text: For … /Contents 53 0 R << Bellman’s equation can be solved by the average-cost exact LP (ELP): 0 (2) 0 @ 9 7 6 Note that the constraints 0 @ 937 6 7can be replaced by 9 7 Y therefore we can think of problem (2) as an LP. /ProcSet [ /PDF /Text ] M�A��N��y��~��n�n� �@h1~t\b�Og�&�ײ)r�{��gR�7$�?��S[e��)�y���n�t���@ �^hB�Z�˦4g��R)��/^ ;������a�Zp6�U�S)i��rU����YR������)�j|�~/Si���1 /T1_2 48 0 R >> Approximate Dynamic Programming full free pdf books Ana Muriel helped me to better understand the connections between my re-search and applications in operations research. /Font << /F35 10 0 R /F15 11 0 R >> ��&V�����2��+p1js��J_��K;��*�qY �y�=4��\Ky�d�Ww H��U�����绡�ǡħ��M�PNQ:*'���C{���:�� a�|�� ��XC�Y����D�0�*sMBP�J��Ib���sJ�Д��,C�k��r?��ÐĐ���VZ�w�L���>�OA�lX�h�|_�ްe�Gd@�5���UK��ʵ���1. 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With an aim of computing a weight vector f E ~K such that If>f is a close approximation to J*, one might pose the following optimization problem: max c'lf>r (2) 6], [3]. /ProcSet [ /PDF /Text /ImageB ] /Filter /FlateDecode /T1_1 36 0 R /XObject << 3 0 obj /Parent 6 0 R We study the case >> 7 0 obj /Resources << Lim-ited understanding also affects the linear programming approach;inparticular,althoughthealgorithmwasintro-duced by Schweitzer and Seidmann more than 15 years ago, there has been virtually no theory explaining its behavior. /T1_0 22 0 R /Parent 1 0 R endobj >> /MediaBox [ 0 0 612 792 ] /Parent 6 0 R of approximate dynamic programming in industry. APPROXIMATE DYNAMIC PROGRAMMING Jennie Si Andy Barto Warren Powell Donald Wunsch IEEE Press John Wiley & sons, Inc. 2004 ISBN 0-471-66054-X-----Chapter 4: Guidance in the Use of Adaptive Critics for Control (pp. 3 0 obj << 8 0 obj << This is the approach broadly taken by Approximate dynamic programming: solving the curses of dimensionality, published by John Wiley and Sons, is the first book to merge dynamic programming and math programming using the language of approximate dynamic programming. /Font << Thus, a decision made at a single state can provide us with information about Praise for the First Edition"Finally, a book devoted to dynamic programming and written using the language of operations research (OR)! 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Given pre-selected basis functions (Pl, .. . and dynamic programming methods using function approximators. << /Title (Approximate Dynamic Programming via Linear Programming) OPTIMIZATION-BASED APPROXIMATE DYNAMIC PROGRAMMING A Dissertation Presented by MAREK PETRIK Submitted to the Graduate School of the University of Massachusetts Amherst in partial ful llment of the requirements for the degree of DOCTOR OF PHILOSOPHY September 2010 Department of Computer Science. << /Type /Catalog >> Praise for the First Edition"Finally, a book devoted to dynamic programming and written using the language of operations research (OR)! Mathematics of Operations Research Published online in Articles in Advance 13 Nov 2017 << 11 0 obj >> Let us now introduce the linear programming approach to approximate dynamic programming. Topaloglu and Powell: Approximate Dynamic Programming INFORMS|New Orleans 2005, °c 2005 INFORMS 3 A= Attribute space of the resources.We usually use a to denote a generic element of the attribute space and refer to a as an attribute vector. << /Im0 40 0 R Approximate Dynamic Programming for Two-Player Zero-Sum Markov Games 1.1. In addition to >> /C0_0 50 0 R Approximate Dynamic Programming. With the growing levels of sophistication in modern-day operations, it is vital for practitioners to understand how to approach, model, and solve complex industrial problems. 8 0 obj Dynamic Programming techniques for MDP ADP for MDPs has been the topic of many studies these last two decades. endobj Approximate Dynamic Programming Dimitri P. Bertsekas Laboratory for Information and Decision Systems Massachusetts Institute of Technology Lucca, Italy June 2017 Bertsekas (M.I.T.) When asking questions, it is desirable to ask as few questions as possible or given a budget of questions asking the most interesting ones. While this sampling method gives desirable statistical properties, trees grow exponentially in the number of time peri-ods, require a model for generation and often sparsely sample the outcome space. Approximate Dynamic Programming With Correlated Bayesian Beliefs Ilya O. Ryzhov and Warren B. Powell Abstract—In approximate dynamic programming, we can represent our uncertainty about the value function using a Bayesian model with correlated beliefs. − This has been a research area of great inter-est for the last 20 years known under various names (e.g., reinforcement learning, neuro-dynamic programming) − Emerged through an enormously fruitfulcross- /XObject << Approximate Dynamic Programming 1 / 24 These algorithms formulate Tetris as a Markov decision process (MDP) in which the state is deﬁned by the current board conﬁguration plus the falling piece, the actions are the /Description (Paper accepted and presented at the Neural Information Processing Systems Conference \050http\072\057\057nips\056cc\057\051) >> Risk-Averse Approximate Dynamic Programming with Quantile-Based Risk Measures Daniel R. Jiang, Warren B. Powell To cite this article: Daniel R. Jiang, Warren B. Powell (2017) Risk-Averse Approximate Dynamic Programming with Quantile-Based Risk Measures. endobj stream /Type /Page Approximate the Policy Alone. Approximate Dynamic Programming for the Merchant Operations of Commodity and Energy Conversion Assets. 1 0 obj /Language (en\055US) This is the approach broadly taken by methods like Policy Search by Dynamic Programming 2 and Conservative Policy 2 J. Praise for the First Edition Finally, a book devoted to dynamic programming and written using the language of operations research (OR)! Approximate dynamic programming (ADP) is an umbrella term for algorithms designed to produce good approximation to this function, yielding a natural ‘greedy’ control policy. /Parent 1 0 R 5 0 obj Approximate Value and Policy Iteration in DP 2 BELLMAN AND THE DUAL CURSES •Dynamic Programming (DP) is very broadly applicable, but it suffers from: –Curse of dimensionality –Curse of modeling •We address “complexity” by using low- dimensional parametric approximations /ProcSet [ /PDF /Text /ImageB ] /T1_1 16 0 R We cover a ﬁnal approach that eschews the bootstrapping inherent in dynamic programming and instead caches policies and evaluates with rollouts. /Length 2655 Get any books you like and read everywhere you want. We cover a ﬁnal approach that eschews the bootstrapping inherent in dynamic programming and instead caches policies and evaluates with rollouts. That is, it … /Font << /Contents 3 0 R xڭYK�����S��^�aI�e��� l�m`Il�msG���4=�_������V;�\,�H����������.-�yQfwOwU��T��j�Yo���W�ޯ�4�&���4|��o3��w��y�����]�Y�6�H6w�. Approximate Dynamic Programming for the Merchant Operations of Commodity and Energy Conversion Assets. /T1_2 56 0 R 1 0 obj << /Type (Conference Proceedings) 2 0 obj << %PDF-1.4 /Count 7 %PDF-1.3 /Length 788 >> /Im0 12 0 R << Reinforcement learning and approximate dynamic programming for feedback control / edited by Frank L. Lewis, Derong Liu. /Contents 11 0 R /MediaBox [0 0 612 792] /Type /Page /C0_0 58 0 R /Font << /XObject << Approximate Dynamic Programming (ADP) is a modeling framework, based on an MDP model, that o ers several strategies for tackling the curses of dimensionality in large, multi- period, stochastic optimization problems (Powell, 2011). /Type /Page /lastpage (695) 97 - 124) George G. 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# Tag Info

4

Assume that your stationary time series (here a daily close-to-close log-returns' series) is modelled as follows $\forall t \in \mathcal{T}=\{1,...,N\}$ \begin{align} r_t &= E_{t-1}[r_t] + \epsilon_t \\ &= E_{t-1}[r_t] + \sigma_t z_t \end{align} with $z_t \sim N(0,1)$ and $\{z_t\}_{t \in \mathcal{T}}$ are IID. The above equations suggest that, ...

2

If your question is: "Given all the information available up to time $t$, if I compute the 1 period ahead forecast $r_{t+1}$, is the conditional volatility over $[t,t+1[$ given by $\sqrt{r_{t+1}}$?", the answer is NO. To compute the 1 period ahead conditional variance, you should use your model equations (see this post which might help you better understand ...

2

These are 2 completely different ways of estimating volatility. GARCH models are calibrated on historical time series i.e. information provided under the real-world measure $\mathbb{P}$. Although you can obviously use them for forecasting, the core information which is used to build the model is backward-looking in nature (historical behaviour of the stock)....

1

let' s define a ARMA-GARCH model: $y_{t} = \mu_{t} + \epsilon_{t}$ where $\mu_{t}$ is the conditional mean process (ARMA(p,q) part, $\mu_{t} = E(y_{t}|\mathcal{F}_{t-1})$) . The errors (or mean residuals) re defined by: $\epsilon_{t} = \sigma_{t} \eta_{t}$ where $\eta_{t}$ is a white noise (0,1) Then : $Var[\epsilon_{t}]= \sigma_{t}^{2}$. next see ...

Only top voted, non community-wiki answers of a minimum length are eligible

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# Types of Numbers

In Math, numbers can be categorized into specific types of number.

Listed below are the main groups that certain types of numbers can fall into.

NATURAL NUMBERS

Sometimes known as the counting numbers. The natural numbers are the group of whole numbers larger than zero.

{ 1 , 2 , 3 , 4 , 5 , …………… }

WHOLE NUMBERS

Very similar to the natural numbers, whole numbers are numbers larger than zero, but also including zero.

{ 0 , 1 , 2 , 3 , 4 , ……………}

INTEGERS

Integers are numbers that are positive and negative whole numbers, also including zero.

{ ……………… , -3 , -2 , -1 , 0 , 1 , 2 , 3 , ……………..}

RATIONAL NUMBERS

Rational numbers, which can also be referred to as quotients.
These are numbers that can be written as a fraction of two integers.

For example  ½ , ¾ , even 3/1 is a quotient.

IRRATIONAL NUMBERS

Irrational numbers are numbers that can’t be written as a fraction/quotient of two integers.

Numbers such as  π  and  √2  are irrational numbers.

REAL NUMBERS

Any real number, all of the number types in the previous groups are real numbers, even the irrational numbers.

Many numbers can be classed as more than just one type.

PRIME NUMBERS

A prime number is any whole number greater than 1 that can only be divided evenly by 1 and itself.

For example:     3, 11 and 17 are prime numbers.

## Special Case, Infinity

A special case that's worth a mention, is Infinity.

Infinity isn't really a number, but is rather an idea or a concept.

The idea of an amount being unlimited, without an end.

The symbol for infinity is .

The Real Numbers for example are infinite.

You can write  1'000'000.........  and keep writing zeroes on and on forever with no end.

The fraction  $\dpi{80}&space;\fn_jvn&space;\frac{1}{3}$  has a decimal form that is infinite.

$\dpi{80}&space;\fn_jvn&space;\frac{1}{3}$  =  0.3333.......,  the three's go on forever.

Infinity does go both ways also, we can have  -.

With numbers going on and on forever in a negative direction.  -1, -2, -3,.......

