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# Matrix element for direct reactions

1. ### Silversonic

130
I apologise since I already have a question on this board, but I've been stuck for a good few hours understanding exactly how this has been done. The differential cross section for a direct reaction from $\alpha$ to $\beta$ is given by

$\frac{d\sigma}{d\Omega} = f(k,k')|T_{\beta \alpha}|^2$

f(k,k') is just a constant dependent on the initial and final momentums. For a one-nucleon transfer, e.g. A(B,C)D where B may be a deuteron and C a proton, $T_{\alpha \beta}$ is given by

$T_{\alpha \beta} \int e^{-i\mathbf{k}\cdot\mathbf{r_{\beta}}} \int \psi^*_C \psi^*_D V \psi_A \psi_B d\tau e^{i\mathbf{k}\cdot\mathbf{r_{\alpha}}} d^3\mathbf{r_{\alpha}} d^3\mathbf{r_{\beta}}$

$\alpha$ represents the relative vector distance between incident particle A and target B
$\beta$ ^^ projectile C and residual nucleus D

V is the potential responsible for the direct reaction process.

$d\tau$ is the integral over all internal coordinates.

For reference Bertulani (nuclear physics in a nutshell) chapter 11.1 and Wong (intro to nuclear) chapter 8.3 focus on this.

There are three approximations then used:
(1) $r_{\alpha} = r_{\beta} = r$ because of the short range of $V$.
(2) $V = V_0\delta(r-R)$ because the interaction is short ranged, but below a certain distance the formation of a compound nucleus is more favourable.
(3) For a (d,p) reaction, which is what we're considering, the neutron is transferred to a single particle state (M=0) of B. i.e. $\psi_D = \psi_B \phi_n = \psi_B f(r) Y_{L0}(\theta,\phi)$

What I'm then confused with is that I'm told these approximations are used to obtain

$T_{\alpha \beta} = V_0\int e^{i (\mathbf{k - k'}) \cdot r} Y_{L0}(\theta,\phi)^*\delta(r-R) d^3\mathbf{r}$

I think some factors were left out because we're interesting in really the angular dependence of the differential cross section. I've tried getting to this myself, but I'm a bit confused as to what is a function of what variables for integration, and I also have no idea how $Y_{L0}$ makes its way outside of the integral for $d\tau$. This is as far as I can get subbing in and using the approximations above;

$T_{\alpha \beta} = V_0 \int e^{i (\mathbf{k - k'})\cdot\mathbf{r}} \int \psi^*_C \psi^*_D \delta(r-R) \psi_A \psi_B d\tau d^3\mathbf{r}$

So I guess all I need is

$\int \psi^*_C \psi^*_D \psi_A \psi_B d\tau = Y_{L0}(\theta,\phi)$

But all I can show is;

$\int \psi^*_C \psi^*_D \psi_A \psi_B = \int \psi^*_C \psi^*_B f(r')Y_{L0}(\theta',\phi') \psi_A \psi_B d\tau$

Then that's really as far as I can get. I don't understand how this results in $Y_{L0}$ when we're integrating over $d\tau$ - all internal coordinates of those wavefunctions. Anyone think they know?

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# Matrix Exponential Jordan Form Linear System

Given the Linear System $$\dot{x}(t)=A x(t)$$ with $$x_0=(x_{01},x_{02})$$ as initial state and $$A=\begin{pmatrix} 0 & 1 \\ -k/M & -h/M \end{pmatrix}$$, when $$h^2=4Mk$$ the matrix A has a single eigenvalue $$s_0=s_{1,2}=-\frac{h}{2M}$$ with algebraic multiplicity 2. In this case A is not diagonalizable and its Jordan Form is $$J=\begin{pmatrix} s_0 & 1 \\ 0 & s_0 \end{pmatrix}$$. My objective is to find the exponential matrix so that $$x(t)=e^{At}x_0$$ using the Jordan normal form. I see that $$e^{Jt}=e^{s_0t}\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$$ So how can I find a matrix $$T$$ such that $$J=TAT^{-1}$$? Why the book suggests to use $$T=\begin{pmatrix} 0 & -1/s_0^2 \\ 1 & -1/s_0 \end{pmatrix}$$?

using the letter $$c,$$
$$\left( \begin{array}{rr} 1&0 \\ c&1 \end{array} \right) \left( \begin{array}{rr} 0&1 \\ -c^2&-2c \end{array} \right) \left( \begin{array}{rr} 1&0 \\ -c&1 \end{array} \right) = \left( \begin{array}{rr} -c&1 \\ 0&-c \end{array} \right)$$
• @Timothy let me call my matrix $G,$ I am going to construct $P$ so that $P^{-1} GP = J$ is the Jordan form. Since $(G+cI)^2 = 0,$ I chose whatever I want for the right column $v$ of $P,$ I chose $v = \left( \begin{array}{rr} 0 \\ 1 \end{array} \right)$ Then the rule just says the left column of $P$ is $u = (G+cI) v$ – Will Jagy Jan 7 at 22:38

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# Bias of method of moments estimator for Pareto distribution with known scale parameter

Let $x$ be a Pareto distribution with a known scale parameter $m>0$, i.e. $x\sim f(x|a)=\frac{am^a}{x^{a+1}}, x>a, a>0$

$\mathrm{E}\left[X\right]=\frac{am}{a-1}$

Using method of moments estimator for the shape parameter, $\frac{\hat{a}m}{\hat{a}-1}=\sum_{i=1}^{n}{\frac{x_i}{n}}, \hat{a}=\frac{\sum{x_i}}{\sum{x_i}-mn}$

How does one calculate the bias of the estimator? What's $\mathrm{E}\left[\hat{a}\right]$? Is there a known distribution for sum of Pareto variables?

If there's no closed form expression for the expectation, is the bias always one way?

• This seems rather like a textbook-style exercise. Is this for some class? – Glen_b -Reinstate Monica Dec 16 '15 at 7:48
• @Glen_b no, I wish. It's for modeling losses associated with business activities, where the threshold is determined by business judgment – C.J. Jackson Dec 16 '15 at 16:24
• I've made some additional comments with that in mind. – Glen_b -Reinstate Monica Dec 16 '15 at 22:37

The distribution of a sum of Pareto variates is not especially simple, but has been done. [1] [2]

Without loss of generality, we can take $m=1$; we can simply divide through by $m$ to work with $X^*=X/m$ and the lower limit for $X^*$ is then $1$. Since $m$ is then just a scale factor applied to the data we can translate any results back to the original data scale.

In this answer I haven't attempted to compute the exact bias from those published results. If it were me faced with this exercise I probably would focus first on using simulation to obtain a clear understanding how the bias relates to the $a$ parameter and the sample size (though I think we can say something about how it should work as a function of sample size).

However, we can do the last part easily enough.

$\bar{x}$ will be unbiased for $E(X) = \frac{a}{a-1}$, but we have

$$\hat{a}=\frac{\bar{x}}{\bar{x}-1}=\frac{1}{{1-\frac{1}{\bar{x}}}}$$

Taking $Y=\bar{X}$, we can show that $\varphi(Y)=\frac{1}{{1-\frac{1}{Y}}}$ is convex.

From Jensen's inequality, we can then show that the estimator is biased and in which direction. Jensen's inequality says that for $\varphi$ convex:

$$\varphi \left(\mathbb {E} [Y]\right)\leq \mathbb {E} \left[\varphi (Y)\right]$$

(The conditions under which equality will hold don't apply here; the inequality will be strict.)

This shows that the estimator will be too high on average.

Here's some results of a simulation with $a=2$

(10000 samples each with n=100, so here we have 10000 $\hat{a}$ values. The blue line is the mean estimate from those 10000 samples.)

This simulation doesn't prove anything, but it shows that we don't contradict the derived result; looks like there was no error in concluding the bias was upward.

Such simulations allow us to see how the bias changes with $a$ -- by doing the same thing across a variety of $a$ values -- and with sample size (again, by using different $n$).

If this is not for a class exercise, I'm very curious why you wouldn't use maximum likelihood in this case:

• it's very simple - for $m=1$ it's the reciprocal of the mean of the logs; if $m$ is not 1, you subtract $log(m)$ from the mean of the logs before taking reciprocals.

