text:
# Extending an incomplete market to generate a complete one

I am asking a question related to some comments and answers I have seen in the site while investigating characteristics of incomplete markets $-$ see for example @AFK 's answer in How to choose a risk-neutral measure when the market is incomplete?.

Let's assume we have a market $\mathcal{M}$ with $n$ tradeable assets $S \equiv (S^{(1)}, \cdots, S^{(n)})$ and $n+1$ sources of risk, e.g. Brownian motions $W \equiv (W^{(1)}, \cdots, W^{(n+1)})$ $-$ for example assets up to $n-1$ might follow a straightforward Geometric Brownian Motion (GBM), while the $n^{th}$ asset has Heston-like stochastic volatility dynamics. This market is incomplete because the number of (tradeable) assets is lower than the number of (sources of) risks.

Now, WLOG let's assimilate risk sources to Brownian motions and let $W^{(n+1)}$ be the "extra" source of risk. Assume we have built the market model to price a derivative security $f \equiv f(W^{(n+1)})$, meaning that the derivative payoff depends on the risk source $W^{(n+1)}$:

• If we now extend our market, let's call it $\mathcal{M}_{+f}$, by including $f$ as a tradeable asset, then is $\mathcal{M}_{+f}$ complete?
• Is it as simple as saying: "by including $f$, the number of assets is equal to the number of risk factors hence the market $\mathcal{M}_{+f}$ is complete"?

For example, as mentioned above, AFK says in his answer:

A stochastic volatility model for a single risky asset can't be complete because you have two sources of randomness. But you can easily make it complete by adding a derivative whose value depends on the volatility. For example, if you add a variance swap in the Heston model then it becomes complete.

Another example is this post, How to prove that markets are incomplete under the Stochastic Volatility model?, see @pbr142 's answer $-$ link to the paper mentioned:

The paper by Marc Romano and Nizar Touzi, Section 3, contains a general proof that a stochastic volatility model cannot be complete in the sense that the addition of the option completes the market (in the sense of Harrison and Pliska) generated by the underlying and risk-free borrowing/lending.

• Have you read @AFK's answer to the question "How to prove that (...) " ? – Quantuple Mar 21 '17 at 22:17

Meta-theorem 8.3.1 Let $$M$$ denote the number of underlying traded assets in the model excluding the risk free asset, and let $$R$$ denote the number of random sources. Generically we then have the following relations:
1. The model is arbitrage free if and only if $$M \leq R$$.
2. The model is complete if and only if $$M \geq R$$.
3. The model is complete and arbitrage free if and only if $$M=R$$.

format: plain


text:
## Found 341 Documents (Results 1–100)

100
MathJax

### Bootstrap inference for weighted nearest neighbors imputation. (English)Zbl 07540813

MSC:  62D05 62G09 62J05
Full Text:

### Handling missing extremes in tail estimation. (English)Zbl 07523469

MSC:  62G32 60G70
Full Text:

MSC:  62Pxx
Full Text:

MSC:  62-XX
Full Text:

### Imputation methods in time series with a trend and a consecutive missing value pattern. (English)Zbl 07508934

MSC:  62D05 62M10
Full Text:

### Computational methods to simultaneously compare the predictive values of two diagnostic tests with missing data: EM-SEM algorithms and multiple imputation. (English)Zbl 07497666

MSC:  62P10 62-XX
Full Text:

MSC:  68T05
Full Text:

### Extremal behaviour of a periodically controlled sequence with imputed values. (English)Zbl 1483.62085

MSC:  62G32 62D10 60G70
Full Text:

Full Text:

### Missing value prediction for qualitative information systems. (English)Zbl 07539025

MSC:  68-XX 90-XX
Full Text:

### Impact of missing data on the prediction of random fields. (English)Zbl 07481407

MSC:  60G60 62M20 62Pxx
Full Text:

### Automated learning of mixtures of factor analysis models with missing information. (English)Zbl 1474.62191

MSC:  62H12 62H25 62H30
Full Text:

### Similarity/diversity indices on incidence matrices containing missing values. (English)Zbl 1479.62036

MSC:  62H12 62D10
Full Text:

### A generalization of stability for families of aggregation operators. (English)Zbl 1464.91037

MSC:  91B06 91B86
Full Text:

MSC:  90Bxx
Full Text:

### Extreme values of linear processes with heavy-tailed innovations and missing observations. (English)Zbl 1468.60063

MSC:  60G70 60G10
Full Text:

### NIS-apriori-based rule generation with three-way decisions and its application system in SQL. (English)Zbl 1456.68201

MSC:  68T37 68P15
Full Text:

### Fuzzy rough set theory-based feature selection. (English)Zbl 07324116

Chakraverty, Snehashish (ed.), Mathematical methods in interdisciplinary sciences. Hoboken, NJ: John Wiley & Sons. 145-166 (2020).
MSC:  74-XX 76-XX 80-XX
Full Text:

Full Text:

MSC:  62-XX
Full Text:

### Best recovery of the solution of the Dirichlet problem in a half-space from inaccurate data. (English. Russian original)Zbl 1457.62054

Comput. Math. Math. Phys. 60, No. 10, 1656-1665 (2020); translation from Zh. Vychisl. Mat. Mat. Fiz. 60, No. 10, 1711-1720 (2020).
MSC:  62D10 62G32 60J46
Full Text:

### Imputation and low-rank estimation with missing not at random data. (English)Zbl 1452.62174

MSC:  62D10 62H12 62P10
Full Text:

### Main effects and interactions in mixed and incomplete data frames. (English)Zbl 1441.62145

MSC:  62H12 62R07
Full Text:

MSC:  62H30
Full Text:

Full Text:

Full Text:

### Control chart for monitoring the Weibull shape parameter under two competing risks. (English)Zbl 07551943

MSC:  62N10 62N05
Full Text:

### Nonparametric product partition models for multiple change-points analysis. (English)Zbl 07551928

MSC:  62F15 62G99 62C99
Full Text:

MSC:  62-XX
Full Text:

MSC:  62D05
Full Text:

### Application of interval-valued aggregation to optimization problem of $$k$$-NN classifiers for missing values case. (English)Zbl 1454.62192

MSC:  62H30 62D10 62C25
Full Text:

Full Text:

Full Text:

### A novel Bayesian approach for latent variable modeling from mixed data with missing values. (English)Zbl 1430.62098

MSC:  62H05 62D10
Full Text:

### Learning causal structure from mixed data with missing values using Gaussian copula models. (English)Zbl 1430.62099

MSC:  62H05 62D10 68T05
Full Text:

Full Text:

Full Text:

Full Text:

MSC:  90B50
Full Text:

### Simple tail index estimation for dependent and heterogeneous data with missing values. (English)Zbl 1414.62420

MSC:  62P05 62G32
Full Text:

MSC:  62-XX
Full Text:

MSC:  62D05
Full Text:

### Transitional modeling of experimental longitudinal data with missing values. (English)Zbl 1414.62013

MSC:  62-07 62P25 62H30
Full Text:

### A DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP. (English)Zbl 1434.90075

MSC:  90B50 90C08
Full Text:

### Markov chain Monte Carlo with the integrated nested Laplace approximation. (English)Zbl 1405.62078

MSC:  62H30 62F15 62J07
Full Text:

### Multiple imputation for multilevel data with continuous and binary variables. (English)Zbl 1397.62265

MSC:  62J15 62H30
Full Text:

### Appraisal of performance of three tree-based classification methods. (English)Zbl 1397.62215

Tez, Müjgan (ed.) et al., Trends and perspectives in linear statistical inference. Proceedings of the LINSTAT2016 meeting held 22–25 August 2016 in Istanbul, Turkey. Cham: Springer (ISBN 978-3-319-73240-4/hbk; 978-3-319-73241-1/ebook). Contributions to Statistics, 41-55 (2018).
MSC:  62H30 62P30
Full Text:

Full Text:

### Multiplicity lists for symmetric matrices whose graphs have few missing edges. (English)Zbl 1380.15009

MSC:  15A18 15B57 05C50
Full Text:

### Rough set analysis of classification data with missing values. (English)Zbl 07542318

Polkowski, Lech (ed.) et al., Rough sets. International joint conference, IJCRS 2017, Olsztyn, Poland, July 3–7, 2017. Proceedings. Part I. Cham: Springer. Lect. Notes Comput. Sci. 10313, 552-565 (2017).
MSC:  68T37 62H30
Full Text:

### Permuting incomplete paired data: a novel exact and asymptotic correct randomization test. (English)Zbl 07191993

MSC:  62G09 62G10
Full Text:

### A simple correction to completer analyses and improvement on baseline observation carried forward. (English)Zbl 1405.62198

MSC:  62P10 62D05
Full Text:

### $$m$$-tables: representing missing data. (English)Zbl 1402.68053

Benedikt, Michael (ed.) et al., 20th international conference on database theory, ICDT 2017, Venice, Italy, March 21–24, 2017. Proceedings. Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik (ISBN 978-3-95977-024-8). LIPIcs – Leibniz International Proceedings in Informatics 68, Article 21, 20 p. (2017).
MSC:  68P15
Full Text:

Full Text:

### Extreme values of the uniform order 1 autoregressive processes and missing observations. (English)Zbl 1387.60085

MSC:  60G70 60G10
Full Text:

Full Text:

MSC:  62G05
Full Text:

### Missing value imputation for gene expression data by tailored nearest neighbors. (English)Zbl 1371.92048

MSC:  92C40 92B15 62P10
Full Text:

