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# Trouble With Springs-2

1 vote
77 views

Question-
A block mass m = 1kg is placed over another block of mass M = 3kg. M is placed over a smooth surface. Both the blocks move with velocity v towards a spring of constant K = 400 N/m and
μ = 0.2 for the two blocks. Then, the maximum value of v, so that there may not be slipping between m and M is -

My Attempt-
Let me call block m and block M together as the system. When the system compresses the spring and momentarily comes to rest, the compression can be found out through conservation of energy.
Elastic Potential Energy Gained = Kinetic Energy Lost (Work Done by Friction is Internal)
1/2 Kx2 = 1/2 (M+m) v2
Plugging in the values,
x = v/10
During this compression, Kinetic Friction Force acts on block m and reduces its velocity to zero.
Deceleration due to Kinetic Friction Force is given by,
2N = 1kg x a
a = 2m/s2
Since this is the maximum frictional force that is acting, the maximum velocity that it can reduce to zero over the same distance x can be obtained from kinematics.
On substituting x = v/10 in the kinematics equation, we get
v = 0.4 m/s

I am told that this answer is wrong. Have I committed any mistake?

Yes! You are Right!

1 vote

Here a two block system moves. Imagine it was colliding with a spring attached and causes a compression. And now it is compressing and a time comes where it has maximum compression , at this time the opposite force to the blocks would be maximum. --------------(1)

Now here the force is variable and hence the acceleration. So what can be the maximum acceleration that friction can provide ? first we need to sort that out.

$\mu = 0.2$
So $a_{max}= 2 m/s^2$

This $a_{max}$ is critical acceleration for which the two blocks move toghether.It is
that acceleration that maximumally friction can provide so as to oppose the relative motion.
Force required to produce this acceleration $F= 4× 2 =8 N$

Update :
Now since a force corresponding to the $a_{max}$ is 8 N , If the spring in the maximum compressed state would exert maximum force and both blocks survive the force together and the top block does not slide and we have counted here the maximum force all other time that is other than maximum compressed state the force would be less and hence friction would be able to provide that.

This maximum force is provided at time as in ---(1)
Hence applying conservation of energy ,

$F^2/ 2k = 1/2m v^2$
$8×8/ (2× 400) = 1/2 × 4 × v^2$

$v = 0.2 m /s$

answered Mar 21, 2017 by (2,290 points)
edited Mar 21, 2017
Which is the force that you mention is variable?
The force due to  the spring
'This a max is critical acceleration for which the two blocks move toghether.' What does it mean?
well;  question asks that the block on top does not slip  $a _{max}$ is that acceleration that maximumally friction can provide so as to oppose the relative motion
Could you tell what mistake I have made?


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# Bayesian Framework

by Chee Yee Lim

Posted on 2021-04-30

Collection of notes on model type - focusing on Bayesian framework (statistical point of view).

## Bayesian Framework

### Overview

• Key concepts
• The main aim of Bayesian method is not to find the single "best" value of the model parameters, but rather to determine the posterior distribution for the model parameters.
• Both parameters $$\beta$$ and output $$y$$ are estimated as distributions.
• Can be explicit about uncertainty.
• Get a distribution per parameter. Can draw random samples from these parameter distributions to get uncertainty bounds around output.
• Sampling process (i.e. model fitting) can be computationally expensive.
• A full Bayesian approach is difficult and often computational intractable, even on a simple problem.

### Model Fitting

• A Bayesian model can be approximate methods like Monte Carlo (random) sampling methods, with the most common being Markov Chain Monte Carlo (MCMC).
• Besides MCMC, there are also Variational Bayes and ABC.
• MCMC and Variational Bayes are expensive to compute, with the cheapest being maximum a posteriori (MAP).
• To get a point estimate from a Bayesian model, MAP can be used. MAP will return the mode of the posterior distribution. It is similar to maximum likelihood estimate (MLE), but with an additional prior distribution. MAP (and MLE) is guaranteed to find one of the models (local maxima).

### Bayesian Linear Regression

• Key concepts
• In the Bayesian viewpoint, we formulate linear regression using probability distributions rather than point estimates.
• A Bayesian linear regression can be formulated as:
• $$P( \beta | y, X ) = \frac{ P( y | \beta, X ) \times P( \beta | X ) }{ P( y | X ) }$$
• Where $$P( \beta | y, X )$$ is the posterior probability distribution of the model parameters given the inputs and outputs, and $$I$$ is the identity matrix.
• This is derived from Bayes Theorem.
• $$Posterior = \frac{ Likelihood \times Prior }{ Normalization }$$

### Bayesian Logistic Regression

• A Bayesian logistic regression can be formulated as a GLM with a binomial link function.
• The prior can be based on prior knowledge or a Gaussian prior, which is a vague prior that will let data speak for themselves.
• The likelihood is the product of n Bernoulli trials, $$\prod_{i=1}^n p_{i}^y (1 - p_i)^{1-y_i}$$,
• Where $$p_i = \frac{1}{1 + e^{-z_i}}, z_i = \beta_0 + \beta_1 x_1 + \beta_2 x_2$$ and $$y_i = \{ 0,1 \}$$.

### Bayesian Networks

• Key concepts
• Each Bayesian network is a directed acyclic graph.
• Each node represents a state and each edge has an associated probability.

format: plain


text:
# Strong law of large numbers - help understanding it

1. Nov 26, 2009

### WillJ

NOTE: This is a long post. If you want, you can skip ahead and start reading where I write "Things get really juicy".

The strong law of large numbers:

The stuff in parentheses is an event. But what is the sample space that that event is drawn from?

Imagine that our experiment is to select a number at random from [0,1], and we repeat this experiment in multiple trials. If we just do the experiment once, the sample space is [0,1]. If we do it twice, the sample space (if I'm not mistaken) is [0,1]X[0,1]. After 4 times, it's [0,1]X[0,1]X[0,1]X[0,1]. And so on. As the number of trials approaches infinity, our sample space approaches an infinite-dimensional sample space. The strong law of large numbers (if I'm not mistaken) says that, in this infinite-dimensional sample space (each outcome in this space being a countably-infinite string of numbers, each drawn from [0,1]), the subset of outcomes in which $$_{n}\stackrel{lim}{\rightarrow}_{\infty} \overline{X}_{n}=\mu$$ is true has probability 1, and the subset of outcomes in which that outcome is not true has probability 0.

Am I correct? Or is my thinking off?

It seems more common to think of an infinite sequence of iid random variables all defined on the same sample space. (Perhaps this is actually the correct interpretation, and my interpretation above is wrong.) Repeated trials of an experiment do not fit this description (at least not directly; see below). In that case, what makes things hard to comprehend is how we can have infinitely many iid r.v.s on the same sample space. The above example doesn't work (or maybe it does; see below). If our experiment is to select a number [0,1], and we do this multiple times, and Xi is the number we select in the ith trial, and we force {Xi} to all be on the same sample space (namely [0,1]), then for any s$$\in$$[0,1], X1(s)=s, X2(s)=s,X3(s)=s, and so on, i.e. the r.v.s are all the same and thus are dependent and the SLOLN does not apply. So we have to think of r.v.s that are all different from each other, yet identically distributed, all on the same sample space ... and infinitely many of them!

This issue was mentioned in a discussion on Terrence Tao's blog. Anonymous writes:

"In theory, we are speaking about a sequence X_1,X_2,... of random variables on the same common probability space. In practice, we think of instantiating a random variable in a sequence of trials. This is a subtle point: each trial is itself a random variable, distributed exactly as the original one, and leaving in an identical, but separate and independent, probability space. When we toss a die 100 times, there are 6^{100} possible outcomes. This means, we are no longer in our original six-element space, but in its cartesian power, with 6^{100} points. That is, tossing a die 100 times is described by the probability space which is the product of 100 copies of the space, corresponding to tossing just one die. So, our X_1,X_2,... actually leave in different spaces.

How does this agree with the theoretical assumption that the X_n’s are defined on the same space? To make this more specific: say, what is the “practical meaning” of pointwise convergence in the Strong law? When we think of trials, the variables \overline{X}_n live in distinct spaces; on what space they pointwise converge?"

Professor Tao responds:

"In elementary (finitary) probability theory, the sample space (or probability space) is often defined in textbooks as simply the set of all possible outcomes, as this is the simplest choice of space to work in for most finitary applications. When it comes to infinitary probability theory, though, it is better to take a more flexible and abstract viewpoint: the sample space is now allowed to be an abstract set, and each outcome corresponds to a separate event inside that set. For instance, if one is studying the flip of a single coin, the sample space \Omega could be a two-element set {Heads, Tails}, but it doesn’t have to have just two elements; it could be a much larger set, partitioned into two disjoint subsets, the “Heads” event and the “Tails” event. For instance, the sample space could be the unit interval [0,1] with Lebesgue measure, and the Heads and Tails events could be the intervals [0,1/2) and (1/2,1] respectively.

For the purposes of probability theory, the exact size of the sample space does not matter; the only thing that matters is the algebra (or more precisely, \sigma-algebra) of events and the probability (or measure) which is assigned to each event; the actual points in the sample space are in fact largely irrelevant to probability theory. (See also the notion of equivalence of measure spaces, as defined for instance in Lecture 11 of my 254A course. Equivalent probability spaces may have very different cardinalities, but are indistinguishable from each other for the purposes of doing probability theory.)"

Unfortunately such talk is rather mysterious to me. He gets around to saying:

To construct a suitable sample space to hold an infinite collection of random variables, such as an infinite sequence of independent die rolls, one can take an inverse limit of the sample spaces associated to finite sequences of die rolls.

Sounds promising, although what exactly does that mean? Can anyone elaborate?

He continues:

One could also resort to more ad hoc devices, such as taking the sample space to be the unit interval {}[0,1] with Lebesgue measure (which is the standard sample space for selecting a random variable x uniformly at random from the unit interval) and then defining the value X_i of the i^th die roll to be the i^th digit of x base 6, plus one. In this space, the law of large numbers has this interpretation: when one selects a number at random from the unit interval, then almost surely, its base 6 digits are uniformly distributed amongst 0,1,2,3,4,5.

