 participants from all over the world with a good blend of more experienced researchers and many younger participants.
During the five days, there were a total of 29 talks with many lively interactions and discussions. The organisers had to exercise some constraint on the participants in order not to overload the programme, and this prompted many discussions and collaborations during the long lunch breaks and in the evenings.
The topics presented in the talks covered a very wide spectrum. There were some major areas with several talks as well as other more individual contributions pointing towards new developments in mathematical finance. The overall tendency went towards more sophisticated and more realistic models of financial markets; the first generation models with frictionless classical semimartingale prices seem largely understood, and one of the major trends is now towards what might be termed second generation modelling. This includes as major topics \emph{transaction costs} and \emph{large investors and liquidity issues} as well as other nonstandard models or ideas related to such developments. A second major topic revolved around \emph{risk measures or monetary utility functions}, and there were several talks on \emph{option pricing}, on \emph{optimisation problems from finance} and on \emph{credit risk}.
We now give a short overview of the topics covered in the talks, roughly ordered into the themes listed above.
\emph{Transaction costs:} The classical Merton problem of optimal investment under transaction costs was reconsidered by \emph{Jan Kallsen} who presented a new approach via shadow prices leading to a simpler way of finding the optimal strategy. \emph{Yuri Kabanov} extended the classical hedging theorem under transaction costs from European to the case of American contingent claims.
\emph{Large investors and liquidity issues:} A model for the optimal liquidation of a large portfolio position was presented by \emph{Alexander Schied}; his results showed that the market impact of such a trader can lead to some unexpected effects. \emph{Mete Soner} studied the problem of superreplicating options in an illiquid market by means of PDE and stochastic control techniques. \emph{Thorsten Rheinl\"ander} developped a new model for utility maximisation by a large trader and showed how this can be modelled via nonlinear stochastic integration theory. A partial equilibrium model for a large investor interacting with other market participants was presented by \emph{Peter Bank}, who emphasised the importance of appropriate financial modelling of gains from trade in continuous time.
\emph{Risk measures or monetary utility functions:} This topic had the largest number of talks. \emph{Michael Kupper} introduced divergence utilities and showed that this fairly large class can be very well manipulated and leads to explicit solutions for optimisation and risk sharing problems. \emph{Damir Filipovi\'c} studied extensions of convex risk measures from the space of all bounded random variables and showed that under law-invariance, the canonical model space consists even of all integrable random variables. \emph{Walter Schachermayer} proved a very new result: He showed that the only time-consistent law-invariant dynamic convex risk measure is the entropic risk measure. \emph{Freddy Delbaen} gave a detailed study of the representation for the penalty function of time-consistent dynamic monetary utility functions with the help of backward stochastic differential equations. Another extension of risk measures to Orlicz hearts was studied by \emph{Patrick Cheridito}, and \emph{Gordan
{\it\check Z}\!
itkovi\'c} introduced maturity-independent risk measures and pointed out some nontrivial existence problems related to this concept.
\emph{Option pricing:} For a class of stochastic volatility models, \emph{David Hobson} showed how to obtain option price comparisons by means of time changes and other purely probabilistic arguments. \emph{Ludger R\"uschendorf} gave a broad overview of methods to obtain comparison results for option prices in large classes of processes and showed several techniques to achieve this goal. \emph{Semyon Malamud} presented a new approach for deriving indifference prices for contingent claims under power utility in discrete-time models having a certain new structure, and pointed out several appealing properties of this class of models. \emph{Vicky Henderson} considered perpetual American options in an incomplete market and determined the optimal exercise strategy when the number of options one owns is infinitely divisible.
\emph{Optimisation problems from finance:} An overview and some new developments for risk-sensitive portfolio optimisation were presented by \emph{Jun Sekine}. \emph{Paolo Guasoni} studied the problem of finding optimal portfolios and risk premia explicitly in the limit of a long time horizon. \emph{Bruno Bouchard} started with quantile hedging and related problems and embedded these into a general stochastic target problem with controlled probability or controlled losses.
\emph{Credit risk:} Valuation of credit-sensitive instruments often involves first passage times, and \emph{Tom Hurd} presented new ideas on how these can be handled more explicitly for a fairly large class of jump-diffusion processes. \emph{Ronnie Sircar} gave an asymptotic analysis of multiscale models for multiname credit risk derivatives and illustrated that his approach leads to computationally tractable and yet fairly accurate results when calibrated to market data.
In addition to the above roughly thematically grouped talks, there were presentations that did not fall readily into a particular area; this illustrates the diversity and multiple facets of the field of mathematical finance. \emph{Yannis Karatzas} started the workshop with a very stimulating talk on so-called diverse financial markets and the idea of finding there optimal arbitrage strategies. \emph{Thaleia Zariphopoulou} presented a new way to look at performance measurement in financial markets and formulated this as a novel and intriguing mathematical problem involving a stochastic partial differential equation. \emph{Jak
\it\check s\!
a Cvitani\'c} gave an overview of recent developments and results on contract theory in continuous time. Motivated by the question of how to model the influence of information on financial markets, \emph{Kostas Kardaras} introduced a topology on
\sigma
-fields and on filtrations and presented some first continuity results. \emph{Josef Teichmann }explained how one can compute moments of affine processes in a very easy way; this was motivated by many examples arising in the valuation of financial derivatives. With the goal of modelling both stock prices and the infinite family of all call options in a joint model, \emph{Ren\'e Carmona} studied dynamic local volatility models and derived the corresponding drift restrictions arising from absence of arbitrage. Another very thought-provoking talk was given by \emph{Denis Talay} who presented some mathematical models and problems connected to so-called technical analysis in financial markets. \emph{Mihai S\^\i rbu} considered a general semimartingale model and gave some necessary and some sufficient conditions for the validity of a mutual fund theorem.
In contrast to the picture shown on the institute homepage, there was no snow around the institute, as remarked (and regretted) by several participants. On the other hand, this allowed on Wednesday a very pleasant excursion to St.~Roman with a very good participation rate.
For us as the organisers, it was (as always) a great pleasure to be at Oberwolfach and to benefit from the excellent infrastructure, support and scientific environment. We thank the Mathematisches Forschungsinstitut Oberwolfach for making this possible, and we are very happy to report that this sentiment was also expressed by all the participants both during and after the conference in many ways. The idea of having a similar workshop in about three years met with very enthusiastic reactions.
Nizar Touzi, Dmitry Kramkov, Martin Schweizer, Stochastic Analysis in Finance and Insurance. Oberwolfach Rep. 5 (2008), no. 1, pp. 173–240
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Ankur borrowed RS 1500 from a financer After 3 and half years he paid Rs 2655 to the - Maths - Comparing Quantities - 10926603 | Meritnation.com
Ankur borrowed RS 1500 from a financer . After 3 and half years he paid Rs 2655 to the financer and cleared the accounts. Find the rate of interest charged
Intereset paid=2655-1500= 1155\phantom{\rule{0ex}{0ex}}SI=\frac{PRT}{100}\phantom{\rule{0ex}{0ex}}1155=\frac{1500×r×3.5}{100}\phantom{\rule{0ex}{0ex}}r=\frac{1155}{15×3.5}\phantom{\rule{0ex}{0ex}}r=22%
Dev Sharma answered this
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Dynamical Systems | EMS Press
This workshop, organised by Helmut Hofer (New York), Jean-Christophe Yoccoz (Paris), and Eduard Zehnder (Z{\"u}rich), continued the biannual series at Oberwolfach on Dynamical Systems that started as the ``Moser--Zehnder meeting'' in 1981. The workshop was attended by more than 50 participants from 12 countries. The main theme of the workshop were the new results and developments in the area of classical dynamical systems, in particular in celestial mechanics and Hamiltonian systems. Among the main topics were new global results on the Reeb dynamics on
3
-manifolds, KAM theory in finite and infinite dimensions, as well as new developments in Floer homology and its applications. High points were the first complete existence proof of quasiperiodic solutions in the planetarian \text{
N
-body} problem, and the solution of a long-standing conjecture of Anosov about the number of closed geodesics on Finsler
2
-spheres. The meeting was held in a very informal and stimulating atmosphere.
Helmut W. Hofer, Jean-Christophe Yoccoz, Eduard Zehnder, Dynamical Systems. Oberwolfach Rep. 2 (2005), no. 3, pp. 1743–1798
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Resolution (algebra) - Wikipedia
(Redirected from Injective resolution)
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution[1]) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the zero-object.[2]
Generally, the objects in the sequence are restricted to have some property P (for example to be free). Thus one speaks of a P resolution. In particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules.
1 Resolutions of modules
1.2 Free, projective, injective, and flat resolutions
1.3 Graded modules and algebras
2 Resolutions in abelian categories
2.1 Abelian categories without projective resolutions in general
3 Acyclic resolution
Resolutions of modulesEdit
Given a module M over a ring R, a left resolution (or simply resolution) of M is an exact sequence (possibly infinite) of R-modules
{\displaystyle \cdots {\overset {d_{n+1}}{\longrightarrow }}E_{n}{\overset {d_{n}}{\longrightarrow }}\cdots {\overset {d_{3}}{\longrightarrow }}E_{2}{\overset {d_{2}}{\longrightarrow }}E_{1}{\overset {d_{1}}{\longrightarrow }}E_{0}{\overset {\varepsilon }{\longrightarrow }}M\longrightarrow 0.}
The homomorphisms di are called boundary maps. The map ε is called an augmentation map. For succinctness, the resolution above can be written as
{\displaystyle E_{\bullet }{\overset {\varepsilon }{\longrightarrow }}M\longrightarrow 0.}
The dual notion is that of a right resolution (or coresolution, or simply resolution). Specifically, given a module M over a ring R, a right resolution is a possibly infinite exact sequence of R-modules
{\displaystyle 0\longrightarrow M{\overset {\varepsilon }{\longrightarrow }}C^{0}{\overset {d^{0}}{\longrightarrow }}C^{1}{\overset {d^{1}}{\longrightarrow }}C^{2}{\overset {d^{2}}{\longrightarrow }}\cdots {\overset {d^{n-1}}{\longrightarrow }}C^{n}{\overset {d^{n}}{\longrightarrow }}\cdots ,}
where each Ci is an R-module (it is common to use superscripts on the objects in the resolution and the maps between them to indicate the dual nature of such a resolution). For succinctness, the resolution above can be written as
{\displaystyle 0\longrightarrow M{\overset {\varepsilon }{\longrightarrow }}C^{\bullet }.}
A (co)resolution is said to be finite if only finitely many of the modules involved are non-zero. The length of a finite resolution is the maximum index n labeling a nonzero module in the finite resolution.
Free, projective, injective, and flat resolutionsEdit
In many circumstances conditions are imposed on the modules Ei resolving the given module M. For example, a free resolution of a module M is a left resolution in which all the modules Ei are free R-modules. Likewise, projective and flat resolutions are left resolutions such that all the Ei are projective and flat R-modules, respectively. Injective resolutions are right resolutions whose Ci are all injective modules.
Every R-module possesses a free left resolution.[3] A fortiori, every module also admits projective and flat resolutions. The proof idea is to define E0 to be the free R-module generated by the elements of M, and then E1 to be the free R-module generated by the elements of the kernel of the natural map E0 → M etc. Dually, every R-module possesses an injective resolution. Projective resolutions (and, more generally, flat resolutions) can be used to compute Tor functors.
Projective resolution of a module M is unique up to a chain homotopy, i.e., given two projective resolutions P0 → M and P1 → M of M there exists a chain homotopy between them.
Resolutions are used to define homological dimensions. The minimal length of a finite projective resolution of a module M is called its projective dimension and denoted pd(M). For example, a module has projective dimension zero if and only if it is a projective module. If M does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative local ring R, the projective dimension is finite if and only if R is regular and in this case it coincides with the Krull dimension of R. Analogously, the injective dimension id(M) and flat dimension fd(M) are defined for modules also.
The injective and projective dimensions are used on the category of right R modules to define a homological dimension for R called the right global dimension of R. Similarly, flat dimension is used to define weak global dimension. The behavior of these dimensions reflects characteristics of the ring. For example, a ring has right global dimension 0 if and only if it is a semisimple ring, and a ring has weak global dimension 0 if and only if it is a von Neumann regular ring.
Graded modules and algebrasEdit
Let M be a graded module over a graded algebra, which is generated over a field by its elements of positive degree. Then M has a free resolution in which the free modules Ei may be graded in such a way that the di and ε are graded linear maps. Among these graded free resolutions, the minimal free resolutions are those for which the number of basis elements of each Ei is minimal. The number of basis elements of each Ei and their degrees are the same for all the minimal free resolutions of a graded module.
If I is a homogeneous ideal in a polynomial ring over a field, the Castelnuovo-Mumford regularity of the projective algebraic set defined by I is the minimal integer r such that the degrees of the basis elements of the Ei in a minimal free resolution of I are all lower than r-i.
A classic example of a free resolution is given by the Koszul complex of a regular sequence in a local ring or of a homogeneous regular sequence in a graded algebra finitely generated over a field.
Let X be an aspherical space, i.e., its universal cover E is contractible. Then every singular (or simplicial) chain complex of E is a free resolution of the module Z not only over the ring Z but also over the group ring Z [π1(X)].
Resolutions in abelian categoriesEdit
The definition of resolutions of an object M in an abelian category A is the same as above, but the Ei and Ci are objects in A, and all maps involved are morphisms in A.
The analogous notion of projective and injective modules are projective and injective objects, and, accordingly, projective and injective resolutions. However, such resolutions need not exist in a general abelian category A. If every object of A has a projective (resp. injective) resolution, then A is said to have enough projectives (resp. enough injectives). Even if they do exist, such resolutions are often difficult to work with. For example, as pointed out above, every R-module has an injective resolution, but this resolution is not functorial, i.e., given a homomorphism M → M' , together with injective resolutions
{\displaystyle 0\rightarrow M\rightarrow I_{*},\ \ 0\rightarrow M'\rightarrow I'_{*},}
there is in general no functorial way of obtaining a map between
{\displaystyle I_{*}}
{\displaystyle I'_{*}}
Abelian categories without projective resolutions in generalEdit
One class of examples of Abelian categories without projective resolutions are the categories
{\displaystyle {\text{Coh}}(X)}
of coherent sheaves on a scheme
{\displaystyle X}
{\displaystyle X=\mathbb {P} _{S}^{n}}
is projective space, any coherent sheaf
{\displaystyle {\mathcal {M}}}
{\displaystyle X}
has a presentation given by an exact sequence
{\displaystyle \bigoplus _{i,j=0}{\mathcal {O}}_{X}(s_{i,j})\to \bigoplus _{i=0}{\mathcal {O}}_{X}(s_{i})\to {\mathcal {M}}\to 0}
The first two terms are not in general projective since
{\displaystyle H^{n}(\mathbb {P} _{S}^{n},{\mathcal {O}}_{X}(s))\neq 0}
{\displaystyle s>0}
. But, both terms are locally free, and locally flat. Both classes of sheaves can be used in place for certain computations, replacing projective resolutions for computing some derived functors.
Acyclic resolutionEdit
In many cases one is not really interested in the objects appearing in a resolution, but in the behavior of the resolution with respect to a given functor. Therefore, in many situations, the notion of acyclic resolutions is used: given a left exact functor F: A → B between two abelian categories, a resolution
{\displaystyle 0\rightarrow M\rightarrow E_{0}\rightarrow E_{1}\rightarrow E_{2}\rightarrow \cdots }
of an object M of A is called F-acyclic, if the derived functors RiF(En) vanish for all i > 0 and n ≥ 0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors vanish on the objects of the resolution.
For example, given a R module M, the tensor product
{\displaystyle \otimes _{R}M}
is a right exact functor Mod(R) → Mod(R). Every flat resolution is acyclic with respect to this functor. A flat resolution is acyclic for the tensor product by every M. Similarly, resolutions that are acyclic for all the functors Hom( ⋅ , M) are the projective resolutions and those that are acyclic for the functors Hom(M, ⋅ ) are the injective resolutions.
Any injective (projective) resolution is F-acyclic for any left exact (right exact, respectively) functor.
The importance of acyclic resolutions lies in the fact that the derived functors RiF (of a left exact functor, and likewise LiF of a right exact functor) can be obtained from as the homology of F-acyclic resolutions: given an acyclic resolution
{\displaystyle E_{*}}
of an object M, we have
{\displaystyle R_{i}F(M)=H_{i}F(E_{*}),}
where right hand side is the i-th homology object of the complex
{\displaystyle F(E_{*}).}
This situation applies in many situations. For example, for the constant sheaf R on a differentiable manifold M can be resolved by the sheaves
{\displaystyle {\mathcal {C}}^{*}(M)}
of smooth differential forms:
{\displaystyle 0\rightarrow R\subset {\mathcal {C}}^{0}(M){\stackrel {d}{\rightarrow }}{\mathcal {C}}^{1}(M){\stackrel {d}{\rightarrow }}\cdots {\mathcal {C}}^{\dim M}(M)\rightarrow 0.}
The sheaves
{\displaystyle {\mathcal {C}}^{*}(M)}
are fine sheaves, which are known to be acyclic with respect to the global section functor
{\displaystyle \Gamma :{\mathcal {F}}\mapsto {\mathcal {F}}(M)}
. Therefore, the sheaf cohomology, which is the derived functor of the global section functor Γ is computed as
{\displaystyle \mathrm {H} ^{i}(M,\mathbf {R} )=\mathrm {H} ^{i}({\mathcal {C}}^{*}(M)).}
Similarly Godement resolutions are acyclic with respect to the global sections functor.
Matrix factorizations (algebra)
^ Jacobson 2009, §6.5 uses coresolution, though right resolution is more common, as in Weibel 1994, Chap. 2
^ projective resolution in nLab, resolution in nLab
^ Jacobson 2009, §6.5
Iain T. Adamson (1972), Elementary rings and modules, University Mathematical Texts, Oliver and Boyd, ISBN 0-05-002192-3
Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 3-540-94268-8, MR 1322960, Zbl 0819.13001
Jacobson, Nathan (2009) [1985], Basic algebra II (Second ed.), Dover Publications, ISBN 978-0-486-47187-7
Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001
Retrieved from "https://en.wikipedia.org/w/index.php?title=Resolution_(algebra)&oldid=1073715287#Free,_projective,_injective,_and_flat_resolutions"
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Samantha thinks that the equation
\left(x − 4\right)^{2} + \left(y − 3\right)^{2} = 25
\left(x − 4\right) + \left(y − 3\right) = 5
. Is she correct? Are the two equations equivalent? Explain how you know. If they are not equivalent, explain Samantha's mistake.
Solve to find a point that makes the second equation true. Does it also make the first equation true?
Look at the graphs of each equation. Are they the same?
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Unsupervised Medical Image Denoising Using CycleGAN - MATLAB & Simulink - MathWorks India
Deep learning techniques can improve the image quality for low-dose CT (LDCT) images. Using a generative adversarial network (GAN) for image-to-image translation, you can convert noisy LDCT images to images of the same quality as regular-dose CT images. For this application, the source domain consists of LDCT images and the target domain consists of regular-dose images. For more information, see Get Started with GANs for Image-to-Image Translation.
CT image denoising requires a GAN that performs unsupervised training because clinicians do not typically acquire matching pairs of low-dose and regular-dose CT images of the same patient in the same session. This example uses a cycle-consistent GAN (CycleGAN) trained on patches of image data from a large sample of data. For a similar approach using a UNIT neural network trained on full images from a limited sample of data, see Unsupervised Medical Image Denoising Using UNIT.
Combine the low-dose and high-dose training data by using a randomPatchExtractionDatastore. When reading from this datastore, augment the data using random rotation and horizontal reflection.
Create each generator network using the cycleGANGenerator function. For an input size of 256-by-256 pixels, specify the NumResidualBlocks argument as 9. By default, the function has 3 encoder modules and uses 64 filters in the first convolutional layer.
Create each discriminator network using the patchGANDiscriminator function. Use the default settings for the number of downsampling blocks and number of filters in the first convolutional layer in the discriminators.
{\mathit{L}}_{\mathrm{Total}}^{}={\mathit{L}}_{\mathrm{Adversarial}}^{}+\lambda \text{\hspace{0.17em}}*{\mathit{L}}_{\mathrm{Cycle}\text{\hspace{0.17em}}\mathrm{consistency}}^{}+{\mathit{L}}_{\mathrm{Fidelity}}^{}
\lambda
To generate new translated images, use the predict (Deep Learning Toolbox) function. Read images from the test data set and use the trained generators to generate new images.
cycleGANGenerator | patchGANDiscriminator | transform | combine | minibatchqueue (Deep Learning Toolbox) | dlarray (Deep Learning Toolbox) | dlfeval (Deep Learning Toolbox) | adamupdate (Deep Learning Toolbox)
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Speed of sound - Simple English Wikipedia, the free encyclopedia
For the roller coaster in Walibi Holland, see Speed of Sound (roller coaster).
The speed of sound is 1,235 kilometres (767 mi) per hour or 330 metres (1,083 ft) per second in dry air in room temperature. It travels at 1500 meters per second through water. Sound moves faster through liquids and solids than air, since they have a larger specific modulus, meaning they are stiffer. Sound cannot travel through a vacuum, which is a space without any air or matter. The speed of sound is affected by temperature. It travels slower at low temperatures, for example in the stratosphere.
You can calculate the speed of sound like this:
{\displaystyle a={\sqrt {\gamma *R*T}}}
{\displaystyle \gamma }
is the ratio of specific heats (1.4 for air) R is the gas constant (
{\displaystyle 287N*m/kg*K}
for air) T is temperature (in Kelvins)
The speed of sound is also known as Mach 1. Things that go faster are supersonic, and things that go five times that speed (Mach 5) are hypersonic.
Calculate the speed of sound in air and the temperature
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Speed_of_sound&oldid=8196350"
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Solving A Nonlinear ODE - MATLAB & Simulink - MathWorks Italia
Solving A Nonlinear ODE
This section discusses these aspects of a nonlinear ODE problem:
You can run this example: “Solving a Nonlinear ODE with a Boundary Layer by Collocation”.
Consider the nonlinear singularly perturbed problem:
\begin{array}{ccc}\epsilon {D}^{2}g\left(x\right)+{\left(g\left(x\right)\right)}^{2}=1& on& \left[0..1\right]\end{array}
Dg\left(0\right)=g\left(1\right)=0
Seek an approximate solution by collocation from C1 piecewise cubics with a suitable break sequence; for instance,
Because cubics are of order 4, you have
Obtain the corresponding knot sequence as
knots = augknt(breaks,k,2);
This gives a quadruple knot at both 0 and 1, which is consistent with the fact that you have cubics, i.e., have order 4.
This implies that you have
i.e., 10 degrees of freedom.
You collocate at two sites per polynomial piece, i.e., at eight sites altogether. This, together with the two side conditions, gives us 10 conditions, which matches the 10 degrees of freedom.
Choose the two Gaussian sites for each interval. For the standard interval [–0.5,0.5] of length 1, these are the two sites
From this, you obtain the whole collection of collocation sites by
With this, the numerical problem you want to solve is to find
y\in {S}_{4,knots}
that satisfies the nonlinear system
\begin{array}{c}Dy\left(0\right)=0\\ {\left(}^{y}+\epsilon {D}^{2}y\left(x\right)=1\text{ for }x\text{ }\in \text{ colsites}\\ y\left(1\right)=0\end{array}
If y is your current approximation to the solution, then the linear problem for the supposedly better solution z by Newton's method reads
\begin{array}{c}Dz\left(0\right)=0\\ {w}_{0}\left(x\right)z\left(x\right)+\epsilon {D}^{2}z\left(x\right)=b\left(x\right)\text{ for }x\text{ }\in \text{ colsites}\\ z\text{(1)=0}\end{array}
with w0(x)=2y(x),b(x)=(y(x))2+1. In fact, by choosing
\begin{array}{l}{w}_{0}\left(1\right):=1,\text{ }{w}_{1}\left(0\right):=1\\ {w}_{1}\left(x\right):=0,\text{ }{w}_{2}\left(x\right):=\epsilon \text{ for }x\in \text{ colsites}\end{array}
and choosing all other values of w0,w1,w2, b not yet specified to be zero, you can give your system the uniform shape
\begin{array}{ccc}{w}_{0}\left(x\right)z\left(x\right)+{w}_{1}\left(x\right)Dz\left(x\right)+{w}_{2}\left(x\right){D}^{2}z\left(x\right)=b\left(x\right),& \text{for}& x\text{ }\in \text{ sites}\end{array}
Because z∊S4,knots, convert this last system into a system for the B-spline coefficients of z. This requires the values, first, and second derivatives at every x∊sites and for all the relevant B-splines. The command spcol was expressly written for this purpose.
Use spcol to supply the matrix
colmat = ...
spcol(knots,k,brk2knt(sites,3));
From this, you get the collocation matrix by combining the row triple of colmat for x using the weights w0(x),w1(x),w2(x) to get the row for x of the actual matrix. For this, you need a current approximation y. Initially, you get it by interpolating some reasonable initial guess from your piecewise-polynomial space at the sites. Use the parabola x2–1, which satisfies the end conditions as the initial guess, and pick the matrix from the full matrix colmat. Here it is, in several cautious steps:
Plot the initial guess, and turn hold on for subsequent plotting:
You can now complete the construction and solution of the linear system for the improved approximate solution z from your current guess y. In fact, with the initial guess y available, you now set up an iteration, to be terminated when the change z–y is small enough. Choose a relatively mild ε = .1.
weights=[0 1 0;
The resulting printout of the errors is:
maxdif = 3.95151e-005
If you now decrease ε, you create more of a boundary layer near the right endpoint, and this calls for a nonuniform mesh.
Use newknt to construct an appropriate finer mesh from the current approximation:
knots = newknt(z, ninterv+1); breaks = knt2brk(knots);
From the new break sequence, you generate the new collocation site sequence:
colpnts = temp(:).';
sites = [0,colpnts,1];
colmat = spcol(knots,k,sort([sites sites sites]));
and use your current approximate solution z as the initial guess:
y = spmak(knots,[0 fnval(z,colpnts) 0]/intmat.');
Thus set up, divide ε by 3 and repeat the earlier calculation, starting with the statements
tolerance=1.e-9;
vtau=fnval(y,colpnts);
Repeated passes through this process generate a sequence of solutions, for ε = 1/10, 1/30, 1/90, 1/270, 1/810. The resulting solutions, ever flatter at 0 and ever steeper at 1, are shown in the example plot. The plot also shows the final break sequence, as a sequence of vertical bars. To view the plots, run the example “Solving a Nonlinear ODE with a Boundary Layer by Collocation”.
In this example, at least, newknt has performed satisfactorily.
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Variable-precision arithmetic (arbitrary-precision arithmetic) - MATLAB vpa - MathWorks Italia
Evaluate Symbolic Inputs with Variable-Precision Arithmetic
Change Precision Used by vpa
Numerically Approximate Symbolic Results
vpa Uses Guard Digits to Maintain Precision
Avoid Hidden Round-off Errors
vpa Restores Precision of Common Double-Precision Inputs
Variable-precision arithmetic (arbitrary-precision arithmetic)
Support for character vectors that do not define a number has been removed. Instead, first create symbolic numbers and variables using sym and syms, and then use operations on them. For example, use vpa((1 + sqrt(sym(5)))/2) instead of vpa('(1 + sqrt(5))/2').
vpa(x)
vpa(x,d)
vpa(x) uses variable-precision floating-point arithmetic (VPA) to evaluate each element of the symbolic input x to at least d significant digits, where d is the value of the digits function. The default value of digits is 32.
vpa(x,d) uses at least d significant digits, instead of the value of digits.
Evaluate symbolic inputs with variable-precision floating-point arithmetic. By default, vpa calculates values to 32 significant digits.
p = sym(pi);
piVpa = vpa(p)
piVpa =
f = a*sin(2*p*x);
fVpa = vpa(f)
0.33333333333333333333333333333333*sin(6.283185307179586476925286766559*x)
Evaluate elements of vectors or matrices with variable-precision arithmetic.
V = [x/p a^3];
M = [sin(p) cos(p/5); exp(p*x) x/log(p)];
vpa(V)
vpa(M)
[ 0.31830988618379067153776752674503*x, 0.037037037037037037037037037037037]
[ 0, 0.80901699437494742410229341718282]
[ exp(3.1415926535897932384626433832795*x), 0.87356852683023186835397746476334*x]
You must wrap all inner inputs with vpa, such as exp(vpa(200)). Otherwise the inputs are automatically converted to double by MATLAB®.
By default, vpa evaluates inputs to 32 significant digits. You can change the number of significant digits by using the digits function.
Approximate the expression 100001/10001 with seven significant digits using digits. Save the old value of digits returned by digits(7). The vpa function returns only five significant digits, which can mean the remaining digits are zeros.
y = sym(100001)/10001;
Check if the remaining digits are zeros by using a higher precision value of 25. The result shows that the remaining digits are in fact a repeating decimal.
Alternatively, to override digits for a single vpa call, change the precision by specifying the second argument.
Find π to 100 significant digits by specifying the second argument.
Restore the original precision value in digitsOld for further calculations.
While symbolic results are exact, they might not be in a convenient form. You can use vpa to numerically approximate exact symbolic results.
Solve a high-degree polynomial for its roots using solve. The solve function cannot symbolically solve the high-degree polynomial and represents the roots using root.
y = solve(x^4 - x + 1, x)
root(z^4 - z + 1, z, 1)
Use vpa to numerically approximate the roots.
0.72713608449119683997667565867496 - 0.43001428832971577641651985839602i
0.72713608449119683997667565867496 + 0.43001428832971577641651985839602i
- 0.72713608449119683997667565867496 + 0.93409928946052943963903028710582i
The value of the digits function specifies the minimum number of significant digits used. Internally, vpa can use more digits than digits specifies. These additional digits are called guard digits because they guard against round-off errors in subsequent calculations.
Numerically approximate 1/3 using four significant digits.
a = vpa(1/3, 4)
Approximate the result a using 20 digits. The result shows that the toolbox internally used more than four digits when computing a. The last digits in the result are incorrect because of the round-off error.
vpa(a, 20)
Hidden round-off errors can cause unexpected results.
Evaluate 1/10 with the default 32-digit precision, and then with the 10 digits precision.
a = vpa(1/10, 32)
b = vpa(1/10, 10)
Superficially, a and b look equal. Check their equality by finding a - b.
The difference is not equal to zero because b was calculated with only 10 digits of precision and contains a larger round-off error than a. When you find a - b, vpa approximates b with 32 digits. Demonstrate this behavior.
a - vpa(b, 32)
Unlike exact symbolic values, double-precision values inherently contain round-off errors. When you call vpa on a double-precision input, vpa cannot restore the lost precision, even though it returns more digits than the double-precision value. However, vpa can recognize and restore the precision of expressions of the form p/q, pπ/q, (p/q)1/2, 2q, and 10q, where p and q are modest-sized integers.
First, demonstrate that vpa cannot restore precision for a double-precision input. Call vpa on a double-precision result and the same symbolic result.
dp = log(3);
s = log(sym(3));
dpVpa = vpa(dp)
sVpa = vpa(s)
d = sVpa - dpVpa
dpVpa =
sVpa =
As expected, the double-precision result differs from the exact result at the 16th decimal place.
Demonstrate that vpa restores precision for expressions of the form p/q, pπ/q, (p/q)1/2, 2q, and 10q, where p and q are modest sized integers, by finding the difference between the vpa call on the double-precision result and on the exact symbolic result. The differences are 0.0 showing that vpa restores lost precision in the double-precision input.
vpa(1/3) - vpa(1/sym(3))
vpa(pi) - vpa(sym(pi))
vpa(1/sqrt(2)) - vpa(1/sqrt(sym(2)))
vpa(2^66) - vpa(2^sym(66))
vpa(10^25) - vpa(10^sym(25))
x — Input to evaluate
number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic multidimensional array | symbolic expression | symbolic function | symbolic character vector
Input to evaluate, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, multidimensional array, expression, function, or character vector.
d — Number of significant digits
Number of significant digits, specified as an integer. d must be greater than 1 and lesser than
{2}^{29}+1
vpa does not convert fractions in the exponent to floating point. For example, vpa(a^sym(2/5)) returns a^(2/5).
vpa uses more digits than the number of digits specified by digits. These extra digits guard against round-off errors in subsequent calculations and are called guard digits.
When you call vpa on a numeric input, such as 1/3, 2^(-5), or sin(pi/4), the numeric expression is evaluated to a double-precision number that contains round-off errors. Then, vpa is called on that double-precision number. For accurate results, convert numeric expressions to symbolic expressions with sym. For example, to approximate exp(1), use vpa(exp(sym(1))).
If the second argument d is not an integer, vpa rounds it to the nearest integer with round.
vpa restores precision for numeric inputs that match the forms p/q, pπ/q, (p/q)1/2, 2q, and 10q, where p and q are modest-sized integers.
Atomic operations using variable-precision arithmetic round to nearest.
The differences between variable-precision arithmetic and IEEE Floating-Point Standard 754 are
Inside computations, division by zero throws an error.
The exponent range is larger than in any predefined IEEE mode. vpa underflows below approximately 10^(-323228496).
Denormalized numbers are not implemented.
Zeroes are not signed.
The number of binary digits in the mantissa of a result may differ between variable-precision arithmetic and IEEE predefined types.
There is only one NaN representation. No distinction is made between quiet and signaling NaN.
No floating-point number exceptions are available.
digits | double | root | vpaintegral
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§ Blazing fast math rendering on the web
So, I've shifted the blog to be static-site-generated using a static-site-generator written by yours truly. The code clocks in at around a thousand lines of C++:
bollu/bollu.github.io/builder/builder.cpp
My generator is a real compiler, so I get errors on math and markdown malformation.
I can write math that loads instantly on your browser, using no MathJax, KaTeX or any client side processing, nor the need to fetch images, which looks like this:
h(x) \equiv \begin{cases} \int_{i=0}^\infty f(x) g(x) dx & x > 0 \\ \sum_{i=0}^\infty f(x) + g(x) & \text{otherwise} \end{cases}
My blog is a single 9000 line markdown file , rendered as a single HTML page , so I need it to compile fast, render fast, render beautiful . Existing tools compromise on one or the other.
§ No seriously, why a single markdown file?
I need a single file to edit, so I can rapidly jot down new ideas. This is the essence of why I'm able to log most of what I study: because it's seamless . Far more importantly, it provides spatio-temporal locality . I add things in chronological order to tbe blog, as I learn thing. If I need to recall something I had studied, go to that location in the blog based on a sense of when . When I do get to a location I want, the scrollbar gives me a sense of where I am in the file. this is important to me, since it hepls me reason spatially about what i know and what I've learnt. It's someting I love about books, and deeply miss when navigtaing the web.I'm determined to keep this spatio-temporal locality on my little slice of the internet.
§ Why is this awful?
As elegant as this model is to edit , it's awful for browsers to render. The file used to take on the order of minutes for all the math to finish rendering. MathJax (and KaTeX) painfully attempt to render each math block. As they do, the page jumps around until everything has settled. As this is happening, your CPU throttles, your lap or hand gets warm, and the page is stuck. Clearly not great UX. I still want math. What do I do? The solution is easy: Approximate the math rendering using ASCII/UTF-8 characters! There are tools that do this --- hevea is one of them. Unfortunately, there is no markdown-based-blogging-platform that uses this, so I had to write my own.
§ The cure
The solution is easy. I wrote the tool. The page you're reading it is rendered using the tool. All the math renders in under a second because it's nothing crazy, it's just text and tables which browsers know how to render. No JavaScript necessary. snappy performance. Whoo!
§ The details: Writing my own Markdown to HTML transpiler.
the final transpiler clocks in at 1300Loc of C++, which is very small for a feature-complete markdown-to-HTML piece of code that's blazing fast, renders math correctly, and provides error messages.
§ Quirks fixed, features gained.
I got quite a bit "for free" as I wrote this, fixing mild annoyances and larger pain points around using github + markdown for publishing on the web:
I really don't want tables, but I do want the ability to write vertical bars | freely in my text. Unfortunately, github insists that those are tables, and completely wrecks rendering.
I get line numbers in code blocks now, which Github Flavoured Markdown did not have.
I get error messages on incorrectly closed bold/italic/code blocks, using heuristics that prevent them from spanning across too many lines.
I get error messages on broken latex, since all my latex passes through hevea. This is awesome, since I no longer need to refresh my browser, wait for mathjax to load, go make myself tea (remember that mathjax was slow?), and then come back to see the errors.
I can get error messages if my internal document links are broken. To be fair, my tool doesn't currently give me these errors, but it can (and soon will).
In general, I get control , which was something I did not have with rendering directly using Github, or using someone else's tool.
§ Choice of language
I choose to write this in C-style-C++, primarily because I wanted the tool to be fast, and I'd missed writing C++ for a while. I really enjoy how stupid-simple C style C++ turns out to be: the C++ papers over some of C's annoyances (like formatted output for custom types), while still preserving the KISS feeling of writing C++. Why not Rust? I freely admit that rust might have been a sane choice as well. unfortunately, asking rust to treat UTF-8 string as a "ball of bytes" is hard, when it's stupidly easy with C. Plus, I wanted to use arena-style-allocation where I make huge allocations in one go and then don't think about memory, something that I don't have control over in Rust. I don't have any segfaults (yet, perhaps), thanks to UBSAN and ASAN. I find Rust to have more impedance than C on small applications, and this was indeed small.
Everything except the latex to HTML is blazing fast. Unfortunately, calling hevea is slow, so I implemented a caching mechanism to make using hevea not-slow. hevea does not have an API, so I need to fork and talk to its process which is understandably flow. I built a "key-value-store" (read: serialize data into a file) with the stupidly-simple approach of writing an append-only log into a file. hevea is a pure function conceptally, since on providing the same latex input it's going to produce the same HTML output, so it's perfectly safe to cache it:
const char DB_PATH[]="./blogcache.txt";
unordered_map<ll, const char *> G_DB;
void loadDB() {
G_DB = {};
FILE *f = fopen(DB_PATH, "rb");
ll k, len;
fread(&k, sizeof(ll), 1, f); if (feof(f)) break;
fread(&len, sizeof(ll), 1, f);
char *buf = (char *)calloc(sizeof(char), len + 2);
const char *lookup_key(ll k) {
unordered_map<ll, const char *>::iterator it = G_DB.find(k);
if (it == G_DB.end()) { return nullptr; } return it->second;
void store_key_value(const ll k, KEEP const char *v, const ll len) {
assert(G_DB.count(k) == 0);
G_DB.insert(make_pair(k, strdup(v)));
FILE *f = fopen(DB_PATH, "ab");
assert(f != nullptr && "unable to open DB file");
fwrite(&k, sizeof(ll), 1, f);
fwrite(&len, sizeof(ll), 1, f);
fwrite(v, sizeof(char), len, f);
§ For the future
I plan to rip out hevea and write my own latex -> HTML converter for the subset of LaTeX I actually use . hevea's strength is its downfall: It can handle all of LaTeX, which means it's really slow. If I can concentrate on a small subset, I don't need to play caching tricks, and I can likely optimise the layout further for my use-cases. I also want colored error messages, because who doesn't? I'll probably gradually improve my static site generator over time. Once it's at a level of polish where I'm happy with it, I'll spin it out as a separate project.
Am I glad I did it? Yes, purely because my chunk of the internet aligns with how I want it to be, and that makes me
\epsilon
more happy. I think of it as an investment into future me, since I can extend the markdown and the transpiler in the way I want it to be.
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Three-winding brushless DC motor with trapezoidal flux distribution - MATLAB - MathWorks Nordic
\frac{d\Phi }{dt}=\frac{\partial \Phi }{\partial \theta }\frac{d\theta }{dt}=\frac{\partial \Phi }{\partial \theta }\omega ,
\frac{\partial \Phi }{\partial \theta }
{\Phi }_{max}=\frac{h}{2}\left({\theta }_{F}+{\theta }_{W}\right),
h=2{\Phi }_{max}/\left({\theta }_{F}+{\theta }_{W}\right).
\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]=\left[\begin{array}{ccc}{R}_{s}& 0& 0\\ 0& {R}_{s}& 0\\ 0& 0& {R}_{s}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}\frac{d{\psi }_{a}}{dt}\\ \frac{d{\psi }_{b}}{dt}\\ \frac{d{\psi }_{c}}{dt}\end{array}\right],
\frac{d{\psi }_{a}}{dt},
\frac{d{\psi }_{b}}{dt},
\frac{d{\psi }_{c}}{dt}
\left[\begin{array}{c}{\psi }_{a}\\ {\psi }_{b}\\ {\psi }_{c}\end{array}\right]=\left[\begin{array}{ccc}{L}_{aa}& {L}_{ab}& {L}_{ac}\\ {L}_{ba}& {L}_{bb}& {L}_{bc}\\ {L}_{ca}& {L}_{cb}& {L}_{cc}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}{\psi }_{am}\\ {\psi }_{bm}\\ {\psi }_{cm}\end{array}\right],
{L}_{aa}={L}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{r}\right),
{L}_{bb}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{r}-2\pi /3\right)\right),
{L}_{cc}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{r}+2\pi /3\right)\right),
{L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{r}+\pi /6\right)\right),
{L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{r}+\pi /6-2\pi /3\right)\right),
{L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{r}+\pi /6+2\pi /3\right)\right),
\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\\ {v}_{0}\end{array}\right]=P\left(\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]-N\omega \left[\begin{array}{c}\frac{\partial {\psi }_{am}}{\partial {\theta }_{r}}\\ \frac{\partial {\psi }_{bm}}{\partial {\theta }_{r}}\\ \frac{\partial {\psi }_{cm}}{\partial {\theta }_{r}}\end{array}\right]\right),
{v}_{d}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}-N\omega {i}_{q}{L}_{q}
{v}_{q}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}+N\omega {i}_{d}{L}_{d},
{v}_{0}={R}_{s}{i}_{0}+{L}_{0}\frac{d{i}_{0}}{dt}
T=\frac{3}{2}N\left({i}_{q}{i}_{d}{L}_{d}-{i}_{d}{i}_{q}{L}_{q}\right)+\left[\begin{array}{ccc}{i}_{a}& {i}_{b}& {i}_{c}\end{array}\right]\left[\begin{array}{c}\frac{\partial {\psi }_{am}}{\partial {\theta }_{r}}\\ \frac{\partial {\psi }_{bm}}{\partial {\theta }_{r}}\\ \frac{\partial {\psi }_{cm}}{\partial {\theta }_{r}}\end{array}\right],
P=2/3\left[\begin{array}{ccc}\mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}-2\pi /3\right)& \mathrm{cos}\left({\theta }_{e}+2\pi /3\right)\\ -\mathrm{sin}{\theta }_{e}& -\mathrm{sin}\left({\theta }_{e}-2\pi /3\right)& -\mathrm{sin}\left({\theta }_{e}+2\pi /3\right)\\ 0.5& 0.5& 0.5\end{array}\right]
\frac{\partial {\psi }_{am}}{\partial {\theta }_{r}},
\frac{\partial {\psi }_{bm}}{\partial {\theta }_{r}},
\frac{\partial {\psi }_{cm}}{\partial {\theta }_{r}}
\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]=P\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]
{V}_{{m}_{rms}}^{}
{P}_{OC}\left({V}_{{m}_{rms}}^{}\right)=\frac{{a}_{h}}{k}{V}_{{m}_{rms}}^{}+\frac{{a}_{j}}{{k}^{2}}{V}_{{m}_{rms}}^{2}+\frac{{a}_{ex}}{{k}^{1.5}}{V}_{{m}_{rms}}^{1.5}
k={V}_{{m}_{rms}}^{}/f
{V}_{{d}_{rms}}^{*}
{P}_{SC}\left({V}_{{d}_{rms}}^{*}\right)=\frac{{b}_{h}}{k}{V}_{{d}_{rms}}^{*}+\frac{{b}_{j}}{{k}^{2}}{V}_{{d}_{rms}}^{*2}+\frac{{b}_{ex}}{{k}^{1.5}}{V}_{{d}_{rms}}^{*1.5}
2\pi /N
2\pi /N
2\pi /N
{L}_{d}={f}_{1}\left({i}_{d},{i}_{q}\right)
{L}_{d}={f}_{2}\left({i}_{d},{i}_{q}\right)
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Arbeitsgemeinschaft mit aktuellem Thema: Twisted K-Theory | EMS Press
The Arbeitsgemeinschaft mit aktuellem Thema ``Twisted K-theory'', organized by Ulrich Bunke (Georg-August-Universit\"at G\"ottingen), Dan Freed (University of Texas at Austin), and Thomas Schick (Georg-August-Universit\"at G\"ottingen), was held from October 8 through October 14, 2006.
Ordinary (co)homology theories always come along with twisted versions of themselves; the most basic example is cohomology twisted by the orientation bundle, which shows up when one discusses Poincar\'e duality (and more general push-forward maps) for non-orientable manifolds.
K
-theory is important from this point of view. Even if a manifold is orientable in the ordinary sense, its
K
-theory does not satisfy Poincar\'e duality. However, this is the case if one considers twisted
K
-theory (one has to twist with the bundle of complex Clifford algebras of the cotangent bundle). In fact, some constructions of twisted cohomology theories are quite classical and can be done in the context of parametrized stable homotopy theory (which on the other hand is itself constantly developing further).
The modern interest in twisted
K
-theory, however, stems from mathematical physics, in particular from string theory. In this theory
D
-branes are objects whose charges are measured, in the presence of a B-field, by twisted
K
-theory. The topological backgrounds of B-fields
\beta
X
are classified by three dimensional integral cohomology classes. Representatives of the B-field (which were called twists during the Arbeitsgemeinschaft) are precisely the data needed to define twisted
K
K^\beta(X)
. In one version of the theory one associates to a twist a non-commutative
C^*
-algebra whose
K
-theory is by definition the twisted
K
K
-theory in mathematics has evolved to an interdisciplinary area which combines elements of topology, non-commutative geometry, functional analysis, representation theory, mathematical physics and other. In the Arbeitsgemeinschaft we presented various aspects of the foundations of twisted
K
-theory and the key calculations. We discussed the construction of twisted equivariant
K
-theory in different contexts (e.g. homotopy theory, non-commutative geometry, groupoids or stacks) and the verification of the basic functorial properties.