### Comparison of the predictive values of multiple binary diagnostic tests in the presence of ignorable missing data. (English)Zbl 1369.62300

MSC:  62P10 62H15
Full Text:

### Multi-view kernel completion. (English)Zbl 1460.62040

MSC:  62G05 62D10 68T05
Full Text:

MSC:  62H25

### MIMCA: multiple imputation for categorical variables with multiple correspondence analysis. (English)Zbl 06697670

MSC:  62H25 62-07 62F40
Full Text:

MSC:  62-XX
Full Text:

### Fuzzy $$c$$-means clustering of incomplete data using dimension-wise fuzzy variances of clusters. (English)Zbl 1457.62199

Carvalho, Joao Paulo (ed.) et al., Information processing and management of uncertainty in knowledge-based systems. 16th international conference, IPMU 2016, Eindhoven, The Netherlands, June 20–24, 2016. Proceedings. Part I. Cham: Springer. Commun. Comput. Inf. Sci. 610, 699-710 (2016).
MSC:  62H30 62H86 62D10
Full Text:

Full Text:

MSC:  62H25
Full Text:

### Accounting for missing values in score-driven time-varying parameter models. (English)Zbl 1400.62191

MSC:  62M10 62P20
Full Text:

### Sparse generalized principal component analysis for large-scale applications beyond Gaussianity. (English)Zbl 1405.62072

MSC:  62H25 62J12 62H12
Full Text:

### An imputation method for missing data based on sigmoid kernel in composition data. (Chinese. English summary)Zbl 1374.62120

MSC:  62N01 62D05

Full Text:

### Estimation of AR(1) models with missing values. (English)Zbl 1359.62366

MSC:  62M10 60G10 62F35
Full Text:

### Lookahead selective sampling for incomplete data. (English)Zbl 1355.93107

MSC:  93C57 93C41
Full Text:

### Weighted norms in subspace-based methods for time series analysis. (English)Zbl 1413.62139

MSC:  62M10 62M20 65F15
Full Text:

### Imputation of missing values for compositional data based on principal component analysis. (Chinese. English summary)Zbl 1363.62056

MSC:  62H25 62H12 62G05
Full Text:

### Treatment of ‘don’t know’ responses in a mixture model for rating data. (English)Zbl 1394.62086

MSC:  62H30 62P25
Full Text:

MSC:  62-XX
Full Text:

### An ensemble learning approach based on missing-valued tables. (English)Zbl 1444.68164

Yao, Yiyu (ed.) et al., Rough sets, fuzzy sets, data mining, and granular computing. 15th international conference, RSFDGrC 2015, Tianjin, China, November 20–23, 2015. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 9437, 310-321 (2015).
MSC:  68T05 68T37
Full Text:

MSC:  62-XX
Full Text:

MSC:  68T37
Full Text:

### Fundamentals of machine learning for predictive data analytics. Algorithms, worked examples, and case studies. (English)Zbl 1393.68007

Cambridge, MA: MIT Press (ISBN 978-0-262-02944-5/hbk; 978-0-262-33172-2/ebook). xxii, 595 p. (2015).
MSC:  68-02 68T05

MSC:  90B50
Full Text:

### Uniform AR(1) processes and maxima on partial samples. (English)Zbl 1333.60106

MSC:  60G70 60F05 60G10
Full Text:

MSC:  62P12
Full Text:

Full Text:

### Estimating with kernel smoothers the mean of functional data in a finite population setting. A note on variance estimation in presence of partially observed trajectories. (English)Zbl 1396.62005

MSC:  62D05 62G07
Full Text:

### An efficient procedure for the avoidance of disconnected incomplete block designs. (English)Zbl 1471.62075

MSC:  62-08 62K10
Full Text:

### Rough fuzzy subspace clustering for data with missing values. (English)Zbl 1413.62084

MSC:  62H30 62H86 68T37

### A new variable importance measure for random forests with missing data. (English)Zbl 1325.62011

MSC:  62-07 68T05
Full Text:

### On inference of statistical regression models for extreme events based on incomplete observation data. (English)Zbl 1328.62209

MSC:  62G05 62G32 65R32 65C50 62F03
Full Text:

MSC:  62M10
Full Text:

Full Text:

Full Text:

### Constrained principal component analysis and related techniques. (English)Zbl 1282.62150

Monographs on Statistics and Applied Probability 129. Boca Raton, FL: CRC Press (ISBN 978-1-4665-5666-9/hbk; 978-1-4665-5668-3/ebook). xvii, 233 p. (2014).
Full Text:

### Maximum likelihood estimation of bivariate circular hidden Markov models from incomplete data. (English)Zbl 1431.62343

MSC:  62M05 62H12 62-08
Full Text:

### Estimation and forecasting of locally stationary processes. (English)Zbl 1397.62319

MSC:  62M10 65C05
Full Text:

### Time series interpolation via global optimization of moments fitting. (English)Zbl 1317.62068

MSC:  62M10 62M20 90C90
Full Text:

### Training energy-based models for time-series imputation. (English)Zbl 1317.68145

MSC:  68T05 62M10
Full Text:

### Model-based classification of clustered binary data with non-ignorable missing values. (English)Zbl 1300.62099

Torelli, Nicola (ed.) et al., Advances in theoretical and applied statistics. Selected papers based on the presentations at the 45th meeting of the Italian Statistical Society (SIS), Padua, Italy, June 16–18, 2010. Berlin: Springer (ISBN 978-3-642-35587-5/hbk; 978-3-642-35588-2/ebook). Studies in Theoretical and Applied Statistics. Selected Papers of the Statistical Societies, 155-165 (2013).
MSC:  62N02 62H30 62P10
Full Text:

### Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates. (English)Zbl 1454.62451

MSC:  62P12 62M05
Full Text:

all top 5

all top 5

all top 5

all top 3

all top 3


text:
Home > TOK > The Allegory of the Cave

## The Allegory of the Cave

Read the allegory of the cave.

What does each of the following represent?

• The cave
• The prisoners


text:
Reserve Bank of Tonga makes further tweaks to exchange controls

The governor of the Reserve Bank of Tonga has announced further changes to the bank's exchange controls, following on from tweaks made in June 2016.

In a statement published on February 3, Sione Ngongo Kioa said that in light of the central bank's "very high level of foreign reserves" and the current monetary policy stance, there was a need to "improve the clarity of the requirements".

Among the changes is a limit of T$100,000 ($46,000) on capital payments, which will include transactions such

Latest issue

Central Banking Journal

Read the latest edition of the Central Banking journal


text:
HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information

# Green up-conversion and infrared emission spectra of U$^{3+}$ in LaCl$_3$/

Abstract : A strong green emission of U/sup 3+/ ions in LaCl/sub 3/ under red excitation has been observed for the first time. The main anti-Stokes fluorescence arises from the U/sup 3+2/K/sub 15/2/ state (around 550 nm). Theoretical analysis and decaytime measurement of the induced fluorescence show that there are two mechanisms for the up-conversion. The first one consists in excited state absorption and the other one is due to energy transfer between two U/sup 3+/ ions in the /sup 4/F/sub 9/2/ state. In addition a detailed study of infrared emission spectra of LaCl/sub 3/:U/sup 3+/ has been carried out and lifetimes of main fluorescent levels were measured at low temperature. From this study three new experimental levels have been assigned.
Document type :
Journal articles

http://hal.in2p3.fr/in2p3-00022975
Contributor : Suzanne Robert Connect in order to contact the contributor
Submitted on : Tuesday, April 18, 2000 - 10:37:55 AM
Last modification on : Wednesday, September 16, 2020 - 4:07:23 PM

### Identifiers

• HAL Id : in2p3-00022975, version 1

### Citation

M. Yin, M.F. Joubert, J.C. Krupa. Green up-conversion and infrared emission spectra of U$^{3+}$ in LaCl$_3$/. Journal of Alloys and Compounds, Elsevier, 1997, 260, pp.64-69. ⟨in2p3-00022975⟩

Record views


text:
# How to solve a simple B-spline with table function?

Determine the natural cubic B-spline for a function in a table form of:

$$x_i\|-1\|0\|1\|2 \\ y_i\|-3\|0\|1\|2$$

So I'm very new to cubic splines and am quite lost on what I should do. I know I should express every point as a function of form $a_ix^3+b_ix^2+c_ix+d_i$ and then solve a system of equations with all of them but how am I supposed to show a pair like $(-1, -3)$ in such a form? I mean: aren't there infinitely many ways of doing it?

Any guidance on how to advance would be welcome :)

-


text:
# Wrapping messages into common header, ANSI C

I have a protocol where I send lots of different messages using two or three different header types (let's call them "Type A" and "Type B" messages).

To simplify unit testing, my idea was to do something like:

typedef struct
{
u32 timestamp; // unix
u32 deviceId;
}
CommonHeader;

void buildMessage(Buffer * output, u16 cookie, CommonHeader * header,
void (*bodySerializerFn)(Buffer*,void*), void * bodySerializerArgs)
{
// create header
Buffer_append_u16(output, cookie);
Buffer_append_u16(output, (u16)0); // dummy length, update at the end
Buffer_append_u32(output, header->timestamp);
Buffer_append_u32(output, header->deviceId);

// invoke callback to fill the body
bodySerializerFn(output, bodySerializerArgs);

// finish wrapping (write length into the header and append crc)
u32 totalLen = Buffer_getLen(output) + 2; // length so far + 16-bit crc
Buffer_write_u16(output, /*offset*/ 2, /*len*/ totalLen);
Buffer_append_crc(output);
}

Buffer is a struct (or ADT) which supports various overflow-safe operations for appending bytes in big endian or little endian format:

typedef struct
{
const u8 * bufferStart; // points to start of  buffer
const u8 * bufferEnd;   // points one byte after end of buffer
u8 * currentPos;  // points to current byte
}
Buffer;

void Buffer_init(Buffer *buffer, u8 * array, u32 arraySize)
{
buffer->bufferStart = array;
buffer->currentPos = array;
buffer->bufferEnd = array + arraySize;
}

void Buffer_append_u8 (Buffer *buffer, u8 singleByte)
{
if (Buffer_isNotFull(buffer))
*buffer->currentPos++ = singleByte;
}

void Buffer_append_u16 (Buffer *buffer, u16 value)
{
Buffer_append_u8(buffer, (u8)(value));
Buffer_append_u8(buffer, (u8)(value >> 8));
}

// etc.