• For this parameterization, it's not unbiased either, but it makes better use of the data, and it will have lower variance (and lower bias, by the look of some simulations).

• The bias is also easy to compute!

[1] Blum, M. (1970),
"On the Sums of Independently Distributed Pareto Variates"
SIAM Journal on Applied Mathematics, 19:1 (Jul.), pp. 191-198

[2] Ramsay, Colin M. (2008)
"The Distribution of Sums of I.I.D. Pareto Random Variables with Arbitrary Shape Parameter"
Communications in Statistics - Theory and Methods, 37:14, pp 2177-2184

• Thanks for the help. I have estimated both MLE in addition to MoM with software. It just that in simulation MoM seem to produce much lower tail losses and I'm trying to figure out why theoretically. If bias on shape parameter is upward, that means tails are thinner, so it makes sense along with what you wrote here. – C.J. Jackson Dec 16 '15 at 23:02
• That would be a good question to post, but briefly, if MoM is more positively biased you'll tend to get less of an upper tail on average (since larger $\hat{a}$ values will be more common). The larger variance of the MoM would be expected to complicate that picture a bit (since it will lead to both more of the larger and smaller $\hat{a}$ values, and the interplay between the way they together impact the distribution of $\hat{a}$ and the way that $\hat{a}$ impacts the tail will determine the overall effect). – Glen_b -Reinstate Monica Dec 16 '15 at 23:08
• The quantity I'm concerned with is covering some high cumulative percentile of losses. The upward bias on $\hat{a}$ would drive it down. But would the higher variance of $\hat{a}$ in this case drive up the cumulative losses? – C.J. Jackson Dec 16 '15 at 23:18
• I'd expect it to, yes, since I expect the extreme upper tail will be more impacted by a slightly lower estimate of the parameter than it will be by a slightly higher one. I can't tell for certain until I actually work it through (calculating $\frac{d^2S(x)}{da^2}$ might be sufficient) or simulate but I'd bet on yes. – Glen_b -Reinstate Monica Dec 17 '15 at 0:05


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# On-time performance

In public transportation, schedule adherence or on-time performance refers to the level of success of the service (such as a bus or train) remaining on the published schedule. On time performance, sometimes referred to as on time running, is normally expressed as a percentage, with a higher percentage meaning more vehicles are on time. The level of on time performance for many transport systems is a very important measure of the effectiveness of the system.

## Background

On time performance is a measure of the ability of transport services to be on time. Almost all transportation systems have timetables, which describe when vehicles are to arrive at scheduled stops. Transport services have a higher utility where services run on time, as anyone planning on making use of the service can align their activities with that of the transport system. On time performance is particularly important where services are infrequent, and people need to plan to meet services.

The ability of different transport modes to meet on time performance statistics depends on the degree to which they are affected by external factors. For example, rail services often operate on a separated right of way, and so are not affected by road congestion. Buses, unless they are separated from road traffic, will be heavily affected by congestion.

To calculate the percentage of vehicles on time, comparison is needed with the timetable. A comparison is often completed for arrival time rather than departure time, but both are common. Departure times are normally more on-time than arrival times, as incidents and breakdowns occur that reduce the on time performance. When the comparison is made between the timetable, and the actual arrival or departure time, a rule is applied to determine how much of a deviation is permitted. For example, if a bus is 4 minutes 20 seconds late, a rules may be applied that a bus is only late when later than 5 minutes, so this service would be counted as on time. The choice of threshold for lateness is an important one, with a low threshold resulting in a lower on time performance statistic. A high threshold will result in more services on time, but may result in the managing corporation or government body being accused on not reporting correctly.

Passengers can be updated on the movement of transport vehicles with passenger information systems. These systems display the arrival time of vehicles to stops, stations or airports, and typical information displayed is the time in minutes to the next scheduled arrival. Some of these systems have been extended to include apps to show the movement of trains/buses/planes/ferries. Where services are delayed, more information can be provided, such as alternative transport options, or estimated time till services resume.

A small number of transport systems operate on the principle of go-when-full. This type of transport system will not have any meaningful measure of on time performance.

### Airlines

Airlines are closely monitored on their on time performance. Numerous websites exist for reporting on punctuality for airlines, often operated by government departments.[1] Virgin Australia, an airline, uses a rule that aircraft that depart within 15 minutes of scheduled departure are on time.[2] The 15 minutes rules for on time performance is commonly applied throughout the airline industry.[3] Airlines typically perform well when their on time performance reaches 90%.[4]

On time performance for aircraft is simple to calculate as aircraft depart and arrive at airports, and these are clear points to complete the on time performance calculation. Things are substantially less clear for bus services, where an on time performance can be calculated for each and every stop. One method is to calculate on time performance for only the start and end of the bus route, which can produce meaningful statistics. One problem with this method is where very large stops or interchanges are not at the start and end of the bus journey, but in the middle, and the start and end of the route are bus depots or other small stops for which on time performance statistics are not as important. In this situation on time performance may be calculated at the major interchange, rather than the start and end.

### Buses

Bus routes are frequently late, given buses mix with road traffic. Bus rapid transit is bus based mass transit system, where road infrastructure has been specifically constructed to allow better on time performance, and more frequent and high speed services. This type of bus system is far less exposed to problem of low on time performance statistics than regular bus services.

### Light rail

Light rail, whilst being a rail system, can be exposed to problems with poor on time performance. This is especially the case where rail service operate in the middle of congested roads, and where a light rail system has a separate right of way the on time performance will be better.

## Transport hubs

Users of transport systems often use several transport modes to complete a journey. They will change transport modes at a transport hub, and all transport modes have some kind of transport hub.

On time performance is important to interchanges. Where on time performance is poor, passengers or goods may miss a connection, and so be forced to wait. In some cases the transit time allowed from one transport mode to another may be very small, and so any lateness can result in services being missed. Where services are infrequent, such as a long distance train trip, or a flight, then the consequences of late services can be high. Transport systems that exist to deliver passengers and services to interchanges should be particularly concerned with achieving a good on time performance.

## Measurement

Typically on time performance is measured by comparing each service with its schedule.[5] A threshold is chosen for how late a service can be before it is determined to be late.

On time Performance ${\displaystyle ={\frac {\text{Services on time}}{\text{Total Services}}}\times 100\%}$

(1)

The scale of delays are often calculated in delay minutes.[6] A delay minute is the number of minutes a service is delayed multiplied by the number of passengers on board the transport vehicle.

Passenger Delay Minutes ${\displaystyle ={\text{Minutes service delayed}}\times {\text{Number of Passengers}}}$

(2)

Network Rail, a large rail infrastructure provider in the UK, uses passenger delay minutes as a mechanism to reward and punish for services being on time and late respectively.[7] Using delay minutes as a measure allows for comparisons between heavily loaded large transport vehicles, and much smaller or lightly loaded vehicles. The economic cost of a large transport vehicle being delayed is much greater than a service that is almost empty.