Quite clever! And going back to the original example of selecting a number from [0,1], we could modify this into selecting a number from {0,1,...,9}, and doing independent trials of this, as represented by selecting a number from [0,1], with the outcome of the ith trial being the ith digit of the number selected. This seems nice, but also a little unsatisfactory to me, because it seems like we're taking the mysteriousness of the strong law of large numbers and explaining it by resorting to the mysteriousness of the real number system - yes, I suppose it's true that, on the interval [0,1], almost surely, the average of the digits of a number goes to 4.5 as one approaches the infinite digit of that number - and, assuming the digits are independent of one another, that's what the SLOLN says should happen - but that's not helping me very much.

Addressing the perhaps unsatisfactory-ness of the above, Tao writes:

Returning specifically to the question of finitary interpretations of the SLLN, these basically have to do with the situation in which one is simultaneously considering multiple averages \overline{X}_n of a single series of empirical samples, as opposed to considering just a single such average (which is basically the situation covered by the WLLN). For instance, if one had some random intensity field of grayscale pixels, and wanted to compare the average intensities at 10 x 10 blocks, 100 x 100 blocks, and 1000 x 1000 blocks, then the SLLN suggests that these intensities would be likely to be simultaneously close to the average intensity. (The WLLN only suggests that each of these spatial averages are individually likely to be close to the average intensity, but does not preclude the possibility that when one considers multiple such spatial averages at once, that a few outlying spatial averages will deviate from the average intensity. In my example with only three different averages, there isn’t much difference here, as the union bound only loses a factor of three at most for the failure probability, but the SLLN begins to show its strength over the WLLN when one is considering a very large number of averages at once.)

Hmm, anyone have any thoughts on that? Sounds promising, but I don't fully understand.

Things get really juicy, though, when Tao writes (explaining the SLLN vs. the WLLN):

Imagine a table in which the rows are all the possible points in the sample space (this is a continuum of rows, but never mind this), and the columns are the number n of trials, and there is a check mark whenever the empirical mean \overline{X}_n deviates significantly from the actual mean {\Bbb E} X. The weak law asserts that the density of check marks in each column goes to zero as one moves off to the right. The strong law asserts that almost all of the rows have only finitely many checkmarks. A double counting argument (or the Lebesgue dominated convergence theorem) then shows that the latter implies the former.

I think that if I manage to understand this, then I'll understand the SLLN in general. But I need some help. I wrote on his blog:

I don’t understand. Each row represents an outcome in the sample space. Each column represents a trial of the experiment. Right? (We assume, for purposes of the illustration, that our sequence of iid random variables is as follows: for any given outcome, all of the r.v.s assign the same number to that outcome; the different r.v.s correspond to the different trials. – Right?) Now where do we put the checkmarks? I can think of two possibilities of what you mean: 1) We imagine repeatedly running the experiment. Each trial gives us an outcome. If trial i gives us outcome j, then put a dot in (row j, column i) in our matrix. Now we take a look at what we have, and at each dot that we put in our matrix, we calculate the empirical mean up to that point (based on where that dot is and where the previous dots are), and if it’s significantly different from the actual mean, then we put a check-mark over that dot. If not, leave the dot be. (We don’t have to ever actually put dots; I just mentioned them for clarification.) But then a column can’t have more than one check-mark, so that must not be what you’re saying. Perhaps, then, you mean: 2) For EVERY cell in our matrix (every [row j, column i] combination), we take a look, do the calculations, and put a check-mark in that cell if, at that point, the empirical mean is significantly different from the actual mean. But that doesn’t make any sense, because the value of the empirical mean at any given cell depends on what the outcomes were in the previous trials, i.e. there’s no unique empirical mean that can be assigned to each cell in our matrix.

Can any of you guys help me out here?

Last edited by a moderator: May 4, 2017
2. Nov 27, 2009

### bpet

There's a few important points you seem to be missing in all this.

Well it is a possible outcome but happens almost never. As T explained *almost all* rows don't look like this.

You need to be clear about whether an experiment/outcome is a single dice roll or an infinite set of dice rolls. For either case you seem to be misinterpreting the table. For his example of real number digit expansions, the (j,i)th entry would be the i'th digit of j where 0<j<1.

More importantly, after a finite number of trials you can't know which specific row to place the dot - only which subset of rows it could be on (those for which the first i entries agree with your first i outcomes). This relates to his earlier point about sigma-algebras - you'll need to study measure theory to understand the subtleties of this.

For any position on the table, the empirical mean is well-defined because all the required information is available in the cells left of it.

Hope this helps clarify things :)

3. Nov 28, 2009

### WillJ

Thanks for the help, bpet. Fortunately, a little while after posting this, I came to see how I was looking at it wrong, and now everything is clear to me. :) Specifically, now I understand it's best to think of running infinitely many trials of an experiment as one big experiment in itself, with each outcome in the sample space specifying the results of the infinitely many trials. Then the table thing makes perfect sense, as do the WLOLN and SLOLN in general. (One can, I suppose, alternatively imagine the infinite-trial experiment as having an infinite-dimensional sample space, which is what I was doing before, but, as I can attest to, that's more taxing on the brain and can trip you up.)


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# Gosper's Mathematics

Sometimes I bump into more of the astonishing results of Gosper (some examples follow) and I gather that a lot of them come from hypergeometrics and special functions.

• Have there been any attempts to try and collect together these?
• What papers/books and such are there which study collections of results like this?

$$\prod_{n=1}^{\infty} \left(\begin{matrix} -\frac{n}{2(2n+1)} & \frac{1}{2n(2n+1)} & \frac{1}{n^4} \\\\ 0 & -\frac{n}{2(2n+1)} & \frac{5}{4n^2} \\\\ 0 & 0 & 1 \end{matrix}\right) = \left(\begin{matrix} 0&0&\zeta(5)\\\\0&0&\zeta(3)\\\\0&0&1 \end{matrix}\right)$$

$$W(x)=a+\sum_{n=0}^\infty \left\{{\sum_{k=0}^n {S_1(n,k)\over \left[{\ln\left({x\over a}\right)-a}\right]^{k-1}(n-k+1)!}}\right\} \left[{1-{\ln\left({x\over a}\right)\over a}}\right]^n$$

from a page about Lambert's W-Function

$$\prod_{n=1}^{\infty} \frac{1}{e}\left(\frac{1}{3n}+1\right)^{3n+1/2}= \sqrt{\frac{\Gamma(\frac{1}{3})}{2 \pi}} \frac{3^{13/24}\exp\left(1+\frac{2\pi^2-3\psi_1\left(\frac{1}{3}\right)}{12 \pi \sqrt{3}}\right)}{A^4}$$

from Mathworld

$$\sum_{n=1}^{\infty} \frac{(-1)^2}{n^2}\cos(\sqrt{n^2 \pi^2 - 9}) = - \frac{\pi^2}{21 e^3}$$

• Ramanujaniacal? – Eric Tressler Oct 5 '10 at 5:11
• SIAM publications...sort of...discuss some. – Unknown Oct 5 '10 at 7:15
• Well... there's HAKMEM: home.pipeline.com/~hbaker1/hakmem/hakmem.html – J. M. isn't a mathematician Oct 5 '10 at 15:14
• What is $A$ in the denominator of the formula for the second infinite product? – John Bentin Oct 5 '10 at 20:03
• There are two typos in the last sum. It should be $\sum_{n=1}^\infty \frac{(-1)^n}{n^2}\cos(\sqrt{n^2\pi^2-9}) = -\frac{\pi^2}{12e^3}$. – Tito Piezas III Dec 30 '12 at 14:38

## 3 Answers

The book 'Concrete Mathematics' by Graham, Knuth, and Patashnik discusses some of the work of Gosper, in particular a lot of detail on the Gosper-Zeilberger algorithm. (It's also a really great book generally.)

The obvious answer to your question is to search for papers authored by Gosper, or papers with Gosper's name in the title, such as Pages from the computer files of R. William Gosper. But surely this answer is so obvious that you already know it.

A good starting point for understanding modern methods for handling hypergeometric and other special function identities (as opposed to just a list of spectacular identities) is the famous book A=B.

Many such identities begin life by being discovered experimentally and then proved afterwards. A good entry point into the experimental mathematics literature is Bailey and Borwein's website.

Some of Gosper's results are studied, and proved, in the article "Pages from the Computer Files of R. William Gosper," by Mourad E. H. Ismail, Yu Takeuchi and Ruiming Zhang (Proceedings of the American Mathematical Society, Volume 119, Number 3, November 1993). Gosper's work is also discussed in Wolfram Koepf's book Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. See also Wikipedia's list of hypergeometric identities.


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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 2, Pages 209–232 (Mi tvp3915)

On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution Tails

A. A. Borovkova, K. A. Borovkovb

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b University of Melbourne, Department of Mathematics and Statistics

Abstract: We establish first-order approximations and asymptotic expansions for probabilities of crossing arbitrary curvilinear boundaries in the large deviations range by random walks with regularly varying distribution tails. In particular, we study the large deviations probabilities for the sums and maxima of partial sums of independent and identically distributed random variables, including the asymptotic behavior of the densities when they exist. Extensions to the "regular exponential" case (when the distribution tail differs from the exponential one by a regularly varying factor) are considered in part II of the paper.

Keywords: large deviations, random walk, regular variation.