So using this I can create data messages:

// append data message to the output buffer
CommonHeader hdr = { .timestamp = getTime(), .deviceId = getId() };
Measurement measurements[100]; // struct Measurement { .timestamp, .value}
buildMessage(output, COOKIE_DATAMSG, &hdr, appendDataBody, &measurements);

or config messages:

// append config message to output
CommonHeader hdr = { .timestamp = getTime(), .deviceId = getId() };
Config config = {
.deviceId = getId(),
.deviceName = "PressureSensor_1",
.reportingFrequency = 10 /* Hz */,
.scalingFactor = 10 /* x0.001 */
};
buildMessage(output, COOKIE_CONFIG, &hdr, appendConfigBody, &config);

appendConfigBody and appendDataBody are functions which fill the buffer using the supplied parameter:

void appendConfigBody(Buffer * output, void * args)
{
Config * cfg = args;
Buffer_append_u16(output, cfg->deviceId);
Buffer_append_string(output, cfg->deviceName, MAX_DEVICE_NAME);
Buffer_append_u16(output, cfg->reportingFrequency);
Buffer_append_u32(output, cfg->scalingFactor);
}

Since there are many messages using the same header, I wanted to decouple the part where the header is being created from the body serialization, which can be tested separately.

This makes testing buildMessage trivial (just create a temporary buffer, pass an empty bodySerializerFn and check the result), and allows me to quickly test each "body serializer" function with a separate buffer (since it simply appends to the buffer), without having to check the header for each individual message.

But my concerns are:

1. Since parameters for the passed function pointer are void*, there is not much compile-time type safety, I need to take care to pass the correct struct type to the function, I am not sure if this is something that can be improved?

2. This will run on an embedded system (ARM based), and the bodies of these messages are often rather small. Perhaps the callback approach is not the most performant way to do it?

3. I am not sure if this code can be considered "idiomatic C code" by people who code in C for living, i.e. future code maintainers?

So, is there a better (i.e. more "idiomatic") way to compose messages which share common parts, which would:

1. Allow me to unit test the whole thing properly?

2. Provide as much compile-time type safety as possible?

3. Ideally have no code duplication?

4. Feel like a natural C code for future maintainers?

• @Edward: sorry, I didn't know about this, I'll fix the question (although it's almost equal to the actual working code). Does this mean I have to post all the Buffer_xxx functions too? Or are you talking about the contents of appendMessageTypeA? – Groo Apr 16 '17 at 19:36
• I don't know whether this is suitable, but have you tried c-faq.com/varargs/vprintf.html? – Neil Edelman Apr 18 '17 at 5:28

## 1 Answer

Here are some thoughts on how to improve this code.

## Provide complete compilable code for review

When the code is missing many pieces, as with this question, it makes it more difficult for reviewers to review the code. When the code is difficult to review, it tends to draw fewer and less thorough reviews. Posting a full, compilable version for review tends to allow for more and better reviews.

## Use standard types

The code uses u32, u8 etc. which I am guessing are equivalent to uint32_t and uint8_t. If that's so, I would strongly recommend using the standard names from <stdint.h> rather than custom synonyms. It will make the code more portable and allow quicker understanding of the code.

## Add error checking and handling

Right now, the code does not appear to provide any signal back to the calling code if the buffer has overflowed within Buffer_append_u8. A useful addition might be to return a bool that indicates success of the operation. This would allow for simple and consistent checks so that calling code would be signaled in the event of a buffer overflow.

## Use const where practical

In a number of cases in this code, such as appendConfigBody some of the pointer arguments are not and should not be modified. For that reason, they should be declared const as in:

void appendConfigBody(Buffer * output, const void *args)
{
const Config *cfg = args;
Buffer_append_u16(output, cfg->deviceId);
Buffer_append_string(output, cfg->deviceName, MAX_DEVICE_NAME);
Buffer_append_u16(output, cfg->reportingFrequency);
Buffer_append_u32(output, cfg->scalingFactor);
}

This also means a change to the declaration of buildMessage:

void buildMessage(Buffer * output, u16 cookie, CommonHeader * header,
void (*bodySerializerFn) (Buffer *, const void *),
const void *bodySerializerArgs)

## Consider a different approach

The approach of passing function pointers and void * for any necessary data is not very idiomatic C in my experience. Instead, what is more often done is to simply call the functions in sequence. That is, instead of your current generic buildMessage, I'd suggest one of two possible approaches. One is to break things into chunks such as buildHeader(), buildDataMsgBody()/buildConfigMsgBody(), and finishMessage() and make three calls where you currently call one function. This would allow better type checking at compile time rather than potentially having a problem only exposed at runtime. Since the bodies of the functions would need to exist anyway, this requires little to no additional ROM space for the code.

The other possible approach, is to create specific functions such as buildDataMessage() and buildConfigMessage() that would create the specific type of message, and would be passed a specific kind of pointer (which would be checked at compile time). This, too, would require little to no additional ROM space but confers the benefit of compile-time type checking.

Either of these approaches are likely to be as fast or faster than the current code because composition of functions at compile time allows for additional optimizations (e.g. register allocations) that are not generally possible when passing function pointers about which the compiler can make no assumptions.

• Reasonable suggestions, thanks. As you suggested, I will probably abandon the fn pointer approach due to possible time/memory constraints and put each sequence of calls (header, body, footer) into a separate function. I still believe the "fn pointer" approach provides looser coupled, is easier to test and to prevent runtime mistakes, but in this case performance and legibility for future maintainers might be more important than some slight code duplication. – Groo Apr 21 '17 at 8:41


text:
# What does the convolution operation do in a convolutional neural network?

What does the convolution operation do in convolutional neural network?

An interviewer told me that you can look at each convolutional filter as a vector and when it convolves with a matrix, it is essentially doing a dot product, giving you a cosine distance. Is that right? I think it is only true when both vectors are normalised to 1, but it is not the case in convolutional neural network, where we can't control the weights in the filter since they are learnt through gradient descent. Also, this doesn't seem to explain why CNN is able to learn well.

## Dot product and cosine similarity

The dot product of two vectors $a, b$ of same length $n$ is defined as $$\sum_{i=1}^n a_i b_i$$

The cosine similarity is defined as $$\frac{ \sum\limits_{i=1}^{n}{a_i b_i} }{ \sqrt{\sum\limits_{i=1}^{n}{a_i^2}} \sqrt{\sum\limits_{i=1}^{n}{b_i^2}} }$$

So the cosine similarity is essentially a normed version of the dot product.

To compute element $(i, j)$ of the feature map resulting from a convolution with a $3 \times 3$ filter $w$ applied to a feature map $F^{(0)}$, one calculates

$$\sum_{x=-1}^1 \sum_{y=-1}^1 F^{(0)}_{x+i,y+j} \cdot w_{x, y}$$

You can, of course, find a flattened version of $F$ / $w$ so that this is a dot product. I'm not too sure how this is implemented in cuDNN. So although it is true what he says, I'm not sure how useful it is to think of convolutional layers like this.

## What Conv-Layers learn

Commonly, people say that CNNs learn edge detectors. For such questions / statements, people often refer to Visualizing and Understanding Convolutional Networks


text:
# All Questions

25 views

### Is ssl_sign safe as it is using OPENSSL_PKCS1_PADDING

I've been hearing a lot of scare talk about PKCS1 padding attacks. Is PKCS1 v1.5 safe in signatures as it is practically transporting only message digest? I'm using it in one of my applications so ...
61 views

### Can RSA-PSS signing be made deterministic without loss of security?

Is it possible to make RSA-PSS signing deterministic without loss of security? Specifically, I would use Blake2b of the key and message to generate the salt.
62 views

### Beavers Triple Vs BGW Multiplication on MPC

Typically MPC protocols that are secure against semi-honest adversaries recommend the use of the revised GMW multiplication protocol by Gennaro et al. This is not the case against Active adversaries ...
55 views

### How does Hash with random cryptography work?

I am new in this world of cryptography and while reading some articles I got stuck into something that I did not understand correctly: the use of hash based on random cryptography. On my knowledge, ...
86 views

### Is Threefish the only cipher with 1024 bits of security?

Threefish has up to a massive 1024 bits of security. Is it the only cipher with such overkill security?
54 views

### Software timing attack using Kocher method

What's the minimum number of random sample points needed in Kocher's timing attack, so that we can determine enough valid measurements of $A_{i,r}$ and $D_{i,r}$? I'm working from this paper: Volker ...
62 views

### Majority encryption algorithm?

Assume that I want to leave an encrypted message to a group of $n$ people in a way that they can only decrypt it if they work together in the following sense: For some fixed $k < n$ every ...
49 views