›   ›  Number Types


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# Tag Info

### When to use GRU over LSTM?

GRUs and LSTMs utilize different approaches toward gating information to prevent the vanishing gradient problem. Here are the main points comparing the two: The GRU unit controls the flow of ...
• 1,929

### When to use GRU over LSTM?

*To complement already great answers above. From my experience, GRUs train faster and perform better than LSTMs on less training data if you are doing language modeling (not sure about other tasks). ...
• 2,117
Accepted

### Time Series prediction using LSTMs: Importance of making time series stationary

In general time series are not really different from other machine learning problems - you want your test set to 'look like' your training set, because you want the model you learned on your training ...
• 2,178
Accepted

### How to feed LSTM with different input array sizes?

The easiest way is to use Padding and Masking. There are three general ways to handle variable-length sequences: Padding and masking (which can be used for (3)), Batch size = 1, and Batch size > 1,...
• 8,967

### Validation loss is not decreasing

The model is overfitting right from epoch 10, the validation loss is increasing while the training loss is decreasing. Dealing with such a Model: Data Preprocessing: Standardizing and Normalizing the ...
• 625

### When to use GRU over LSTM?

FULL GRU Unit $\tilde{c}_t = \tanh(W_c [G_r * c_{t-1}, x_t ] + b_c)$ $G_u = \sigma(W_u [ c_{t-1}, x_t ] + b_u)$ $G_r = \sigma(W_r [ c_{t-1}, x_t ] + b_r)$ \$ c_t = G_u * \tilde{c}_t + (1 - ...
• 341
Accepted

### Multi-dimentional and multivariate Time-Series forecast (RNN/LSTM) Keras

The input shape for an LSTM must be (num_samples, num_time_steps, num_features). In your example case, combining both cities as input, ...
• 14.4k

### Dropout on which layers of LSTM?

I prefer not to add drop out in LSTM cells for one specific and clear reason. LSTMs are good for long terms but an important ...
• 13.6k

### What's the difference between the cell and hidden state in LSTM?

In short: Cell state: Long term memory of the model, only part of LSTM models Hidden state: Working memory, part of LSTM and RNN models Additional Information RNN and vanishing/exploding gradients ...
• 1,918

### When to use GRU over LSTM?

This answer actually lies on the dataset and the use case. It's hard to tell definitively which is better. GRU exposes the complete memory unlike LSTM, so applications which that acts as advantage ...
• 2,296
Accepted

### What is LSTM, BiLSTM and when to use them?

RNN architectures like LSTM and BiLSTM are used in occasions where the learning problem is ...
• 13.6k
Accepted

### Sliding window leads to overfitting in LSTM?

Although the previous answer by @Imran is correct, I feel it necessary to add a caveat: there are applications out there where people do feed a sliding window in to an LSTM. For example, here, for ...
• 1,971

### What is the best method for classification of time series data? Should I use LSTM or a different method?

I would not use the word "best" but LSTM-RNN are very powerful when dealing with timeseries, simply because they can store information about previous values and exploit the time dependencies ...
• 3,870
Accepted

### Activation function between LSTM layers

Simple explanation with images We know that an activation is required between matrix multiplications to afford a neural network the ability to model non-linear processes. A classical LSTM cell ...
• 14.4k
Accepted

### Prediction interval around LSTM time series forecast

Directly, this is not possible. However, if you model it in a different way you can get out confidence intervals. You could instead of a normal regression approach it as estimating a continuous ...
• 9,248

### So what's the catch with LSTM?

You are right that LSTMs work very well for some problems, but some of the drawbacks are: LSTMs take longer to train LSTMs require more memory to train LSTMs are easy to overfit Dropout is much ...
• 2,341
Accepted

### Keras LSTM with 1D time series

LSTM layers require data of a different shape. From your description, I understand the starting dataset to have 3125 rows and 1000 columns, where each row is one time-step. The target variable should ...
• 14.4k

### What does the output of model.predict function from Keras mean?

The output of a neural network will never, by default, be binary - i.e. zeros or ones. The network works with continuous values (not discrete) in order to optimise the loss more freely in the ...
• 14.4k
Accepted

### Checking for stationarity in LSTM

In principle no you do not need to check for stationarity nor correct for it when you are using an LSTM. The thing about stationarity is that it makes prediction tasks much more efficient, and stable....
• 702
Accepted

### Batch Size of Stateful LSTM in keras

I solved the problem this way: I realized that I needed to find the HCF (highest common factor) of both the length of x_train ...
• 421
Accepted

### What's the difference of stateless LSTM and a normal feed-forward NN?

Theoretically a stateless LSTM gives the same result as a statefull LSTM, but there are few pros and cons between them. A stateless LSTM requires you to structure your data in a particular way, in ...
Accepted

### What is the job of "RepeatVector" and "TimeDistributed"?

tf.keras.layers.RepeatVector According to the docs : Repeats the input n times. They have also provided an example : ...
• 2,055
Accepted

### Value Error: Operands could not be broadcast together with shapes - LSTM

The answer that OP provided is correct, yet I would like to elaborate a little more on it, in an attempt to shed more light. First of all, you have to understand what is performed by the call ...
• 3,870

### When to use Stateful LSTM?

As for stateful LSTM and its understanding, refer to here. Quoting an answer from there: "I’m given a big sequence (e.g. Time Series) and I split it into smaller sequences to construct my input ...
• 3,870

### Binary classification of every time series step based on past and future values

You are facing a very common problem: handling imbalanced data. For neural networks, typical procedures are: Having the proper metrics: global accuracy should not be used. Oversampling the minority ...
• 1,618

### What is the job of "RepeatVector" and "TimeDistributed"?

For encoder-decoder, your input is squashed into a single feature vector, if you want your output to regenerate the same dimension as the original input, you are "artificially" converting this feature ...
• 191
Accepted

### How to use Embedding() with 3D tensor in Keras?

I'm not entirely sure if this is the cleanest solution but I stitched everything together. Each of the 10 word positions get their own input but that shouldn't be too much of a problem. The idea is to ...
• 9,248

From What are the advantages of stacking multiple LSTMs? (I'll only update the answer there): From {1}: While it is not theoretically clear what is the additional power gained by the ...
• 5,478


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# Numerical Solution of the KdV

Nonlinear PDE's Course
Current Topic Numerical Solution of the KdV
Next Topic Conservation Laws for the KdV
Previous Topic Introduction to KdV

## Introduction

We present here a method to solve the KdV equation numerically. There are many different methods to solve the KdV and we use here a spectral method which has been found to work well. Spectral methods work by using the Fourier transform (or some varient of it) to calculate the derivative.

## Fourier Transform

Recall that the Fourier transform is given by

$\displaystyle{ \mathcal{F}\left[ f(x)\right] =\hat{f}\left( k\right) =\int_{-\infty }^{\infty }f\left( x\right) \mathrm{e}^{-\mathrm{i}kx}\mathrm{d}x }$

and the inverse Fourier transform is given by

$\displaystyle{ f\left( x\right) =\mathcal{F}^{-1}\left[ \hat{f}\left( k\right) \right] = \frac{1}{2\pi }\int_{-\infty }^{\infty }\hat{f}\left( k\right) \mathrm{e}^{\mathrm{i}kx}\mathrm{d}k }$

(note that there are other ways of writing this transform). The most important property of the Fourier transform is that

$\displaystyle{ \begin{matrix} \int_{-\infty }^{\infty }\left( \partial _{x}f\left( x\right) \right) \mathrm{e}^{-\mathrm{i}kx}\mathrm{d}x &=&-\int_{-\infty }^{\infty }f\left( x\right) \left( \partial_{x}\mathrm{e}^{-\mathrm{i}kx}\right) \mathrm{d}x \\ &=&ik\int_{-\infty }^{\infty }f\left( x\right) \mathrm{e}^{-\mathrm{i}kx}\mathrm{d}x \end{matrix} }$

where we have assumed that the function $\displaystyle{ f\left( x\right) }$ vanishes at $\displaystyle{ \pm \infty . }$ This means that the Fourier transform converts differentiation to multiplication by $\displaystyle{ k }$.

## Solution for the Linearized KdV

We begin with a simple example. Suppose we want to solve the linearized KdV equation

$\displaystyle{ \partial _{t}u+\partial _{x}^{3}u=0,\ \ -\infty \lt x \lt \infty }$

subject to solve initial conditions

$\displaystyle{ u\left( x,0\right) =f\left( x\right) }$

We can solve this equation by taking the Fourier transform. and obtain

$\displaystyle{ \partial _{t}\hat{u}-ik^{3}\hat{u}=0 }$

So that

$\displaystyle{ \hat{u}\left( k,t\right) =\hat{f}\left( k\right) \mathrm{e}^{\mathrm{i}k^{3}t} }$

Therefore

$\displaystyle{ u\left( x,t\right) = \frac{1}{2\pi} \int_{-\infty}^{\infty }\hat{f}\left( k\right) \mathrm{e}^{\mathrm{i}k^{3}t}\mathrm{e}^{\mathrm{i}kx}\mathrm{d}k }$

## Numerical Implementation Using the FFT

The Fast Fourier Transform (FFT) is a method to calculate the fourier transform efficiently for discrete sets of points. These points need to be evenly spaced in the $\displaystyle{ x }$ plane and are given by

$\displaystyle{ x_{n}=x_{0}+n\Delta x,\ \ 0\leq n\leq N-1 }$

(note they can start at any $\displaystyle{ x }$ value). For the FFT to be as efficient as possible $\displaystyle{ N }$ should be a power of $\displaystyle{ 2. }$ Corresponding to the discrete set of points in the $\displaystyle{ x }$ domain is a discrete set of points in the $\displaystyle{ k }$ plane given by

$\displaystyle{ k_{n}=\left\{ \begin{matrix} n\Delta k,\ \ 0\leq n\leq \frac{N}{2} \\ \left( n-N\right) \Delta k,\ \ \frac{N}{2}+1\leq n\leq N-1 \end{matrix} \right. }$

where $\displaystyle{ \Delta k=2\pi /(N\Delta x). }$ Note that this numbering seems slighly odd and is due to aliasing. We are not that interested in the frequency domain solution but we need to make sure that we select the correct values of $\displaystyle{ k }$ for our numerical code.

## Numerical Code for the Linear KdV

Here is the code to solve the linear KdV using MATLAB

N = 1024;

t=0.1;

x = linspace(-10,10,N);

delta_x = x(2) - x(1);

delta_k = 2*pi/(N*delta_x);

k = [0:delta_k:N/2*delta_k,-(N/2-1)*delta_k:delta_k:-delta_k];

f = exp(-x.^2);

f_hat = fft(f);

u = real(ifft(f_hat.*exp(i*k.^3*t)));

## Numerical Solution of the KdV

It turns out that a method to solve the KdV equation can be derived using spectral methods. We begin with the KdV equation written as

$\displaystyle{ \partial _{t}u+3\partial _{x}\left( u\right) ^{2}+\partial _{x}^{3}u=0. }$

The Fourier transform of the KdV is therefore

$\displaystyle{ \partial _{t}\hat{u}+3ik\widehat{\left( u^{2}\right)} -ik^{3}\hat{u}=0 }$

We solve this equation by a split step method. We write the equation as

$\displaystyle{ \partial _{t}\hat{u}=-3ik\widehat{\left( u^{2}\right)} +ik^{3}\hat{u} }$

We can solve

$\displaystyle{ \partial _{t}\hat{u}=ik^{3}\hat{u} }$

exactly while the equation

$\displaystyle{ \partial _{t}\hat{u}=-3ik\widehat{\left( u^{2}\right)} }$

needs to be solved by time stepping. The idea of the split step method is to solve alternatively each of these equations when stepping from $\displaystyle{ t }$ to $\displaystyle{ t+\Delta t. }$ Therefore we solve

$\displaystyle{ \begin{matrix} \hat{u}_{1}\left( k,t+\Delta t\right) &=&\hat{u}\left( k,t\right) \mathrm{e}^{\mathrm{i}k^{3}\Delta t} \\ \hat{u}\left( k,t+\Delta t\right) &=&\hat{u}_{1}\left( k,t+\Delta t\right) -3ik \Delta t\widehat{\left( u_{1}^{2}\right)} \end{matrix} }$

Note that we are using Euler's method to time step and that the solution could be improved by using a better method, such as the Runge-Kutta 4 method.