DOI: https://doi.org/10.4213/tvp3915

Full text: PDF file (2155 kB)

English version:
Theory of Probability and its Applications, 2002, 46:2, 193–213

Bibliographic databases:

Citation: A. A. Borovkov, K. A. Borovkov, “On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution Tails”, Teor. Veroyatnost. i Primenen., 46:2 (2001), 209–232; Theory Probab. Appl., 46:2 (2002), 193–213

Citation in format AMSBIB
\Bibitem{BorBor01} \by A.~A.~Borovkov, K.~A.~Borovkov \paper On Probabilities of Large Deviations for Random Walks. I. Regularly Varying Distribution Tails \jour Teor. Veroyatnost. i Primenen. \yr 2001 \vol 46 \issue 2 \pages 209--232 \mathnet{http://mi.mathnet.ru/tvp3915} \crossref{https://doi.org/10.4213/tvp3915} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1968683} \zmath{https://zbmath.org/?q=an:1006.60020} \transl \jour Theory Probab. Appl. \yr 2002 \vol 46 \issue 2 \pages 193--213 \crossref{https://doi.org/10.1137/S0040585X97978877} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000176400600001}

• http://mi.mathnet.ru/eng/tvp3915
• https://doi.org/10.4213/tvp3915
• http://mi.mathnet.ru/eng/tvp/v46/i2/p209

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
Cycle of papers

This publication is cited in the following articles:
1. Borovkov A.A., “Large deviations probabilities for random walks in the absence of finite expectations of jumps”, Probab. Theory Related Fields, 125:3 (2003), 421–446
2. Foss S., Zachary S., “The maximum on a random time interval of a random walk with long-tailed increments and negative drift”, Ann. Appl. Probab., 13:1 (2003), 37–53
3. A. A. Borovkov, “Kolmogorov and boundary problems of probability theory”, Russian Math. Surveys, 59:1 (2004), 91–102
4. A. A. Borovkov, K. A. Borovkov, “On probabilities of large deviations for random walks. II. Regular exponentially decaying distributions”, Theory Probab. Appl., 49:3 (2005), 189–206
5. Tang Qihe, “Uniform estimates for the tail probability of maxima over finite horizons with subexponential tails”, Probab. Engrg. Inform. Sci., 18:1 (2004), 71–86
6. A. A. Borovkov, K. A. Borovkov, “Large Deviations Probabilities for Generalized Renewal Processes with Regularly Varying Jump Distributions”, Siberian Adv. Math., 16:1 (2006), 1–65
7. A. A. Borovkov, “Asymptotic analysis for random walks with nonidentically distributed jumps having finite variance”, Siberian Math. J., 46:6 (2005), 1020–1038
8. Foss S., Palmowski Z., Zachary S., “The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk”, Ann. Appl. Probab., 15:3 (2005), 1936–1957
9. Barbe P., Mccormick W.P., “Asymptotic expansions of convolutions of regularly varying distributions”, J. Aust. Math. Soc., 78:3 (2005), 339–371
10. A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations of sums of independent random vectors under Cramér's condition. II”, Theory Probab. Appl., 51:4 (2007), 567–594
11. A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations for sums of independent random vectors under the Cramer condition. I”, Theory Probab. Appl., 51:2 (2007), 227–255
12. A. A. Borovkov, A. A. Mogul'skii, L. V. Rozovskii, A. I. Sakhanenko, “On Zhulev's paper “On large deviations. II””, Theory Probab. Appl., 51:2 (2007), 398–400
13. A. A. Borovkov, A. A. Mogul'skii, “On Large Deviations of Sums of Independent Random Vectors on the Boundary and Outside of the Cramér Zone. I”, Theory Probab. Appl., 53:2 (2009), 301–311
14. A. A. Borovkov, A. A. Mogul'skii, “On Large Deviations of Sums of Independent Random Vectors on the Boundary and Outside of the Cramér Zone. II”, Theory Probab. Appl., 53:4 (2009), 573–593
15. Blanchet J.H., Liu Jingchen, “State-dependent importance sampling for regularly varying random walks”, Adv. in Appl. Probab., 40:4 (2008), 1104–1128
16. Barbe Ph., McCormick W.P., “Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications”, Mem. Amer. Math. Soc., 197, no. 922, 2009, viii+117 pp.
17. Kortschak D., Albrecher H., “An asymptotic expansion for the tail of compound sums of Burr distributed random variables”, Stat. Probab. Lett., 80:7-8 (2010), 612–620
18. Fushiya H., Kusuoka Sh., “Uniform Estimate for Distributions of the Sum of i.i.d. Random Variables with Fat Tail”, Journal of Mathematical Sciences-the University of Tokyo, 17:1 (2010), 79–121
19. Albrecher H., Hipp Ch., Kortschak D., “Higher-order expansions for compound distributions and ruin probabilities with subexponential claims”, Scandinavian Actuarial Journal, 2010, no. 2, 105–135
20. V. R. Fatalov, “Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method”, Izv. Math., 75:4 (2011), 837–868
21. Mao T., Hu T., “Second-Order Properties of Risk Concentrations Without the Condition of Asymptotic Smoothness”, Extremes, 16:4 (2013), 383–405
22. Lin J., “Second Order Tail Behaviour For Heavy-Tailed Sums and Their Maxima With Applications To Ruin Theory”, Extremes, 17:2 (2014), 247–262
23. Blanchet J. Murthy K.R.A., “Tail Asymptotics For Delay in a Half-Loaded Gi/Gi/2 Queue With Heavy-Tailed Job Sizes”, Queueing Syst., 81:4 (2015), 301–340


text: @Article{JPDE-5-53, author = {Wang Lianju}, title = {The Dirichlet Problems for a Class of Fully Nonlinear Elliptic Equations Relative to the Eigenvalues of the Hessian}, journal = {Journal of Partial Differential Equations}, year = {1992}, volume = {5}, number = {2}, pages = {53--68}, abstract = { ln this paper we discuss the Dirichlet problems for a class of fully nonliucar elliptic equations F(D² u) = ψ(x, u)(ψ(x, u, ∇u)) \quad in Q u = φ(x) \quad on ∂Ω where F is represented by a symmetric function f(λ_1, …, λ_n) of the eigenvalues (λ_1, …,λ_n) of the Hessian D²u. This result extends the works of Caffarelli L., Nirenberg L., Spruck L. [2] to more general cases.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5737.html} }


text:
# Need some exercises w/ solns for compound interest, present value, nominal rate, etc

#### Janahi

##### New member
Hi All,

Can anyone provide me some problems with solutions for following topics:

Compound interest
Amount & present value
Nominal & Effective Rates
Annuity due and ordinary deferred annuity
needed for my final exam revision

#### Subhotosh Khan

##### Super Moderator
Staff member
Hi All,

Can anyone provide me some problems with solutions for following topics:

Compound interest
Amount & present value
Nominal & Effective Rates
Annuity due and ordinary deferred annuity
needed for my final exam revision

worksheets "compound interest" - you'll find many websites with problem sets. Try it and let us know what you found.....

#### Denis

##### Senior Member
If you lend me some money, I'll gladly show
you how to calculate the monthly payment

#### Jomo

##### Elite Member
If you lend me some money, I'll gladly show
you how to calculate the monthly payment
I'll lend you some money, say $1,000. Simple interest. At the end of the week you owe me$1000 in interest plus the initial $1000. If you don't pay me the initial$1000, then at the end of the 2nd week you owe me $1000 in interest plus the initial$1000.


text:
## Precalculus: Mathematics for Calculus, 7th Edition

$k = \ln m$
$Write$ $the$ $equation$ $in$ $logarithmic$ $form:$ $e^k = m$ Exponential form to natural log form: $e^b = a \rightarrow \ln a = b$ [e stands for euler's number] Plug in $k$ for b and $m$ for a $$e^k = m \rightarrow \ln m = k$$ $$k = \ln m$$


text:
# Can rocket candy explode during the preparation?

I would like to follow the steps in this video: https://www.youtube.com/watch?v=hyPysthIRss

After I mixed the Potassium nitrate KNO3 and the sugar, I start to cook it in a pot, and stir it very often. After 25-30 min, the KNO3 and the sugar will melt together, and become a brown mass.

My question is: Can this mixture explode during the heating somehow? Or it can just burn, not explode? How to prevent it, and do it the most secure way?

• If it's brown, it may be too hot. Do you use a temperature probe? in my case it didn't get significantly dark. You are caramelizing the sugar... that's too hot. – FarO Aug 18 '16 at 8:33

## 1 Answer

See James Yawn's website ${\it Recrystalized\; Rocketry}$, which is all about solid propellant rocketry of the type you're interested in. It's really an exceptionally comprehensive site.

Can this mixture explode during the heating somehow?

It appears that it's more likely to catch fire, and do so rather vigorously, as in cases where it's exposed to a spark or an open flame. Because the system (pan and mixture of $\ce{KNO3}$ and sugar) would be open to the atmosphere (and not confined to a fixed volume), there's more of a risk of fire than of an explosion. As reported by James Yawn in the subsection on "Skillet Rcandy":

This stuff looks innocent enough in the pan. But if you have ever seen a quantity this size burn you will have more respect for it. This batch could send a tongue of flame 20 feet in the air and engulf the person stirring it, if only briefly. Burns to the body would likely be second degree, and I am willing to tolerate that level of risk. But eyeballs don't recover from this kind of trauma easily.

Or it can just burn, not explode?

Yes, it can just burn: that's essentially what's going on in the rocket engine. The exhaust from the burn is channeled in one direction (${\it via}$ the hollow core in the engine itself), resulting in thrust in the other direction.

How to prevent it, and do it the most secure way?

From James Yawn's site:

Basic Safety Precautions

At present, I do NOT recommend making this propellant in a household kitchen.

Make small batches only! No more than one standard recipe, which yields about 160 grams of propellant.

Wear protective gear. You should have a full-face shield, long sleeved heavy cotton shirt, heavy apron and heat-resistant gloves. These must be worn at appropriate points in the process.

Do not allow children or rambunctious pets in the room where you are working.

Do not allow distracting, inattentive, inebriated or otherwise incompetent adults in the room, especially if they smoke.

Be vigilant of possible sources of sparks, flame, or high temperature that could ignite propellant on contact.

If you smoke, place your cigarettes and lighter in another room so you will not light up unconsciously.

Clear your work-area of any flammable materials.

Have a good fire extinguisher available. Also a 5-gallon bucket of water nearby and a garden hose outside.

Make sure there is good ventilation, and a quick escape route to fresh air.

Avoid stupid rocket tricks.

Avoid excessive exposure to Potassium Nitrate.

All of the above points are elaborated upon at the linked page.