In order to get used to the definitions we made example calculations of twisted
K
-theory using methods from algebraic topology (Mayer-Vietoris sequences and some spectral sequences). The Umkehr- (or integration or Gysin map) for twisted
K
-theory is of particular importance and was illustrated through an interpretation of the classical Borel-Weil-Bott theorem. The culmination of the Arbeitsgemeinschaft was the calculation of the equivariant twisted
K
-theory of compact Lie groups due to Freed-Hopkins-Teleman and the interpretation of this result in the context of representation theory of loop groups.
In the string theory context, some aspects of mirror symmetry are reflected in twisted
K
-theory; under certain situations a non-commutative space-time (i.e. a space with
B
-field) will have a dual with isomorphic
K
-theory, possibly in shifted degrees. The mathematical formulation of this is
T
-duality, which was worked out at the AG, and was studied also as a computational tool.
T
-duality, but also in the equivariant situation not only ordinary spaces but more singular objects naturally show up. This results in the need to work out equivariant twisted
K
-theory, twisted
K
-theory for orbifolds, and for even more singular spaces. A convenient framework to develop this in the necessary generality is the language of stacks, which was introduced and used during the AG.
A Chern character was constructed which relates twisted
K
-theory to twisted (de Rham) cohomology. During the AG, it was worked out that sheaf theory is important here and developments in twisted
K
-theory are in fact topological versions of similar results and constructions in algebraic geometry.
An interesting feature of the definition of twisted
K
-theory in terms of cycles and relations is that those cycles appear naturally in geometric and analytic constructions. As mentioned above, starting from representations of compact Lie groups, an explicit construction of cycles given by families of Dirac type operators was discussed in connection with the Borel-Weil-Bott theorem. In a more elaborate way, this also works for (projective) representations of loop groups, as was first discovered in the physics related literature. Using this idea the calculation of the equivariant twisted
K
-theory of a compact Lie group
G
(acting on itself by conjugation) by Freed-Hopkins-Teleman (FHT) can be explicitly interpreted in terms of cycles. In this way the twisted
K
-theory is identified with the
K
-group of projective positive energy representations of the loop group
\mathcal{L}G
. The twist corresponds to the ``level'' of the representation. It was one of the major goals of the AG to prove this FHT theorem.
It turns out that the twisted
K
-theory on the one side, and the
K
-group of projective positive energy representations on the other side, both have a subtle products (in
K
-theory the Pontryagin product induced by the multiplcation map, and the Fusion product (Verlinde algebra) on the other side), and the FHT-isomorphism respects these multiplications. The construction of the
K
-theoretic product was addressed, but because of lack of time we were not able to prove multiplicativity of the FHT-isomorphism. In the Arbeitsgemeinschaft we actually discussed a related product for the twisted
K
-theory of orbifolds with is closely related to the quantum product in orbifold cohomology.
Altogether, there were 17 talks by the participants, two sessions where questions left open during the talks were discussed, and ample free interaction between the participants.
The conference was attended by 45 participants coming mainly from all over Europe, Northern America and Australia. It is a pleasure to thank the institute for providing a pleasant and stimulating atmosphere.
Thomas Schick, Ulrich Bunke, Daniel S. Freed, Arbeitsgemeinschaft mit aktuellem Thema: Twisted K-Theory. Oberwolfach Rep. 3 (2006), no. 4, pp. 2757–2804
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Extended Kalman filter for object tracking - MATLAB - MathWorks 日本
The function must return bounds as an M-by-2 real-valued matrix, where M is the size of z(k). In each row, the first and second elements specify the lower and upper bounds, respectively, for the corresponding measurement variable. You can use −Inf or Inf to represent that the variable does not have a lower or upper bound.
The second argument is the wrapping bounds, returned as an M-by-2 real-valued matrix, where M is the dimension of the measurement. In each row, the first and second elements are the lower and upper bounds for the corresponding measurement variable. You can use −Inf or Inf to represent that the variable does not have a lower or upper bound.
xpred = 4×1
Ppred = 4×4
{x}_{k+1}=f\left({x}_{k},{u}_{k},{w}_{k},t\right)
{x}_{k+1}=f\left({x}_{k},{u}_{k},t\right)+{w}_{k}
{z}_{k}=h\left({x}_{k},{v}_{k},t\right)
{z}_{k}=h\left({x}_{k},t\right)+{v}_{k}
[2] Kalman, R. E. “A New Approach to Linear Filtering and Prediction Problems.†Transactions of the ASME–Journal of Basic Engineering. Vol. 82, Series D, March 1960, pp. 35–45.
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Electron Behavior and Periodic Properties of Elements - Vocabulary - Course Hero
General Chemistry/Electron Behavior and Periodic Properties of Elements/Vocabulary
height of a wave, measured as the distance from the point of equilibrium to a crest or a trough
principle that states that electrons fill orbitals in order of increasing energy
idea that matter may behave as a wave
wavelength (
\lambda
) of a particle having mass m and traveling at velocity v using Planck's constant, h; given as
\lambda=\frac h{mv}
one of two or more orbitals with the same energy
wave of energy produced by the movement of particles through space
entire range of electromagnetic waves, defined by their energy, wavelengths, and frequencies
orbital filling of electrons in an atom based on their quantum numbers in increasing energies
area of an atom in which an electron has the greatest probability of being located. Each orbital can contain at most two electrons.
one or more electron subshells that have the same quantum number
n
electron subshell
group of electron energy levels with the same size and shape that have the same quantum numbers
n
\ell
electron state that has a higher energy level than the lowest energy state (ground state)
number of oscillations of a wave that occur in a given period of time, usually a second, measured in hertz (Hz)
electron state that is lowest in energy
principle that it is impossible to simultaneously measure the position and the momentum of a particle
rule that states that when filling degenerate orbitals (for example, the 3p orbital), electrons must first singly occupy all the empty orbitals in the subshell before pairing within the same orbital
pattern of colors on a dark background produced by an element when it gives off light (emission) or a pattern of dark lines on a background of the electromagnetic spectrum produced by an element when it absorbs light (absorption)
point on a standing wave that stays fixed and does not oscillate
principle that states that paired electrons may never have the same spin value, which means they cannot have the same four quantum numbers
emission of electrons when light strikes a material
fundamental particle that has a fixed amount of energy stored as electromagnetic radiation
6.62607\times 10^{-34}\;\rm{J}\cdot\rm{s}
; describes the ratio of energy to frequency of a photon
exists only as discrete values
branch of science that deals with subatomic particles, their behaviors, and their interactions
one of four numbers,
n
\ell
m
s
, that together describe the orbital state of subatomic electrons
periodic oscillation of energy or matter
mathematical expression that gives information about measurable properties of a system, such as energy, momentum, and position
having properties of both particles and waves
distance between two identical parts of a wave
<Overview>Electromagnetic Energy
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Probability ContentsและInterpretations[edit]
The word probability derives from the Latin probabilitas, which can also mean "probity", a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility. In a sense, this differs much from the modern meaning of probability, which in contrast is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.[9]
The sixteenth-century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes[14]). Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.[15] Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics.[16] See Ian Hacking's The Emergence of Probability[9] and James Franklin's The Science of Conjecture[17] for histories of the early development of the very concept of mathematical probability.
{\displaystyle \phi (x)=ce^{-h^{2}x^{2}},}
{\displaystyle h}
{\displaystyle c}
is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's (1850).[citation needed] Gauss gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W.F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula[clarification needed] for r, the probable error of a single observation, is well known.
{\displaystyle \Omega }
{\displaystyle P(A)}
{\displaystyle p(A)}
{\displaystyle {\text{Pr}}(A)}
{\displaystyle A',A^{c}}
{\displaystyle {\overline {A}},A^{\complement },\neg A}
{\displaystyle {\sim }A}
{\displaystyle =1-{\tfrac {1}{6}}={\tfrac {5}{6}}}
{\displaystyle P(A\cap B)}
{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=P(A)P(B).}
{\displaystyle {\tfrac {1}{2}}\times {\tfrac {1}{2}}={\tfrac {1}{4}}}
{\displaystyle P(A\cap B)}
{\displaystyle P(A{\mbox{ and }}B)=P(A\cap B)=0}
{\displaystyle P(A\cup B)}
{\displaystyle P(A{\mbox{ or }}B)=P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-0=P(A)+P(B)}
{\displaystyle P(1{\mbox{ or }}2)=P(1)+P(2)={\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{3}}.}
{\displaystyle P\left(A{\hbox{ or }}B\right)=P(A\cup B)=P\left(A\right)+P\left(B\right)-P\left(A{\mbox{ and }}B\right).}
{\displaystyle {\tfrac {13}{52}}+{\tfrac {12}{52}}-{\tfrac {3}{52}}={\tfrac {11}{26}}}
{\displaystyle P(A\mid B)}
{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}.\,}
{\displaystyle P(B)=0}
{\displaystyle P(A\mid B)}
is formally undefined by this expression. However, it is possible to define a conditional probability for some zero-probability events using a σ-algebra of such events (such as those arising from a continuous random variable).[citation needed]
{\displaystyle 1/2}
{\displaystyle 1/3}
, since only 1 red and 2 blue balls would have been remaining.
{\displaystyle A_{1}}
{\displaystyle A_{2}}
{\displaystyle B}
{\displaystyle A_{1}}
{\displaystyle A_{2}}
{\displaystyle A}
{\displaystyle P(A|B)\propto P(A)P(B|A)}
{\displaystyle A}
{\displaystyle B}
{\displaystyle P(A)\in [0,1]\,}
{\displaystyle P(A^{\complement })=1-P(A)\,}
{\displaystyle {\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&=P(A)+P(B)\qquad {\mbox{if A and B are mutually exclusive}}\\\end{aligned}}}
{\displaystyle {\begin{aligned}P(A\cap B)&=P(A|B)P(B)=P(B|A)P(A)\\P(A\cap B)&=P(A)P(B)\qquad {\mbox{if A and B are independent}}\\\end{aligned}}}
{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}\,}
^ Shoesmith, Eddie (November 1985). "Thomas Simpson and the arithmetic mean". Historia Mathematica. 12 (4): 352–355. doi:10.1016/0315-0860(85)90044-8.
^ "Data: Data Analysis, Probability and Statistics, and Graphing". archon.educ.kent.edu. Retrieved 28 May 2017.
^ Gorman, Michael F. (2010). "Management Insights". Management Science. 56: iv–vii. doi:10.1287/mnsc.1090.1132.
^ Burgin, Mark (2010). "Interpretations of Negative Probabilities". p. 1. arXiv:1008.1287v1 [physics.data-an].
vteLogic
vteGlossaries of science and engineering
|
Brillouin function - zxc.wiki
for different values of J
The Brillouin function (after the French-American physicist Léon Brillouin (1889–1969)) is a special function that emerges from the quantum mechanical description of a paramagnet :
{\ displaystyle B (x)}
{\ displaystyle {\ begin {alignedat} {2} B_ {J} (x) & = {\ frac {2J + 1} {2J}} \ cdot \ coth \ left ({\ frac {2J + 1} {2J }} \, x \ right) && - {\ frac {1} {2J}} \ cdot \ coth \ left ({\ frac {1} {2J}} \, x \ right) \\ & = \ left ( 1 + {\ frac {1} {2J}} \ right) \ cdot \ coth \ left [\ left (1 + {\ frac {1} {2J}} \ right) x \ right] && - {\ frac { 1} {2J}} \ cdot \ coth \ left ({\ frac {1} {2J}} \, x \ right) \ end {alignedat}}}
The symbols represent the following quantities :
{\ displaystyle J}
in the physical application for the total angular momentum quantum number
{\ displaystyle \ coth}
for the hyperbolic cotangent .
With the Brillouin function, the magnetization of a paramagnet of the amount of substance in an external magnetic field can be formulated:
{\ displaystyle M}
{\ displaystyle N}
{\ displaystyle {\ begin {aligned} M & = NmB_ {J} (\ xi) \\\ Leftrightarrow B_ {J} (\ xi) & = {\ frac {M} {Nm}}. \ end {aligned}} }
the magnetic moment of a particle
{\ displaystyle m}
{\ displaystyle \ xi = {\ frac {mB} {k _ {\ mathrm {B}} \, T}} = {\ frac {g \ mu _ {\ mathrm {B}} \, JB} {k _ {\ mathrm {B}} \, T}}}
the magnitude of the magnetic flux density of the applied external magnetic field
{\ displaystyle B}
the Boltzmann constant
{\ displaystyle k _ {\ mathrm {B}}}
the absolute temperature
{\ displaystyle T}
the Landé factor
{\ displaystyle g}
the Bohr magneton .
{\ displaystyle \ mu _ {\ mathrm {B}}}
Another, semi-classical description of a paramagnet occurs with the help of the Langevin function , which results in the Limes and at the same time from the Brillouin function (whereby the total magnetic moment remains constant):
{\ displaystyle L}
{\ displaystyle J \ to \ infty}
{\ displaystyle g \ mu _ {\ mathrm {B}} \ to 0}
{\ displaystyle {\ begin {aligned} M & = NmL (\ xi) \\\ Leftrightarrow L (\ xi) & = {\ frac {M} {Nm}}. \ end {aligned}}}
Torsten Fließbach: Statistical Physics - Textbook on Theoretical Physics IV . Elsevier Spectrum Academic Publishing House, Heidelberg 2006.
Free spins in the magnetic field
Magnetism and Thermodynamics
This page is based on the copyrighted Wikipedia article "Brillouin-Funktion" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Litepaper - Socean
Socean's vision
Socean aims to solve the following pain points:
Billions of dollars worth of staked SOL sit idle instead of serving as collateral or as liquidity in financial markets.
The lack of good tools and infrastructure to monitor and manage stake means that most users delegate to a small handful of nodes--typically the most prominent--and hardly change their delegations thereafter. This comes at the expense of balancing staking rewards and risk and incentivising better network performance.
The Socean stake pool aims to solve these pain points via the following:
The scnSOL token, representing ownership of staked SOL in the Socean pool, will allow holders to participate and profit in the DeFi ecosystem while earning full staking rewards. We think it is likely that scnSOL will replace the use of SOL in many DeFi use-cases, including market making, margin trading, lending, yield farming, and further DeFi innovations in the Solana ecosystem.
Socean will provide a suite of tools to make it easy to monitor stake allocation and performance metrics. The protocol makes decisions about its dynamic allocation via a principled, unbiased delegation strategy that takes into account historical data and network health. We aim to maximise staking rewards, reduce specific risk and increase decentralization, and meritocratically support well-performing, lesser-known validators.
Socean's long term goal is to become a fully-autonomous stake management system. This objective will be achieved in stages, gradually replacing managerial roles with on-chain and permissionless infrastructure.
Socean will launch with an initial delegation strategy, then transition into a more sophisticated optimal delegation strategy.
Initial delegation strategy
Socean will start by delegating in equal proportions to the top-performing N (~50) validators that are not in either the minimal security group or the minimal data center security group.
Top performing: validators with the highest staking returns for the past 5 epochs
Minimal security group: minimum number of validators whose cumulative stake exceeds 33.3% of all staked SOL
Minimal data center security group: minimum number of data centers hosting validators whose cumulative stake exceeds 33.3% of all staked SOL
The significance of 33.3% is that it corresponds to the amount of stake required to halt the network.
Future delegation strategy
The Socean team will base its eventual delegation strategy on Bayesian mean-variance optimisation and utility theory. We give a simplified explanation below. The full explanation can be found in our technical whitepaper below
We first define our utility function, which expresses how much we care about APY, risk, and decentralisation:
U = -e^{-\beta \cdot TotalSOL - \alpha \cdot NetworkHealth(\mathbf{w})}
\beta
is the degree of risk aversion, and
\mathbf{w}
is a weighted allocation of stake.
The network health is a function of stake concentration in individual nodes as well as datacenters and jurisdictions.
We use recent work on Bayesian statistics (Bauder et al. 2020) to express the expected utility as a function of historical data, and use gradient descent to find a weighted allocation corresponding to the approximate maximum utility.
Socean will transition over time into becoming fully autonomous. Holders of our governance tokens will be able to vote on pool parameters, including parameters such as
\alpha
\beta
You can read more details in our technical whitepaper below:
socean_delegation_strategy-2021-07-30.pdf
Delegation strategy technical whitepaper
Socean aims to transition into a fully autonomous, decentralised stake management system. We plan to introduce a governance token that allows tokenholders to vote on the delegation strategy, the management fees, and redeem their tokens for the fees that Socean earns. We will provide more details at a later date.
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Comparing Quantities, Popular Questions: CBSE Class 8 ENGLISH, English Grammar - Meritnation
A boat which has a speed of 5 km/hr in still water crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water in km/hr is ?
Question (Also give the process to solve):
Q. Maya has Rs. M with her and her friend Chanda has Rs. C with her. Maya spends 12 % of her money and Chand also spends the same amount as Maya did. What percentage of her money did chanda spend?
\frac{18M}{C}
\frac{18C}{M}
\frac{12M}{C}
\frac{12C}{M}
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Letxbe the cost of the shirt without including VAT.
Cost of shirt including VAT = Selling price + VAT
Rs 896 =x+ 12% ofx
Rs 896 =
Thus, the selling price of the shirt without including VAT is Rs 800.
this is given in maths comparing quantities but i dint get how
cause 896*100/112 = 200
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Plz solve urgently dear experts with proper explanation and steps plz:
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Limit of a sequence - Citizendium
The mathematical concept of limit of a sequence provides a rigorous definition of the idea of a sequence converging towards a point called the limit.
Suppose x1, x2, ... is a sequence of real numbers. We say that the real number L is the limit of this sequence and we write
{\displaystyle \lim _{n\to \infty }x_{n}=L}
if and only if for every real number ε > 0 there exists a natural number n0 such that for all n > n0 we have |xn − L| < ε. The number n0 will in general depend on ε.
Retrieved from "https://citizendium.org/wiki/index.php?title=Limit_of_a_sequence&oldid=345990"
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Quantum computation - Citizendium
Throughout this article the Many Worlds Interpretation (MWI) of quantum mechanics is used.
1 Differences with classical computation
2 Quantum computers and information theory
3 Interference & a simple computation
4.2 Dynamics of quantum gates in Schrödinger picture
4.2.2 Controlled Not, CNOT
4.2.3 Hadamard gate, H
Differences with classical computation
In classical computation there is the concept of a discrete bit, taking only one of two values. However , the world which classical physics describes is that of continua. Thus this is obviously not an ideal way of attempting to describe or simulate the world in which we live. Feynman was the first to consider the idea of a quantum computer being necessary to simulate the quantum mechanical world in which we live.[1]
Quantum computers and information theory
The quantum mechanical analogue of the classical bit is the qubit. A qubit is an actual physical system, all of whose observables are Boolean.
Interference & a simple computation
An algorithm is a hardware-independent recipe for performing a particular computation. A program is a way of preparing a specific computer to do such a task. Algorithms for quantum computers offer a wider range of computational tasks which may be solved by the use of interference. Some of the new possibilities which are opened up may prove to have drastic consequences for the future: e.g. Shor's algorithm relevance to cryptography.
An oracle is a black box which it is impossible to look inside and performs a particular function f on the input qubits in a time which is independent of the particular input. It is not possible. They are used to simplify the analysis of algorithms as the specific nature of how a task is performed is irrelevant due to the criterion of hardware-independence. The oracle must of course be reversible as the laws of QM do not make a distinction between forwards and backwards time (i.e. time does not have 'an arrow').
Dynamics of quantum gates in Schrödinger picture
NB In the Schrödinger picture the state vector evolves in the following manner:
{\displaystyle \left|\psi (t+1)\right\rangle =U\left|\psi (t)\right\rangle }
{\displaystyle U}
is the characteristic unitary matrix of the gate.
{\displaystyle \left|a\right\rangle \to \left|-a\right\rangle ,\qquad a=\pm 1}
{\displaystyle \left|a\right\rangle =\left|\psi (0)\right\rangle }
{\displaystyle \left|-a\right\rangle =\left|\psi (1)\right\rangle }
Controlled Not, CNOT
This i a two input gate where whether the NOT operation is performed on the second is dependant (i.e. controlled) by the value of the first input bit which is unchanged.
{\displaystyle \left|x\right\rangle \left|y\right\rangle \equiv \left|x,y\right\rangle \to \left|x,xy\right\rangle }
Hadamard gate, H
The Hadamard gate, normally denoted as H, creates an equally weighted superposition of the
{\displaystyle \left|1\right\rangle }
{\displaystyle \left|-1\right\rangle }
states. There is no classical analogue of it.
{\displaystyle \left|1\right\rangle \to {\frac {1}{\sqrt {2}}}(\left|1\right\rangle +\left|-1\right\rangle )}
{\displaystyle \left|-1\right\rangle \to {\frac {1}{\sqrt {2}}}(\left|1\right\rangle -\left|-1\right\rangle )}
{\displaystyle H^{2}=I}
i.e the Hadamard gate is self-inverse.
An oracle which performs the function
{\displaystyle f(x)}
has the following dynamics in the Schrödinger picture. The value of y is normally set to one so that the output is
{\displaystyle x}
{\displaystyle f(x)}
{\displaystyle \left|x,y\right\rangle \to \left|x,yf(x)\right\rangle }
Let us create an oracle which performs the following function:
{\displaystyle f:\{-1,1\}\mapsto \{-1,1\}}
There are four possibilities for this function:
{\displaystyle x\mapsto x}
not:
{\displaystyle x\mapsto -x}
output -1:
{\displaystyle x\mapsto -1}
{\displaystyle x\mapsto 1}
Computational task: To determine if
{\displaystyle f(1)=f(-1)}
This is equivalent to trying to determine
{\displaystyle f(1)f(-1)}
without looking inside the oracle above. Classically this may be done by consulting the oracle twice. [diagram of classical situation] However, using quantum computation the oracle need only be consulted once.
(N.B. For the analysis of algorithms, the Schrödinger picture is often preferable and thus shall be used here. It is of course still possible to use the Heisenberg picture.
{\displaystyle \left|\psi (0)\right\rangle =\left|1,1\right\rangle }
{\displaystyle \left|\psi (1)\right\rangle =\left|1,-1\right\rangle }
{\displaystyle \left|\psi (2)\right\rangle ={\frac {1}{2}}(\left|1\right\rangle +\left|-1\right\rangle )(\left|1\right\rangle -\left|-1\right\rangle )={\frac {1}{2}}(\left|1,1\right\rangle +\left|-1,1\right\rangle -\left|1,-1\right\rangle -\left|-1,-1\right\rangle )}
{\displaystyle \left|\psi (3)\right\rangle ={\frac {1}{2}}(\left|1,f(1)\right\rangle +\left|-1,f(1)\right\rangle -\left|1,-f(-1)\right\rangle -\left|-1,-f(-1)\right\rangle )}
Now we must examine the two possibilities 1)
{\displaystyle f(1)=f(-1)}
{\displaystyle f(1)=-f(-1)}
{\displaystyle f(1)=f(-1)=f}
{\displaystyle \Rightarrow \left|\psi (3)\right\rangle ={\frac {1}{2}}(\left|1,f\right\rangle +\left|-1,f\right\rangle -\left|1,-f\right\rangle -\left|-1,-f\right\rangle )={\frac {1}{2}}(\left|1\right\rangle +\left|-1\right\rangle )(\left|f\right\rangle -\left|-f\right\rangle )}
{\displaystyle f(1)=-f(-1)=f}
{\displaystyle \Rightarrow \left|\psi (3)\right\rangle ={\frac {1}{2}}(\left|1\right\rangle -\left|-1\right\rangle )(\left|f\right\rangle -\left|-f\right\rangle )}
In each of these cases the state vector when t=3 is a superposition of two pure states:
{\displaystyle {\frac {1}{\sqrt {2}}}(\left|1\right\rangle \pm \left|-1\right\rangle )}
{\displaystyle {\frac {1}{\sqrt {2}}}(\left|f\right\rangle \pm \left|-f\right\rangle )}
. The purpose of the final Hadamard gate is to differentiate between the states
{\displaystyle {\frac {1}{\sqrt {2}}}(\left|1\right\rangle \pm \left|-1\right\rangle )}
and thus determine whether
{\displaystyle f(1)=f(-1)}
{\displaystyle H{\frac {1}{\sqrt {2}}}(\left|1\right\rangle +\left|-1\right\rangle )=\left|1\right\rangle }
{\displaystyle H{\frac {1}{\sqrt {2}}}(\left|1\right\rangle -\left|-1\right\rangle )=\left|-1\right\rangle }
{\displaystyle \left|\psi (4)\right\rangle ={\frac {1}{\sqrt {2}}}\left|f(1)f(-1)\right\rangle (\left|f\right\rangle -\left|-f\right\rangle }
Shor's algorithm is used to find the factors of a number. It is particularly important because of the use in cryptography of multiplying together two large prime numbers. Factoring this number into its prime factors allows the cracking of the code.
The algorithm itself has two parts: one quantum and one classical. The latter can be done in
{\displaystyle O(x^{n})}
time. The use of a quantum algorithm makes this true for the former part as well.
Let us wish to factor a number n. The first part of the algorithm wishes to find the period, r, of the function:
{\displaystyle \scriptstyle F(a)=x^{a}{\pmod {n}}}
, i.e. find
{\displaystyle r}
{\displaystyle \scriptstyle F(a)=F(a+r)\,}
{\displaystyle r}
has been found by use of quantum parallelism, the second part of the algorithm may be performed:
{\displaystyle x^{0}{\pmod {n}}=1}
{\displaystyle \Rightarrow x^{r}mod\,n=1,x^{2r}{\pmod {n}}=1}
{\displaystyle \Rightarrow x^{r}\equiv 1{\pmod {n}}}
{\displaystyle \Rightarrow (x^{\frac {r}{2}})^{2}=x^{r}\equiv 1{\pmod {n}}}
{\displaystyle \Rightarrow (x^{\frac {r}{2}})^{2}-1\equiv 0{\pmod {n}}}
{\displaystyle \Rightarrow (x^{\frac {r}{2}}-1)(x^{\frac {r}{2}}+1)\equiv 0{\pmod {n}}}
{\displaystyle r}
Based on a talk given by Charles Blackham to 6P at Winchester College, UK on 7/3/07
Cambridge Centre for Quantum Computation
↑ R.P. Feynman International Journal of Theoretical Physics 21(6/7) 1982
Retrieved from "https://citizendium.org/wiki/index.php?title=Quantum_computation&oldid=24617"
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2017 Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function
We introduce a generalized sigmoidal transformation
{w}_{m}\left(r;x\right)
on a given interval
\left[a,b\right]
with a threshold at
x=r\in \left(a,b\right)
{w}_{m}\left(r;x\right)
, we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples.
Beong In Yun. "Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function." Abstr. Appl. Anal. 2017 1 - 7, 2017. https://doi.org/10.1155/2017/1364914
Received: 1 September 2017; Revised: 16 October 2017; Accepted: 19 October 2017; Published: 2017
Beong In Yun "Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function," Abstract and Applied Analysis, Abstr. Appl. Anal. 2017(none), 1-7, (2017)
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Arbeitsgemeinschaft mit aktuellem Thema: Algebraic Cobordism | EMS Press
Over the years, many different types and flavors of cohomology theories for algebraic varieties have been constructed. Theories like \'etale cohomology or de Rham cohomology provide algebraic versions of the topological theory of singular cohomology. The Chow ring and algebraic
K_0
are other (partial) examples, more directly tied to algebraic geometry. The partial theory
K_0^{alg}
was extended to a full theory with the advent of Quillen's higher algebraic
K
-theory. It took considerably longer for the Chow ring to be extended to motivic cohomology. In the process of doing so, Voevodsky developed his category of motives, and this construction was put in a more general setting with the development by Morel-Voevodskyof of
\mathbb{A}^1
homotopy theory. This enabled a systematic construction of cohomology theories on algebraic varieties, with algebraic
K
-theory and motivic cohomology being only two fundamental examples. These two cohomology theories have in common the existence of a good theory of push-forward maps for projective morphisms. Not all cohomology theories have this structure, those that do are called {\em oriented}. In the Morel-Voevodsky stable homotopy category, the universal oriented theory is represented by the
\mathbb{P}^1
-spectrum
\MGL
, an algebraic version of the classical Thom spectrum
MU
. The corresponding cohomology theory
\MGL^{*,*}
is called {\em higher algebraic cobordism}. In an attempt to better understand the theory
\MGL^{*,*}
, Levine and Morel constructed a theory of {\em algebraic cobordism}
\Omega^*
. This is (conjecturally) related to
\MGL^{*,*}
as the classical Chow ring
\CH^*
is to motivic cohomology and like
\CH^*
\Omega^*
has a purely algebro-geometric description. In addition to giving some insight into
\MGL^{*,*}
\Omega^*
gives a simultaneous presentation of both
\CH^*
K_0
, exhibiting
K_0
as a deformation of
\CH^*
\Omega^*
has also been used to give conceptually simple proofs of various "degree formulas" first formulated by Rost. These degree formulas have been used in the study of Pfister quadrics and norm varieties, properties of which are used in the proofs of the Milnor conjecture and the Bloch-Kato conjecture. In this workshop, we describe aspects of the topological theory of complex cobordism which are important for algebraic cobordism (Lectures 1-3) and give the construction of
\Omega^*
and proofs of its fundamental properties (Lectures 4-7). In lectures 8-11, we show how
K_0
\CH^*
\Omega^*
, how
\Omega^*
recovers the universal formal group law, give the proof the generalized degree formula for
\Omega^*
and use this to proof the degree formula for the Segre class. Additional applications to Steenrod operations, further degree formulas and the use of these in the study of quadrics and other varietes is given in lectures 12 and 13. Lectures 14 and 15 concern the construction of funtorial pull-backs in algebraic cobordism. The two concluding lectures (16 and 17) give a quick sketch of the Morel-Voevodsky
\mathbb{A}^1
stable homotopy category and describe what we know about
\MGL
and its relation to motivic cohomology and
\Omega^*
. The workshop \emph{Algebraic Cobordism},organised by Marc Levin (Boston) and Fabien Morel (M\"unchen) was held April 4th--April 8th, 2005. This meeting was well attended with 55 participants.
Fabien Morel, Marc Levine, Arbeitsgemeinschaft mit aktuellem Thema: Algebraic Cobordism. Oberwolfach Rep. 2 (2005), no. 2, pp. 881–924
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44Axx Integral transforms, operational calculus
{L}_{p}
{L}_{q}
A characterization of probability measures by f-moments
Given a real-valued continuous function ƒ on the half-line [0,∞) we denote by P*(ƒ) the set of all probability measures μ on [0,∞) with finite ƒ-moments
{ʃ}_{0}^{\infty }ƒ\left(x\right){\mu }^{*n}\left(dx\right)
(n = 1,2...). A function ƒ is said to have the identification propertyif probability measures from P*(ƒ) are uniquely determined by their ƒ-moments. A function ƒ is said to be a Bernstein function if it is infinitely differentiable on the open half-line (0,∞) and
{\left(-1\right)}^{n}{ƒ}^{\left(n+1\right)}\left(x\right)
is completely monotone for some nonnegative integer n. The purpose of this paper...
A characterization of the generalized Meijer transform.
Deeba, E.Y., Koh, E.L. (1992)
A Convolution Connected with the Kontorovich-Lebedev Transform.
Hans-Jürgen, Heß, Albrecht Glaeske (1986)
A distribution Hardy transformation.
Pathak, R.S., Pandey, J.N. (1979)
A family of heat functions as solutions of indeterminate moment problems.
Gómez, Ricardo, López-García, Marcos (2007)
A finite-interval uniqueness theorem for bilateral Laplace transforms.
Chareka, Patrick (2007)
A general scheme for constructing inversion algorithms for cone beam CT.
Katsevich, Alexander (2003)
A generalized Hankel convolution on Zemanian spaces.
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A combinatorial problem in infinite groups.
Abdollahi, Alireza (2002)
Wituła, Roman, Słota, Damian (2007)
Abelian Extensions of Arbitrary Fields.
W. Kuyk, H.W. jr. Lenstra (1975)
Alcune proprietà gruppali invarianti per semi-isomorfismi
Federico Menegazzo (1971)
An application of Ramsey's theory to partition in groups. - II
Zvi Arad, Gideon Ehrlich, Otto H. Kegel (1993)
Bicartesian Squares of Nilpotent Groups.
Guido Mislin, Peter Hilton (1975)
Circles in (B, N)-Groups.
Thomas F. Bickel (1973)
Complete metacyclic groups
N. J. Mutio (1975)
Computing Adams operations on the Burnside ring of a finite group.
C.D. Gay, G.C. Morris, I. Morris (1983)
Irina Gelbukh (2015)
For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group's rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive...
Corrigendum to: "Aspherical four-manifolds and the centres of two-knot groups".
Die maximale lokale Erklärung einer gesättigten Formation
Klaus Doerk (1973)
Endliche minkowskische Ebenen von Charakteristik ... 2.
Rolf Lingenberg (1975)
Extensions of Certain Homomorphisms of Subsemi-groups to Homomorphisms of Groups
D.Z. DJOKOVIC, J.A. BAKER, J. ACZÉL (1971)
Extensions of homomorphisms of subsemigroups of a group.
K.E. Osondu (1977/1978)
Amel Dilmi (2007)
𝒳
is a class of groups, then a group
G
is said to be minimal non
𝒳
-group if all its proper subgroups are in the class
𝒳
G
itself is not an
𝒳
-group. The main result of this note is that if
c>0
is an integer and if
G
is a minimal non
\left(\mathrm{ℒℱ}\right)𝒩
\left(\mathrm{ℒℱ}\right){𝒩}_{c}
)-group, then
G
is a finitely generated perfect group which has no non-trivial finite factor and such that
G/Frat\left(G\right)
is an infinite simple group; where
𝒩
{𝒩}_{c}
\mathrm{ℒℱ}
) denotes the class of nilpotent (respectively, nilpotent of class at most
c
, locally...
Homology theory in the alternative set theory I. Algebraic preliminaries
Jaroslav Guričan (1991)
The notion of free group is defined, a relatively wide collection of groups which enable infinite set summation (called commutative
\pi
-group), is introduced. Commutative
\pi
-groups are studied from the set-theoretical point of view and from the point of view of free groups. Commutativity of the operator which is a special kind of inverse limit and factorization, is proved. Tensor product is defined, commutativity of direct product (also a free group construction and tensor product) with the special...
Homology theory in the AST III. Comparison with homology theories of Čech and Vietoris
The isomorphism between our homology functor and these of Vietoris and Čech is proved. Introductory result on dimension is proved.
Fares Gherbi, Tarek Rouabhi (2007)
The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group
G
is finite-by-nilpotent if and only if every infinite subset contains two distinct elements
x
y
{\gamma }_{n}\left(〈x\text{,}\phantom{\rule{4pt}{0ex}}{x}^{y}〉\right)
xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> = γ n ...
Inert actions on periodic points.
Kim, K.H., Roush, F.W., Wagoner, J.B. (1997)
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A Generalization of the Schwarz Alternating Method to an Arbitrary Number of Subdomains.
Lori Badea (1987)
Bernard Bialecki, Andreas Karageorghis (2000)
A Legendre spectral collocation method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplacian of the solution on the two vertical sides of the square. The Schur complement system is solved by a...
A Multigrid Method for a Parameter Dependent Problem in Solid Mechanics.
D. Braess, C. Blömer (1990)
A multilevel method with correction by aggregation for solving discrete elliptic problems
Radim Blaheta (1986)
The author studies the behaviour of a multi-level method that combines the Jacobi iterations and the correction by aggragation of unknowns. Our considerations are restricted to a simple one-dimensional example, which allows us to employ the technique of the Fourier analysis. Despite of this restriction we are able to demonstrate differences between the behaviour of the algorithm considered and of multigrid methods employing interpolation instead of aggregation.
Mary Fanett Wheeler, Guangri Xue, Ivan Yotov (2012)
In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cell-centered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid...
A new numerical scheme for non uniform homogenized problems: application to the nonlinear Reynolds compressible equation.
Buscaglia, Gustavo C., Jai, Mohammed (2001)
A Note on the Approximation of Mildly Nonlinear Dirichlet Problems by Finite Differences.
R. Meyer-Spasche (1979)
Andrea Toselli, Xavier Vasseur (2006)
In this paper, we present extensive numerical tests showing the performance and robustness of a Balancing Neumann-Neumann method for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in three dimensions. The numerical results are in good agreement with the theoretical bound for the condition number of the preconditioned operator derived in [Toselli and Vasseur, IMA J. Numer. Anal.24 (2004)...
A posteriori error estimators for the Stokes equations II non-conforming discretizations.
R. Verfürth (1991/1992)
A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization
Chunmei Wang (2014)
In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by
{\left(1+log\left(H/h\right)\right)}^{2}
H
h
are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.
|
Apply decibel gain - Simulink - MathWorks Nordic
Apply decibel gain
Math Functions / Math Operations
dspmathops
The dB Gain block multiplies the input by the decibel values specified in the Gain parameter. For an M-by-N input matrix u with elements uij, the Gain parameter can be a real M-by-N matrix with elements gij to be multiplied element-wise with the input, or a real scalar.
{y}_{ij}={u}_{ij}{10}^{\left({g}_{ij}/k\right)}
The value of k is 10 for power signals (select Power as the Input signal parameter) and 20 for voltage signals (select Amplitude as the Input signal parameter).
The value of the equivalent linear gain
{g}_{ij}^{lin}={10}^{\left({g}_{ij}/k\right)}
is displayed in the block icon below the dB gain value. The output is the same size as the input.
The dB Gain block supports real and complex floating-point and fixed-point data types.
The following diagram shows the data types used within the dB Gain subsystem block for fixed-point signals.
The settings for the fixed-point parameters of the Gain block in the diagram above are as follows:
Saturate on integer overflow — unselected
Parameter data type mode — Inherit via internal rule
Output data type mode — Inherit via internal rule
See the Gain (Simulink) reference page for more information.
The dB gain to apply to the input, a scalar or a real M-by-N matrix. Tunable (Simulink).
The type of input signal: Power or Amplitude. Tunable (Simulink).
This block does not support tunability in generated code.
dB Conversion DSP System Toolbox
Math Function (Simulink) Simulink
|
What Does the EMA Tell You?
Exponential Moving Average FAQs
An exponential moving average (EMA) is a type of moving average (MA) that places a greater weight and significance on the most recent data points. The exponential moving average is also referred to as the exponentially weighted moving average. An exponentially weighted moving average reacts more significantly to recent price changes than a simple moving average simple moving average (SMA), which applies an equal weight to all observations in the period.
Formula for Exponential Moving Average (EMA)
\begin{aligned} &\begin{aligned} EMA_{\text{Today}}=&\left(\text{Value}_{\text{Today}}\ast\left(\frac{\text{Smoothing}}{1+\text{Days}}\right)\right)\\ &+EMA_{\text{Yesterday}}\ast\left(1-\left(\frac{\text{Smoothing}}{1+\text{Days}}\right)\right)\end{aligned}\\ &\textbf{where:}\\ &EMA=\text{Exponential moving average} \end{aligned}
EMAToday=(ValueToday∗(1+DaysSmoothing))where:
The 12- and 26-day exponential moving averages (EMAs) are often the most quoted and analyzed short-term averages. The 12- and 26-day are used to create indicators like the moving average convergence divergence (MACD) and the percentage price oscillato (PPO). In general, the 50- and 200-day EMAs are used as indicators for long-term trends. When a stock price crosses its 200-day moving average, it is a technical signal that a reversal has occurred.
Traders who employ technical analysis find moving averages very useful and insightful when applied correctly. However, they also realize that these signals can create havoc when used improperly or misinterpreted. All the moving averages commonly used in technical analysis are lagging indicators.
More specifically, the EMA gives higher weights to recent prices, while the SMA assigns equal weights to all values. The two averages are similar because they are interpreted in the same manner and are both commonly used by technical traders to smooth out price fluctuations.
Since EMAs place a higher weighting on recent data than on older data, they are more responsive to the latest price changes than SMAs. That makes the results from EMAs more timely and explains why they are preferred by many traders.
What Is a Good Exponential Moving Average?
The longer-day EMAs (i.e. 50 and 200-day) tend to be used more by long-term investors, while short-term investors tend to use 8- and 20-day EMAs.
Is Exponential Moving Average Better Than Simple Moving Average?
The EMA focused more on recent price moves, which means it tends to respond more quickly to price changes than the SMA.
How Do You Read Exponential Moving Averages?
Investors tend to interpret a rising EMA as a support to price action and a falling EMA as a resistance. With that interpretation, investors look to buy when the price is near the rising EMA and sell when the price is near the falling EMA.
CME Group. "Understanding Moving Averages."
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EUDML | Range of density measures EuDML | Range of density measures
Range of density measures
Martin Sleziak; Miloš Ziman
Acta Mathematica Universitatis Ostraviensis (2009)
We investigate some properties of density measures – finitely additive measures on the set of natural numbers
ℕ
extending asymptotic density. We introduce a class of density measures, which is defined using cluster points of the sequence
\left(\frac{A\left(n\right)}{n}\right)
as well as cluster points of some other similar sequences. We obtain range of possible values of density measures for any subset of
ℕ
. Our description of this range simplifies the description of Bhashkara Rao and Bhashkara Rao [Bhaskara Rao, K. P. S., Bhaskara Rao, M., Theory of Charges – A Study of Finitely Additive Measures, Academic Press, London–New York, 1983.] for general finitely additive measures. Also the values which can be attained by the measures defined in the first part of the paper are studied.
Sleziak, Martin, and Ziman, Miloš. "Range of density measures." Acta Mathematica Universitatis Ostraviensis 17.1 (2009): 33-50. <http://eudml.org/doc/35196>.
@article{Sleziak2009,
abstract = {We investigate some properties of density measures – finitely additive measures on the set of natural numbers $\text\{$\mathbb \{N\}$\}$ extending asymptotic density. We introduce a class of density measures, which is defined using cluster points of the sequence $\bigl (\frac\{A(n)\}\{n\}\bigr )$ as well as cluster points of some other similar sequences. We obtain range of possible values of density measures for any subset of $\text\{$\mathbb \{N\}$\}$. Our description of this range simplifies the description of Bhashkara Rao and Bhashkara Rao [Bhaskara Rao, K. P. S., Bhaskara Rao, M., Theory of Charges – A Study of Finitely Additive Measures, Academic Press, London–New York, 1983.] for general finitely additive measures. Also the values which can be attained by the measures defined in the first part of the paper are studied.},
author = {Sleziak, Martin, Ziman, Miloš},
journal = {Acta Mathematica Universitatis Ostraviensis},
keywords = {asymptotic density; density measure; finitely additive measure; asymptotic density; density measure; finitely additive measure},
publisher = {University of Ostrava},
title = {Range of density measures},
AU - Sleziak, Martin
AU - Ziman, Miloš
TI - Range of density measures
JO - Acta Mathematica Universitatis Ostraviensis
PB - University of Ostrava
AB - We investigate some properties of density measures – finitely additive measures on the set of natural numbers $\text{$\mathbb {N}$}$ extending asymptotic density. We introduce a class of density measures, which is defined using cluster points of the sequence $\bigl (\frac{A(n)}{n}\bigr )$ as well as cluster points of some other similar sequences. We obtain range of possible values of density measures for any subset of $\text{$\mathbb {N}$}$. Our description of this range simplifies the description of Bhashkara Rao and Bhashkara Rao [Bhaskara Rao, K. P. S., Bhaskara Rao, M., Theory of Charges – A Study of Finitely Additive Measures, Academic Press, London–New York, 1983.] for general finitely additive measures. Also the values which can be attained by the measures defined in the first part of the paper are studied.
KW - asymptotic density; density measure; finitely additive measure; asymptotic density; density measure; finitely additive measure
Ašić, M. D., Adamović, D. D., 10.2307/2316738, Amer. Math. Monthly 77 (1970) 613–616 (1970) MR0264599DOI10.2307/2316738
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Banach, S., Theory of Linear Operations, North Holland, Amsterdam, 1987 (1987) Zbl0613.46001MR0880204
Bhaskara Rao, K. P. S., Bhaskara Rao, M., Theory of Charges – A Study of Finitely Additive Measures, Academic Press, London–New York, 1983 (1983) Zbl0516.28001MR0751777
Blass, A., Frankiewicz, R., Plebanek, G., Ryll-Nardzewski, C., 10.1090/S0002-9939-01-05941-X, Proc. Amer. Math. Soc. 129 (11) (2001) 3313–3320 (2001) Zbl0992.28002MR1845008DOI10.1090/S0002-9939-01-05941-X
Blümlinger, M., 10.1090/S0002-9947-96-01779-5, Trans. Amer. Math. Soc. 348 (12) (1996) 5087–5111 (1996) MR1390970DOI10.1090/S0002-9947-96-01779-5
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Fey, M., May’s theorem with an infinite population, Social Choice and Welfare 23 (2004) 275–293 (2004) Zbl1090.91020MR2084902
Fey, M., Problems (Density measures), Tatra Mnt. Math. Publ. 31 (2005) 177–181 (2005) MR2208785
Fuchs, A., Giuliano Antonini, R., Théorie générale des densités, Rend. Acc. Naz. delle Scienze detta dei XL, Mem. di Mat. 108 (1990) Vol. XI, fasc. 14, 253–294 (1990) Zbl0726.60004MR1106580
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Giuliano Antonini, R., Grekos, G., Mišík, L., 10.1007/s10587-007-0087-z, Czechosl. Math. J. 57 (3) (2007) 947–962 (2007) Zbl1195.11018MR2356932DOI10.1007/s10587-007-0087-z
Grekos, G., On various definitions of density, (survey), Tatra Mt. Math. Publ. 31 (2005) 17–27 (2005) Zbl1150.11339MR2208784
Grekos, G., Šalát, T., Tomanová, J., Gaps and densities, Bull. Math. Soc. Sci. Math. Roum. 46 (3–4) (2003–2004) 121–141 (2004) MR2094181
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, Topology and its Applications 47 (1992) 223–268 (1992) MR1192311
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asymptotic density, density measure, finitely additive measure, asymptotic density, density measure, finitely additive measure
Infinite automorphism groups
Density, gaps, topology
Articles by Martin Sleziak
Articles by Miloš Ziman
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Physics - Nobel Prize—Tools for Quantum Tinkering
Nobel Prize—Tools for Quantum Tinkering
David Wineland and Serge Haroche, who studied photons and atoms in new ways, have won the 2012 Nobel Prize in Physics.
Nature (London) 446, 275 (2007)
Reflecting well on the physicist. Haroche and his colleagues used highly reflective cavities like this one to hold small numbers of photons for controlled interactions with single atoms. It is several centimeters across and is “opened up” for the photo.