## Factors

There are many factors that can affect on-time performance.[8] Depending on the situation, the service may face regular delays or a service that usually performs on time may be occasionally behind schedule. Some of these factors include:

• Traffic congestion:
• Bus transport: The volume of automobile traffic on the bus's route can affect the bus's ability to keep on the schedule. While a route that is free of obstruction can remain on schedule or even move ahead of schedule, heavy traffic can slow down the bus behind its schedule. This can be due to regular conditions at the time of day, or an occasional or one-time event, such as a sports game or festival that draws a large crowd.
• Rail transport: The volume of traffic on the route can cause even more delays for rail vehicles, because of tight scheduling at sidings and other meet points. On passenger railroads, freight traffic is especially problematic because of the length and slow acceleration of the freight trains.
• Ferries are often affected by congestion at quays and piers. One report reported a third of all delays are caused by congestion.[9]
• Accident: A collision that obstructs traffic flow can also tie up vehicles on the route until the accident is cleared and therefore cause them to be behind schedule.
• Breakdown: A disabled bus or train, besides being unable to complete its own route, may slow down others that follow on the schedule. As riders scramble to board a subsequent bus or train on the schedule, the higher ridership levels on that run may slow it down. Additionally, on a rail line, a disabled train may block other trains from passing, or may require track-sharing with opposing trains, thereby slowing service in both directions.
• Detour: A road closure that forces the bus to temporarily deviate from its regular route may cause it take more time than planned to reach its destination.
• Passenger load: When a special event takes place, the service may experience higher ridership levels than usual, leading to more time the bus or train may have to spend allowing passengers to board and depart.
• Unrealistic scheduling: Many public transportation agencies are tax-subsidized, and therefore are often cash-strapped while attempting to maximize service provided to customers. In such cases, schedules that are written do not provide enough time for operators to travel along the route in the time allotted, and layovers are too short to allow enough recovery time, thereby seriously delaying the service on a regular basis.[10]
• Weather: Inclement weather may cause a bus that usually remains on schedule to be behind as a result of precautions that must be taken by the vehicle's operator and other vehicles on the road. Trains might have to operate slower as a result of slippery rail during the autumn season, and high wind can affect signal systems.
• Crewing issues. Crew may be late to work, or be at the wrong place at the wrong time. Alternatively, they may make mistakes that delay services.[11]

## Effects of poor on-time performance

The effect of delays to a transport system are normally calculated as costs in dollars, either to the passengers, or to the transport provider. The transport provider will incur costs of the additional use of the vehicle, crew costs, and fuel.[12] In many cases the cost is calculated as a dollar value per minute. The total cost of delays for an entire transport system for one year can be very large.[13]

In many publications the effect of poor on time performance is equated to lost money. As passengers are delayed, this delay is equated to dollars, and this is the amount that is lost. Calculations are performed for the total amount "lost" per year for different countries, for example, for the US it was estimated that the cost to the country was $32.9 billion in 2007.[14] The total amount of cost per country per year can be very large. An economic cost is calculated through the following formula: Economic cost ${\displaystyle ={\text{Minutes service delayed}}\times {\text{Cost per minute delay}}}$ (3) The delay cost per minute is calculated as the addition of direct costs to a service provider, and the economic cost to passengers through lost time.[15] This is: Delay cost per minute ${\displaystyle ={\text{Direct cost to service provider per minute}}+{\text{Economic cost to late customers per minute}}}$ (4) The economic cost per minute for passengers is often expressed as a percentage of the average in the area.[16] For different transport modes the cost per minute may be different, as those using buses may have a lower income than those using trains. Buses when late may experience a problem known as bus bunching. On some bus lines with a more frequent service, if one bus falls behind schedule passenger numbers waiting at bus stops may grow, required a longer layover time. One or more subsequent buses on the published schedule may pass these already cleared stops and have a nearly empty run, and may actually jump ahead of their scheduled time to the point that two or more buses are within close sight of one another. In some cases, one bus is able to pass another.[17] This phenomenon is sometimes known as clumping or bunching.[18] When this occurs, the even spacing of buses on the schedule may be severely disrupted, leading to extremely long waits for those attempting to catch a bus, and multiple buses arriving at once.[19] Bus bunching serves to reduce the effectiveness of buses as a transport mode. ## Improving schedule adherence Transit agencies often take the following measures in attempts to improve schedule adherence on their routes: • Providing better information to drivers on their schedule and on time performance. Bus and rail drivers may not know if they are on time as this information is not commonly provided. A driver advisory system (DAS) can provide better information, and can inform drivers of their correct departure and arrival time.[20] • Adding additional route capacity, so the effect of bottlenecks is reduced. Capacity constraints are common in many transport systems, and adding capacity is normally an effective way to reduce delays.[21] • Modify schedules by adding running time, known as schedule padding. This is the most common solution, but agencies often add running time by cutting layover time, which adversely affects the ability to recover from unplanned incidents. If layover time is to remain the same, it may require an increased expenditure in the system's budget or a slight reduction in the number of trips. • Increasing the number of reserve vehicles.[22] All transport vehicles can experience engineering failures, which cause them to be unavailable for service. Increasing the number of reserve vehicles increases costs, but allows vehicles to be replaced when needed. For this strategy to work vehicles need to be placed at strategic locations. • Modifying routes to a route where buses can be less obstructed, when this does not interfere with the ability of riders to reach their buses. • Building bus and rail lines which are grade separated from road traffic. • For airlines, delays are often caused by ground processing at airports, so improving the airport plane and passenger handling can have a big effect of on time performance.[23] • Splitting a long route into two or more shorter ones, as shorter routes are more likely to remain on schedule.Shorter routes generally are exposed to fewer problems.[24] • Some cities have introduced bus rapid transit services or limited stop lines on long, overcrowded routes. This involves the use of part of the route's budget to operate another line on the same route that stops only at key points. While the overall frequency of the original route is reduced, riders traveling over a longer distance have the option of obtaining a quicker trip on the limited-stop line. • Some agencies and transport companies have installed GPS devices on their buses and trains to monitor the locations of the vehicles. By linking this real-time information with stop locations and predicted travel-times, the estimated time of arrival of the next service is then displayed at some stops. • Customer interfaces: Some agencies and companies have made the GPS tracking data described above available to public transport users online, by a toll free number, or through the mobile web. ## Real-world examples The following chart shows some examples of real-world on-time performance. The figures are always (unless stated otherwise) per vehicle, not per customer. Operator Mode of transport Period < 1' < 2' < 3' < 5' < 10' < 15' < 20' < 30' Comments Source SBB All passenger rail 2014 87.7% [25] Freight 77.2% Network Rail National Railway 2014 89.3% Commuter services are late >5 min, intercity >10 min [26] Taiwan High Speed Rail High Speed Rail 2013 99.44% [27] Lufthansa All flights 2014 84.7% [28] Qantas Domestic Air Travel 2015 88.6% [29] Cathay Pacific International Air Travel 2014 70.7% [30] Yarra Trams Trams Jan 2016 86.7% [31] Hong Kong MTR Metro (rail) 2015 to October 99.9% The threshold for on time is high at 8 minutes [32] GO Transit Commuter Rail 2014-15 Fiscal Year 94% [33] Buses 95% SMRT Metro (rail) FY 2014 92.6% [34] New York City Subway Metro (rail) 2014 74% [35] ## See also ## References 1. ^ "Airlines on time statistics delays and causes". Bureau of Transport Statistics. Retrieved 6 March 2016. 2. ^ "Virgin Australia Website". Retrieved February 21, 2016. 3. ^ "Punctuality:How Airlines can Improve On time performance" (PDF). Booz Allen Hamilton. Retrieved 6 March 2016. 4. ^ "Australian Business Traveller Website". Retrieved March 4, 2016. 5. ^ "Why Amtrak on time performance is so much worse this year". Retrieved February 10, 2016. 6. ^ "Flight delays cost$39.2 billion, passengers foot half the bill". Berkeley News. University of Berkeley. Retrieved 12 March 2016.
7. ^ "Payments for disruption on the railway". Network Rail. Retrieved 12 March 2016.
8. ^ Strathman, james (April 1993). "An Empirical Analysis of Bus Transit On Time Performance". Transportation Research Part A: Policy and Practice 27 (2): 93–100.
9. ^ "Ferries: On Time performance Reporting" (PDF). Washington State Transport Commission. Retrieved 12 March 2016.
10. ^ "Metra Website". Retrieved March 1, 2016.
11. ^ "Ferries On-Time Performance Reporting" (PDF). Washington State Department of Transport. Retrieved 12 March 2016.
12. ^ "Per minute cost of delays to US airlines". Retrieved 11 March 2016.
13. ^ "Annual US Impact of Flight Delays". Airlines for America. Airlines for America. Retrieved 11 March 2016.
14. ^ "Total Delay Impact Study" (PDF). Nextor. The National Centre for Excellence for Aviation Operations Research. Retrieved 12 March 2016.
15. ^ "Standard Economic Value Guidelines" (PDF). Civil Aviation Safety Authority. Retrieved 12 March 2016.
16. ^ "Standard Economic Value Guidelines" (PDF). Civil Aviation Safety Authority. Retrieved 12 March 2016.
17. ^ "A Cure for Bus Bunching". ITS Berkeley. Retrieved 6 March 2016.
18. ^ UOR_1.2
19. ^ Debora MacKenzie (29 October 2009) “Why three buses come at once, and how to avoid it”, New Scientist. Retrieved 9 December 2014
20. ^ Yang, Li. "Achieving energy efficiency and on time performance with driver advisory systems" (PDF). IEEE Explore. IEEE International Conference on Intelligent Rail. Retrieved 6 March 2016.
21. ^ "Punctuality:How Airlines can Improve On time performance" (PDF). Booz Allen Hamilton. Retrieved 6 March 2016.
22. ^ "Punctuality:How Airlines can improve on time performance" (PDF). Booz Allen Hamilton. Retrieved 6 March 2016.
23. ^ "Punctuality:How Airlines can Improve Performance" (PDF). Booz Allen Hamilton. Retrieved 11 March 2016.
24. ^ "Transit Capacity and Quality of Service Manual" (PDF). Retrieved 11 March 2016.
25. ^ "2014 Annual Report" (PDF). Swiss Federal Railways.
26. ^ "Network Rail websiter". Network Rail.
27. ^ "2013 Annual report". Taiwan High Speed Rail Corp.
28. ^
29. ^ "Domestic Airline On Time Performance 2015" (PDF). Australian Government, Department of Infrastructure and Regional Development.
30. ^ "Cathay Pacific Interim 2014 Report" (PDF). Cathay Pacific.
31. ^ "Yarra Trams Monthly Report". Yarra Trams.
32. ^ "Hong Kong MTR Press Release" (PDF). Hong Kong MTR.
33. ^ "2015 July to Sept Quarterly report" (PDF). GO Transit.
34. ^ "Singapore Financial year 2014 Annual Report" (PDF). SMRT.
35. ^
• The compilation of statistics for on time performance is possible only for transport providers that publish these statistics. Where the performance is poor, they tend to be reluctant to publish anything. For example, Amtrak was having legal issues in 2013 with track access, and so on time performance dropped. It was very difficult to find any recent figures on the performance for Amtrak.