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text:
Contributors:

Date created: | Last Updated:

: DOI | ARK

Category: Project

Description: This website reposts data previously posted on the American Arbitration Association's website: https://www.adr.org/consumer. The AAA describes the data on their website: "The AAA maintains an online Consumer Arbitration Statistics report based on consumer cases filed with the AAA for at least the last five years. This report is made available pursuant to state statutes such as the California Code of Civil Procedure §1281.96 and Maryland Commercial Law §§ 14-3901 to 3905 and updated quarterly, as required by law." In practice, each time the AAA adds the latest quarter of data, it takes down the earliest quarter. This website aids researchers by retaining the data each quarter in exactly the format in which it was originally posted. We are retaining and posting this data because we have found it useful in trying to understand the effect of mandates for consumers to arbitrate. Caveats are in order. A first limitation of the data is the absence of access to the underlying materials, which are held privately. As the AAA explains, it does not independently verify what arbitrators report to it. A second problem is that coding errors can occur at both individual and aggregate levels. For example, when researching consumer arbitration between 2015 and 2016, we identified sixty-two cases in the set that were described as seeking the same amount ($607,525.40) and in which each consumer was listed as having received the same award ($585.71). AAA research staff responded to our inquiries, identified a computer coding error affecting these cases as well as other cases, and posted corrected data. But no red flags told other researchers that the data had been corrected. Thus, a vivid example of a potential error may be found through culling thousands of entries and then seeking clarification, but the general public has no systematic method of checking the accuracy of the data posted by AAA. Several authors have used this data, see, e.g., David Horton & Andrea Cann Chandrasekher, After the Revolution: An Empirical Study of Consumer Arbitration, 104 Geo. L.J. 57 (2015); Judith Resnik, Diffusing Disputes: The Public in the Private of Arbitration, the Private in Courts, and the Erasure of Rights, 124 Yale L.J. 2804 (2015). Another site that has usable AAA data is Level Playing Field (http://levelplayingfield.io), and there could be other sites as well. If you have ideas about or corrections to this data please email ylsarbitrationdataarchive@gmail.com. Thanks are due to the Yale Law Library for their help conceiving and constructing this website.


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# nLab E-∞ geometry

## Theorems

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

What may be called $E_\infty$-geometry, or spectral algebraic geometry is the “full” version of derived algebraic geometry where the spaces are locally modelled on E-infinity rings (as opposed to simplicial commutative rings or dg-algebras).

(In view of existing terms “arithmetic geometry” and “higher geometry” one might also tend call this higher arithmetic geometry,but notice that the term higher arithmetic geometry already has a different traditional usage. Maybe $E_\infty$-arithmetic geometry works well.)

That is to say, it is the higher geometry in the (∞,1)-topos over the (∞,1)-site of formal duals of E-∞ rings, equipped with the etale topology.

An E-∞ scheme ( spectral scheme ) in this sense is a structured (∞,1)-topos in $E_\infty$-geometry.

## Terminology

The term spectral algebraic geometry is used in the literature. On the nLab we use the term $E_\infty$-geometry due to a possible terminology clash discussed at spectral geometry.

## Properties

### Higher analogs of Fermat derivations and Frobenius maps

The refinement to $E_\infty$-arithmetic geometry of the power map $(-)^p$ which gives the Fermat quotient p-derivation and the Frobenius homomorphism in plain arithmetic geometry is the power operations on multiplicative cohomology theory (Lurie “Adic”, remark 2.2.9).

### Cohesion

cohesion modalitysymbolinterpretation
flat modality$\flat$formal completion at
shape modality$ʃ$torsion approximation
dR-shape modality$ʃ_{dR}$localization away
dR-flat modality$\flat_{dR}$adic residual
$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$

## References

The basic definitions are in

Fundamental properties of $E_\infty$-geometry are discussed in

See also the overview

• Clark Barwick, Applications of derived algebraic geometry to homotopy theory, lecture notes, mini-course in Salamanca, 2009, pdf.

Last revised on February 5, 2016 at 06:54:11. See the history of this page for a list of all contributions to it.


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Hide

As Easy as CAB

We all know how to alphabetize a list of distinct words when you know the alphabet: One word may be a prefix of another longer word, in which case the shorter word always comes before the longer word. With any other two words there must be a first place in the words where their letters differ. Then the order of the words is determined by the lexicographical order of these first differing letters.

How about the reverse problem: Can you find the lexicographic order of the alphabet from an ordered list of words? Suppose an alphabet exists where the following list of word strings is given in lexicographical order:

cab
cda
ccc

It is clear that c comes before b in the underlying alphabet because cab comes before badca. Similarly, we know a comes before d, because cab < cda, a comes before c because cab < ccc, and d comes before c because cda < ccc. The only ordering of the 4 alphabet characters that is possible is adcb.

However, it may be that a list contains inconsistencies that make it impossible to be ordered under any proposed alphabet. For example, in the following list it must be that a comes before b in the alphabet since abc < bca, yet it also must be that b comes before a in the alphabet since bca < aca.

abc
bca
cab
aca

Finally, some lists may not provide enough clues to derive a unique alphabet order, such as the following:

dea
cfb

In this list, d comes before c but we don’t know about the relative positions of any of the other letters, so we are unable to uniquely discern the order of the alphabet characters.

Input

The first line of input will contain $L$ and $N$, separated by a space, where $L$ is a lowercase character $\texttt{b} \le L \le \texttt{z}$ representing the highest character in the English alphabet that appears in the derived alphabet, and $N$ is an integer $1 \leq N \leq 1\, 000$ that is equal to the number of strings in the list. Each of the next $N$ lines will contain a single nonempty string of length at most $1\, 000$, consisting only of characters in the derived alphabet. No two strings will be the same.

Output

If the input is consistent with a unique ordering of the alphabet, output a string that designates that ordered alphabet. If the data is inconsistent with any ordering, output IMPOSSIBLE. If the data is consistent with multiple orderings, output AMBIGUOUS.

Sample Input 1 Sample Output 1
d 4
cab
cda
ccc

adcb

Sample Input 2 Sample Output 2
c 4
abc
bca
cab
aca

IMPOSSIBLE

Sample Input 3 Sample Output 3
f 2
dea
cfb

AMBIGUOUS

Sample Input 4 Sample Output 4
e 5
ebbce
dbe

edabc


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# Can mairlist scripts include other scripts?

Hi,

I have code that I would like to use in more than one mls script.

Can I import code like this (functions & procedures) into the mls scripts where I need to use it
or do I need to cut and paste the same code into every mls script I write?

An example is this function that generates a somewhat random id.

function generate_ext_id(i: integer): string;

// generate a good enough random external playlist id
var s: string;
j: integer;
begin
s := '#E';
for j := 1 to 5 do
s := s + chr(65 + random(25));
s := s + '-' + inttostr(i);
result := s;
end;

Inclusion works using a “special” comment syntax:

{$Include path\to\somescript.mls} You can also defend against multiple inclusion using a preprocessor constant. If, for example, you put the following inside somescript.mls: {$Define SOME_SCRIPT_LOADED}

You can then put this in any script that needs the functions and procedures defined in somescript.mls:

{$Ifndef SOME_SCRIPT_LOADED} {$Include path\to\somescript.mls}
{\$Endif}

Cameron

Interesting information; never relalized the defines and ifdef/ifndefs worked in scripts. (The scripting engine is not my work but a 3rd party library.)


text:
Engineering Mechanics: Statics & Dynamics (14th Edition)

$\omega=28.9~rad/s \circlearrowright$ $\alpha=470~rad/s^2\circlearrowleft$
The required angular velocity and angular acceleration can be determined as follows: We know that $\frac{dx}{dt}=V=-0.2sin\theta\frac{d\theta}{dt}$ $\omega=\frac{d\theta}{dt}$ $\implies V=-0.2\omega sin\theta$ We plug in the known values to obtain: $-5=-0.2\omega sin 60$ This simplifies to: $\omega=28.9~rad/s \circlearrowright$ As $V=-0.2\omega sin\theta$ Differentiating with respect to $t$, we obtain: $\frac{dV}{dt}=a=-0.2sin\theta\frac{d\omega}{dt}-0.2\omega cos\theta \frac{d\theta}{dt}$ This simplifies to: $a=-0.2\alpha sin\theta-0.2\omega^2 cos\theta$ We plug in the known values to obtain: $-2=-0.2\alpha sin60-0.2(28.8675)^2cos60$ $\implies \alpha=470~rad/s^2\circlearrowleft$


text:
# User profile URL design

I've noticed that some sites are moving away from root-directory user profiles. For example, YouTube channel URLs are now /channel/<id> and /c/<username> instead of simply /<username>.

For a site with user profiles (or in YouTube's case, channels), which is preferred?

For the most part, Google doesn't really care how you structure your URLs (as long as they're reasonably stable & crawlable; with the exception of country-targeting). Think about what you'd want out of your URL structure instead:

• need to do country-targeting? Use subdomains or high-level folders, e.g., uk.domain.com/... , domain.com/uk/... (this is the main place where Google's guidelines come into play)
• need to track or control access by type of content? Folders are useful for that, e.g., /users/... , /blog/... , /private/... , /shop/shoes/... (site:-queries give you some insight & many tools allow separating out directories, they're also easy to find in log files)
• need to remove content by user quickly? Put their user-id in the path too, e.g., /users/user1234/content123 (the removal tools let you do directory removals)
• need temporary identifiers, like referral-Ids? Put them in the query-string so they can be easily recognized & stripped, e.g., /bluesuedeshoes.htm?ref=123 (using the standard key=value format makes it easy for most tools to recognize).

At any rate, I would not blindly follow some big site's structure, in the hope that they must have planned everything in detail. All sites are different, and you'd be surprised at how often big sites manage to get things wrong :).

Using "detail" identifiers (user-name, channel-name, product-name, etc) at the root level makes recognizing (for serving, tracking, blocking, etc) specific types of content impossible, and can result in clashes between different types of content. For example, is /chicken the user "chicken", the channel "chicken", or the product category & what if multiple of these exist? No need to make running a site harder than it already is.