To understand the quantum world, researchers have developed lab-scale tools to manipulate microscopic objects without disturbing them. The 2012 Nobel Prize in Physics recognizes two of these quantum tinkerers: David Wineland, of the National Institute of Standards and Technology and the University of Colorado in Boulder, and Serge Haroche, of the Collège de France and the Ecole Normale Supérieure in Paris. Two of their papers, published in 1995 and ‘96 in Physical Review Letters, exemplify their contributions. The one by Wineland and collaborators showed how to use atomic states to make a quantum logic gate, the first step toward a superfast quantum computer. The other, by Haroche and his colleagues, demonstrated one of the strange predictions of quantum mechanics—that measuring a quantum system can pull the measuring device into a weird quantum state which then dissipates over time.
A quantum system can exist in two distinct states at the same time. The challenge in studying this so-called superposition of quantum states is that any nudge from the environment can quickly push the system into one state or the other. Wineland and Haroche both designed experiments that isolate particles—ions or photons—from the environment, so that they can be carefully controlled without losing their quantum character.
Since the 1980s, Haroche has been one of the pioneers in the field of cavity quantum electrodynamics, where researchers observe a single atom interacting with a few photons inside a reflective cavity. Haroche and his colleagues can keep a photon bouncing back and forth in a centimeter-sized cavity billions of times before it escapes. But only photons of specific wavelengths determined by the cavity size can survive. Haroche’s group was one of the first to show that this wavelength selectivity could amplify [1] or suppress [2] the emission from an atom inside the cavity. Haroche was later able to tune a cavity so that the allowed wavelengths were close to, but not equal to, those associated with transitions in an atom, so that the photons and atom did not exchange energy. Instead, they incurred a phase change that could carry information about, for example, the number of photons in the cavity [3].
In 1996, Haroche’s group used such a system to study the process by which a quantum superposition settles into a single state. The researchers placed a highly excited rubidium atom in a superposition of two energy states and then sent it through a cavity containing about ten photons. The matter-light interaction “entangled” the photons and atom together, so that the photons entered their own superposition of two states (a “Schrödinger cat” state, in the team’s language), which acted as a “measurement” of the atom’s superposition state. Measuring devices don’t ordinarily remain in two states; instead, they give up their quantum nature almost immediately through interactions with the environment. However, this so-called decoherence process was expected to take longer for a “small” device with only a few particles (photons in this case).
To see this effect, the team arranged for a second atom to enter the cavity shortly after the first. Separate observations of the atoms after each passed through the cavity showed that the superposition in the photons survived for several microseconds. This was the first experimental exploration of the quantum measurement process at the so-called “mesoscopic” boundary between the macroscopic and the microscopic world, says coauthor Jean-Michel Raimond of the Pierre and Marie Curie University in Paris. “The experiment is even now described in a few standard quantum mechanics textbooks,” he says.
Wineland performed similar sorts of quantum-probing experiments through his own pioneering work with trapped ions [4,5]. The tight confinement of ions in these electric field traps causes ion motion to be restricted to distinct quantum states, each of which represents a different frequency of bouncing back-and-forth between the electric field “walls.” These motional, or “vibrational,” states are typically independent of the internal, electronic energy states of the ion, but Wineland and others showed that laser light could transfer energy from one set of states to the other. The researchers used this laser coupling to cool an ion to the state with the slowest motion [6] and to make the world’s most precise clocks [7].
In their 1995 paper, Wineland and his colleagues demonstrated the first quantum logic gate, the basic building block of a quantum computer. They trapped a single beryllium ion and prepared it with two quantum bits (quantum two-state systems, or “qubits”): one corresponding to the two lowest vibrational states and the other to a pair of electronic states. A series of laser pulses would either have no effect on the electronic qubit or would switch its value—say, from the lower- to the higher-energy state—depending on the vibrational qubit’s state. This “controlled NOT” operation did not measure either qubit, so the quantum nature of the states was preserved. “It was a simple gate, but it was interesting because it was clear how to scale the system up,” says coauthor Chris Monroe of the University of Maryland in College Park. Since then, researchers have succeeded in performing more complicated logic operations with as many as 14 ions.
“There is a beautiful duality between the two techniques,” Raimond says. Wineland traps matter particles (ions) and studies them with laser beams, while Haroche traps photons and studies them with a matter beam. “I think the match by the Nobel committee is quite perfect: Same generation, similar achievements, same global objectives,” says Raimond, “and two excellent friends.”
P. Goy, J. M. Raimond, M. Gross, and S. Haroche, “Observation of Cavity-Enhanced Single-Atom Spontaneous Emission,” Phys. Rev. Lett. 50, 1903 (1983)
W. Jhe, A. Anderson, E. A. Hinds, D. Meschede, L. Moi, and S. Haroche, “Suppression of Spontaneous Decay at Optical Frequencies: Test of Vacuum-Field Anisotropy in Confined Space,” Phys. Rev. Lett. 58, 666 (1987)
S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deléglise, U. Busk Hoff, M. Brune, J. M. Raimond, and S. Haroche, “Quantum Jumps of Light Recording the Birth and Death of a Photon in a Cavity,” Nature 446, 297 (2007)
D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-Pressure Cooling of Bound Resonant Absorbers,” Phys. Rev. Lett. 40, 1639 (1978)
D.J. Wineland and Wayne M. Itano, “Spectroscopy of a Single
{\text{Mg}}^{+}
Ion,” Phys. Lett. A 82, 75 (1981)
F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser Cooling to the Zero-Point Energy of Motion,” Phys. Rev. Lett. 62, 403 (1989)
Synopsis: Better timing with aluminum ions, http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.104.070802
Nobel Prize website with “advanced” and “popular” information
Short video explaining Nobel Prize by minutephysics
Short video explaining Schrödinger’s cat by minutephysics
Focus: Schrödinger’s Drum
Focus: Ultracool Atoms in a Quantum Cavity
Atomic and Molecular PhysicsQuantum PhysicsQuantum Information
|
Martinez, Patrick ; Vancostenoble, Judith
Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback
a\left(t\right){u}_{t}
. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions
a
: typically
a
1
\left(0,T\right)
0
\left(T,qT\right)
qT
-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases, we prove that there are explicit exceptional values of
T
for which the energy of some solutions remains constant with time. If
T
is different from those exceptional values, the energy of all solutions decays exponentially to zero. This number of exceptional values is countable in the boundary case and finite in the distributed case. When the feedback is acting on the boundary, we also study the case of postive-negative feedbacks:
a\left(t\right)={a}_{0}>0
\left(0,T\right)
a\left(t\right)=-{b}_{0}<0
\left(T,qT\right)
, and we give the necessary and sufficient condition under which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity. The proofs of these results are based on congruence properties and on a theorem of Weyl in the boundary case, and on new observability inequalities for the undamped wave equation, weakening the usual “optimal time condition” in the locally distributed case. These new inequalities provide also new exact controllability results.
Classification : 35L05, 35B35, 35B40, 11A07
Mots clés : damped wave equation, asymptotic behavior, on-off feedback, congruences, observability inequalities
author = {Martinez, Patrick and Vancostenoble, Judith},
title = {Stabilization of the wave equation by on-off and positive-negative feedbacks},
AU - Martinez, Patrick
AU - Vancostenoble, Judith
TI - Stabilization of the wave equation by on-off and positive-negative feedbacks
Martinez, Patrick; Vancostenoble, Judith. Stabilization of the wave equation by on-off and positive-negative feedbacks. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 335-377. doi : 10.1051/cocv:2002015. http://www.numdam.org/articles/10.1051/cocv:2002015/
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[15] V. Komornik and E. Zuazua A direct method for boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33-54. | MR 1054123 | Zbl 0636.93064
[16] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method. John Wiley, Chicester and Masson, Paris (1994). | MR 1359765 | Zbl 0937.93003
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[19] I. Lasiecka and R. Triggiani, Uniform exponential decay in a bounded region with
{L}_{2}\left(0,T;{L}_{2}\left(\Sigma \right)\right)
-feedback control in the Dirichlet boundary condition. J. Differential Equations 66 (1987) 340-390. | MR 876804 | Zbl 0629.93047
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|
Brownian interpolation of stochastic differential equations (SDEs) for SDE, BM, GBM, CEV, CIR, HWV, Heston, SDEDDO, SDELD, or SDEMRD models - MATLAB interpolate - MathWorks Switzerland
Stochastic Interpolation Without Refinement
Simulation of Conditional Gaussian Distributions
Brownian interpolation of stochastic differential equations (SDEs) for SDE, BM, GBM, CEV, CIR, HWV, Heston, SDEDDO, SDELD, or SDEMRD models
[XT,T] = interpolate(MDL,Times,Paths)
[XT,T] = interpolate(___,Name,Value)
[XT,T] = interpolate(MDL,Times,Paths) performs a Brownian interpolation into a user-specified time series array, based on a piecewise-constant Euler sampling approach.
[XT,T] = interpolate(___,Name,Value) adds optional name-value pair arguments.
Many applications require knowledge of the state vector at intermediate sample times that are initially unavailable. One way to approximate these intermediate states is to perform a deterministic interpolation. However, deterministic interpolation techniques fail to capture the correct probability distribution at these intermediate times. Brownian (or stochastic) interpolation captures the correct joint distribution by sampling from a conditional Gaussian distribution. This sampling technique is sometimes referred to as a Brownian Bridge.
The default stochastic interpolation technique is designed to interpolate into an existing time series and ignore new interpolated states as additional information becomes available. This technique is the usual notion of interpolation, which is called Interpolation without refinement.
Alternatively, the interpolation technique may insert new interpolated states into the existing time series upon which subsequent interpolation is based, by that means refining information available at subsequent interpolation times. This technique is called interpolation with refinement.
Interpolation without refinement is a more traditional technique, and is most useful when the input series is closely spaced in time. In this situation, interpolation without refinement is a good technique for inferring data in the presence of missing information, but is inappropriate for extrapolation. Interpolation with refinement is more suitable when the input series is widely spaced in time, and is useful for extrapolation.
The stochastic interpolation method is available to any model. It is best illustrated, however, by way of a constant-parameter Brownian motion process. Consider a correlated, bivariate Brownian motion (BM) model of the form:
\begin{array}{l}d{X}_{1t}=0.3dt+0.2d{W}_{1t}-0.1d{W}_{2t}\\ d{X}_{2t}=0.4dt+0.1d{W}_{1t}-0.2d{W}_{2t}\\ E\left[d{W}_{1t}d{W}_{2t}\right]=\rho dt=0.5dt\end{array}
Create a bm object to represent the bivariate model:
mu = [0.3; 0.4];
sigma = [0.2 -0.1; 0.1 -0.2];
rho = [1 0.5; 0.5 1];
obj = bm(mu,sigma,'Correlation',rho);
Assuming that the drift (Mu) and diffusion (Sigma) parameters are annualized, simulate a single Monte Carlo trial of daily observations for one calendar year (250 trading days):
rng default % make output reproducible
dt = 1/250; % 1 trading day = 1/250 years
[X,T] = simulate(obj,250,'DeltaTime',dt);
It is helpful to examine a small interval in detail.
Interpolate into the simulated time series with a Brownian bridge:
t = ((T(1) + dt/2):(dt/2):(T(end) - dt/2));
x = interpolate(obj,t,X,'Times',T);
Plot both the simulated and interpolated values:
plot(T,X(:,1),'.-r',T,X(:,2),'.-b')
plot(t,x(:,1),'or',t,x(:,2),'ob')
title('Bi-Variate Brownian Motion: \rho = 0.5')
axis([0.4999 0.6001 0.25 0.4])
The solid red and blue dots indicate the simulated states of the bivariate model.
The straight lines that connect the solid dots indicate intermediate states that would be obtained from a deterministic linear interpolation.
Open circles indicate interpolated states.
Open circles associated with every other interpolated state encircle solid dots associated with the corresponding simulated state. However, interpolated states at the midpoint of each time increment typically deviate from the straight line connecting each solid dot.
You can gain additional insight into the behavior of stochastic interpolation by regarding a Brownian bridge as a Monte Carlo simulation of a conditional Gaussian distribution.
This example examines the behavior of a Brownian bridge over a single time increment.
Divide a single time increment of length dt into 10 subintervals:
rng default; % make output reproducible
n = 125; % index of simulated state near middle
times = (T(n):(dt/10):T(n + 1));
nTrials = 25000; % # of Trials at each time
In each subinterval, take 25000 independent draws from a Gaussian distribution, conditioned on the simulated states to the left, and right:
average = zeros(length(times),1);
variance = zeros(length(times),1);
for i = 1:length(times)
t = times(i);
x = interpolate(obj,t(ones(nTrials,1)),...
X,'Times',T);
average(i) = mean(x(:,1));
variance(i) = var(x(:,1));
Plot the sample mean and variance of each state variable:
The following graph plots the sample statistics of the first state variable only, but similar results hold for any state variable.
plot([T(n) T(n + 1)],[X(n,1) X(n + 1,1)],'.-b')
plot(times, average, 'or')
title('Brownian Bridge without Refinement: Sample Mean')
axis([T(n) T(n + 1) limits(3:4)]);
plot(T(n),0,'.-b',T(n + 1),0,'.-b')
plot(times, variance, '.-r')
title('Brownian Bridge without Refinement: Sample Variance')
The Brownian interpolation within the chosen interval, dt, illustrates the following:
The conditional mean of each state variable lies on a straight-line segment between the original simulated states at each endpoint.
The conditional variance of each state variable is a quadratic function. This function attains its maximum midway between the interval endpoints, and is zero at each endpoint.
The maximum variance, although dependent upon the actual model diffusion-rate function G(t,X), is the variance of the sum of NBrowns correlated Gaussian variates scaled by the factor dt/4.
The previous plot highlights interpolation without refinement, in that none of the interpolated states take into account new information as it becomes available. If you had performed interpolation with refinement, new interpolated states would have been inserted into the time series and made available to subsequent interpolations on a trial-by-trial basis. In this case, all random draws for any given interpolation time would be identical. Also, the plot of the sample mean would exhibit greater variability, but would still cluster around the straight-line segment between the original simulated states at each endpoint. The plot of the sample variance, however, would be zero for all interpolation times, exhibiting no variability.
All MDL parameters are assumed piecewise constant, evaluated from the most recent observation time in Times that precedes a specified interpolation time in T. This is consistent with the Euler approach of Monte Carlo simulation.
Times — Interpolation times
Interpolation times, specified as a NTimes element vector. The length of this vector determines the number of rows in the interpolated output time series XT.
Paths — Sample paths of correlated state variables
time series array
Sample paths of correlated state variables, specified as a NPeriods-by-NVars-by-NTrials time series array.
For a given trial, each row of this array is the transpose of the state vector Xt at time t. Paths is the initial time series array into which the interpolate function performs the Brownian interpolation.
Example: [XT,T] = interpolate(MDL,T,Paths,'Times',t)
Times — Observation times associated with the time series input Paths
zero-based, unit-increment column vector of length NPeriods (default) | column vector
Observation times associated with the time series input Paths, specified as the comma-separated pair consisting of 'Times' and a column vector.
Refine — Flag that indicates whether interpolate uses the interpolation times you request
False (interpolate bases the interpolation only on the state information specified in Paths) (default) | logical with values True or False
Flag that indicates whether interpolate uses the interpolation times you request (see T) to refine the interpolation as new information becomes available, specified as the comma-separated pair consisting of 'Refine' and a logical with a value of True or False.
Processes — Sequence of background processes or state vector adjustments
interpolate makes no adjustments and performs no processing (default) | function | cell array of functions
Sequence of background processes or state vector adjustments, specified as the comma-separated pair consisting of 'Processes' and a function or cell array of functions of the form
{X}_{t}=P\left(t,{X}_{t}\right)
The interpolate function runs processing functions at each interpolation time. They must accept the current interpolation time t, and the current state vector Xt, and return a state vector that may be an adjustment to the input state.
If you specify more than one processing function, interpolate invokes the functions in the order in which they appear in the cell array. You can use this argument to specify boundary conditions, prevent negative prices, accumulate statistics, plot graphs, and so on.
XT — Interpolated state variables
Interpolated state variables, returned as a NTimes-by-NVars-by-NTrials time series array.
For a given trial, each row of this array is the transpose of the interpolated state vector Xt at time t. XT is the interpolated time series formed by interpolating into the input Paths time series array.
T — Interpolation times associated with the output time series XT
Interpolation times associated with the output time series XT, returned as a NTimes-by-1 column vector.
If the input interpolation time vector Times contains no missing observations (NaNs), the output of T is the same time vector as Times, but with the NaNs removed. This reduces the length of T and the number of rows of XT.
This function performs a Brownian interpolation into a user-specified time series array, based on a piecewise-constant Euler sampling approach.
Consider a vector-valued SDE of the form:
d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}
X is an NVars-by-1 state vector.
F is an NVars-by-1 drift-rate vector-valued function.
G is an NVars-by-NBrowns diffusion-rate matrix-valued function.
Given a user-specified time series array associated with this equation, this function performs a Brownian (stochastic) interpolation by sampling from a conditional Gaussian distribution. This sampling technique is sometimes called a Brownian bridge.
Unlike simulation methods, the interpolation function does not support user-specified noise processes.
The interpolate function assumes that all model parameters are piecewise-constant, and evaluates them from the most recent observation time in Times that precedes a specified interpolation time in T. This is consistent with the Euler approach of Monte Carlo simulation.
When an interpolation time falls outside the interval specified by Times, a Euler simulation extrapolates the time series by using the nearest available observation.
The user-defined time series Paths and corresponding observation Times must be fully observed (no missing observations denoted by NaNs).
The interpolate function assumes that the user-specified time series array Paths is associated with the sde object. For example, the Times and Paths input pair are the result of an initial course-grained simulation. However, the interpolation ignores the initial conditions of the sde object (StartTime and StartState), allowing the user-specified Times and Paths input series to take precedence.
sde | simulate
|
14E16 McKay correspondence
14E22 Ramification problems
A Counterexample to a Conjecture of Zariski.
Ignacio Luengo (1984)
A counterexample to the theorem of Beppo Levi in three dimensions.
Mark Spivakovsky (1989)
\le 1
ℋ=\bigcup {N}_{i}
la décomposition canonique de l’espace des arcs
ℋ
passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que
{N}_{i}
n’est pas inclus dans
{N}_{j}
. On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.
A resolution theorem for homology cycles of real algebraic varieties.
Selman Akbulut, Henry King (1985)
We give a simpler and more conceptual proof of toroidalization of morphisms of 3-folds to surfaces, over an algebraically closed field of characteristic zero. A toroidalization is obtained by performing sequences of blow ups of nonsingular subvarieties above the domain and range, to make a morphism toroidal. The original proof of toroidalization of morphisms of 3-folds to surfaces is much more complicated.
A simplified proof of desingularization and applications.
Ana María Bravo, Santiago Encinas, Orlando Villamayor Uriburu (2005)
This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of...
A simply connected surface of general type with pg = 0.
Rebecca Barlow (1985)
A Smooth Four-Dimensional G-Hilbert Scheme
Sebestean, Magda (2004)
2000 Mathematics Subject Classification: 14C05, 14L30, 14E15, 14J35.When the cyclic group G of order 15 acts with some specific weights on affine four-dimensional space, the G-Hilbert scheme is a crepant resolution of the quotient A^4 / G. We give an explicit description of this resolution using G-graphs.
Frank Olaf Schreyer (1991)
Lorenzo Robbiano (1979)
Additive vector fields, algebraicity and rationality.
Jun-Muk Hwang (1996)
Altérations de variétés algébriques
Pierre Berthelot (1995/1996)
Ample line bundles on blown up surfaces.
Oliver Küchle (1996)
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A 3D Broadband Seismometer Array Experiment at the Homestake Mine | Seismological Research Letters | GeoScienceWorld
A 3D Broadband Seismometer Array Experiment at the Homestake Mine
Vuk Mandic;
School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, Minnesota 55455 U.S.A., vuk@umn.edu
Victor C. Tsai;
Seismological Laboratory, California Institute of Technology, 1200 E. California Boulevard, MS 252‐21, Pasadena, California 91125 U.S.A.
Gary L. Pavlis;
Department of Geological Sciences, Indiana University, 1001 E. 10th Street, Bloomington, Indiana 47405 U.S.A.
Tanner Prestegard;
Daniel C. Bowden;
Patrick Meyers;
Vuk Mandic, Victor C. Tsai, Gary L. Pavlis, Tanner Prestegard, Daniel C. Bowden, Patrick Meyers, Ross Caton; A 3D Broadband Seismometer Array Experiment at the Homestake Mine. Seismological Research Letters 2018;; 89 (6): 2420–2429. doi: https://doi.org/10.1785/0220170228
Seismometer deployments are often confined to near the Earth’s surface for practical reasons, despite the clear advantages of deeper seismometer installations related to lower noise levels and more homogeneous conditions. Here, we describe a 3D broadband seismometer array deployed at the inactive Homestake Mine in South Dakota, which takes advantage of infrastructure originally setup for mining and is now used for a range of scientific experiments. The array consists of 24 stations, of which 15 were underground, with depths ranging from 300 ft (91 m) to 4850 ft (1478 m), and with a 3D aperture of
∼1.5 km
in each direction, thus spanning a 3D volume of about
3.4 km3
. We describe unique research opportunities and challenges related to the 3D geometry, including the generally low ambient noise levels, the strong coherency between observed event waveforms across the array, and the technical challenges of running the network. This article summarizes preliminary results obtained using data acquired by the Homestake array, illustrating the range of possible studies supported by the data.
Apparent Wave Velocity and Site Amplification at the California Strong Motion Instrumentation Program Carquinez Bridge Geotechnical Arrays during the 2014 M6.0 South Napa Earthquake
MEMS Accelerometer Mini-Array (MAMA): A Low-Cost Implementation for Earthquake Early Warning Enhancement
Resolving Shallow Shear‐Wave Velocity Structure beneath Station CBN by Waveform Modeling of the Mw 5.8 Mineral, Virginia, Earthquake Sequence
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is it true that a right angled triangle has two acute angle is it true that an acute angled - Maths - The Triangle and its Properties - 9794973 | Meritnation.com
is it true that a right angled triangle has two acute angle
is it true that an acute angled triangle has three acute angles
is it true that an obtuse angled triangle has one obtuse angle
1 ) is it true that a right angled triangle has two acute angle .
We know from angle sum property of triangle that sum of all three internal angles of any triangle is 180
°
Here one of the angle is 90
°
, So sum of rest of two angle must be 90
°
It is true that a right angled triangle has two acute angle .
2 ) Is it true that an acute angled triangle has three acute angles .
We know acute angle is angles less than 90
°
and acute angle triangle have all three angles less than 90
°
It is true that an acute angled triangle has three acute angles .
3 ) Is it true that an obtuse angled triangle has one obtuse angle .
°
We know obtuse angle is angles more than 90
°
Here one of the angle is more than 90
°
, So sum of rest of two angle must be less than 90
°
It is true that an obtuse angled triangle has one obtuse angle .
Hope this information will clear your doubts about The Triangle and its Properties .
Yes, all the statements are true....
|
A direct proof of van der Vaart's theorem
J. Bourgain, H. Sato (1986)
Antoine Delcroix (2008)
A note on fusion Banach frames
S. K. Kaushik, Varinder Kumar (2010)
For a fusion Banach frame
\left(\left\{{G}_{n},{v}_{n}\right\},S\right)
for a Banach space
E
\left(\left\{{v}_{n}^{*}\left({E}^{*}\right),{v}_{n}^{*}\right\},T\right)
is a fusion Banach frame for
{E}^{*}
\left(\left\{{G}_{n},{v}_{n}\right\},S;\left\{{v}_{n}^{*}\left({E}^{*}\right),{v}_{n}^{*}\right\},T\right)
is called a fusion bi-Banach frame for
E
. It is proved that if
E
has an atomic decomposition, then
E
also has a fusion bi-Banach frame. Also, a sufficient condition for the existence of a fusion bi-Banach frame is given. Finally, a characterization of fusion bi-Banach frames is given.
A note on the convolution theorem for the Fourier transform
In this paper we characterize those bounded linear transformations
Tf
{L}^{1}\left({ℝ}^{1}\right)
into the space of bounded continuous functions on
{ℝ}^{1}
, for which the convolution identity
T\left(f*g\right)=Tf·Tg
holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.
A note on weak estimates for oscillating kernels
G. Sampson (1981)
A note to the Fourier method of solving partial second-order differential equations
Jaroslav Hančl (1988)
There are two grounds the spline theory stems from - the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We use the general variational approach called smooth interpolation introduced by Talmi and Gilat and show that it covers not only the cubic spline and its 2D and 3D analogues but also the well known tension spline...
\mu
\mathbf{R}
ϵ>0
\left\{x\in \mathbf{R}|\phantom{\rule{0.166667em}{0ex}}\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\left(\mu \left(x\right)\right)>ϵ\right\}
ϵ=0
A relation between Fourier transforms in one and two variables
Per Sjölin (1979)
A stable method for the inversion of the Fourier transform in
{\mathbb{R}}^{N}
Leonede De Michele, Delfina Roux (1993)
A general method is given for recovering a function
f:{\mathbb{R}}^{N}\to C
N\ge 1
, knowing only an approximation of its Fourier transform.
Absolute convergence of multiple Fourier integrals
Yurii Kolomoitsev, Elijah Liflyand (2013)
Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of
{L}_{p}
integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.
Jean-Pierre Conze, Michael Lin (2013)
It is well-known that a probability measure
\mu
𝕋
\parallel {\mu }^{n}{*f-\int f\phantom{\rule{0.166667em}{0ex}}\mathrm{d}m\parallel }_{p}\to 0
f\in {L}_{p}
, every (some)
p\in \left[1,\infty \right)
|\stackrel{^}{\mu }\left(n\right)|lt;1
for every non-zero
n\in ℤ
\mu
is strictly aperiodic). In this paper we study the a.e. convergence of
{\mu }^{n}*f
f\in {L}_{p}
pgt;1
. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of
\mu
, for the strong sweeping out property (existence of a Borel set
B
lim sup{\mu }^{n}*{1}_{B}=1
lim inf{\mu }^{n}*{1}_{B}=0
a.e.). The results are extended to general compact Abelian groups
G
with Haar...
|
p
A Frobenius Theorem for Blocks.
Michel Broué, Lluis Puig (1980)
A Remark on Blocks with Dihedral Defect Groups in Solvable Groups.
Shigeo Koshitani (1982)
A semigroup approach to wreath-product extensions of Solomon's descent algebras.
Hsiao, Samuel K. (2009)
Abelian p-adic Group Rings.
Eugene Spiegel (1975)
Algebras and Quaternion Defect Groups. I.
Karin Erdmann (1988)
Algebras and Quaternion Defect Groups. II.
An algorithm for the decomposition of ideals of the group ring of a symmetric group.
Fiedler, B. (1997)
An analogue of the Duistermaat-van der Kallen theorem for group algebras
Wenhua Zhao, Roel Willems (2012)
Let G be a group, R an integral domain, and V G the R-subspace of the group algebra R[G] consisting of all the elements of R[G] whose coefficient of the identity element 1G of G is equal to zero. Motivated by the Mathieu conjecture [Mathieu O., Some conjectures about invariant theory and their applications, In: Algèbre non Commutative, Groupes Quantiques et Invariants, Reims, June 26–30, 1995, Sémin. Congr., 2, Société Mathématique de France, Paris, 1997, 263–279], the Duistermaat-van der Kallen...
Another counterexample to a conjecture of Zassenhaus.
Hertweck, M. (2002)
Augmentation quotients for Burnside rings of generalized dihedral groups
Shan Chang (2016)
H
be a finite abelian group of odd order,
𝒟
be its generalized dihedral group, i.e., the semidirect product of
{C}_{2}
H
by inverting elements, where
{C}_{2}
is the cyclic group of order two. Let
\Omega \left(𝒟\right)
be the Burnside ring of
𝒟
\Delta \left(𝒟\right)
be the augmentation ideal of
\Omega \left(𝒟\right)
{\Delta }^{n}\left(𝒟\right)
{Q}_{n}\left(𝒟\right)
n
\Delta \left(𝒟\right)
n
th consecutive quotient group
{\Delta }^{n}\left(𝒟\right)/{\Delta }^{n+1}\left(𝒟\right)
, respectively. This paper provides an explicit
ℤ
{\Delta }^{n}\left(𝒟\right)
and determines the isomorphism class of
{Q}_{n}\left(𝒟\right)
n
Binary Polyhedral Groups and Euclidean Diagrams.
D. Happel, U. Preiser (1980)
Blocks, Vertices and Normal Subgroups.
Reinhard Knörr (1976)
Alex Bartel, Tim Dokchitser (2015)
G
is a non-cyclic finite group, non-isomorphic
G
X,Y
may give rise to isomorphic permutation representations
ℂ\left[X\right]\cong ℂ\left[Y\right]
. Equivalently, the map from the Burnside ring to the rational representation ring of
G
has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of
p
Certain normal subgroups of units in group rings.
Jürgen Ritter, Sudarshan K. Sehgal (1987)
Characterization of a radical of group rings over finite prime fields.
Kornev, A.I., Pavlova, T.V. (2004)
Clifford theory for group-graded rings.
Everett C. Dade (1986)
Clifford theory for group-graded rings. II.
Clifford theory for p-sections of finite groups
Marton Harris (1988)
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Pascal's Wager and the Space of Possible Gods | Refute me
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Home Thought things. Pascal’s Wager and the Space of Possible Gods
Pascal’s Wager and the Space of Possible Gods
June 27, 2018 , Matthew Pearce , Comments Off on Pascal’s Wager and the Space of Possible Gods
Julia Galef spoke with Amanda Askell about Pascal’s Wager on Rationally Speaking (podcast, blog). Pascal’s wager being that if you’re among the faithful you shall be infinitely rewarded, if that faith is the right model of the world. Whereas if you’re an atheist you get nowt, if you’re correct about how the world is. So the expected utility of being numbered among the faithful is argued to be higher than that of not being.
Askell makes a number of defences of Pascal’s Wager. I’m going to make two arguments in response. The first is that the appeal to expected utility theory is flawed; in principle because the space of possible gods is not defined and, in the general case, will not support Askell’s proposed defence. The second is that because reward and punishment from a jealous god is asymmetric the calculus could well point the other way. Afterwards I sketch out why the first problem seems pretty hard to avoid.
Gods of the continuum
Askell’s first defence (about 3 mins 45 into the podcast) relates to a many gods argument. Criticism: the Wager is unconvincing as adherents of other faiths can make similar claims, so being presented with the wager does not help guide our action. Defence: perhaps, but if a god does exist then you have a higher chance of picking the right faith by picking a faith rather than none, and therefore of enjoying infinite utils. Hence you should just pick one. Furthermore if we have any information which makes one deity more likely than another (e.g. we had a good vibe about in a dream, a fortune cookie etc) then the symmetry is broken and we should pick the most likely god to believe in.
The mathematical issue is that this relies on the space of possible gods being finite. In the interest of brevity, this section contains a quick version of a counter argument; with further elaboration below.
Suppose that for every real number, [r]
r
, in the interval [(0,1) \in \mathcal R]
(0,1) \in \mathcal R
the god [G_r]
G_r
holds the number [r]
r
to be the Real Number. [G_r]
G_r
has decreed that on the Day of Observation believers in [r]
r
shall be risen to spend eternity in heaven. Unbelievers and heretics shall be annihilated. How do we assess the expected utility?
If we are completely ignorant of the Real Number, then a uniform distribution on [(0,1)]
(0,1)
would be appropriate for our model. Let [u(s,r)]
u(s,r)
be the utility we receive by believing in [s]
s
when [r]
r
is the Real Number. Now compute the expected utility of arbitrarily picking a god [G_s]
G_s
to believe in. Here we’re even assuming that there really is a god [G_r]
G_r
(though the result would obviously still hold with some mass on an atheist universe):
$$\mathbb E[u(s,r)] = \mathbb E[u(s,r)|r \neq s] P(r \neq s) + \mathbb E[u(s,r) | r=s]P(r=s) = 0 \cdot 1 + \infty \cdot 0 = 0$$
Where the first expectation was equivalent to the integral of a constant, [0]
0
, over the domain of a random variable, [(0,1)]
(0,1)
, since the removal of a set of measure zero [\{r\}]
\{r\}
from the domain has no effect on the integral. While [P(r=s)=0]
P(r=s)=0
since again [\{r\}]
\{r\}
was a set of measure zero. Let [a(r)]
a(r)
represent the utility of atheism if [r]
r
is the Real Number, so that the utility of being a non-believer then is:
$$\mathbb E[a(r)] = 0$$
Hence, in a case of uncountably many possible gods, picking a god randomly to believe in has no higher utility than atheism. Ignoring all short run costs and benefits of either policy. It’s worth repeating that this was not a hard atheist argument. We not only supposed a non-zero probability density for every possible god, but even assumed one of the theologies was correct!
Next, Askell’s asymmetry argument. Suppose we have information that we are more likely to find the Real Number in one place than another. Then decision theory directs us to choose the wager for a god [G_r]
G_r
which we believe maximises our chances. This defence breaks on a modified example also. For suppose that instead of being uniformly ignorant of the Real Number, we instead believed it to lie in the vicinity of [\frac{1}{2}]
\frac{1}{2}
. We could model this with a Beta([2,2]
2,2
) distribution and exactly the same outcome would obtain. For the same reason as above regarding the measure of a particular point in a continuum. Indeed the result should hold up for any continuous density on [(0,1)]
(0,1)
that was non-zero on every element.
Jealousy on Olympus
4. What about the atheist-loving god? [IC] The basic idea: Suppose there’s a god that sends all non-believers to heaven and all believers to hell. Given the logic of Pascal’s wager, I ought not to believe in God. Answer: If it’s rational for you to think that disbelief in God (or cars, or hands) will maximize your chance of getting into heaven, then that’s what you ought to do under PW. What’s the evidence for the belief-shunning God? Possibly: ‘Divine hiddenness’ plus God making us capable of evidentialism. The evidence against? God making us capable of performing expected utility calculations, all the historical testimonial evidence for belief-loving Gods. I suspect the latter will outweigh the former. But if you’re making this objection you’re already on my side really: we’re now just quibbling about what God wants us to do.
The problem here stems from an assumption that God is not jealous. In addition to, “all the historical testimonial evidence for belief-loving Gods,” there is a good deal of historical testimonial evidence for heresy hating gods. For example let us modify our previous set of gods to become jealous in a first approximation of Dante’s Inferno. There, god really hates heretics (among many other groups). Virtuous atheists sit about in Limbo.
So let’s suppose that each god, [G_r]
G_r
declares that on the Day of Outcomes believers in the Real Number [r]
r
shall be infinitely rewarded ([+\infty]
+\infty
). Heretics (adherents to [s \neq r]
s \neq r
) shall be consigned to watch I’m a Celebrity! Get Me Out of Here on loop eternally ([- \infty]
- \infty
). While atheists shall be left in Limbo ([0]
0
What’s the expected utility of different belief policies for these jealous gods? If we assume all numbers are symmetric then a uniform distribution is appropriate. Well, for the atheist it goes unchanged. For randomly picking a god [V_r]
V_r
$$\mathbb E[u(s,r)] = \mathbb E[u(s,r)|r \neq s] P(r \neq s) + \mathbb E[u(s,r) | r=s]P(r=s) = -\infty \cdot 1 + \infty \cdot 0 = -\infty$$
Because the probability of picking exactly the right god is zero, whereas there is a continuum of ways to end up watching reruns of I’m a Celebrity! until long after the stars go out and the universe turns cold. So, while this might be characterised as quibbling about what god wants us to do, it really matters if we’re taking the wager seriously.
Discrete gods
The root cause of the first issue is that we don’t actually know what the space of all possible theologies looks like. So, for a start, we can’t actually evaluate the expectation as supposed by Askell. Really, I should just object that the space is not well defined, and then shoot down any attempt to define it satisfactorily However, if we suppose that we’re satisfied with the definition of the space we can look again at the mechanics of Askell’s many gods defence. We’re asked to choose one belief from among the union, [U]
U
, of a (presumably) countable set of theologies, [G]
G
, and a set containing atheism, [A]
A
. It is assumed that a PMF exists over this union such if we restrict our choice of beliefs to the subset [G]
G
, the redistribution of probability mass from [A]
A
to [G]
G
increases the probability mass of each god in [G]
G
. The problem is that if any individual item in [G]
G
has no mass to begin with such a rescaling achieves nothing. So, this issue is a generalisation of the false dichotomy: we’re asked to choose an option from a finite set when the actual set of choices is actually much larger.
In case the example I chose with gods labelled by real numbers seems too artificial, we can consider a more ‘realistic’ scenario that gets us to the same place. Suppose a religion consists of a set of beliefs. What is the space of all possible beliefs? Well as a first stab, we could say that it is the union of all sentences in all languages. There exist plenty of formal languages with a countably infinite number of possible sentences. For example, the number of correct C++ programs is at least countably infinite. Therefore the union of all sentences in all languages is at least countably infinite.
Next, let’s say that a theology is a set of beliefs. Then the space of all possible theologies therefore has a cardinality like the powerset of all possible beliefs. Moreover, the powerset of a countably infinite set is uncountable. An uncountable set has no probability mass function that is nonzero for all of its elements. Hence no probability mass function exists for the set of all possible theologies (defined in this manner) which is non-zero over all of its elements. Therefore redistributing whatever mass we gave to atheism over possible theologies has no effect on their mass and Askell’s defence fails again as in the case of the real numbers.
We have backed the proposed defences of Pascal’s wager based on expected utility theory into something of a corner. The wriggle room that I can see left is to either:
Assert that the space of all possible theologies is countable, requiring by extension an assertion that the space of all possible beliefs is finite. However this has a zero prior for me, since, for example, I already have a several beliefs about every real number [r]
r
, e.g. that a larger real number [s]
s
Assert that some arbitrary subset of possible theologies deserves to be treated as atoms with positive mass (e.g. the historically observed religions). But this is just begging the question (what about the historically unobserved but possible religions).
Assert that gods will reward beliefs that are ‘close enough’ to being correct in some sense to be defined. But without a proper framework (e.g. some weird filtration) it will not be possible to evaluate the expected utility of the decision. However, it seems that this would resolve as deciding which belief sets have a higher likelihood of reward. This would seem to look something like smaller belief sets having a higher likelihood than larger ones by virtue of being closer to a larger number of other belief sets. Which would lead us again to the same sample space, but with a non-uniform distribution over it, and still no mass function.
Neither is it obvious that the historical religions fit the ‘close enough’ model c.f. the Inferno. Indeed we also have to think about which belief sets would be ‘close enough’ to satisfy deities that we can have no knowledge of, e.g. all religions occurring in the (possibly infinite) universe outside our light cone. We should then reason towards a set of beliefs maximally likely to be rewarded. It’s not clear that such a process would end up anywhere different than secular reasoning about morality would lead us to; particularly if we take the Inferno‘s treatment of virtuous pagans as a model. I’d be interested to see someone try.
Also, everything I’ve said here is agnostic with respect to what ‘god’ actually means. E.g. the simulation hypothesis being correct, and the extra-galactic grad student rewarding all simulants with some particular belief set.
We’ve made it through to the end without griping about infinite expected utils too!
Posted in Thought things..
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AMM | APWine Finance
The AMM.sol contract is the main contract of the AMM. It exposes all traders and liquidity providers oriented methods. It also contains methods to control the pools - e.g. creating the initial liquidity pausing, unpausing the AMM, switching periods. These functions can be invoked using web3 libraries and browser wallets such as Metamask.
PairID#
The _pairID argument is used to select the pool. At V1 launch two pools are available per AMM instance, the PT/Underlying pool and the PT/FYT pool.
_pairID
0 PT/Underlying
1 PT/FYT
n \in \mathbb{N}, n > 1
Pairs (or Pools) that might be added in the future...
For more information on the AMM design architecture check this section.
switchPeriod#
function switchPeriod()
This function is called to reinitialize all pools at the end of a future period. Once a future expires, new FYTs are minted for each Principal Token. Old FYTs are no longer traded as their price is fixed.
getPTWeightInPair#
function getPTWeightInPair()
Get the PT weight in the PT/Underlying pair.
withdrawExpiredToken#
function withdrawExpiredToken(address _user, uint256 _pairID)
Withdraw all redeemable expired tokens in pool _pairID of user _user.
ℹ️ Expired Tokens are IBTs and Underlying assets. Liquidity Providers can claim the expired liquidity burning their LP Tokens:
Interest Bearing Tokens (IBTs): PT and FYT liquidity is converted into IBT at the end of a period. Yield is claimed from expired FYTs and PTs are unlocked from the protocol
Underlying: The underlying liqudity is removed from the pool at the end of the period.
getRedeemableExpiredTokens#
function getRedeemableExpiredTokens(address _user, uint256 _pairID)
(uint256)
Get the amount of expired tokens a liquidity provider can withdraw from the pool. It checks when was the last time user _user withdrew their expired tokens from the pool with ID _pairID. It applies compounding per period to take into account the yield generated between the last period renewed and now.
swapExactAmountIn#
function swapExactAmountIn( uint256 _pairID, uint256 _tokenIn, uint256 _tokenAmountIn, uint256 _tokenOut, uint256 _minAmountOut)
returns (uint256 tokenAmountOut, uint256 spotPriceAfter)
Swap _tokenAmountIn of _tokenIn for at least _minAmountOut of _tokenOut in the pool with _pairID. Returns the tokenAmountOut the user will get and the resulting spotPriceAfter after the trade.
calcOutAndSpotGivenIn#
function calcOutAndSpotGivenIn( uint256 _pairID, uint256 _tokenIn, uint256 _tokenAmountIn, uint256 _tokenOut, uint256 _minAmountOut)
Get the amount of _tokenOut and the spotPriceAfter resulting in trading exactly _tokenAmountIn of _tokenIn for at least _minAmountOut of _tokenOut in the pool with _pairID.
swapExactAmountOut#
function swapExactAmountOut( uint256 _pairID, uint256 _tokenIn, uint256 _maxAmountIn, uint256 _tokenOut, uint256 _tokenAmountOut,)
returns (uint256 tokenAmountIn, uint256 spotPriceAfter)
Swap at most _maxAmountIn of _tokenIn for exactly _tokenAmountOut of _tokenOut in the pool with _pairID. Returns the tokenAmountIn the user needed to perform the trade and the resulting spotPriceAfter after the trade.
calcInAndSpotGivenOut#
function calcInAndSpotGivenOut( uint256 _pairID, uint256 _tokenIn, uint256 _maxAmountIn, uint256 _tokenOut, uint256 _tokenAmountOut,)
Get the amount tokenAmountIn and the spotPriceAfter resulting in trading at most _maxAmountIn of _tokenIn for exactly _tokenAmountOut of _tokenOut in the pool with _pairID.
joinSwapExternAmountIn#
function joinSwapExternAmountIn( uint256 _pairID, uint256 _tokenIn, uint256 _tokenAmountIn, uint256 _minPoolAmountOut)
returns (uint256 poolAmountOut)
Pay _tokenAmountIn of token _tokenIn to join the pool _pairID, getting poolAmountOut of the pool shares.
The transaction completes only if poolAmountOut
\geq
_minPoolAmountOut
joinSwapPoolAmountOut#
function joinSwapPoolAmountOut( uint256 _pairID, uint256 _tokenIn, uint256 _poolAmountOut, uint256 _maxAmountIn)
returns (uint256 tokenAmountIn)
Specify the amount _poolAmountOut of pool shares that you want to get by providing at most _maxAmountIn of _tokenIn liquidity in the pool _pairID. Returns the amount tokenAmountIn of tokens needed to get the shares.
exitSwapPoolAmountIn#
function exitSwapPoolAmountIn( uint256 _pairID, uint256 _tokenOut, uint256 _poolAmountIn, uint256 _minAmountOut)
returns (uint256 tokenAmountOut)
Pay _poolAmountIn pool shares into the pool, getting at least _minAmountOut tokenAmountOut of the given token _tokenOut out of the pool _pairID.
exitSwapExternAmountOut#
function exitSwapExternAmountOut( uint256 _pairID, uint256 _tokenOut, uint256 _tokenAmountOut, uint256 _maxPoolAmountIn)
returns (uint256 poolAmountIn)
Specify the amount _tokenAmountOut of token _tokenOut that you want to get out of the pool pairID by providing at most _maxPoolAmountIn of LPTokens liquidity from the pool _pairID. Returns the amount poolAmountIn of LP tokens needed to get the shares.
createLiquidity#
function createLiquidity(uint256 _pairID, uint256[2] memory _tokenAmounts)
This function is called to initiate the liquidity at the beginning of a period. The first liquidity provider specify the pool with the _pairID and the _tokenAmounts of tokens to provide.
_tokenAmounts is an array containing the amount of each tokens of the pair. (Element 0 being the amount of Principal tokens and 1 the amount of either underlying or Future Yield Tokens depending on the pair)
At the beginning of each period both pool start with 50/50 weights (see this section for more informations about the AMM architecture). Therefore the spotPrice of each asset for the other can be computed as
P_{AB} = \frac{Amount_A}{Amount_B}
function addLiquidity( uint256 _pairID, uint256 _poolAmountOut, uint256[2] memory _maxAmountsIn)
Join the pool _pairID, getting _poolAmountOut pool tokens. This will pull some of each of the currently trading tokens in the pool, meaning you must have called approve for each token for this pool. These values are limited by the array of _maxAmountsIn in the order of the pool tokens.
function removeLiquidity( uint256 _pairID, uint256 _poolAmountIn, uint256[2] memory _minAmountsOut)
Exit the pool _pairID, paying _poolAmountIn pool tokens and getting some of each of the currently trading tokens in return. These values are limited by the array of _minAmountsOut in the order of the pool tokens.
getSpotPrice#
function getSpotPrice( uint256 _pairID, uint256 _tokenIn, uint256 _tokenOut)
Get the spot price of asset _tokenIn for _tokenOut based on the liquidity of pool _pairID.
getSwappingFees#
function getSwappingFees()
Returns the swapping fees currently set up in this AMM instance.
getAMMState#
function getAMMState()
returns (AMMGlobalState)
Returns the state of the AMM.