### Does OpenSSL apply ASN1 encoding to the hash before signing using ECDSA?

I read on stack overflow that OpenSSL performs ASN1 encoding to the hash before signing it for, for ECDSA. In other words, OpenSSL performs the following steps when for an Elliptic curve key ...
172 views

### What is the meaning of IND-CCA secure under standard model? [duplicate]

I notice that in many research papers (viz. "Universal hash proofs and a paradigm for adaptive chosen ciphertext secure public-key encryption" by Cramer and Shoup) the authors showed that their ...
27 views

### Weaknesses of RSA for $e_A\neq e_B: \gcd(e_A,e_B)=1$ [duplicate]

If two people, A and B, are using a common RSA modulus N, and they select distinct exponents $e_A$ and $e_B$, such that they are coprime, why does this mean that someone can decrypt (in polynomial ...
32 views

### Increase in security with increase in size of salt [duplicate]

Does increasing or doubling the size of salt result in improved security? It will require some extra time to brute force the passwords, but any added advantage?
75 views

### Why is 128-bit considered “medium term” security?

Why is 128-bit encryption considered good enough for medium term security only? How is expected to be eventually broken? Quantum computing or brute force attack?
95 views

### Confusion about definition of homomorphic encryption

I am trying to better understand homomorphic encryption, but I feel like I keep getting inconsistent information in the papers that I am reading. One of the papers I am reading says the following: ...
38 views

### What is the definition of Selective Security?

What exactly is the definition of "selective security"? For instance, it is used in [KSW'13] (Predicate Encryption Supporting Disjunctions, Polynomial Equations, and Inner Products), but is there a ...
71 views

### Inverting RSA using an oracle

Say you are given an efficient deterministic algorithm 'I' that can invert the RSA function on 1% of the points in $Z^*_{N}$. That is to say that if y $\in$ $Z^*_{N}$ is a "good" point for 'I', ...
99 views

### ElGamal - What if the Decisional Diffie-Hellmann problem could be solved?

I have read that if the DDH problem could be efficiently solved, the IND-CPA would not hold for ElGamal. I don't see why it makes ElGamal less secure if you have $g^a$, $g^b$ and $g^c$
29 views

56 views

### What is the security strength of an n-bit HMAC?

HMACs depend on preimage resistance, or so I have read. The security strength of hash function with an $n$-bit output against collision attacks is $2^{\frac{n}{2}}$. Against preimage attacks, it is ...

15 30 50 per page


text:
$$\require{cancel}$$

# 7.2: Electric Potential Energy

Learning Objectives

By the end of this section, you will be able to:

• Define the work done by an electric force
• Define electric potential energy
• Apply work and potential energy in systems with electric charges

When a free positive charge q is accelerated by an electric field, it is given kinetic energy (Figure $$\PageIndex{1}$$). The process is analogous to an object being accelerated by a gravitational field, as if the charge were going down an electrical hill where its electric potential energy is converted into kinetic energy, although of course the sources of the forces are very different. Let us explore the work done on a charge q by the electric field in this process, so that we may develop a definition of electric potential energy.

The electrostatic or Coulomb force is conservative, which means that the work done on q is independent of the path taken, as we will demonstrate later. This is exactly analogous to the gravitational force. When a force is conservative, it is possible to define a potential energy associated with the force. It is usually easier to work with the potential energy (because it depends only on position) than to calculate the work directly.

To show this explicitly, consider an electric charge $$+q$$ fixed at the origin and move another charge $$+Q$$ toward q in such a manner that, at each instant, the applied force $$\vec{F}$$ exactly balances the electric force $$\vec{F}_e$$ on Q (Figure $$\PageIndex{2}$$). The work done by the applied force $$\vec{F}$$ on the charge Q changes the potential energy of Q. We call this potential energy the electrical potential energy of Q.

The work $$W_{12}$$ done by the applied force $$\vec{F}$$ when the particle moves from $$P_1$$ to $$P_2$$ may be calculated by

$W_{12} = \int_{P_1}^{P_2} \vec{F} \cdot d\vec{l}.$

Since the applied force $$\vec{F}$$ balances the electric force $$\vec{F}_e$$ on Q, the two forces have equal magnitude and opposite directions. Therefore, the applied force is

$\vec{F} = -\vec{F}_e = - \dfrac{kqQ}{r^2} \hat{r},$

where we have defined positive to be pointing away from the origin and r is the distance from the origin. The directions of both the displacement and the applied force in the system in Figure $$\PageIndex{2}$$ are parallel, and thus the work done on the system is positive.

We use the letter U to denote electric potential energy, which has units of joules (J). When a conservative force does negative work, the system gains potential energy. When a conservative force does positive work, the system loses potential energy, $$\Delta U = - W$$. In the system in Figure $$\PageIndex{3}$$, the Coulomb force acts in the opposite direction to the displacement; therefore, the work is negative. However, we have increased the potential energy in the two-charge system.

Example $$\PageIndex{1}$$: Kinetic Energy of a Charged Particle

A $$+3.0-nC$$ charge Q is initially at rest a distance of 10 cm ($$r_1$$) from a $$+5.0-nC$$ charge q fixed at the origin (Figure $$\PageIndex{3}$$). Naturally, the Coulomb force accelerates Q away from q, eventually reaching 15 cm ($$r_2$$).

1. What is the work done by the electric field between $$r_1$$ and $$r_2$$?
2. How much kinetic energy does Q have at $$r_2$$?

Strategy

Calculate the work with the usual definition. Since Q started from rest, this is the same as the kinetic energy.

Solution

Integrating force over distance, we obtain

\begin{align} W_{12} &= \int_{r_1}^{r_2} \vec{F} \cdot d\vec{r} \nonumber \\[4pt] &= \int_{r_1}^{r_2} \dfrac{kqQ}{r^2}dr \nonumber \\[4pt] &= \left. - \dfrac{kqQ}{r} \right|_{r_1}^{r_2} \nonumber \\[4pt] &= kqQ \left[\dfrac{-1}{r_2} + \dfrac{1}{r_1}\right] \nonumber \\[4pt] &= (8.99 \times 10^9 \, Nm^2/C^2)(5.0 \times 10^{-9} C)(3.0 \times 10^{-9} C) \left[ \dfrac{-1}{0.15 \, m} + \dfrac{1}{0.10 \, m}\right] \nonumber \\[4pt] &= 4.5 \times 10^{-7} \, J. \nonumber \end{align} \nonumber

This is also the value of the kinetic energy at $$r_2$$.

Significance

Charge Q was initially at rest; the electric field of q did work on Q, so now Q has kinetic energy equal to the work done by the electric field.

Exercise $$\PageIndex{1}$$

If Q has a mass of $$4.00 \, \mu g$$, what is the speed of Q at $$r_2$$?

Solution

$$K = \frac{1}{2}mv^2$$, $$v = \sqrt{2\frac{K}{m}} = \sqrt{2\frac{4.5 \times 10^{-7}J}{4.00 \times 10^{-9}kg}} = 15 \, m/s.$$

In this example, the work W done to accelerate a positive charge from rest is positive and results from a loss in U, or a negative $$\Delta U$$. A value for U can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point.

Electric Potential Energy

Work W done to accelerate a positive charge from rest is positive and results from a loss in U, or a negative $$\Delta U$$. Mathematically,

$W = - \Delta U. \label{7.1}$

Gravitational potential energy and electric potential energy are quite analogous. Potential energy accounts for work done by a conservative force and gives added insight regarding energy and energy transformation without the necessity of dealing with the force directly. It is much more common, for example, to use the concept of electric potential energy than to deal with the Coulomb force directly in real-world applications.

In polar coordinates with q at the origin and Q located at r, the displacement element vector is $$d\vec{l} = \hat{r} dr$$ and thus the work becomes

\begin{align} W_{12} &= kqQ \int_{r_1}^{r_2} \dfrac{1}{r^2} \hat{r} \cdot \hat{r} dr \nonumber \\[4pt] &= \underbrace{kqQ \dfrac{1}{r_2}}_{final \, point} - \underbrace{kqQ \dfrac{1}{r_1}}_{initial \,point}. \end{align}

Notice that this result only depends on the endpoints and is otherwise independent of the path taken. To explore this further, compare path $$P_1$$ to $$P_2$$ with path $$P_1 P_3 P_4 P_2$$ in Figure $$\PageIndex{4}$$.

The segments $$P_1P_3$$ and $$P_4P_2$$ are arcs of circles centered at q. Since the force on Q points either toward or away from q, no work is done by a force balancing the electric force, because it is perpendicular to the displacement along these arcs. Therefore, the only work done is along segment $$P_3P_4$$ which is identical to $$P_1P_2$$.

One implication of this work calculation is that if we were to go around the path $$P_1P_3P_4P_2P_1$$, the net work would be zero (Figure $$\PageIndex{5}$$). Recall that this is how we determine whether a force is conservative or not. Hence, because the electric force is related to the electric field by $$\vec{F} = g\vec{E}$$, the electric field is itself conservative. That is,

$\oint \vec{E} \cdot d\vec{l} = 0.$

Note that Q is a constant.