When having the username at the beginning of the path, you would have to make sure to avoid name clashes with non-user pages (and, for sites that allow custom templates and forms, phishing potentials), and you lose some usability.

Examples:

• Reserved filenames: A user could choose the username robots.txt, favicon.png, .well-known, etc.

• Internal pages: Your contact/about/etc. page vs. a user named contact/about/etc.

• Usability: users can’t no longer be sure what the URL will be about without visiting it:

https://twitter.com/StackOverflow/media is the "media" page of a user named "StackOverflow"
https://twitter.com/hashtag/media is the "media" hashtag, not the "media" page of a user named "hashtag"

• Phishing (if you allow custom templates and forms): when /login is your login page, a user named "signin" could bring people to login on their user page /signin.

When having a path prefix for usernames (e.g., /users/<username>), you don’t have these problems, and you get a nice (because browsable) URL for showing the user list: /users.

Additional benefits of such a path prefix:

• It allows you to easily block the crawling of user profiles in robots.txt:

Disallow: /users

If you don’t have such a "namespace", you would have to list each username.

• Users can use an external search engine to search all user profile pages, e.g. with site:example.com/users.


text:
### Term Structure of Yield Volatility and Interest Rate Risk

Time-horizon is a very important aspect in understanding interest rate risk and the return characteristics of a fixed-rate investment. The primary concern for an investor is the change in the price of a bond given a sudden change in its yield-to-maturity. Reinvestment of coupon interest is a key factor in the investor’s horizon yield.

Prices of fixed income securities are affected not just by the level of interest rates but also the volatility of interest rates. The risk of a default-free security emanates from two sources: interest rate shift and the risk of changes in interest rate volatility.

## Risk of Interest Rate Shift

To manage this type of risk, analysts have to measure it first. In this regard, duration becomes a particularly important tool since it helps the analyst establish the sensitivity of a portfolio to changes in the level of interest rate changes.

## Risk of Changes in Interest Rate Volatility

This type of risk is less obvious but nonetheless represents a major component of the total risk of a fixed income security. For a given duration and current value of a position, interest rate increases as the expected yield volatility increases.

## Investment Horizon and Interest Rate Risk

When the government par-curve is shifted up or down by some amount to calculate effective duration and effective convexity, it is described as “parallel” yield curve shifts. Yield curves are rarely straight lines, so this shift may also be described as “shape-preserving” shift to the yield curve.

Given a parallel shift to the yield curve, the yield-to-maturity and coupon reinvestment rates are assumed to change by the same amount and in the same direction. Macaulay duration is particularly important because it identifies the investment horizon such that losses (gains) from coupon reinvestment offset gains (losses) from changes in market prices.

The term structure of yield volatility is the relationship between the volatility of bond yields-to-maturity and times-to-maturity. The term structure of bond yields (also called the “term structure of interest rates”) is typically upward sloping. However, it can be different depending on some factors, including government policy.

For instance, a central bank engaging in an expansionary monetary policy might cause the yield curve to steepen by reducing short-term interest rates. However, this policy might result in greater volatility in short-term bond yields to maturity than in longer-term bonds. Longer-term bond yields are mainly determined by future inflation and economic growth expectations. Such expectations are often less volatile.

## Question

An investment bank needs to rank 3 bonds in terms of interest rate risk.

$$\begin{array}{l|c|c|c} \textbf{Bond} & \textbf{Modified Duration} & \textbf{Convexity} & \textbf{Yield} \\ \hline \text{A} & \text{3.50} & \text{100} & \text{25} \\ \text{B} & \text{5.25} & \text{350} & \text{15} \\ \text{C} & \text{5.30} & \text{250} & \text{10} \\ \end{array}$$

The modified duration and convexity statistics are annualized. Given the predicted change in yield in each of these bonds, how should the bonds be ranked in terms of interest rate risk (from the highest risk to the lowest)?

A. Bond C has the highest degree of interest rate risk, followed by Bond A, and Bond B

B. Bond B has the highest degree of interest rate risk, followed by Bond C, and Bond A

C. Bond C has the highest degree of interest rate risk, followed by Bond B, and Bond A

Solution

Based on the assumed changes in the yields and given measures of modified duration and convexity, Bond C has the highest degree of interest rate risk followed by Bond B, and then Bond C.

Bond A: %ΔPVFULL ≈ (-3.50×0.0025) + (1/2×100×(0.0025)2) ≈ -0.0084375

Bond B: %ΔPVFULL ≈ (-5.25×0.0015) + (1/2×350×(0.0015)2) ≈ -0.0074813

Bond C: %ΔPVFULL ≈ (-5.30×0.0010) + (1/2×250×(0.0010)2) ≈ -0.005175

Describe how the term structure of yield volatility affects the interest rate risk of a bond

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text:
New Similarity Measures of Simplified Neutrosophic Sets and Their Applications

Chunfang Liu*

Abstract

Abstract: The simplified neutrosophic set (SNS) is a generalization of fuzzy set that is designed for some practical situations in which each element has truth membership function, indeterminacy membership function and falsity membership function. In this paper, we propose a new method to construct similarity measures of single valued neutrosophic sets (SVNSs) and interval valued neutrosophic sets (IVNSs), respectively. Then we prove that the proposed formulas satisfy the axiomatic definition of the similarity measure. At last, we apply them to pattern recognition under the single valued neutrosophic environment and multi-criteria decision-making problems under the interval valued neutrosophic environment. The results show that our methods are effective and reasonable.

Keywords: Multi-Criteria Decision-Making , Pattern Recognition , Similarity Measure , Simplified Neutrosophic Sets

1. Introduction

Zadeh [1] proposed fuzzy set (FS) to solve uncertain information. After that, it has been successfully used in different areas [2-7]. Atanassov [8] extended FS to the intuitionistic fuzzy set (IFS) which is very effective to deal with vagueness of the information. The IFS has the ability to express fuzziness and uncertainty of practical situations which each element concludes membership degree, non-membership degree and hesitation degree and they are presented by exact numbers. The sum of these three functions is number 1. In general, the degrees may be not exact numbers but interval values. Then the theory of IFS was extended to interval-valued IFS [9]. With the development of the theory of FS and its extension, they have been handled many uncertainties in different real-life problems. But, there are many phenomena cannot be dealt by the FS and its extension. For example, when we ask an expert about a complex statement, he or she may not give the exact answer of the problem and say that the possibility that the statement is true is 0.6, that is false is 0.5 and the degree that he or she is not sure is 0.3. This situation cannot be expressed by FS and IFS, and some new set is needed to express the condition. Neutrosophic set (NS) was first proposed by Smarandache [10,11] from philosophical point of view. A NS is a set that each element has truth membership degree, indeterminacy membership degree and falsity membership degree.

As we known, it was difficult to apply the NS in real science and engineering fields. Wang et al. [12,13] proposed single valued neutrosophic set (SVNS) and interval valued neutrosophic set (IVNS). SVNS and IVNS are the subclasses of simplified neutrosophic set which was proposed by Ye [14] and he presented the operators and relationships of the sets and gave their different properties. Therefore, some new results of SNS appeared [15,16]. Similarity measure is used to estimate the degree of similarity of two sets. The similarity measures have been applied to various areas, such as personnel assessment, ecology, medical diagnosis, psychology and clustering analysis, and so on. Broumi and Smarandache [17] defined several similarity measures based on the Hausdorff distance measures of neutrosophic sets. Majumdar and Samanta [18] presented several similarity measures of SVNSs based on distance measures. Ye [14,19-22] also presented the similarity measures of IVNSs and SVNSs and applied them to multi-attribute decision-making problems. Based on tangent function, Prarnaik and Mondal [23] proposed the weighted fuzzy similarity measure and used it to the area of medical diagnosis. On the basis of existing similarity measures, we propose the cotangent similarity measures of SNSs and the weighted cotangent similarity measures of SNSs. Then we apply the proposed similarity measures to pattern recognition in the framework of SVNS and multi-criteria decision-making in the framework of IVNS. The results show that our proposed similarity measure methods are effective and reasonable.

The remainder of the paper is constructed as follows: Section 2 gives some preliminaries. Section 3 gives new cotangent similarity measures of SVNSs. Section 4 gives new cotangent similarity measures of IVNSs. Section 5 gives the applications of the proposed similarity measures in pattern recognition and multi-criteria decision-making. Section 6 concludes the paper.

2. Preliminaries

2.1 Neutrosophic Set (NS)

Suppose X is a universe of discourse, and its generic element is x . A neutrosophic set A in X is expressed by truth-membership function TA(x) , indeterminacy-membership function IA(x) , and falsity-membership function FA(x) , respectively. The functions TA(x), IA(x), FA(x) in X are real standard or nonstandard subsets of ]-0,1+[, i.e.,

[TeX:] $$T _ { A } ( x ) : X \rightarrow ] { ^{-} 0,1^{ + } [ } \\ I _ { A } ( x ) : X \rightarrow ] { ^{-} 0,1 ^{ + }[ } \\ F _ { A } ( x ) : X \rightarrow ] { ^{-} 0,1 ^{ + } [ }$$

where ]-0,1+[ is the nonstandard unit interval, which is an extension of the standard interval [0,1], [TeX:] $$- 0 = 0 - \varepsilon , 1 ^ { + } = 1 + \varepsilon , \varepsilon > 0$$. Then, the sum of [TeX:] $$T _ { A } ( x ) , I _ { A } ( x ) , F _ { A } ( x )$$ has no restriction, that is [TeX:] $$- 0 \leq \sup T _ { A } ( x ) + \sup I _ { A } ( x ) + \sup F _ { A } ( x ) \leq 3 ^ { + }$$.

Since it is difficult to apply NSs to practical problems, Ye [20] reduced NSs of non-standard intervals into standard intervals that still remain the operations of NSs. If the functions [TeX:] $$T _ { A } ( x ) , I _ { A } ( x ) , F _ { A } ( x )$$ are single subintervals/subsets in the real standard [0,1]. Then, a simplification of the NS A is denoted by

[TeX:] $$A = \left\{ < x , T _ { A } ( x ) , I _ { A } ( x ) , F _ { A } ( x ) > | \quad x \in X \right\}$$

It is a subclass of the NS and contains SVNS and IVNS.