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Cryptology ePrint Archive: Report 2022/556

Resumable Zero-Knowledge for Circuits from Symmetric Key Primitives

Handong Zhang and Puwen Wei and Haiyang Xue and Yi Deng and Jinsong Li and Wei Wang and Guoxiao Liu

Abstract: Consider the scenario that the prover and the verifier perform the zero-knowledge (ZK) proof protocol for the same statement multiple times sequentially, where each proof is modeled as a session. We focus on the problem of how to resume a ZK proof efficiently in such scenario. We introduce a new primitive called resumable honest verifier zero-knowledge proof of knowledge (resumable HVZKPoK) and propose a general construction of the resumable HVZKPoK for circuits based on the MPC-in-the-head" paradigm, where the complexity of the resumed session is less than that of the original ZK proofs. To ensure the knowledge soundness for the resumed session, we identify a property called extractable decomposition. Interestingly, most block ciphers satisfy this property and the cost of resuming session can be reduced dramatically when the underlying circuits are implemented with block ciphers. As a direct application of our resumable HVZKPoK, we construct a post quantum secure stateful signature scheme, which makes Picnic3 suitable for blockchain protocol. Using the same parameter setting of Picnic3, the sign/verify time of our subsequent signatures can be reduced to 3.1%/3.3% of Picnic3 and the corresponding signature size can be reduced to 36%. Moreover, by applying a parallel version of our method to the well known Cramer, Damgaard and Schoenmakers (CDS) transformation, we get a compressed one-out-of-$N$ proof for circuits, which can be further used to construct a ring signature from symmetric key primitives only. When the ring size is less than $2^4$, the size of our ring signature scheme is only about 1/3 of Katz et al.'s construction.

Category / Keywords: public-key cryptography / Resumable, honest verifier zero-knowledge, MPC-in-the-head, stateful signature, ring signature, blockchain.

Date: received 6 May 2022, last revised 11 May 2022

Contact author: hdzhang at mail sdu edu cn, pwei at sdu edu cn

Available format(s): PDF | BibTeX Citation

Note: references typos

Short URL: ia.cr/2022/556

[ Cryptology ePrint archive ]


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Subscribe
Issue No.06 - June (1981 vol.3)
pp: 626-633
Tsvi Dubitzki , Computer Vision Laboratory, Computer Science Center, University of Maryland, College Park, MD 20742; ELEX Control Systems, Israel.
Angela Y. Wu , Computer Vision Laboratory, Computer Science Center, University of Maryland, College Park, MD 20742; Department of Mathematics, Statistics, and Computer Science, American Universit
Azriel Rosenfeld , FELLOW, IEEE, Computer Vision Laboratory, Computer Science Center, University of Maryland, College Park, MD 20742.
ABSTRACT
Given a binary image stored in a cellular array, a local reconfiguration process can be used to reconnect some of the cells into a quadtree network representing the image. This quadtree can also be roped,'' i.e., nodes representing adjacent image blocks of the same size can be joined. Using the roped quadtree network as a parallel (cellular) computer, image properties such as perimeter and genus, as well as the quadtree distance transform, can be computed in O(tree height) = O(log image diameter) time. The area and centroid of the image can be computed in O(height) time without the need for roping.
CITATION
Tsvi Dubitzki, Angela Y. Wu, Azriel Rosenfeld, "Parallel Region Property Computation by Active Quadtree Networks", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.3, no. 6, pp. 626-633, June 1981, doi:10.1109/TPAMI.1981.4767163


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# Finite-state machine problem

livcon

## Homework Statement

Construct a finite-state machine that changes every other bit, starting with the second bit, of an input string, and leaves the other bits unchanged.

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## The Attempt at a Solution

Say I have the string 1001101, then I am supposed to make a finite-state machine that turns it into 1100111 (every other bit changed, starting from the 2nd). I know how to change the bits in a machine, what I need help for is how to determine the index of the input, that is how to let the finite-state machine know i.e. that 0 which happens to be the second bit of the string, is in fact the second.
Thanks.

I realized that this problem is quite easy and requires only two states, $$s_{0}$$ and $$s_{1}$$, using the first one as initial state. When the machine receives either 0 or 1 in the initial state, the output function simply returns the same bit as the input as it changes state from $$s_{0}$$ to $$s_{1}$$. From this state the bit changes on its way back to state $$s_{0}$$.


text:
# Question marks in a reference to a table?

I have this code and when I compile (even several times) some question marks appear on the pdf in the place where the reference should be.

Actually if we perform an Augmented Dickey-Fuller test on the in-sample we refuse the null hypothesis of explosive process with a p-value of 0.8226.
This applies to all the models here presented, AR(1), AR(3) and
ARMA(1,1). Model coefficients can be inspected in Table
\ref{tab:daily_model_coefficients}
\medskip

\begin{figure}[H]
\centering
\caption{Model coefficients of the Daily Component Models\label{tab:daily_model_coefficients}}
\begin{tabular}{c|c|c|c|c|c}
\hline \hline
$Model$ & \phi_{0} & \phi_{1} & \phi_{2} & \phi_{3} & \theta_{1}\\
\hline
AR(1)   & 6.5397    & 0.8988 &- &- & - \\
AR(3) & 6.2932  & 0.7257    & -0.0099   & 0.2162 &- \\
ARMA(1,1) & 6.0340  & 0.9741    & -&    -   & -0.4991 \\
\hline
\end{tabular}
\end{table}

So the point is that when I compile several times I get the following:

Actually if we perform an Augmented Dickey-Fuller test on the in-sample we refuse the null hypothesis of explosive process with a p-value of 0.8226. This applies to all the models here presented, AR(1), AR(3) and ARMA(1,1). Model coefficients can be inspected in Table ??

I am using Sharelatex now. I will appreciate any suggestion

• I have already tried writing the \caption before \label and it doesn't work... – adrian1121 Jun 6 '16 at 21:02
• If you get a tex error you should just look at the log, tex never tries to make sensible typeset output after an error, it just recovers enough to syntax-check the rest of the document. – David Carlisle Jun 6 '16 at 21:07

In order for your references to work properly (on ShareLaTeX), you need to make the document compile without error. You'll note that in your example you started with a figure but ended with a table:

\begin{figure}[H]
% ...
\end{table}

Also, tabular header row contains math content without you inserting math mode properly.