The only slighly tricky thing is that we have both $\displaystyle{ u }$ and $\displaystyle{ \hat{u} }$ in the equation, but we can simply use the Fourier transform to connect these. The equation then becomes

$\displaystyle{ \begin{matrix} \hat{u}_{1}\left( k,t+\Delta t\right) &=&\hat{u}\left( k,t\right) \mathrm{e}^{\mathrm{i}k^{3}\Delta t} \\ \hat{u}\left( k,t+\Delta t\right) &=&\hat{u}_{1}\left( k,t+\Delta t\right) - 3ik\Delta t\left( \mathcal{F}\left( \left( \mathcal{F}^{-1}\left[ \hat{u}_{1}\left( k,t+\Delta t\right)\right] \right) ^{2}\right) \right) \end{matrix} }$

## Numerical Code to solve the KdV by the split step method

Here is some code to solve the KdV using MATLAB

N = 256;

x = linspace(-10,10,N);

delta_x = x(2) - x(1);

delta_k = 2*pi/(N*delta_x);

k = [0:delta_k:N/2*delta_k,-(N/2-1)*delta_k:delta_k:-delta_k];

c=16;

u = 1/2*c*(sech(sqrt(c)/2*(x+8))).^2;

delta_t = 0.4/N^2;

tmax = 0.1; nmax = round(tmax/delta_t);

U = fft(u);

for n = 1:nmax

% first we solve the linear part

U = U.*exp(1i*k.^3*delta_t);

%then we solve the non linear part

U = U - delta_t*(3i*k.*fft(real(ifft(U)).^2));

end

## Example Calculations

We cosider the evolution of the KdV with two solitons as initial condition as given below.

$\displaystyle{ u(x,0) = 8\,\mathrm{sech}^{2}\left(2(x+8)\right) + \, 2 \mathrm{sech}^{2}\left(\left( x +1\right) \right) }$
Animation Three-dimensional plot.
Evolution of $\displaystyle{ u(x,t) }$ for two solitons.
Evolution of $\displaystyle{ u(x,t) }$ for two solitons. Note the phase shift which occurs in the soliton interaction.


text:
# Calc. Find the limit using L'Hospital's Rule. lim x--->0 (x)/(tan^-1 (4x))

You need to use l'Hospital's theorem if the limit proves to be indeterminate, hence you should check if it is the case such that:

`lim_(x-gt0) x/(tan^(-1)4x) = 0/(tan^(-1) 0) = 0/0`

Since the limit is indeterminate, you may use l'Hospital's theorem such that:

`lim_(x-gt0) x/(tan^(-1)4x) = lim_(x-gt0) (x')/((tan^(-1)4x)')`

`lim_(x-gt0) (x')/((tan^(-1)4x)') = lim_(x-gt0) 1/(((4x)')/(1+16x^2))`

`lim_(x-gt0) 1/(((4x)')/(1+16x^2))= lim_(x-gt0)(1+16x^2)/4`

`lim_(x-gt0)(1+16x^2)/4 = (1+0)/4 = 1/4`

Hence, evaluating the limit to the given function yields `lim_(x-gt0) x/(tan^(-1)4x) =1/4.`

Approved by eNotes Editorial Team


text:
# zbMATH — the first resource for mathematics

## Li, Xin

Compute Distance To:
 Author ID: li.xin.13 Published as: Li, Xin; Li, X.
 Documents Indexed: 323 Publications since 1983, including 2 Books
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#### Co-Authors

 41 single-authored 17 Feng, Yuehong 16 Wang, Shu 9 Zhang, Luming 6 Chattopadhyay, Eshan 6 Dong, Weisheng 6 Shi, Guangming 5 Arnoldus, Henk F. 5 Chen, Chaoping 4 Ye, Chengqing 4 Zhang, Mingwang 3 Goyal, Vipul 3 Gu, Xianfeng 3 Guo, Jiming 3 Kang, Shin Min 3 Keer, Leon M. 3 Liu, Zeqing 3 Steigmann, David J. 3 Yan, Zhenya 3 Zhang, Baoxue 3 Zuckerman, David 2 Alam, Md Nur 2 Chen, Xin 2 Chen, Yan 2 Chen, Yong 2 Cheng, Chuntian 2 Cheng, Kuan 2 Cornelissen, Gunther 2 Dodis, Yevgeniy 2 Fang, Xiaochun 2 Ganie, Hilal Ahmad 2 Golberg, Michael A. 2 Guo, Pengfei 2 Houkes, Zweitze 2 Hu, Qingping 2 Ishai, Yuval 2 Ji, Ping 2 Jiang, Weihua 2 Kobayashi, Naoki 2 Kwun, Young Chel 2 Lai, Jingang 2 Li, Jing 2 Li, Ming 2 Lian, Zhaotong 2 Liao, HongLin 2 Liu, Tian 2 Liu, Zihui 2 Ma, Bin 2 Ma, Xueying 2 Ma, Yi 2 Marcolli, Matilde 2 Mikusiński, Piotr 2 Pirzada, Shariefuddin 2 Qi, Feng 2 Qin, Hong 2 Smit, Harry 2 Tang, Ruoli 2 Taylor, Michael D. 2 Vitányi, Paul M. B. 2 Wang, Guoyu 2 Wang, Zhiwen 2 Wooley, Trevor D. 2 Xu, Shuangshuang 2 Zhang, Xiaohong 2 Zheng, Bing 2 Zheng, Fu 2 Zhu, Jingfu 2 Zou, Xingfu 1 Amin, Moeness G. 1 An, Zhanzhou 1 Aristidou, Michael 1 Avilés L., J. 1 Bai, Liang 1 Bai, Yanqin 1 Baschuk, J. J. 1 Basu, Amitabh 1 Bazargan, Kia 1 Belianinov, Alex 1 Ben-Aroya, Avraham 1 Berg, Matthew J. 1 Bi, Xiaolei 1 Boukpeti, Nathalie 1 Bracciali, Cleonice F. 1 Buddhakulsomsiri, Jirachai 1 Cai, Erjuan 1 Cai, Jing 1 Cao, Xiaohui 1 Cao, Xun 1 Cao, Zhixian 1 Chai, Guangying 1 Chang, Zhe 1 Charlier, Robert 1 Chat, Bilal Ahmad 1 Che, Tao 1 Chen, Jiwen 1 Chen, Kun 1 Chen, Qian 1 Chen, Yanfei 1 Chen, Yangkui 1 Chen, Yanhua 1 Cheng, Xiangguo ...and 330 more Co-Authors
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#### Serials

 9 Advances in Computational Mathematics 9 IEEE Transactions on Image Processing 6 Physics Letters. A 5 Applied Mathematics and Computation 5 Mathematics in Practice and Theory 5 Communications in Theoretical Physics 4 Numerical Algorithms 4 Mathematical Problems in Engineering 3 Discrete Mathematics 3 International Journal of Solids and Structures 3 Mathematical Methods in the Applied Sciences 3 Journal of Computational and Applied Mathematics 3 Operations Research Letters 3 Acta Automatica Sinica 3 Journal of Zhejiang University. Engineering Science 3 Journal of Jilin University. Science Edition 3 Journal of Beijing University of Technology 2 Modern Physics Letters B 2 Acta Mechanica 2 Computer Methods in Applied Mechanics and Engineering 2 International Journal of Non-Linear Mechanics 2 International Journal of Theoretical Physics 2 Journal of the Franklin Institute 2 Journal of Mathematical Physics 2 Linear and Multilinear Algebra 2 Physica A 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 IEEE Transactions on Computers 2 Proceedings of the American Mathematical Society 2 SIAM Journal on Computing 2 International Journal of Production Research 2 Constructive Approximation 2 Acta Scientiarum Naturalium Universitatis Pekinensis 2 CAD. Computer-Aided Design 2 Archive of Applied Mechanics 2 Pure and Applied Mathematics 2 Complexity 2 Engineering Analysis with Boundary Elements 2 INFORMS Journal on Computing 2 Abstract and Applied Analysis 2 Journal of Zhengzhou University. Natural Science Edition 2 Italian Journal of Pure and Applied Mathematics 2 Communications in Nonlinear Science and Numerical Simulation 2 Journal of Systems Science and Complexity 2 Acta Mathematica Scientia. Series A. (Chinese Edition) 2 EURASIP Journal on Applied Signal Processing 2 Communications in Information and Systems 2 Journal of Applied Mathematics and Computing 2 Journal of Software 2 Journal of Shandong University. Natural Science 2 Control and Decision 2 Discrete and Continuous Dynamical Systems. Series S 2 International Journal of Nonlinear Science 2 Fuzzy Systems and Mathematics 2 Journal of Applied Analysis and Computation 1 Biological Cybernetics 1 Journal of Beijing Institute of Technology. English Edition 1 Computers & Mathematics with Applications 1 Communications in Mathematical Physics 1 Computers and Structures 1 Discrete Applied Mathematics 1 Houston Journal of Mathematics 1 IEEE Transactions on Information Theory 1 IMA Journal of Applied Mathematics 1 International Journal of Engineering Science 1 International Journal for Numerical and Analytical Methods in Geomechanics 1 Journal of Fluid Mechanics 1 Journal of Mathematical Analysis and Applications 1 Lithuanian Mathematical Journal 1 Quarterly Journal of Mechanics and Applied Mathematics 1 Ukrainian Mathematical Journal 1 Zeitschrift fuer Naturforschung, A: Physical Sciences 1 Mathematics of Computation 1 The Annals of Statistics 1 Biometrical Journal 1 Biometrics 1 IEEE Transactions on Automatic Control 1 Mathematica Japonica 1 Mathematics Seminar Notes 1 Mechanics Research Communications 1 Naval Research Logistics 1 Tamkang Journal of Mathematics 1 Theoretical Computer Science 1 Journal of Nanjing University. Natural Sciences 1 Insurance Mathematics & Economics 1 Bulletin of the Korean Mathematical Society 1 Journal of Xiamen University. Natural Science 1 Journal of East China Normal University. Natural Science Edition 1 Journal of Northeast Normal University. Natural Science Edition 1 Advances in Mathematics 1 Computer Aided Geometric Design 1 Journal of Computational Mathematics 1 Journal of Nanjing University. Mathematical Biquarterly 1 Parallel Computing 1 Applied Numerical Mathematics 1 Journal of Systems Science and Mathematical Sciences 1 Acta Mathematicae Applicatae Sinica. English Series 1 Northeastern Mathematical Journal 1 Journal of Northwest University. Natural Sciences Edition 1 Computational Mechanics ...and 94 more Serials
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#### Fields

 64 Computer science (68-XX) 53 Partial differential equations (35-XX) 51 Numerical analysis (65-XX) 40 Information and communication theory, circuits (94-XX) 35 Operations research, mathematical programming (90-XX) 24 Systems theory; control (93-XX) 19 Mechanics of deformable solids (74-XX) 19 Fluid mechanics (76-XX) 17 Statistics (62-XX) 14 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 13 Combinatorics (05-XX) 12 Optics, electromagnetic theory (78-XX) 12 Statistical mechanics, structure of matter (82-XX) 11 Approximations and expansions (41-XX) 11 Quantum theory (81-XX) 10 Linear and multilinear algebra; matrix theory (15-XX) 10 Ordinary differential equations (34-XX) 10 Biology and other natural sciences (92-XX) 9 Dynamical systems and ergodic theory (37-XX) 7 Mathematical logic and foundations (03-XX) 7 Real functions (26-XX) 7 Special functions (33-XX) 6 Operator theory (47-XX) 5 Functional analysis (46-XX) 5 Probability theory and stochastic processes (60-XX) 4 Group theory and generalizations (20-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 4 Classical thermodynamics, heat transfer (80-XX) 3 Number theory (11-XX) 3 Functions of a complex variable (30-XX) 3 Differential geometry (53-XX) 3 Geophysics (86-XX) 2 General and overarching topics; collections (00-XX) 2 Algebraic geometry (14-XX) 2 Nonassociative rings and algebras (17-XX) 2 Topological groups, Lie groups (22-XX) 2 Mechanics of particles and systems (70-XX) 2 Relativity and gravitational theory (83-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Associative rings and algebras (16-XX) 1 Measure and integration (28-XX) 1 Potential theory (31-XX) 1 Difference and functional equations (39-XX) 1 Sequences, series, summability (40-XX) 1 Integral equations (45-XX) 1 General topology (54-XX) 1 Manifolds and cell complexes (57-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Mathematics education (97-XX)

