text:
User talk:Zebulon pike~enwikibooks

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.– Mike.lifeguard  | talk 02:06, 29 October 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Zebulon pike. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Zebulon pike~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:55, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:40, 19 April 2015 (UTC)

title: User talk:Zebulon pike~enwikibooks


text: User talk:68.224.239.145

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:68.224.239.145


text:
Talk:Solving Integrals by Trigonometric substitution/Procedure

This example is not appropriate...
The problem given

$$\int_{0}^{1} x\, \sqrt{4-3x}\, dx$$ is better solved using integration by parts. It is much too complicated to be used as a first example. I am going to change it to:

$$\int_{0}^{1} x\, \sqrt{4-3x^{2}}\, dx$$ --Hamsterpoop 03:13, 29 October 2007 (UTC)

 * Actually, I think it would be much easier if this wikibook is merged into Calculus. --Hamsterpoop 04:09, 29 October 2007 (UTC)


text:
User talk:Hamsterpoop

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.

I suggested merging the book Solving Integrals by Trigonometric substitution into Calculus. I think it would likely be a better fit in that location. Let me know what you think about this. --Whiteknight (Page) (Talk) 03:28, 29 October 2007 (UTC)

 * Thanks for the quick reply! I wouldnt say that anything is "keeping" the merger from happening, I guess nobody has ever suggested it before. If you want the books to be merged, I can take care of the hard work later this afternoon. It shouldnt cause any problems, so there is no sense waiting any longer then that. --Whiteknight (Page) (Talk) 12:40, 29 October 2007 (UTC)

title: User talk:Hamsterpoop


text:
User:Hayesr

This is my page.

title: User:Hayesr


text: Semiconductor Electronics/Diode Applications/Half Wave Rectifier


text: Semiconductor Electronics/Diode Applications/Full Wave Rectifier


text:
Semiconductor Electronics/DC Biasing-BJTs

 * /Introduction/
 * /Operating point/
 * /Fixed-Bias Circuit/
 * /Emitter-Stabilized Bias Circuit/
 * /Voltage-Divider Bias/
 * /DC Bias with Voltage Feedback/
 * /Miscellaneous Bias Configurations/
 * /Design Operations/
 * /Transistor Switching Networks/
 * /Troubleshooting Technique/
 * /PNP Transistors/
 * /Bias Stabilization/
 * /Software/


text:
Wikijunior:Ancient Civilizations/Saxons

Where did the Saxons live?
Before the 7th Century, the Saxons lived in Central Europe (especially Germany). During the 7th Century the Saxons seeking new lands invaded England. For 400 years the Saxons (along with other tribes) became part of Great Britan. In 1066, William the Conqueror (a Norman King) seized control ending Saxon rule.

<-- The Saxons first lived in central Europe (specifically Germany) but invaded Great Britain in the 7th century. They lived there for 400 years as the lords of the land, but were taken over by William the Conqueror in 1066. -->

Archeologists discovered that the ruins of villages were small one room houses surrounding a great hall in the center. Saxons lived in large one-room houses called heals or halls. They were made of wood and had thatched roofs made of a layer of bark with grassy sod on top of it. The roof was low to the ground so that animals could graze on it. There was a window in the top of the roof to let light and air in. It also let the smoke out from the big fire used for heating and cooking.

The Saxons first invaded Britain 1,500 years ago. Angles and Jutes also invaded Britain at the same time. The groups mixed and established the Anglo Saxon Kingdom which lasted until the Norman Conquest of 1066. According to Petolmey (a Germanic author during the AD 100s) the Saxons lived in what is now Schleswig-Holstein, Germany. During the 400s and 500s, the Saxons invaded Britain, and settled in southern and eastern Britain. By the 500s the Saxons had settled in most of Northern Germany between the Elbe and Rhine Rivers. In the 700s Charlemange conquered the Saxons and made their land part of his empire.

Who are the Saxons?
The Angles, Saxons and Jutes are known collectively as the Anglo-Saxons and they were the largest attacking tribes.

What did they believe?
Prior to their acceptance of Christianity under Charlemange, the various Saxon tribes held many pagan or Nordic beliefs. The Saxons were forced to accept Christianity by Charlemange when he conquered their country.