This is what you need:

\documentclass{article}

\begin{document}

Actually if we perform an Augmented Dickey-Fuller test on the in-sample we
refuse the null hypothesis of explosive process with a p-value of 0.8226.
This applies to all the models here presented, AR(1), AR(3) and
ARMA(1,1). Model coefficients can be inspected in Table
\ref{tab:daily_model_coefficients}.

\begin{table}
\centering
\caption{Model coefficients of the Daily Component Models\label{tab:daily_model_coefficients}}
\begin{tabular}{c|c|c|c|c|c}
\hline \hline
Model & $\phi_0$ & $\phi_1$ & $\phi_2$ & $\phi_3$ & $\theta_1$ \\
\hline
AR(1)   & 6.5397    & 0.8988 &- &- & - \\
AR(3) & 6.2932  & 0.7257    & -0.0099   & 0.2162 &- \\
ARMA(1,1) & 6.0340  & 0.9741    & -&    -   & -0.4991 \\
\hline
\end{tabular}
\end{table}

\end{document}

Consider using booktabs for your tabular representations.

• thank you very much for your help!! this worked perfectly!! – adrian1121 Jun 6 '16 at 21:09

A nuance, perhaps, but I am under the impression that the recommended practice is to place the label after the caption:

\documentclass{article}

\begin{document}

Actually if we perform an Augmented Dickey-Fuller test on the in-sample we refuse the null hypothesis of explosive process with a p-value of 0.8226.
This applies to all the models here presented, AR(1), AR(3) and
ARMA(1,1). Model coefficients can be inspected in Table
\ref{tab:daily_model_coefficients}
\medskip
\begin{table}[htbp]
\centering
\begin{tabular}{c|c|c|c|c|c}
\hline \hline
$Model$ & $\phi_{0}$ & $\phi_{1}$ & $\phi_{2}$ & $\phi_{3}$ & $\theta_{1}$ \\
\hline
AR(1)   & 6.5397    & 0.8988 &- &- & - \\
AR(3) & 6.2932  & 0.7257    & -0.0099   & 0.2162 &- \\
ARMA(1,1) & 6.0340  & 0.9741    & -&    -   & -0.4991 \\
\hline
\end{tabular}
\caption{Model coefficients of the Daily Component Models}
\label{tab:daily_model_coefficients}
\end{table}

\end{document}

Which produces this using ShareLaTeX:

• no actually it is better to put the label in the caption. – David Carlisle Jun 6 '16 at 21:11
• @DavidCarlisle interesting, i hadn't heard that. if that is the case then i stand corrected. – aeroNotAuto Jun 6 '16 at 21:13
• I was just finding this answer... tex.stackexchange.com/a/264115/1090 – David Carlisle Jun 6 '16 at 21:14
• I had the problem and writing \label after \caption fixed the issue. Weird... – MehmedB Jul 16 at 19:15


text:
## Duke Mathematical Journal

### Nonclassical multidimensional viscous and inviscid shocks

#### Abstract

Extending our earlier work on Lax-type shocks of systems of conservation laws (see [GM+1], [GM+2], [GM+4]), we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general Lax- or undercompressive-type shock waves of nonconservative hyperbolic systems with parabolic regularization. The hyperbolic equations may be of variable multiplicity, and the parabolic regularization may be of “real,” or partially parabolic, type. We prove an existence result for inviscid nonconservative shocks which extends a one-dimensional result of X to multidimensional shocks. Lin [L] proved by quite different methods. In addition, we construct families of smooth viscous shocks converging to a given inviscid shock as viscosity goes to zero, thereby justifying the small viscosity limit for multidimensional nonconservative shocks.

In our previous work on shocks, we made use of conservative form, especially in parts of the low-frequency analysis. Thus, most of the new analysis of this article is concentrated in this area. By adopting the more general nonconservative viewpoint, we are able to shed new light on both the viscous and inviscid theories. For example, we can now provide a clearer geometric motivation for the low-frequency analysis in the viscous case. Also, we show that one may, in the treatment of inviscid stability of nonclassical and/or nonconservative shocks, remove an apparently restrictive technical assumption made by Mokrane [Mo] and Coulombel [C] in their work on, respectively, shock-type nonconservative boundary problems and conservative undercompressive shocks. Another advantage of the nonconservative perspective is that Lax and undercompressive shocks can be treated by exactly the same analysis

#### Article information

Source
Duke Math. J., Volume 142, Number 1 (2008), 1-110.

Dates
First available in Project Euclid: 27 March 2008

https://projecteuclid.org/euclid.dmj/1206642064

Digital Object Identifier
doi:10.1215/00127094-2008-001

Mathematical Reviews number (MathSciNet)
MR2397883

Zentralblatt MATH identifier
1162.35050

Subjects
Secondary: 76N17: Viscous-inviscid interaction

#### Citation

Guès, Olivier; Métivier, Guy; Williams, Mark; Zumbrun, Kevin. Nonclassical multidimensional viscous and inviscid shocks. Duke Math. J. 142 (2008), no. 1, 1--110. doi:10.1215/00127094-2008-001. https://projecteuclid.org/euclid.dmj/1206642064