Another implication is that we may define an electric potential energy. Recall that the work done by a conservative force is also expressed as the difference in the potential energy corresponding to that force. Therefore, the work $$W_{ref}$$ to bring a charge from a reference point to a point of interest may be written as

$W_{ref} = \int_{r_{ref}}^r \vec{F} \cdot d\vec{l}$

and, by Equation \ref{7.1}, the difference in potential energy ($$U_2 - U_1$$) of the test charge Q between the two points is

$\Delta U = - \int_{r_{ref}}^r \vec{F} \cdot d\vec{l}.$

Therefore, we can write a general expression for the potential energy of two point charges (in spherical coordinates):

$\Delta U = - \int_{r_{ref}}^r \dfrac{kqQ}{r^2}dr = -\left[-\dfrac{kqQ}{r}\right]_{r_{ref}}^r = kqQ\left[ \dfrac{1}{r} - \dfrac{1}{r_{ref}}\right].$

We may take the second term to be an arbitrary constant reference level, which serves as the zero reference:

$U(r) = k\dfrac{qQ}{r} - U_{ref}.$

A convenient choice of reference that relies on our common sense is that when the two charges are infinitely far apart, there is no interaction between them. (Recall the discussion of reference potential energy in Potential Energy and Conservation of Energy.) Taking the potential energy of this state to be zero removes the term $$U_{ref}$$ from the equation (just like when we say the ground is zero potential energy in a gravitational potential energy problem), and the potential energy of Q when it is separated from q by a distance r assumes the form

$\underbrace{U(r) = k\dfrac{qQ}{r}}_{zero \, reference \, at \, r = \infty}.$

This formula is symmetrical with respect to $$q$$ and $$Q$$, so it is best described as the potential energy of the two-charge system.

Example $$\PageIndex{2}$$: Potential Energy of a Charged Particle

A $$+3.0-nC$$ charge Q is initially at rest a distance of 10 cm $$(r_1)$$ from a $$+5.0-nC$$ charge q fixed at the origin (Figure $$\PageIndex{6}$$). Naturally, the Coulomb force accelerates Q away from q, eventually reaching 15 cm $$(r_2)$$.

What is the change in the potential energy of the two-charge system from $$r_1$$ to $$r_2$$?

Strategy

Calculate the potential energy with the definition given above:

$$\Delta U_{12} = -\int_{r_1}^{r_2} \vec{F} \cdot d\vec{r}$$. Since Q started from rest, this is the same as the kinetic energy.

Solution

We have

\begin{align} \Delta U_{12} &= - \int_{r_1}^{r_2} \vec{F} \cdot d\vec{r} \nonumber \\[4pt] &= - \int_{r_1}^{r_2} \dfrac{kqQ}{r^2}dr \nonumber \\[4pt] &= - \left[ - \dfrac{kqQ}{r}\right]_{r_1}^{r_2} \nonumber \\[4pt] &=kqQ \left[ \dfrac{1}{r_2} - \dfrac{1}{r_1} \right] \nonumber \\[4pt] &= (8.99 \times 10^9 \, Nm^2/C^2)(5.0 \times 10^{-9} C)(3.0 \times 10^{-9} C) \left[ \dfrac{1}{0.15 \, m} - \dfrac{1}{0.10 \, m}\right] \nonumber \\[4pt] &= - 4.5 \times 10^{-7} \, J. \end{align}

Significance

The change in the potential energy is negative, as expected, and equal in magnitude to the change in kinetic energy in this system. Recall from Example $$\PageIndex{1}$$ that the change in kinetic energy was positive.

Exercise $$\PageIndex{2}$$

What is the potential energy of Q relative to the zero reference at infinity at $$r_2$$ in the above example?

Solution

It has kinetic energy of $$4.5 \times 10^{-7} \, J$$ at point $$r_2$$ and potential energy of $$9.0 \times 10^{-7} \, J$$, which means that as Q approaches infinity, its kinetic energy totals three times the kinetic energy at $$r_2$$, since all of the potential energy gets converted to kinetic.

Due to Coulomb’s law, the forces due to multiple charges on a test charge $$Q$$ superimpose; they may be calculated individually and then added. This implies that the work integrals and hence the resulting potential energies exhibit the same behavior. To demonstrate this, we consider an example of assembling a system of four charges.

Example $$\PageIndex{3}$$: Assembling Four Positive Charges

Find the amount of work an external agent must do in assembling four charges $$+2.0-\mu C$$, $$+3.0-\mu C$$, $$+4.0-\mu C$$ and $$+5.0-\mu C$$ at the vertices of a square of side 1.0 cm, starting each charge from infinity (Figure $$\PageIndex{7}$$).

Strategy

We bring in the charges one at a time, giving them starting locations at infinity and calculating the work to bring them in from infinity to their final location. We do this in order of increasing charge.

Solution

Step 1. First bring the $$+2.0-\mu C$$ charge to the origin. Since there are no other charges at a finite distance from this charge yet, no work is done in bringing it from infinity,

$W_1 = 0.$

Step 2. While keeping the $$+2.0-\mu C$$ charge fixed at the origin, bring the $$+3.0-\mu C$$ charge to $$(x,y,z) = (1.0 \, cm, \, 0, \, 0)$$ (Figure $$\PageIndex{8}$$). Now, the applied force must do work against the force exerted by the $$+2.0-\mu C$$ charge fixed at the origin. The work done equals the change in the potential energy of the $$+3.0-\mu C$$ charge:

\begin{align} W_2 &= k\dfrac{q_1q_2}{r{12}} \nonumber \\[4pt] &= \left(9.0 \times 10^9 \frac{N \cdot m^2}{C^2}\right) \dfrac{(2.0 \times 10^{-6} C)(3.0 \times 10^{-6}C)}{1.0 \times 10^{-2} m} \nonumber \\[4pt] &= 5.4 \, J.\nonumber \end{align} \nonumber

Step 3. While keeping the charges of $$+2.0-\mu C$$ and $$+3.0-\mu C$$ fixed in their places, bring in the $$+4.0-\mu C$$ charge to $$(x,y,z) = (1.0 \, cm, \, 1.0 \, cm, \, 0)$$ (Figure)$$\PageIndex{9}$$. The work done in this step is

\begin{align} W_3 &= k\dfrac{q_1q_3}{r_{13}} + k \dfrac{q_2q_3}{r_{23}} \nonumber \\[4pt] &= \left(9.0 \times 10^9 \frac{N \cdot m^2}{C^2}\right) \left[ \dfrac{(2.0 \times 10^{-6}C)(4.0 \times 10^{-6}C)}{\sqrt{2} \times 10^{-2}m} + \dfrac{(3.0 \times 10^{-6} C)(4.0 \times 10^{-6}C)}{1.0 \times 10^{-2} m}\right] \nonumber \\[4pt] &= 15.9 \, J. \nonumber \end{align} \nonumber

Step 4. Finally, while keeping the first three charges in their places, bring the $$+5.0-\mu C$$ charge to $$(x,y,z) = (0, \, 1.0 \, cm, \, 0)$$ (Figure $$\PageIndex{10}$$). The work done here is

\begin{align} W_4 &= kq_4 \left[ \dfrac{q_1}{r_{14}} + \dfrac{q_2}{r_{24}} + \dfrac{q_3}{r_{34}}\right], \nonumber \\[4pt] &= \left(9.0 \times 10^9 \frac{N \cdot m^2}{C^2}\right)(5.0 \times 10^{-6}C) \left[ \dfrac{(2.0 \times 10^{-6}C)}{1.0 \times 10^{-2}m} + \dfrac{(3.0 \times 10^{-6} C)} {\sqrt{2} \times 10^{-2} m} + \dfrac{(4.0 \times 10^{-6}C)}{1.0 \times 10^{-2}m} \right] \nonumber \\[4pt] &= 36.5 \, J. \nonumber \end{align} \nonumber

Hence, the total work done by the applied force in assembling the four charges is equal to the sum of the work in bringing each charge from infinity to its final position:

\begin{align} W_T &= W_1 + W_2 + W_3 + W_4 \nonumber \\[4pt] &= 0 + 5.4 \, J + 15.9 \, J + 36.5 \, J \nonumber \\[4pt] &= 57.8 \, J. \nonumber \end{align} \nonumber

Significance

The work on each charge depends only on its pairwise interactions with the other charges. No more complicated interactions need to be considered; the work on the third charge only depends on its interaction with the first and second charges, the interaction between the first and second charge does not affect the third.

Exercise $$\PageIndex{3}$$

Is the electrical potential energy of two point charges positive or negative if the charges are of the same sign? Opposite signs? How does this relate to the work necessary to bring the charges into proximity from infinity?

Solution

positive, negative, and these quantities are the same as the work you would need to do to bring the charges in from infinity

Note that the electrical potential energy is positive if the two charges are of the same type, either positive or negative, and negative if the two charges are of opposite types. This makes sense if you think of the change in the potential energy $$\Delta U$$ as you bring the two charges closer or move them farther apart. Depending on the relative types of charges, you may have to work on the system or the system would do work on you, that is, your work is either positive or negative. If you have to do positive work on the system (actually push the charges closer), then the energy of the system should increase. If you bring two positive charges or two negative charges closer, you have to do positive work on the system, which raises their potential energy. Since potential energy is proportional to 1/r, the potential energy goes up when r goes down between two positive or two negative charges.

On the other hand, if you bring a positive and a negative charge nearer, you have to do negative work on the system (the charges are pulling you), which means that you take energy away from the system. This reduces the potential energy. Since potential energy is negative in the case of a positive and a negative charge pair, the increase in 1/r makes the potential energy more negative, which is the same as a reduction in potential energy.