What did they eat?
The Saxons ate porridge made from barley or oats. They used wheat and rye for bread. They ate vegetables such as carrots, parsnips, cabbages, beans, peas, and onions. They ate fruits such as apples, cherries, and plums. Exotic foods such as potatoes, tomatoes, bananas, and pineapples would not have been available to them, as these grew only in the Americas at that time. They drank a weak beer made from barley. Wine was only for the rich. Deer and wild boar were only hunted by people who owned the land, so not everyone had these meats. Fish, such as Bottom Herring, Salmon, Eel, Perch, Pike, and Plaice, were caught in rivers and seas.

title: Wikijunior:Ancient Civilizations/Saxons


text:
User talk:Jerome Nugent-Smith

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.– Mike.lifeguard  | talk 21:35, 29 October 2007 (UTC)

title: User talk:Jerome Nugent-Smith


text: User talk:193.164.107.130

title: User talk:193.164.107.130


text:
Artificial Neural Networks

are a computational tool, based on the properties of biological neural systems. Neural networks excel in a number of problem areas where conventional von Neumann computer systems have traditionally been slow and inefficient. This book is going to discuss the creation and use of artificial neural networks.

Table of Contents

 * Introduction

Overview

 * Neural Network Basics
 * Biological Neural Networks
 * History
 * MATLAB Neural Networking Toolbox
 * Activation Functions

ANN Models

 * Feed-Forward Networks
 * Radial Basis Function Networks
 * Recurrent Networks
 * Echo State Networks
 * Hopfield Networks
 * Self-Organizing Maps
 * Competitive Models
 * ART Models
 * Boltzmann Machines
 * Committee of Machines
 * Autoencoders
 * Convolutional Neural Networks

Teaching and Learning

 * Learning Paradigms
 * Error-Correction Learning
 * Hebbian Learning
 * Competitive Learning
 * Boltzmann Learning
 * ART Learning
 * Self-Organizing Maps

Applications

 * Pattern Recognition
 * Clustering
 * Feature Detection
 * Series Prediction
 * Data Compression
 * Curve Fitting
 * Optimization
 * Control

Future Work

 * Criticisms and Problems
 * Artificial Intelligence

Resources

 * Resources
 * Licensing

title: Artificial Neural Networks


text:
Template:Artificial Neural Networks/Page

 Artificial Neural Networks


text:
Artificial Neural Networks/Licensing

License
The text of this book is released under the following license:

title: Artificial Neural Networks/Licensing


text:
Artificial Neural Networks/Resources

Wikibooks

 * Artificial Intelligence
 * Linear Algebra
 * Abstract Algebra
 * Calculus
 * Signals and Systems
 * Engineering Analysis
 * MATLAB Programming

Commons

 * Artificial neural network
 * Commons:Category:Neural networks

title: Artificial Neural Networks/Resources


text: Category:Book:Artificial Neural Networks


text:
Artificial Neural Networks/Introduction

Introduction
 are one of the most popular and promising areas of research. Artificial Neural Networks are abstract computational models, roughly based on the organizational structure of the human brain. There are a wide variety of network architectures and methods that can be combined to produce neural networks with different computational abilities.

What is This Book About?
This book is going to serve as a general-purpose overview of artificial neural networks, including network construction, use, and applications.

Who is This Book For?
This book is going to be aimed at advanced undergraduates and graduate students in the areas of computer science, mathematics, engineering, and the sciences.

What Are The Prerequisites?
Readers of this book are going to require a solid mathematical background that includes, but may not be limited to:

 * Linear Algebra
 * Abstract Algebra
 * Calculus

Students may also find some benefit in the following engineering texts:

 * Signals and Systems
 * Engineering Analysis

Students who wish to implement the lessons learned in this book should be familiar with at least one general-purpose programming language or have a background in:

 * MATLAB Programming
 * Programmable Logic

title: Artificial Neural Networks/Introduction


text:
Artificial Neural Networks/Neural Network Basics

Artificial Neural Networks
Artificial Neural Networks, also known as “Artificial neural nets”, “neural nets”, or ANN for short, are a computational tool modeled on the interconnection of the neuron in the nervous systems of the human brain and that of other organisms. Biological Neural Nets (BNN) are the naturally occurring equivalent of the ANN. Both BNN and ANN are network systems constructed from atomic components known as “neurons”. Artificial neural networks are very different from biological networks, although many of the concepts and characteristics of biological systems are faithfully reproduced in the artificial systems. Artificial neural nets are a type of non-linear processing system that is ideally suited for a wide range of tasks, especially tasks where there is no existing algorithm for task completion. ANN can be trained to solve certain problems using a teaching method and sample data. In this way, identically constructed ANN can be used to perform different tasks depending on the training received. With proper training, ANN are capable of generalization, the ability to recognize similarities among different input patterns, especially patterns that have been corrupted by noise.