#### References

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text: If S is positive definite, there exists a unique tensor U such that U 2 = S → U = S. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): It has been recently pointed out that a definition of geometric entropy using a partition function in a conical space does not in general lead to positive definite quantity. If a symmetrica In dynamical systems, a real-valued, continuously differentiable function f can be called positive-definite on a neighborhood D of the origin if Positive Deﬁnite Matrices and the SVD 397 Positive energy is equivalent to positive eigenvalues, when S is symmetric. is positive semi-definite (which requires A to be Hermitian; therefore f(−x) is the complex conjugate of f(x)). Main article: Bochner's theorem. DEFINE_COM_1608981449.html. This is just one way to show that it's positive definite. I) dIiC fifl/-, Juste une remarque que, dans la semi-définie positive cas, numériquement parlant, on peut aussi ajouter un peu de l'identité de la matrice (donc changer toutes les valeurs propres d'une petite quantité par exemple un peu de temps à la machine de précision), puis utilisez la méthode de cholesky comme d'habitude. Comments. If the quadratic form is positive for all values of x and y, then our stationary point must be a minimum, and we say that the (Hessian) matrix is positive definite. Typically, n scalar measurements of some scalar value at points in In this context, Fourier terminology is not normally used and instead it is stated that f(x) is the characteristic function of a symmetric probability density function (PDF). See also: Positive But for D to be positive definite, we have to show that x transpose D*x is bigger than 0. L'influence de la télévision sur les jeunes. The matrix in the middle of expression [3] is known as the Hessian. Hence there is no … Positive quantity synonyms, Positive quantity pronunciation, Positive quantity translation, English dictionary definition of Positive quantity. 11. [2][3] In physics, the requirement that An n × n complex matrix M is positive definite if ℜ(z*Mz) > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z and ℜ(c) is the real part of a complex number c. An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z. The only way in which the volume integral of a positive definite quantity can be zero is if that quantity itself is zero throughout the volume. Numbers or symbols proceeded by the sign ‘+’ or no sing are called positive quantities. f the energy density acquires It has been recently pointed out that a definition of the geometric entropy using the partition function in a conical space does not in general lead to a positive-definite quantity. {\displaystyle f(x)>0} 28 sentence examples: 1. definite quantity synonyms, definite quantity pronunciation, definite quantity translation, English dictionary definition of definite quantity. This lecture covers how to tell if a matrix is positive deﬁnite, what it means for it to be positive deﬁnite, and some geometry. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. Positive definite matrix occupies a very important position in matrix theory, and has great value in practice. This quantity is an example of what is known as a quadratic form (in that the highest power of x or y present is two). And we see that we get a sum of squares. 0 And my question is--is that quantity positive or not? See, in particular: Index of articles associated with the same name, "Positive definite kernels: Past, present and future", https://en.wikipedia.org/w/index.php?title=Positive_definiteness&oldid=961643038, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, This page was last edited on 9 June 2020, at 16:48. The quantity z*Mz is always real because Mis a Hermitian matrix. 3. The converse result is Bochner's theorem, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure. f d Before we prove this theorem, it’s worth noting a few points that are immediately intuitive from its statement. S is definite positive iff its principal values are positive, so iff the principal invariants of S are positive.. Theorem. Note that is a positive definite quantity. Definition of electric charge. It cannot be positive definite if P is singular since then a may be chosen such that Pa = o and, hence, aTPTVPa = 0 for a # 0. The flux is in general not a scalar quantity, because it is described by the magnitude and the direction as well. GOOD: BAD: SERIOUS: CRITICAL: NEUTRAL: Definite Quantity . Prove that every diagonal element of a symmetric positive-definite matrix is positive. Definition of "Definite Quantity" at Define.com Simple Psychedelic Plain Text English Dictionary with Hyperlinks to The Free World Bank - A BIG Thinking Scientific Save the World High Level Concept on Amazon S3. https://ocw.mit.edu/.../lecture-25-symmetric-matrices-and-positive-definiteness D Positive quantity synonyms, Positive quantity pronunciation, Positive quantity translation, English dictionary definition of Positive quantity. positive if a # o and equal to zero if a = o since V is positive definite. So the system does have a definite position, a definite momentum, definite energy and so forth. To see this, let us recall that a function K: X × X → C, where X is any set, is called a positive definite kernel if for any points x 1, …, x m ∈ X and any c 1, …, c m ∈ C we have that (2) ∑ j, k = 1 m K (x j, x k) c j c k ¯ ≥ 0. Let S be a second order positive definite tensor, i.e. = A positive-definite function of a real variable x is a complex-valued function {\displaystyle x\in D} Positive definition: If you are positive about things, you are hopeful and confident , and think of the good... | Meaning, pronunciation, translations and examples The system has always definite values of all physical quantities. An automatic pet food dispenser at preset times and for a definite quantity includes a dispenser body, a food storing tank, a funnel-shaped receiver, a transporting mechanism. Comments. Positive definite matrices and the S-divergence. may be dropped (see, e.g., Corney and Olsen[4]). x ) In plain English, this theorem states that, assuming $$A$$ and $$B$$ are both positive definite, $$A-B$$ is positive semi-definite if and only if the inverse of $$B$$ minus the inverse of $$A$$ is positive semi-definite. An automatic pet food dispenser at preset times and for a definite quantity includes a dispenser body, a food storing tank, a funnel-shaped receiver, a transporting mechanism. $\begingroup$ Interesting, so if both A and B are semi-positive definite, does that rearrangement guarantee that quantity is >= 0. having fixed limits; bounded with precision: a definite area. Positive values of h X help create nonaffine rearrangements away from the reference configuration. Bochner's theorem. an affirmative quantity, or one affected by the sign plus . positive definite quantity. #top. 2. Whilst those proceeded by the sign ‘-‘ are called negative quantities. {\displaystyle f(0)=0} If this quantity is positive, if, if, if, it's positive for all x's and y's, all x1 x2s, then I call them--then that's the matrix is positive definite… self-adjoint) operator such that $\langle Ax, x\rangle > 0$ for all $x \neq 0$. Because it looks like you are running a 1-sample ttest (which generally has null hypothesis that the mean of the inputs is zero, and the alternative hypothesis being that they are *not* zero), and mALFF is a positive definite quantity whos values should always be positive-- so I wouldn't see it ever having a chance to be "zero mean" in any group. such that (u, S.u) ≥ 0, ∀u, and (u,S.u) = 0 ⇔ u = 0.It turns out that only the symmetrical part of S plays a role.. Theorem. {\displaystyle R^{d}} {\displaystyle f(0)=0} 7.2. > ( See, in particular: Positive-definite bilinear form; Positive-definite quadratic form; Positive-definite matrix; Positive-definite function; Positive-definite kernel; Positive-definite function on a group; References In positive and negative quantities, quantity is used in the sense of numbers. locally compact abelian topological group, "Non-Gaussian pure states and positive Wigner functions", https://en.wikipedia.org/w/index.php?title=Positive-definite_function&oldid=998568480, Articles with empty sections from August 2017, Creative Commons Attribution-ShareAlike License. ∈ : Buy Positive Definite Matrices (Princeton Series in Applied Mathematics) by Bhatia, Rajendra (ISBN: 9780691129181) from Amazon's Book Store. The first step is to show that the energy spectrum of a R + R2 theory is qualitatively the same as in the purely linear theory, i.e. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. In particular, it is necessary (but not sufficient) that, (these inequalities follow from the condition for n = 1, 2.). R 28.3 Symmetric positive-definite matrices and least-squares approximation 28.3-1. : a definite quantity of electricity, either negative or positive, usually regarded as a more or less localized population of electrons separated or considered separately from their corresponding protons or vice versa : the quantity of electricity held by a body and construed as an excess or deficiency of electrons. ( Since $ab$ denotes the flux from $A$ to $B$, then the information of direction is encoded in the ordering of the characters $a$ and $b$. Let me test the energy xTSx in all three examples. ) Positive deﬁnite matrices and minima Studying positive deﬁnite matrices brings the whole course together; we use pivots, determinants, eigenvalues and stability. Let me test the energy xTSx in all three examples. The new quantity here is xTAx; watch for it. the theory of unitary representations). 7.2. such that for any real numbers x1, …, xn the n × n matrix. Therefore, PTVP is positive definite if P is nonsingular. A matrix is positive definite fxTAx > Ofor all vectors x 0. Positive definition: If you are positive about things, you are hopeful and confident , and think of the good... | Meaning, pronunciation, translations and examples and A real valued function $f: X \rightarrow \mathbb{R}$ on an arbitrary set $X$ is called positive-definite if $f(x)>0, \forall x \in \mathcal{X}$. De très nombreux exemples de phrases traduites contenant "definite quantity" – Dictionnaire français-anglais et moteur de recherche de traductions françaises. Positive definite matrices abound in a dazzling variety of applications. an affirmative quantity, or one affected by the sign plus . ) 0 A function is semidefinite if the strong inequality is replaced with a weak (≤, ≥ 0). Frequently in physics the energy of a system in state x is represented as XTAX (or XTAx) and so this is frequently called the energy-baseddefinition of a positive definite matrix. Everyday low prices and free delivery on eligible orders. Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0. . rn … Positive-definite functions on groups occur naturally in the representation theory of groups on Hilbert spaces (i.e. x In mathematics, a positive-definite function is, depending on the context, either of two types of function. noun. Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.. In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. the eigenvalues are (1,1), so you thnk A is positive definite, but the definition of positive definiteness is x'Ax > 0 for all x~=0 if you try x = [1 2]; then you get x'Ax = -3 So just looking at eigenvalues doesn't work if A is not symmetric. The matrix A can be positive definite only if n+n≤m, where m is the first dimension of K.” (Please could you refer me to an articles or books where I can find such property above). In statistics, and especially Bayesian statistics, the theorem is usually applied to real functions. Example-Prove if A and B are positive definite then so is A + B.) The present demonstration will ultimately rely on Witten's proof given in [2]. positive; certain; sure: It is definite that he will take the job. One strategy is to define a correlation matrix A which is then multiplied by a scalar to give a covariance matrix: this must be positive-definite. Positive semi - definite matrices are positive definite if and only if they are nonsingular. This includes the diagonal operator, which acts on a basis $( e _ {n} ) _ {n=} 1 ^ \infty$ of a Hilbert space as $Ae _ {n} = n ^ {-} 1 e _ {n}$. Specific rearrangements, such as a slip by a lattice spacing, map the crystal onto itself and do not change lattice symmetry but still contribute to the energy H for nonzero h X. → Positive Deﬁnite Matrices and the SVD 397 Positive energy is equivalent to positive eigenvalues, when S is symmetric. One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context. = This completes the proof. Thus each of the expressions, 4, +6, , are positive quantities and -4, -6, , are negative quantities. Define definite quantity. More generally, a positive-definite operator is defined as a bounded symmetric (i.e. That's--for every x1 and x2, that is my new definition--that's my definition of a positive definite matrix. Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g ( y) ≥ 0. See Kriging. ( Positive definite functions constitute a big class of positive definite kernels. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function f), then function f must be positive-definite to ensure the covariance matrix A is positive-definite. Sorry B can't be PSD $\endgroup$ – user2879934 Jul 13 '19 at 16:12 This definition makes some properties of positive definite matrices much easier to prove. clearly defined or determined; not vague or general; fixed; precise; exact: a definite quantity; definite directions. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manif Informationsquelle Autor NPE. The absolute value of a positive or a negative quantity is its value considered a part of its sign. So if I write x out using components, x_1, x_2, dot, dot, dot, to x_n--I'll write it like this--then you can work out the quantity x transpose D*x. R From WordNet (r) 3.0 (2006) [wn]: definite quantity. 2. 1 definition found. It only takes a minute to sign up. ∙ Max Planck Society ∙ 0 ∙ share . More generally, a positive-definite operator is defined as a bounded symmetric (i.e. Any positive-definite operator is a positive operator. Christian Berg, Christensen, Paul Ressel. self-adjoint) operator such that $\langle Ax, x\rangle > 0$ for all $x \neq 0$. are taken and points that are mutually close are required to have measurements that are highly correlated. 10/08/2011 ∙ by Suvrit Sra, et al. Many translated example sentences containing "definite quantity" – French-English dictionary and search engine for French translations. Any positive-definite operator is a positive operator. This page was last edited on 6 January 2021, at 00:26. See also: Positive Webster's Revised Unabridged Dictionary, published 1913 by G. & C. Merriam Co In practice, one must be careful to ensure that the resulting covariance matrix (an n × n matrix) is always positive-definite. The following definition conflict with the one above. A function is negative definite if the inequality is reversed. for every non-zero Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Thus if stands for 5 and stands for 3, and have the same absolute value, which is 15 3… f It's not the only way. Note: 1. Define Positive quantity. 0 In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. For a scalar field model with a non-minimal coupling we clarify the origin of the anomalous behaviour from the … Action, généralement continue, qu'exerce quelque chose sur quelque chose ou sur quelqu'un : L'influence du climat sur la végétation. C 0 The converse result is Bochner's theorem, stating that any continuous positive-definite function on the real line is the Fourier transform of a (positive) measure.[1]. {\displaystyle f:\mathbb {R} \to \mathbb {C} } 0 The “energy in a small disturbance” in a viscous compressible heat-conductive medium is defined as a positive definite quantity characterizing the mean level of fluctuation in the disturbance which, in the absence of heat transfer at the boundaries and of work done by boundary forces or body forces, and in the absence of heat and material sources, is a monotone non-increasing function of time. f Définitions de influence. But both experimentally and theoretically we can't access this data. An n × n real matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries (), where zT denotes the transpose of z. ( 2006 ) [ wn ]: definite quantity ou sur quelqu'un: L'influence du climat sur la.. 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text:
# a) Find all vectors v such that (1, 2, 1) \times v = (3, 1, -5). Describe the shape these vectors...