#### Citations contained in zbMATH Open

144 Publications have been cited 787 times in 2 Documents Cited by Year
Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model. Zbl 1267.35236
Li, Xin; Jiang, Weihua; Shi, Junping
2013
The similarity metric. Zbl 1316.68052
Li, Ming; Chen, Xin; Li, Xin; Ma, Bin; Vitányi, Paul M. B.
2004
Function projectve synchronization between two identical chaotic systems. Zbl 1139.37301
Chen, Yong; Li, Xin
2007
Nonlocally centralized sparse representation for image restoration. Zbl 1373.94104
Dong, Weisheng; Zhang, Lei; Shi, Guangming; Li, Xin
2013
Fixed point theorems for mappings satisfying contractive conditions of integral type and applications. Zbl 1271.54084
Liu, Zeqing; Li, Xin; Kang, Shin Min; Cho, Sun Young
2011
On BCH-algebras. Zbl 0579.03047
Hu, Qingping; Li, Xin
1983
Finite plane twist of an annular membrane. Zbl 0785.73041
Li, X.; Steigmann, D. J.
1993
Linear regression with doubly censored data. Zbl 0898.62042
Zhang, Cun-Hui; Li, Xin
1996
Compressive sensing via nonlocal low-rank regularization. Zbl 1374.94085
Dong, Weisheng; Shi, Guangming; Li, Xin; Ma, Yi; Huang, Feng
2014
A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator. Zbl 1309.65099
Li, Xin; Zhang, Luming; Wang, Shanshan
2012
The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels. Zbl 1220.76025
Zhang, T. T.; Jia, L.; Wang, Z. C.; Li, X.
2008
Nonlocal image restoration with bilateral variance estimation: a low-rank approach. Zbl 1373.94102
Dong, Weisheng; Shi, Guangming; Li, Xin
2013
Modified projective synchronization of a new hyperchaotic system via nonlinear control. Zbl 1180.37039
Li, Xin
2009
Certificateless signature and proxy signature schemes from bilinear pairings. Zbl 1076.94029
Li, X.; Chen, K.; Sun, L.
2005
On solving boundary value problems of modified Helmholtz equations by plane wave functions. Zbl 1099.65118
Li, Xin
2006
Approximation by radial bases and neural networks. Zbl 0976.65014
Li, Xin; Micchelli, Charles A.
2000
Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer. Zbl 0861.41013
Li, Xin
1996
On proper BCH-algebras. Zbl 0583.03050
Hu, Qingpin; Li, Xin
1985
On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation. Zbl 1110.65100
Li, Xin
2005
Polycube splines. Zbl 1206.65038
Wang, Hongyu; He, Ying; Li, Xin; Gu, Xianfeng; Qin, Hong
2008
Real orthogonal polynomials in frequency analysis. Zbl 1061.42013
Bracciali, C. F.; Li, Xin; Sri Ranga, A.
2005
Interior-point algorithm for linear optimization based on a new trigonometric kernel function. Zbl 1408.90318
Li, Xin; Zhang, Mingwang
2015
The asymptotic behavior of globally smooth solutions of bipolar nonisentropic compressible Euler-Maxwell system for plasma. Zbl 1296.35134
Wang, Shu; Feng, Yuehong; Li, Xin
2012
On the instability of plane liquid sheets in two gas streams of unequal velocities. Zbl 0855.76020
Li, X.
1994
A mesh-free method using hyperinterpolation and fast Fourier transform for solving differential equations. Zbl 1081.65110
Li, Xin; Chen, C. S.
2004
On simultaneous approximations by radial basis function neural networks. Zbl 1085.82503
Li, Xin
1998
Convergence of the method of fundamental solutions for solving the boundary value problem of modified Helmholtz equation. Zbl 1079.65107
Li, Xin
2004
2-source extractors under computational assumptions and cryptography with defective randomness. Zbl 1292.94088
Kalai, Yael Tauman; Li, Xin; Rao, Anup
2009
Improved non-malleable extractors, non-malleable codes and independent source extractors. Zbl 1370.94527
Li, Xin
2017
Point loads on a hemispherical elastic membrane. Zbl 0835.73031
Li, X.; Steigmann, D. J.
1995
Computational test of approximation of functions and their derivatives by radial basis functions. Zbl 0998.65020
Li, Xin; Ho, C. H.; Chen, C. S.
2002
Interpolation by ridge polynomials and its application in neural networks. Zbl 0999.41003
Li, Xin
2002
Stability of non-constant steady-state solutions for non-isentropic Euler-Maxwell system with a temperature damping term. Zbl 1348.35134
Feng, Yue-Hong; Wang, Shu; Li, Xin
2016
Monotonicity result for generalized logarithmic means. Zbl 1132.26326
Li, Xin; Chen, Ch.-P.; Qi, F.
2007
Non-malleable codes and extractors for small-depth circuits, and affine functions. Zbl 1370.68083
2017
Non-malleable extractors and codes, with their many tampered extensions. Zbl 1377.94042
Chattopadhyay, Eshan; Goyal, Vipul; Li, Xin
2016
Fine-granularity and spatially-adaptive regularization for projection-based image deblurring. Zbl 1372.94152
Li, Xin
2011
Two-dimensional coupled mathematical modeling of fluvial processes with intense sediment transport and rapid bed evolution. Zbl 1160.86309
Yue, Zhiyuan; Cao, Zhixian; Li, Xin; Che, Tao
2008
Finite deformation of a pressurized toroidal membrane. Zbl 0836.73035
Li, X.; Steigmann, D. J.
1995
On the crosscorrelation of sequences with the decimation factor $$d=\frac{p^n+1}{p+1}-\frac{p^n-1}{2}$$. Zbl 1015.94006
Hu, Z.; Li, X.; Mills, D.; Müller, E.; Sun, W.; Willems, W.; Yang, Y.; Zhang, Z.
2001
Training multilayer perceptrons via minimization of sum of ridge functions. Zbl 1002.90069
Wu, Wei; Feng, Guorui; Li, Xin
2002
The similarity metric. Zbl 1092.68683
Li, Ming; Chen, Xin; Li, Xin; Ma, Bin; Vitányi, Paul
2003
Epidemic spreading on complex networks with overlapping and non-overlapping community structure. Zbl 1395.92165
Shang, Jiaxing; Liu, Lianchen; Li, Xin; Xie, Feng; Wu, Cheng
2015
Hyperspectral image super-resolution via non-negative structured sparse representation. Zbl 1408.94139
Dong, Weisheng; Fu, Fazuo; Shi, Guangming; Cao, Xun; Wu, Jinjian; Li, Guangyu; Li, Xin
2016
A numerical method for axisymmetric wave propagation problem of anisotropic solids. Zbl 0894.73217
Liu, Kaishin; Li, Xin; Sun, Xiannian
1997
Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere. Zbl 1138.65097
Li, Xin
2008
Moment Lyapunov exponent and stochastic stability for a binary airfoil driven by an ergodic real noise. Zbl 1281.76023
Li, X.; Liu, X. B.
2013
Generalized projective synchronization using nonlinear control method. Zbl 1187.93106
Li, Xin
2009
Image restoration. Fundamentals and advances. Zbl 1252.68033
Gunturk, Bahadir Kursat (ed.); Li, Xin (ed.)
2012
Upper bounds for the sum of Laplacian eigenvalues of a graph and Brouwer’s conjecture. Zbl 1410.05125
Ganie, Hilal A.; Pirzada, S.; Ul Shaban, Rezwan; Li, X.
2019
On the topology distortion in self-organizing feature maps. Zbl 0797.92006
Li, X.; Gasteiger, J.; Zupan, J.
1993
On the convergence of the dual reciprocity method for Poisson’s equation. Zbl 1045.65110
Li, X.; Golberg, M. A.
2003
Asymptotic behavior of global smooth solutions for bipolar compressible Navier-Stokes-Maxwell system from plasmas. Zbl 1349.35270
Feng, Yuehong; Wang, Shu; Li, Xin
2015
Image restoration via simultaneous sparse coding: where structured sparsity meets Gaussian scale mixture. Zbl 1398.94029
Dong, Weisheng; Shi, Guangming; Ma, Yi; Li, Xin
2015
The asymptotic behavior of globally smooth solutions of non-isentropic Euler-Maxwell equations for plasmas. Zbl 1410.82029
Wang, Shu; Feng, Yue-Hong; Li, Xin
2014
Using Lyapunov function to design optimal controller for AQM routers. Zbl 1140.93384
Zhang, Peng; Ye, Chengqing; Ma, Xueying; Chen, Yanhua; Li, Xin
2007
Topology optimization of structures under multiple load cases using a fiber-reinforced composite material model. Zbl 1102.74036
Zhou, K.; Li, X.
2006
Characterization of global fields by Dirichlet $$L$$-series. Zbl 1460.11137
Cornelissen, Gunther; de Smit, Bart; Li, Xin; Marcolli, Matilde; Smit, Harry
2019
An interior-point algorithm for $$P_*(\kappa)$$-LCP based on a new trigonometric kernel function with a double barrier term. Zbl 1391.90590
Li, Xin; Zhang, Mingwang; Chen, Yan
2017
Sharp $$H^1$$-norm error estimate of a cosine pseudo-spectral scheme for 2D reaction-subdiffusion equations. Zbl 1440.65092
Li, Xin; Zhang, Luming; Liao, Hong-Lin
2020
On the Lagrange interpolation for a subset of $$C^ k$$ functions. Zbl 0830.41001
Li, Xin
1995
Stability of non-constant steady-state solutions for bipolar non-isentropic Euler-Maxwell equations with damping terms. Zbl 1362.35039
Li, Xin; Wang, Shu; Feng, Yue-Hong
2016
A new numerical scheme for the nonlinear Schrödinger equation with wave operator. Zbl 1379.65064
Li, Xin; Zhang, Luming; Zhang, Ting
2017
Appell series $$\mathbb{F}_1$$ over finite fields. Zbl 1390.33026
Li, Long; Li, Xin; Mao, Rui
2018
Coefficient formula and matrix of nonuniform B-spline functions. Zbl 0783.65003
Grabowski, H.; Li, X.
1992
Robust pseudorandom generators. Zbl 1336.94053
Ishai, Yuval; Kushilevitz, Eyal; Li, Xin; Ostrovsky, Rafail; Prabhakaran, Manoj; Sahai, Amit; Zuckerman, David
2013
Design extractors, non-malleable condensers and privacy amplification. Zbl 1286.94077
Li, Xin
2012
Interacting spin pairs in rotational magnetic fields and geometric phase. Zbl 1221.81032
Li, Xin
2008
On a reaction-diffusion model for sterile insect release method with release on the boundary. Zbl 1255.35210
Li, Xin; Zou, Xingfu
2012
Sharp upper bounds of the spectral radius of a graph. Zbl 1416.05173
Guo, Ji-Ming; Wang, Zhi-Wen; Li, Xin
2019
Some integration-by-parts formulas involving 2-copulas. Zbl 1322.60008
Li, X.; Mikusiński, P.; Taylor, M. D.
2002
Data mining via minimal spanning tree clustering for prolonging lifetime of wireless sensor networks. Zbl 1135.68546
Huang, Guangyan; Li, Xiaowei; He, Jing; Li, Xin
2007
Rate of convergence of the method of fundamental solutions and hyperinterpolation for modified Helmholtz equations on the unit ball. Zbl 1157.65058
Li, Xin
2008
Two novel classes of linear high-order structure-preserving schemes for the generalized nonlinear Schrödinger equation. Zbl 1437.65213
Li, Xin; Gong, Yuezheng; Zhang, Luming
2020
Radial basis approximation and its application to biharmonic equation. Zbl 1187.65133
Li, Xin
2010
Instability of cylindrical compressible gas jets in viscous liquid streams. Zbl 0912.76018
Shen, J.; Li, X.
1998
Finite element analysis of DAM-reservoir systems using an exact far-boundary condition. Zbl 0918.73210
Li, X.; Romo O., M. P.; Avilés L., J.
1996
No graph with nullity $$\eta(G) = | V(G) | - 2 m(G) + 2 c(G) - 1$$. Zbl 1419.05131
Li, Xin; Guo, Ji-Ming
2019
Remarks on convergence of Markov operators. Zbl 1021.60007
Li, Xin; Mikusiński, P.; Taylor, M. D.
2002
Padé approximant related to asymptotics for the gamma function. Zbl 1359.33001
Li, Xin; Chen, Chao-Ping
2017
Dynamic environmental/economic scheduling for microgrid using improved MOEA/D-M2M. Zbl 1400.90172
Li, Xin; Fang, Yanjun
2016
On the second relative greedy weight. Zbl 1362.94063
Li, Xin; Liu, Zihui
2017
Connected component model for multi-object tracking. Zbl 1408.94243
He, Zhenyu; Li, Xin; You, Xinge; Tao, Dacheng; Tang, Yuan Yan
2016
Reconstructing global fields from dynamics in the abelianized Galois group. Zbl 1431.11110
Cornelissen, Gunther; Li, Xin; Marcolli, Matilde; Smit, Harry
2019
A direct method for solving crack growth problems. I. Zbl 0771.73067
Li, X.; Keer, L. M.
1992
Integral formulas and inequalities for rational functions. Zbl 0878.30001
Li, Xin
1997
Inequalities for the gamma function. Zbl 1143.33003
Li, Xin; Chen, Chao-Ping
2007
Transformed partial least squares for multivariate data. Zbl 1132.62100
Zhu, Li-Xing; Zhu, Li-Ping; Li, Xin
2007
The structure of 3-solvable non-degenerate Lie algebra and its derivation Lie algebra. Zbl 1146.17005
Hu, Caihui; Li, Xin
2006
Geometric phase subject to stochastic noise. Zbl 1211.81066
Li, Xin; Li, Zhongfang
2011
Cooperative dynamics of snowdrift game on spatial distance-dependent small-world networks. Zbl 1189.91184
Shang, L. H.; Li, X.; Wang, X. F.
2006
Searching geometry-aware pants decomposition in different isotopy classes. Zbl 1328.68264
Zhang, Kang; Li, Xin
2014
Non-malleable condensers for arbitrary min-entropy, and almost optimal protocols for privacy amplification. Zbl 1359.94653
Li, Xin
2015
Privacy amplification and nonmalleable extractors via character sums. Zbl 1302.94043
Dodis, Yevgeniy; Li, Xin; Wooley, Trevor D.; Zuckerman, David
2014
Static flexural analysis of elliptic Reissner-Mindlin plates on a Pasternak foundation by the triangular differential quadrature method. Zbl 1119.74633
Zhong, Hongzhi; Li, X.; He, Yuhong
2005
Conformal spherical parametrization for high genus surfaces. Zbl 1171.30311
Zeng, Wei; Li, Xin; Yau, Shing-Tung; Gu, Xianfeng
2007
An efficient spectral-collocation difference method for two-dimensional Schrödinger equation with Neumann boundary conditions. Zbl 1437.65214
Li, Xin; Zhang, Luming
2020
Transmission of electric dipole radiation through an interface. Zbl 1331.78009
Arnoldus, Henk F.; Berg, Matthew J.; Li, Xin
2014
Indecomposable positive linear maps constructed from permutations. Zbl 1335.15041
Li, Xin; Fang, Xiaochun
2016
Quantum bidirectional teleportation $$2 \leftrightarrow 2$$ or $$2 \leftrightarrow 3$$ qubit teleportation protocol via 6-qubit entangled state. Zbl 1433.81047
Zhou, Ri-Gui; Li, Xin; Qian, Chen; Ian, Hou
2020
Sharp $$H^1$$-norm error estimate of a cosine pseudo-spectral scheme for 2D reaction-subdiffusion equations. Zbl 1440.65092
Li, Xin; Zhang, Luming; Liao, Hong-Lin
2020
Two novel classes of linear high-order structure-preserving schemes for the generalized nonlinear Schrödinger equation. Zbl 1437.65213
Li, Xin; Gong, Yuezheng; Zhang, Luming
2020
An efficient spectral-collocation difference method for two-dimensional Schrödinger equation with Neumann boundary conditions. Zbl 1437.65214