AMM can be:
Created: When it has been initialized
Activated: When it has been finalized
Paused: When it has been paused (for emergency situations or when there is a period switch)
getFutureAddress#
function getFutureAddress()
Returns the Future Vault address corresponding to the future of this AMM instance.
getAPWIBTAddress#
function getAPWIBTAddress()
Returns the address of the Principal Token corresponding to the future of this AMM instance.
getUnderlyingOfIBTAddress#
function getUnderlyingOfIBTAddress()
Returns the address of the underlying asset of the iBT deposited in the future corresponding to this AMM instance.
getIBTAddress#
function getIBTAddress()
Returns the address of the iBT deposited in the future corresponding to this AMM instance.
getFYTAddress#
function getFYTAddress()
Returns the address of the Future Yield Token corresponding to the ongoing period of the future of this AMM instance.
getPoolTokenAddress#
function getPoolTokenAddress()
Returns the address of the LPToken contract.
getPairWithID#
function getPairWithID(uint256 _pairID)
returns (Pair memory)
Returns the Pair corresponding to the _pairID. The Pair contains the tokenAddress of the second token of the pair (the first one is always the Principal Token), the weights, the balances and a boolean flag indicating if the liquidity for this pair has been initialized.
struct Pair { address tokenAddress; // first is always PT uint256[2] weights; uint256[2] balances; bool liquidityIsInitialized;}
getPairIDForToken#
function getPairIDForToken(address _tokenAddress)
Returns the _pairID corresponding to the token with _tokenAddress. 0 for underlying, 1 for FYT.
getLPTokenId#
function getLPTokenId( uint256 _ammId, uint256 _periodIndex, uint256 _pairID)returns returns (uint256)
Construct the LPToken ID based on the _ammId, _periodIndex and _pairID.
« ⚙️ AMM Architecture Overview
LPToken »
switchPeriod
getPTWeightInPair
withdrawExpiredToken
getRedeemableExpiredTokens
calcOutAndSpotGivenIn
calcInAndSpotGivenOut
createLiquidity
getSwappingFees
getAMMState
getFutureAddress
getAPWIBTAddress
getUnderlyingOfIBTAddress
getIBTAddress
getFYTAddress
getPoolTokenAddress
getPairWithID
getPairIDForToken
getLPTokenId
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11B50 Sequences (mod
m
11B57 Farey sequences; the sequences
{1}^{k},{2}^{k},\cdots
11B65 Binomial coefficients; factorials;
q
0\text{-}1
\left(1\text{-}1\right)
a
0-1
{\delta }^{\left(a\right)}
a
\left(1-1\right)
0-1
a,b
b
{\delta }^{\left(a\right)}={\delta }^{\left(b\right)}
A binomial representation of the 3x + 1 problem
Maurice Margenstern, Yuri Matiyasevich (1999)
A certain power series associatedwith a Beatty sequence
Takao Komatsu (1996)
A chaotic cousin of Conway's recursive sequence.
Pinn, Klaus (2000)
A characterization of Büchi's integer sequences of length 3
Pablo Sáez, Xavier Vidaux (2011)
A class of 1-additive sequences and quadratic recurrences.
Cassaigne, Julien, Finch, Steven R. (1995)
1/\pi
A combinatorial interpretation of the eigensequence for composition.
A common generating function for Catalan numbers and other integer sequences.
Cossali, G.E. (2003)
A complete annotated bibliography of work related to Sidon sequences.
O'Bryant, Kevin (2004)
A Connell-like sequence.
Stevens, Gary E. (1998)
A construction for sets of integers with distinct subset sums.
Bohman, Tom (1998)
János Barát, Péter P. Varjú (2005)
{\left\{{d}_{i}n+{b}_{i}:n\in ℤ\right\}}_{i\in I}
{d}_{i}={p}^{k}{q}^{l}
p,q
k,l\ge 0
j\ne i
{d}_{i}|{d}_{j}
. We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to
1
{d}_{i}
1254
A density estimate for the
3x+1
Ivan Korec (1994)
A few new facts about the EKG sequence.
Hofman, Piotr, Pilipczuk, Marcin (2008)
A function related to the central limit theorem
A number of properties of a function which originally appeared in a problem proposed by Ramanujan are presented. Several equivalent representations of the function are derived. These can be used to evaluate the function. A new derivation of an expansion in inverse powers of the argument of the function is obtained, as well as rational expressions for higher order coefficients.
A further note on a class of
{I}_{0}
David Grow (1987)
2
Martin Helm (1994)
𝒯
be a system of disjoint subsets of
{ℕ}^{*}
. In this paper we examine the existence of an increasing sequence of natural numbers,
A
, that is an asymptotic basis of all infinite elements
{T}_{j}
𝒯
simultaneously, satisfying certain conditions on the rate of growth of the number of representations
{𝑟}_{𝑛}\left(𝐴\right);{𝑟}_{𝑛}\left(𝐴\right):=\left|\left\{\left({𝑎}_{𝑖},{𝑎}_{𝑗}\right):{𝑎}_{𝑖}<{𝑎}_{𝑗};{𝑎}_{𝑖},{𝑎}_{𝑗}\in 𝐴;𝑛={𝑎}_{𝑖}+{𝑎}_{𝑗}\right\}\right|
, for all sufficiently large
n\in {T}_{j}
j\in {ℕ}^{*}
A theorem of P. Erdös is generalized.
A generalization of a theorem of Erdős-Rényi to m-fold sums and differences
Kathryn E. Hare, Shuntaro Yamagishi (2014)
Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define
{r}_{N}^{\left(m\right)}\left(\omega \right)
to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that
{r}_{N}^{\left(m\right)}\left(\omega \right)<K
for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.
A generalization of Beatty's theorem.
Holshouser, Arthur, Reiter, Harold (2001)
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[ x^n + y^n = z^n ]
\displaystyle \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }(ϕ5−ϕ)e52π1=1+1+1+1+1+⋯e−8πe−6πe−4πe−2π
\displaystyle \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)(k=1∑nakbk)2≤(k=1∑nak2)(k=1∑nbk2)
\displaystyle {1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots }= \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for }\lvert q\rvert<1.1+(1−q)q2+(1−q)(1−q2)q6+⋯=j=0∏∞(1−q5j+2)(1−q5j+3)1,for ∣q∣<1.
This is an example of an inline equation:
y = mx + b
500ms + 80ms + 500ms = 1080ms = 1.08 seconds
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
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Dynamic Stability of a Class of Second-Order Distributed Structural Systems With Sinusoidally Varying Velocities | J. Appl. Mech. | ASME Digital Collection
Dynamic Stability of a Class of Second-Order Distributed Structural Systems With Sinusoidally Varying Velocities
W. D. Zhu 1
Manuscript received August 31, 2012; final manuscript received December 4, 2012; accepted manuscript posted February 12, 2013; published online August 19, 2013. Assoc. Editor: Wei-Chau Xie.
Zhu, W. D., and Wu, K. (August 19, 2013). "Dynamic Stability of a Class of Second-Order Distributed Structural Systems With Sinusoidally Varying Velocities." ASME. J. Appl. Mech. November 2013; 80(6): 061008. https://doi.org/10.1115/1.4023638
Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. While parametric excitation of lumped-parameter systems has been extensively studied, that of distributed-parameter systems has been traditionally analyzed by applying Floquet theory to their spatially discretized equations. In this work, parametric instability regions of a second-order nondispersive distributed structural system, which consists of a translating string with a constant tension and a sinusoidally varying velocity, and two boundaries that axially move with a sinusoidal velocity relative to the string, are obtained using the wave solution and the fixed point theory without spatially discretizing the governing partial differential equation. There are five nontrivial cases that involve different combinations of string and boundary motions: (I) a translating string with a sinusoidally varying velocity and two stationary boundaries; (II) a translating string with a sinusoidally varying velocity, a sinusoidally moving boundary, and a stationary boundary; (III) a translating string with a sinusoidally varying velocity and two sinusoidally moving boundaries; (IV) a stationary string with a sinusoidally moving boundary and a stationary boundary; and (V) a stationary string with two sinusoidally moving boundaries. Unlike parametric instability regions of lumped-parameter systems that are classified as principal, secondary, and combination instability regions, the parametric instability regions of the class of distributed structural systems considered here are classified as period-1 and period-
i
i>1
) instability regions. Period-1 parametric instability regions are analytically obtained; an equivalent total velocity vector is introduced to express them for all the cases considered. While period-
i
i>1
) parametric instability regions can be numerically calculated using bifurcation diagrams, it is shown that only period-1 parametric instability regions exist in case IV, and no period-
i
i>1
) parametric instability regions can be numerically found in case V. Unlike parametric instability in a lumped-parameter system that is characterized by an unbounded displacement, the parametric instability phenomenon discovered here is characterized by a bounded displacement and an unbounded vibratory energy due to formation of infinitely compressed shock-like waves. There are seven independent parameters in the governing equation and boundary conditions, and the parametric instability regions in the seven-dimensional parameter space can be projected to a two-dimensional parameter plane if five parameters are specified. Period-1 parametric instability occurs in certain excitation frequency bands centered at the averaged natural frequencies of the systems in all the cases. If the parameters are chosen to be in the period-
i
i≥1
) parametric instability region corresponding to an integer
k
, an initial smooth wave will be infinitely compressed to
k
shock-like waves as time approaches infinity. The stable and unstable responses of the linear model in case I are compared with those of a corresponding nonlinear model that considers the coupled transverse and longitudinal vibrations of the translating string and an intermediate linear model that includes the effect of the tension change due to axial acceleration of the string on its transverse vibration. The parametric instability in the original linear model can exist in the nonlinear and intermediate linear models.
parametric instability, sinusoidally varying velocity, wave solution, fixed point theory, bifurcation diagram, shock-like wave, wave pattern
String, Waves
Wei-Chau
On a Peculiar Class of Acoustical Figure; and on Certain Forms Assumed by Groups of Particles Upon Vibrating Elastic Surfaces
On Maintained Vibrations
On the Parametric Excitation of a Dynamic System Having Multiple Degrees of Freedom
Further Results on Parametric Excitation of a Dynamic System
On the Stability of Hill’s Equation With Four Independent Parameters
Stability Analysis of Systems With Periodic Coefficients: An Approximate Approach
Experimental Investigation of a Parametrically Excited Nonlinear System
,” Proceedings of the European Nonlinear Dynamics Conference (ENDC 2011), Rome, July 24–29.
Stability of an Axially Accelerating String
Asymptotic Behavior for the Vibrating String With a Moving Boundary
Long-Time Behavior and Energy Growth for Electromagnetic Waves Reflected by a Moving Boundary
Exact Response of a Translating String With Arbitrary Varying Length Under General Excitation
Free and Forced Vibration of an Axially Moving String With an Arbitrary Velocity Profile
Dynamic Stability of a Translating String With a Sinusoidally Varying Velocity
Energetics and Stability of Translating Media With an Arbitrary Varying Length
Chaos: An Introduction to Dynamical Systems
Free, Periodic, Nonlinear Oscillation of an Axially Moving Strip
Parametric Instability in a Taut String With a Periodically Moving Boundary
Dynamics Stability of a Translating String With a Sinusoidally Varying Velocity
Higher Order Boundary Element Method Applied to the Hydrofoil Beneath the Free Surface
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The quantity theory of money is a theory that variations in price relate to variations in the money supply. It is most commonly expressed and taught using the equation of exchange and is a key foundation of the economic theory of monetarism.
The quantity theory of money is a framework to understand price changes in relation to the supply of money in an economy.
It argues that an increase in money supply creates inflation and vice versa.
The Irving Fisher model is most commonly used to apply the theory. Other competing models were formulated by British economist John Maynard Keynes, Swedish economist Knut Wicksell, and Austrian economist Ludwig von Mises.
The other models are dynamic and posit an indirect relationship between money supply and price changes in an economy.
The most common version, sometimes called the "neo-quantity theory" or Fisherian theory, suggests there is a mechanical and fixed proportional relationship between changes in the money supply and the general price level. This popular, albeit controversial, formulation of the quantity theory of money is based upon an equation by American economist Irving Fisher.
The Fisher equation is calculated as:
\begin{aligned} &\text{M} \times \text{V} = \text{P} \times \text{T} \\ &\textbf{where:} \\ &\text{M} = \text{money supply} \\ &\text{V} = \text{velocity of money} \\ &\text{P} = \text{average price level} \\ &\text{T} = \text{volume of transactions in the economy} \\ \end{aligned}
M×V=P×Twhere:M=money supplyV=velocity of moneyP=average price levelT=volume of transactions in the economy
Generally speaking, the quantity theory of money explains how increases in the quantity of money tends to create inflation, and vice versa. In the original theory, V was assumed to be constant and T is assumed to be stable with respect to M, so that a change in M directly impacts P. In other words, if the money supply increases then the average price level will tend to rise in proportion (and vice versa), with little effect on real economic activity.
For example, if the Federal Reserve (Fed) or European Central Bank (ECB) doubled the supply of money in the economy, the long-run prices in the economy would tend to increase dramatically. This is because more money circulating in an economy would equal more demand and spending by consumers, driving prices up.
Criticism of Fisher's Quantity Theory of Money
Economists disagree about how quickly and how proportionately prices adjust after a change in the quantity of money, and about how stable V and T actually are with respect to time and to M.
The classical treatment in most economic textbooks is based on the Fisher Equation, but competing theories exist.
The Fisher model has many strengths, including simplicity and applicability to mathematical models. However, it uses some assumptions that other economists have questioned to generate its simplicity, including the neutrality of the money supply and transmission mechanism, the focus on aggregate and average variables, the independence of the variables, and the stability of V.
Monetarist economics, usually associated with Milton Friedman and the Chicago school of economics, advocate the Fisher model, albeit with some modifications. In this view, V may not be constant or stable, but it does vary predictably enough with business cycle conditions that its variation can be adjusted for by policymakers and mostly ignored by theorists.
From their interpretation, monetarists often support a stable or consistent increase in money supply. While not all economists accept this view, more economists accept the monetarist claim that changes in the money supply cannot affect the real level of economic output in the long run.
Keynesians more or less use the same framework as monetarists, with few exceptions. John Maynard Keynes rejected the direct relationship between M and P, as he felt it ignored the role of interest rates. Keynes also argued the process of money circulation is complicated and not direct, so individual prices for specific markets adapt differently to changes in the money supply.
His theory emphasized that velocity (V) is not constant or stable, but can swing widely based on optimism or fear and uncertainty about the future, which drives liquidity preference. Keynes believed inflationary policies could help stimulate aggregate demand and boost short-term output to help an economy achieve full employment.
Knut Wicksell and the Austrians
The most serious challenge to Fisher came from Swedish economist Knut Wicksell, whose theories developed in continental Europe, while Fisher's grew in the United States and Great Britain. Wicksell, along with Austrian economists such as Ludwig von Mises and Joseph Schumpeter, agreed that increases in the quantity of money led to higher prices.
In their view, however, an artificial stimulation of the money supply through the banking system would distort prices unevenly, particularly in the capital goods sectors. This, in turn, shifts real wealth unevenly and could even cause business cycles.
The dynamic Wicksellian, Austrian, and Keynesian models stand in contrast to the static Fisherian model. Unlike the monetarists, adherents to the later models don't advocate a stable price level in monetary policy.
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Sets, Popular Questions: CBSE Class 11-humanities ENGLISH, English Grammar - Meritnation
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A and B are two sets then prove that A minus B union B is equal to A union B.
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Shailja & 1 other asked a question
26. Out of 1600 students in a college, 390 played cricket, 450 played hockey and 580 played basketball, 90 played both cricket and hockey, 125 played all three games.
a) How many students did not play any game?
b) How many played only cricket?
c) How many played only one game?
d) How many played only two games?
Aradhya & 1 other asked a question
Logarithm comes under which chapter of maths class 11???
Shuvi Dobhal asked a question
Let A ={x:x3+1=0};B={x:x2-x+1=0} .Find A n(intersection) B when (i) x is real (ii) x is not real.
Show that AUB(A union B) = A Intersection B implies A=B.
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1. In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B and 10% buy C. 5% buy AB, 3% buy BC and 4% buy AC. If 2% families buy all the three, find the number of families which buy i) A only ii) B only iii) none of A,BC.
a survey shows that 63% of americans like cheese whereas 76% like apples.If x% of americans like both cheese and apples,find the value of x.
plz explain in very a easy step.....
WRITE THE POWER SET OF A= {0,{0,1}]
Akankshya Sahoo asked a question
3.In a survey of 25 students, it was found that 15 had taken mathematics,12 had taken Physics and 11 had taken chemistry.5 had takenMathematics and Chemistry,9 had taken mathematics and Physics,4 had taken Chemistry and Physics and 3 had taken all the three subjects. Find the number of students that had
i)only Chemistry ii)only Physics iii)only Mathematics
iv)Physics and Maths but not Chemistry
v)Mathematics and Chemistry but not Physics
vi)only one of the subjects
vii)atleast one of the subjects
viii)none of the subjects
for what value of x {x=x^2;x belongs to R}
for that your answer is
We have, x 2 = x ⇒ x 2 - x = 0 ⇒ x(x - 1) = 0 ⇒ x = 0 or 1
So there are two values of x that is 0 and 1 as both of them belongs to R
What if we take 1=x/x^2 and if we substitute x=0 then it becomes 1/0 which is undefined.
Please explain why x=0
Vasudha Venkat asked a question
sir can you prove the de morgan's law using venn diagram?
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Shanmuga Priya asked a question
7. In a survey of 100 people it was found that 28 read magazine A, 30 read magazine B, 42 read magazine C, 8 read magazines A B, 10 read magazine AC 5 read magazine B and C and 3 read all the three. Find:i) How many read none of the magazines?ii) How many read magazine C only?
Complaints about a students hostel mess fell into three categories. Complaints about mess(M), food(F) and services(S). the total number of complaints received were 173 and were as follows: n(M) = 110, n(F) = 55, n(S) = 67, n( M F S!) = 20, n( M S F!) = 11, n( F S M!) = 16. Determine the number of complaints about i) all the three ii) about two or more than two
Best books for cbse class 11 commerce students?
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sir this is 3 mark question
Q.25. A survey was conducted of the T.V programmes watched by 120 students of a school hostel revealed that 70 students watched 'Discovery channel' and 56 students watched 'sports channel' where as 24 watched both the programmes. Find the number of students who did not watch T.V on that day.
pronomita.das11... asked a question
suppose A1, A2,....,A30 are 30 sets each with 5 elements and B1, B2,...., Bn are n sets each with 3 elements. let union ofA1, A2,....,A30 = union ofB1, B2,...., Bn =S. Each element of S belongs to 10 of A's and 9 of B's. find n.
WHAT DOES IT MEAN????????????
Dishi Sibbal & 1 other asked a question
wat are d uses of sets in our daily life?
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difference between finite and infinite set in detail and question based on it
Sumrah Shakeel asked a question
At break in a school, 123 students go to canteen which sells cakes, ice-cream and buns. 42 students buy ice-cream, 36 buy buns, 10 buy only cakes. 15 students buy ice-cream and buns. 10 buy ice-cream and cakes, and 4 buy cakes and buns but not ice-cream and 11 buy ice-cream and buns but not cakes. Draw Venn diagram to illustrate the above information and find:
How many students buy nothing at all
How many students buy at least two items
How many students buy all three items
Number of ordered pairs of integers (n,m) for which n^2-m^2=14 is. (A)0 (B)1 (C)2 (4)
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IF P(A)=P(B) SHOW THAT A=B
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Prove That : cos6A = 32cos^6A - 48cos^4A + 18cos^2A - 1.
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does {phi} belong to A={1,2, {3,4},5}
Suma Dadarkar asked a question
There are 20 students in a chemistry class and 30 students in a physics class. find the number of students which are either in physics or in chemistry class in the following cases:
a)two classes meet at the same hour. (hint: n(C∩P) =Φ)
b)The two classes meet at different hours and 10 students are enrolled in both the courses. (hint: n(C∩P)=10)
Amrita S asked a question
Let A={x:x=2n, n∈Z}and B={x:x=3n, n∈Z}then find A∩B.
In a survey of 400 students in a school ,100 were listed as taking apple juice ,150 were listed as taking orange juice and 75 were listed as taking both apple as well as orange juice .Find how many students were taking neither apple juice nor orange juice.
Arathy Venu & 1 other asked a question
In a class, 36 students offered Physics, 46 offered Chemistry and 48 offered Mathematics. Of these, 26 are in both Chemistry and Mathematics, 24 are in both Physics and Chemistry, 22 are in both Physics and Mathematics. 12 have offered all the three subjects. Find:
a) the number of students in the class.
b) the number of students who have offered Mathematics but not Chemistry.
c) the number of students who have offered exactly one subject.
Verify De Morgan's law by using venn's diagram.
Rajat Ranjan asked a question
Write the following in set-builder form :-
1. C = {1, -1, i, -i}
Write the following in roster form :-
1. D = {x : x = cube roots of unity}.
prove that A intersection (B-C) = (A intersection B) - (A intersection C)
a and b are two sets such that
n(a-b)=20+x
n(b-a)=3x
n(a∩b) =x+1 if n(a)= n(b)
Avinash Jadhao asked a question
The complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements i.e.,
These are also known as De Morgan’s law.
Please explain this in detail
Naina Sharma asked a question
show that if ais A subset B,Then C-B is subset of C-A
If A = { 2x : x element of N}, B = { 3x : x element of N}, C = { 5x : x element of N}, then find (A intersection B) intersection C.
Q1if A,B,C are three sets then proove that A∩(B C)=(A∩B) (A∩C)
Q2let A,B, and C be the sets such that AUB=AUC and A∩B=A∩C show that B=C
Nikita Bordoloi asked a question
A and B are two finite sets such that n(A)=m1 and n(A)=m2, then find the least and greatest values of n(AUB).
Let A,B,and C be the sets such that AUB
= AUC and A intersection B and A intersection B = A intesection C show that
Varsha Ajaykumar asked a question
if u=(aeiou) a=(aei) b=(eou) c=(aiu) then verify that aintersection bminusc is equal to aintersection b minus aintersectionc
Prove A-(B-C) = (A-B) U (A intersect C) for any sets A,B and C.
How do we prove it without using the Venn diagram?
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In a beauty contest, half the number of judges voted for Miss A, 2/3 voted for Miss B, 20 voted for both and 6 did not vote for neither Miss A nor Miss B. Find how many judges , in all, were present there?
12) If A = { (x,y0 : y=1/x ,x?0 x?R} and B= {(x,y) : y=-x ,x?R } then write A?B =?..
abhishek241096 asked a question
bhak ye sab kya satta batta kar raha hai?
\left(A\cap B\right)\cup \left(\left(X-A\right)\cap \left(X-B\right)\right).
\subset
\ne
\subset
\ne
In a class 18 students took physics, 23 took chemistry, 24 took maths, out of them 13 took chemistry and mathematics, 12 took physics and chemistry, 11 took physics and mathmetics. if 6 were offered all the three subjects find :
2. How many took maths but not chemistrty
3. How many took exactly one of the three subjects
What is the difference between belongs to and subset?
1. Prove A intersection (A U B)' = phi
2. Prove that if A U B = C and A intersection B = phi, then C - B = A.
3. In ao town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper, 10% families buy newspaper C, 5% buy A and B, 3% buy B and C, 4% buy A and C, If 2% buy all three papers, then find the number of families which buy newspaper A only.
Bhumika Hasija asked a question
what is meaning of this De Morgans law.
Sheril asked a question
in a class of 60 students,23 play hockey,15 play basketball,20 play cricket and 7 play hockey and basketball,5 play cricket and basketball,4 play hockeyand cricket,15 do not play any of the three games. find the number of students who
1- play all the three games
2-play hockey but not cricket
3-play hockey and cricket both,but not basketball
Raunak Tuw asked a question
Out of 2000 employees in a office 48 % preferred cofee, 54% liked tea , 64% used to smoke . Out of th etotal employee 28% used cofee and tea , 32% used tea and smoke and 30% prefered cofee and smoke and only 6% did none of these . The number having all th ethrree is ?
A and B are two sets such that n(A-B)=20+x, n(B-A)=3x and n(A ∩B)=x+1.Draw a Venn diagram to illustrate this information.Find (i)the value of x (ii)n(AUB)
if n(U)=25, n(A)=15 , n(A intersectionB)=6 and n(AUB)' = 8 then find
1. n(B) 2. n(B-A) with proper steps
Using properties of sets prove the statements given:-
(i) For all sets A and B, AU(B-A) = AUB
(ii) For all sets A and B, A-(A-B) = AintersectionB
(iii) For all sets A and B, (AUB)-B = A-B
n({fi})=?
tushal_gupta... asked a question
(AnB )'=A'nB' prove that
David Patel & 1 other asked a question
in a class of 25 student , 12 have taken maths,8 have taken maths but not biology.find the number of student who are taken maths and biology and the number of those who have taken biology but not maths .each student have taken either maths or biology or both?
Two finite sets have m and n elements. The total no. of subsets of the first set is 56 more than that of the total no. of subsets of the second set. Find the values of m and n.
Write in roaster form:
{x : x = 3p, p belongs to W, p lesser and equal to 3}
At a certain conference of 100 people there are 29 Indian women and 23 Indian men . Out of these Indian people 4 are doctor and 24 are either men or doctors . There are no foriegn doctors. The number of women doctors attending the conference is ?
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Probability and Statistics | Mathematics or Quantitative Aptitude | CAT Entrance Exams Online Objective Test Probability and Statistics | Mathematics or Quantitative Aptitude | CAT Entrance Exams Online Objective Test
Probability and Statistics - Online Test
CAT Entrance Exams Mathematics or Quantitative Aptitude Probability and Statistics TEST : 1
# CAT Entrance Exams # Mathematics or Quantitative Aptitude # Probability and Statistics # CBSE 11th Mathematics Prepare / Learn
Q1. A batsman scores runs in 10 innings as 38,70,48,34,42,55,63,46,54 and 44 , then the mean score is
Mean (x) =
Q2. A coin is tossed three times, if E : head on third toss , F : heads on first two tosses. Find P(E|F)
# CAT Entrance Exams # Mathematics or Quantitative Aptitude # Probability and Statistics # General Aptitude Prepare / Learn
Direction: Study the given information carefully and answer the questions that follow:
A basket contains 5 red, 4 blue, 3 green stones.
If two stones are picked at random, what is the probability that both are blue?
Answer : Option B Explaination / Solution: Probability if both is blue
4C2/12C2 = 6/66 → 1/11
Q4. The S.D. of the observations 22,26,28,20,24,30 is
Q5. A coin is tossed three times, E : at most two tails , F : at least one tail. Find P(E|F)
If four balls are picked at random, what is the probability that two are blue and two are red?
Answer : Option A Explaination / Solution: Probability if two are Blue and two are Red
[(4C2*5C2)/12C4] = (6*10)/495 → 4/33
Q7. If the angle between two lines of regression is , then it represents
B. no linear correlation
C. perfect negative correlation
hence r = 0, therefore there is no linear correlation.
Q8. A black and a red dice are rolled. Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.
Answer : Option C Explaination / Solution: n(S)=36.
Let A = event of getting sum greater than 9.
= {(4,6),(5,5),(6,4),(5,6),(6,5),(6,6)}
And B = event of getting 5 on black die.
={(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)}
If three balls are picked at random, what is the probability that at least one is green?
Probability if at least one is Green
[1-(9C3/12C3)] = 84/220 → 34/55
Q10. The two lines of regression are 2x - 7y + 6 = 0 and 7x – 2y +1 = 0. What is correlation coefficient between x and y ?
A. - 2/ 7
but sign of
\rho
is same as the sign of
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Cointegration and Error Correction Analysis - MATLAB & Simulink - MathWorks 日本
An n-dimensional time series yt is cointegrated if some linear combination β1y1t + … + βnynt of the component variables is stationary. The combination is called a cointegrating relation, and the coefficients β = (β1 , … , βn)′ form a cointegrating vector. Cointegration is usually associated with systems of I(1) variables, since any I(0) variables are trivially cointegrated with other variables using a vector with coefficient 1 on the I(0) component and coefficient 0 on the other components. The idea of cointegration can be generalized to systems of higher-order variables if a linear combination reduces their common order of integration.
Cointegration is distinguished from traditional economic equilibrium, in which a balance of forces produces stable long-term levels in the variables. Cointegrated variables are generally unstable in their levels, but exhibit mean-reverting “spreads†(generalized by the cointegrating relation) that force the variables to move around common stochastic trends. Cointegration is also distinguished from the short-term synchronies of positive covariance, which only measures the tendency to move together at each time step. Modification of the VAR model to include cointegrated variables balances the short-term dynamics of the system with long-term tendencies.
The tendency of cointegrated variables to revert to common stochastic trends is expressed in terms of error-correction. If yt is an n-dimensional time series and β is a cointegrating vector, then the combination β′yt−1 measures the “error†in the data (the deviation from the stationary mean) at time t−1. The rate at which series “correct†from disequilibrium is represented by a vector α of adjustment speeds, which are incorporated into the VAR model at time t through a multiplicative error-correction term αβ′yt−1.
In general, there may be multiple cointegrating relations among the variables in yt, in which case the vectors α and β become matrices A and B, with each column of B representing a specific relation. The error-correction term becomes AB′yt−1 = Cyt−1. Adding the error-correction term to a VAR model in differences produces the vector error-correction (VEC) model:
\mathrm{Î}{y}_{t}=C{y}_{tâ1}+\underset{i=1}{\overset{q}{â}}{B}_{i}\mathrm{Î}{y}_{tâi}+{\mathrm{ε}}_{t}.
If the variables in yt are all I(1), the terms involving differences are stationary, leaving only the error-correction term to introduce long-term stochastic trends. The rank of the impact matrix C determines the long-term dynamics. If C has full rank, the system yt is stationary in levels. If C has rank 0, the error-correction term disappears, and the system is stationary in differences. These two extremes correspond to standard choices in univariate modeling. In the multivariate case, however, there are intermediate choices, corresponding to reduced ranks between 0 and n. If C is restricted to reduced rank r, then C factors into (nonunique) n-by-r matrices A and B with C = AB′, and there are r independent cointegrating relations among the variables in yt.
{y}_{t}={A}_{1}{y}_{tâ1}+...+{A}_{p}{y}_{tâp}+{\mathrm{ε}}_{t}.
\begin{array}{l}{A}_{1}=C+{I}_{n}\text{+ }{B}_{1}\hfill \\ {A}_{i}={B}_{i}â{B}_{iâ1}\text{, }i=2,...,q\hfill \\ {A}_{p}=â{B}_{q}\hfill \end{array}\right\}\text{ VEC(}q\text{) to VAR(}p=q+1\right)\text{ (using }vec2var\text{)}
\begin{array}{l}C=\underset{i=1}{\overset{p}{â}}{A}_{i}â{I}_{n}\hfill \\ {B}_{i}=â\underset{j=i+1}{\overset{p}{â}}{A}_{j}\hfill \end{array}\right\}\text{ VAR(}p\text{) to VEC(}q=pâ1\text{) (using }var2vec\text{)}
\mathrm{Î}{y}_{t}=A{B}^{â²}{y}_{tâ1}+\underset{i=1}{\overset{q}{â}}{B}_{i}\mathrm{Î}{y}_{tâi}+Dx+{\mathrm{ε}}_{t}.
Variables in x may include seasonal or interventional dummies, or terms representing deterministic trends in the levels of the data. Since the model is expressed in differences ∆yt, constant terms in x represent deterministic linear trends in the levels of yt and linear terms represent deterministic quadratic trends. In contrast, constant and linear terms in the cointegrated series have the usual interpretation as intercepts and linear trends, although restricted to the stationary variable formed by the cointegrating relation. Johansen [110] considers five cases for AB´yt−1 + Dx which cover the majority of observed behaviors in macroeconomic systems:
Form of Cyt−1 + DX
AB´yt−1. There are no intercepts or trends in the cointegrated series and there are no deterministic trends in the levels of the data.
A(B´yt−1+c0). There are intercepts in the cointegrated series and there are no deterministic trends in the levels of the data.
A(B´yt−1+c0)+c1. There are intercepts in the cointegrated series and there are deterministic linear trends in the levels of the data. This is the default value.
'H*' A(B´yt−1+c0+d0t)+c1. There are intercepts and linear trends in the cointegrated series and there are deterministic linear trends in the levels of the data.
'H' A(B´yt−1+c0+d0t)+c1+d1t. There are intercepts and linear trends in the cointegrated series and there are deterministic quadratic trends in the levels of the data.
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q
\left(m,r\right)
m>r\ge 0
\left(m,r\right)
G=\left(d,h\right)={\left({d}_{n,k}\right)}_{n,k\in ℕ}
{d}_{mn+r,\left(m-1\right)n+r}
n=0,1,2,\cdots
\left(m,r\right)
G
{G}^{\left(m,r\right)}={\left({d}_{mn+r,\left(m-1\right)n+k+r}\right)}_{n,k\in ℕ}.
\left(m,r\right)
G=\left(d,h\right)
h\left(0\right)=0
d\left(0\right),{h}^{\text{'}}\left(0\right)\ne 0
\left(m,r\right)
\left(m,r\right)
{G}^{\left(m,r\right)}
G
A
A binomial coefficient identity associated to a conjecture of Beukers.
Ahlgren, Scott, Ekhad, Shalosh B., Ono, Ken, Zeilberger, Doron (1998)
A combinatorial approach to hyperharmonic numbers.
Benjamin, Arthur T., Gaebler, David, Gaebler, Robert (2003)
A combinatorial interpretation for an identity of Barrucand.
A combinatorial interpretation of the numbers
6\left(2n\right)!/n!\left(n+2\right)!
Gessel, Ira M., Xin, Guoce (2005)
A combinatorial proof of Sun's “curious” identity.
Voigt, B. (1984)
A Contribution to the Theory of Permutations
Josef Kaucký (1971)
A curious identity involving binomial coefficients.
Sun, Zhiwei (2002)
A divisibility property of binomial coefficients viewed as an elementary sieve.
A Generalization of Functions !n and n!
Zoran Šami (1996)
Gábor Kallós (2006)
In this paper we generalize the Pascal triangle and examine the connections among the generalized triangles and powering integers respectively polynomials. We emphasize the relationship between the new triangles and the Pascal pyramids, moreover we present connections with the binomial and multinomial theorems.
A generating functions proof of a curious identity.
Panholzer, Alois, Prodinger, Helmut (2002)
A natural series for the natural logarithm.
Dasbach, Oliver T. (2008)
A note on nested sums.
Butler, Steve, Karasik, Pavel (2010)
q
-binomial rational root theorem.
Lin, Ying-Jie (2009)
A proof of a conjecture of Knuth.
Paule, Peter (1996)
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Perform rational fitting to complex frequency-dependent data - MATLAB - MathWorks í•œêµ
\begin{array}{l}F\left(s\right)=\underset{k=1}{\overset{n}{â}}\frac{{C}_{k}}{sâ{A}_{k}}+D\\ \text{where, }s=jÃ2\mathrm{Ï}f\\ Câ\text{residues}\\ Aâ\text{poles}\\ Dâ\text{direct term}\end{array}
z=2×2 cell array
p=2×2 cell array
k = 2×2
dcgain = 2×2
[1] Nakatsukasa, Yuji, Olivier Sète, and Lloyd N. Trefethen. “The AAA Algorithm for Rational Approximation.†SIAM Journal on Scientific Computing 40, no. 3 (January 2018): A1494–1522. https://doi.org/10.1137/16M1106122.
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What the CAGR Can Tell You
Example of How to Use CAGR
Additional CAGR Uses
How Investors Use the CAGR
Modifying the CAGR Formula
Smooth Rate of Growth Limitation
Other CAGR Limitations
Formula and Calculation of the Compound Annual Growth Rate (CAGR)
\begin{aligned}&CAGR= \left ( \frac{EV}{BV} \right ) ^{\frac{1}{n}}-1\times 100\\&\textbf{where:}\\&EV = \text{Ending value}\\&BV = \text{Beginning value}\\&n = \text{Number of years}\end{aligned}
CAGR=(BVEV)n1−1×100where:EV=Ending valueBV=Beginning valuen=Number of years
Multiply by 100 to convert the answer into a percentage.
In reality, this sort of performance is unlikely. However, the CAGR can be used to smooth returns so that they may be more easily understood compared to alternative methods.
Imagine you invested $10,000 in a portfolio with the returns outlined below:
From Jan. 1, 2018, to Jan. 1, 2019, your portfolio grew to $13,000 (or 30% in year one).
On Jan. 1, 2020, the portfolio was $14,000 (or 7.69% from January 2019 to January 2020).
On Jan. 1, 2021, the portfolio ended with $19,000 (or 35.71% from January 2020 to January 2021).
We can see that on an annual basis, the year-to-year growth rates of the investment portfolio were quite different as shown in the parentheses.
On the other hand, the compound annual growth rate smooths the investment’s performance and ignores the fact that 2018 and 2020 were vastly different from 2019. The CAGR over that period was 23.86% and can be calculated as follows:
CAGR=\left(\frac{\$19,000}{\$10,000}\right )^{\frac{1}{3}}-1\times100=23.86\%
CAGR=($10,000$19,000)31−1×100=23.86%
The CAGR of 23.86% over the three-year investment period can help an investor compare alternatives for their capital or make forecasts of future values. For example, imagine an investor is comparing the performance of two uncorrelated investments.
In any given year during the period, one investment may be rising while the other falls. This could be the case when comparing high-yield bonds to stocks, or a real estate investment to emerging markets. Using CAGR would smooth the annual return over the period so the two alternatives would be easier to compare.
As another example, let’s say an investor bought 55 shares of Amazon.com (AMZN) stock in December 2017 at $1,180 per share, for a total investment of $64,900. After three years, in December 2020, the stock has risen to $3,200 per share, and the investor’s investment is now worth $176,000. What is the CAGR?
Using the CAGR formula, we know that we need the:
Beginning Balance: $64,900
So to calculate the CAGR for this simple example, we would enter that data into the formula as follows: [($176,000 / $64,900) ^ (1/3)] - 1 = 39.5%.
The CAGR can be used to calculate the average growth of a single investment. As we saw in our example above, due to market volatility, the year-to-year growth of an investment will likely appear erratic and uneven.
For example, an investment may increase in value by 8% in one year, decrease in value by -2% the following year, and increase in value by 5% in the next. CAGR helps smooth returns when growth rates are expected to be volatile and inconsistent.
The CAGR can be used to compare different investment types with one another. For example, suppose that in 2015, an investor placed $10,000 into an account for five years with a fixed annual interest rate of 1% and another $10,000 into a stock mutual fund. The rate of return in the stock fund will be uneven over the next few years, so a comparison between the two investments would be difficult.
Assume that at the end of the five-year period, the savings account’s balance is $10,510.10 and, although the other investment has grown unevenly, the ending balance in the stock fund was $15,348.52. Using the CAGR to compare the two investments can help an investor understand the difference in returns:
\text{Savings Account CAGR} =\, \left ( \frac{\$ 10,510.10}{\$ 10,000} \right )^{\frac{1}{5}}-1 \times 100= 1.00\%
Savings Account CAGR=($10,000$10,510.10)51−1×100=1.00%
\text{Stock fund CAGR} =\, \left ( \frac{\$ 15,348.52}{\$ 10,000} \right )^{\frac{1}{5}}-1 \times 100= 8.95\%
Stock fund CAGR=($10,000$15,348.52)51−1×100=8.95%
On the surface, the stock fund may look like a better investment, with nearly nine times the return of the savings account. On the other hand, one of the drawbacks of the CAGR is that by smoothing the returns, The CAGR cannot tell an investor how volatile or risky the stock fund was.
The CAGR can also be used to track the performance of various business measures of one or multiple companies alongside one another. For example, over a five-year period, Big-Sale Stores’ market share CAGR was 1.82%, but its customer satisfaction CAGR over the same period was -0.58%. In this way, comparing the CAGRs of measures within a company reveals strengths and weaknesses.
Detect Weaknesses and Strengths
Comparing the CAGRs of business activities across similar companies will help evaluate competitive weaknesses and strengths. For example, Big-Sale’s customer satisfaction CAGR might not seem so low compared with SuperFast Cable’s customer satisfaction CAGR of -6.31% during the same period.
Understanding the formula used to calculate CAGR is an introduction to many other ways that investors evaluate past returns or estimate future profits. The formula can be manipulated algebraically into a formula to find the present value or future value of money, or to calculate a hurdle rate of return.
For example, imagine that an investor knows that they need $50,000 for a child’s college education in 18 years and they have $15,000 to invest today. How much does the average rate of return need to be to reach that objective? The CAGR calculation can be used to find the answer to this question as follows:
\text{Required Return} =\, \left ( \frac{\$ 50,000}{\$ 15,000} \right )^{\frac{1}{18}}-1 \times 100= 6.90\%
Required Return=($15,000$50,000)181−1×100=6.90%
An investment is rarely made on the first day of the year and then sold on the last day of the year. Imagine an investor who wants to evaluate the CAGR of a $10,000 investment that was entered on June 1, 2013, and sold for $16,897.14 on Sept. 9, 2018.
Before the CAGR calculation can be performed, the investor will need to know the fractional remainder of the holding period. They held the position for 213 days in 2013, a full year in 2014, 2015, 2016, and 2017, and 251 days in 2018. This investment was held for 5.271 years, which is calculated by the following:
The total number of days that the investment was held was 1,924 days. To calculate the number of years, divide the total number of days by 365 (1,924/365), which equals 5.271 years.
The total number of years that the investment was held can be placed in the denominator of the exponent inside CAGR’s formula as follows:
\text{Investment CAGR} =\, \left ( \frac{\$ 16,897.14}{\$ 10,000} \right )^{\frac{1}{5.271}}-1 \times 100= 10.46\%
Investment CAGR=($10,000$16,897.14)5.2711−1×100=10.46%
The most important limitation of the CAGR is that because it calculates a smoothed rate of growth over a period, it ignores volatility and implies that the growth during that time was steady. Returns on investments are uneven over time, except bonds that are held to maturity, deposits, and similar investments.
Also, the CAGR does not account for when an investor adds funds to a portfolio or withdraws funds from the portfolio over the period being measured.
For example, if an investor had a portfolio for five years and injected funds into the portfolio during the five-year period, then the CAGR would be inflated. The CAGR would calculate the rate of return based on the beginning and ending balances over the five years, and would essentially count the deposited funds as part of the annual growth rate, which would be inaccurate.
Besides the smoothed rate of growth, the CAGR has other limitations. A second limitation when assessing investments is that no matter how steady the growth of a company or investment has been in the past, investors cannot assume that the rate will remain the same in the future. The shorter the time frame used in the analysis, the less likely it will be for the realized CAGR to meet the expected CAGR when relying on historical results.
A third limitation of the CAGR is a limitation of representation. Say that an investment fund was worth $100,000 in 2016, $71,000 in 2017, $44,000 in 2018, $81,000 in 2019, and $126,000 in 2020. If the fund managers represented in 2021 that their CAGR was a whopping 42.01% over the past three years, they would be technically correct. They would, however, be omitting some very important information about the fund’s history, including the fact that the fund’s CAGR over the past five years was a modest 4.73%.
The CAGR measures the return on an investment over a certain period of time. The internal rate of return (IRR) also measures investment performance but is more flexible than the CAGR.
The most important distinction is that the CAGR is straightforward enough that it can be calculated by hand. In contrast, more complicated investments and projects, or those that have many different cash inflows and outflows, are best evaluated using IRR. To back into the IRR, a financial calculator, Excel, or portfolio accounting system is ideal.
Those interested in learning more about CAGR and other financial topics may want to consider enrolling in one of the best investing courses currently available.
What Is an Example of Compound Annual Growth Rate (CAGR)?
The CAGR is a measurement used by investors to calculate the rate at which a quantity grew over time. The word “compound” denotes the fact that the CAGR takes into account the effects of compounding, or reinvestment, over time. For example, suppose you have a company with revenue that grew from $3 million to $30 million over a span of 10 years. In that scenario, the CAGR would be approximately 25.89%.
What Is Considered a Good CAGR?
What counts as a good CAGR will depend on the context. But generally speaking, investors will evaluate this by thinking about their opportunity cost as well as the riskiness of the investment. For example, if a company grew by 25% in an industry with an average CAGR closer to 30%, then its results might seem lackluster by comparison. But if the industry-wide growth rates were lower, such as 10% or 15%, then its CAGR might be very impressive.
What Is the Difference Between the CAGR and a Growth rate?
The main difference between the CAGR and a growth rate is that the CAGR assumes the growth rate was repeated, or “compounded,” each year, whereas a traditional growth rate does not. Many investors prefer the CAGR because it smooths out the volatile nature of year-by-year growth rates. For instance, even a highly profitable and successful company will likely have several years of poor performance during its life. These bad years could have a large effect on individual years’ growth rates but would have a relatively small impact on the company’s CAGR.
Can the CAGR be Negative?
Yes. A negative CAGR would indicate losses over time rather than gains.
To compare the performance and risk characteristics among various investment alternatives, investors can use a risk-adjusted CAGR. A simple method for calculating a risk-adjusted CAGR is to multiply the CAGR by one minus the investment’s standard deviation. If the standard deviation (i.e., its risk) is zero, then the risk-adjusted CAGR is unaffected. The larger the standard deviation, the lower the risk-adjusted CAGR will be.
Yahoo Finance. “Amazon.com, Inc. (AMZN) Historical Data.” Accessed Sept. 7, 2021.
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Look at the representations shown in the Math Notes box for this lesson (“Representations of Portions”). Copy the diagrams below and write a fraction and a percent for the shaded portion of each one.
A fraction is a number that represents a part of the whole. How many parts are shaded and how many total parts are there?
The shaded pieces make up the numerator while the total number of pieces makes up the denominator.
\text{Fraction}: \frac{1}{4}
To convert from fraction to percent, use standard long division to get the decimal form, and then multiply by
100
or shift the decimal point twice to the right. For example,
\frac{3}{20} \rightarrow 0.15 \rightarrow 15 \text{ percent}
Partial Answer 2 (a):
25\%
Think of each triangle as two pieces. Then use the method from part (a).
\space
\space
\space
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EUDML | A sharp estimate of the number of integral points in a 4-dimensional tetrahedra. EuDML | A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.
A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.