What Are Neural Nets?
The term “Neural Net” refers to both the biological and artificial variants, although typically the term is used to refer to artificial systems only. Mathematically, neural nets are nonlinear. Each layer represents a non-linear combination of non-linear functions from the previous layer. Each neuron is a multiple-input, multiple-output (MIMO) system that receives signals from the inputs, produces a resultant signal, and transmits that signal to all outputs. Practically, neurons in an ANN are arranged into layers. The first layer that interacts with the environment to receive input is known as the input layer. The final layer that interacts with the output to present the processed data is known as the output layer. Layers between the input and the output layer that do not have any interaction with the environment are known as hidden layers. Increasing the complexity of an ANN, and thus its computational capacity, requires the addition of more hidden layers, and more neurons per layer.

Biological neurons are connected in very complicated networks. Some regions of the human brain such as the cerebellum are composed of very regular patterns of neurons. Other regions of the brain, such as the cerebrum have less regular arrangements. A typical biological neural system has millions or billions of cells, each with thousands of interconnections with other neurons. Current artificial systems cannot achieve this level of complexity, and so cannot be used to reproduce the behavior of biological systems exactly.

Processing Elements
In an artificial neural network, neurons can take many forms and are typically referred to as Processing Elements (PE) to differentiate them from the biological equivalents. The PE are connected into a particular network pattern, with different patterns serving different functional purposes. Unlike biological neurons with chemical interconnections, the PE in artificial systems are electrical only, and may be either analog, digital, or a hybrid. However, to reproduce the effect of the synapse, the connections between PE are assigned multiplicative weights, which can be calibrated or “trained” to produce the proper system output.

McCulloch-Pitts Model
Processing Elements are typically defined in terms of two equations that represent the McCulloch-Pitts model of a neuron:

 * $$\zeta = \sum_i w_ix_i$$
 * $$y = \sigma(\zeta)$$

Where ζ is the weighted sum of the inputs (the inner product of the input vector and the tap-weight vector), and σ(ζ) is a function of the weighted sum. If we recognize that the weight and input elements form vectors w and x, the ζ weighted sum becomes a simple dot product:

 * $$\zeta = \mathbf{w} \cdot \mathbf {x}$$

This may be called either the activation function (in the case of a threshold comparison) or a transfer function. The image to the right shows this relationship diagrammatically. The dotted line in the center of the neuron represents the division between the calculation of the input sum using the weight vector, and the calculation of the output value using the activation function. In an actual artificial neuron, this division may not be made explicitly.

The inputs to the network, x, come from an input space and the system outputs are part of the output space. For some networks, the output space Y may be as simple as {0, 1}, or it may be a complex multi-dimensional space. Neural networks tend to have one input per degree of freedom in the input space, and one output per degree of freedom in the output space.

The tap weight vector is updated during training by various algorithms. One of the more popular of which is the backpropagation algorithm which we will discuss in more detail later.

Why Use Neural Nets?
Artificial neural nets have a number of properties that make them an attractive alternative to traditional problem-solving techniques. The two main alternatives to using neural nets are to develop an algorithmic solution, and to use an expert system.

Algorithmic methods arise when there is sufficient information about the data and the underlying theory. By understanding the data and the theoretical relationship between the data, we can directly calculate unknown solutions from the problem space. Ordinary von Neumann computers can be used to calculate these relationships quickly and efficiently from a numerical algorithm.

Expert systems, by contrast, are used in situations where there is insufficient data and theoretical background to create any kind of a reliable problem model. In these cases, the knowledge and rationale of human experts is codified into an expert system. Expert systems emulate the deduction processes of a human expert, by collecting information and traversing the solution space in a directed manner. Expert systems are typically able to perform very well in the absence of an accurate problem model and complete data. However, where sufficient data or an algorithmic solution is available, expert systems are a less than ideal choice.