Li, Xin; Zhang, Luming
2020
Quantum bidirectional teleportation $$2 \leftrightarrow 2$$ or $$2 \leftrightarrow 3$$ qubit teleportation protocol via 6-qubit entangled state. Zbl 1433.81047
Zhou, Ri-Gui; Li, Xin; Qian, Chen; Ian, Hou
2020
Upper bounds for the sum of Laplacian eigenvalues of a graph and Brouwer’s conjecture. Zbl 1410.05125
Ganie, Hilal A.; Pirzada, S.; Ul Shaban, Rezwan; Li, X.
2019
Characterization of global fields by Dirichlet $$L$$-series. Zbl 1460.11137
Cornelissen, Gunther; de Smit, Bart; Li, Xin; Marcolli, Matilde; Smit, Harry
2019
Sharp upper bounds of the spectral radius of a graph. Zbl 1416.05173
Guo, Ji-Ming; Wang, Zhi-Wen; Li, Xin
2019
No graph with nullity $$\eta(G) = | V(G) | - 2 m(G) + 2 c(G) - 1$$. Zbl 1419.05131
Li, Xin; Guo, Ji-Ming
2019
Reconstructing global fields from dynamics in the abelianized Galois group. Zbl 1431.11110
Cornelissen, Gunther; Li, Xin; Marcolli, Matilde; Smit, Harry
2019
Black and gray solitons in holographic superfluids at zero temperature. Zbl 1416.83101
Gao, Meng; Jiao, Yuqiu; Li, Xin; Tian, Yu; Zhang, Hongbao
2019
Appell series $$\mathbb{F}_1$$ over finite fields. Zbl 1390.33026
Li, Long; Li, Xin; Mao, Rui
2018
Stability of non-constant equilibrium solutions for two-fluid non-isentropic Euler-Maxwell systems arising in plasmas. Zbl 1394.76159
Feng, Yue-Hong; Li, Xin; Wang, Shu
2018
Stability of non-constant equilibrium solutions for bipolar full compressible Navier-Stokes-Maxwell systems. Zbl 1403.35157
Li, Xin; Wang, Shu; Feng, Yue-Hong
2018
Advanced implicit meshless approaches for the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative. Zbl 1404.76010
Bi, Xiaolei; Mu, Shanjun; Liu, Qingxia; Liu, Quanzhen; Liu, Baoquan; Zhuang, Pinghui; Gao, Jian; Jiang, Hui; Li, Xin; Li, Bochen
2018
Error estimates of a trigonometric integrator sine pseudo-spectral method for the extended Fisher-Kolmogorov equation. Zbl 1462.65163
Li, Xin; Zhang, Luming
2018
Stability of nonconstant steady-state solutions for 2-fluid nonisentropic Euler-Poisson equations in semiconductor. Zbl 1394.35268
Li, Xin; Wang, Shu; Feng, Yuehong
2018
Improved non-malleable extractors, non-malleable codes and independent source extractors. Zbl 1370.94527
Li, Xin
2017
Non-malleable codes and extractors for small-depth circuits, and affine functions. Zbl 1370.68083
2017
An interior-point algorithm for $$P_*(\kappa)$$-LCP based on a new trigonometric kernel function with a double barrier term. Zbl 1391.90590
Li, Xin; Zhang, Mingwang; Chen, Yan
2017
A new numerical scheme for the nonlinear Schrödinger equation with wave operator. Zbl 1379.65064
Li, Xin; Zhang, Luming; Zhang, Ting
2017
Padé approximant related to asymptotics for the gamma function. Zbl 1359.33001
Li, Xin; Chen, Chao-Ping
2017
On the second relative greedy weight. Zbl 1362.94063
Li, Xin; Liu, Zihui
2017
An extended cohesive damage model for simulating arbitrary damage propagation in engineering materials. Zbl 1439.74334
Li, X.; Chen, J.
2017
A hybrid constraints handling strategy for multiconstrained multiobjective optimization problem of microgrid economical/environmental dispatch. Zbl 1386.90137
Li, Xin; Lai, Jingang; Tang, Ruoli
2017
The $$t$$-wise intersection of relative three-weight codes. Zbl 1387.94116
Li, Xin; Liu, Zihui
2017
Stability, integrability, and nonlinear dynamics of $$\mathcal{P} T$$-symmetric optical couplers with cubic cross-interactions or cubic-quintic nonlinearities. Zbl 1403.34040
Li, Xin; Yan, Zhenya
2017
Price and capacity decisions of service systems with boundedly rational customers. Zbl 1411.90098
Li, Xin; Guo, Pengfei; Lian, Zhaotong
2017
Stability of non-constant steady-state solutions for non-isentropic Euler-Maxwell system with a temperature damping term. Zbl 1348.35134
Feng, Yue-Hong; Wang, Shu; Li, Xin
2016
Non-malleable extractors and codes, with their many tampered extensions. Zbl 1377.94042
Chattopadhyay, Eshan; Goyal, Vipul; Li, Xin
2016
Hyperspectral image super-resolution via non-negative structured sparse representation. Zbl 1408.94139
Dong, Weisheng; Fu, Fazuo; Shi, Guangming; Cao, Xun; Wu, Jinjian; Li, Guangyu; Li, Xin
2016
Stability of non-constant steady-state solutions for bipolar non-isentropic Euler-Maxwell equations with damping terms. Zbl 1362.35039
Li, Xin; Wang, Shu; Feng, Yue-Hong
2016
Dynamic environmental/economic scheduling for microgrid using improved MOEA/D-M2M. Zbl 1400.90172
Li, Xin; Fang, Yanjun
2016
Connected component model for multi-object tracking. Zbl 1408.94243
He, Zhenyu; Li, Xin; You, Xinge; Tao, Dacheng; Tang, Yuan Yan
2016
Indecomposable positive linear maps constructed from permutations. Zbl 1335.15041
Li, Xin; Fang, Xiaochun
2016
Semigroup $$C^{\ast}$$-algebras. Zbl 1375.46040
Li, Xin
2016
Computing approximate PSD factorizations. Zbl 1398.65076
Basu, Amitabh; Dinitz, Michael; Li, Xin
2016
Robust dynamic energy use and climate change. Zbl 1396.91415
Li, Xin; Narajabad, Borghan; Temzelides, Ted
2016
A Hotelling queue model with probabilistic service. Zbl 1408.90103
Xu, Xiaoya; Lian, Zhaotong; Li, Xin; Guo, Pengfei
2016
Extractors for sumset sources. Zbl 1377.94014
2016
Interior-point algorithm for linear optimization based on a new trigonometric kernel function. Zbl 1408.90318
Li, Xin; Zhang, Mingwang
2015
Epidemic spreading on complex networks with overlapping and non-overlapping community structure. Zbl 1395.92165
Shang, Jiaxing; Liu, Lianchen; Li, Xin; Xie, Feng; Wu, Cheng
2015
Asymptotic behavior of global smooth solutions for bipolar compressible Navier-Stokes-Maxwell system from plasmas. Zbl 1349.35270
Feng, Yuehong; Wang, Shu; Li, Xin
2015
Image restoration via simultaneous sparse coding: where structured sparsity meets Gaussian scale mixture. Zbl 1398.94029
Dong, Weisheng; Shi, Guangming; Ma, Yi; Li, Xin
2015
Non-malleable condensers for arbitrary min-entropy, and almost optimal protocols for privacy amplification. Zbl 1359.94653
Li, Xin
2015
Hopf bifurcation in a new four-dimensional hyperchaotic system. Zbl 1321.34057
Li, Xin; Yan, Zhen-Ya
2015
Compressive sensing via nonlocal low-rank regularization. Zbl 1374.94085
Dong, Weisheng; Shi, Guangming; Li, Xin; Ma, Yi; Huang, Feng
2014
The asymptotic behavior of globally smooth solutions of non-isentropic Euler-Maxwell equations for plasmas. Zbl 1410.82029
Wang, Shu; Feng, Yue-Hong; Li, Xin
2014
Searching geometry-aware pants decomposition in different isotopy classes. Zbl 1328.68264
Zhang, Kang; Li, Xin
2014
Privacy amplification and nonmalleable extractors via character sums. Zbl 1302.94043
Dodis, Yevgeniy; Li, Xin; Wooley, Trevor D.; Zuckerman, David
2014
Transmission of electric dipole radiation through an interface. Zbl 1331.78009
Arnoldus, Henk F.; Berg, Matthew J.; Li, Xin
2014
On a reaction-diffusion model for sterile insect release method on a bounded domain. Zbl 1302.35211
Jiang, Weihua; Li, Xin; Zou, Xingfu
2014
Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model. Zbl 1267.35236
Li, Xin; Jiang, Weihua; Shi, Junping
2013
Nonlocally centralized sparse representation for image restoration. Zbl 1373.94104
Dong, Weisheng; Zhang, Lei; Shi, Guangming; Li, Xin
2013
Nonlocal image restoration with bilateral variance estimation: a low-rank approach. Zbl 1373.94102
Dong, Weisheng; Shi, Guangming; Li, Xin
2013
Moment Lyapunov exponent and stochastic stability for a binary airfoil driven by an ergodic real noise. Zbl 1281.76023
Li, X.; Liu, X. B.
2013
Robust pseudorandom generators. Zbl 1336.94053
Ishai, Yuval; Kushilevitz, Eyal; Li, Xin; Ostrovsky, Rafail; Prabhakaran, Manoj; Sahai, Amit; Zuckerman, David
2013
New independent source extractors with exponential improvement. Zbl 1293.68059
Li, Xin
2013
Numerical study of a single drop impact onto a liquid film up to the consequent formation of a crown. Zbl 1297.76175
Jiang, T.; Ouyang, J.; Li, X.; Ren, J.; Wang, X.
2013
A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator. Zbl 1309.65099
Li, Xin; Zhang, Luming; Wang, Shanshan
2012
The asymptotic behavior of globally smooth solutions of bipolar nonisentropic compressible Euler-Maxwell system for plasma. Zbl 1296.35134
Wang, Shu; Feng, Yuehong; Li, Xin
2012
Image restoration. Fundamentals and advances. Zbl 1252.68033
2012
Design extractors, non-malleable condensers and privacy amplification. Zbl 1286.94077
Li, Xin
2012
On a reaction-diffusion model for sterile insect release method with release on the boundary. Zbl 1255.35210
Li, Xin; Zou, Xingfu
2012
Positive periodic solutions for first-order neutral functional differential equations with periodic delays. Zbl 1245.34073
Liu, Zeqing; Li, Xin; Kang, Shin Min; Kwun, Young Chel
2012
Gravitational deflection of light in Rindler-type potential as a possible resolution to the observations of bullet cluster 1E0657-558. Zbl 1247.83153
Li, Xin; Chang, Zhe
2012
Approximate symmetry reduction to the perturbed coupled KdV equations derived from two-layer fluids. Zbl 1264.35197
Li, Xin; Qian, Suping
2012
Fixed point theorems for mappings satisfying contractive conditions of integral type and applications. Zbl 1271.54084
Liu, Zeqing; Li, Xin; Kang, Shin Min; Cho, Sun Young
2011
Fine-granularity and spatially-adaptive regularization for projection-based image deblurring. Zbl 1372.94152
Li, Xin
2011
Geometric phase subject to stochastic noise. Zbl 1211.81066
Li, Xin; Li, Zhongfang
2011
An architecture for fault-tolerant computation with stochastic logic. Zbl 1367.65242
Qian, Weikang; Li, Xin; Riedel, Marc D.; Bazargan, Kia; Lilja, David J.
2011
Privacy amplification and non-malleable extractors via character sums. Zbl 1292.94169
Dodis, Yevgeniy; Li, Xin; Wooley, Trevor D.; Zuckerman, David
2011
Radial basis approximation and its application to biharmonic equation. Zbl 1187.65133
Li, Xin
2010
Reversal of the dipole vortex in a negative index of refraction material. Zbl 1238.78005
Li, Xin; Arnoldus, Henk F.
2010
On exponential time lower bound of Knapsack under backtracking. Zbl 1194.68127
Li, Xin; Liu, Tian
2010
Modified projective synchronization of a new hyperchaotic system via nonlinear control. Zbl 1180.37039
Li, Xin
2009
2-source extractors under computational assumptions and cryptography with defective randomness. Zbl 1292.94088
Kalai, Yael Tauman; Li, Xin; Rao, Anup
2009
Generalized projective synchronization using nonlinear control method. Zbl 1187.93106
Li, Xin
2009
Sufficient conditions for $$\lambda_k$$-optimality and super-$$\lambda_k$$ in graphs. Zbl 1212.05224
Gao, Jingzhen; Li, Xin
2009
A clustering algorithm based on multiple kernel SVM. Zbl 1212.68174
Zhao, Lifeng; Li, Xin; Wang, Dong
2009
Using data assimilation method to calibrate a heterogeneous conductivity field and improve solute transport prediction with an unknown contamination source. Zbl 1416.86019
Huang, Chunlin; Hu, Bill X.; Li, Xin; Ye, Ming
2009
The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels. Zbl 1220.76025
Zhang, T. T.; Jia, L.; Wang, Z. C.; Li, X.
2008
Polycube splines. Zbl 1206.65038
Wang, Hongyu; He, Ying; Li, Xin; Gu, Xianfeng; Qin, Hong
2008
Two-dimensional coupled mathematical modeling of fluvial processes with intense sediment transport and rapid bed evolution. Zbl 1160.86309
Yue, Zhiyuan; Cao, Zhixian; Li, Xin; Che, Tao
2008
Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere. Zbl 1138.65097
Li, Xin
2008
Interacting spin pairs in rotational magnetic fields and geometric phase. Zbl 1221.81032
Li, Xin
2008
Rate of convergence of the method of fundamental solutions and hyperinterpolation for modified Helmholtz equations on the unit ball. Zbl 1157.65058
Li, Xin
2008
Function projective synchronization of discrete-time chaotic and hyperchaotic systems using backstepping method. Zbl 1392.93039
Jin, Yi-Liang; Li, Xin; Chen, Yong
2008
Function projectve synchronization between two identical chaotic systems. Zbl 1139.37301
Chen, Yong; Li, Xin
2007
Monotonicity result for generalized logarithmic means. Zbl 1132.26326
Li, Xin; Chen, Ch.-P.; Qi, F.
2007
Using Lyapunov function to design optimal controller for AQM routers. Zbl 1140.93384
Zhang, Peng; Ye, Chengqing; Ma, Xueying; Chen, Yanhua; Li, Xin
2007
Data mining via minimal spanning tree clustering for prolonging lifetime of wireless sensor networks. Zbl 1135.68546
Huang, Guangyan; Li, Xiaowei; He, Jing; Li, Xin
2007
Inequalities for the gamma function. Zbl 1143.33003
Li, Xin; Chen, Chao-Ping
2007
Transformed partial least squares for multivariate data. Zbl 1132.62100
Zhu, Li-Xing; Zhu, Li-Ping; Li, Xin
2007
Conformal spherical parametrization for high genus surfaces. Zbl 1171.30311
Zeng, Wei; Li, Xin; Yau, Shing-Tung; Gu, Xianfeng
2007
Improved exponential time lower bound of Knapsack problem under BT model. Zbl 1194.68128
Li, Xin; Liu, Tian; Peng, Han; Qian, Liyan; Sun, Hongtao; Xu, Jin; Xu, Ke; Zhu, Jiaqi
2007
Function projective synchronization between two different chaotic systems. Zbl 1203.37134
Chen, Yong; Li, Xin
2007
On solving boundary value problems of modified Helmholtz equations by plane wave functions. Zbl 1099.65118
Li, Xin
2006
Topology optimization of structures under multiple load cases using a fiber-reinforced composite material model. Zbl 1102.74036
Zhou, K.; Li, X.
2006
The structure of 3-solvable non-degenerate Lie algebra and its derivation Lie algebra. Zbl 1146.17005
Hu, Caihui; Li, Xin
2006
Cooperative dynamics of snowdrift game on spatial distance-dependent small-world networks. Zbl 1189.91184
Shang, L. H.; Li, X.; Wang, X. F.
2006
...and 44 more Documents