Stephen S.-T. Yau; Yi-Jing Xu
Yau, Stephen S.-T., and Xu, Yi-Jing. "A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.." Journal für die reine und angewandte Mathematik 473 (1995): 1-24. <http://eudml.org/doc/153802>.
author = {Yau, Stephen S.-T., Xu, Yi-Jing},
keywords = {number of integral points in a 4-dimensional tetrahedra; analytic number theory; primality testing; factoring; germ of complex analytic functions; Milnor number; Durfee conjecture; weighted homogeneous polynomial; upper estimate},
title = {A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.},
AU - Yau, Stephen S.-T.
AU - Xu, Yi-Jing
TI - A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.
KW - number of integral points in a 4-dimensional tetrahedra; analytic number theory; primality testing; factoring; germ of complex analytic functions; Milnor number; Durfee conjecture; weighted homogeneous polynomial; upper estimate
number of integral points in a 4-dimensional tetrahedra, analytic number theory, primality testing, factoring, germ of complex analytic functions, Milnor number, Durfee conjecture, weighted homogeneous polynomial, upper estimate
Convex sets in
Factorization; primality
Articles by Stephen S.-T. Yau
Articles by Yi-Jing Xu
|
(Created page with " <!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: ...")
m (→Construction and examples)
== Construction and examples ==
For a manifold $X$, $\widetilde X$ denotes ''the deleted product'' of $X$, i.e. $X^2$ minus an open tubular neighborhood of the diagonal. It is a manifold with boundary and has the standard free involution.
{{theorem|Definition}}[of the Haefliger-Wu invariant $\alpha$]
The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case.
<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; ... </wikitex> == Construction and examples == <wikitex>; For a manifold $X$, $\widetilde X$ denotes ''the deleted product'' of $X$, i.e. $X^2$ minus an open tubular neighborhood of the diagonal. It is a manifold with boundary and has the standard free involution. {{theorem|Definition}}[of the Haefliger-Wu invariant $\alpha$] \label{DefHaef} The Haefliger-Wu invariant $\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})$ is induced by the Gauss map, also denoted by $\alpha$. The Gauss map assigns to an individual embedding $f:N\to\R^{k}$ an equivariant map $\widetilde{N}\to S^{k-1}$ defined by the formula $$ (x,y)\mapsto \frac{f(x)-f(y)} {\|f(x)-f(y)\|}, \quad (x,y)\in\widetilde{N}\subset N\times N. $$ The Haefliger-Wu invariant and the Gauss map are analogously defined for $N_0$; we will denote them by $\alpha_0$ in this case. {{definition}} </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> <!-- == Acknowledgments == ... == Footnotes == <references/> --> == References == {{#RefList:}} <!-- == External links == * The Wikipedia page about [[Wikipedia:Page_name|link text]]. --> <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]X
\widetilde X
X
X^2
\alpha
\alpha:\mathrm{Emb}^{k}N\to \pi_{\mathrm{eq}}^{k-1}(\widetilde{N})
\alpha
f:N\to\R^{k}
\widetilde{N}\to S^{k-1}
N_0
\alpha_0
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Aliyu, I. and Sani, B. (2018) An Inventory Model for Deteriorating Items with Generalised Exponential Decreasing Demand, Constant Holding Cost and Time-Varying Deterioration Rate. American Journal of Operations Research, 8, 1-16. doi: 10.4236/ajor.2018.81001.
I\left(t\right)
0\le t\le T
D\left(t\right)
D\left(t\right)=K{\text{e}}^{h-\beta t},K>0,\beta >0,h>0
a+bt
{I}_{O}
{I}_{o}^{*}
T{C}^{*}
{h}_{0}\left(t\right)={h}_{1}+{h}_{2}t
\left[0,T\right]
\frac{\text{d}I\left(t\right)}{\text{d}t}+\left(a+bt\right)I\left(t\right)=-D\left(t\right),\text{}0\le t\le T
D\left(t\right)=K{\text{e}}^{h-\beta t}
I\left(t\right)=\frac{-K}{-\beta +a+bt}{\text{e}}^{h-\beta t}+C{\text{e}}^{-at-\frac{1}{2}b{t}^{2}}
I\left(t\right)=0
t=T
\begin{array}{l}I\left(0\right)=0=\frac{-K}{-\beta +a+bT}{\text{e}}^{h-\beta T}+C{\text{e}}^{-aT-\frac{1}{2}b{T}^{2}}\\ ⇒\frac{K}{-\beta +a+bT}{\text{e}}^{h-\beta T}=C{\text{e}}^{-aT-\frac{1}{2}b{T}^{2}}\text{or}C=\frac{K}{-\beta +a+bT}{\text{e}}^{h-\beta T}\cdot {\text{e}}^{aT+\frac{1}{2}b{T}^{2}}\end{array}
\begin{array}{l}I\left(t\right)=\frac{-K}{-\beta +a+bt}{\text{e}}^{h-\beta t}+\frac{K}{-\beta +a+bT}{\text{e}}^{h-\beta T}\cdot {\text{e}}^{aT+\frac{1}{2}b{T}^{2}}\cdot {\text{e}}^{-at-\frac{1}{2}b{t}^{2}}\\ \text{}=K{\text{e}}^{h}\left[-\frac{{\text{e}}^{-\beta t}}{-\beta +a+bt}+\frac{{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-at-\frac{1}{2}b{t}^{2}}}{-\beta +a+bT}\right]\\ \text{}=\frac{K{\text{e}}^{h}}{\left(-\beta +a+bt\right)\left(-\beta +a+bT\right)}\left[-{\text{e}}^{-\beta t}\left(-\beta +a+bT\right)+\left(-\beta +a+bt\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-at-\frac{1}{2}b{t}^{2}}\right],\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le t\le T\end{array}
I\left(0\right)={I}_{0}
\begin{array}{l}I\left(0\right)={I}_{0}=\frac{K{\text{e}}^{h}}{\left(-\beta +a+b\left(0\right)\right)\left(-\beta +a+bT\right)}\\ \text{}\times \left[-{\text{e}}^{-\beta \left(0\right)}\left(-\beta +a+bT\right)+\left(-\beta +a+b\left(0\right)\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-a\left(0\right)-\frac{1}{2}b{\left(0\right)}^{2}}\right]\\ \text{}=\frac{K{\text{e}}^{h}}{\left(-\beta +a\right)\left(-\beta +a+bT\right)}\left[-\left(-\beta +a+bT\right)+\left(-\beta +a\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\right]\end{array}
\left[0,T\right]
\begin{array}{c}{\int }_{0}^{T}D\left(t\right)\text{d}t={\int }_{0}^{T}K{\text{e}}^{h-\beta t}\text{d}t=\frac{K}{-\beta }{\left[{\text{e}}^{h-\beta t}\right]}_{0}^{T}\\ =\frac{K}{-\beta }\left[{\text{e}}^{h-\beta T}-{\text{e}}^{h}\right]=\frac{K{e}^{h}}{-\beta }\left[{\text{e}}^{-\beta T}-1\right]\end{array}
\left[0,T\right]
\begin{array}{l}{I}_{0}-{\displaystyle {\int }_{0}^{T}D\left(t\right)\text{d}t}\\ =\frac{K{\text{e}}^{h}}{\left(-\beta +a\right)\left(-\beta +a+bT\right)}\left[-\left(-\beta +a+bT\right)+\left(-\beta +a\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\right]-\frac{K{\text{e}}^{h}}{-\beta }\left[{\text{e}}^{-\beta T}-1\right]\\ =K{\text{e}}^{h}\left[\left\{\frac{-\left(-\beta +a+bT\right)}{\left(-\beta +a\right)\left(-\beta +a+bT\right)}+\frac{\left(-\beta +a\right)}{\left(-\beta +a\right)\left(-\beta +a+bT\right)}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\right\}-\frac{1}{-\beta }\left\{{\text{e}}^{-\beta T}-1\right\}\right]\\ =\frac{K{\text{e}}^{h}}{-\beta \left(-\beta +a\right)\left(-\beta +a+bT\right)}\\ \text{}\times \left[\beta \left(-\beta +a+bT\right)-\beta \left(-\beta +a\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}-\left(-\beta +a\right)\left(-\beta +a+bT\right)\left\{{\text{e}}^{-\beta T}-1\right\}\right]\end{array}
\begin{array}{l}=\frac{K{\text{e}}^{h}}{-\beta \left(-\beta +a\right)\left(-\beta +a+bT\right)}\\ \text{}\left[{\beta }^{2}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}-a\beta {\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}-{\beta }^{2}{\text{e}}^{-\beta T}+2a\beta {\text{e}}^{-\beta T}\\ \underset{}{\overset{{}^{}}{}}+b\beta T{\text{e}}^{-\beta T}-{a}^{2}{\text{e}}^{-\beta T}-abT{\text{e}}^{-\beta T}-a\beta +{a}^{2}+abT\right]\end{array}
\left[O,T\right]=Cx
\begin{array}{l}=\frac{CK{\text{e}}^{h}}{-\beta \left(-\beta +a\right)\left(-\beta +a+bT\right)}\\ \text{}\left[{\beta }^{2}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}-a\beta {\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}-{\beta }^{2}{\text{e}}^{-\beta T}+2a\beta {\text{e}}^{-\beta T}\\ \underset{}{\overset{{}^{}}{}}+b\beta T{\text{e}}^{-\beta T}-{a}^{2}{\text{e}}^{-\beta T}-abT{\text{e}}^{-\beta T}-a\beta +{a}^{2}+abT\right]\end{array}
\left[O,T\right]
\begin{array}{l}={\displaystyle {\int }_{0}^{T}iCI\left(t\right)\text{d}t}\\ =\frac{iCK{\text{e}}^{h}}{\left(-\beta +a+bt\right)\left(-\beta +a+bT\right)}\\ \text{}\times \left[{\displaystyle {\int }_{0}^{T}\left\{\left(-\beta +a+bt\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-at-\frac{1}{2}b{t}^{2}}-{\text{e}}^{-\beta t}\left(-\beta +a+bT\right)\right\}\text{d}t}\right]\\ =\frac{iCK{\text{e}}^{h}}{\left(-\beta +a+bT\right)}\\ \text{}\times \left[{\displaystyle {\int }_{0}^{T}\left\{{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-at-\frac{1}{2}b{t}^{2}}-{\text{e}}^{-\beta t}\left(-\beta +a+bT\right){\left(-\beta +a+bt\right)}^{-1}\right\}\text{d}t}\right]\end{array}
{\int }_{0}^{T}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-at-\frac{1}{2}b{t}^{2}}\text{d}t
{\int }_{0}^{T}{\text{e}}^{-\beta t}\left(-\beta +a+bT\right){\left(-\beta +a+bt\right)}^{-1}\text{d}t
{\int }_{0}^{T}iCI\left(t\right)\text{d}t=\frac{iCK{\text{e}}^{h}}{\left(-\beta +a+bT\right)}\text{}\left[\text{SolutionofD-SolutionofE}\right]
D={\int }_{0}^{T}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-at-\frac{1}{2}b{t}^{2}}\text{d}t
\begin{array}{c}{\int }_{0}^{T}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-at-\frac{1}{2}b{t}^{2}}\text{d}t={\frac{1}{-a-bt}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}-at-\frac{1}{2}b{t}^{2}}|}_{0}^{T}\\ =\frac{1}{-a-bT}{\text{e}}^{-\beta T}+\frac{1}{a}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\end{array}
\begin{array}{c}E={\int }_{0}^{T}{\text{e}}^{-\beta t}\left(-\beta +a+bT\right){\left(-\beta +a+bt\right)}^{-1}\text{d}t\\ =\left(-\beta +a+bT\right){\int }_{0}^{T}{\text{e}}^{-\beta t}{\left(-\beta +a+bt\right)}^{-1}\text{d}t\\ =\left(-\beta +a+bT\right)\\ \text{}{\left[\frac{1}{b}In\left(-\beta +a+bT\right){\text{e}}^{-\beta t}-\frac{1}{b}In\left(-\beta +a+bT\right){\text{e}}^{-\beta t}+\frac{E}{\left(-\beta +a+bT\right)}\right]}_{0}^{T}\\ =\left(-\beta +a+bT\right){\left[\frac{K}{\left(-\beta +a+bT\right)}\right]}_{0}^{T}=0\end{array}
\begin{array}{l}\therefore {\int }_{0}^{T}iCI\left(t\right)\text{d}t=\frac{iCK{\text{e}}^{h}}{\left(-\beta +a+bT\right)}\left[\text{SolutionofD-SolutionofE}\right]\\ =\frac{iCK{\text{e}}^{h}}{\left(-\beta +a+bT\right)}\left[-\frac{1}{a+bT}{\text{e}}^{-\beta T}+\frac{1}{a}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}-0\right]\\ =\frac{iCK{\text{e}}^{h}}{\left(-\beta +a+bT\right)}\left[-\frac{1}{a+bT}{\text{e}}^{-\beta T}+\frac{1}{a}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\right]\end{array}
\begin{array}{l}TC\left(T\right)=\frac{{N}_{0}}{T}+\frac{CK{\text{e}}^{h}}{-\beta \left(-\beta +a\right)\left(-\beta +a+bT\right)T}\\ \text{}\left[{\beta }^{2}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}-a\beta {\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}-{\beta }^{2}{\text{e}}^{-\beta T}+2a\beta {\text{e}}^{-\beta T}\right]\\ \text{}\underset{}{\overset{{}^{}}{}}+b\beta T{\text{e}}^{-\beta T}-{a}^{2}{\text{e}}^{-\beta T}-abT{\text{e}}^{-\beta T}-a\beta +{a}^{2}+abT\right]\\ \text{+}\frac{iCK{\text{e}}^{h}}{\left(-\beta +a+bT\right)T}\left[-\frac{1}{a+bT}{\text{e}}^{-\beta T}+\frac{1}{a}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\right]\end{array}
\begin{array}{l}=\frac{{N}_{0}}{T}+\frac{CK{\text{e}}^{h}}{-\beta \left(-\beta +a\right)}\\ \times [\frac{{\beta }^{2}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{\left(-\beta T+aT+b{T}^{2}\right)}-\frac{a\beta {\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{\left(-\beta T+aT+b{T}^{2}\right)}-\frac{{\beta }^{2}{\text{e}}^{-\beta T}}{\left(-\beta T+aT+b{T}^{2}\right)}\\ +\frac{2a\beta {\text{e}}^{-\beta T}}{\left(-\beta T+aT+b{T}^{2}\right)}+\frac{b\beta {\text{e}}^{-\beta T}}{\left(-\beta T+aT+b{T}^{2}\right)}-\frac{{a}^{2}{\text{e}}^{-\beta T}}{\left(-\beta T+aT+b{T}^{2}\right)}\\ -\frac{ab{\text{e}}^{-\beta T}}{\left(-\beta +a+bT\right)}-\frac{a\beta }{\left(-\beta T+aT+b{T}^{2}\right)}+\frac{{a}^{2}}{\left(-\beta T+aT+b{T}^{2}\right)}+\frac{ab}{\left(-\beta +a+bT\right)}]\\ +iCK{e}^{h}\left[-\frac{{\text{e}}^{-\beta T}}{\left(a+bT\right)\left(-\beta T+aT+b{T}^{2}\right)}+\frac{{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{\left(-\beta T+aT+b{T}^{2}\right)}\right]\end{array}
\frac{\text{d}TC\left(T\right)}{\text{d}T}=0\text{and}\frac{{\text{d}}^{2}TC\left(T\right)}{\text{d}{T}^{2}}>0
\begin{array}{l}\frac{\text{d}TC\left(T\right)}{\text{d}T}=-\frac{{N}_{0}}{{T}^{2}}+\frac{CK{\text{e}}^{h}}{-\beta \left(-\beta +a\right)}\\ \left[\frac{{\beta }^{2}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{T}-\frac{{\beta }^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{{\left(-\beta T+aT+b{T}^{2}\right)}^{2}}-\frac{a\beta {\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{T}\\ +\frac{a\beta \left(-\beta +a+2bT\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{{\left(-\beta T+aT+b{T}^{2}\right)}^{2}}+\frac{{\beta }^{3}{\text{e}}^{-\beta T}}{\left(-\beta T+aT+b{T}^{2}\right)}+\frac{{\beta }^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T}}{{\left(-\beta T+aT+b{T}^{2}\right)}^{2}}\\ -\frac{2a{\beta }^{2}{\text{e}}^{-\beta T}}{\left(-\beta T+aT+b{T}^{2}\right)}-\frac{2a\beta \left(-\beta +a+2bT\right){\text{e}}^{-\beta T}}{{\left(-\beta T+aT+b{T}^{2}\right)}^{2}}-\frac{b{\beta }^{2}{\text{e}}^{-\beta T}}{\left(-\beta +a+bT\right)}\end{array}
\begin{array}{l}-\frac{{b}^{2}\beta {\text{e}}^{-\beta T}}{{\left(-\beta +a+bT\right)}^{2}}+\frac{{a}^{2}\beta {\text{e}}^{-\beta T}}{\left(-\beta T+aT+b{T}^{2}\right)}+\frac{{a}^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T}}{{\left(-\beta T+aT+b{T}^{2}\right)}^{2}}\\ +\frac{ab\beta {\text{e}}^{-\beta T}}{\left(-\beta +a+bT\right)}+\frac{a{b}^{2}{\text{e}}^{-\beta T}}{{\left(-\beta +a+bT\right)}^{2}}+\frac{a\beta \left(-\beta +a+2bT\right)}{{\left(-\beta T+aT+b{T}^{2}\right)}^{2}}\\ -\frac{{a}^{2}\left(-\beta +a+2bT\right)}{{\left(-\beta T+aT+b{T}^{2}\right)}^{2}}-\frac{a{b}^{2}}{{\left(-\beta +a+bT\right)}^{2}}\right]\end{array}
\begin{array}{l}+ick{\text{e}}^{h}\left[+\frac{\beta {\text{e}}^{-\beta T}}{\left(a+bT\right)\left(-\beta T+aT+b{T}^{2}\right)}+\frac{b{\text{e}}^{-\beta T}}{{\left(a+bT\right)}^{2}\left(-\beta T+aT+b{T}^{2}\right)}\\ +\frac{\left(-\beta +a+2bT\right){\text{e}}^{-\beta T}}{\left(a+bT\right){\left(-\beta T+aT+b{T}^{2}\right)}^{2}}+\frac{\left(-\beta +a+bT\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{a\left(-\beta T+aT+b{T}^{2}\right)}\\ -\frac{\left(-\beta +a+2bT\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}}{a{\left(-\beta T+aT+b{T}^{2}\right)}^{2}}\right].\end{array}
-{T}^{2}a\beta \left(-\beta +a\right){\left(-\beta T+aT+b{T}^{2}\right)}^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}
\begin{array}{l}{N}_{0}a\beta \left(-\beta +a\right){\left(-\beta T+aT+b{T}^{2}\right)}^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\\ +CK{\text{e}}^{h}aT{\beta }^{2}{\left(-\beta T+aT+b{T}^{2}\right)}^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\\ -CK{\text{e}}^{h}a{T}^{2}{\beta }^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\\ -CK{\text{e}}^{h}{a}^{2}\beta T{\left(-\beta T+aT+b{T}^{2}\right)}^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\\ +{\text{e}}^{h}{a}^{2}\beta {T}^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\end{array}
\begin{array}{l}+CK{\text{e}}^{h}a{T}^{2}{\beta }^{3}\left(-\beta T+aT+b{T}^{2}\right){\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}{\text{e}}^{-\beta T}\\ +CK{\text{e}}^{h}a{T}^{2}{\beta }^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T}\\ -2CK{\text{e}}^{h}{a}^{2}{T}^{2}{\beta }^{2}\left(-\beta T+aT+b{T}^{2}\right){\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}{\text{e}}^{-\beta T}\\ -2CK{\text{e}}^{h}{a}^{2}{T}^{2}\beta {\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T}\\ -CK{\text{e}}^{h}ab{T}^{2}{\beta }^{2}{\left(-\beta T+aT+b{T}^{2}\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+bT\right){\text{e}}^{-\beta T}\end{array}
\begin{array}{l}-CK{\text{e}}^{h}a{b}^{2}{T}^{2}\beta {\left(-\beta T+aT+b{T}^{2}\right)}^{2}{\left(a+bT\right)}^{2}{\text{e}}^{-\beta T}\\ +CK{\text{e}}^{h}{a}^{3}{T}^{2}\beta \left(-\beta T+aT+b{T}^{2}\right){\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}{\text{e}}^{-\beta T}\\ +CK{\text{e}}^{h}{a}^{3}{T}^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T}\\ +CK{\text{e}}^{h}{a}^{2}b{T}^{2}\beta {\left(-\beta T+aT+b{T}^{2}\right)}^{2}\left(-\beta +a+bT\right){\left(a+bT\right)}^{2}{\text{e}}^{-\beta T}\\ +CK{\text{e}}^{h}{a}^{2}{b}^{2}{T}^{2}{\left(-\beta T+aT+b{T}^{2}\right)}^{2}{\text{e}}^{h}{a}^{2}{b}^{2}{T}^{2}{\left(-\beta T+aT+b{T}^{2}\right)}^{2}\end{array}
\begin{array}{l}+CK{\text{e}}^{h}{a}^{2}\beta {T}^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+2bT\right)\\ -CK{\text{e}}^{h}{a}^{3}{T}^{2}{\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T}\\ -CK{\text{e}}^{h}{a}^{2}{b}^{2}{T}^{2}{\left(-\beta T+aT+b{T}^{2}\right)}^{2}{\left(a+bT\right)}^{2}\\ -iCK{\text{e}}^{h}a{T}^{2}{\beta }^{2}\left(-\beta +a\right)\left(-\beta T+aT+b{T}^{2}\right){\left(-\beta +a+bT\right)}^{2}\left(a+bT\right){\text{e}}^{-\beta T}\\ -iCK{\text{e}}^{h}{T}^{2}ab\beta \left(-\beta +a\right)\left(-\beta T+aT+b{T}^{2}\right){\left(-\beta +a+bT\right)}^{2}{\text{e}}^{-\beta T}\end{array}
\begin{array}{l}-iCK{\text{e}}^{h}{T}^{2}a\beta \left(-\beta +a\right){\left(-\beta +a+bT\right)}^{2}\left(a+bT\right)\left(-\beta +a+2bT\right){\text{e}}^{-\beta T}\\ -iCK{\text{e}}^{h}{T}^{2}\beta \left(-\beta +a\right)\left(-\beta T+aT+b{T}^{2}\right){\left(-\beta +a+bT\right)}^{3}{\left(a+bT\right)}^{2}{\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}\\ +iCK{\text{e}}^{h}{T}^{2}\beta \left(-\beta +a\right){\left(-\beta +a+bT\right)}^{2}{\left(a+bT\right)}^{2}\left(-\beta +a+2bT\right){\text{e}}^{-\beta T+aT+\frac{1}{2}b{T}^{2}}=0.\end{array}
\frac{{\text{d}}^{2}TC\left(T\right)}{\text{d}{T}^{2}}>0
K=500
\beta =0.02
a=0.2
b=0.01
i=0.1
h=2
{I}_{0}^{*}
\frac{{\text{d}}^{2}TC\left(T\right)}{\text{d}{T}^{2}}>0
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
{I}_{0}^{*}
\frac{\text{d}I\left(t\right)}{\text{d}t}+\left(a+bt\right)I\left(t\right)=-D\left(t\right),0\le t\le T
D\left(t\right)=K{\text{e}}^{h-\beta t}
I\left(0\right)={I}_{0}
I\left(T\right)=0
\frac{\text{d}I\left(t\right)}{\text{d}t}+I\left(t\right)\left(a+bt\right)=-K{\text{e}}^{h-\text{β}t}
IF={\text{e}}^{{\int }^{\text{}}\left(a+bt\right)\text{d}t}={\text{e}}^{at+\frac{1}{2}b{t}^{2}}
\begin{array}{l}I\left(t\right){\text{e}}^{at+\frac{1}{2}b{t}^{2}}=-K{\int }^{\text{}}{\text{e}}^{h-\text{β}t}\cdot {\text{e}}^{at+\frac{1}{2}b{t}^{2}}\text{d}t\\ =\frac{-K}{-\beta +a+bt}{\text{e}}^{h-\text{β}t+at+\frac{1}{2}b{t}^{2}}+C\\ \therefore I\left(t\right)=\frac{-K}{-\beta +a+bt}{\text{e}}^{h-\beta t}+C{\text{e}}^{-at-\frac{1}{2}b{t}^{2}}\end{array}
[18] Aliyu, I. and Sani, B. (2016) On an Inventory Model for Deteriorating Items with Generalised Exponential Decreasing Demand and Time-Varying Holding Cost. Journal of Nigerian Association of Mathematical Physics, 36, 193-202.
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Predictive driver controller to track longitudinal speed and lateral path - Simulink - MathWorks 日本
y=\frac{{K}_{ff}}{{v}_{nom}}{v}_{ref}+\frac{{K}_{p}{e}_{ref}}{{v}_{nom}}+â«\left(\frac{{K}_{i}{e}_{ref}}{{v}_{nom}}+{K}_{aw}{e}_{out}\right)dt+{K}_{g}\mathrm{θ}
y=\frac{{K}_{ff}\left(v\right)}{{v}_{nom}}{v}_{ref}+\frac{{K}_{p}\left(v\right){e}_{ref}}{{v}_{nom}}+â«\left(\frac{{K}_{i}\left(v\right){e}_{ref}}{{v}_{nom}}+{K}_{aw}{e}_{out}\right){e}_{ref}dt+{K}_{g}\left(v\right)\mathrm{θ}
\begin{array}{l}\text{where:}\\ \\ {e}_{ref}={v}_{ref}âv\\ {e}_{out}={y}_{sat}ây\\ \\ {y}_{sat}=\left\{\begin{array}{cc}â1& y<â1\\ y& â1â¤yâ¤1\\ 1& 1<y\end{array}\end{array}
H\left(s\right)=\frac{1}{{\mathrm{Ï}}_{err}s+1}\text{ for }{\mathrm{Ï}}_{err}>0
\begin{array}{l}{y}_{acc}=\left\{\begin{array}{cc}0& {y}_{sat}<0\\ {y}_{sat}& 0â¤{y}_{sat}â¤1\\ 1& 1<{y}_{sat}\end{array}\\ \\ {y}_{dec}=\left\{\begin{array}{cc}0& {y}_{sat}>0\\ â{y}_{sat}& â1â¤{y}_{sat}â¤0\\ 1& {y}_{sat}<â1\end{array}\end{array}
Ï„err
\begin{array}{l}{x}_{1}=U\\ {\stackrel{Ë}{x}}_{1}={x}_{2}=\frac{{K}_{pt}}{m}+ vrâg\text{sin}\left(\mathrm{γ}\right)+{F}_{r}{x}_{1}\\ \stackrel{Ë}{y}=v+U\mathrm{Ï}\\ \stackrel{Ë}{v}=\left[â\frac{2\left({C}_{\mathrm{α}F}+{C}_{\mathrm{α}R}\right)}{mU}\right]v+\left[\frac{2\left(b{C}_{\mathrm{α}R}âa{C}_{\mathrm{α}F}\right)}{mU}âU\right]r+\left(\frac{2{C}_{\mathrm{α}F}}{m}\right){\mathrm{δ}}_{F}\\ \stackrel{Ë}{r}=\left[\frac{2\left(b{C}_{\mathrm{α}R}âa{C}_{\mathrm{α}F}\right)}{IU}\right]v+\left[â\frac{2\left({a}^{2}{C}_{\mathrm{α}F}+{b}^{2}{C}_{\mathrm{α}R}\right)}{IU}\right]r+\left(\frac{2a{C}_{\mathrm{α}F}}{I}\right){\mathrm{δ}}_{F}\\ \stackrel{Ë}{\mathrm{Ï}}=r\end{array}
\begin{array}{l}\stackrel{Ë}{x}=Fx+gu\\ \\ \text{where:}\\ \\ x=\left[\begin{array}{c}\begin{array}{c}{x}_{1}\\ {x}_{2}\\ y\\ v\\ r\end{array}\\ \mathrm{Ï}\end{array}\right]\\ \\ F=\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ \frac{{F}_{r}}{m}& 0& 0& 0& v& 0\\ 0& 0& 0& 1& 0& U\\ 0& 0& 0& â\frac{2\left({C}_{\mathrm{α}F}+{C}_{\mathrm{α}R}\right)}{mU}& \frac{2\left(b{C}_{\mathrm{α}R}âa{C}_{\mathrm{α}F}\right)}{mU}âU& 0\\ 0& 0& 0& \frac{2\left(b{C}_{\mathrm{α}R}âa{C}_{\mathrm{α}F}\right)}{IU}& â\frac{2\left({a}^{2}{C}_{\mathrm{α}F}+{b}^{2}{C}_{\mathrm{α}R}\right)}{IU}& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right]\\ \\ g=\left[\begin{array}{c}\begin{array}{cc}0& 0\\ \frac{{K}_{pt}}{m}& 0\\ 0& 0\\ 0& \frac{2{C}_{\mathrm{α}F}}{m}\\ 0& \frac{2a{C}_{\mathrm{α}F}}{I}\\ 0& 0\end{array}\end{array}\right]\\ \\ u=\left[\begin{array}{c}\begin{array}{c}\stackrel{¯}{u}\\ {\mathrm{δ}}_{F}\end{array}\end{array}\right] \\ \\ \stackrel{¯}{u}=uâ \frac{{m}^{2}}{{K}_{pt}}g\text{sin}\left(\mathrm{γ}\right)\end{array}
\begin{array}{l}{a}^{*}=\left({T}^{*}\right){m}^{T}\left[I+\underset{n=1}{\overset{\infty }{â}}\frac{{F}^{n}{\left({T}^{*}\right)}^{n}}{\left(n+1\right)!}\right]g\\ {b}^{*}={m}^{T}\left[I+\underset{n=1}{\overset{\infty }{â}}\frac{{F}^{n}{\left({T}^{*}\right)}^{n}}{n!}\right]\\ {m}^{T}=\left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\end{array}\right]\end{array}
CÉ‘F
CÉ‘R
δ, δF
Æ’(t+T*)
J=\frac{1}{T}{â«}_{t}^{t+T}{\left[f\left(\mathrm{η}\right)ây\left(\mathrm{η}\right)\right]}^{2}d\mathrm{η}
\frac{dJ}{du}=0
{u}^{o}\left(t\right)=u\left(t\right)+\frac{e\left(t+{T}^{*}\right)}{{a}^{*}}
{T}^{*}=\frac{L}{U}
H\left(s\right)={e}^{âs\mathrm{Ï}}
Vehicle yaw angle, Ψo, in the inertial reference frame, in units specified by Angular units, angUnits.
Road grade angle, γ, in deg.
[x, y, Θ] vector
Reference pose, specified as an [x, y, Θ] vector. x and y are in meters, and Θ are in units specified by Angular units, angUnits.
x and y specify the reference point to steer the vehicle toward. Θ specifies the orientation angle of the path at this reference point and is positive in the counterclockwise direction.
Current pose of the vehicle, specified as an [x, y, Θ] vector. x and y are in meters, and Θ is in units specified by Angular units, angUnits.
Steer δF
\underset{0}{\overset{t}{â«}}{e}_{ref}{}^{2}dt
\mathrm{max}\left({e}_{ref}\left(t\right)\right)
\mathrm{min}\left({e}_{ref}\left(t\right)\right)
\underset{0}{\overset{t}{â«}}{e}_{ref}{}^{2}dt
\mathrm{max}\left({e}_{ref}\left(t\right)\right)
\mathrm{min}\left({e}_{ref}\left(t\right)\right)
Commanded steer angle, δF.
Error filter time constant, Ï„err, in s. To disable the filter, enter 0.
Driver response time, Ï„, in s.
Rolling and driveline resistance coefficient, bR, in N·s/m. Block uses the parameter to estimate the linear velocity-dependent acceleration or braking effort.
Aerodynamic drag coefficient, cR, in N·s^2/m^2. Block uses the parameter to estimate the quadratic velocity-dependent acceleration or braking effort.
Cornering stiffness coefficient, CαF , in N/rad.
Cornering stiffness coefficient, CαR , in N/rad.
Vehicle rotational inertia, I, about the vehicle yaw axis, in N·m·s^2.
Tire wheel angle limit, θ, in rad.
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Properties of Real Numbers - Course Hero
College Algebra/Algebraic Expressions, Equations, and Inequalities/Properties of Real Numbers
The associative, commutative, and identity properties of addition can be used to simplify expressions with real numbers. The additive identity is zero, and the additive inverse, or opposite, of a real number
a
-a
Three important properties of addition can be used to simplify expressions.
The commutative property of addition states that when adding real numbers, the sum does not change based on the order of the numbers:
a+b=b+a
The associative property of addition states that when adding three or more numbers, the sum does not change based on the way the numbers are grouped:
a+(b+c)=(a+b)+c
The identity property of addition states that when adding zero to any any given number, the sum is the given number itself:
a+0=a
The additive identity is the number zero, which has the property that
a+0=a
a
. This is stated as the identity property of addition, which is the sum of zero and any number is the given number:
a+0=a
a
, the additive inverse is the number
-a
. It is also called the opposite of
a
. The sum of a number and its additive inverse is the additive identity, zero. This value is called the inverse, or opposite, because it reverses the addition. For any number, when adding
and then adding
-a
, the result is the original number:
b+a+(-a)=b
Order does not affect the sum. Grouping does not affect the sum. Adding zero does not affect the sum.
a+b=b+a
a+(b+c)=(a+b)+c
a+0=a
9+5=5+9
5+(7+6)=(5+7)+6
26+0=26
Number Plus Inverse
a
-a
a+(-a)=0
97
-97
97+(-97)=0
-\frac{4}{5}
-\left(-\frac{4}{5}\right)=\frac{4}{5}
-\frac{4}{5}+\frac{4}{5}=0
0
-0=0
0+0=0
The associative, commutative, and identity properties of multiplication can be used to simplify expressions with real numbers. The multiplicative identity is 1, and the multiplicative inverse, or reciprocal, of a real number
a
\frac{1}{a}
The properties of multiplication are similar to the properties of addition and are also used to simplify expressions.
The commutative property of multiplication states that when multiplying real numbers, the product does not change based on the order of the numbers:
ab=ba
The associative property of multiplication states that when multiplying three or more numbers, the product does not change based on the way the numbers are grouped:
a(bc)=(ab)c
The multiplicative identity is the number 1, which has the property that
a\cdot1=a
a
. This is stated as the identity property of multiplication, which is the product of 1 and any number is the given number:
a\cdot1=a
For a nonzero real number
a
, the multiplicative inverse is the number
\frac{1}{a}
. It is also called the reciprocal of
a
. The product of a number and its multiplicative inverse is the multiplicative identity, 1. This value is called the inverse because it reverses the multiplication. For any number, when multiplying by
and then multiplying by
\frac{1}{a}
b(a)\left(\frac{1}{a}\right)=b
The multiplicative inverse can be applied to any value except zero, which produces an undefined multiplicative inverse. In addition, multiplying any value by zero does not result in 1, which means that the multiplicative identity does not apply to zero.
Order does not affect the product. Grouping does not affect the product. Multiplying by 1 does not affect the product.
ab=ba
a(bc)=(ab)c
a\cdot1=a
4\cdot12=12\cdot4
3(2\cdot5)=(3\cdot2)5
7\cdot1=7
Number Times Inverse
a
\frac{1}{a}
a\cdot\frac{1}{a}
8
\frac{1}{8}
8\cdot\frac{1}{8}=1
-3
-\frac{1}{3}
-3\cdot\left(-\frac{1}{3}\right)=1
\frac{2}{3}
\frac{1}{\left(\frac{2}{3}\right)}=\frac{3}{2}
\frac{2}{3}\cdot\frac{3}{2}=1
0
\frac{1}{0}
is undefined. So, there is no multiplicative inverse of zero. There is no number that can be multiplied by zero to produce 1.
The distributive property of multiplication over addition can be used to simplify expressions with real numbers.
The distributive property states that multiplying an expression by a sum is the same as multiplying the expression by each term in the sum and then adding the products.
The value that is multiplied can be on the left side of the expression, such as:
a(b+c)
It can also be on the right side, such as:
(a+b)c
Multiplying an expression by one or more terms in a sum is called distributing. For example:
a(b+c)
In the expression, the
a
can be distributed over the sum, resulting in:
ac+bc
For instance, simplify the expression:
2(4+6)
The addition can be performed first, or the 2 can be distributed over the sum. The result is the same.
Add, then multiply.
Multiply, then add.
\begin{gathered}2(4+6)\\2(10)\\20\end{gathered}
\begin{gathered}2(4+6)\\2(4)+2(6)\\8+12\\20\end{gathered}
3x(4x+5y-7)
Apply the distributive property. Multiply
3x
by each term in the sum.
\begin{gathered}3x(4x+5y-7)\\3x(4x)+3x(5y)-3x(7)\end{gathered}
Simplify each product.
\begin{gathered}3x(4x)+3x(5y)-3x(7)\\12x^2+15xy-21x\end{gathered}
12x^2+15xy-21x
To simplify an expression of the form
(a+b)(c+d)
, the distributive property is applied twice. First, distribute
c+d
over the sum
a+b
\begin{gathered}(a+b)(c+d) \\ a(c+d)+b(c+d)\end{gathered}
Next, distribute
and
b
c+d
\begin{gathered} a(c+d)+b(c+d)\\ac+ad+bc+bd\end{gathered}
The memory aid FOIL can be used to remember the order of multiplying terms in expressions using the form:
(a+b)(c+d)
ac
is the product of the First terms of
({\color{#c42126}a}+b)({\color{#c42126}c}+d)
ad
is the product of the Outside terms of
({\color{#c42126}a}+b)(c+{\color{#c42126}d})
bc
is the product of the Inside terms of
(a+{\color{#c42126}b})({\color{#c42126}c}+d)
bd
is the product of the Last terms of
(a+{\color{#c42126}b})(c+{\color{#c42126}d})
Distributing an Expression to Another Expression
(x+4)(x+1)
x+1
x+4
\begin{gathered}({\color{#c42126} x+4})(x+1)\\{\color{#c42126} {x}}(x+1){\color{#c42126} {\;+\;4}}(x+1)\end{gathered}
x
4
x+1
. The result is the products of the first, outside, inside, and last terms of the expressions
\begin{gathered}x(x+1)+4(x+1)\\x^2+x+4x+4\end{gathered}
Simplify by combining like terms
x
4x
\begin{gathered}x^2+x+4x+1\\x^2+5x+4\end{gathered}
The distributive property can also be used to rewrite expressions by factoring out a common factor.
\begin{gathered}6y-24\\({\color{#c42126} 6})(y)-({\color{#c42126} 6})(4)\\{\color{#c42126}{6}}(y-4)\end{gathered}
The common factor in the expression is 6. Note that the original expression, which is also the simplified form, and the factored form of the expression are equivalent:
\begin{gathered}{6y-24}\\{6(y-4)}\end{gathered}
Factoring is not the same as simplifying an expression, but it may be used as a step in simplifying or in finding solutions of an equation.
Expressions can be simplified by combining the properties of operations with the order of operations.
If an expression has more than one operation, performing the operations in a different order can produce different results. So, there is an agreed-upon order to ensure the result is always the same. The order of operations is a set of rules indicating which calculations to perform first to simplify a mathematical expression.
1. Simplify expressions within grouping symbols, such as parentheses
( )
[ ]
, or braces
\{ \}
, or expressions in the numerator or denominator of a fraction or under a radical. For nested grouping symbols
\{ [ ( ) ] \}
2. Simplify any exponents or radicals in the expression.
3. Multiply and divide, working from left to right across the expression.
4. Add and subtract, working from left to right across the expression.
Properties of operations can be used with the order of operations to simplify expressions.
Simplifying Expressions with Properties of Operations
\frac{1}{2}+5(2\cdot11)+\frac{\sqrt{9}}{2}
The first step in the order of operations is to evaluate within grouping symbols. However, the associative property of multiplication can be used to rewrite the expression in a way that is easier to simplify.
\begin{gathered}\frac{1}{2}+{\color{#c42126}{5(2\cdot11)}}+\frac{\sqrt{9}}{2}\\\frac{1}{2}+{\color{#c42126}{(5\cdot2)11}}+\frac{\sqrt{9}}{2}\end{gathered}
Now, evaluate the expression within the grouping symbols.
\begin{gathered}\frac{1}{2}+({\color{#c42126}{5\cdot2}})11+\frac{\sqrt{9}}{2}\\\frac{1}{2}+{\color{#c42126}{10}}\cdot11+\frac{\sqrt{9}}{2}\end{gathered}
Simplify exponents and radicals.
\begin{gathered}\frac{1}{2}+10\cdot11+\frac{{\color{#c42126}{\sqrt{9}}}}{2}\\\frac{1}{2}+10\cdot11+\frac{{\color{#c42126}{3}}}{2}\end{gathered}
The next step is to multiply.
\begin{gathered}\frac{1}{2}+{\color{#c42126}{10\cdot11}}+\frac{3}{2}\\\frac{1}{2}+{\color{#c42126}{110}}+\frac{3}{2}\end{gathered}
Use the commutative property of addition to rewrite the expression.
\begin{gathered}\frac{1}{2}+{\color{#c42126}{110+\frac{3}{2}}}\\\frac{1}{2}+{\color{#c42126}{\frac{3}{2}+110}}\end{gathered}
\begin{gathered}{\color{#c42126}{\frac{1}{2}+\frac{3}{2}}}+110\\{\color{#c42126}{2}}+110\\112\end{gathered}
<Vocabulary>Algebraic Expressions
|
Forms of Linear Equations - Course Hero
College Algebra/Graphing Lines/Forms of Linear Equations
m
y
b
y = mx + b
The slope-intercept form of a line is a linear equation, where
m
b
y
y=mx+b
As the name suggests, the slope-intercept form gives two important pieces of information: the slope and the
y
-intercept. This information can be used to graph the line without making a table and plotting points.
Graphing Positive Slope in Slope-Intercept Form
Graph the line:
y=\frac{3}{2}x+5
The equation is in slope-intercept form, where
m
b
y
y=mx+b
y
-intercept of the equation.
y
-intercept is 5, so plot one point at
(0, 5)
Then use the rise and run of the slope,
m
, to locate another point.
\begin{aligned}m&=\frac{\text{Rise}}{\text{Run}}\\&=\frac{3}{2}\end{aligned}
The rise is 3, so move up 3 units from the first point
(0, 5)
. The run is 2, so move right 2 units. This means that another point on the line is
(2, 8)
Connect the two points to graph the line.
Graphing with Negative Slope in Slope-Intercept Form
y=-2x-3
m
b
y
-intercept. Identify the slope and
y
-intercept of the equation:
\begin{aligned}y&=mx+b\\y&=-2x-3\end{aligned}
y
y
(0,-3)
m
, to locate another point. Write the slope as a fraction with a denominator of 1.
\begin{aligned}m&=\frac{\text{Rise}}{\text{Run}}\\&=\frac{-2}{1}\end{aligned}
The rise is –2, so move 2 units down. The run is 1, so move 1 unit to the right. So, another point on the line is
(1,-5)
m
(x_1, y_1)
y-y_1=m(x-x_1)
The point-slope form of a line is a linear equation, where
m
(x_1, y_1)
y-y_1=m(x-x_1)
As the name suggests, the point-slope form is convenient to use in writing the equation of a line when the slope and the coordinates of a point on the line are given.
Writing an Equation with a Given Slope and Point
Write the equation of a line with a slope of 3 that passes through the point
(-4,1)
m
x_1
y_1
\begin{aligned}y-y_1&=m(x-x_1)\\y-1&=3(x-(-4))\\y-1&=3(x+4)\end{aligned}
Although the point-slope form of a line is a correct form, it is common to rewrite the equation of a line in slope-intercept form. First, distribute the 3 on the right side.
\begin{aligned}y-1&=3(x+4)\\y-1&=3x+12\end{aligned}
\begin{aligned}y-1&=3x+12\\\underline{\phantom{y}+\ 1}&\phantom{=\:}\underline{\phantom{3x\ }+\ \: 1}\\y&=3x+13\end{aligned}
Graphing a Line in Point-Slope Form
Graph the line of the equation:
y-4=-3(x+2)
Identify the slope and a point on the line.
The equation uses the point-slope form, where
m
x_1
y_1
are coordinates of a point on the line, or
(x_1, y_1)
y-y_1=m(x-x_1)
x_1
y_1
are being subtracted in the equation, the coordinates are positive.
x_1
y_1
are being added in the equation, the coordinates are negative.
Plot the point at
(-2, 4)
. Then use the rise and run of the slope,
m
, to locate another point. Write the slope as:
\begin{aligned}m&=\frac{\text{Rise}}{\text{Run}}\\&=\frac{-3}{1}\end{aligned}
The rise is –3, so move down 3 units. The run is 1, so move right 1 unit.
Determining the Equation of a Line through Two Points
Identify the equation of a line that passes through point
(3, 1)
(5, 9)
Use the coordinates of the two points to identify the slope.
\begin{aligned}m &=\frac{y_2-y_1}{x_2-x_1}\\&=\frac{9-1}{5-3}\\&=\frac{8}{2}\\&=4\end{aligned}
Choose one of the points on the line to write an equation in point-slope form. Note that using either point will result in correct equations.
(x_1,y_1)=(3,1)
Use the slope and one of the points on the line to write the point-slope form:
y-1=4(x-3)
It is common to rewrite the equation in slope-intercept form. First, distribute the 4. Then add 1 to both sides.
\begin{aligned}y-1&=4(x-3)\\y-1&=4x-12\\y-1+1&=4x-12+1\\y&=4x-11\end{aligned}
Ax + By = C
x
y
The standard form of a line is a linear equation, where
A
B
C
A
A
B
Ax+By=C
Although the standard form of a line is not the most commonly used for graphing, it is called standard because it is used in other applications, such as solving systems of equations. It is also a useful form for finding the intercepts of a line. The same process is used to find the intercepts of a line in any form.
To calculate the
x
y=0
x
y
x=0
y
Identifying the Intercepts of a Line in Standard Form
x
-intercept and the
y
3x + 4y = 12
x
y
x
\begin{aligned}3x + 4y &= 12\\3x+4(0)&=12\\3x&=12\\x&=4\end{aligned}
y
x
y
\begin{aligned}3x + 4y &= 12\\3(0)+4y&=12\\4y&=12\\y&=3\end{aligned}
x
(4,0)
y
(0,3)
Although it is possible to graph a line in standard form, it is sometimes convenient to rewrite the equation in slope-intercept form. One reason is to determine the slope. Another reason is that it is easier to compare two different lines when they are both written in slope-intercept form.