Artificial neural nets are useful for situations where there is an abundance of data, but little underlying theory. The data, which typically arises through extensive experimentation may be non-linear, non-stationary, or chaotic, and so may not be easily modeled. Input-output spaces may be so complex that a reasonable traversal with an expert system is not a satisfactory option. Importantly, neural nets do not require any a priori assumptions about the problem space, not even information about statistical distribution. Though such assumptions are not required, it has been found that the addition of such a priori information as the statistical distribution of the input space can help to speed training. Many mathematical problem models tend to assume that data lies in a standard distribution pattern, such as Gaussian or Maxwell-Boltzmann distributions. Neural networks require no such assumption. During training, the neural network performs the necessary analytical work, which would require non-trivial effort on the part of the analyst if other methods were to be used.

Learning
Learning is a fundamental component to an intelligent system, although a precise definition of learning is hard to produce. In terms of an artificial neural network, learning typically happens during a specific training phase. Once the network has been trained, it enters a production phase where it produces results independently. Training can take on many different forms, using a combination of learning paradigms, learning rules, and learning algorithms. A system which has distinct learning and production phases is known as a static network. Networks which are able to continue learning during production use are known as dynamical systems.

A learning paradigm is supervised, unsupervised or a hybrid of the two, and reflects the method in which training data is presented to the neural network. A method that combines supervised and unsupervised training is known as a hybrid method. A learning rule is a model for the types of methods to be used to train the system, and also a goal for what types of results are to be produced. The learning algorithm is the specific mathematical method that is used to update the inter-neuronal synaptic weights during each training iteration. Under each learning rule, there are a variety of possible learning algorithms for use. Most algorithms can only be used with a single learning rule. Learning rules and learning algorithms can typically be used with either supervised or unsupervised learning paradigms, however, and each will produce a different effect.

Overtraining is a problem that arises when too many training examples are provided, and the system becomes incapable of useful generalization. This can also occur when there are too many neurons in the network and the capacity for computation exceeds the dimensionality of the input space. During training, care must be taken not to provide too many input examples and different numbers of training examples could produce very different results in the quality and robustness of the network.

Network Parameters
There are a number of different parameters that must be decided upon when designing a neural network. Among these parameters are the number of layers, the number of neurons per layer, the number of training iterations, et cetera. Some of the more important parameters in terms of training and network capacity are the number of hidden neurons, the learning rate and the momentum parameter.

Number of neurons in the hidden layer
Hidden neurons are the neurons that are neither in the input layer nor the output layer. These neurons are essentially hidden from view, and their number and organization can typically be treated as a black box to people who are interfacing with the system. Using additional layers of hidden neurons enables greater processing power and system flexibility. This additional flexibility comes at the cost of additional complexity in the training algorithm. Having too many hidden neurons is analogous to a system of equations with more equations than there are free variables: the system is over specified, and is incapable of generalization. Having too few hidden neurons, conversely, can prevent the system from properly fitting the input data, and reduces the robustness of the system.

Data type: Integer Domain: [1, ∞) Typical value: 8

Meaning: Number of neurons in the hidden layer (additional layer to the input and output layers, not connected externally).

Learning Rate
Data type: Real Domain: [0, 1] Typical value: 0.3

Meaning: Learning Rate. Training parameter that controls the size of weight and bias changes in learning of the training algorithm.

Momentum
Data type: Real Domain: [0, 1] Typical value: 0.9

Meaning: Momentum simply adds a fraction m of the previous weight update to the current one. The momentum parameter is used to prevent the system from converging to a local minimum or saddle point. A high momentum parameter can also help to increase the speed of convergence of the system. However, setting the momentum parameter too high can create a risk of overshooting the minimum, which can cause the system to become unstable. A momentum coefficient that is too low cannot reliably avoid local minima, and can also slow down the training of the system.

Training type
Data type: Integer Domain: [0, 1] Typical value: 1

Meaning: 0 = train by epoch, 1 = train by minimum error

Epoch
Data type: Integer Domain: [1, ∞] Typical value: 5000000

Meaning: Determines when training will stop once the number of iterations exceeds epochs. When training by minimum error, this represents the maximum number of iterations.