The result from Example $$\PageIndex{2}$$ may be extended to systems with any arbitrary number of charges. In this case, it is most convenient to write the formula as

$W_{12 . . . N} = \dfrac{k}{2} \sum_i^N \sum_j^N \dfrac{q_iq_j}{r_{ij}} \, for \, i \neq j.$

The factor of 1/2 accounts for adding each pair of charges twice.


text:
# AIB4 (AiB2+A1Bs+B4) (—Bs—A1B2B3 —ATB4)

Online Encyclopedia
Originally appearing in Volume V01, Page 641 of the 1911 Encyclopedia Britannica.
 AIB4 (AiB2+A1Bs+B4) (—Bs—A1B2B3 —ATB4) =0. Selecting the product A;B4B3BQ, we find the simplest perpetuant (14).(4322)b — (13). (43221)6+(12),(432212)6 — (1).(432213)1, +ao(432214)b, and thence the general form (1A1+4), (4µa+13.3+1201}2) b —... t a0(4'44+l3µ3+l2µ2+21A1+4)1,, due to the generating function so that +µ2X y a1x—a2 Y= = x Y'X y' Xi+µ1X µ2—µ1x it is seen that the two lines, on which lie (x, y), (X, Y), have a definite projective correspondence. The linear transformation replaces points on lines through the origin by corresponding points on projectively corresponding lines through the origin; it therefore replaces a pencil of lines by another pencil, which corresponds projectively, and harmonic and other properties of pencils which are unaltered by linear transformation we may expect to find indicated in the invariant system. Or, instead of looking upon a linear substitution as replacing a pencil of lines by a projectively corresponding pencil retaining the same axes of co-ordinates, we may look upon the substitution as changing the axes of co-ordinates retaining the same pencil. Then a binary n", equated to zero, represents n straight lines through the origin, and the x, y of any line through the origin are given constant multiples of the sines of the angles which that line makes with two fixed lines, the axes of co-ordinates. As new axes of co-ordinates we may take any other pair of lines through the origin, and for the X, Y corresponding to x, y any new constant multiples of the sines of the angles which the line makes with the new axes. The substitution for x, y in terms of X, Y is the most general linear substitution in virtue of the four degrees of arbitrariness introduced, viz. two by the choice of axes, two by the choice of multiples. If now the it" denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a covariant expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples. Besides the invariants and covariants, hitherto studied, there are others which appertain to particular cases of the general linear substitution. Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution tt xl= XItt Ei+111 , x2 = P,26. Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w'=/3—a, and inclined at angles a, 13 to the old axis of x, without change of origin, are x—sinw-a)X+sin(w—li)y sinw sin w __sin aX y sin w a transformation of modulus sin co' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry. Of the quadratic ax2+2bxy+cy2, he discovered the two invariants ac—b2, a—2b cosw+c, and it may be verified that, if the transformed of the quadratic be AX2-2BXY+CY2, (sin 0 2 AC—B2= \co) (ac—b2), A--2Bcosw'+C= (sin w'1 ?(a—2bcosw+c). \sin w f The fundamental fact that he discovered was the invariance of x2+2 cos co xy+y2, viz. x2+2 cos w xy+y2=X2+2 cos w'XY+Y2, from which it appears that the Boolian invariants of axe+2bxy—y2 are nothing more than the full invariants of the simultaneous quadratics ax2+ 2bxy+y2, x2+2 cos w xy±y2, the word invariant including here covariant. In general the Boolian system, of the general n'°, is coincident with the simultaneous system of the n"° and the quadratic x2+2 cos w xy+y2. Orthogonal System.—In particular, if we consider the transformation from one pair of rectangular axes to another pair of rectangular axes we obtain an orthogonal system which we will now briefly inquire into. We have cos w' = cos w=o and the substitution xi = cos 8X1—sin OX2 x2 = sin 8X1 +cos 8X2, with modulus unity. This is called the direct orthogonal substitu- tion, because the sense of rotation from the axis of XI to the axis of X2 is the same as that from that of x1 to that of x2. If the senses of rotation be opposite we have the skew orthogonal substitution j xl =coseXi+sinOX2, x2 = sin 8X1—cos8X2, of modulus—1. In both cases dal and dal are cogredient with xi and x2; for, in the case of direct substitution, d axi = cos 8aX 1— sin BdX 2' d dx2 = sin BaXI +cos BdX 2' and for skew substitution adl=cos0-Xl+sinead 2X d dx2=sin 0 —cosed- 2 Hence, in both cases, contragrediency and cogrediency are identical, and contravariants are included in covariants. 215 215 1—z.1—z2.1—z3.1—z4' The series may be continued, but the calculations soon become very laborious. V. RESTRICTED SUBSTITUTIONS We may regard the factors of a binary n'° equated to zero as denoting n straight lines through the origin, the co-orstinates being Cartesian and the axes inclined at any angle. Taking the variables to be x, y and effecting the linear transformation x = XiX+µlY, y = X2X+µ2Y, sin l3 +sin wY' Consider the binary na. (aixi+a2x2)^=a=, and the direct substitution x, = )X1 —µX2, x2=µXi+XX2, where a2+-µ2 =1; X, µ replacing cos 0, sin 0 respectively. In the notation a: =71x1+72x2, observe that as = +a2, ab =aibi+a2b2. Suppose that a:==c,=... is transformed into Ax=BX=Cx=... then of course (AB)=(ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A, = +µa2, A2 = +Xaz, AA=Ai+Al= (X2+) (4+a2) =aa; AB = A, B,+A2B2 = (X2+µ2) (a1 b, +a2b2) = ab; (RA) = X1A2 —R2A1= (Xxi+µx2) ( —µa,+Xa2) - (-µxi +axe) (aai +µa2) = (a2+µ2) (xias - x2ai) = (xa) ; showing that, in the present theory, ¢a, ab, and (xa) possess the invariant property. Since x;+xi=x. we have six types of symbolic factors which may be used to form invariants and covariants, viz. (ab), a,,, ab, (xa), as, xa. The general form of covariant is therefore (ab) (ac) b2 (bc) ba...a~1b2c`a aa1a7g6~a a b c c •• X (xa) k1 (xb) k2 (xc) k' ...a." = (AB) 61 (AC) k2 (BC) ba...AABBC...A ACB..: X (RA) 'hi (RB) k2 (RC) ha...A7[Bj1Cj[...XI. If this be of order a and appertain to an 00 Xk+El+2m = e, hi +h2+...+2i, +j,+j2+... +ki+li = n, hi+ha+...+2i2+ji+is+...+k2+12=n, h2+h3+... +2'la+j2+ia+...+ka+la =n ; viz., the symbols a, b, c,... must each occur it times. It may denote a simultaneous orthogonal invariant of forms of orders n1, n2, na,...; the symbols must then present themselves n2, na...times respectively. The number of different symbols a, b, c,...denotes the degree 0 of the covariant in the coefficients. The coefficients of the covariants are homogeneous, but not in generalisobaric fuhctions, of the coefficients of the original form or forms. Of the above general form of covariant there are important transformations due to the symbolic identities: (ab) s 2 2 2 as a consequence any even power of a determinant factor may be expressed in terms of the other symbolic factors, and any uneven power may be expressed as the product of its first power and a function of the other symbolic factors. Hence in the above general form of covariant we may suppose the exponents h1, h2, ha,...k1, k2, ka,... if the determinant factors to be, each of them, either zero or unity. Or, if we please, we may leave the determinant factors untouched and consider the exponents ji, j2, j3,•••11, 12, 18,... to be, each of them, either zero or unity. Or, lastly, we may leave the exponents h, k, j, 1, untouched and consider the product .1 t2 :a m to be reduced either to the form g, where g is a symbol of the series a, b, c,... or to a power of x,. To assist us in handling the symbolic products we have not only the identity (ab)c,+(bc)a.+(ca)b: =0, but also (ab)x,+(bx)a2+(xa)b==0, (ab)a,+(bc)aa+(ca)ab=0, and many others which may be derived from these in the manner which will be familiar to students of the works of Aronhold, Clebsch and Gordan. Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms. For the linear forms aox,+alx2==b. there are four fundamental forms (i.) ate=70x1+7,x2 of degree-order (1, 1), (ii.) x = x1 +x2 „ (0, 2), (iii.) (xa) =—aox2 „ (1, 1), (iv.) ab=a02+ai (2, 0), (iii.) and (iv.) being the linear covariant and the quadrinvariant respectively. Every other concomitant is a rational integral function of these four forms. The linear covariant, obviously the Jacobian of aZ and xx is the line perpendicular to as, and the vanishing of the quadrinvariant ab is the condition that a. passes through one of the circular points at infinity. In general any pencil of lines, connected with the line as by descriptive or metrical properties, has for its equation a rational integral function of the four forms equated to zero. For the quadratic aoxi+2axx1x2+72x9, we have (i.) as =aixi+2a,xix2+a.,xi, (ii.) xz=xl+x9, (iii.) (ab)2=2(¢oaa—ai), (iv.) a,,=ao+a2, (v.) (xa) as = a1x + (a2 -70 xi x2 — a1x Z . This is the fundamental system; we may, if we choose, replace (ab)2 by 4 =74+2—4 +—a; since the identity aabb -4 b = (ab)2 shows the syzygetic relation (ao+72)2—1Io+2ai+4) =2(¢oa~—ai)• There is no linear covariant, since it is impossible to form a symbolic product which will contain x once and at the same time appertain to a quadratic. (v.) is the Jacobian; geometrically it denotes the bisectors of the angles between the lines ay, or, as we may say, the common harmonic conjugates of the lines ay and the lines xy. The linear invariant as is such that, when equated to zero, it determines the lines a, as harmonically conjugate to the lines xe; or, in other words, it is the condition that al may denote lines at right angles. End of Article: AIB4 (AiB2+A1Bs+B4) (—Bs—A1B2B3 —ATB4) [back]AHVAZ [next]AIBONITO


text:
Welcome guest
You're not logged in.
176 users online, thereof 0 logged in

## Proposition: Multiplication of Integers

According the definition of integers, we can identify integers $$x,y \in \mathbb Z$$ with equivalence classes $$x:=[a,b]$$, $$y:=[c,d]$$ for some natural numbers $$a,b,c,d\in\mathbb N$$.