## Question:

a) Find all vectors {eq}v {/eq} such that {eq}(1, 2, 1) \times v = (3, 1, -5) {/eq}. Describe the shape these vectors form.

b) Explain why there is no vector {eq}v {/eq} such that {eq}(1, 2, 1) \times v = (3, 1, 5) {/eq}.

## Operations of Vectors

The cross product of two vectors is a vector orthogonal on both vectors involved in the operation and defined as

{eq}\displaystyle \langle a_1,b_1,c_1 \rangle \times \langle a_2,b_2,c_2\rangle= \begin{array}{|ccc|} \mathbf{i} &\mathbf{j}&\mathbf{k}\\ a_1&b_1&c_1\\ a_2&b_2&c_2 \end{array} =\langle b_1c_2-b_2c_1, -a_1c_2+a_2c_1, a_1b_2-a_2b_1\rangle. {/eq}

To check if two vectors are orthogonal, we can check if their dot product is zero,

{eq}\displaystyle \langle a_1,b_1,c_1\rangle \cdot \langle a_2,b_2,c_2\rangle =a_1a_2+b_1b_2+c_1c_2\\ \displaystyle \text{ or }\vec{u} \cdot \vec{v} =|\vec{u}||\vec{v}|\cos \theta, \text{ where }\theta \text{ is the angle between the two vectors}. {/eq}

a) To find the vector {eq}\displaystyle \vec{v}=\langle a, b, c \rangle {/eq} such that {eq}\displaystyle \langle 1,2,1\rangle \times \langle a,b,c\rangle=\langle 3,1,-5\rangle {/eq}

we will use the cross product rule, as below.

{eq}\displaystyle \begin{align}\langle 1,2,1\rangle \times \langle a,b,c\rangle&=\langle 3,1,-5\rangle\\ \left|\begin{array}{ccc} 1 &2&1\\ a&b&c \end{array}\right|&=\langle 3,1,-5\rangle\\ \langle 2c-b, -c+a, b-2a\rangle&=\langle 3,1,-5\rangle\\ \implies &\begin{cases}&-b&+2c&=3\\ a& &-c&=1\\ -2a&+b& & =-5 \end{cases}\iff &\begin{cases} a& &-c&=1\\ &-b&+2c&=3\\ -2a&+b& & =-3 \end{cases}\iff &\begin{cases} a& &-c&=1\\ &-b&+2c&=3\\ &b&-2c & =-3 \end{cases}\iff \begin{cases} a& &-c&=1\\ &-b&+2c&=3\\ && 0& =0 \end{cases}\\ \implies &b=-3+2c \text{ and }a=1+c \text{ for any }c\\ \implies &\boxed{\vec{v}=\langle 1+c, -3+2c, c \rangle, c\in\mathbb{R}}. \end{align} {/eq}

So, we obtain a set of vectors {eq}\displaystyle \vec{v} {/eq} that are perpendicular on the two vectors, {eq}\displaystyle \langle 1,2,1\rangle, \langle 3,1,-5\rangle, {/eq}

therefore, the set of vectors are in a plane perpendicular on the plane determined by the two vectors.

b) There is no vector {eq}\displaystyle \vec{v} {/eq} such that {eq}\displaystyle \langle 1,2,1\rangle \times \langle a,b,c\rangle=\langle 3,1,5\rangle {/eq}

because the cross product is perpendicular on the two vectors in the cross product

and {eq}\displaystyle \langle 3,1,5\rangle, \langle 1,2,1\rangle {/eq} are not orthogonal, since their dot product is nonzero {eq}\displaystyle \langle 3,1,5\rangle \cdot \langle 1,2,1\rangle=3+2+5\neq 0. {/eq}


text:
The enzyme carbonic anhydrase catalyzes the following reaction

CO2+ H20 --- HCO3- + H+

This reaction is important in the transport of carbon dioxide by red blood cells. Carbonic anhydrase has a molecular weight of 30,000g/mol and a turnover number of approximately 1 X 106 per second. Assume you are given a solution that is saturated with CO2 containing 2�g of carbonic anhydrase. At what rate ( in millimoles of CO2 consumed per second) will this reaction proceed under optimal conditions?


text:
# 600 followers problem

Algebra Level 2

Evaluate

$\large{ \frac { { (1+i) }^{ 85 } }{ { (1-i) }^{ 83 } }}$

Hint: There's an extremely fast way to solve this if you know why
$$\large 1+i = \sqrt{2}e^{i{\pi}/{4}}$$

×


text:
# Related to the Muscular System Sliding Filament Theory, what molecule must bind to the cross bridge in order for it to disconnect with actin?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Maxwell Share
Jul 21, 2017

$\text{ATP}$

#### Explanation:

$\text{ATP}$ plays a key role in allowing the detachment of the myosin-actin linkage.

Once detached, $\text{ATP}$ is hydrolyzed into $\text{P"_"i" + "ADP}$ which allows the myosin head to recock back into its high-energy position for a whole new cycle.