For one thing, the membership functions [TeX:] $$T _ { A } ( x ) , I _ { A } ( x ) , F _ { A } ( x )$$ in a SNS A are exact numbers in the real unit interval [0,1], and the sum of the three functions satisfies the inequality [TeX:] $$0 \leq T _ { A } ( x ) + I _ { A } ( x ) + F _ { A } ( x ) \leq 3$$, For every x in X . In this case, the SNS changes to the SVNS. Let A and B be two SVNSs, A is contained in B and denoted by A ⊆ B, if and only if

[TeX:] $$T _ { A } ( x ) \leq T _ { B } ( x ) , I _ { A } ( x ) \geq I _ { B } ( x ) , F _ { A } ( x ) \geq F _ { B } ( x )$$

for every x in X .

For another, the membership functions [TeX:] $$T _ { A } ( x ) , I _ { A } ( x ) , F _ { A } ( x )$$ in SNS A are the subunit intervals of the real unit interval [0,1]. We have that

[TeX:] $$T _ { A } ( x ) = \left[ \inf T _ { A } ( x ) , \sup T _ { A } ( x ) \right] \\ I _ { A } ( x ) = \left[ \inf I _ { A } ( x ) , \sup I _ { A } ( x ) \right] \\ F _ { A } ( x ) = \left[ \inf F _ { A } ( x ) , \sup F _ { A } ( x ) \right]$$

for any point x in X. In this case, the SNS reduces to the IVNS.

For two IVNSs A and B, A is contained in B, that is A ⊆ B, if and only if

[TeX:] $$\inf T _ { A } ( x ) \leq \inf T _ { B } ( x ) \sup T _ { A } ( x ) \leq \sup T _ { B } ( x ), \\ \inf I _ { A } ( x ) \geq \inf I _ { B } ( x ) , \sup I _ { A } ( x ) \geq \sup I _ { B } ( x ), \\ \inf F _ { A } ( x ) \geq \inf F _ { B } ( x ) , \sup F _ { A } ( x ) \geq \sup F _ { B } ( x )$$

for every x in X.

2.2 Similarity Measure of SNSs

Similarity measure is supposed to depict the similarity of a set of alternatives. We use fuzzy evaluation theory to determine the evaluating criteria and get the similarity measure of each alternative and obtain the best alternative. Let A, B, C be three SNSs, a real valued function S: SNS(X)XSNS(X)→[0,1], if it satisfies the following axiomatic conditions:

[TeX:] $$( \mathrm { S } 1 ) 0 \leq S ( A , B ) \leq 1;$$
[TeX:] $$( \mathrm { S } 2 ) S ( A , B ) = 1 \text { if and only if } A = B;$$
[TeX:] $$( \mathrm { S } 3 ) S ( A , B ) = S ( B , A );$$
[TeX:] $$( S 4 ) A \subseteq B \subseteq C , \text { then } S ( A , C ) \leq S ( A , B ) \text { and } S ( A , C ) \leq S ( B , C ).$$

Then we call S a similarity measure of SNS(X).

3. Cotangent Similarity Measure of SVNSs

The cosine similarity measure [14] is based on the inner product of two vectors divided by the product of their models. The similarity measure from [22] is based on the tangent function. We first give these two formulas, and give examples to show these results are unreasonable.

(1)
[TeX:] $$C ( A , B ) = \frac { 1 } { n } \sum _ { j = 1 } ^ { n } \frac { T _ { A } \left( x _ { j } \right) T _ { B } \left( x _ { j } \right) + I _ { A } \left( x _ { j } \right) I _ { B } \left( x _ { j } \right) + F _ { A } \left( x _ { j } \right) F _ { B } \left( x _ { j } \right) } { \sqrt { T _ { A } \left( x _ { j } \right) ^ { 2 } + I _ { A } \left( x _ { j } \right) ^ { 2 } + F _ { A } \left( x _ { j } \right) ^ { 2 } } \sqrt { T _ { B } \left( x _ { j } \right) ^ { 2 } + I _ { B } \left( x _ { j } \right) ^ { 2 } + F _ { B } \left( x _ { j } \right) ^ { 2 } } }$$

(2)
[TeX:] $$T ( A , B ) = 1 - \frac { 1 } { n } \sum _ { j = 1 } ^ { n } \tan \left[ \frac { \pi } { 4 } \max \left\{ \left| T _ { A } \left( x _ { j } \right) - T _ { B } \left( x _ { j } \right) \right| , \left| I _ { A } \left( x _ { j } \right) - I _ { B } \left( x _ { j } \right) \right| , \left| F _ { A } \left( x _ { j } \right) - F _ { B } \left( x _ { j } \right) \right| \right\} \right]$$

Example 1. Let [TeX:] $$A _ { 1 } = \langle 0.4,0.2,0.6 \rangle , B _ { 1 } = \langle 0.2,0.1,0.3 \rangle$$ be two SVNSs, we use the Eq. (1) and (2) to calculate similarity measure of A1 and B1.

By computing, we get [TeX:] $$C \left( A _ { 1 } , B _ { 1 } \right) = 1 , T \left( A _ { 1 } , B _ { 1 } \right) = 0.6897$$, the result is quite different from each other. We find the membership functions of the two sets exist some differences, but the result shows that their similarity measure is 1. If the similarity measure is 1, it shows that the sets are almost the same sets, which contradict with our intuition.

Example 2. Let [TeX:] $$A _ { 2 } = \langle 0.3,0.2,0.4 \rangle , B _ { 2 } = \langle 0.4,0.2,0.3 \rangle$$ be two SVNSs, we use the Eq. (1) and (2) to calculate similarity measure of A2 and B2.

By computing, we get [TeX:] $$C \left( A _ { 2 } , B _ { 2 } \right) = 0.9945 , T \left( A _ { 2 } , B _ { 2 } \right) = 0.9958$$, the result is almost the same with each other. We found the membership functions of the two sets exist some differences, but the result shows that their similarity measure is very similar, it shows that the sets are almost the same sets, which contradict with our intuition.

From the above examples, we see that the existing similarity measure of SVNSs sometimes will not consistent with our idea. Motivated by the existing similarity measures, we propose a cotangent similarity measure as follows.

Let

[TeX:] $$A = \left\{ \left\langle x _ { j } ; T _ { A } \left( x _ { j } \right) , I _ { A } \left( x _ { j } \right) , F _ { A } \left( x _ { j } \right) \right\rangle | x _ { j } \in X \right\} \\ B = \left\{ \left\langle x _ { j } ; T _ { B } \left( x _ { j } \right) , I _ { B } \left( x _ { j } \right) , F _ { B } \left( x _ { j } \right) \right\rangle | x _ { j } \in X \right\}$$

be two SVNSs in [TeX:] $$X = \left\{ x _ { 1 } , x _ { 2 } , \cdots x _ { n } \right\}$$. We propose the cotangent similarity measure as follows:

(3)
[TeX:] $$S ( A , B ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \cot \frac { \pi } { 4 } \left[ 1 + \left( \begin{array} { l } { \left| T _ { A } \left( x _ { j } \right) - T _ { B } \left( x _ { j } \right) \right| Ｖ } \\ { \left| I _ { A } \left( x _ { j } \right) - I _ { B } \left( x _ { j } \right) \right| Ｖ \left| F _ { A } \left( x _ { j } \right) - F _ { B } \left( x _ { j } \right) \right| } \end{array} \right) \right]$$

where the symbol “Ｖ” is the maximum operation. Then, we can easily prove Eq. (3) satisfies definition of similarity measure. (S1), (S2), (S3) are easily get. We just prove (S4).

If A ⊆ B ⊆ C , then

[TeX:] $$T _ { A } \left( x _ { j } \right) \leq T _ { B } \left( x _ { j } \right) \leq T _ { C } \left( x _ { j } \right) \\ I _ { A } \left( x _ { j } \right) \geq I _ { B } \left( x _ { j } \right) \geq I _ { C } \left( x _ { j } \right) \\ F _ { A } \left( x _ { j } \right) \geq F _ { B } \left( x _ { j } \right) \geq F _ { C } \left( x _ { j } \right)$$

then we have the following inequalities:

[TeX:] $$\left| T _ { A } \left( x _ { j } \right) - T _ { B } \left( x _ { j } \right) \right| \leq \left| T _ { A } \left( x _ { j } \right) - T _ { c } \left( x _ { j } \right) \right|, \left| T _ { B } \left( x _ { j } \right) - T _ { C } \left( x _ { j } \right) \right| \leq \left| T _ { A } \left( x _ { j } \right) - T _ { C } \left( x _ { j } \right) \right| \\ \left| I _ { A } \left( x _ { j } \right) - I _ { B } \left( x _ { j } \right) \right| \leq \left| I _ { A } \left( x _ { j } \right) - I _ { c } \left( x _ { j } \right) \right|, \left| I _ { B } \left( x _ { j } \right) - I _ { C } \left( x _ { j } \right) \right| \leq \left| I _ { A } \left( x _ { j } \right) - I _ { c } \left( x _ { j } \right) \right| \\ \left| F _ { A } \left( x _ { j } \right) - F _ { B } \left( x _ { j } \right) \right| \leq \left| F _ { A } \left( x _ { j } \right) - F _ { C } \left( x _ { j } \right) \right|, \left| F _ { B } \left( x _ { j } \right) - F _ { C } \left( x _ { j } \right) \right| \leq \left| F _ { A } \left( x _ { j } \right) - F _ { C } \left( x _ { j } \right) \right|$$

Combining these inequalities, since the cotangent function is a decreasing function within the interval [0, π/2]. Hence, S(A,C)≤S(A,B) and S(A,C)≤S(B,C). Therefore we complete the proof.

Usually, the differences of elements’ importance are considered. Thus, we need to take the weight of each element xi(i=1,2,...,n) into account.