For a line that is given in standard form
Ax+By=C
, to rewrite the equation in slope-intercept form, first subtract
-Ax
from both sides. Then divide both sides by
B
Rewriting the Standard Form of a Line in Slope-Intercept Form
Write the linear equation in slope-intercept form:
3x + 4y = 12
The equation is in standard form:
\begin{aligned}Ax+By&=C\\3x+4y&=12\end{aligned}
Rewrite the equation to the slope-intercept form:
y=mx+b
-3x
from both sides to isolate the
y
-term. Since the slope-intercept form of a line has the
x
-term before the constant, write the
x
-term first on the right side of the equation.
\begin{aligned}3x+4y&=12\\3x+4y-3x&=12-3x\\4y&=-3x+12\end{aligned}
\begin{aligned}4y&={-3x+12}\\{\frac{4y}{4}}&=\frac{-3x+12}{4}\end{aligned}
\begin{aligned}\frac{4y}{4}&=\frac{-3x+12}{4}\\y&=-\frac{3}{4}x+3\end{aligned}
-\frac{3}{4}
y
<Linear Equations>Types of Lines
|
Radar Architecture: Test Automation and Requirements Traceability (Part 2) - MATLAB & Simulink - MathWorks í•œêµ
This example is the second part of a two-part series on how to design and test a radar system in Simulink® based on a set of performance requirements. It discusses testing of the model developed in Part 1 and verification of the initial requirements. It shows how to use Simulink Test™ for setting up test suites to verify requirements linked to the components of the system. The example also explores a scenario when the stated requirements have been revised leading to the changes in the design and tests.
Part 1 of this example starts with a set of performance requirements. It develops an architecture model of a radar system using Simulink System Composer™. This architecture model is employed as a virtual test bed for testing and verifying the radar system designs. Part 1 shows how to use Requirements Toolbox™ to link the requirements to the components of the architecture. It also shows how to implement the individual components of the architecture using Simulink.
Prior to setting up the tests, load the model constructed in the Part 1 of the example.
Simulink Test Manager is a tool for creating tests suites for the model. To access Test Manager click on Simulink Test in the Apps tab, then navigate to Tests tab and click Simulink Test Manager. To get started with the tests, create a new test file for the model by clicking on New Test File. Then add two separate test suites, one for each requirement. Further configure the test suites by:
Adding a description to each test suite to shortly describe what functionality is being tested.
Linking the test suite to one or multiple requirements. The tests in the test suite must pass in order for the requirements to be verified.
Adding callbacks for setup before and cleanup after the test run. This example requires a global variable in the base workspace in order to aggregate the results of multiple Monte Carlo runs within a single test suite.
Next configure the tests within the test suites. The changes are made only in the System Under Test, Parameter Overrides, Iterations, and Custom Criteria sections.
In the System Under Test section, set the Model field to the name of the model, which in this example is slexRadarArchitectureExample.
The Parameter Overrides section is used to assign different values to the parameters in the base workspace during a test execution. Use this section to specify the targets parameters for the maximum range test and the range resolution test.
For the maximum range test, specify a single target with 1 m
{}^{2}
radar cross section (RCS) at the range of 6000 m from the radar as stated in R1.
For the range resolution test, specify two targets with different RCS separated in range by 70 m as required by R2.
Because of the random noise and the target fluctuation effects, it is possible to verify only the averaged radar system performance collected over multiple test runs. The Iterations section of the test can be used to configure the test to run multiple times to implement Monte Carlo simulations. This example adds a custom script to the Scripted Iterations subsection to set up Monte Carlo. The script performs only ten iterations. To robustly verify the performance of the system more iterations are required.
The Custom Criteria section allows you to specify a custom rule that verifies the test results at the end of each iteration. Configure it to run the helperslexRadarArchitectureTestCriteria helper function that processes results of each test iteration and stores them in the detectionResults variable in the base workspace. This function computes the number of detection threshold crossings. If this number is equal to the number of targets in the test, the system passes the test iteration, otherwise the iteration is declared as failed. In the last iteration, helperslexRadarArchitectureTestCriteria computes the total number of passed iterations. The second argument to this helper function is the percentage of the iterations that must pass for the entire test to pass. The maximum range test requires that at least 90% of all iterations pass. Since the range resolution test models two independent targets, it requires that at least 80% of all test iterations are successful.
Open this test suite in Test Manager.
open('slexRadarArchitectureTests.mldatx')
After adding the tests and linking them to the requirements, the status of the requirements in the Requirements Editor indicates that the verification has been added but the tests have not yet been executed.
Now the tests can be launched. After running both test suites, inspect the results of each individual iteration using the Data Inspector. The custom criteria helper function also prints the status of each iteration to the Command Window.
Since both tests passed, Requirements Editor now shows that both requirements have been implemented and verified.
It is common that during a design process the initial requirements are revised and changed. This example assumes that the new maximum range requirement is 8000 m and the new range resolution requirement is 35 m. The updated requirements are:
{}^{2}
R2: When returns are detected from two Swerling 1 Case targets separated in range by 35 m, with the same azimuth, the radar must resolve the two targets and generate two unique target reports 80 percent of the time.
Making changes to requirements in Requirements Editor will generate change issues and highlight the Summary status of the corresponding requirement in red. The links to the components that implement the changed requirement and to the tests that verify it are also highlighted. This way it is easy to identify which components of the design and which tests need to be updated in order to address the changes in the requirement and to test them.
To monitor the changes in the requirements or in the implementations of the system components use the requirements Traceability Matrix.
Updated System Parameters
The new maximum range requirement is beyond the current unambiguous range of the system that equals 7494.8 m. To satisfy the new requirement, increase the unambiguous range. This can be accomplished by lowering the PRF. Setting the PRF to 16 kHz results in the unambiguous range of 9368.5 m, which is well beyond the required maximum range of 8000 m.
Since the current radar design transmits unmodulated rectangular pulses, the resolution limit of the system is determined by the pulse width. The current range resolution limit is 60 m. The new requirement of 35 m is almost two times lower. A rectangular pulse which satisfies this requirement would have to be twice as short, reducing the available power at the same range by half. The requirement analysis using the Radar Designer app shows that this system cannot reach the required detection performance at the maximum range of 8000 m. To achieve the required maximum range and range resolution, without increasing the peak transmitted power or the antenna gain, adopt a new waveform with the time-bandwidth product that is larger than 1. Setting the pulse width to 1
\mathrm{μ}s
and the bandwidth to 5 MHz will provide the desired resolution.
radarDesigner('RadarDesigner_LFMWaveform.mat')
The Pulse Waveform Analyzer app can be used to select a radar waveform from several alternatives. This example uses the LFM waveform.
pulseWaveformAnalyzer('PulseWaveformAnalyzer_LFMWaveform.mat')
A convenient way to modify the behavior of a component of the system is to add an alternative design by creating a variant. This is done by right clicking on the component and selecting Add Variant Choice. Add a variant to Waveform Generator and add Simulink behavior to it to implement the LFM waveform generation.
Configure the Linear FM block by setting the pulse width to the new value of 1
\mathrm{μ}s
. Set the sweep bandwidth to 5 MHz and the PRF property to the updated PRF value of 16 kHz. Run the model with the LFM waveform.
% Update the model parameters to use the LFM waveform
helperslexRadarArchitectureParametersLFM;
simOut = sim('slexRadarArchitectureExample.slx');
Before verifying that the radar system with LFM can satisfy the updated requirements, make corresponding modifications to the tests by updating the targets positions.
Set the target range in the maximum range test to 8000 m
Change target ranges in the range resolution test so the targets are positioned 35 m from each other
After updating the tests, clear all change issues in Requirements Editor. Click Show Links in the Requirements tab, then select the links and click on Clear All button in Change Information section of the Details panel on the right. Launch the test when the issues are cleared. The new design will pass the updated tests and verify that the system satisfies the updated requirements confirming the predictions made by the Radar Designer app.
This example is the second part of a two-part series on how to design and test a radar system in Simulink based on a set of performance requirements. It shows how to use Simulink Test to test the model developed in Part 1, how to link the test to the requirements, and how to verify that the requirements are satisfied by running Monte Carlo simulations. The example also illustrates how to trace changes in the requirements to the corresponding components and how to create alternative designs by adding variants to the model. Part 1 of this example starts with the requirements that must be satisfied by the final design. It uses System Composer to develop an architecture model of a radar system that can serve as a virtual test bed. Part 1 also shows how to use Requirements Toolbox to link the requirements to the components and how to implement the individual components of the architecture using Simulink.
|
Analyses of Convection Heat Transfer From Discrete Heat Sources in a Vertical Rectangular Channel | J. Electron. Packag. | ASME Digital Collection
H. Bhowmik,
H. Bhowmik
, Gazipur 1700, Bangladesh
e-mail: pp0504835@ntu.edu.sg
C. P. Tso,
, 50 Nanyang Avenue, Sinapore 639798
K. W. Tou
, Jalan Ayer Keroh Lama, Melaka 75450, Malaysia
Bhowmik, H., Tso, C. P., and Tou, K. W. (August 9, 2004). "Analyses of Convection Heat Transfer From Discrete Heat Sources in a Vertical Rectangular Channel." ASME. J. Electron. Packag. September 2005; 127(3): 215–222. https://doi.org/10.1115/1.1938207
Steady-state experiments are performed to study the convection heat transfer from four in-line simulated chips in a vertical rectangular channel using water as the working fluid. The experimental data cover a wide range for laminar flow under natural, mixed, and forced convection conditions with the Reynolds number based on the channel hydraulic diameter ranging from 40 to 2220 and on the heat source length ranging from 50 to 2775. The heat flux ranges from
0.1W∕cm2to0.6W∕cm2
. The effects of heat flux, flow rates, and chip number are investigated and results indicate that the Nusselt number is strongly affected by the Reynolds number. To develop empirical correlations, the appropriate value of the exponent
n
ReD
is determined to collapse all the lines into a single line to show the independence of heat flux. Based on experimental results, the empirical correlations are developed for relations using
Nuℓ
ReD
GrD
. The results are compared to predictions from a three-dimensional numerical simulation, and a numerical correlation is also developed.
thermal management (packaging), forced convection, natural convection, channel flow, laminar flow, water
Computer simulation, Convection, Flow (Dynamics), Forced convection, Heat, Water, Heat flux, Fluids, Laminar flow, Reynolds number, Natural convection, Steady state, Temperature
Analysis of Laminar Mixed Convection in Shrouded Arrays of Heated Blocks
Mixed Convection in Vertical Channels With a Discrete Heat Source
Heat Transfer and Pressure Loss Characterization in a Channel With Discrete Flush-Mounted and Protruding Heat Sources
Local Convective Heat Transfer From an Array of Wall-Mounted Cubes
Mixed Convection Heat Transfer in Open Ended Channels With Protruding Heaters
An Experimental Study on Forced Convection Heat Transfer From Flush-Mounted Discrete Heat Sources
Direct Liquid Cooling of Electronic Chips by Single-Phase Forced Convection of FC-72
General Convection Heat Transfer From Discrete Heat Sources in a Rectangular Channel
Proc. 4th European Thermal Science Conf.
, FCH 3, pp.
Convection Heat Transfer From Discrete Heat Sources in a Rectangular Channel
Forced Convection Boiling and Critical Heat Flux From a Linear Array of Discrete Heat Sources
Liquid Immersion Cooling of a Longitudinal Array of Discrete Heat Sources in Protruding Substrates: 2-Forced Convection Boiling
Greankoplis
, Supplement No. 1,
Combined Natural and Forced Convection Flow on Vertical Surfaces
Natural Convection Through Vertical Plane Layers: Moderate and High Prandtl Number Fluids
|
Physics - Insulating behavior in topological insulators
Insulating behavior in topological insulators
A new topological insulator that approaches true insulating behavior in the bulk is synthesized.
Credit: Z. Ren et al., Phys. Rev. B (2010)
3D topological insulators represent a unique quantum state for matter that is supposed to show insulating behavior in the bulk and spin-dependent metallic conduction on the surface. In practice, the best-known exemplars of materials that show a topologically protected metallic surface state, such as and , are also conducting in the bulk due to the presence of vacancies. Significant efforts in trying to find a topological insulator that is truly insulating in the bulk have met with little success.
Presenting their results as a Rapid Communication in Physical Review B, Zhi Ren and colleagues from Osaka University, Japan, have synthesized a new topological insulator, , that approaches insulating behavior in the bulk with a high resistivity. Ren et al. demonstrate variable-range hopping that is the hallmark of an insulator in high-quality single crystals of and Shubnikov-de Haas oscillations coming from the 2D surface metallic state. Surface contribution to the total conductance of the crystal at is the largest ever achieved in a topological insulator. From a detailed study of the Hall effect, the authors also determine the transport mechanism in the bulk that reveals an impurity band in the band gap along with hopping conduction of localized electrons. These results pave the way for exploiting the unique surface conduction properties of topological insulators. – Sarma Kancharla
Large bulk resistivity and surface quantum oscillations in the topological insulator
{\text{Bi}}_{2}{\text{Te}}_{2}\text{Se}
Zhi Ren, A. A. Taskin, Satoshi Sasaki, Kouji Segawa, and Yoichi Ando
{\text{Bi}}_{2}{\text{Te}}_{2}\text{Se}
|
K3
surfaces and Enriques surfaces
3
4
n
-folds (
n>4
ℚ
Roberto Muñoz, Luis E. Solá Conde, Gianluca Occhetta (2014)
ℰ
{H}^{2}\left(X,ℤ\right)\cong {H}^{4}\left(X,ℤ\right)\cong ℤ
ℙ\left(ℰ\right)
X
ℰ
Hajime Ono, Yuji Sano, Naoto Yotsutani (2012)
Donaldson proved that if a polarized manifold
\left(V,L\right)
has constant scalar curvature Kähler metrics in
{c}_{1}\left(\phantom{\rule{-0.166667em}{0ex}}L\right)
and its automorphism group
\mathrm{Aut}\left(V\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}},\phantom{\rule{-0.166667em}{0ex}}L\right)
is discrete,
\left(V,L\right)
is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case where
\mathrm{Aut}\left(V,L\right)
is not discrete.
Stefan Kebekus, Sándor J. Kovács (2004)
X
be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of
X
is contained in at most one rational curve of minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.
Bilinear forms and extremal Kähler vector fields associated with Kähler classes.
Akito Futaki, Toshiki Mabuchi (1995)
Birational automorphisms of higher-dimensional algebraic varieties.
Pukhlikov, Aleksandr V. (1998)
Classification des varietes coplexes projectives de dimension trois dont une section hyperplane générale est une surface d'Enriques.
Lionel Bayle (1994)
Compactifications of C2 x C*.
Computing the quantum cohomology of some Fano threefolds and its semisimplicity
Gianni Ciolli (2004)
We compute explicit presentations for the small Quantum Cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from
{\mathbb{P}}^{3}
or the smooth quadric. Systematic usage of the associativity property of quantum product implies that only a very small and enumerative subset of Gromov- Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of Quantum Cohomology is proven by checking the computed Quantum Cohomology rings and by showing...
Connexité rationnelle des variétés de Fano
F. Campana (1992)
X
be a Fano manifold with
{b}_{2}=1
different from the projective space such that any two surfaces in
X
have proportional fundamental classes in
{H}_{4}\left(X,\mathbf{C}\right)
f:Y\to X
be a surjective holomorphic map from a projective variety
Y
. We show that all deformations of
Y
X
fixed, come from automorphisms of
X
. The proof is obtained by studying the geometry of the integral varieties of the multi-valued foliation defined by the variety of minimal rational tangents of
X
Duality and integrability on contact Fano manifolds.
Buczynski, Jaroslaw (2010)
Erratum: Rational points of bounded height on Fano varieties. Invent. Math. 95, 421-435 (1989).
J. Franke, Y.I. Manin, Y. Tschinkel (1990)
Exceptional linear systems on curves on Del Pezzo surfaces.
Giuseppe Pareschi (1991)
Fano bundles of rank 2 on surfaces
Michał Szurek, Jarosław A. Wisniewski (1990)
Fano hypersurfaces in weighted projective 4-spaces.
Johnson, Jennifer M., Kollár, János (2001)
Fano manifolds and quadric bundles.
Jaroslaw A. Wisniewski (1993)
Fano manifolds of degree ten and EPW sextics
Atanas Iliev, Laurent Manivel (2011)
O’Grady showed that certain special sextics in
{ℙ}^{5}
called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.
Fano Schemes of Linear Spaces on Hypersurfaces.
Adrian Langer (1997)
General elephants of
ℚ
-Fano 3-folds
Valery Alexeev (1994)
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49Kxx Optimality conditions
49Mxx Numerical methods
49Rxx Variational methods for eigenvalues of operators
49Sxx Variational principles of physics
{ℒ}^{2,\text{Φ}}
Natal'ya Pavlova (2007)
\Gamma
\epsilon
{\Omega }_{\epsilon }=\omega ×\left(-\epsilon ,\epsilon \right)
\omega \subset {ℝ}^{2}
{ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W\left(\overline{m},\nabla \overline{m}\right)+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x
\text{div}\left(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }}\right)=0\phantom{\rule{1.0em}{0ex}}\text{on}{ℝ}^{3},
|\overline{m}|=1\text{on}{\Omega }_{\epsilon },
W
p
p>1
p=1
W
{\Omega }_{\epsilon }=\omega ×\left(-\epsilon ,\epsilon \right)
\omega \subset {ℝ}^{2}
{ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W\left(\overline{m},\nabla \overline{m}\right)+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x
\text{div}\left(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }}\right)=0\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{ℝ}^{3},
|\overline{m}|=1\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{\Omega }_{\epsilon },
G. Barles, A. Briani, E. Chasseigne (2013)
This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly...
A Bolza optimal synthesis problem for singular estimate control systems
Irena Lasiecka, Amjad Tuffaha (2009)
{C}^{1,1}
functions via lower directional derivatives
Dušan Bednařík, Karel Pastor (2009)
\stackrel{˜}{\ell }
-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of
\stackrel{˜}{\ell }
-stable functions coincides with the class of C
{}^{1,1}
functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp. 383–387.
{R}^{N}
Antonio Corbo Esposito, Riccardo De Arcangelis (1994)
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6-manifolds: 2-connected - Manifold Atlas
5.1 Topological 2-connected 6-manifolds
5.2 Mapping class groups
5.3 Diffeomorphism groups
\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\mathcal{M}_6(0)
be the set of diffeomorphism classes of closed smooth simply-connected 2-connected 6-manifolds
M
(the notation is used to be consistent with 6-manifolds: 1-connected).
\mathcal{M}_6(0)
was one of Smale's first applications of the h-cobordism theorem [Smale1962a, Corollary 1.3]. The is a precise 6-dimensional analogue of the classification of orientable surfaces: every 2-connected 6-manifold
M
is diffeomorphic to a connected-sum
\#_0(S^3 \times S^3) = S^6
r
is determined by the formula for the Euler characteristic of
M
For the more general case where
H_2(M) \neq 0
, see 6-manifolds: 1-connected.
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
S^6
, the standard 6-sphere.
\#_b(S^3 \times S^3)
b
S^3 \times S^3
M
is diffeomorphic to
\#_b(S^3 \times S^3)
\pi_3(M) \cong H_3(M) \cong \Zz^{2b}
the third Betti-number of
M
b_3(M) = 2b
the Euler characteristic of
M
\chi(M) = 2 - 2b
the intersection form of
M
is isomorphic to the sum of b-copies of
H_{-}(\Zz)
, the standard skew-symmetric hyperbolic form on
\Zz^2
[edit] 4 Classification
Recall that the following theorem was stated in other words in the introduction:
Theorem 4.1 [Smale1962a, Corolary 1.3]. The semi-group of 2-connected 6-manifolds is generated by
S^3 \times S^3
\Nn
denotes the natural numbers we obtain a bijection
[edit] 5.1 Topological 2-connected 6-manifolds
\mathcal{M}^{\Top}_6(0)
be the set of homeomorphism classes of topological 2-connected 6-manifolds.
Theorem 5.1. Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. In particular, there is a bijection
Proof. For any such manifold
M
H^4(M; \Zz/2) \cong 0
M
is smoothable (see 6-manifolds: 1-connected). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem 4.1 are diffeomorphic.
\square
[edit] 5.2 Mapping class groups
\pi_0\Diff_+(M)
denote the group of isotopy classes of diffeomorphisms
f \colon M \to M
2
6
M
\Aut(M)
denote the group of isomorphisms of
H_3(M)
perserving the intersection form:
\Aut(M) \cong Sp_{2b}(\Zz)
is the symplectic group when
M = \#_b(S^3 \times S^3)
. By [Cerf1970] the forgetful map to the group of orientation preserving pseudo-isotopy classes of
M
is an isomorphism. Applying Cerf's theorem Kreck proved in [Kreck1979] that there are exact sequences
\pi_0\SDiff(M)
is the subgroup of isotopy classes induced the identity on
H_*(M)
\Theta_7 \cong \pi_0(\Diff(D^6, \partial))
is the group of homotopy
7
\pi_0(\Diff_+(S^6)) \cong \Zz/28 \cong \Theta_7
For more information about the extensions in
(\ast)
above, see [Krylov2003], [Johnson1983] and [Crowley2009].
[edit] 5.3 Diffeomorphism groups
\mathcal{D}_b = \Diff(\#_b S^3 \times S^3, D^6)
denote group of diffeomorphisms of
\#_b S^3 \times S^3
which are the identity inside a marked disc, and
B\mathcal{D}_b
denote the classifying spaces of this topological group. Connect-sum inside the marked disc gives homomorphisms
\mathcal{D}_b \to \mathcal{D}_{b+1}
, and so continuous maps
\mathcal{S} : B\mathcal{D}_b \to B\mathcal{D}_{b+1}
. The homology of these classifying spaces is approachable in a range of degrees, by the following theorem.
Theorem 5.2 [Galatius&Randal-Williams2012, Theorem 1.2]. The map
b \geq 2k+4
B\mathrm{Spin}(6)
denote the classifying space of the group
\mathrm{Spin}(6)
\gamma_6^{\mathrm{Spin}}
denote its universal vector bundle. We write
MT\mathrm{Spin}(6)
for the Thom spectrum of the virtual bundle
-\gamma_6^{\mathrm{Spin}}
. Pontrjagin--Thom theory provides a map
to the basepoint component of the infinite loop space of the spectrum
MT\mathrm{Spin}(6)
, and these fit together under the maps
\mathcal{S}
to give a map
Theorem 5.3 [Galatius&Randal-Williams2012a, Corollary 1.2]. The map
\alpha
induces an isomorphism on (co)homology.
It is not difficult to calculate the rational cohomology of
\Omega^\infty_\bullet MT\mathrm{Spin}(6)
, and find that it is a polynomial algebra with generators in degrees
2,2,4,6,6,6,8,8,10,10,10,12,12,\ldots
, which can be given an explicit description in terms of generalised Miller-Morita-Mumford classes. By the stability theorem, this calculates the rational cohomology of
B\mathcal{D}_b
* \leq (b-4)/2
[Cerf1970] J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970), no.39, 5–173. MR0292089 (45 #1176) Zbl 0213.25202
[Crowley2009] D. Crowley, On the mapping class groups of
\#_r(S^p \times S^p)
p = 3, 7
, (2009). Available at the arXiv:0905.0423.
[Galatius&Randal-Williams2012] S. Galatius and O. Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds, (2012). Available at the arXiv:1203.6830.
[Galatius&Randal-Williams2012a] S. Galatius and O. Randal-Williams, Stable moduli spaces of high dimensional manifolds, (2012). Available at the arXiv:1201.3527.
[Johnson1983] D. Johnson, A survey of the Torelli group, Low-dimensional topology (San Francisco, Calif., 1981), Amer. Math. Soc. (1983), 165–179. MR718141 (85d:57009) Zbl 0553.57002
[Kreck1979] M. Kreck, Isotopy classes of diffeomorphisms of
(k-1)
-connected almost-parallelizable
2k
-manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Springer (1979), 643–663. MR561244 (81i:57029) Zbl 0421.57009
[Krylov2003] N. A. Krylov, On the Jacobi group and the mapping class group of
S^3\times S^3
, Trans. Amer. Math. Soc. 355 (2003), no.1, 99–117 (electronic). MR1928079 (2003i:57039) Zbl 1015.57020
[Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103
Retrieved from "http://www.map.mpim-bonn.mpg.de/index.php?title=6-manifolds:_2-connected&oldid=8848"
Highly-connected manifolds
|
Fermat's factorization method - Wikipedia
Find sources: "Fermat's factorization method" – news · newspapers · books · scholar · JSTOR (February 2022) (Learn how and when to remove this template message)
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares:
{\displaystyle N=a^{2}-b^{2}.}
That difference is algebraically factorable as
{\displaystyle (a+b)(a-b)}
; if neither factor equals one, it is a proper factorization of N.
Each odd number has such a representation. Indeed, if
{\displaystyle N=cd}
is a factorization of N, then
{\displaystyle N=\left({\frac {c+d}{2}}\right)^{2}-\left({\frac {c-d}{2}}\right)^{2}}
Since N is odd, then c and d are also odd, so those halves are integers. (A multiple of four is also a difference of squares: let c and d be even.)
In its simplest form, Fermat's method might be even slower than trial division (worst case). Nonetheless, the combination of trial division and Fermat's is more effective than either.
2 Fermat's and trial division
3 Sieve improvement
4 Multiplier improvement
6 Factorization with rectangles
7 Factorization with cubes
8 Factorization with cuboids
One tries various values of a, hoping that
{\displaystyle a^{2}-N=b^{2}}
, a square.
FermatFactor(N): // N should be odd
a ← ceiling(sqrt(N))
b2 ← a*a - N
repeat until b2 is a square:
// b2 ← b2 + 2*a + 1
// a ← a + 1
return a - sqrt(b2) // or a + sqrt(b2)
For example, to factor
{\displaystyle N=5959}
, the first try for a is the square root of 5959 rounded up to the next integer, which is 78. Then,
{\displaystyle b^{2}=78^{2}-5959=125}
. Since 125 is not a square, a second try is made by increasing the value of a by 1. The second attempt also fails, because 282 is again not a square.
The third try produces the perfect square of 441. So,
{\displaystyle a=80}
{\displaystyle b=21}
, and the factors of 5959 are
{\displaystyle a-b=59}
{\displaystyle a+b=101}
Suppose N has more than two prime factors. That procedure first finds the factorization with the least values of a and b. That is,
{\displaystyle a+b}
is the smallest factor ≥ the square-root of N, and so
{\displaystyle a-b=N/(a+b)}
is the largest factor ≤ root-N. If the procedure finds
{\displaystyle N=1\cdot N}
, that shows that N is prime.
{\displaystyle N=cd}
, let c be the largest subroot factor.
{\displaystyle a=(c+d)/2}
, so the number of steps is approximately
{\displaystyle (c+d)/2-{\sqrt {N}}=({\sqrt {d}}-{\sqrt {c}})^{2}/2=({\sqrt {N}}-c)^{2}/2c}
If N is prime (so that
{\displaystyle c=1}
), one needs
{\displaystyle O(N)}
steps. This is a bad way to prove primality. But if N has a factor close to its square root, the method works quickly. More precisely, if c differs less than
{\displaystyle {\left(4N\right)}^{1/4}}
{\displaystyle {\sqrt {N}}}
, the method requires only one step; this is independent of the size of N.[citation needed]
Fermat's and trial division[edit]
Consider trying to factor the prime number N = 2345678917, but also compute b and a − b throughout. Going up from
{\displaystyle {\sqrt {N}}}
, we can tabulate:
In practice, one wouldn't bother with that last row until b is an integer. But observe that if N had a subroot factor above
{\displaystyle a-b=47830.1}
, Fermat's method would have found it already.
Trial division would normally try up to 48,432; but after only four Fermat steps, we need only divide up to 47830, to find a factor or prove primality.
This all suggests a combined factoring method. Choose some bound
{\displaystyle c>{\sqrt {N}}}
; use Fermat's method for factors between
{\displaystyle {\sqrt {N}}}
{\displaystyle c}
. This gives a bound for trial division which is
{\displaystyle c-{\sqrt {c^{2}-N}}}
. In the above example, with
{\displaystyle c=48436}
the bound for trial division is 47830. A reasonable choice could be
{\displaystyle c=55000}
giving a bound of 28937.
In this regard, Fermat's method gives diminishing returns. One would surely stop before this point:
Sieve improvement[edit]
When considering the table for
{\displaystyle N=2345678917}
, one can quickly tell that none of the values of
{\displaystyle b^{2}}
are squares:
It is not necessary to compute all the square-roots of
{\displaystyle a^{2}-N}
, nor even examine all the values for a. Squares are always congruent to 0, 1, 4, 5, 9, 16 modulo 20. The values repeat with each increase of a by 10. In this example, N is 17 mod 20, so subtracting 17 mod 20 (or adding 3),
{\displaystyle a^{2}-N}
produces 3, 4, 7, 8, 12, and 19 modulo 20 for these values. It is apparent that only the 4 from this list can be a square. Thus,
{\displaystyle a^{2}}
must be 1 mod 20, which means that a is 1, 9, 11 or 19 mod 20; it will produce a
{\displaystyle b^{2}}
which ends in 4 mod 20 and, if square, b will end in 2 or 8 mod 10.
This can be performed with any modulus. Using the same
{\displaystyle N=2345678917}
modulo 16: Squares are 0, 1, 4, or 9
{\displaystyle a^{2}}
can only be 9
and a must be 3 or 5 or 11 or 13 modulo 16
modulo 9: Squares are 0, 1, 4, or 7
{\displaystyle a^{2}}
and a must be 4 or 5 modulo 9
One generally chooses a power of a different prime for each modulus.
Given a sequence of a-values (start, end, and step) and a modulus, one can proceed thus:
FermatSieve(N, astart, aend, astep, modulus)
a ← astart
do modulus times:
if b2 is a square, modulo modulus:
FermatSieve(N, a, aend, astep * modulus, NextModulus)
a ← a + astep
But the recursion is stopped when few a-values remain; that is, when (aend-astart)/astep is small. Also, because a's step-size is constant, one can compute successive b2's with additions.
Multiplier improvement[edit]
Fermat's method works best when there is a factor near the square-root of N.
If the approximate ratio of two factors (
{\displaystyle d/c}
) is known, then a rational number
{\displaystyle v/u}
can be picked near that value.
{\displaystyle Nuv=cv\cdot du}
, and Fermat's method, applied to Nuv, will find the factors
{\displaystyle cv}
{\displaystyle du}
quickly. Then
{\displaystyle \gcd(N,cv)=c}
{\displaystyle \gcd(N,du)=d}
. (Unless c divides u or d divides v.)
Generally, if the ratio is not known, various
{\displaystyle u/v}
values can be tried, and try to factor each resulting Nuv. R. Lehman devised a systematic way to do this, so that Fermat's plus trial division can factor N in
{\displaystyle O(N^{1/3})}
The fundamental ideas of Fermat's factorization method are the basis of the quadratic sieve and general number field sieve, the best-known algorithms for factoring large semiprimes, which are the "worst-case". The primary improvement that quadratic sieve makes over Fermat's factorization method is that instead of simply finding a square in the sequence of
{\displaystyle a^{2}-n}
, it finds a subset of elements of this sequence whose product is a square, and it does this in a highly efficient manner. The end result is the same: a difference of square mod n that, if nontrivial, can be used to factor n.
Factorization with rectangles[edit]
The method can be modified to use rectangles instead of squares, by adding a constant c:
{\displaystyle (a+c)\times a-(b+c)\times b=(a-b)\times (a+b+c)}
Factorization with cubes[edit]
{\displaystyle a^{3}-b^{3}=(a-b)\times ((a+b)\times a+b\times b)}
Factorization with cuboids[edit]
{\displaystyle (a+c)\times a^{2}-(b+c)\times b^{2}=(a-b)\times ((a+b+c)\times a+(b+c)\times b)}
^ Lehman, R. Sherman (1974). "Factoring Large Integers" (PDF). Mathematics of Computation. 28 (126): 637–646. doi:10.2307/2005940.
Fermat (1894), Oeuvres de Fermat, vol. 2, p. 256
McKee, J (1999). "Speeding Fermat's factoring method". Mathematics of Computation (68): 1729–1737.
Fermat's factorization running time, at blogspot.in
Fermat's Factorization Online Calculator, at windowspros.ru
Retrieved from "https://en.wikipedia.org/w/index.php?title=Fermat%27s_factorization_method&oldid=1078039557"
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Rank features for unsupervised learning using Laplacian scores - MATLAB fsulaplacian - MathWorks Benelux
{S}_{i,j}=\mathrm{exp}\left(-{\left(\frac{Dis{t}_{i,j}}{\sigma }\right)}^{2}\right)
S={\left({S}_{i,j}\right)}_{i,j=1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}n}
{D}_{g}\left(i,i\right)=\sum _{j=1}^{n}{S}_{i,j}.
L={D}_{g}-S.
Dis{t}_{i,j}
{S}_{i,j}=\mathrm{exp}\left(-{\left(\frac{Dis{t}_{i,j}}{\sigma }\right)}^{2}\right)
{\stackrel{˜}{x}}_{r}={x}_{r}-\frac{{x}_{r}^{T}{D}_{g}1}{{1}^{T}{D}_{g}1}1,
{1}^{T}={\left[1,\cdots ,1\right]}^{T}
{s}_{r}=\frac{{\stackrel{˜}{x}}_{r}^{T}S{\stackrel{˜}{x}}_{r}}{{\stackrel{˜}{x}}_{r}^{T}{D}_{g}{\stackrel{˜}{x}}_{r}}.
{L}_{r}=\frac{{\stackrel{˜}{x}}_{r}^{T}L{\stackrel{˜}{x}}_{r}}{{\stackrel{˜}{x}}_{r}^{T}{D}_{g}{\stackrel{˜}{x}}_{r}}=1-\frac{{\stackrel{˜}{x}}_{r}^{T}S{\stackrel{˜}{x}}_{r}}{{\stackrel{˜}{x}}_{r}^{T}{D}_{g}{\stackrel{˜}{x}}_{r}},
\frac{{\sum }_{i,j}{\left({x}_{ir}-{x}_{jr}\right)}^{2}{S}_{i,j}}{Var\left({x}_{r}\right)},
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Constant Elasticity of Variance (CEV) model - MATLAB - MathWorks Switzerland
Create a cev Object
Constant Elasticity of Variance (CEV) model
Creates and displays cev objects, which derive from the sdeld (SDE with drift rate expressed in linear form) class.
Use cev objects to simulate sample paths of NVars state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, approximating continuous-time stochastic processes.
This model allows you to simulate any vector-valued CEV of the form:
d{X}_{t}=\mu \left(t\right){X}_{t}dt+D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)d{W}_{t}
μ is an NVars-by-NVars (generalized) expected instantaneous rate of return matrix.
CEV = cev(Return,Alpha,Sigma)
CEV = cev(___,Name,Value)
CEV = cev(Return,Alpha,Sigma) creates a default CEV object.
CEV = cev(___,Name,Value) creates a CEV object with additional options specified by one or more Name,Value pair arguments.
The CEV object has the following Properties:
If you specify Sigma as an array, it must be an NVars-by-NBrowns matrix of instantaneous volatility rates. In this case, each row of Sigma corresponds to a particular state variable. Each column corresponds to a particular Brownian source of uncertainty, and associates the magnitude of the exposure of state variables with sources of uncertainty.
Although cev does not enforce restrictions on the signs of these input arguments, each argument is specified as a positive value.
If StartState is a scalar, cev applies the same initial value to all state variables on all trials.
If StartState is a column vector, cev applies a unique initial value to each state variable on all trials.
If StartState is a matrix, cev applies a unique initial value to each state variable on each trial.
F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}
Diffusion rate component of continuous-time stochastic differential equations (SDEs), specified as an object or function accessible by (t, Xt.
G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)
Create a univariate cev object to represent the model:
d{X}_{t}=0.25{X}_{t}+0.3{X}_{t}^{\frac{1}{2}}d{W}_{t}
cev objects display the parameter B as the more familiar Return
When you invoke these parameters with inputs, they behave like functions, giving the impression of dynamic behavior. The parameters accept the observation time t and a state vector Xt, and return an array of appropriate dimension. Even if you originally specified an input as an array, cev treats it as a static function of time and state, by that means guaranteeing that all parameters are accessible by the same interface.
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Physics - Challenging a hole to move through an ordered insulator
Challenging a hole to move through an ordered insulator
Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
A theoretical framework to explain how a hole moves through an antiferromagnetically and orbitally ordered lattice could also provide insight into the interplay between these two ordered phases.
Figure 1: (Top) The string of defects (emphasized by blue halos) that are generated by hole motion in an antiferromagnetic background can be “healed” by quantum fluctuations. (Bottom) A similar string of defects (blue halos) is generated by zigzag hole motion in an alternating-orbital-order background (the green and purple lobes represent orbitals with
x\phantom{\rule{0}{0ex}}z
y\phantom{\rule{0}{0ex}}z
symmetry, respectively). These cannot be healed by quantum fluctuations and instead the hole has to do it by retracing its steps. Overall hole motion is enabled by three-site processes (red halos), consisting of directional tunneling through a doubly occupied site without generating a defect.(Top) The string of defects (emphasized by blue halos) that are generated by hole motion in an antiferromagnetic background can be “healed” by quantum fluctuations. (Bottom) A similar string of defects (blue halos) is generated by zigzag hole motion ... Show more
This is a difficult question, and enormous theoretical effort has been (and still is) devoted to answering it. On one hand, we know from experimental data that these holes are mobile, since the doped systems quickly become metallic. On the other hand, it is hard to see how the hole can move at all given the type of order of the parent compounds: in cuprates, the spins form an antiferromagnetic arrangement at low temperatures (as in the top part of Fig. 1), while in manganites, the spatial distribution of the orbitals on the sites forms a pattern of alternating orbital order (similar, though not identical, to the bottom of Fig. 1). Since in both cases, the proper alternation of the spins or orbitals is disrupted as the hole moves through the lattice, one might expect that the hole cannot go too far from its original location.
Writing in Physical Review B, Krzysztof Wohlfeld, Andrzej Oleś, and Peter Horsch of the Max Planck Institute in Stuttgart, Germany, take the next step and for the first time address the issue of a hole moving in an insulator that has, simultaneously, both antiferromagnetic and alternating orbital order [1]. Beyond the direct implications of their results for real materials that possess both types of order, such as , their study begins to address the effects of the interplay between two different types of “alternating” orders—a long-standing question in the study of a number of other materials, including some of the manganites.
Before discussing their findings, it is instructive to review what is known about the motion of a charge carrier in materials with simpler types of order. The only case with a known exact solution at zero temperature is long-range ferromagnetic order with no orbital degeneracy (that is, there is only one orbital that can be occupied) [2,3]. This occurs, for example, in the ferromagnetic insulator . A charge carrier with a spin parallel to this ferromagnetic background propagates without any hindrance since no spin-flipping processes are possible. If, on the other hand, the charge carrier’s spin is antiparallel to the background, it oscillates between propagating freely and flipping its spin to be parallel to the background, at the price of a simultaneous flip of a spin that make up the ferromagnetic background. (The reason this process can occur is rooted in the fact that the magnetic exchange between the charge carrier’s spin and the spins in the background has full rotational—Heisenberg—symmetry.) This results in the excitation of a magnon: a collective excitation of the ferromagnetic background. In favorable circumstances, the magnon becomes bound to the charge carrier and together they form a so-called spin-polaron that propagates coherently. The spin-polaron is a true quasiparticle with an infinite lifetime and a mass somewhat, though minimally, heavier than that of a free electron or hole, because of the bound magnon.
The situation for cuprates is more complex, as the spins on the valence electrons on the copper sites are now ordered antiferromagnetically. Upon hole doping (removal of one electron), it seems that the hole can no longer move freely, in essence, through the system, as in the simpler case of the ferromagnet. This is because, as sketched in the top panel of Fig. 1, as the hole hops around, it reshuffles the spins it visits and creates a “string” of defects that costs significant magnetic (or “exchange”) energy.
Figure 2: Wohlfeld et al. ask the question: Is it possible for a hole to move coherently in an antiferromagnetic and alternating-orbital-order background, and if yes, how?
An exact solution for how the hole succeeds in being mobile in antiferromagnets like the cuprates is not known, not least because we are missing an exact description of the antiferromagnetic undoped state itself. Two simplifications make this problem easier to solve. One is to forbid having two electrons at the same site because the Coulomb repulsion is so large. In this case, it is possible to describe the collective excitations of the antiferromagnetic background with the linear spin-wave approximation, and the problem reduces to a simpler polaronlike problem [4]. This permits the calculation of the hole’s propagator, which is the amplitude of the probability that the hole evolves from a specified initial state to a specified final state. From the propagator we can determine whether the hole is mobile or not, and its imaginary part—the spectral weight—can be measured by both angle-resolved photoemission and scanning tunneling microscopy.
Even after all these simplifications, the propagator still cannot be calculated exactly. A second simplification is therefore the application of the self-consistent Born approximation, which leads to an accurate solution that compares well with numerical simulations [5]. This is meaningful because this approximation only includes processes where the hole reabsorbs the magnons in the inverse order it emitted them. In other words, the hole indeed retraces its steps and “heals” the string of defects it created. This is supplemented by the fact that two neighboring incorrectly oriented spins can flip each other back to the correct antiferromagnetic order, through their Heisenberg exchange. Thus, we can picture the mobile quasiparticle as consisting of the hole surrounded by a cloud of magnons (flipped spins), which it continuously emits and reabsorbs as it tries to move away and is forced to come back. Every now and then “self-healing” removes some of the defects from this string of defects tying the hole to its initial position and allows it to progress to a new starting point, explaining why it is mobile even though rather slow and heavy.
The motion of a hole in a background of alternating orbital order but with ferromagnetic spin order is somewhat different. Consider a material such as undoped with one electron per site occupying either of the two orbitals with and symmetry (the other orbitals have different energies and their occupation numbers are assumed to be fixed). By symmetry, the occupied orbital determines the allowed hopping: electrons in the and orbitals can only hop along the and axes, respectively. This suggests that a hole doped in this type of ordered background moves in zigzags, leaving a string of orbital “defects” behind as sketched in the bottom panel of Fig. 1. Unlike in cuprates, however, there is no “self-healing” possible for these defects because of the Ising-like symmetry of their interactions. So, again one expects the hole to be trapped. This is not the case, as demonstrated recently in Ref. [6], due to “three-site terms,” which appear when the approximation of no double occupancy is enforced, and which allow the hole to tunnel through a doubly occupied configuration without creating a defect (Fig. 1, bottom panel). Such processes have well-defined directionality and result in an even heavier quasiparticle with one-dimensional motion. One can picture it again as the hole creating and undoing strings of orbital defects, and changing its starting point every now and then through directional three-site tunneling.
The new question (see Fig. 2) is what happens if the insulator has both antiferromagnetic and alternating orbital order, as is the case in ? Wohlfeld et al. use the same simplifications as for cuprates and manganates to reduce the problem to being polaronlike, but now with simultaneous scattering of the hole from both spin waves and orbital waves. They then calculate the spectral weight using the self-consistent Born approximation. This reveals again the appearance of a low-energy mobile quasiparticle, with properties superficially similar to those of a quasiparticle in a background that only has alternating orbital order. As the hole moves along the zigzags allowed by the orbital order, it creates strings of both orbital and magnetic defects. Unlike holes moving in antiferromagnetic-only backgrounds, the defects formed in this more complex case cannot “self-heal” through quantum fluctuations because of the Ising-like nature of the orbitals. So the hole is forced to fully retrace its steps and repair the defects it created. Overall motion is only enabled, as for the case of hole motion in the alternating-orbital-only background, by three-site processes. This results in a heavy quasiparticle with a quasi-one-dimensional dispersion.
These considerations explain the superficial resemblance between this solution and the orbital-only polarons. But Wohlfeld et al. show that the spins also play a key part. The spin dynamics is effectively confined to being Ising-like because of the simultaneous coupling to the alternating orbital order; however, they contribute significantly to the energetics of the system since the strings generated by the hole have joint spin-orbital character.
This work predicts that holes find a way to propagate coherently even through such challenging backgrounds as a lattice with both antiferromagnetic and alternating orbital order. Once this conclusion is confirmed by photoemission experiments, it will open the way to understanding the low-doping part of the vanadate phase diagrams, as well as other materials with similarly complex ordered states.
K. Wohlfeld, A. M. Oleś, and P. Horsch, Phys. Rev. B 79, 224433 (2009)
B. S. Shastry and D. C. Mattis, Phys. Rev. B 24, 5340 (1981)
M. Berciu and G. A. Sawatzky, Phys. Rev. B 79, 195116 (2009)
See, for instance, G. Martinez and P. Horsch, Phys. Rev. B 44, 317 (1991)
See, for instance, A. S. Mishchenko, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. B 64, 033101 (2001)
M. Daghofer, K. Wohlfeld, A. M. Oleś, E. Arrigoni, and P. Horsch, Phys. Rev. Lett. 100, 066403 (2008)
Mona Berciu received her B.Sc. from the University of Bucharest in Romania in 1994 and her Ph.D. from the University of Toronto in 1999. After postdoctoral work at Princeton University, in 2002 she moved to the University of British Columbia where she is currently an Associate Professor.
Krzysztof Wohlfeld, Andrzej M. Oleś, and Peter Horsch
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Physics - Diamonds are a scientist’s best friend
February 19, 2009 • Physics 2, s18
Diamond is famous for its exceptional hardness and structural stability. Researchers are exploring different ways to push these mechanical properties beyond their current limits.
Two papers appearing in Physical Review Letters take different routes toward investigating the legendary hardness of diamond. Diamond is used to investigate the high-pressure behavior of solids, through diamond-anvil cells. However, these cells can typically only withstand pressures of up to 300 due to the mechanical strength limit of diamond. While this may seem high enough, it is believed that the pressures exceed 1 at the core of planets like Jupiter and Saturn. The nature of solids at such pressures remains a mystery.
To study solids at higher pressures, a group of researchers at the Lawrence Livermore National Laboratory present a novel ramp-wave compression technique that allows them to study diamond at 800 . The diamond sample is ablated with x-ray lasers that are ramped up monotonically until a uniform compression wave is produced. This wave propagates faster than the thermal wave caused by the laser ablation, resulting in compression without heating.