Based on the addition of natural numbers and the multiplication of natural numbers, we define a new multiplication operation “$$\cdot$$” for all integers by setting

$\begin{array}{ccl} x\cdot y:=[a,b] \cdot [c,d] &:=& [a\cdot c + b\cdot d,~ a\cdot d + c\cdot b]=[ac + bd,~ ad + bc], \end{array}$

where $$[ac + bd,~ ad + bc]$$ is also an integer, called the product of the integers $$x$$ and $$y$$. The product exists and is well-defined, i.e. it does not depend on the specific representatives $$[a,b]$$ and $$[c,d]$$ of $$x$$ and $$y$$.

| | | | | created: 2014-09-21 14:52:36 | modified: 2019-04-17 06:09:15 | by: bookofproofs | references: [696]

(none)


text:
# When is a metric space isometrically embeddable into some Banach space?

EDIT

Oops---I found the answer to the first question of mine here on Wikipedia---this is really classic material. I'll leave the question open for a bit, in case someone tells me something interesting for my second question.

(A version of the above was mentioned by Michael at about the same time as my "oops", so I accepted his answer)

Sorry for asking a basic and naive question---if this is textbook material somewhere, please let me know, so that I may close this question.

My question is:

When is a metric space $(X,d)$ isometrically embeddable into some Banach space?

Can one say something "stronger", if we know that the closure of $X$ is actually a convex cone?

(By "stronger" one of the things I mean is whether we can actually obtain an embedding that is easy to compute...)

-

Arbitrarily fix $y_0\in X$. Then, with every $y\in X$ you can associate a bounded continuous function from X to R defined by $$f_y(x)=d(y,x)-d(y_0,x).$$ It is easy to show that $$\max_x |f_y(x)-f_z(x)|=d(y,z),$$ with the maximum assumed if $x=y$ or $x=z$. Hence $X$ is isometrically embedded in the Banach space $C_b(X)$.
One useful embedding is into the "free space" (or "Arens-Eels space") over $X$, which is the predual of the space $Lip(X)$ of Lipschitz functions from the (pointed) metric space $(X,0)$ into the real line that map zero to zero. The free space of $X$ is just the closed span in $Lip(X)^*$ of the pointwise evaluations functionals. This millennium Godefroy and Kalton used this embedding to good effect in their paper Lipschitz-free Banach spaces, Studia Math. 159 (2003), 121–141. The embedding is described and discussed in the book N. Weaver, Lipschitz algebras, World Scientific Publishing Co. Inc., River Edge, NJ, 1999.


text:
"Had to be me. Someone else might have gotten it wrong." - Mordin Solus - Mass Effect 3 San and crasX rule! Fk droolz (but is working on gamrs.co 4.0)

Welcome to gamrs.co - 15+ years of friendly PC and Console gaming!
Registering for a forum account is pretty painless and you'll never see this again. If you are looking to become a contributing GC member and get a reserve across all our servers you need to start here.

Have you been banned from one of our servers? You might have deserved it but sometimes people get caught in the crossfire. Either way, you'll need to read this first, then register on the forums so you can post up in our banned forums and state your case.

See you in-game or on the forums and again, welcome to GC!

Login to Account Create an Account

# L4D2 player/member commands list, & BINDS

96 replies to this topic

### #76 Vilkys Posted 17 March 2016 - 01:29 PM

Vilkys
• 215 posts      1 topics
Joined April 11, 2014

does the suicide bind work?

Yes, don't use it as survivor, you might end up buying something by mistake.

### #77 Beelzebub Posted 18 March 2016 - 09:47 AM

Beelzebub
• 53 posts      3 topics
Joined September 6, 2012

it doesnt work as survivor so you dont end up buying things you didnt wanted to buy ! Have fun and see ya all

### #78 Beelzebub Posted 19 March 2016 - 11:30 PM

Beelzebub
• 53 posts      3 topics
Joined September 6, 2012

had do correct my jointeam bind because it didnt worked

new version :

bind "v" "say !jointeam"

have fun with it

### #79 InspectorCallahan Posted 25 July 2016 - 02:47 PM

InspectorCallahan
• 6 posts      0 topics
Joined July 25, 2016

Color Coding:

Main Categories:

Weapons

Health Items

Sub-Categories

Sub-Sub-Categories ( "Next" )

Binds (Entire section highlighted in Green)

1. Weapons

1. Melee

.

8. Back

.

9. Next

.

8. Previous

.

0. Exit

0. Exit

2. Snipers

.

8. Back

.

0. Exit

3. Rifles

.

8. Back

.

0. Exit

4. Shotguns

.

8. Back

.

0. Exit

5. SMGs

.

8. Back

.

0. Exit

6. Throwables

.

8. Back

.

0. Exit

7. Misc

.

8. Back

9. Next

.

8. Previous

.

0. Exit

0. Exit

.

8. Back

.

0. exit

3. Health Items

.

8. Back

.

0. Exit

.

0. Exit

Edited by InspectorCallahan, 25 July 2016 - 06:10 PM.

### #80 ctrlfrk Posted 26 July 2016 - 02:29 AM

ctrlfrk
• 124 posts      25 topics
Joined September 3, 2012

Hi InspectorCallahan, I was the guy who originally made those aliases you're using. The problem of the menu remaining onscreen is a tricky one because it seems to require timing things perfectly. There was a comment at the top of the original config file that kind of explained it, but basically the wait 123; slot10 is ping dependent and was meant to be changed by the person using the config using find/replace.

The value of 123 was pretty high because I have a 200ms+ ping on these servers. You can try change it down to wait 10; or so and see if that helps you. You can also try changing slot10 to menuselect 0

Let me know how you get on.

### #81 InspectorCallahan Posted 26 July 2016 - 12:15 PM

InspectorCallahan
• 6 posts      0 topics
Joined July 25, 2016

Hi InspectorCallahan, I was the guy who originally made those aliases you're using. The problem of the menu remaining onscreen is a tricky one because it seems to require timing things perfectly. There was a comment at the top of the original config file that kind of explained it, but basically the wait 123; slot10 is ping dependent and was meant to be changed by the person using the config using find/replace.

The value of 123 was pretty high because I have a 200ms+ ping on these servers. You can try change it down to wait 10; or so and see if that helps you. You can also try changing slot10 to menuselect 0

Let me know how you get on.

Hi ctrlfrk, ty for the response. At first I was a bit confused but realizing first that the delay number is in milliseconds (ms), (50 = .5 sec, 100  = 1 sec etc.) and after some searching, I combed through my server config and voila! Sitting under my nose the whole time  The server convar below which I long had forgotten about fixes obviously the menu screen remaining on after binds are executed and I've hence adjusted my delay times accordingly and all is 5x5. The command below for the M60 is a fun one but I'm sure with aliasing it could be cleaned up for aesthetics.

sv_allow_wait_command 1

Buy an M60 with Explosive ammo + Laser (Tank Killer):

Edited by InspectorCallahan, 27 July 2016 - 10:20 AM.

### #82 Soda Posted 15 January 2017 - 09:11 PM

Soda
• 53 posts      4 topics
Joined November 1, 2014

this one's for my boy general

Edited by oGSoda, 23 February 2017 - 03:26 AM.

[GC] Elias Was kicked from the server for exceeding admin warnings.

### #83 ituly Posted 20 February 2017 - 01:18 PM

ituly
• 61 posts      16 topics
Joined May 24, 2014

Where do we insert the binds that InspectorCallahan mentioned ?

### #84 InspectorCallahan Posted 20 February 2017 - 01:45 PM

InspectorCallahan
• 6 posts      0 topics
Joined July 25, 2016

Where do we insert the binds that InspectorCallahan mentioned ?

In the autoexec.cfg file in your game's /cfg directory, or in that same directory you can create a new file e.g. mybinds.cfg and then in your autoexxec.cfg you execute it like this:

exec mybinds.cfg

you could, for example, to keep things tidy, create a sub directory under /cfg, call it /binds so it would read like this:

/cfg/binds

and in this new sub dir, you can keep all of your custom binds etc. Now in the autoexec.cfg you would then execute these binds as follows:

exec binds\mybinds.cfg

exec binds\mybinds2.cfg

and so on...

### #85 anzul Posted 20 February 2017 - 09:14 PM

anzul
• 337 posts      1 topics
Joined July 17, 2014

hey thats neat, thanks for the info inspector!

### #86 MOONFACE Posted 26 February 2017 - 07:28 PM

MOONFACE
• 4 posts      1 topics
Joined February 12, 2017

So I have followed the instructions and my binds will not work. Ex: bind "l" "laser"  like this in console today and it would not work.

Any help would be great!!

Thanks.

### #87 Elias Posted 26 February 2017 - 08:46 PM

Elias
• 87 posts      3 topics
Joined May 7, 2015

Laser Sight. bind "x" "sm_buy; wait 2; menuselect 2; wait 2; menuselect 1; wait 2; menuselect 1; wait 123; slot10". Copy exactly the way it is. Look up at anzuls post to see what other binds you want.

Edited by EliasPerez, 26 February 2017 - 08:48 PM.