#### Cited by 3 Authors

 2 Arnoldus, Henk F. 1 Berg, Matthew J. 1 Li, Xin

#### Cited in 2 Serials

 1 Physics Letters. A 1 Journal of Modern Optics

#### Cited in 2 Fields

 2 Optics, electromagnetic theory (78-XX) 1 Partial differential equations (35-XX)


text:
# zbMATH — the first resource for mathematics

An efficient approach to nominal equalities in hybrid logic tableaux. (English) Zbl 1242.03029
Summary: Basic hybrid logic extends modal logic with the possibility of naming worlds by means of a distinguished class of atoms (called nominals) and the so-called satisfaction operator, that allows one to state that a given formula holds at the world named $$a$$, for some nominal $$a$$. Hence, in particular, hybrid formulae include “equality” assertions, stating that two nominals are distinct names for the same world. The treatment of such nominal equalities in proof systems for hybrid logics may induce many redundancies. This paper introduces an internalized tableau system for basic hybrid logic, significantly reducing such redundancies. The calculus enjoys a strong termination property: tableau construction terminates without relying on any specific rule application strategy, and no loop-checking is needed. The treatment of nominal equalities specific of the proposed calculus is briefly compared to other approaches. Its practical advantages are demonstrated by empirical results obtained by use of implemented systems. Finally, it is briefly shown how to extend the calculus to include the global and converse modalities.