In another study on diamond, researchers from Université Paris Nord, Université Blaise Pascal, and Université Pierre et Marie Curie have investigated the solubility of boron in diamond at the European Synchrotron Radiation Facility in Grenoble, France. They have synthesized cubic boron carbide ( ), a diamondlike phase with the highest boron content ever achieved. Compared to diamond, the Vickers hardness of is slightly less, but it has a comparable hardness at the nanoscale and nearly twice the fracture toughness. This makes an exceptional super-abrasive and a promising material for high-temperature electronics applications. – Daniel Ucko
Diamond at 800 GPa
D. K. Bradley, J. H. Eggert, R. F. Smith, S. T. Prisbrey, D. G. Hicks, D. G. Braun, J. Biener, A. V. Hamza, R. E. Rudd, and G. W. Collins
Ultimate Metastable Solubility of Boron in Diamond: Synthesis of Superhard Diamondlike
{\mathrm{BC}}_{5}
Vladimir L. Solozhenko, Oleksandr O. Kurakevych, Denis Andrault, Yann Le Godec, and Mohamed Mezouar
{\mathrm{BC}}_{5}
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A characterisation of dilation-analytic operators
E. Balslev, A. Grossmann, T. Paul (1986)
Given a domain
\Omega
{C}^{k,1}
k\in ℕ
, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of
\Omega
\left(\partial -\partial {x}_{n}\right)\alpha \left({x}^{\text{'}},0\right)=-N\left({x}^{\text{'}}\right)
and that still is of class
{C}^{k,1}
. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to
k
on domains of class
{C}^{k,1}
. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.
A commutant lifting theorem on analytic polyhedra
Calin Ambrozie, Jörg Eschmeier (2005)
In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in ℂⁿ is proved. Let T be the compression of the multiplication tuple
{M}_{z}
to a *-invariant closed subspace of the underlying functional Hilbert space. Our main result characterizes those operators in the commutant of T which possess a lifting to a multiplier with Schur class symbol. As an application we obtain interpolation results of Nevanlinna-Pick and Carathéodory-Fejér type...
A continuation theorem for holomorphic mapping into a Hilbert space
M. Skwarczyński (1970)
A dilation theorem for operators on Banach spaces
Elena Stroescu (1972)
A note on general dilation theorems
A survey on dilations of projective isometric representations.
Costache, T.L. (2008)
A Szegö type property for two doubly commuting contractions
Marek Słociński (1979)
Adjoining inverses to noncommutative Banach algebras and extensions of operators
An abstract model for compressions
Vlastimil Pták, Pavla Vrbová (1988)
An operator-theoretic approach to truncated moment problems
Raúl Curto (1997)
We survey recent developments in operator theory and moment problems, beginning with the study of quadratic hyponormality for unilateral weighted shifts, its connections with truncated Hamburger, Stieltjes, Hausdorff and Toeplitz moment problems, and the subsequent proof that polynomially hyponormal operators need not be subnormal. We present a general elementary approach to truncated moment problems in one or several real or complex variables, based on matrix positivity and extension. Together...
Antisymmetric operator algebras, II
Wacław Szymański (1980)
Boundedness in dilation theory
F. Szafraniec (1982)
Pascal Auscher, Philippe Tchamitchian (1995)
T=b\left(x\right)D\left(a\left(x\right)D\right)
a\left(x\right)
b\left(x\right)
D=-i\frac{d}{dx}
T
{\partial }_{t}^{2}u-Tu=0
ℝ×\left[0,+\infty \left[
{T}^{1/2}{D}^{-1}
\left(a,b\right)
\mathrm{exp}\left(-t{L}^{1/2}\right)
et le calcul fonctionnel permettent de développer une théorie...
Characterizations of nonnegative selfadjoint extensions.
Sandovici, Adrian (2008)
Common extensions for linear operators
Rodica-Mihaela Dăneţ (2011)
The main meaning of the common extension for two linear operators is the following: given two vector subspaces G₁ and G₂ in a vector space (respectively an ordered vector space) E, a Dedekind complete ordered vector space F and two (positive) linear operators T₁: G₁ → F, T₂: G₂ → F, when does a (positive) linear common extension L of T₁, T₂ exist? First, L will be defined on span(G₁ ∪ G₂). In other results, formulated in the line of the Hahn-Banach extension theorem, the common...
Conditionally positive definite functions on linear spaces
Contractions similar to isometries
Vladimír Kordula (1993)
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A boundary value problem with a discontinuous coefficient and containing a spectral parameter in the boundary condition.
Darwish, A.A. (1995)
A characterization of exponential stability for periodic evolution families in terms of lower semicontinuous functionals.
Buşe, Constantin (2003)
A characterization of some subsets of S-essential spectra of a multivalued linear operator
Teresa Álvarez, Aymen Ammar, Aref Jeribi (2014)
We characterize some S-essential spectra of a closed linear relation in terms of certain linear relations of semi-Fredholm type.
A characterization of the essential spectrum and applications
Aref Jeribi (2002)
In this article the essential spectrum of closed, densely defined linear operators is characterized on a large class of spaces, which possess the Dunford-Pettis property or which isomorphic to one of the spaces
{L}_{p}\left(\mathrm{\Omega }\right)
p>1
. A practical...
We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1-β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup
{\left(Tⁿ\right)}_{n=1,2,...}
by the continuous semigroup
{\left({e}^{-t\left(I-T\right)}\right)}_{t\ge 0}
. Moreover, we give a stronger quadratic form inequality which ensures that
supn\parallel Tⁿ-{T}^{n+1}\parallel :n=1,2,...<\infty
. The results apply to large classes of Markov operators on countable spaces or on locally compact groups.
A connection between spectral radius and trace
J. Długosz (1981)
A formula for the inner spectral radius.
Manjegani, S. Mahmoud (2004)
A formula to calculate the spectral radius of a compact linear operator.
Garibay Bonales, Fernando, Vera Mendoza, Rigoberto (1997)
A Generalization of the Stein-Rosenberg Theorem to Banach Spaces.
J.P. Milaszewicz (1980)
A generalization of the uniform ergodic theorem to poles of arbitrary order
Laura Burlando (1997)
A new approach to inverse spectral theory. II: General real potentials and the connection to the spectral measure.
Gesztesy, Fritz, Simon, Barry (2000)
A note on semicharacters
Z. Słodkowski, W. Żelazko (1982)
A note on spectra of weighted composition operators on weighted Banach spaces of holomorphic functions.
Wolf, Elke (2008)
A note on the
a
-Browder’s and
a
-Weyl’s theorems
M. Amouch, H. Zguitti (2008)
T
be a Banach space operator. In this paper we characterize
a
-Browder’s theorem for
T
by the localized single valued extension property. Also, we characterize
a
-Weyl’s theorem under the condition
{E}^{a}\left(T\right)={\pi }^{a}\left(T\right),
{E}^{a}\left(T\right)
is the set of all eigenvalues of
T
which are isolated in the approximate point spectrum and
{\pi }^{a}\left(T\right)
is the set of all left poles of
T.
Some applications are also given.
A note on the almost left and almost right joint spectra of R. Harte
Andrzej Sołtysiak (1989)
|
A family of M-surfaces whose automorphism groups act transitively on the mirrors.
Adnan Melekoglu (2000)
Let X be a compact Riemmann surface of genus g > 1. A symmetry T of X is an anticonformal involution. The fixed point set of T is a disjoint union of simple closed curves, each of which is called a mirror of T. If T fixes g +1 mirrors then it is called an M-symmetry and X is called an M-surface. If X admits an automorphism of order g + 1 which cyclically permutes the mirrors of T then we shall call X an M-surface with the M-property. In this paper we investigate those M-surfaces with the...
Fuertes, Yolanda, González-Diez, Gabino (2003)
A Modular Group and Riemann Surfaces of Genus 2.
Linda Keen (1975)
A natural graded Lie algebra sheaf over Riemann surfaces
Paolo Teofilatto (1991)
A note on isolated points in the branch locus of the moduli space of compact Riemann surfaces.
Bujalance, Emilio, Costa, Antonio F., Izquierdo, Milagros (1998)
A4, A5, S4 and S5 of Schottky type.
Rubén A. Hidalgo (2002)
Let H be a group of conformal automorphisms of a closed Riemann surface S, isomorphic to either of the alternating groups A4 or A5 or the symmetric groups S4 or S5. We provide necessary and sufficient conditions for the existence of a Schottky uniformization of S for which H lifts. In particular, togheter with the previous works in Hidalgo (1994,1999), we exhaust the list of finite groups of Möbius transformations of Schottky type.
Almost Abelian regular dessins d'enfants
Ruben A. Hidalgo (2013)
A regular dessin d'enfant, in this paper, will be a pair (S,β), where S is a closed Riemann surface and β: S → ℂ̂ is a regular branched cover whose branch values are contained in the set {∞,0,1}. Let Aut(S,β) be the group of automorphisms of (S,β), that is, the deck group of β. If Aut(S,β) is Abelian, then it is known that (S,β) can be defined over ℚ. We prove that, if A is an Abelian group and Aut(S,β) ≅ A ⋊ ℤ₂, then (S,β) is also definable over ℚ. Moreover, if A ≅ ℤₙ, then we provide explicitly...
An Algorithm To Compute Odd Orders and Ramification Indices of Cyclic Actions on Compact Surfaces.
E. Bujalance, J.M. Gamboa, A.F. Costa, J. Lafuente (1994)
An example of degeneration on the noded Schottky space.
In these notes we construct explicit examples of degenerations on the noded Schottky space of genus g ≥ 3. The particularity of these degenerations is the invariance under the action of a dihedral group of order 2g. More precisely, we find a two-dimensional complex manifold in the Schottky space such that all groups (including the limit ones in the noded Schottky space) admit a fixed topological action of a dihedral group of order 2g as conformal automorphisms.
Analytic Geometry on Real Algebraic Curves.
Norman L. Alling (1974)
Analytic mappings between two ultrahyperelliptic surfaces.
I.N. Baker (1976)
Analytic mappings between two ultrahyperelliptic surfaces. (Short Communication).
K{U}^{*}
R. Seroul (1972)
Hidalgo, Rubén A., Costa, Anotnio F. (2001)
|
Inverse discrete cosine transform - MATLAB idct - MathWorks India
Signal Reconstruction Using Inverse Discrete Cosine Transform
DCT Orthogonality
x = idct(y)
x = idct(y,n)
x = idct(y,n,dim)
y = idct(___,'Type',dcttype)
x = idct(y) returns the inverse discrete cosine transform of input array y. The output x has the same size as y. If y has more than one dimension, then idct operates along the first array dimension with size greater than 1.
x = idct(y,n) zero-pads or truncates the relevant dimension of y to length n before transforming.
x = idct(y,n,dim) computes the transform along dimension dim. To input a dimension and use the default value of n, specify the second argument as empty, [].
y = idct(___,'Type',dcttype) specifies the type of inverse discrete cosine transform to compute. See Inverse Discrete Cosine Transform for details. This option can be combined with any of the previous syntaxes.
Generate a signal that consists of a 25 Hz sinusoid sampled at 1000 Hz for 1 second. The sinusoid is embedded in white Gaussian noise with variance 0.01.
x = sin(2*pi*25*t) + randn(size(t))/10;
Compute the discrete cosine transform of the sequence. Determine how many of the 1000 DCT coefficients are significant. Choose 1 as the threshold for significance.
y = dct(x);
sigcoeff = abs(y) >= 1;
howmany = sum(sigcoeff)
Reconstruct the signal using only the significant components.
y(~sigcoeff) = 0;
z = idct(y);
Verify that the different variants of the discrete cosine transform are orthogonal, using a random signal as a benchmark.
Start by generating the signal.
Verify that DCT-1 and DCT-4 are their own inverses.
dct1 = dct(s,'Type',1);
idt1 = idct(s,'Type',1);
max(abs(dct1-idt1))
Verify that DCT-2 and DCT-3 are inverses of each other.
y — Input discrete cosine transform
Input discrete cosine transform, specified as a real-valued or complex-valued vector, matrix, N-D array, or gpuArray object.
Example: dct(sin(2*pi*(0:255)/4)) specifies the discrete cosine transform of a sinusoid.
Example: dct(sin(2*pi*[0.1;0.3]*(0:39))') specifies the discrete cosine transform of a two-channel sinusoid.
Inverse transform length, specified as a positive integer scalar.
dcttype — Inverse discrete cosine transform type
Inverse discrete cosine transform type, specified as a positive integer scalar from 1 to 4.
x — Inverse discrete cosine transform
Inverse discrete cosine transform, returned as a real-valued or complex-valued vector, matrix, N-D array, or gpuArray object.
The inverse discrete cosine transform reconstructs a sequence from its discrete cosine transform (DCT) coefficients. The idct function is the inverse of the dct function.
The DCT has four standard variants. For a transformed signal y of length N, and with δkℓ the Kronecker delta, the inverses are defined by:
Inverse of DCT-1:
x\left(n\right)=\sqrt{\frac{2}{N-1}}\sum _{k=1}^{N}y\left(k\right)\frac{1}{\sqrt{1+{\delta }_{k1}+{\delta }_{kN}}}\frac{1}{\sqrt{1+{\delta }_{n1}+{\delta }_{nN}}}\mathrm{cos}\left(\frac{\pi }{N-1}\left(k-1\right)\left(n-1\right)\right)
x\left(n\right)=\sqrt{\frac{2}{N}}\sum _{k=1}^{N}y\left(k\right)\frac{1}{\sqrt{1+{\delta }_{k1}}}\mathrm{cos}\left(\frac{\pi }{2N}\left(k-1\right)\left(2n-1\right)\right)
x\left(n\right)=\sqrt{\frac{2}{N}}\sum _{k=1}^{N}y\left(k\right)\frac{1}{\sqrt{1+{\delta }_{n1}}}\mathrm{cos}\left(\frac{\pi }{2N}\left(2k-1\right)\left(n-1\right)\right)
x\left(n\right)=\sqrt{\frac{2}{N}}\sum _{k=1}^{N}y\left(k\right)\mathrm{cos}\left(\frac{\pi }{4N}\left(2k-1\right)\left(2n-1\right)\right)
All variants of the DCT are unitary (or, equivalently, orthogonal): To find the forward transforms, switch k and n in each definition. DCT-1 and DCT-4 are their own inverses. DCT-2 and DCT-3 are inverses of each other.
dct | dct2 (Image Processing Toolbox) | idct2 (Image Processing Toolbox) | ifft
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Hydraulic_press Knowpia
A hydraulic press is a machine press using a hydraulic cylinder to generate a compressive force.[1] It uses the hydraulic equivalent of a mechanical lever, and was also known as a Bramah press after the inventor, Joseph Bramah, of England.[2] He invented and was issued a patent on this press in 1795. As Bramah (who is also known for his development of the flush toilet) installed toilets, he studied the existing literature on the motion of fluids and put this knowledge into the development of the press.[3]
Hydraulic force increase
Pressure of fluid due to the application force F1
{\displaystyle P={F_{1} \over A_{1}}}
Resulting force F2 on the larger cylinder due to the pressure of the fluid. With A1 and A2 being the areas of cylinder 1 and 2 respectively.
{\displaystyle F_{2}=PA_{2}=F_{1}{A_{2} \over A_{1}}}
{\displaystyle {F_{2} \over F_{1}}={A_{2} \over A_{1}}}
A small effort force acts on a small piston. This creates a pressure which is transferred through the hydraulic fluid to apply a greater force on the larger piston.[4]
Hydraulic presses are commonly used for forging, clinching, moulding, blanking, punching, deep drawing, and metal forming operations.[5][6] The hydraulic press is advantageous in manufacturing, it gives the ability to create more intricate shapes and can be economical with materials.[7] A hydraulic press will take up less space compared to a mechanical press of the same capability.[8]
In geology a tungsten carbide coated hydraulic press is used in the rock crushing stage of preparing samples for geochemical analyses in topics such as understanding the origins of volcanism.[9]
440-ton compression hydraulic molding press
The room featured in Fermat's Room has a design similar to that of a hydraulic press.[10] Boris Artzybasheff also created a drawing of a hydraulic press, in which the press was created out of the shape of a robot. In 2015, the Hydraulic Press Channel, a YouTube channel dedicated to crushing objects with a hydraulic press, was created by Lauri Vuohensilta, a factory owner from Tampere, Finland.[11] The Hydraulic Press Channel has since grown to over 3 million subscribers on YouTube.
There are numerous other YouTube channels that publish videos involving hydraulic presses that are tasked with crushing many different items, such as bowling balls, soda cans, plastic toys and metal tools.
A hydraulic press features prominently in the Sherlock Holmes story "The Adventure of the Engineer's Thumb".
^ "What is a Hydraulic Press?". XRF. 2018-02-08. Retrieved 2019-09-16.
^ Phelan, Richard M. (2014). "Hydraulic press". Access Science. doi:10.1036/1097-8542.327000.
^ Hydraulic Press Demo, archived from the original on 2021-12-22, retrieved 2019-09-16
^ Nakagawa, Takeo; Nakamura, Kazubiko; Amino, Hiroyuki (1997-11-01). "Various applications of hydraulic counter-pressure deep drawing". Journal of Materials Processing Technology. 71 (1): 160–167. doi:10.1016/S0924-0136(97)00163-5. ISSN 0924-0136.
^ "How It Works With The Hydraulic Press". www.hydraulicmania.com. Retrieved 2019-09-16.
^ Garcia, Michael O.; Swinnard, Lisa; Weis, Dominique; Greene, Andrew R.; Tagami, Taka; Sano, Hiroki; Gandy, Christian E. (2010). "Petrology, Geochemistry and Geochronology of Kaua‘i Lavas over 4.5 Myr: Implications for the Origin of Rejuvenated Volcanism and the Evolution of the Hawaiian Plume". Journal of Petrology 51(7): 1507-1540. doi: "10.1093/petrology/egq027". Retrieved 15 May 2021.
Media related to Hydraulic presses at Wikimedia Commons
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André Lichnerowicz (1979)
Takashi Tsuboi (1992)
We investigate the natural domain of definition of the Godbillon-Vey 2- dimensional cohomology class of the group of diffeomorphisms of the circle. We introduce the notion of area functionals on a space of functions on the circle, we give a sufficiently large space of functions with nontrivial area functional and we give a sufficiently large group of Lipschitz homeomorphisms of the circle where the Godbillon-Vey class is defined.
B\Gamma
Francis Sergeraert (1977/1978)
M. De Wilde, P. B. A. Lecomte (1983)
Differential equations, Spencer cohomology, and computing resolutions.
Lambe, Larry A., Seiler, Werner M. (2002)
Marc De Wilde, P. B. A. Lecomte (1985)
It is shown that if a manifold admits an exact symplectic form, then its Poisson Lie algebra has non trivial formal deformations and the manifold admits star-products. The non-formal derivations of the star-products and the deformations of the Poisson Lie algebra of an arbitrary symplectic manifold are studied.
Fibre bundles associated with fields of geometric objects and the structure tensor
J. J. Konderak (1991)
Tong Van Duc (1985)
Isomorphism of intransitive linear Lie equations.
Veloso, Jose Miguel Martins (2009)
Losik cohomology of the Lie algebra of infinitesimal automorphisms of a
G
Vojtěch Bartík, Jiří Vanžura (1985)
G
-structure. II
K. S. Sarkaria (1984)
Each Lie algebra
ℱ
of vector fields (e.g. those which are tangent to a foliation) of a smooth manifold
M
définies, in a natural way, a spectral sequence
{E}_{k}\left(ℱ\right)
which converges to the de Rham cohomology of
M
in a finite number of steps. We prove e.g. that for all
k\ge 0
there exists a foliated compact manifold with
{E}_{k}\left(ℱ\right)
infinite dimensional.
Note on two compatibility criteria: Jacobi-Mayer bracket vs. differential Gröbner basis.
Kruglikov, Boris (2006)
Yuly Billig, Karl-Hermann Neeb (2008)
In the present paper we determine for each parallelizable smooth compact manifold
M
the second cohomology spaces of the Lie algebra
{𝒱}_{M}
of smooth vector fields on
M
with values in the module
\overline{\Omega }{\phantom{\rule{0.166667em}{0ex}}}_{M}^{p}={\Omega }_{M}^{p}/d{\Omega }_{M}^{p-1}
p=1
is of particular interest since the gauge algebra of functions on
M
with values in a finite-dimensional simple Lie algebra has the universal central extension with center
{\overline{\Omega }}_{M}^{1}
, generalizing affine Kac-Moody algebras. The second cohomology
{H}^{2}\left({𝒱}_{M},{\overline{\Omega }}_{M}^{1}\right)
classifies twists of the semidirect product of
{𝒱}_{M}
with the...
On the first homology of automorphism groups of manifolds with geometric structures
Kōjun Abe, Kazuhiko Fukui (2005)
Hermann and Thurston proved that the group of diffeomorphisms with compact support of a smooth manifold M which are isotopic to the identity is a perfect group. We consider the case where M has a geometric structure. In this paper we shall survey on the recent results of the first homology of the diffeomorphism groups which preserve a smooth G-action or a foliated structure on M. We also work in Lipschitz category.
In this paper we study the real secondary classes of transversely holomorphic foliations. We define a homomorphism from the space
{H}^{*}\left({\mathrm{WO}}_{2q}\right)
of the real secondary classes to the space
{H}^{*}\left({\mathrm{WU}}_{q}\right)
of the complex secondary classes that corresponds to forgetting the transverse holomorphic structure. By using this homomorphism we show, for example, the decomposition of the Godbillon-Vey class into the imaginary part of the Bott class and the first Chern class of the complex normal bundle of the foliation. We show also...
Katsuyuki Shibata (1980)
Claude Roger (1980)
Nous démontrons la finitude de la cohomologie de l’algèbre de Lie des champs de vecteurs formels à
2n+1
variables, respectant la forme de contact universelle
w=d{x}_{0}+{\sum }_{i=1}^{n}{x}_{i}d{\stackrel{‾}{x}}_{i}
Paul Ver Eecke (1984)
Translation of natural operators on manifolds with AHS-structures
Andreas Čap (1996)
We introduce an explicit procedure to generate natural operators on manifolds with almost Hermitian symmetric structures and work out several examples of this procedure in the case of almost Grassmannian structures.
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16W22 Actions of groups and semigroups; invariant theory
16W50 Graded rings and modules
16W55 ``Super'' (or ``skew'') structure
16W60 Valuations, completions, formal power series and related constructions
16W70 Filtered rings; filtrational and graded techniques
A characterization of the norm of an Azumaya algebra of constant rank through the divided powers algebra of an algebra
Dieter Ziplies (1986)
1/2\left(\phi \left(a\right)+{\phi }^{-1}\left(a\right)\right)=a
A note on commutativity of automorphisms.
Samman, M.S., Chaudhry, M.Anwar, Thaheem, A.B. (1998)
À propos des théories de Galois finies et infinies
R. Moors (1974)
Abbildungen auf Ringen insbesondere mit Involution
Walter Streb (1984)
An extension of Zassenhaus' theorem on endomorphism rings
Let R be a ring with identity such that R⁺, the additive group of R, is torsion-free. If there is some R-module M such that
R\subseteq M\subseteq ℚR\left(=ℚ{\otimes }_{ℤ}R\right)
En{d}_{ℤ}\left(M\right)=R
, we call R a Zassenhaus ring. Hans Zassenhaus showed in 1967 that whenever R⁺ is free of finite rank, then R is a Zassenhaus ring. We will show that if R⁺ is free of countable rank and each element of R is algebraic over ℚ, then R is a Zassenhaus ring. We will give an example showing that this restriction on R is needed. Moreover, we will show that a ring due to A....
An identity on partial generalized automorphisms of prime rings
Shuliang Huang (2013)
An identity related to centralizers in semiprime rings
The purpose of this paper is to prove the following result: Let
R
2
-torsion free semiprime ring and let
T:R\to R
be an additive mapping, such that
2T\left({x}^{2}\right)=T\left(x\right)x+xT\left(x\right)
x\in R
T
is left and right centralizer.
Anti-Isomorphisms of Endomorphism Rings of Locally Free Modules.
Kenneth G. Wolfson (1989)
Manfred Dugas, Shalom Feigelstock (2003)
A ring R is called an E-ring if every endomorphism of R⁺, the additive group of R, is multiplication on the left by an element of R. This is a well known notion in the theory of abelian groups. We want to change the "E" as in endomorphisms to an "A" as in automorphisms: We define a ring to be an A-ring if every automorphism of R⁺ is multiplication on the left by some element of R. We show that many torsion-free finite rank (tffr) A-rings are actually E-rings. While we have an example of a mixed...
Associated prime ideals of skew polynomial rings.
Bhat, V.K. (2008)
Automorphism group of representation ring of the weak Hopf algebra
\stackrel{˜}{{H}_{8}}
Dong Su, Shilin Yang (2018)
{H}_{8}
be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra
\stackrel{˜}{{H}_{8}}
{H}_{8}
, then we investigate the structure of the representation ring of
\stackrel{˜}{{H}_{8}}
. Finally, we prove that the automorphism group of
r\left(\stackrel{˜}{{H}_{8}}\right)
is just isomorphic to
{D}_{6}
{D}_{6}
is the dihedral group with order 12.
Automorphismen und Antiautomorphismen von Tensoralgebren.
Heinz-Georg Quebbemann (1978)
Automorphismes de certains completés du corps de Weyl quantique.
J. Alev, F. Dumas (1995)
Automorphisms and f-simplicity in skew polynomial rings
Michael G. Voskoglou (1996)
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Bernt Oksendal, Daniel W. Stroock (1982)
The exit distribution for open sets of a path-continuous, strong Markov process in
{\mathbf{R}}^{n}
is characterized as a weak star limit of successive spherical sweepings of measures, starting with the unit point mass. Then this is used to prove that two path-continuous strong Markov processes with identical exit distributions from balls when starting form the center, have identical exit distributions from all opens sets, provided they both exit a.s. from bounded sets. This implies that the only path-continuous,...
A Construction of Representing Measures for Elliptic and Parabolic Differential Equations.
Peter A. Loeb (1982)
A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains.
Peter Stollmann (1995)
A family of L 2-spaces associated to the jumps of a Markov process
Valentin Grecea (2011)
Given the (canonical) Markov process associated with a sufficiently general semigroup (P t), we establish a result concerning the uniform completeness of a family of L 2-spaces naturally associated with the jumps of the process. An application of this result is presented.
A fine domination principle for excessive measures.
P.J. Fitzsimmons, R.K. Getoor (1991)
Angel, Omer (2000)
R. R. London, H. P. Mc Kean, L. C. G. Rogers, David Williams (1982)
R. R. London, Henry P. Mc Kean, L. C. G. Rogers, David Williams (1982)
Nicole El Karoui, Hans Föllmer (2005)
A Note on Quasicontinuous Kernels Representing Quasi-Linear Positive Maps.
Sergio Albeverio, Zhi-Ming Ma (1991)
A note on Revuz measure
Murali Rao (1980)
A note on the density of the parabolic area integral.
Ileana Iribarren (2001)
The density of the area integral for parabolic functions is defined in analogy with the case of harmonic functions. We prove its equivalence with the local time of the associated martingale. Using probabilistic methods, we show its equivalence in L p -norm with the parabolic area function for p>1.
{L}_{\infty }
K. A. Astbury (1976)
A potential-theoretic note on the quadratic Wiener-Hopf equation for Q-matrices
A probabilistic approach to the trace at the boundary for solutions of a semilinear parabolic partial differential equation.
Le Gall, Jean-François (1996)
A recurrence condition for some subordinated strongly local Dirichlet forms.
Ivor McGillivray (1997)
A representation for non-colliding random walks.
O'Connell, Neil, Yor, Marc (2002)
A short proof of the Hausdorff dimension formula for Lévy processes.
Yang, Ming (2006)
|
p
A gauge theoretical approach to space-time structures
Folkert Müller-Hoissen (1984)
Monique Combescure (1992)
A recurrence relation approach to higher order quantum superintegrability.
Kalnins, Ernie G., Kress, Jonathan M., Miller, Willard jun. (2011)
A representation independent propagator I : compact Lie groups
Wolfgang A. Tomé (1995)
Wolfgang Tomé (1996)
A theory of relativistic unstable particles
R. Raczka (1973)
A unified model of phantom energy and dark matter.
Chaves, Max, Singleton, Douglas (2008)
Affine Poisson groups and WZW model.
Klimc̆ík, Ctirad (2008)
Jean-Louis Verdier (1980/1981)
Algèbres et faisceaux d'algèbres de Lie graduées, associées à des espaces spinoriels et fibrations spinorielles par un principe de trialité
A. Crumeyrolle (1982)
All linear representations of the Poincaré group up to dimension 8
Stephen M. Paneitz (1984)
An Operational Calculus for the Euclidean Motion Group with Applications in Robotics and Polymer Science.
G.S. Chirikjian, A.B. Kyatkin (2000)
An optimal control problem on the Heisenberg Lie group
H\left(3\right)
Pop, Camelia (1997)
Application of the Gel'fand matrix method to the missing label problem in classical kinematical Lie algebras.
Campoamor-Stursberg, Rutwig (2006)
A. Huckleberry, H. Sebert (2013)
|{\varphi }_{n}{|}^{2}=|{s}_{N}{|}^{2}/||{s}_{N}{||}_{{L}^{2}}^{2}
{s}_{N}\in \Gamma \left(X,{L}^{N}\right)
X
L
is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate and Zelditch....
Automorphism Groups of Classical Mechanical Systems.
Berezin-Weyl quantization for Cartan motion groups
Benjamin Cahen (2011)
We construct adapted Weyl correspondences for the unitary irreducible representations of the Cartan motion group of a noncompact semisimple Lie group by using the method introduced in [B. Cahen, Weyl quantization for semidirect products, Differential Geom. Appl. 25 (2007), 177--190].
Class Operators for Compact Lie Groups
N. B. Backhouse, J. Rembielinski, W. Tybor (1996)
6j
-symbols and the tetrahedron.
Roberts, Justin (1999)
|
q
A Generalization of Pachpatte Difference Inequalities
Stevo Stevic (2000)
A global description of the positive solutions of sublinear second-order discrete boundary value problems.
Ma, Ruyun, Xu, Youji, Gao, Chenghua (2009)
A KAM theorem for infinite-dimensional discrete systems.
Perfetti, Paolo (2003)
A note on asymptotic stability of delay difference systems.
Matsunaga, Hideaki (2005)
A note on the global attractivity of a discrete model of Nicholson's blowflies.
Zhang, B.G., Xu, H.X. (1999)
A positive solution for singular discrete boundary value problems with sign-changing nonlinearities.
Lü, Haishen, O'Regan, Donal, Agarwal, Ravi P. (2006)
The aim of this contribution is to study the role of the coefficient
r
in the qualitative theory of the equation
{\left(r\left(t\right)\Phi \left({y}^{\Delta }\right)\right)}^{\Delta }+p\left(t\right)\Phi \left({y}^{\sigma }\right)=0
\Phi \left(u\right)={|u|}^{\alpha -1}\mathrm{sgn}u
\alpha >1
. We discuss sign and smoothness conditions posed on
r
, (non)availability of some transformations, and mainly we show how the behavior of
r
, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati...
A sharpening of discrete analogues of Wirtinger's inequality
A simple system of discrete two-scale difference equations.
Berg, L., Krüppel, M. (2000)
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Functional dependency - Wikipedia
This article is about a concept in relational database theory. For function dependencies in the Haskell programming language, see type class.
Find sources: "Functional dependency" – news · newspapers · books · scholar · JSTOR (October 2012) (Learn how and when to remove this template message)
In relational database theory, a functional dependency is a constraint between two sets of attributes in a relation from a database. In other words, a functional dependency is a constraint between two attributes in a relation. Given a relation R and sets of attributes
{\displaystyle X,Y\subseteq R}
, X is said to functionally determine Y (written X → Y) if and only if each X value in R is associated with precisely one Y value in R; R is then said to satisfy the functional dependency X → Y. Equivalently, the projection
{\displaystyle \Pi _{X,Y}R}
In other words, a dependency FD: X → Y means that the values of Y are determined by the values of X. Two tuples sharing the same values of X will necessarily have the same values of Y.
The determination of functional dependencies is an important part of designing databases in the relational model, and in database normalization and denormalization. A simple application of functional dependencies is Heath's theorem; it says that a relation R over an attribute set U and satisfying a functional dependency X → Y can be safely split in two relations having the lossless-join decomposition property, namely into
{\displaystyle \Pi _{XY}(R)\bowtie \Pi _{XZ}(R)=R}
{\displaystyle \Sigma }
{\displaystyle \Gamma }
{\displaystyle \Sigma }
{\displaystyle \Gamma }
{\displaystyle \Sigma \models \Gamma }
1.3 Employee department model
4.2 Equivalence of two sets of FDs
Suppose one is designing a system to track vehicles and the capacity of their engines. Each vehicle has a unique vehicle identification number (VIN). One would write VIN → EngineCapacity because it would be inappropriate for a vehicle's engine to have more than one capacity. (Assuming, in this case, that vehicles only have one engine.) On the other hand, EngineCapacity → VIN is incorrect because there could be many vehicles with the same engine capacity.
1234 6 Numerical Methods John
1221 4 Numerical Methods Smith
1234 6 Visual Computing Bob
1201 2 Physics II Simon
Note that if a row was added where the student had a different value of semester, then the functional dependency FD would no longer exist. This means that the FD is implied by the data as it is possible to have values that would invalidate the FD.
Employee department model[edit]
A classic example of functional dependency is the employee department model.
0001 John Doe 1 Human Resources
0002 Jane Doe 2 Marketing
0003 John Smith 1 Human Resources
0004 Jane Goodall 3 Sales
This case represents an example where multiple functional dependencies are embedded in a single representation of data. Note that because an employee can only be a member of one department, the unique ID of that employee determines the department.
Employee ID → Employee Name
Employee ID → Department ID
In addition to this relationship, the table also has a functional dependency through a non-key attribute
Department ID → Department Name
This example demonstrates that even though there exists a FD Employee ID → Department ID - the employee ID would not be a logical key for determination of the department ID. The process of normalization of the data would recognize all FDs and allow the designer to construct tables and relationships that are more logical based on the data.
Properties and axiomatization of functional dependencies[edit]
Main article: Armstrong's axioms
{\displaystyle X\rightarrow \varnothing }
{\displaystyle \vdash X\rightarrow \varnothing }
{\displaystyle X\rightarrow Y\vdash XZ\rightarrow YZ}
{\displaystyle X\rightarrow Y,Y\rightarrow Z\vdash X\rightarrow Z}
By applying augmentation and transitivity, one can derive two additional rules:
Pseudotransitivity: If X → Y and YW → Z, then XW → Z[3]
Composition: If X → Y and Z → W, then XZ → YW[6]
One can also derive the union and decomposition rules from Armstrong's axioms:[3][7]
X → Y and X → Z if and only if X → YZ
Closure of functional dependency[edit]
The closure is essentially the full set of values that can be determined from a set of known values for a given relationship using its functional dependencies. One uses Armstrong's axioms to provide a proof - i.e. reflexivity, augmentation, transitivity.
{\displaystyle R}
{\displaystyle F}
{\displaystyle R}
{\displaystyle F}
{\displaystyle R}
{\displaystyle F}
+) is the set of all FDs that are logically implied by
{\displaystyle F}
Closure of a set of attributes[edit]
{\displaystyle F}
{\displaystyle F}
Imagine the following list of FDs. We are going to calculate a closure for A from this relationship.
a) A → A (by Armstrong's reflexivity)
b) A → AB (by 1. and (a))
c) A → ABD (by (b), 3, and Armstrong's transitivity)
d) A → ABCD (by (c), and 2)
Covers and equivalence[edit]
{\displaystyle F}
{\displaystyle G}
{\displaystyle G}
{\displaystyle F}
{\displaystyle F}
{\displaystyle G}
{\displaystyle G}
{\displaystyle F}
Equivalence of two sets of FDs[edit]
{\displaystyle F}
{\displaystyle G}
{\displaystyle R}
{\displaystyle F}
{\displaystyle G}
{\displaystyle F}
{\displaystyle G}
{\displaystyle F}
{\displaystyle G}
{\displaystyle F}
{\displaystyle G}
Non-redundant covers[edit]
{\displaystyle F}
{\displaystyle F'}
{\displaystyle F}
{\displaystyle F'}
{\displaystyle F}
{\displaystyle F'}
{\displaystyle F}
{\displaystyle F}
{\displaystyle G}
{\displaystyle F}
{\displaystyle G}
{\displaystyle F}
{\displaystyle F}
{\displaystyle F}
{\displaystyle F}
{\displaystyle \models }
{\displaystyle F}
redundant in
{\displaystyle F}
{\displaystyle F}
{\displaystyle \models }
X → Y.
Applications to normalization[edit]
Heath's theorem[edit]
{\displaystyle R=\Pi _{XY}(R)\bowtie \Pi _{XZ}(R)}
{\displaystyle \Pi _{XY}(R)\bowtie \Pi _{XZ}(R)}
) ensuring that when the two parts are joined back no data is lost, i.e. a functional dependency provides a simple way to construct a lossless join decomposition of R in two smaller relations. This fact is sometimes called Heaths theorem; it is one of the early results in database theory.[9]
Heath's theorem effectively says we can pull out the values of Y from the big relation R and store them into one,
{\displaystyle \Pi _{XY}(R)}
, which has no value repetitions in the row for X and is effectively a lookup table for Y keyed by X and consequently has only one place to update the Y corresponding to each X unlike the "big" relation R where there are potentially many copies of each X, each one with its copy of Y which need to be kept synchronized on updates. (This elimination of redundancy is an advantage in OLTP contexts, where many changes are expected, but not so much in OLAP contexts, which involve mostly queries.) Heath's decomposition leaves only X to act as a foreign key in the remainder of the big table
{\displaystyle \Pi _{XZ}(R)}
Functional dependencies however should not be confused with inclusion dependencies, which are the formalism for foreign keys; even though they are used for normalization, functional dependencies express constraints over one relation (schema), whereas inclusion dependencies express constraints between relation schemas in a database schema. Furthermore, the two notions do not even intersect in the classification of dependencies: functional dependencies are equality-generating dependencies whereas inclusion dependencies are tuple-generating dependencies. Enforcing referential constraints after relation schema decomposition (normalization) requires a new formalism, i.e. inclusion dependencies. In the decomposition resulting from Heath's theorem, there is nothing preventing the insertion of tuples in
{\displaystyle \Pi _{XZ}(R)}
{\displaystyle \Pi _{XY}(R)}
Normal forms[edit]
Normal forms are database normalization levels which determine the "goodness" of a table. Generally, the third normal form is considered to be a "good" standard for a relational database.[citation needed]
Normalization aims to free the database from update, insertion and deletion anomalies. It also ensures that when a new value is introduced into the relation, it has minimal effect on the database, and thus minimal effect on the applications using the database.[citation needed]
Irreducible function depending set[edit]
A set S of functional dependencies is irreducible if the set has the following three properties:
Sets of functional dependencies with these properties are also called canonical or minimal. Finding such a set S of functional dependencies which is equivalent to some input set S' provided as input is called finding a minimal cover of S': this problem can be solved in polynomial time.[10]
^ Terry Halpin (2008). Information Modeling and Relational Databases (2nd ed.). Morgan Kaufmann. p. 140. ISBN 978-0-12-373568-3.
^ Chris Date (2012). Database Design and Relational Theory: Normal Forms and All That Jazz. O'Reilly Media, Inc. p. 21. ISBN 978-1-4493-2801-6.
^ a b c Abraham Silberschatz; Henry Korth; S. Sudarshan (2010). Database System Concepts (6th ed.). McGraw-Hill. p. 339. ISBN 978-0-07-352332-3.
^ a b M. Y. Vardi. Fundamentals of dependency theory. In E. Borger, editor, Trends in Theoretical Computer Science, pages 171–224. Computer Science Press, Rockville, MD, 1987. ISBN 0881750840
^ Abiteboul, Serge; Hull, Richard B.; Vianu, Victor (1995), Foundations of Databases, Addison-Wesley, pp. 164–168, ISBN 0-201-53771-0
^ S. K. Singh (2009) [2006]. Database Systems: Concepts, Design & Applications. Pearson Education India. p. 323. ISBN 978-81-7758-567-4.
^ Hector Garcia-Molina; Jeffrey D. Ullman; Jennifer Widom (2009). Database systems: the complete book (2nd ed.). Pearson Prentice Hall. p. 73. ISBN 978-0-13-187325-4. This is sometimes called the splitting/combining rule.
^ Saiedian, H. (1996-02-01). "An Efficient Algorithm to Compute the Candidate Keys of a Relational Database Schema". The Computer Journal. 39 (2): 124–132. doi:10.1093/comjnl/39.2.124. ISSN 0010-4620.
^ Heath, I. J. (1971). "Unacceptable file operations in a relational data base". Proceedings of the 1971 ACM SIGFIDET (now SIGMOD) Workshop on Data Description, Access and Control - SIGFIDET '71. pp. 19–33. doi:10.1145/1734714.1734717. cited in:
Ronald Fagin and Moshe Y. Vardi (1986). "The Theory of Data Dependencies - A Survey". In Michael Anshel and William Gewirtz (ed.). Mathematics of Information Processing: [short Course Held in Louisville, Kentucky, January 23-24, 1984]. American Mathematical Soc. p. 23. ISBN 978-0-8218-0086-7.
C. Date (2005). Database in Depth: Relational Theory for Practitioners. O'Reilly Media, Inc. p. 142. ISBN 978-0-596-10012-4.
^ Meier, Daniel (1980). "Minimum covers in the relational database model". Journal of the ACM. doi:10.1145/322217.322223.
Codd, E. F. (1972). "Further Normalization of the Data Base Relational Model" (PDF). ACM Transactions on Database Systems. San Jose, California: Association for Computing Machinery.
Gary Burt (Summer 1999). "CS 461 (Database Management Systems) lecture notes". University of Maryland Baltimore County Department of Computer Science and Electrical Engineering.
Jeffrey D. Ullman. "CS345 Lecture Notes" (PostScript). Stanford University.
Osmar Zaiane (June 9, 1998). "Chapter 6: Integrity constraints". CMPT 354 (Database Systems I) lecture notes. Simon Fraser University Department of Computing Science.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Functional_dependency&oldid=1083809006"
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40 CFR § 63.765 - Glycol dehydration unit process vent standards. | CFR | US Law | LII / Legal Information Institute
40 CFR § 63.765 - Glycol dehydration unit process vent standards.
(a) This section applies to each glycol dehydration unit subject to this subpart that must be controlled for air emissions as specified in either paragraph (c)(1)(i) or paragraph (d)(1)(i) of § 63.764.
(b) Except as provided in paragraph (c) of this section, an owner or operator of a glycol dehydration unit process vent shall comply with the requirements specified in paragraphs (b)(1) and (b)(2) of this section.
(1) For each glycol dehydration unit process vent, the owner or operator shall control air emissions by either paragraph (b)(1)(i), (ii), or (iii) of this section.
(i) The owner or operator of a large glycol dehydration unit, as defined in § 63.761, shall connect the process vent to a control device or a combination of control devices through a closed-vent system. The closed-vent system shall be designed and operated in accordance with the requirements of § 63.771(c). The control device(s) shall be designed and operated in accordance with the requirements of § 63.771(d).
(ii) The owner or operator of a large glycol dehydration unit shall connect the process vent to a control device or combination of control devices through a closed-vent system and the outlet benzene emissions from the control device(s) shall be reduced to a level less than 0.90 megagrams per year. The closed-vent system shall be designed and operated in accordance with the requirements of § 63.771(c). The control device(s) shall be designed and operated in accordance with the requirements of § 63.771(d), except that the performance levels specified in § 63.771(d)(1)(i) and (ii) do not apply.
(iii) You must limit BTEX emissions from each existing small glycol dehydration unit process vent, as defined in § 63.761, to the limit determined in Equation 1 of this section. You must limit BTEX emissions from each new small glycol dehydration unit process vent, as defined in § 63.761, to the limit determined in Equation 2 of this section. The limits determined using Equation 1 or Equation 2 must be met in accordance with one of the alternatives specified in paragraphs (b)(1)(iii)(A) through (D) of this section.
{\mathrm{EL}}_{\mathrm{BTEX}}=3.28\phantom{\rule{0ex}{0ex}}x{\phantom{\rule{0ex}{0ex}}10}^{-4}*\text{Throughput}*{C}_{i,\mathrm{BTEX}}*365\phantom{\rule{0ex}{0ex}}\frac{\mathrm{days}}{\mathrm{yr}}*\frac{1\mathrm{Mg}}{1x{10}^{4}\text{grams}}
ELBTEX = Unit-specific BTEX emission limit, megagrams per year;
3.28 × 10−4 = BTEX emission limit, grams BTEX/standard cubic meter-ppmv;
Throughput = Annual average daily natural gas throughput, standard cubic meters per day.
Ci,BTEX = average annual BTEX concentration of the natural gas at the inlet to the glycol dehydration unit, ppmv.
{\mathrm{EL}}_{\mathrm{BTEX}}=4.66\phantom{\rule{0ex}{0ex}}x{\phantom{\rule{0ex}{0ex}}10}^{-4}*\text{Throughput}*{C}_{i,\mathrm{BTEX}}*365\phantom{\rule{0ex}{0ex}}\frac{\mathrm{days}}{\mathrm{yr}}*\frac{1\mathrm{Mg}}{1x{10}^{4}\text{grams}}
(A) Connect the process vent to a control device or combination of control devices through a closed-vent system. The closed vent system shall be designed and operated in accordance with the requirements of § 63.771(c). The control device(s) shall be designed and operated in accordance with the requirements of § 63.771(f).
(B) Meet the emissions limit through process modifications in accordance with the requirements specified in § 63.771(e).
(C) Meet the emissions limit for each small glycol dehydration unit using a combination of process modifications and one or more control devices through the requirements specified in paragraphs (b)(1)(iii)(A) and (B) of this section.
(D) Demonstrate that the emissions limit is met through actual uncontrolled operation of the small glycol dehydration unit. Document operational parameters in accordance with the requirements specified in § 63.771(e) and emissions in accordance with the requirements specified in § 63.772(b)(2).
(2) One or more safety devices that vent directly to the atmosphere may be used on the air emission control equipment installed to comply with paragraph (b)(1) of this section.
(c) As an alternative to the requirements of paragraph (b) of this section, the owner or operator may comply with one of the requirements specified in paragraphs (c)(1) through (3) of this section.