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text:
# Wrong result when realizing fixed-point biquad IIR filter

I'm implementing a fixed-point direct form I biquad filter with Q1.15 precision. I use fraction saving to improve quantization error. However, my function gives me the different results compared by MATLAB's filter. I didn't find anything wrong, please help. My code is as follows

void iir_biquad_df1_16(
const short* pCoeffs,             /* b0, b1, b2, a1, a2 */
short* pStates,                   /* x[n-1], x[n-2], y[n-1], y[n-2], fraction_state */
const short* pSrc,                /* Input buffer */
short* pDst,                      /* Output buffer */
unsigned int blockSize,           /* Frame length */
const unsigned int postShift,     /* Post shift when coefficients out of range [-1,1) */
const unsigned int nSections      /* Number of cascaded sections */
)
{
short* pIn = pSrc;                              /* Source pointer */
short* pOut = pDst;                             /* Destination pointer */
short* pState = pStates;                        /* State pointer */
const short* pCoeff = pCoeffs;                  /* Coefficient pointer */
int acc;                                        /* Accumulator */
int b0, b1, b2, a1, a2;                         /* Filter coefficients */
int x1, x2, y1, y2;                             /* Filter state */
int xIn;                                        /* Temporary input */
unsigned int shift = 15U - postShift;           /* Post shift for output */
unsigned int mask = (1 << shift) - 1;           /* Fraction saving */
int saturation = (0x00008000 << shift) - 1;
unsigned int samples, section = nSections;      /* Loop counters */

do
{
/* Read the coefficients */
b0 = (int)(*pCoeff++);
b1 = (int)(*pCoeff++);
b2 = (int)(*pCoeff++);
a1 = (int)(*pCoeff++);
a2 = (int)(*pCoeff++);

/* Read the state values */
x1 = (int)(*pState++);
x2 = (int)(*pState++);
y1 = (int)(*pState++);
y2 = (int)(*pState++);
acc = (int)(*pState);
pState -= 4;

samples = blockSize;

while (samples > 0U)
{
xIn = (int)(*pIn++);
/* acc =  b0*x[n] + b1*x[n-1] + b2*x[n-2] - a1*y[n-1] - a2*y[n-2] */
acc += b0*xIn + b1*x1 + b2*x2 - a1*y1 - a2*y2;

if (acc >  saturation) acc =  saturation;         /* saturate if necessary */
if (acc < -saturation) acc = -saturation;

x2 = x1;
x1 = xIn;
y2 = y1;
y1 = acc >> shift;

*pOut++ = (short)y1;
acc &= mask;

samples--;
}

/* Store the updated state variables back into the pState array */
*pState++ = (short)x1;
*pState++ = (short)x2;
*pState++ = (short)y1;
*pState++ = (short)y2;
*pState++ = (short)acc;

/* Subsequent sections take previous output buffer as input */
pIn = pDst;
pOut = pDst;

} while (--section);
}

I have 100 samples for input signal, and two cascaded biquad IIR filters. The block size is 50 so I have 2 blocks to process in order. The main function for test is given by

int main()
{
short x[100] = { 10313,13297,-12223,13546,4337,-13188,-7258,1536,14992,15233,-11219,15420,14980,-479,9840,-11735,-2564,13623,9575,15057,5103,-15214,11440,14221,5857,8446,7967,-3532,5095,-10775,6752,-15341,-7310,-14871,-13201,10599,6384,-5993,14753,-15255,-2007,-3881,8700,9673,-10261,-335,-1783,4794,6860,8346,-7339,5888,5082,-11056,-12485,-54,15065,-5230,2794,-9050,8234,-8025,195,6523,12809,15050,1547,-11842,-11492,-7946,11165,-8052,10298,-8404,14066,-4916,-9942,-8156,3803,-875,-4861,10841,2794,1629,13671,-7018,8428,8314,-3918,2222,-13898,-14616,1009,9148,14222,-12127,2255,-1003,-15994,-5337 };
short y[100] = { 0 };
short coeffs[5 * 2] = { 575, -1150, 575, -31232, 14931, 575, 1150, 575, -32621, 16238 };
short states[5 * 2] = { 0 };
unsigned int blockSize = 50;
unsigned int postShift = 1;
unsigned int nSections = 2;
// the first block of 50 samples
iir_biquad_df1_16(coeffs, states, x, y, blockSize, postShift, nSections);
// the second block of 50 samples
iir_biquad_df1_16(coeffs, states, x + blockSize, y + blockSize, blockSize, postShift, nSections);

return 0;
}

The input signal is random noise generated in MATLAB and the filter coefficients is also designed in MATLAB. The MATLAB validation code is

% clear
%% filter design
fs = 48000;
f1 = 50;
f2 = 600;
w1 = f1 / (fs / 2);
w2 = f2 / (fs / 2);
N = 2;
[b_bp, a_bp] = butter(N, [w1, w2], 'bandpass');

%% convert to cascade biquad
[sos,g] = tf2sos(b_bp, a_bp);
b1 = sos(1,1:3) * sqrt(g); a1 = sos(1, 4:6);
b2 = sos(2,1:3) * sqrt(g); a2 = sos(2, 4:6);

postShift = 1;
b1_q14 = int16(b1*2^14);
b2_q14 = int16(b2*2^14);
a1_q14 = int16(a1*2^14);
a2_q14 = int16(a2*2^14);

%% input signal generation
rng(0)
x = rand(1, 100) - 0.5; % uniform distribution in range (-0.5, 0.5)
x_q15 = int16(x*2^15);
x = x';

%% output signal
y1 = filter(b1, a1, x);
y2 = filter(b2, a2, y1);
y1_q15 = int16(y1*2^15);
y2_q15 = int16(y2*2^15);

Since some of the filter coefficients are out of range of $$[-1, 1)$$, I impose a post shift that shrinks all filter coefficients in the range of $$[-1, 1)$$. Thus the conversion to fixed-point is multiplied by $$2^{14}$$ and gives a Q2.14 precision.

After debugging, I found that the output signal of the first biquad section seems to be correct, but after cascading the second section, the final output is not the same as MATLAB gives. This problem has bothered me for two days. Any advice would be greatly appreciated.

Edit: I plot the signal before and after the second section of biquad filter. The input signal is random noise

and the filtered signal after the first secion of filter is

We can see that the C output is nearly identical with the MATLAB output. However, after the second biquad, they are quite different.

Does it mean that 16-bit word length is not long enough for my application (audio processing)?

• If the first block works fine, then the code probably works. Is your output on the same format as your input? It should be...
– Ben
Nov 3, 2021 at 13:32
• @PeterK. No, I got the same result. The filter states are just two previous samples of input and output, I didn't find anything wrong with the states. Nov 4, 2021 at 1:35
• Instead of b0, b1, b2, a1, and a2 being short and you casting them to int in the inner computation (which is repeated), why not declare those coef variables as int and cast them when loading them from the coef array? Nov 5, 2021 at 4:11
• @robertbristow-johnson and btw, fraction saving is learned from one of your answers : ) I've learned a lot on this site! Nov 5, 2021 at 6:12
• I think it's coefficient quantization. Your input signal is $\pm$0.5 and your output signal is about $\frac{1}{30}$ of the size. Why not scale the b0, b1, b2 coefficients 20 times bigger before quantizing them to the nearest integer and see if you get better agreement between the MATLAB and C filtering? Nov 6, 2021 at 6:21

## 1 Answer

The C Implementation uses first biquad states for the second. Whereas the matlab implementation has initial states zero for both the biquads. Try using two separate state variables for each of the biquads.

Thanks!

• Thank you for response! When I complete filtering and then store the filter states to the buffer, I use *pState++. So I don't think I'm using the first states for the second section. The pState array has a length of 5 * nSections. Nov 5, 2021 at 1:39
• the *pState++ variable that you are using is local to the biquad function - iir_biquad_df1_16. So the states variable in the calling function's value does not get changed. Request you to try printing the states pointer value before calling the biquad function call. Thanks! Nov 5, 2021 at 13:00
• One more way to verify this is by adding a memset statement as below just before the second biquad call. memset(&states[0], 0, sizeof(states)); This resets the states variable and is expected to make the C implementation in line with matlab. Thanks! Nov 5, 2021 at 13:04
• // the *pState++ variable that you are using is local to the biquad function - iir_biquad_df1_16 // that pState variable is local to the biquad function, but it is correctly inited to pStates value passed to it from main() which is the array states. Nov 6, 2021 at 6:15
• Found the mistake in my understanding. Apologies for the confusion and thanks for helping me understand @robertbristow-johnson Nov 7, 2021 at 9:02