### #88 anzul Posted 27 February 2017 - 12:41 AM

anzul
• 337 posts      1 topics
Joined July 17, 2014

So I have followed the instructions and my binds will not work. Ex: bind "l" "laser"  like this in console today and it would not work.

Any help would be great!!

Thanks.

thats not it works, what you did there was bind "l" to execute the comand "laser". you should follow what the other post say THOROUGHLY.

just so you know what you are doing, here's a brief explanation on how the binds work and what you need to do get the laser bind working.

first, binds are just a shorcut to execute one or more of the game's engine comands

second, the server has its own comands, they are only executable trough the chat.

third, you can either create all your binds trough the console, or by creating a .cfg file which you can edit anytime you want. the later is ideal if you plan on having 20+ binds

so, what you need to do is bind a key to sayin the chat the server's comand that gives you the laser, its goes like this:

sm_      is the source/game engine comand that says something in the game chat whithout it actualy apearing in it, you need to add what you want it to say

!laser    is the server's chat comand that you use during the game to get the laser

so what you need to do is this:

bind "l" "sm_!laser"

hope that helps you understand how it works!

Edited by anzul, 27 February 2017 - 12:42 AM.

### #89 ituly Posted 10 March 2017 - 11:49 AM

ituly
• 61 posts      16 topics
Joined May 24, 2014

How do I do this mac? x)

### #90 Soda Posted 10 March 2017 - 03:28 PM

Soda
• 53 posts      4 topics
Joined November 1, 2014

Laser Sight. bind "x" "sm_buy; wait 2; menuselect 2; wait 2; menuselect 1; wait 2; menuselect 1; wait 123; slot10". Copy exactly the way it is. Look up at anzuls post to see what other binds you want.

lol

[GC] Elias Was kicked from the server for exceeding admin warnings.

### #91 MOONFACE Posted 12 March 2017 - 10:53 PM

MOONFACE
• 4 posts      1 topics
Joined February 12, 2017

I will give it a try and see what it does.

is that the correct place?

### #92 Soda Posted 12 March 2017 - 11:17 PM

Soda
• 53 posts      4 topics
Joined November 1, 2014

bind "l" "sm_!laser"

isn't it just:

bind l sm_laser

same as sm_heal, sm_buy, sm_sp, etc.

[GC] Elias Was kicked from the server for exceeding admin warnings.

### #93 MOONFACE Posted 19 March 2017 - 02:13 PM

MOONFACE
• 4 posts      1 topics
Joined February 12, 2017

this is where I put the binds in my cfg directory. went in today to see why there werent working and they were gone.
Again the path I used to get here is: I:\Steam\SteamApps\common\Left 4 Dead 2\left4dead2\cfg
Im not well versed on this kinda stuff. Ask me anything about your eyes and various diseases and treatments I can help you with that.but this stuff, well. Its beyond me.
Im trying,really I am.
bind "0" "slot10"
bind "1" "slot1"
bind "2" "slot2"
bind "3" "slot3"
bind "4" "slot4"
bind "5" "slot5"
bind "6" "slot6"
bind "7" "slot7"
bind "8" "slot8"
bind "9" "slot9"
bind "a" "+moveleft"
bind "c" "+voicerecord"
bind "d" "+moveright"
bind "e" "+use"
bind "f" "impulse 100"
bind "h" "motd"
bind "l" "sm_!laser"
bind "m" "chooseteam"
bind "q" "lastinv"
bind "s" "+back"
bind "t" "impulse 201"
bind "u" "messagemode2"
bind "v" "tankbuster"
bind "w" "+forward"
bind "y" "messagemode"

### #94 Vilkys Posted 19 March 2017 - 07:44 PM

Vilkys
• 215 posts      1 topics
Joined April 11, 2014

Grab the autoexec.cfg on the first post and place it in your cfg folder.
After that just edit it in notepad or the text editor of your choice.

First post has examples and it's easy to understand.

### #95 DAlighieri Posted 20 March 2017 - 03:02 PM

DAlighieri
• 158 posts      10 topics
Joined August 13, 2015

this one's for my boy general

Don't you mean:

### #96 Soda Posted 21 March 2017 - 11:12 PM

Soda
• 53 posts      4 topics
Joined November 1, 2014

this one's for my boy general

Don't you mean:

Can't, mouse 1 already bound to propane tanks
[GC] Elias Was kicked from the server for exceeding admin warnings.

### #97 DAlighieri Posted 08 April 2017 - 01:09 PM

DAlighieri
• 158 posts      10 topics
Joined August 13, 2015

Public service announcement:

If you want to bind the plus key (+), you can't directly, as it says "invalid key".

bind + "sm_heal"

You have to use the "name" of the plus key, kp_plus.

Thus:

bind kp_plus "sm_heal"

#### 0 user(s) are reading this topic

0 members, 0 guests, 0 anonymous users


text:
## On generating sets of the clone of aggregation functions on finite lattices.(English)Zbl 1446.06010

Summary: In a recent paper [the first and the third author, Inf. Sci. 329, 381–389 (2016; Zbl 1390.06006)] we have shown that aggregation functions on a bounded lattice $$L$$ form a clone, i.e., the set of functions closed under projections and composition of functions. Moreover, for any finite lattice $$L$$ we gave a finite set of unary and binary aggregation functions on $$L$$ from which the aggregation clone is generated. In this paper, a general method for constructing generating sets of the aggregation clone on $$L$$ is presented. Our approach is based on extending of $$L$$-valued capacities leading to so-called full systems of aggregation functions. Several full systems on $$L$$ are presented (including singleton ones) and their arities are discussed.

### MSC:

 06B05 Structure theory of lattices 06A15 Galois correspondences, closure operators (in relation to ordered sets) 08A40 Operations and polynomials in algebraic structures, primal algebras

Zbl 1390.06006
Full Text:

### References:

 [1] Anderson, I., Combinatorics of Finite Sets (2011), Dover Publications: Dover Publications Mineola [2] Beliakov, G.; Pradera, A.; Calvo, T., Studies fuzziness on soft computing, Aggregation Functions: A Guide for Practitioners, 221 (2007), Springer [3] Botur, M.; Halaš, R.; Mesiar, R.; Pócs, J., On generating of idempotent aggregation functions on finite lattices, Inf. Sci. (NY), 430-431, 39-45 (2018) [4] Bronevich, A.; Mesiar, R., Invariant continuous aggregation functions, Int. J. Gen. Syst., 39, 2, 177-188 (2010) · Zbl 1193.93009 [5] Burris, S.; Sankappanavar, H. P., A Course in Universal Algebra (1981), Springer-Verlag · Zbl 0478.08001 [6] Ghédira, K.; Dubuisson, B., Constrained Satisfaction Problems (2013), Wiley [7] Grabisch, M., Set Functions, Games and Capacities in Decision Making (2016), Springer Verlag: Springer Verlag Berlin · Zbl 1339.91003 [8] Grabisch, M.; Marichal, J. L.; Mesiar, R.; Pap, E., Aggregation Functions (2009), Cambridge University Press: Cambridge University Press Cambridge [9] Grätzer, G., Lattice Theory: Foundation (2011), Birkhäuser: Birkhäuser Basel · Zbl 1233.06001 [10] Halaš, R.; Mesiar, R.; Pócs, J., Generators of aggregation functions and fuzzy connectives, IEEE Trans. Fuzzy Syst., 24, 6, 1690-1694 (2016) [11] Halaš, R.; Pócs, J., On lattices with a smallest set of aggregation functions, Inf. Sci. (NY), 325, 316-323 (2015) · Zbl 1387.06006 [12] Halaš, R.; Pócs, J., On the clone of aggregation functions on bounded lattices, Inf. Sci. (NY), 329, 381-389 (2016) · Zbl 1390.06006 [13] Jablonskii, S. W.; Gawrilov, G. P.; Kudrjavcev, W. B., Boolesche funktionen und postsche klassen (1970), Braunschweig: Braunschweig Vieweg · Zbl 0193.29601 [14] Klement, E. P.; Mesiar, R., On the expected value of fuzzy events, Int. J. Uncertain. Fuzziness Knowl. Based Syst., 23, 57-74 (2015) · Zbl 1377.60012 [15] Lau, D., Function Algebras on Finite Sets (2006), Springer-Verlag: Springer-Verlag Berlin [16] Ovchinnikov, S.; Dukchovny, A., Integral representation of invariant functionals, J. Math. Anal. Appl., 244, 1, 228-232 (2000) · Zbl 0954.28010 [17] Torra, V.; Narukawa, Y., Modeling Decisions: Information Fusion and Aggregation Operators (2007), Springer-Verlag
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
## You are here

View all publications

# Appendix A: Time Averages (NES for Air Quality, Schedule 1)

1-hour mean
1. a mean calculated every hour on the hour for the preceding hour; and
2. b) in relation to a contaminant at a particular location for a particular hour, means the mean of not more than 10-minute means, collected not less than once every 10 seconds, for the contaminant at that location during that hour

24-hour mean

1. a mean calculated every 24 hours at midnight for the preceding 24 hours; and
2. b) in relation to a contaminant at a particular location for a particular 24-hour period, means
1. the mean level at which the contaminant is recorded in the air, by continuous sampling of the air at that location, throughout that 24-hour period; or
2. the mean of the 1-hour means for that contaminant at that location for the preceding 24 hours

running 8-hour mean

1. a mean calculated every hour on the hour for that hour and the preceding 7 hours to give 1 running 8-hour mean per hour; and
2. b) in relation to a contaminant at a particular location for a particular hour, means the mean of the 1-hour means for that contaminant at that location for that hour and the preceding 7 hours.