(1) The owner or operator shall control air emissions by connecting the process vent to a process natural gas line.
(2) The owner or operator shall demonstrate, to the Administrator's satisfaction, that the total HAP emissions to the atmosphere from the large glycol dehydration unit process vent are reduced by 95.0 percent through process modifications, or a combination of process modifications and one or more control devices, in accordance with the requirements specified in § 63.771(e).
(3) Control of HAP emissions from a GCG separator (flash tank) vent is not required if the owner or operator demonstrates, to the Administrator's satisfaction, that total emissions to the atmosphere from the glycol dehydration unit process vent are reduced by one of the levels specified in paragraph (c)(3)(i) through (iv) of this section, through the installation and operation of controls as specified in paragraph (b)(1) of this section.
(i) For any large glycol dehydration unit, HAP emissions are reduced by 95.0 percent or more.
(ii) For any large glycol dehydration unit, benzene emissions are reduced to a level less than 0.90 megagrams per year.
(iii) For each existing small glycol dehydration unit, BTEX emissions are reduced to a level less than the limit calculated by Equation 1 of paragraph (b)(1)(iii) of this section.
(iv) For each new small glycol dehydration unit, BTEX emissions are reduced to a level less than the limit calculated by Equation 2 of paragraph (b)(1)(iii) of this section.
[64 FR 32628, June 17, 1999, as amended at 66 FR 34551, June 29, 2001; 72 FR 38, Jan. 3, 2007; 77 FR 49570, Aug. 16, 2012]
|
Sign of real or complex value - MATLAB sign - MathWorks Australia
Signs of Real Numbers
Signs of Matrix Elements
Sign of Symbolic Expression
Sign of real or complex value
sign(z) returns the sign of real or complex value z. The sign of a complex number z is defined as z/abs(z). If z is a vector or a matrix, sign(z) returns the sign of each element of z.
Find the signs of these symbolic real numbers:
[sign(sym(1/2)), sign(sym(0)), sign(sym(pi) - 4)]
Find the signs of the real and complex elements of matrix A:
[ 5^(1/2)*(1/5 + 2i/5), -1]
[ 2^(1/2)*(- 1/2 + 1i/2), 5^(1/2)*18^(1/2)*(1/30 - 1i/10)]
Find the sign of this expression assuming that the value x is negative:
sign(5*x^3)
Input specified as a symbolic number, variable, expression, function, vector, or matrix.
The sign function of any number z is defined via the absolute value of z:
\mathrm{sign}\left(z\right)=\frac{z}{|z|}
Thus, the sign function of a real number z can be defined as follows:
\mathrm{sign}\left(z\right)=\left\{\begin{array}{c}-1\text{ if }x<0\\ \text{ }0\text{ if }x=0\\ \text{ }1\text{ if }x>0\end{array}
Calling sign for a number that is not a symbolic object invokes the MATLAB® sign function.
abs | angle | imag | real | signIm
|
65N06 Finite difference methods
65N08 Finite volume methods
65N12 Stability and convergence of numerical methods
65N20 Ill-posed problems
65N25 Eigenvalue problems
65N35 Spectral, collocation and related methods
65N38 Boundary element methods
65N40 Method of lines
65N50 Mesh generation and refinement
65N75 Probabilistic methods, particle methods, etc.
65N80 Fundamental solutions, Green's function methods, etc.
65N85 Fictitious domain methods
𝐴
\mathrm{𝑃𝑂𝑆𝑇𝐸𝑅𝐼𝑂𝑅𝐼}
\mathrm{𝑉𝐼𝐴}
We give details of the theory of primal domain decomposition (DD) methods for a 2-dimensional second order elliptic equation with homogeneous Dirichlet boundary conditions and jumping coefficients. The problem is discretized by the finite element method. The computational domain is decomposed into triangular subdomains that align with the coefficients jumps. We prove that the condition number of the vertex-based DD preconditioner is
O\left({\left(1+log\left(H/h\right)\right)}^{2}\right)
, independently of the coefficient jumps, where
H
h
denote...
{ℝ}^{3}
A boundary multivalued integral “equation” approach to the semipermeability problem
Jaroslav Haslinger, Charalambos C. Baniotopoulos, Panagiotis D. Panagiotopoulos (1993)
The present paper concerns the problem of the flow through a semipermeable membrane of infinite thickness. The semipermeability boundary conditions are first considered to be monotone; these relations are therefore derived by convex superpotentials being in general nondifferentiable and nonfinite, and lead via a suitable application of the saddlepoint technique to the formulation of a multivalued boundary integral equation. The latter is equivalent to a boundary minimization problem with a small...
Shangyou Zhang (2008)
A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic...
A class of nonparametric DSSY nonconforming quadrilateral elements
Youngmok Jeon, Hyun NAM, Dongwoo Sheen, Kwangshin Shim (2013)
A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747–770.] The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence [Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Calcolo...
Segeth, Karel (2010)
A lot of papers and books analyze analytical a posteriori error estimates from the point of view of robustness, guaranteed upper bounds, global efficiency, etc. At the same time, adaptive finite element methods have acquired the principal position among algorithms for solving differential problems in many physical and technical applications. In this survey contribution, we present and compare, from the viewpoint of adaptive computation, several recently published error estimation procedures for...
A comparison of two FEM-based methods for the solution of the nonlinear output regulation problem
Branislav Rehák, Sergej Čelikovský, Javier Ruiz, Jorge Orozco-Mora (2009)
The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations (PDE) combined with an algebraic equation. The classical approach to its solution is to use the Taylor series with undetermined coefficients. In this contribution, another path is followed: the equation is solved using the finite-element method which is, nevertheless, suitable to solve PDE part only. This paper...
A computational domain decomposition approach for solving coupled flow-structure-thermal interaction problems.
Aulisa, Eugenio, Manservisi, Sandro, Seshaiyer, Padmanabhan (2009)
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Quantize fixed-point numbers - MATLAB - MathWorks Nordic
fixed.Quantizer is not recommended
fixed.Quantizer is not recommended. Use cast, zeros, ones, eye, or subsasgn instead. For more information, see Compatibility Considerations.
The fixed.Quantizer object describes data type properties to use for quantization. After you create a fixed.Quantizer object, use quantize to quantize fi values.
q = fixed.Quantizer(nt,rm,oa)
q = fixed.Quantizer(s,wl,fl,rm,oa)
q = fixed.Quantizer(Name,Value)
q = fixed.Quantizer creates a quantizer object q that quantizes fixed-point numbers using the fixed-point settings of q.
q = fixed.Quantizer(nt,rm,oa) creates a fixed-point quantizer object with numerictype nt, rounding method rm, and overflow action oa.
The numerictype, rounding method, and overflow action apply only during the quantization. The output q does not have an attached fimath.
q = fixed.Quantizer(s,wl,fl,rm,oa) creates a binary-point scaled fixed-point quantizer object with signedness s, word length wl, fraction length fl, rounding method rm, and overflow action oa.
q = fixed.Quantizer(Name,Value) creates a quantizer object with the property options specified by one or more property Name,Value arguments.
numerictype object that describes a binary-point scaled or a slope-bias scaled fixed-point data type, specified as a numerictype object.
If fixed.Quantizer uses a numerictype object that has either a Signedness of Auto or unspecified Scaling, an error occurs.
rm — Rounding method
When the values of data to be quantized lie outside the range of the largest and smallest representable numbers as specified by the numeric type properties, these values are quantized to the value of either the largest or smallest representable value, depending on which is closest.
When the values of data to be quantized lie outside the range of the largest and smallest representable numbers as specified by the numeric type properties, these values are wrapped back into that range using modular arithmetic relative to the smallest representable number.
s — Whether output is signed
Whether output is signed, specified as one of the following:
1 or true — Signed
0 or false — Unsigned
Word length of the stored integer value of the output data in bits, specified as a positive scalar integer.
Fraction length of the stored integer value of the output data in bits, specified as a scalar integer.
Bias associated with the quantizer object, specified as a scalar integer.
The bias is a part of the numerical representation used to interpret a fixed-point number. Along with the slope, the bias forms the scaling of the number. For more information, see Fixed-point numbers.
-15 (default) | scalar integer
Fixed-point exponent associated with the quantizer object, specified as a scalar integer. The exponent is part of the numerical representation used to interpret a fixed-point number. The exponent of a fixed-point number is equal to the negative of the fraction length. For more information, see Fixed-point numbers.
Fraction length of the stored integer value of the object, in bits, specified as a scalar integer.
The fraction length automatically defaults to the best precision possible based on the value of the word length and the real-world value of the fi object being quantized.
Signed — Whether output is signed
Although the Signed property is still supported, the Signedness property always appears in the fixed.Quantizer object display. If you choose to change or set the signedness of your fixed.Quantizer object using the Signed property, MATLAB® updates the corresponding value of the Signedness property.
Signedness — Whether output is signed
'Signed' (default) | 'Unsigned'
Whether output is signed, specified as 'Signed' or 'Unsigned'.
Slope — Slope associated with object
2^-15 (default) | positive scalar
Slope associated with the object. The slope is part of the numerical representation used to express a fixed-point number. Along with the bias, the slope forms the scaling of a fixed-point number. For more information, see Fixed-point numbers.
SlopeAdjustmentFactor — Slope adjustment associated with object
1 (default) | scalar greater than or equal to 1 and less than 2
Slope adjustment associated with the object, specified as a scalar greater than or equal to 1 and less than 2. The slope adjustment is equivalent to the fractional slope of a fixed-point number. The fractional slope is part of the numerical representation used to express a fixed-point number. For more information, see Fixed-point numbers.
Word length of the stored integer value of the output data, in bits, specified as a positive scalar integer.
real\text{-}worldvalue=\left(slope×storedinteger\right)+bias
slope=fractionalslope×{2}^{fixedexponent}
Use y = quantize(q,x) to quantize input array x using the fixed-point settings of the quantizer object q. x can be any fixed-point fi number, except a Boolean value. If x is a scaled double, the x and y data will be the same, but y will have fixed-point settings. If x is a double or single, then y = x. This functionality lets you share the same code for both floating-point data types and fi objects when quantizers are present.
Use n = numerictype(q) to get a numerictype for the current settings of the quantizer object q.
Use clone(q) to create a quantizer object with the same property values as q.
fixed.Quantizer is a handle object and must be declared as persistent in code generation.
R2013a: fixed.Quantizer is not recommended
fixed.Quantizer is not recommended. Use cast, zeros, ones, eye, or subsasgn instead. There are no plans to remove fixed.Quantizer.
quantize | fi | numerictype
|
Home : Support : Online Help : Connectivity : Maple T.A. : MapleTA Package : Builtin : rand
generate random real numbers
rand(m, n)
rand(m, n, k)
The rand command returns a random real number between m and n (inclusive). For example, to generate random numbers between 0.5 and 9.5, use the formula: rand(0.5, 9.5)
Using the three-argument form, rand returns a random real number between m and n expressed to k significant digits. For example, to generate numbers between 2.73 and 7.91 to two significant figures, use: rand(2.73, 7.91, 2).
Note: In this definition, the dummy variable k represents the number of significant figures, not the step size (as it indicates in defining range and rint). To generate random numbers that vary an order of magnitude and have the same number of decimal digits, use the decimal command in conjunction with the rand command. For example, to generate a random number between 2 and 20 with one decimal place, use: decimal(1, rand(2, 20, 3) )
\mathrm{MapleTA}:-\mathrm{Builtin}:-\mathrm{rand}\left(0.5,9.5\right)
\textcolor[rgb]{0,0,1}{6.871587673}
\mathrm{MapleTA}:-\mathrm{Builtin}:-\mathrm{rand}\left(2.73,7.91,2\right)
\textcolor[rgb]{0,0,1}{7.1}
The MapleTA[Builtin][rand] command was introduced in Maple 18.
MapleTA,Builtin,decimal
MapleTA,Builtin,int
MapleTA,Builtin,range
MapleTA,Builtin,sig
|
Semantic compression - Wikipedia
In natural language processing, semantic compression is a process of compacting a lexicon used to build a textual document (or a set of documents) by reducing language heterogeneity, while maintaining text semantics. As a result, the same ideas can be represented using a smaller set of words.
In most applications, semantic compression is a lossy compression, that is, increased prolixity does not compensate for the lexical compression, and an original document cannot be reconstructed in a reverse process.
1 By generalization
2 Implicit semantic compression
3 Applications and advantages
By generalization[edit]
Semantic compression is basically achieved in two steps, using frequency dictionaries and semantic network:
determining cumulated term frequencies to identify target lexicon,
replacing less frequent terms with their hypernyms (generalization) from target lexicon.[1]
Step 1 requires assembling word frequencies and information on semantic relationships, specifically hyponymy. Moving upwards in word hierarchy, a cumulative concept frequency is calculating by adding a sum of hyponyms' frequencies to frequency of their hypernym:
{\displaystyle cumf(k_{i})=f(k_{i})+\sum _{j}cumf(k_{j})}
{\displaystyle k_{i}}
is a hypernym of
{\displaystyle k_{j}}
. Then, a desired number of words with top cumulated frequencies are chosen to build a targed lexicon.
In the second step, compression mapping rules are defined for the remaining words, in order to handle every occurrence of a less frequent hyponym as its hypernym in output text.
The below fragment of text has been processed by the semantic compression. Words in bold have been replaced by their hypernyms.
They are both nest building social insects, but paper wasps and honey bees organize their colonies
in very different ways. In a new study, researchers report that despite their differences, these insects rely on the same network of genes to guide their social behavior.The study appears in the Proceedings of the Royal Society B: Biological Sciences. Honey bees and paper wasps are separated by more than 100 million years of
evolution, and there are striking differences in how they divvy up the work of maintaining a colony.
The procedure outputs the following text:
They are both facility building insect, but insects and honey insects arrange their biological groups
in very different structure. In a new study, researchers report that despite their difference of opinions, these insects act the same network of genes to steer their party demeanor. The study appears in the proceeding of the institution bacteria Biological Sciences. Honey insects and insect are separated by more than hundred million years of
organic processes, and there are impinging differences of opinions in how they divvy up the work of affirming a biological group.
Implicit semantic compression[edit]
A natural tendency to keep natural language expressions concise can be perceived as a form of implicit semantic compression, by omitting unmeaningful words or redundant meaningful words (especially to avoid pleonasms).[2]
Applications and advantages[edit]
In the vector space model, compacting a lexicon leads to a reduction of dimensionality, which results in less computational complexity and a positive influence on efficiency.
Semantic compression is advantageous in information retrieval tasks, improving their effectiveness (in terms of both precision and recall).[3] This is due to more precise descriptors (reduced effect of language diversity – limited language redundancy, a step towards a controlled dictionary).
As in the example above, it is possible to display the output as natural text (re-applying inflexion, adding stop words).
^ Ceglarek, D.; Haniewicz, K.; Rutkowski, W. (2010). "Semantic Compression for Specialised Information Retrieval Systems". Advances in Intelligent Information and Database Systems. Studies in Computational Intelligence. 283: 111–121. doi:10.1007/978-3-642-12090-9_10. ISBN 978-3-642-12089-3.
^ Percova, N.N. (1982). "On the types of semantic compression of text". COLING '82 Proceedings of the 9th Conference on Computational Linguistics. Vol. 2. pp. 229–231. doi:10.3115/990100.990155. ISBN 0-444-86393-1. S2CID 33742593.
^ Ceglarek, D.; Haniewicz, K.; Rutkowski, W. (2010). "Quality of semantic compression in classification". Proceedings of the 2nd International Conference on Computational Collective Intelligence: Technologies and Applications. Vol. 1. Springer. pp. 162–171. ISBN 978-3-642-16692-1.
Semantic compression on Project SENECA (Semantic Networks and Categorization) website
Retrieved from "https://en.wikipedia.org/w/index.php?title=Semantic_compression&oldid=1034646674"
|
{E}^{2}
2
3
{E}^{3}
{S}^{3}
{E}^{4}
4
{E}^{n}
n
4<n<\infty
{S}^{n-1}\subset {E}^{n}
A LIP Immersion of Lipschitz Manifolds Modelled on Some Banach Spaces
Radu Miculescu (2000)
A note on closed
N
-cells in
{ℝ}^{N}
A set of moves for Johansson representation of 3-manifolds
Rubén Vigara (2006)
A Dehn sphere Σ in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The Dehn sphere Σ fills M if it defines a cell decomposition of M. The inverse image in S² of the double curves of Σ is the Johansson diagram of Σ and if Σ fills M it is possible to reconstruct M from the diagram. A Johansson representation of M is the Johansson diagram of a filling Dehn sphere of M. Montesinos proved that every closed 3-manifold has a Johansson representation...
{E}^{n}
{S}^{3}
About a problem of Ulam concerning flat sections of manifolds.
Luis Montejano (1990)
An annulus theorem for suspension spheres
Ronald Rosen (1976)
Approximating continous maps of metric spaces into manifolds by embeddings.
Approximation L2-invariants by Their Finite-Dimensional Analogues.
W. Lück (1994)
Arcs in the Hilbert cube
\left({S}^{n}\right)
whose complements have different fundamental groups
R. J. Daverman, S. Singh (1983)
Chapman's Classification of Shapes. A Proof Using Collapsing.
Larry Siebenmann (1975)
{S}^{n}
{S}^{n}
{S}^{1}
Jean Cerf (1998)
On définit le bicomplexe
{C}_{•,•}
, extension naturelle du complexe
C
engendré par un ensemble simplicial
\Gamma
. Ceci permet de définir la notion de ruban de base un cycle de
C
. La somme directe de l’homologie des colonnes de
{C}_{•,•}
contient, outre l’homologie de
C
, des groupes dans lesquels se trouvent les obstructions à l’existence de rubans. Si
\Gamma
est un sous-ensemble simplicial, stable par subdivision, de l’ensemble des simplexes singuliers d’un espace topologique, l’existence de rubans entraîne l’invariance...
Compacta with the shape finite complexes
Ross Geoghegan, R. Lacher (1976)
Computing immersed normal surfaces in the figure-eight knot complement.
Rannard, Richard (1999)
Correction to "Locally flat 2-spheres in simply connected 4-manifolds".
Ronnie Lee, D.M. Wilczynski (1992)
Reckziegel, H., Schaaf, M. (1997)
Deformation of Homeomorphismus on Stratified Sets
L.C. Siebenmann (1972)
Deloopings of the spaces of long embeddings
Keiichi Sakai (2014)
The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give a delooping of the long embedding space, which can be regarded as a version of Morlet-Burghelea-Lashof's delooping of the diffeomorphism group of the disk relative to the boundary....
|
What is the length of the marked portion of each line segment? Copy the segment onto your paper before finding the missing length. Assume that the entire line segment is subdivided into equal sections.
How many segments is
75
broken into? Use this to figure out the value of each segment.
75
45
. Be sure to show why this answer is correct.
How is this problem the same or different than parts (a) and (b)?
|
76T15 Dusty-gas two-phase flows
Shawn W. Walker (2014)
Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking...
A new approach to the problem of removal of soluble gaseous pollutants from the atmosphere and its correspondence with the existing approach.
Chattopadhyay, Soumitra, Mukherjee, A.K. (1994)
Frédéric Coquel, Jean-Marc Hérard, Khaled Saleh, Nicolas Seguin (2014)
We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing...
A Second Order Splitting Method for the Cahn-Hilliard Equation.
C.M. Elliott, D.A. French, F.A Milner (1989)
{\rho }_{i}
{u}^{\left(i\right)}
{\rho }_{i}{|}_{\infty }={\rho }_{i\infty }>0
{u}^{\left(i\right)}{|}_{\infty }=0
{\rho }_{i}\equiv {\rho }_{i\infty }
{u}^{\left(i\right)}\equiv 0
i=1,2
C. Bourdarias, M. Ersoy, Stéphane Gerbi (2013)
In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme...
An investigation on gas lift performance curve in an oil-producing well.
Saepudin, Deni, Soewono, Edy, Sidarto, Kuntjoro Adji, Gunawan, Agus Yodi, Siregar, Septoratno, Sukarno, Pudjo (2007)
Analyse mathématique de modèles variationells en simulation pétrolière. Le cas du modèle black-oil pseudo-compositionnel standard isoterme.
Gérard Gagneux, Ann-Marie Lefevere, Monique Madaune-Tort (1989)
The aim of the paper is an analytical and numerical approach to the pseudo-compositional black-oil model for simulating a 3-D isothermal constrained polyphasic flow in porous media, taking into account realistic boundary conditions. The handling of the component conservation laws leads to a strongly coupled system including parabolic quasilinear degenerated equations and first-order hyperbolic inequalities: the introduction of unilateral problems arises from the nature of the thermodynamical equilibrium...
Application of a multiphase flow code for investigation of influence of capillary pressure parameters on two-phase flow
Jiří Mikyška, Tissa H. Illangasekare (2007)
We have developed a multiphase flow code that has been applied to study the behavior of non-aqueous phase liquids (NAPL) in the subsurface. We describe model formulation, discretization, and use the model for numerical investigation of sensitivity of the NAPL plume with respect to capillary parameters of the soil. In this paper the soil is assumed to be spatially homogeneous. A 2-D reference problem has been chosen and has been recomputed repeatedly with modified parameters of the Brooks–Corey capillary...
Basic singularities in the theory of internal waves with surface tension.
Gorgui, M.A., Faltas, M.S. (1986)
Calculation of the critical Stokes number for wide-stream impaction of potential flow over symmetric arc-nosed collectors.
Lesnic, Daniel (2005)
Characterization of the speed of a two-phase interface in a porous medium.
Abbassi, Adil, Namah, Gawtum (2005)
Kimie Nakashima, Kazunaga Tanaka (2003)
Smadar Karni, Eduard Kirr, Alexander Kurganov, Guergana Petrova (2004)
Continuous-time finite element analysis of multiphase flow in groundwater hydrology
Zhangxin Chen, Magne Espedal, Richard E. Ewing (1995)
A nonlinear differential system for describing an air-water system in groundwater hydrology is given. The system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. A continuous-time version of the finite element method is developed and analyzed for the approximation of the saturation and pressure. The saturation equation is treated by a Galerkin finite element method, while the pressure equation is treated by a mixed finite element method. The analysis...
Diffusion models of multicomponent mixtures in the lung*
L. Boudin, D. Götz, B. Grec (2010)
In this work, we are interested in two different diffusion models for multicomponent mixtures. We numerically recover experimental results underlining the inadequacy of the usual Fick diffusion model, and the importance of using the Maxwell-Stefan model in various situations. This model nonlinearly couples the mole fractions and the fluxes of each component of the mixture. We then consider a subregion of the lower part of the lung, in which we compare...
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Probability/Probability Spaces - Wikibooks, open books for an open world
Probability/Probability Spaces
Probability Spaces Conditional Probability
4 Probability definition
5 Properties of probability
5.1 Basic properties of probability
5.2 More advanced properties of probability
Probability space[edit | edit source]
{\displaystyle \Omega }
{\displaystyle S}
{\displaystyle U}
{\displaystyle U}
{\displaystyle \omega }
{\displaystyle \Omega }
{\displaystyle \Omega =\{1,2,3,4,5,6\}}
{\displaystyle {\mathcal {F}}={\mathcal {P}}(\Omega )}
{\displaystyle \varnothing ,\{1,2,3\},\Omega }
{\displaystyle \{0\}}
{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
{\displaystyle \Omega }
{\displaystyle {\mathcal {F}}}
{\displaystyle \mathbb {P} }
{\displaystyle \Pr ,P}
{\displaystyle p}
{\displaystyle \mathbb {P} }
{\displaystyle P}
{\displaystyle p}
{\displaystyle \Omega }
{\displaystyle {\mathcal {F}}}
{\displaystyle \mathbb {P} }
{\displaystyle \mathbb {P} }
Terminologies[edit | edit source]
{\displaystyle E_{1},\dotsc ,E_{n}}
{\displaystyle E_{1}\cup \dotsb \cup E_{n}=\Omega }
Example. When we are rolling a six-faced dice, and we are considering the number coming up as the outcome, the events
{\displaystyle \{1,2,3,4\}}
{\displaystyle \{3,4,5,6\}}
{\displaystyle \varnothing }
{\displaystyle \{1,2,3,4,5\}}
{\displaystyle E_{1},\dotsc ,E_{n}}
{\displaystyle \Omega }
Example. When we are rolling a six-faced dice, and we are considering the number coming up as the outcome, the group of events
{\displaystyle \varnothing }
{\displaystyle \Omega }
{\displaystyle \{1,2,3,4\}}
{\displaystyle \{3,4,5,6\}}
Probability definition[edit | edit source]
{\displaystyle \mathbb {P} }
{\displaystyle 0}
{\displaystyle 1}
{\displaystyle 1}
{\displaystyle 0}
{\displaystyle \Omega }
{\displaystyle E}
{\displaystyle \mathbb {P} (E)={\frac {{\text{no. of outcomes in }}E}{{\text{no. of outcomes in }}\Omega }}.}
{\displaystyle \omega _{1},\omega _{2},...,\omega _{n}}
{\displaystyle \Omega }
{\displaystyle \mathbb {P} (\omega _{i})=p_{i},i=1,2,...,n}
{\displaystyle p_{i}}
{\displaystyle E}
{\displaystyle \omega _{i},\omega _{j},...,\omega _{m}}
{\displaystyle \mathbb {P} (E)=\mathbb {P} (\omega _{i})+\mathbb {P} (\omega _{j})+\dotsb +\mathbb {P} (\omega _{m})=p_{i}+p_{j}+\dotsb +p_{m}}
{\displaystyle E}
However, the probability of each of the not equally likely outcomes is often unknown. If this is the case, the above method does not work, and we cannot apply combinatorial probability in this context. So, the combinatorial probability definition does not work well when we are encountering outcomes that are not equally likely.
{\displaystyle {\frac {1}{6\times 6}}={\frac {1}{36}}}
{\displaystyle \underbrace {6} _{\text{red}}\times \underbrace {6} _{\text{blue}}=36}
{\displaystyle \Box }
{\displaystyle {\frac {1}{36}}}
{\displaystyle {\frac {1}{21}}}
{\displaystyle {\frac {1}{18}}}
{\displaystyle {\frac {1}{15}}}
{\displaystyle {\frac {1}{6}}}
{\displaystyle N}
{\displaystyle k\leq N}
{\displaystyle n\geq r}
{\displaystyle \leq N}
{\displaystyle r\leq k}
{\displaystyle n}
{\displaystyle {\frac {{\binom {k}{r}}\times {\binom {N-k}{n-r}}}{\binom {N}{n}}}}
{\displaystyle N}
{\displaystyle {\binom {N}{n}}}
{\displaystyle n}
{\displaystyle N}
{\displaystyle {\binom {k}{r}}}
{\displaystyle r}
{\displaystyle k}
{\displaystyle {\binom {N-k}{n-r}}}
{\displaystyle n-r}
{\displaystyle N-k}
{\displaystyle r}
{\displaystyle \Box }
{\displaystyle {\frac {1}{28}}}
{\displaystyle {\frac {3}{28}}}
{\displaystyle {\frac {1}{9}}}
{\displaystyle {\frac {1}{3}}}
{\displaystyle {\frac {2}{7}}}
{\displaystyle {\frac {5}{9}}}
{\displaystyle {\frac {5}{7}}}
{\displaystyle {\frac {5}{6}}}
{\displaystyle n}
{\displaystyle {\frac {1}{3}}}
{\displaystyle n}
{\displaystyle n}
{\displaystyle n}
{\displaystyle r}ed balls are drawn and
{\displaystyle b}lue balls are drawn if
{\displaystyle k}
{\displaystyle r}
{\displaystyle b}
{\displaystyle k}
{\displaystyle {\frac {{\binom {3}{r}}{\binom {2}{b}}}{\binom {9}{k}}}}
{\displaystyle {\frac {{\binom {3}{r}}{\binom {2}{b}}}{\binom {b+r+k}{k}}}}
{\displaystyle {\frac {{\binom {3}{r}}{\binom {2}{b}}}{\binom {9}{9-b-r}}}}
{\displaystyle {\frac {{\binom {9-b-k}{r}}{\binom {2}{b}}}{\binom {9}{k}}}}
{\displaystyle {\frac {{\binom {3}{r}}{\binom {9-r-k}{b}}}{\binom {9}{k}}}}
{\displaystyle n(E)}
{\displaystyle E}
{\displaystyle n}
{\displaystyle E}
{\displaystyle \mathbb {P} (E)=\lim _{n\to \infty }{\frac {n(E)}{n}}.}
{\displaystyle {\frac {700102}{1000000}}=0.700102}
{\displaystyle {\mathcal {F}}}
{\displaystyle ={\mathcal {P}}(\Omega )}
{\displaystyle \mathbb {P} (E)}
{\displaystyle E}
{\displaystyle E\in {\mathcal {F}}}
{\displaystyle \mathbb {P} (E)\geq 0}
{\displaystyle \mathbb {P} (\Omega )=1}
{\displaystyle E_{1},E_{2},\dotsc }
{\displaystyle \mathbb {P} \left(\bigcup _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }\mathbb {P} (E_{i})}
{\displaystyle \Box }
Properties of probability[edit | edit source]
Basic properties of probability[edit | edit source]
{\displaystyle \mathbb {P} (\varnothing )=0}
{\displaystyle E_{i}=\varnothing }
{\displaystyle i}
{\displaystyle E_{1},E_{2},\dotsc }
{\displaystyle E_{1}\cup E_{2}\cup \dotsb =\varnothing \cup \varnothing \cup \dotsb =\varnothing }
{\displaystyle {\begin{aligned}&&\mathbb {P} (\varnothing )&=\mathbb {P} (E_{1}\cup E_{2}\cup \dotsb )\\&&&{\overset {\text{ P3 }}{=}}\mathbb {P} (E_{1})+\mathbb {P} (E_{2})+\dotsb \\&\Rightarrow &\underbrace {\mathbb {P} (\varnothing )-\mathbb {P} (E_{1})} _{0}&=\mathbb {P} (E_{1})+\mathbb {P} (E_{2})+\dotsb \\&\Rightarrow &\mathbb {P} (E_{2})+\dotsb &=0\\&\Rightarrow &\mathbb {P} (E_{2})&\leq \mathbb {P} (E_{2})+\dotsb =0.\end{aligned}}}
{\displaystyle \mathbb {P} (E_{2})\geq 0}
{\displaystyle \mathbb {P} (\varnothing )=\mathbb {P} (E_{2})=0}
{\displaystyle \Box }
{\displaystyle k}
{\displaystyle A_{1},\dotsc ,A_{k}}
{\displaystyle A_{k+1}=\varnothing ,A_{k+2}=\varnothing ,\dotsc }
{\displaystyle \mathbb {P} \left(\bigcup _{i=1}^{k}A_{i}\right)=\mathbb {P} \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\mathbb {P} (A_{i})=\sum _{i=1}^{k}\mathbb {P} (A_{i})}
{\displaystyle \sum _{i=k+1}^{\infty }\mathbb {P} (A_{i})=\mathbb {P} (\varnothing )+\dotsb =0}
{\displaystyle \Box }
{\displaystyle A,B}
{\displaystyle \mathbb {P} (B)=\mathbb {P} (B\cap A)+\mathbb {P} (B\setminus A)}
{\displaystyle \mathbb {P} (B)=\mathbb {P} (B\cap (\underbrace {A\cup A^{c}} _{\Omega }))=\mathbb {P} {\big (}(B\cap A)\cup (\underbrace {B\cap A^{c}} _{:=B\setminus A}){\big )}{\overset {\text{ ext. P3 }}{=}}\mathbb {P} (B\cap A)+\mathbb {P} (B\setminus A)}
{\displaystyle \Box }
{\displaystyle A}
{\displaystyle B}
{\displaystyle \mathbb {P} (A\cup B)=\mathbb {P} (A)+\mathbb {P} (B)-\mathbb {P} (A\cap B)}
{\displaystyle A}
{\displaystyle B\setminus A}
{\displaystyle \mathbb {P} (A\cup (B\setminus A))=\mathbb {P} (A)+\mathbb {P} (B\setminus A)=\mathbb {P} (A)+(\mathbb {P} (B)-\mathbb {P} (B\cap A))=\mathbb {P} (A)+\mathbb {P} (B)-\mathbb {P} (A\cap B)}
{\displaystyle \mathbb {P} (B)=\mathbb {P} (B\cap A)+\mathbb {P} (B\setminus A)\Rightarrow \mathbb {P} (B\setminus A)=\mathbb {P} (B)-\mathbb {P} (B\cap A)}
{\displaystyle \Box }
{\displaystyle \mathbb {P} (A\cup B)=\mathbb {P} ({\text{I}})+\mathbb {P} ({\text{II}})+\mathbb {P} (A\cap B)=\underbrace {\mathbb {P} ({\text{I}})+\mathbb {P} (A\cap B)} _{\mathbb {P} (A)}+\underbrace {\mathbb {P} ({\text{II}})+\mathbb {P} (A\cap B)} _{\mathbb {P} (B)}-\mathbb {P} (A\cap B)}
{\displaystyle E}
{\displaystyle \mathbb {P} (E)=1-\mathbb {P} (E^{c}).}
{\displaystyle \mathbb {P} (E)=\mathbb {P} (E)+\mathbb {P} (E^{c})-\mathbb {P} (E^{c}){\overset {\text{ ext. P3 }}{=}}\mathbb {P} (\underbrace {E\cup E^{c}} _{\Omega })-\mathbb {P} (E^{c}){\overset {\text{ P2 }}{=}}1-\mathbb {P} (E^{c})}
{\displaystyle \Box }
{\displaystyle E}
{\displaystyle 0\leq \mathbb {P} (E)\leq 1}
{\displaystyle \mathbb {P} (E)\geq 0}
{\displaystyle \mathbb {P} (E^{c})\geq 0}
{\displaystyle \mathbb {P} (E)\leq \mathbb {P} (E)+\mathbb {P} (E^{c})=\mathbb {P} (E)+(1-\mathbb {P} (E))=1}
{\displaystyle \Box }
{\displaystyle A\subseteq B}
{\displaystyle \mathbb {P} (A)\leq \mathbb {P} (B)}
{\displaystyle \mathbb {P} (B)=\mathbb {P} (\underbrace {B\cap A} _{A})+\mathbb {P} (B\setminus A){\overset {\text{ P1 }}{\geq }}\mathbb {P} (A){\cancel {+0}}.}
{\displaystyle \Box }
{\displaystyle A\subsetneq B}
{\displaystyle C}
{\displaystyle F}
{\displaystyle C\subseteq F}
{\displaystyle C\Rightarrow ({\text{implies}})\;F}
{\displaystyle \mathbb {P} (C)\leq \mathbb {P} (F)}
{\displaystyle \Box }
{\displaystyle A=B}
{\displaystyle \mathbb {P} (A)=\mathbb {P} (B)}
{\displaystyle \mathbb {P} {\big (}A\setminus (B\cup C){\big )}=\mathbb {P} (A)+\mathbb {P} (A\cap B)+\mathbb {P} (A\cap C)-\mathbb {P} (A\cap B\cap C)}
{\displaystyle {\frac {\mathbb {P} (A\cap B)}{\mathbb {P} (B)}}=1}
{\displaystyle A\subseteq B}
{\displaystyle \mathbb {P} (B)>0}
{\displaystyle 0\leq {\frac {\mathbb {P} (A\cap B)}{\mathbb {P} (B)}}\leq 1}
{\displaystyle \mathbb {P} (B)>0}
More advanced properties of probability[edit | edit source]
{\displaystyle n=3}
{\displaystyle E_{1},\dotsc ,E_{n}}
{\displaystyle {\begin{aligned}\mathbb {P} (E_{1}\cup \dotsb \cup E_{n})&=\mathbb {P} (E_{1})+\dotsb +\mathbb {P} (E_{n})\\&\;-{\big (}\mathbb {P} (E_{1}\cap E_{2})+\mathbb {P} (E_{1}\cap E_{3})+\dotsb +\mathbb {P} (E_{n-1}\cap E_{n}){\big )}\\&\;+{\big (}\mathbb {P} (E_{1}\cap E_{2}\cap E_{3})+\mathbb {P} (E_{1}\cap E_{2}\cap E_{4})+\dotsb +\mathbb {P} (E_{n-2}\cap E_{n-1}\cap E_{n}){\big )}\\&\;-\dotsb \\&\;+(-1)^{n+1}\mathbb {P} (E_{1}\cap \dotsb \cap E_{n}).\end{aligned}}}
{\displaystyle n=2}
{\displaystyle n=2}
{\displaystyle n}
{\displaystyle {\begin{aligned}\mathbb {P} ((E_{1}\cup \dotsb \cup E_{n-1})\cup {\color {darkgreen}E_{n}})&=\mathbb {P} (E_{1}\cup \dotsb \cup E_{n-1})+\mathbb {P} ({\color {darkgreen}E_{n}})-\mathbb {P} {\big (}(E_{1}\cup \dotsb \cup E_{n-1})\cap {\color {darkgreen}E_{n}}{\big )}\\&=\mathbb {P} (E_{1}\cup \dotsb \cup E_{n-1})+\mathbb {P} ({\color {darkgreen}E_{n}})-\mathbb {P} {\big (}(E_{1}\cap {\color {darkgreen}E_{n}})\cup \dotsb \cup (E_{n-1}\cap {\color {darkgreen}E_{n}}){\big )}\\&=\dotsb \end{aligned}}}
{\displaystyle \Box }
{\displaystyle \mathbb {P} (E_{1}\cup \dotsb \cup E_{n})=\sum _{j=1}^{n}(-1)^{j+1}\sum _{i_{1}<\dotsb <i_{j}}^{}\mathbb {P} (E_{i_{1}}\cap \dotsb \cap E_{i_{j}})}
{\displaystyle n=3}
{\displaystyle A,B}
{\displaystyle C}
{\displaystyle \mathbb {P} (A\cup B\cup C)=\mathbb {P} (A)+\mathbb {P} (B)+\mathbb {P} (C)-\mathbb {P} (A\cap B)-\mathbb {P} (A\cap C)-\mathbb {P} (B\cap C)+\mathbb {P} (A\cap B\cap C).}
{\displaystyle \mathbb {P} (A\cup B\cup C\cup D)}
{\displaystyle A,B,C}
{\displaystyle D}
{\displaystyle {\begin{aligned}\mathbb {P} (A)+\mathbb {P} (B)+\mathbb {P} (C)+\mathbb {P} (D)-\mathbb {P} (A\cap B)-\mathbb {P} (A\cap C)-\mathbb {P} (A\cap D)-\mathbb {P} (B\cap C)-\mathbb {P} (B\cap D)\\-\mathbb {P} (C\cap D)+\mathbb {P} (A\cap B\cap C)+\mathbb {P} (A\cap B\cap D)+\mathbb {P} (B\cap C\cap D)-\mathbb {P} (A\cap B\cap C\cap D)\end{aligned}}}
{\displaystyle {\begin{aligned}\mathbb {P} (A)+\mathbb {P} (B)+\mathbb {P} (C)+\mathbb {P} (D)-\mathbb {P} (A\cap B)-\mathbb {P} (A\cap C)-\mathbb {P} (A\cap D)-\mathbb {P} (B\cap C)-\mathbb {P} (B\cap D)\\-\mathbb {P} (C\cap D)+\mathbb {P} (A\cap B\cap C)+\mathbb {P} (A\cap B\cap D)+\mathbb {P} (B\cap C\cap D)+\mathbb {P} (A\cap B\cap C\cap D)\end{aligned}}}
{\displaystyle {\begin{aligned}\mathbb {P} (A)+\mathbb {P} (B)+\mathbb {P} (C)+\mathbb {P} (D)+\mathbb {P} (A\cap B)+\mathbb {P} (A\cap C)+\mathbb {P} (A\cap D)+\mathbb {P} (B\cap C)+\mathbb {P} (B\cap D)\\+\mathbb {P} (C\cap D)-\mathbb {P} (A\cap B\cap C)-\mathbb {P} (A\cap B\cap D)-\mathbb {P} (B\cap C\cap D)+\mathbb {P} (A\cap B\cap C\cap D)\end{aligned}}}
{\displaystyle \mathbb {P} (A)+\mathbb {P} (B)+\mathbb {P} (C)+\mathbb {P} (D)-\mathbb {P} (A\cap B)-\mathbb {P} (A\cap C)-\mathbb {P} (B\cap C)+\mathbb {P} (A\cap B\cap C)-\mathbb {P} {\big (}(A\cup B\cup C)\cap D{\big )}}
{\displaystyle 160{\big (}1-(0.4+0.55+0.3-0.2-0.15-0.2+0.1){\big )}=160(1-0.8)=32}
{\displaystyle M,S,A}
{\displaystyle {\begin{aligned}\mathbb {P} (M^{c}\cap S^{c}\cap A^{c})&=\mathbb {P} {\big (}(M\cup S\cup A)^{c}{\big )}=1-\mathbb {P} (M\cup S\cup A)\\&=1-(\mathbb {P} (M)+\mathbb {P} (S)+\mathbb {P} (A)-\mathbb {P} (M\cap S)-\mathbb {P} (M\cap A)-\mathbb {P} (S\cap A)+\mathbb {P} (M\cap S\cap A))\\&=1-(0.4+0.55+0.3-0.2-0.15-0.2+0.1)\\&=1-0.8=0.2.\end{aligned}}}
{\displaystyle {\frac {\text{no. of students that do not have any of those majors}}{\text{no. of students}}}=0.2\Rightarrow {\text{no. of students that do not have any of those majors}}=0.2(160)=32.}
{\displaystyle \mathbb {P} (M\cup S\cup A)=0.05+0.05+0.15+0.1+0.1+0.1+0.25=0.8}
{\displaystyle \Box }
{\displaystyle \mathbb {P} }
{\displaystyle N}
{\displaystyle N(E)}
{\displaystyle E}
{\displaystyle E_{1},E_{2},\dotsc }
{\displaystyle \mathbb {P} \left(\bigcup _{i=1}^{\infty }E_{i}\right){\overset {\text{ def }}{=}}\mathbb {P} \left(\lim _{n\to \infty }\bigcup _{i=1}^{n}E_{i}\right)=\lim _{n\to \infty }\mathbb {P} \left(\bigcup _{i=1}^{n}E_{i}\right).}
{\displaystyle \lim _{n\to \infty }\mathbb {P} \left(\bigcup _{i=1}^{n}E_{i}\right){\overset {\text{ext. P3}}{=}}\lim _{n\to \infty }\sum _{i=1}^{n}\mathbb {P} (E_{i}){\overset {\text{ def }}{=}}\sum _{i=1}^{\infty }\mathbb {P} (E_{i}){\overset {\text{ P3 }}{=}}\mathbb {P} \left(\bigcup _{i=1}^{\infty }E_{i}\right).}
{\displaystyle \Box }
{\displaystyle E_{1},E_{2},\ldots }
{\displaystyle \mathbb {P} \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mathbb {P} (E_{i})}
{\displaystyle A}
{\displaystyle B}
{\displaystyle \mathbb {P} (A\cap B)=\mathbb {P} (A)+\mathbb {P} (B)-\mathbb {P} (A\cap B){\overset {\text{ P1 }}{\leq }}\mathbb {P} (A)+\mathbb {P} (B)}
{\displaystyle \mathbb {P} \left(\bigcup _{i=1}^{n}E_{i}\right)\leq \mathbb {P} (E_{1})+\mathbb {P} \left(\bigcup _{i=2}^{n}E_{i}\right)\leq \mathbb {P} (E_{1})+\mathbb {P} (E_{2})+\mathbb {P} \left(\bigcup _{i=3}^{n}E_{i}\right)\leq \dotsb \leq \mathbb {P} (E_{1})+\mathbb {P} (E_{2})+\dotsb +\mathbb {P} (E_{n})=\sum _{i=1}^{n}\mathbb {P} (E_{i}).}
{\displaystyle \mathbb {P} \left(\bigcup _{i=1}^{\infty }E_{i}\right)=\lim _{n\to \infty }\mathbb {P} \left(\bigcup _{i=1}^{n}E_{i}\right){\overset {\text{from above}}{\leq }}\lim _{n\to \infty }\sum _{i=1}^{n}\mathbb {P} (E_{i}){\overset {\text{ def }}{=}}\sum _{i=1}^{\infty }\mathbb {P} (E_{i}).}
{\displaystyle \Box }
↑ e.g. the sample space of throwing a dice may include the six numbers, or may only include two outcomes: odd number and even number
↑ e.g. it is given that a coin is biased, such that it is more likely that head comes up
↑ ext. stands for 'extended'
↑ Given:
{\displaystyle \mathbb {P} (M\cap S\cap A)=0.1}
| | | | <---- M
| |0.1 | | |
| | | | | <---- S
{\displaystyle \mathbb {P} ((M\cap A)\setminus S)=\underbrace {\mathbb {P} (M\cap A)} _{0.15}-\underbrace {\mathbb {P} (M\cap A\cap S)} _{0.1}=0.05}
by observing the Venn digram and using the given information
| |0.05| | <---- M
{\displaystyle \mathbb {P} ((M\cap S)\setminus A)=\underbrace {\mathbb {P} (M\cap S)} _{0.2}-\underbrace {\mathbb {P} (M\cap S\cap A)} _{0.1}=0.1}
{\displaystyle \mathbb {P} (M\setminus (S\cup A))=\underbrace {\mathbb {P} (A)} _{0.3}-\underbrace {\mathbb {P} {\big (}M\cap (S\cup A){\big )}} _{0.05+0.1+0.1}=0.15}
by observing the Venn diagram and using the given information
| |0.05|0.15| <---- M
{\displaystyle \mathbb {P} ((S\cap A)\setminus M)=\underbrace {\mathbb {P} (S\cap A)} _{0.2}-\underbrace {\mathbb {P} (S\cap A\cap M)} _{0.1}=0.1}
using the given information
| 0.1 | | | | <---- S
{\displaystyle \mathbb {P} (A\setminus (S\cup M))=\underbrace {\mathbb {P} (A)} _{0.3}-\underbrace {\mathbb {P} (A\cap (S\cup M))} _{0.05+0.1+0.1}=0.05}
using the given information and observing the Venn digram
|0.05 |0.05|0.15| <---- M
{\displaystyle \mathbb {P} (S\setminus (A\cup M))=\underbrace {\mathbb {P} (S)} _{0.55}-\underbrace {\mathbb {P} (S\cap (A\cup M))} _{0.1+0.1+0.1}=0.25}
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