In the following, we develop the weighted similarity measure between SNSs. Let [TeX:] $$\omega _ { i } , ( i = 1,2 , \cdots , n )$$be the weight of each element [TeX:] $$x _ { i } ( i = 1,2 , \cdots , n ) , \omega _ { i } \in [ 0,1 ] , \sum _ { i = 1 } ^ { n } \omega _ { i } = 1$$.Then, the weighted similarity measure is obtained as follows:

(4)
[TeX:] $$W S ( A , B ) = \sum _ { i = 1 } ^ { n } w _ { j } \cot \frac { \pi } { 4 } \left[ 1 + \left( \begin{array} { l } { \left| T _ { A } \left( x _ { j } \right) - T _ { B } \left( x _ { j } \right) \right| \vee } \\ { \left| I _ { A } \left( x _ { j } \right) - I _ { B } \left( x _ { j } \right) \right| \vee \left| F _ { A } \left( x _ { j } \right) - F _ { B } \left( x _ { j } \right) \right| } \end{array} \right]\right.$$

Especially if [TeX:] $$\omega _ { j } = \frac { 1 } { n } , j = 1,2 , \cdots , n$$.Eq. (4) reduces to (3).

Now we use the proposed Eq. (3) to calculate the similarity measure of Example 1, then we get S(A1, B1) = 0.6128.

Similarly, we use the proposed Eq. (3) to calculate the similarity measure of Example 2, then we get S(A2, B2) = 0.8540 .

From the two examples, we found that the proposed formula is coincided with our intuition and effective to calculate the similarity measure of SVNSs.

4. Cotangent Similarity Measure of IVNSs

With the same method, we present the formula of similarity measures of IVNSs.

Let

[TeX:] $$A = \left\{ \left\langle x _ { j } ; T _ { A } \left( x _ { j } \right) , I _ { A } \left( x _ { j } \right) , F _ { A } \left( x _ { j } \right) \right\rangle | x _ { j } \in X \right\} \\ B = \left\{ \left\langle x _ { j } ; T _ { B } \left( x _ { j } \right) , I _ { B } \left( x _ { j } \right) , F _ { B } \left( x _ { j } \right) \right\rangle | x _ { j } \in X \right\}$$

be two IVNSs in X = {x1, x2,...,xn}. Then we propose the new similarity measure of IVNSs.

[TeX:] $$I S ( A , B ) = \frac{1}{n} \sum _ { j = 1 } ^ { n } \cot \frac { \pi } { 4 } \left[ 1 + \left( \begin{array} { c } { | \inf T _ { A } \left( x _ { j } \right) - \inf T _ { B } \left( x _ { j } \right) | \vee | \inf I _ { A } \left( x _ { j } \right) - \inf I _ { B } \left( x _ { j } \right) | } \\ { \vee \left| \inf F _ { A } \left( x _ { j } \right) - \inf F _ { B } \left( x _ { j } \right) \right| \vee \left| \sup T _ { A } \left( x _ { j } \right) - \sup T _ { B } \left( x _ { j } \right) \right| } \\ { \vee \left| \sup I _ { A } \left( x _ { j } \right) - \sup I _ { B } \left( x _ { j } \right) \right| \vee \left| \sup F _ { A } \left( x _ { j } \right) - \sup F _ { B } \left( x _ { j } \right) \right| } \end{array} \right]\right.$$

where the symbol "Ｖ" is the maximum operator. Then, we can easily prove Eq. (5) satisfies the definition of the similarity measure. Here we omitted.

Let [TeX:] $$\omega _ { i } , ( i = 1,2 , \cdots , n )$$ be the weight of each element [TeX:] $$x _ { i } ( i = 1,2 , \cdots , n ) , \omega _ { i } \in [ 0,1 ] , \sum _ { i = 1 } ^ { n } \omega _ { i } = 1$$. Then, we have the following weighted similarity measure.

(6)
[TeX:] $$W I S ( A , B ) = \sum _ { j = 1 } ^ { n } \omega _ { j } \cot \frac { \pi } { 4 } \left[ 1 + \left( \begin{array} { c } { | \inf T _ { A } \left( x _ { j } \right) - \inf T _ { B } \left( x _ { j } \right) | \vee | \inf I _ { A } \left( x _ { j } \right) - \inf I _ { B } \left( x _ { j } \right) | } \\ { \vee \left| \inf F _ { A } \left( x _ { j } \right) - \inf F _ { B } \left( x _ { j } \right) \right| \vee \left| \sup T _ { A } \left( x _ { j } \right) - \sup T _ { B } \left( x _ { j } \right) \right| } \\ { \vee \left| \sup I _ { A } \left( x _ { j } \right) - \sup I _ { B } \left( x _ { j } \right) \right| \vee \left| \sup F _ { A } \left( x _ { j } \right) - \sup F _ { B } \left( x _ { j } \right) \right| } \end{array} \right]\right.$$

5. Applications

SNS is a very suitable tool to process the incomplete, uncertainty and inconsistent information. Similarity measure is an important mathematical tool to deal with pattern recognition, medical diagnosis, clustering analysis and decision-making. In the following part, we will use the proposed similarity measures to pattern recognition under the SVN environment and multi-attribute decisionmaking under the IVN environment.

5.1 Application to Pattern Recognition under the SVN Environment

Here is a pattern recognition problem. Suppose that there are m patterns and they are expressed by SVNSs. Suppose [TeX:] $$A _ { i } = \left\{ \left\langle x _ { j } ; T _ { A } \left( x _ { j } \right) , I _ { A } \left( x _ { j } \right) , F _ { A } \left( x _ { j } \right) \right\rangle \right\} , ( i = 1,2 , \cdots , m )$$ are m patterns in the feature space [TeX:] $$X = \left\{ x _ { 1 } , x _ { 2 } , \cdots , x _ { n } \right\}$$. Let [TeX:] $$B = \left\{ \left\langle x _ { j } ; T _ { B } \left( x _ { j } \right) , I _ { B } \left( x _ { j } \right) , F _ { B } \left( x _ { j } \right) \right\rangle \right\}$$ be a sample needed to be recognized. Our aim is to classify the pattern B to one of the patterns A1,A2,...,Am Aaccording to the principle of the maximum similarity measures. The bigger the similarity measure of Ai and B is, the more similar Ai and B is. Now, we give the steps for the pattern recognitions. First, calculate the similarity measure of Ai and B,that is S(Ai,B),i=1,2,...,m. Second, choose the largest one S(Ak,B) from S(Ai,B),i=1,2,...,m.Third, conclude that the sample B belongs to the pattern Ak.

In the following, a pattern recognition problem about the classification of building material [24] is used to illustrate effective of the proposed similarity measures. Suppose that there are five classes of building materials which are represented by SVNSs Ai,i = 1,2,...,5, the feature space is [TeX:] $$X = \left\{ x _ { 1 } , x _ { 2 } , \cdots , x _ { 5 } \right\}$$, and B is an unknown material. They are listed as follows:

[TeX:] $$\begin{array} { r } { A _ { 1 } = \left\{ \left\langle x _ { 1 } , 0.4,0.6,0.0 \right\rangle , \left\langle x _ { 2 } , 0.3,0.2,0.5 \right\rangle , \left\langle x _ { 3 } , 0.1,0.3,0.7 \right\rangle \right. } \\ { \left\langle x _ { 4 } , 0.4,0.3,0.3 \right\rangle , \left\langle x _ { 5 } , 0.1,0.2,0.7 \right\rangle \} } \end{array}$$
[TeX:] $$\begin{array} { r } { A _ { 2 } = \left\{ \left\langle x _ { 1 } , 0.7,0.3,0.0 \right\rangle , \left\langle x _ { 2 } , 0.2,0.2,0.6 \right\rangle , \left\langle x _ { 3 } , 0.0 .0 .1,0.9 \right\rangle \right. } \\ { \left\langle x _ { 4 } , 0.7,0.3,0.0 \right\rangle , \left\langle x _ { 5 } , 0.1,0.1,0.8 \right\rangle \} } \end{array}$$
[TeX:] $$\begin{array} { r } { A _ { 3 } = \left\{ \left\langle x _ { 1 } , 0.3,0.4,0.3 \right\rangle , \left\langle x _ { 2 } , 0.6,0.3,0.1 \right\rangle , \left\langle x _ { 3 } , 0.2,0.1,0.7 \right\rangle \right. } \\ { \left\langle x _ { 4 } , 0.2,0.2,0.6 \right\rangle , \left\langle x _ { 5 } , 0.1,0,0.9 \right\rangle \} } \end{array}$$
[TeX:] $$\begin{array} { r } { A _ { 4 } = \left\{ \left\langle x _ { 1 } , 0.1,0.2,0.7 \right\rangle , \left\langle x _ { 2 } , 0.4,0.4,0.4 \right\rangle , \left\langle x _ { 3 } , 0.8,0.2,0 \right\rangle \right. } \\ { \left\langle x _ { 4 } , 0.2,0.2,0.7 \right\rangle , \left\langle x _ { 5 } , 0.2,0.1,0.7 \right\rangle \} } \end{array}$$
[TeX:] $$\begin{array} { c } { A _ { 5 } = \left\{ \left\langle x _ { 1 } , 0.1,0.1,0.8 \right\rangle , \left\langle x _ { 2 } , 0,0.2,0.8 \right\rangle , \left\langle x _ { 3 } , 0.2,0,0.8 \right\rangle \right. } \\ { \left\langle x _ { 4 } , 0.2,0,0.8 \right\rangle , \left\langle x _ { 5 } , 0.8,0.1,0.1 \right\rangle \} } \end{array}$$
[TeX:] $$\begin{array} { r l } { B = \left\{ \left\langle x _ { 1 } , 0.8,0.2,0.1 \right\rangle , \left\langle x _ { 2 } , 0.6,0.3,0.1 \right\rangle , \left\langle x _ { 3 } , 0.2,0.1,0.8 \right\rangle \right. } \\ { \left\langle x _ { 4 } , 0.6,0.5,0.1 \right\rangle , \left\langle x _ { 3 } , 0.1,0.4,0.6 \right\rangle \} } \end{array}$$

We use Eq. (3) to calculate the similarity measure of Ai and B, i = 1,2,...,5

[TeX:] \begin{aligned} S \left( A _ { 1 } , B \right) = 0.6397 , S \left( A _ { 2 } , B \right) = 0.7150 , S \left( A _ { 3 } , B \right) = 0.6383 \\ S \left( A _ { 4 } , B \right) = 0.3211 , S \left( A _ { 5 } , B \right) = 0.3628 \end{aligned}

Since S(A2,B) is the biggest, we conclude that the unknown material B belongs to A2.