Minimum Error
Data type: Real Domain: [0, 0.5] Typical value: 0.01

Meaning: Minimum mean square error of the epoch. Square root of the sum of squared differences between the network targets and actual outputs divided by number of patterns (only for training by minimum error).

Examples and realizations

 * 1) Repository on the github.


text:
Artificial Neural Networks/Biological Neural Networks

Biological Neural Nets
In the case of a biological neural net, neurons are living cells with axons and dendrites that form interconnections through electro-chemical synapses. Signals are transmitted through the cell body (soma), from the dendrite to the axon as an electrical impulse. In the pre-synaptic membrane of the axon, the electrical signal is converted into a chemical signal in the form of various neurotransmitters. These neurotransmitters, along with other chemicals present in the synapse form the message that is received by the post-synaptic membrane of the dendrite of the next cell, which in turn is converted to an electrical signal.

This page is going to provide a brief overview of biological neural networks, but the reader will have to find a better source for a more in-depth coverage of the subject.

Synapses

The figure above shows a model of the synapse showing the chemical messages of the synapse moving from the axon to the dendrite. Synapses are not simply a transmission medium for chemical signals, however. A synapse is capable of modifying itself based on the signal traffic that it receives. In this way, a synapse is able to “learn” from its past activity. This learning happens through the strengthening or weakening of the connection. External factors can also affect the chemical properties of the synapse, including body chemistry and medication.

Neurons
Cells have multiple dendrites, each receives a weighted input. Inputs are weighted by the strength of the synapse that the signal travels through. The total input to the cell is the sum of all such synaptic weighted inputs. Neurons utilize a threshold mechanism, so that signals below a certain threshold are ignored, but signals above the threshold cause the neuron to fire. Neurons follow an “all or nothing” firing scheme, and are similar in this respect to a digital component. Once a neuron has fired, a certain refraction period must pass before it can fire again.

Biological Networks
Biological neural systems are heterogeneous, in that there are many different types of cells with different characteristics. Biological systems are also characterized by macroscopic order, but nearly random interconnection on the microscopic layer. The random interconnection at the cellular level is rendered into a computational tool by the learning process of the synapse, and the formation of new synapses between nearby neurons.


text:
Artificial Neural Networks/History

Early History
The history of neural networking arguably started in the late 1800s with scientific attempts to study the workings of the human brain. In 1890, William James published the first work about brain activity patterns. In 1943, McCulloch and Pitts produced a model of the neuron that is still used today in artificial neural networking. This model is broken into two parts: a summation over weighted inputs and an output function of the sum.

Artificial Neural Networking
In 1949, Donald Hebb published The Organization of Behavior, which outlined a law for synaptic neuron learning. This law, later known as Hebbian Learning in honor of Donald Hebb is one of the simplest and most straight-forward learning rules for artificial neural networks.

In 1951, Marvin Minsky created the first ANN while working at Princeton.

In 1958 The Computer and the Brain was published posthumously, a year after John von Neumann’s death. In that book, von Neumann proposed many radical changes to the way in which researchers had been modeling the brain.

Mark I Perceptron
The Mark I Perceptron was also created in 1958, at Cornell University by Frank Rosenblatt. The Perceptron was an attempt to use neural network techniques for character recognition. The Mark I Perceptron was a linear system, and was useful for solving problems where the input classes were linearly separable in the input space. In 1960, Rosenblatt published the book Principles of Neurodynamics, containing much of his research and ideas about modeling the brain.

The Perceptron was a linear system with a simple input-output relationship defined as a McCulloch-Pitts neuron with a step activation function. In this model, the weighted inputs were compared to a threshold θ. The output, y, was defined as a simple step function:

 * $$y = \left\{ \begin{matrix}1 & \mbox{ if } \zeta \ge \theta \\ 0 & \mbox{ if } \zeta < \theta\end{matrix}\right.$$

Despite the early success of the Perceptron and artificial neural network research, there were many people who felt that there was limited promise in these techniques. Among these were Marvin Minsky and Seymore Papert, whose 1969 book Perceptrons was used to discredit ANN research and focus attention on the apparent limitations of ANN work. One of the limitations that Minsky and Papert pointed out most clearly was the fact that the Perceptron was not able to classify patterns that are not linearly separable in the input space. Below, the figure on the left shows an input space with a linearly separable classification problem. The figure on the right, in contrast, shows an input space where the classifications are not linearly separable.