##### MSC:
 03B35 Mechanization of proofs and logical operations 03B62 Combined logics 68T27 Logic in artificial intelligence
##### Keywords:
tableaux; hybrid logic; modal logic
##### Software:
HyLoTab; Sibyl; Spartacus; HTab; OCaml
Full Text:
##### References:
 [1] Alechina N., Journal of Logic and Computation 13 (6) pp 936– (2003) · Zbl 1093.68032 [2] Areces C., The Journal of Symbolic Logic 66 (3) pp 977– (2001) · Zbl 0984.03018 [3] Areces C., Proceedings of the 13th International Workshop and 8th Annual Conference of the EACSL on Computer Science Logic, number 1683 in LNCS pp 307– [4] Areces C., Journal of Logic and Computation 11 (5) pp 717– (2001) · Zbl 0993.03007 [5] Areces C., Proceedings of CADE-18 pp 156– [6] Areces C., Methods for Modalities 3 (2003) [7] Balsiger P., Journal of Automated Reasoning 24 (3) pp 297– (2000) · Zbl 0955.03013 [8] DOI: 10.3166/jancl.14.447-475 · Zbl 1185.68290 [9] Blackburn P., Journal of Logic and Computation 10 (1) pp 137– (2000) · Zbl 0953.03018 [10] DOI: 10.1017/CBO9781107050884 [11] Blackburn P., Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2002), number 2381 in LNAI pp 38– (2002) [12] Bolander T., Journal of Logic and Computation 17 (3) pp 517– (2007) · Zbl 1140.03005 [13] Bolander T., Proceedings of the 5th Workshop on Methods for Modalities, 2007 231 pp 21– [14] Bolander T., Journal of Logic and Computation 16 (6) pp 737– (2006) · Zbl 1122.03006 [15] Braüner T., Journal of Logic and Computation 14 (3) pp 329– (2004) · Zbl 1060.03036 [16] Cerrito S., Terminating tableaux for HL(@) without loop-checking (2007) [17] Cerrito S., Tableaux with Substitution for Hybrid Logic with the Global and Converse Modalities (2009) [18] Cialdea Mayer M., Two tableau provers for basic hybrid logic (2009) · Zbl 1291.68333 [19] Franceschet M., Methods for Modalities 3 pp 109– (2003) [20] Franceschet M., Proceedings of Computer Science Logic 2005 3634 pp 339– [21] Götzmann D., Spartacus: A Tableau Prover for Hybrid Logic (2009) [22] Hoffmann G., Proceedings of the 5th Workshop on Methods for Modalities, 2007 231 pp 3– [23] DOI: 10.1093/logcom/9.3.385 · Zbl 0940.03039 [24] Horrocks I., Journal of Automated Reasoning 39 (3) pp 249– (2007) · Zbl 1132.68734 [25] Kaminski M., Proc. of IJCAR 2008 5195 pp 210– [26] Kaminski M., Proceedings of the 5th Workshop on Methods for Modalities, 2007 231 pp 241– [27] Kaminski M., Journal of Logic, Language and Information 18 (4) pp 437– (2009) · Zbl 1188.03013 [28] Leroy X., The Objective Caml system, release 3.11 (2008) [29] Tzakova M., Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 1999) 1617 pp 278– (1999) [30] van Eijck J., HyLoTab – Tableau-based theorem proving for hybrid logics (2002) [31] van Eijck J., Constraint tableaux for hybrid logics (2002)
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:
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# The Tigers beat the Sharks 2 out of the first 3 times they played. The

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The Tigers beat the Sharks 2 out of the first 3 times they played. The  [#permalink]

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18 Mar 2019, 10:02
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The Tigers beat the Sharks 2 out of the first 3 times they played. They then played N more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for N?

(A) 35
(B) 37
(C) 39
(D) 41
(E) 43
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Re: The Tigers beat the Sharks 2 out of the first 3 times they played. The  [#permalink]

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18 Mar 2019, 10:38
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Noshad wrote:
The Tigers beat the Sharks 2 out of the first 3 times they played. They then played N more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for N?

(A) 35
(B) 37
(C) 39
(D) 41
(E) 43

Currently the winning percent of Sharks is 1/3*100= 33.33 %.Since we have to find the minimum possible value of N for which Shark's winning percentage becomes at least 95% . So sharks has to win all the N matches so that N is minimized.

Hence total matches = N+3. When shark will have winning percent of 95% then that of tigers will be 5%. And they will win only total of 2 matches

So 2 matches correspond to 5% of the total matches.
Therefore total number of matches = 2/5*100 = 40.

So N+3 = 40 => N= 37

Alternatively, you can solve the following equation and find the answer .

2/(N+3) = 5/100

Solving you will get N = 37. Hence B)
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Re: The Tigers beat the Sharks 2 out of the first 3 times they played. The  [#permalink]

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28 Feb 2020, 00:00
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Noshad wrote:
The Tigers beat the Sharks 2 out of the first 3 times they played. They then played N more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for N?

(A) 35
(B) 37
(C) 39
(D) 41
(E) 43

Solution:

• Tiger beats sharks in 2 out of the first 3 matches.
• Sharks ended up winning at least $$95\%$$ matches
o Sharks lose at most $$5\%$$ match
• Now, total matches = $$N+3$$
o For the minimum number of matches, we can assume that sharks lost only 2 matches
• By above inferences
o $$2 ≤ 5\%\ of\ (N+3)$$
o $$2 ≤\frac{5}{100}*(N+3)$$
o $$2 ≤ \frac{1}{20}*(N+3)$$
o $$40 ≤ N+3$$
o $$40-3 ≤ N$$
o $$37 ≤ N$$
• The minimum possible value of N is 37.
Hence, the correct answer is Option B.
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Re: The Tigers beat the Sharks 2 out of the first 3 times they played. The  [#permalink]

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28 Feb 2020, 00:32
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total game played among them= N+3

game won by sharks = N+3-2

(because we have to minimize the total no. of games....so assume that all other games were won by sharks...)

given...N+1/N+3 * 100 = 95

N= 37..

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Re: The Tigers beat the Sharks 2 out of the first 3 times they played. The  [#permalink]

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28 Feb 2020, 00:46
Initial 3 matches, tiger wins 2 and shark wins 1, so if they play n more matches
Total matches =n+3
As we have to consider minimum value it is possible when shark losses no further match
So lost match by shark = 2 which is 5% of total matches played
I.e. 2=5%x (X is total matches played n+3)
=40
N=37

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Re: The Tigers beat the Sharks 2 out of the first 3 times they played. The   [#permalink] 28 Feb 2020, 00:46
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# The Tigers beat the Sharks 2 out of the first 3 times they played. The

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# Determinant after some modification

Algebra Level pending

If $$A$$ is a $$4\times 4$$ matrix with $$\text{det}(A)=7,$$ and matrix $$B$$ is obtained from $$A$$ by multiplying the second column by $$5$$ (and everything else stays the same), what is $$\text{det}(B)?$$

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# Prove that $\lim\limits_{x\to+\infty}f'(x)=\lim\limits_{x\to+\infty}f''(x)=0$

Given $$f \in C^3(\mathbb{R}).$$ Suppose that $$\lim\limits_{x\to+\infty}f(x)=a \in \mathbb{R}$$ and $$\lim\limits_{x\to+\infty}f'''(x)=0$$. Prove that $$\lim\limits_{x\to+\infty}f'(x)=\lim\limits_{x\to+\infty}f''(x)=0$$.

My solution: Since $$\lim\limits_{x\to+\infty}f(x)=a$$, there exists $$M>0$$ that for all $$x, x' \in \mathbb{R}$$, if $$x,x'>M$$ then $$f(x), f(x') \in \left(a-\varepsilon^2/2, a+\varepsilon^2/2\right)$$. This yields $$\vert f(x)-f(x') \vert < \varepsilon^2.$$

Therefore $$\frac{\vert f(x)-f(x') \vert}{\varepsilon} < \varepsilon$$. So, for all $$x>M$$, $$x'=x+\varepsilon$$ we have $$0\le\frac{\vert f(x+\varepsilon)-f(x) \vert}{\varepsilon} < \varepsilon.$$ Let $$\varepsilon \to 0$$ we have $$0 \le f'(x) \le 0 \text{ or } f'(x)=0.$$

That means $$\lim\limits_{x\to+\infty}f'(x)=0$$.

Doing similarly as above we have $$\lim\limits_{x\to+\infty}f''(x)=0.$$

My question: $$\lim\limits_{x\to+\infty}f'''(x)=0$$ is really necessary or not (because I don't use it in my proof)?

Here's a proof that shows how the condition on the 3rd derivative could be utilized.

Take Taylor's remainder theorem to 3rd order and apply it to the intervals $$(x,x+h)$$, $$(x,x+k)$$, $$h\neq k$$. Now consider only the first interval since the second one can be done similarly.

There exists $$\xi\in(x,x+h)$$ such that $$f(x+h)=f(x)+f'(x)h+\frac{f''(x)}{2!}h^2+\frac{f'''(\xi)}{3!}h^3$$

Now since $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f(x+h)=a$$, and $$\lim_{x\to\infty}f'''(\xi(x,h))=0$$ (why?) we conclude that $$\lim_{x\to\infty}\left(f(x+h)-f(x)-\frac{f'''(\xi)}{3!}h^3\right)=h\lim_{x\to\infty}\left(f'(x)+\frac{h}{2}f''(x)\right)=0 ~~~~~~~~~~(1)$$ Similarly we conclude $$\lim_{x\to\infty}\left(f'(x)+\frac{k}{2}f''(x)\right)=0~~~~~~~~~~~~~(2)$$

Taking the linear combinations $$(2)-(1)$$ and $$h(2)-k(1)$$ and exploiting the linearity of the limit operation when two limits exist, we obtain the desired result. The result is generalizable to a $$C^n$$ function whose $$n$$th derivative goes to zero for any $$n$$.

• – Martin R Jan 18 at 19:11
• +1 Thanks to the limit of $f^{(n)}$ being $0$, your proof is simple. The result holds even when $f^{(n)}$ is bounded but the proof gets a bit complicated. See math.stackexchange.com/a/731013/72031 – Paramanand Singh Jan 19 at 1:17
• There is a typo in the third derivative in the Taylor polynomial, also the denominator is 6, not 3. – edm Jan 19 at 5:55

First, consider $$f(x)=\frac{\sin(x^2)}{x}.$$ As $$x\rightarrow\infty$$, this approaches $$0$$. On the other hand, $$f'(x)=\frac{2x^2\cos(x^2)-\sin(x^2)}{x^2}=2\cos(x^2)-\frac{\sin(x^2)}{x^2}$$ does not have a limit as $$x\rightarrow\infty$$. Therefore, the condition on $$f'''$$ is necessary for the result.

The problem with your setup is that you never specify when you're quantifying $$\varepsilon$$. You're actually quantifying it before computing $$M$$, so we'll write $$M=M(\varepsilon)$$. Next, you consider $$\frac{|f(x+\varepsilon)-f(x)|}{(x+\varepsilon)-x}.$$ At this point, you let $$\varepsilon$$ go to $$0$$, but here's your problem, as $$\varepsilon$$ goes to $$0$$, $$M(\varepsilon)$$ may go to infinity, so you don't get a bound on the derivative when $$\varepsilon$$ is too small, so the squeeze theorem cannot be applied.


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# Area of union of random circles in a plane

If $n$ circles of radius $r$ are drawn at random in the plane such that their centers lie in some region $S$ (we could take $S$ to be a square), what is the expected area $E$ of their union?

Edit: In light of Noam's comment, it seems like the answer is

$\displaystyle \int \int 1- (1-f(x,y))^n dx dy$

where $f(x,y) = a(B_r(x,y) \cap S)/a(s)$; the probability a random circle contains $(x,y)$.

If the region $S$ is large compared to $r$ and it has a reasonably "out of the way" boundary, then $f(x,y)$ is approximately constant over all of $S$, and is about $\pi r^2/ a(S)$, and so the expected value $E$ for a fixed $n$ is $n\pi r^2 + O(r^4)$.

However, the more interesting cases occur when $r$ is not small or when $\partial S$ gets "in the way" significantly.

Suppose $S$ is a square of side length $s$ and the circles have radius $r=s/2$. How does $E$ grow with $n$ asymptotically? What if $S$ is a circle of radius $s$? A regular $k$-gon inscribed in a circle of radius $s$?

-
Compute the probability $p$ that a given point of $S$ is not hit by any of the circles (which is constant away from the edge), and multiply $1-p$ by the area of $S$. –  Noam D. Elkies Sep 3 '12 at 15:58
@Noam's comment is obviously the answer, with the one caveat that a non-asymptotic result will be a little tricky. –  Igor Rivin Sep 3 '12 at 16:13
You will probably get a more "natural" answer if you choose a "torus", i.e., identify opposite edges of a square, to eliminate edge effects. –  Dick Palais Sep 3 '12 at 23:24

## 1 Answer

The question is already interesting for general $r$ when $S=[0,1]$. Then the area of $E$ is $\pi r^2 + 2 r$. Because the area defect to $E$ of two circles being $d$ apart is $\frac{d^3}{12~ r}$, and $d\approx n^{-1}$, for large $n$ only the distance between minimum $a$ and maximum $b$ of the circle centers in $S$ determines the asymptotics. The independent expectation values for $a$ and $b$ are $1/(n+1)$ and $1-1/(n+1)$ giving an asymptotic area defect to $E$ of $4r/(n+1)$.