In this section, we use the similarity measure to the application of pattern recognition, by the similarity measure method, we classify the unknown material B to A2. Now we use the Method of reference [14] and [22] to show that our result is more feasible.

Method of reference [14]: If we use Eq. (1) to calculate the similarity measure, we get

[TeX:] $$\begin{array} { r l } { S \left( A _ { 1 } , B \right) } { = 0.8692 , S \left( A _ { 2 } , B \right) = 0.8661 , S \left( A _ { 3 } , B \right) = 0.8206 } \\ { } { S \left( A _ { 4 } , B \right) = 0.5621 , S \left( A _ { 5 } , B \right) = 0.4244 } \end{array}$$

Since S(A1,B) is the biggest, we conclude that the unknown material B belongs to A2.

Method of reference [22]: If we use Eq. (2) to calculate the similarity measure, we get

[TeX:] $$\begin{array} { r l } { T \left( A _ { 1 } , B \right) = } { 0.9863 , T \left( A _ { 2 } , B \right) = 0.9961 , T \left( A _ { 3 } , B \right) = 0.9860 } \\ { } { T \left( A _ { 4 } , B \right) = 0.9926 , T \left( A _ { 5 } , B \right) = 0.9920 } \end{array}$$

Since S(A2,B) is the biggest, we conclude that the unknown material B belongs to A2.

Comparison and analysis: The result of method of reference [14] is different from ours, In Section 3, Example 1 showed its drawbacks. The result of the method of reference [22] is the same as ours. But the calculation of T(Ai,B) is very close to each other which showed that the unknown material B is almost close to all Ai. The result of our given formula shows that our method is effective and reasonable.

5.2 Application to Multi-Criteria Decision-Making under IVN Environment

In this section, we apply the similarity measures to multi-criteria decision-making problems in IVNSs.

Let [TeX:] $$X = \left\{ a _ { 1 } , a _ { 2 } , \cdots , a _ { n } \right\}$$ be a set of alternatives and [TeX:] $$C = \left\{ c _ { 1 } , c _ { 2 } , \cdots , c _ { m } \right\}$$ be a set of criteria. Assume that the weight of the criterion [TeX:] $$c _ { j } \text { is } w _ { j } , w _ { j } \geq 0 , \sum _ { j = 1 } ^ { m } w _ { i } = 1$$ The characteristic of the alternative ai is represented by IVNS.

[TeX:] $$a _ { i } = \left\{ \left\langle c _ { j } , T _ { a _ { i } } \left( c _ { j } \right) , I _ { a _ { i } } \left( c _ { j } \right) , F _ { a _ { i } } \left( c _ { j } \right) \right\rangle | c _ { j } \in C \right\} \\ T _ { a _ { i } } \left( c _ { j } \right) = \left[ \inf T _ { a _ { i } } \left( c _ { j } \right) , \sup T _ { a _ { i } } \left( c _ { j } \right) \right] \\ I _ { a _ { i } } \left( c _ { j } \right) = \left[ \inf I _ { a _ { i } } \left( c _ { j } \right) , \sup I _ { a _ { i } } \left( c _ { j } \right) \right] \\ F _ { a _ { i } } \left( c _ { j } \right) = \left[ \inf F _ { a _ { i } } \left( c _ { j } \right) , \sup F _ { a _ { i } } \left( c _ { j } \right) \right]$$

If there is only one element in the IVNS, for the sake of simplicity, the interval neutrosophic set is denoted by the interval valued neutrosophic value (IVNV), we denote it as:

[TeX:] $$\alpha _ { i j } = \left\langle \left[ a _ { i j } , b _ { i j } \right] , \left[ c _ { i j } , d _ { i j } \right] , \left[ e _ { i j } , f _ { i j } \right] \right\rangle$$

In multi-criteria decision-making problems, we suppose the ideal point is existed and use it to help identifying the best alternative in the decision set. Then we construct the ideal point to evaluate the alternatives. Generally speaking, the evaluation criteria can be classified by benefit criteria and cost criteria. If the criterion belongs to benefit criterion, we set the ideal point as below:

[TeX:] $$a ^ { * } = \left\langle \left[ \max _ { i } \left( a _ { i j } \right) , \max _ { i } \left( b _ { i j } \right) \right] \left[ \min _ { i } \left( c _ { i j } \right) , \min _ { i } \left( d _ { i j } \right) \right] , \left[ \min _ { i } \left( e _ { i j } \right) , \min _ { i } \left( f _ { i j } \right) \right] \right\rangle$$

If the criterion belongs to cost criterion, we set the ideal point as below:

[TeX:] $$a ^ { * } = \left\langle \left[ \min _ { i } \left( a _ { i j } \right) , \min _ { i } \left( b _ { i j } \right) \right] \left[ \max _ { i } \left( c _ { i j } \right) , \max _ { i } \left( d _ { i j } \right) \right] \cdot \left[ \max _ { i } \left( e _ { i j } \right) , \max _ { i } \left( f _ { i j } \right) \right] \right\rangle$$

We calculate the similarity measure S(ai,a*) by Eq. (5) and (6), the ranking order of the alternatives can be determined and the best one can be chosen as well.

Here an example adapted from [19] is utilized to illustrate the applicability and validity of the proposed similarity measure. There is a financial company that wants to invest four companies and they are considered as the four potential alternatives: (1) car company a1; (2) food company a2; (3) computer company a3; (4) arms company a4. The invest company must make a decision making considering the following three criteria: c1(risk analysis); c2(growth analysis); c3(environmental impact analysis). Among the three criteria, both c1 and c2 are benefit criteria, and c3 is a cost criterion. Assume that the criteria weight vector is [TeX:] $$w = [ 0.35,0.25,0.40 ] ^ { T }$$ The four possible alternatives are to be evaluated under these three criteria and are in the form of IVNSs, as shown in the following neutrosophic decision matrix D.

[TeX:] $$D = \left( \begin{array}{c} \langle [ 0.4,0.5 ] , [ 0.2,0.3 ] , [ 0.3,0.4 ] \rangle \langle [ 0.4,0.6 ] , [ 0.1,0.3 ] , [ 0.2,0.4 ] \rangle \langle [ 0.7,0.9 ] , [ 0.2,0.3 ] , [ 0.4,0.5 ] \rangle \\ \langle [ 0.6,0.7 ] , [ 0.1,0.2 ] , [ 0.2,0.3 ] \rangle \langle [ 0.6,0.7 ] , [ 0.1,0.2 ] , [ 0.2,0.3 ] \rangle \langle [ 0.3,0.6 ] , [ 0.3,0.5 ] , [ 0.8,0.9 ] \rangle\\ \langle [ 0.3,0.6 ] , [ 0.2,0.3 ] , [ 0.3,0.4 ] \rangle \langle [ 0.5,0.6 ] , [ 0.2,0.3 ] , [ 0.3,0.4 ] \rangle \langle [ 0.4,0.5 ] , [ 0.2,0.4 ] , [ 0.7,0.9 ] \rangle\\ \langle [ 0.7,0.8 ] , [ 0,0.1 ] , [ 0.1,0.2 ] \rangle \langle [ 0.6,0.7 ] , [ 0.1,0.2 ] , [ 0.1,0.3 ] \rangle \langle [ 0.6,0.7 ] , [ 0.3,0.4 ] , [ 0.8,0.9 ]\rangle \end{array}\right)$$

From the interval valued neutrosophic matrix, we get the ideal alternative as below:

[TeX:] $$a ^ { * } = \{ \langle [ 0.7,0.8 ] , [ 0,0.1 ] , [ 0.1,0.2 ] \rangle, \langle [ 0.6,0.7 ] , [ 0.1,0.2 ] , [ 0.1,0.3 ] \rangle,\\ \langle [ 0.3,0.5 ] , [ 0.3,0.5 ] , [ 0.8,0.9 ] \rangle \}$$

According to Eq. (6), we get [TeX:] $$S \left( a _ { 1 } , a ^ { * } \right) = 0.6000 , S \left( a _ { 2 } , a ^ { * } \right) = 0.8906 , S \left( a _ { 3 } , a ^ { * } \right) = 0.7016 , S \left( a _ { 4 } , a ^ { * } \right) = 0.8451$$.

Then we rank the alternatives as[TeX:] $$a _ { 2 } > a _ { 4 } > a _ { 3 } > a _ { 1 }$$. Obviously, a2 is the best alternative.

Comparison and analysis: In this section, we have proposed a method to solve the MADM problem expressed with IVNSs. From the above example and comparison with the method in [18], we see that their result is the same as ours. In [19], they use two formulas to calculate the similarity measure which is given by the Hamming and Euclidean distance measures; while our method is based on the cotangent functions which satisfy the definition of similarity measure. We use cotangent functions to calculate the similarity measure and the result shows that our method is effective and reasonable.

6. Conclusion

In this paper, we commented on the similarity measure of SVNSs and IVNSs. We gave the new cotangent functions to calculate the similarity measures. They all satisfied the definition of similarity measures. Then the weighted cotangent similarity measures were introduced by considering the importance of each element. At last, we applied the similarity measure to pattern recognition under the single valued neutrosophic environment and multi-criteria decision-making under the interval valued neutrosophic environment. The results show that our methods are effective and reasonable.

Acknowledgement

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2572017CB29) and Harbin Science Technology Innovation Talent Research Fund (No. 2016RQQXJ230).

Biography

Chunfang Liu
https://orcid.org/0000-0003-0396-4115

She received M.S. degree in School of Harbin Institute of Technology, China, in 2003. She is studying for her doctorate in Harbin Engineering University. She is currently working at College of Science, Northeast Forestry University. Her research interests includes system engineering, intelligent control and fuzzy systems.

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