Despite the failure of the Mark I Perceptron to handle non-linearly separable data, it was not an inherent failure of the technology, but a matter of scale. The Mark I was a two layer Perceptron, Hecht-Nielsen showed in 1990 that a three layer machine (multi layer Perceptron, or MLP) was capable of solving nonlinear separation problems. Perceptrons ushered in what some call the “quiet years”, where ANN research was at a minimum of interest. It wasn’t until the rediscovery of the backpropagation algorithm in 1986 that the field gained widespread interest again.

Backpropagation and Rebirth
The backpropagation algorithm, originally discovered by Werbos in 1974 was rediscovered in 1986 with the book Learning Internal Representation by Error Propagation by Rumelhart, Hinton and Williams. Backpropagation is a form of the gradient descent algorithm used with artificial neural networks for minimization and curve-fitting.

In 1987 the IEEE annual international ANN conference was started for ANN researchers. In 1987 the International Neural Network Society (INNS) was formed, along with the INNS Neural Networking journal in 1988.

title: Artificial Neural Networks/History


text:
Artificial Neural Networks/Feed-Forward Networks

Feedforward Systems
Feed-forward neural networks are the simplest form of ANN. Shown below, a feed-forward neural net contains only forward paths. A Multilayer Perceptron (MLP) is an example of feed-forward neural networks. The following figure below show a feed-forward networks with two hidden layers.

Figure: Feed-forward networks with two hidden layers

Connection Weights
In a feed-forward system PE are arranged into distinct layers with each layer receiving input from the previous layer and outputting to the next layer. There is no feedback. This means that signals from one layer are not transmitted to a previous layer. This can be stated mathematically as:

 * $$w_{ij} = 0 \mbox{ if } i = j$$
 * $$w_{ij} = 0 \mbox{ if } layer(i) \le layer(j)$$

Weights of direct feedback paths, from a neuron to itself, are zero. Weights from a neuron to a neuron in a previous layer are also zero. Notice that weights for the forward paths may also be zero depending on the specific network architecture, but they do not need to be. A network without all possible forward paths is known as a sparsely connected network, or a non-fully connected network. The percentage of available connections that are utilized is known as the connectivity of the network.

Mathematical Relationships
The weights from each neuron in layer l - 1 to the neurons in layer l are arranged into a matrix wl. Each column corresponds to a neuron in l - 1, and each row corresponds to a neuron in l. The input signal from l - 1 to l is the vector xl. If ρl is a vector of activation functions [σ1 σ2 … σn] that acts on each row of input and bl is an arbitrary offset vector (for generalization) then the total output of layer l is given as:

 * $$\mathbf{y}_l = \rho_l(\mathbf{w}_l\mathbf{x}_l + \mathbf{b}_l)$$

Two layers of output can be calculated by substituting the output from the first layer into the input of the second layer:

 * $$\mathbf{y}_l = \rho_l(\mathbf{w}_l\rho_{l-1}(\mathbf{w}_{l-1}\mathbf{x}_{l-1} + \mathbf{b}_{l-1})+ \mathbf{b}_l)$$

This method can be continued to calculate the output of a network with an arbitrary number of layers. Notice that as the number of layers increases, so does the complexity of this calculation. Sufficiently large neural networks can quickly become too complex for direct mathematical analysis.


text:
Artificial Neural Networks/Radial Basis Function Networks

In a radial basis function (RBF) networks are neural nets with three layers. The first input layer feeds data to a hidden intermediate layer. The hidden layer processes the data and transports it to the output layer. Only the tap weights between the hidden layer and the output layer are modified during training. Each hidden layer neuron represents a basis function of the output space, with respect to a particular center in the input space. The activation function chosen is commonly a Gaussian kernel:

 * $$\sigma(\zeta) = e^{-\beta \zeta^2}$$

This kernel is centered at the point in the input space specified by the weight vector. The closer the input signal is to the current weight vector, the higher the output of the neuron will be. Radial basis function networks are used commonly in function approximation and series prediction.


text:
Artificial Neural Networks/Recurrent Networks

Recurrent Networks
In a recurrent network, the weight matrix for each layer l contains input weights from all other neurons in the network, not just neurons from the previous layer. The additional complexity from these feedback paths can have a number of advantages and disadvantages in the network.