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Satisfiability Problem: Theory and Applications
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The satisfiability (SAT) problem is central in mathematical logic, computing theory, and many industrial applications. There has been a strong relationship between the theory, the algorithms, and the applications of the SAT problem. This book aims to bring together work by the best theorists, algorithmists, and practitioners working on the SAT problem and on industrial applications, as well as to enhance the interaction between the three research groups. The book features the application of theoretical/algorithmic results to practical problems and presents practical problems for theoretical/algorithmic study.

Major topics covered in the book include practical and industrial SAT problems and benchmarks, significant case studies and applications of the SAT problem and SAT algorithms, new algorithms and improved techniques for satisfiability testing, specific data structures and implementation details of the SAT algorithms, and the theoretical study of the SAT problem and SAT algorithms.

Features:

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Co-published with the Center for Discrete Mathematics and Theoretical Computer Science beginning with Volume 8. Volumes 1-7 were co-published with the Association for Computer Machinery (ACM).

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• S. A. Cook and D. G. Mitchell -- Finding hard instances of the satisfiability problem: A survey
• J. Gu, P. W. Purdom, J. Franco, and B. W. Wah -- Algorithms for the satisfiability (SAT) problem: A survey
• P. W. Purdom and G. N. Haven -- Backtracking and probing
• J. Franco -- Relative size of certain polynomial time solvable subclasses of satisfiability
• M. V. Marathe, H. B. Hunt III, R. E. Stearns, and V. Radhakrishnan -- Complexity of hierarchically and 1-dimensional periodically specified problems. I: Hardness results
• O. Kullmann -- Worst-case analysis, 3-SAT decision, and lower bounds: Approaches for improved SAT algorithms
• K. Iwama and K. Takaki -- Satisfiability of 3CNF formulas with small clause/variable-ratio
• D. A. Plaisted and G. D. Alexander -- Propositional search efficiency and first-order theorem proving
• J. Wang -- Branching rules for propositional satisfiability test
• B. W. Wah and Y. Shang -- A discrete Lagrangian-based global-search method for solving satisfiability problems
• M. G. C. Resende, L. S. Pitsoulis, and P. M. Pardalos -- Approximate solution of weighted MAX-SAT problems using GRASP
• J. Gu -- Multispace search for satisfiability and NP-hard problems
• S. Joy, J. Mitchell, and B. Borchers -- A branch and cut algorithm for MAX-SAT and weighted MAX-SAT
• A. Løkketangen and F. Glover -- Surrogate constraint analysis--new heuristics and learning schemes for satisfiability problems
• H. Kautz, B. Selman, and Y. Jiang -- A general stochastic approach to solving problems with hard and soft constraints
• J. Hsiang and G. S. Huang -- Some fundamental properties of Boolean ring normal forms
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• R. Battiti and M. Protasi -- Solving MAX-SAT with non-oblivious functions and history-based heuristics
• E. Speckenmeyer, M. Böhm, and P. Heusch -- On the imbalance of distributions of solutions of CNF-formulas and its impact on satisfiability solvers
• H. van Maaren -- On the use of second order derivatives for the satisfiability problem
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text:
## K in Rate Laws [ENDORSED]

$\frac{d[R]}{dt}=-k[R]^{2}; \frac{1}{[R]}=kt + \frac{1}{[R]_{0}}; t_{\frac{1}{2}}=\frac{1}{k[R]_{0}}$

Jared Smith 1E
Posts: 51
Joined: Fri Sep 29, 2017 7:04 am

### K in Rate Laws

I noticed that depending on the reaction order, the slope of the graph can be equal to positive or negative k. For first and zero-order reactions, the slope of the graph is equal to -k, but for second order reactions, the slope is equal to +k. Can someone please explain to me why one is different from the other two?

Nisarg Shah 1C
Posts: 54
Joined: Sat Jul 22, 2017 3:00 am

### Re: K in Rate Laws

It depends on the derivation of the integrated rate laws. When you integrate the zero and first-order laws you get a -kt term, and when you integrate the second order law you get a positive kt term. In this case, where we relate these integrated laws to y = mx+b, m (the slope) is equal to positive or negative k.

Justin Folk 3I
Posts: 43
Joined: Wed Sep 21, 2016 2:56 pm

### Re: K in Rate Laws  [ENDORSED]

Because for second order reactions the y-axis is 1/[A]. As time progresses the concentration of [A] decreases (assuming A is reactant) and thus 1/A increases, therefore slope of graph >0


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predict.fahrmeier_94

0th

Percentile

Predict function for the result of ddhazard

Predict function for the result of ddhazard

Usage
# S3 method for fahrmeier_94
predict(object, new_data, type = c("response", "term"),
tstart = "start", tstop = "stop", use_parallel = F, sds = F,
max_threads = getOption("ddhazard_max_threads"), ...)
Arguments
object

Result of a ddhazard call

new_data

New data to base predictions on

type

Either "response" for predicted probability of death or "term" for predicted terms in the linear predictor

tstart

Name of the start time column in new_data. It must corresponds to tstart used in the Surv(tstart, tstop, event) in the formula passed to ddhazard

tstop

same as tstart for the stop argument

use_parallel

TRUE if computation for type = "response" should be computed in parallel with the parallel package

sds

TRUE if point wise standard deviation should be computed. Convenient if you use functions like ns and you only want one term per term in the right hand site of the formula used in ddhazard

Maximum number of threads to use. -1 if it should be determined by a call to detectCores

...

Not used

Term

The result of type = "term" is a list with the following elements

terms

Is a 3D array. The first dimension is the number of bins, the second dimension is rows in new_data and the last dimension is the state space terms

sds

Similar to terms for the point wise confidence intervals using the smoothed co-variance matrices

fixed_terms

Vector of the fixed effect terms for each observation

Response

The result of type = "response" is a list with the elements below. The function check if there are columns in new_data which's names match tstart and tstop. If not, then each row in new data will get a predicted probability of dying in every bin.

fits

Fitted probability of dying

istart

Vector with the index of the first bin the elements in fits is in

istop

Vector with the index of the last bin the elements in fits is in

Aliases
• predict.fahrmeier_94
Documentation reproduced from package dynamichazard, version 0.3.1, License: GPL-2

Community examples

Looks like there are no examples yet.


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# window for short time fourier transform

When we implement the 1D-short time fourier transform, the formula is as such:

$$Sf[m,l]=\sum_{n=0}^{N-1}f[n]g[n-m]\exp\left(\frac{-i2\pi l n}{N}\right).$$

the window $g$ used is normally symmetric, why is this so? (Hint: Use the phase of Short time fourier transform.

• Isn't Gabor transform a Gaussian window by definition? Any other window is just an STFT. – endolith Nov 7 '13 at 22:30
• Ok, STFT in general, but why the window must be symmetric for 1D signals? @endolith – meta_warrior Nov 8 '13 at 0:11

You are talking about symmetry in two contexts :-

1. Gaussian
2. Window

I will talk about these two contexts one by one

Gaussian

A symmetric 2-D gaussian (I assume 1-D as in 1-D the question of symmetry does not arise) means that it does not have any directional preference (dictated by equal variances $\sigma_x$ and $\sigma_y$) . Now the job of Gabor Transform is to give us a joint time-frequency distribution (i.e telling us about frequency and phase content of a local image patch) Now consider the following two cases :-

1. Symmetric Guassian ($\sigma_x$ = $\sigma_y$) In this case the gaussian does not have any directional preference.

In the above image, suppose you are centered on the central pixel and consider a patch around it which is indicated in the above image by a black rectangle in the center. Following points are important for this image :

1. Doing a Gabor Transform over that patch would yield a time-frequency representation, which involves all the pixels in the patch accordingly weighted by the symmetric gaussian.
2. All the directions at a certain distance from the origin get the same weigthage.

However, if a gaussian oriented at an angle of $135^{\circ}$, then :-

1. the white principle diagonal pixels would get almost all the weightage while the other pixels will get little weightage.
2. The time-frequency representation will be mainly a time-frequency representation around the principle diagonal.
3. If the values of $\sigma_x$ and $\sigma_y$ are sufficiently small, the gabor transform will be very very close to the time-frequency representation for the straight line which marks the principle diagonal.

Thus, for the Gaussian function,

• Symmetric gaussian is non-directional.
• Non-Symmetric gaussian is directionally oriented.
• Non-Symmetric gaussian will be useful in cases where the time-frequency analysis of a directionally oriented feature in a patch has to be carried out.

Window

The Window size basically represents the time-resolution (or space-resolution in the case of 2-D signals like images). A narrow window means that the focus on some local patch of the image is high and hence a more faithful time resolution of the Time-frequency analysis will be possible.

However, this is limited by the Heisenberg's Uncertainty principle. Although, Gabor Transform attains the theoretical lower bound for the uncertainty principle, it stays within the bounds of the principle. Thus, changing the window size is akin to changing your time resolution.

A Non-symmetrical window (like $3\times7$) is generally not chosen because generally we do not have a reason or a preference for changing our space-resolution in one direction (x or y) relative to other. In case there is a concern for the same, it can be done. However, I have never till now seen such a situation.

• Hi, thanks for your contribution. Do you have any idea what happens if for the 1D case, the window is asymmetric? – meta_warrior Nov 10 '13 at 12:38
• In the case 1D signals, the window is also 1-D. So, as far as the window is concerned its shape cannot be asymmetric, unless you specify a center point. In the 1-D Case, the only likely case can be that the window is not symmetric with respect to the central point with respect to which the window is translated. So, In this case, as in the 2D case, one side of the sequence will have less weightage than the other side. It becomes important when analysing actuarial data, where past is known but future can only be estimated. So more weight is given to past samples and hence window is asymmetric. – Ujjwal Aryan Nov 11 '13 at 9:54
• Ok, can you make your explanation more mathematical? – meta_warrior Nov 11 '13 at 12:22
• No mathematics can clear this up than intuition. Read my answer in the 2-D Case and take it to the 1-D Case understanding that now the asymmetry cannot be talked about in the sense of the window size but rather simply as that point of the window which is centered at the signal point where the Time-freq. analysis is to be done. If that point is the center of the window then it is symmetric (provided symmetry is there in the window shape itself). – Ujjwal Aryan Nov 12 '13 at 8:00
• Does the symmetry of the window speed up computations? – meta_warrior Nov 14 '13 at 2:42


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# Difference between revisions of "2003 AMC 12A Problems/Problem 4"

The following problem is from both the 2003 AMC 12A #4 and 2003 AMC 10A #4, so both problems redirect to this page.

## Problem

It takes Anna $30$ minutes to walk uphill $1$ km from her home to school, but it takes her only $10$ minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?

$\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 3.125\qquad \mathrm{(C) \ } 3.5\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 4.5$

## Solution 1

Since she walked $1$ km to school and $1$ km back home, her total distance is $1+1=2$ km.

Since she spent $30$ minutes walking to school and $10$ minutes walking back home, her total time is $30+10=40$ minutes = $\frac{40}{60}=\frac{2}{3}$ hours.

Therefore her average speed in km/hr is $\frac{2}{\frac{2}{3}}=\boxed{\mathrm{(A)}\ 3}$.

## Solution 2

The average speed of two speeds that travel the same distance is the harmonic mean of the speeds, or $\dfrac{2}{\dfrac{1}{x}+\dfrac{1}{y}}=\dfrac{2xy}{x+y}$ (for speeds $x$ and $y$). Mary's speed going to school is $2\,\text{km/hr}$, and her speed coming back is $6\,\text{km/hr}$. Plugging the numbers in, we get that the average speed is $\dfrac{2\times 6\times 2}{6+2}=\dfrac{24}{8}=\boxed{\mathrm{(A)}\ 3}$.


text:
The function $g(x) = 2^x$ is graphed below, along with the tangent line to the graph at $x=0$.

Use the definition of the derivative, with $h=0.01$, to estimate the value of $g'(0)$.
$g'(0) \approx$

Use the definition of the derivative, with $h=0.001$, to estimate the value of $g'(0)$.
$g'(0) \approx$

Which of the following constants appears to be closest to $g'(0)$?