Simple Recurrent Networks
Recurrent networks, in contrast to feed-forward networks, do have feedback elements that enable signals from one layer to be fed back to a previous layer. A basic recurrent network is shown in figure 6. A simple recurrent network is one with three layers, an input, an output, and a hidden layer. A set of additional context units are added to the input layer that receive input from the hidden layer neurons. The feedback paths from the hidden layer to the context units have a fixed weight of unity.

A fully recurrent network is one where every neuron receives input from all other neurons in the system. Such networks cannot be easily arranged into layers. A small subset of neurons receives external input, and another small subset produce system output. A recurrent network is known as symmetrical network if:

 * $$w_{ij} = w_{ji} \forall i, j$$

Elman Network
An Elman Network is a special case of a Simple Recurrent Network (SRN) with four layers: An input layer, an output layer, a hidden layer and a context layer. The context layer feeds the hidden layer at iteration N with a value computed from the output of the hidden layer at iteration N-1, providing a short memory effect. Elman networks are used, for example, for predicting series of values.


text:
Artificial Neural Networks/Hopfield Networks

Hopfield Networks
Hopfield networks are one of the oldest and simplest networks. Hopfield networks utilize a network energy function. The activation function of a binary Hopfield network is given by the signum function of a biased weighted sum:

 * $$y_i = \sgn(\zeta_i - \theta_i)$$

Hopfield networks are frequently binary-valued, although continuous variants do exist. Binary networks are useful for classification and clustering purposes.

Energy Function
The energy function for the network is given as:

 * $$E = -\frac{1}{2}\sum_i\sum_j w_{ij}y_i y_j$$

Here, the y parameters are the outputs of the ith and jth units. During training the network energy should decrease until it reaches a minimum. This minimum is known as the attractor of the network. As a Hopfield network progresses, the energy minimizes itself. This means that mathematical minimization or optimization problems can be solved automatically by the Hopfield network if that problem can be formulated in terms of the network energy.

Associative Memory
Hopfield networks can be used as an associative memory network for data storage purposes. Each attractor represents a different data value that is stored in the network, and a range of associated patterns can be used to retrieve the data pattern. The number of distinct patterns p that can be stored in such a network is given approximately as:

 * $$p_{max} \approx 0.15n$$

Where n is the number of neurons in the network.


text:
Artificial Neural Networks/Echo State Networks

Echo State Networks, part of the Reservoir Computing paradigm, are an efficient type of Recurrent Neural Network where the recurrent neurons only have a partial connectivity between themselves. Such networks where only a subset of all possible connections is made are known as sparsely connected networks. In Echo State Networks, only the weights from the hidden layer to the output layer may be altered during training.

Echo State Networks are useful for matching and reproducing specific input patterns. Since the only tap weights modified during training are the output layer tap weights, training is typically quick and computationally efficient in comparison to other multi-layer networks that are not sparsely connected.


text:
Artificial Neural Networks/ART Models

In adaptive resonance theory (ART) networks, an overabundance of neurons leads some neurons to be committed (active) and others to be uncommitted (inactive). The weight vector, also known as the prototype, is said to resonate with the input vector if the two are sufficiently similar. Weights are only updated if they are resonating in the current iteration. ART networks commit an uncommitted neuron when a new input pattern is detected that does not resonate with any of the existing committed neurons. ART networks are fully-connected networks, in that all possible connections are made between all nodes.

title: Artificial Neural Networks/ART Models


text:
Artificial Neural Networks/Committee of Machines

Artificial neural networks can have very different properties depending on how they are constructed and how they are trained. Even in the case where two networks are trained on the same set of input data, different training algorithms can produce different systems with different characteristics. By combining multiple ANN into a single system, a committee of machines is formed. The result of a committee system is a combination of the results of the various component systems. For instance, the most common answer among a discrete set of answers in the committee can be taken as the overall answer, or the average answer can be taken.

Committee of machines (COM) systems tend to be more robust then the individual component systems, but they can also lose some of the “expertise” of the individual systems when answers are averaged out.


text:
Artificial Neural Networks/Learning Paradigms

Learning Paradigms
There are three different learning paradigms that can be used to train a neural network. Supervised and unsupervised learning are the most common, with hybrid approaches between the two becoming increasingly common as well.

Supervised Learning
Supervised learning is a technique where the input and expected output of the system are provided, and the ANN is used to model the relationship between the two. Given an input set x, and a corresponding output set y, an optimal rule is to be determined such that:

 * $$y = f(x) + e$$

Here, e is an approximation error that needs to be minimized. The input values are provided to the network which produces a result. This result is compared to the desired result, and this error signal is used to update the network weight vectors. Supervised learning is useful when we want the network to reproduce the characteristics of a certain relationship

Unsupervised Learning
In unsupervised learning, the data and a cost function are provided that is a function of the system input and output. The ANN is trained to minimize the cost function by finding a suitable input-output relationship.

Given an input set x, and a cost function g(x, y) of the input and output sets, the goal is to minimize the cost function through a proper selection of f (the relationship between x, and y). At each training iteration, the trainer provides the input to the network, and the network produces a result. This result is put into the cost function, and the total cost is used to update the weights. Weights are continually updated until the system output produces a minimal cost. Unsupervised learning is useful in situations where a cost function is known, but a data set is not know that minimizes that cost function over a particular input space.


text:
Artificial Neural Networks/Error-Correction Learning

Error-Correction Learning
Error-Correction Learning, used with supervised learning, is the technique of comparing the system output to the desired output value, and using that error to direct the training. In the most direct route, the error values can be used to directly adjust the tap weights, using an algorithm such as the backpropagation algorithm. If the system output is y, and the desired system output is known to be d, the error signal can be defined as:

 * $$e = d - y$$

Error correction learning algorithms attempt to minimize this error signal at each training iteration. The most popular learning algorithm for use with error-correction learning is the backpropagation algorithm, discussed below.

Gradient Descent
The gradient descent algorithm is not specifically an ANN learning algorithm. It has a large variety of uses in various fields of science, engineering, and mathematics. However, we need to discuss the gradient descent algorithm in order to fully understand the backpropagation algorithm. The gradient descent algorithm is used to minimize an error function g(y), through the manipulation of a weight vector w. The cost function should be a linear combination of the weight vector and an input vector x. The algorithm is:

 * $$w_{ij}[n+1] = w_{ij}[n] + \eta g(w_{ij}[n])$$

Here, η is known as the step-size parameter, and affects the rate of convergence of the algorithm. If the step size is too small, the algorithm will take a long time to converge. If the step size is too large the algorithm might oscillate or diverge.

The gradient descent algorithm works by taking the gradient of the weight space to find the path of steepest descent. By following the path of steepest descent at each iteration, we will either find a minimum, or the algorithm could diverge if the weight space is infinitely decreasing. When a minimum is found, there is no guarantee that it is a global minimum, however.

Backpropagation
The backpropagation algorithm, in combination with a supervised error-correction learning rule, is one of the most popular and robust tools in the training of artificial neural networks. Back propagation passes error signals backwards through the network during training to update the weights of the network. Because of this dependence on bidirectional data flow during training, backpropagation is not a plausible reproduction of biological learning mechanisms. When talking about backpropagation, it is useful to define the term interlayer to be a layer of neurons, and the corresponding input tap weights to that layer. We use a superscript to denote a specific interlayer, and a subscript to denote the specific neuron from within that layer. For instance:

 * $$\zeta^l_j = \sum_{i=1}^{N^{l-1}} w^l_{ij}x^{l-1}_i$$ (1)
 * $$x_j^l = \sigma(\zeta^l_j)$$ (2)

Where xil-1 are the outputs from the previous interlayer (the inputs to the current interlayer), wijl is the tap weight from the i input from the previous interlayer to the j element of the current interlayer. Nl-1 is the total number of neurons in the previous interlayer.

The backpropagation algorithm specifies that the tap weights of the network are updated iteratively during training to approach the minimum of the error function. This is done through the following equation:

 * $$w_{ij}^l [n] = w_{ij}^l [n-1]+\delta w_{ij}^l [n]$$ (3)
 * $$w_{ij}^{l-1} [n] = \eta \delta_j^l x_i^{l-1}[n] + \mu \Delta w_{ij}^l[n-1]$$ (3)

The relationship between this algorithm and the gradient descent algorithm should be immediately apparent. Here, η is known as the learning rate, not the step-size, because it affects the speed at which the system learns (converges). The parameter μ is known as the momentum parameter. The momentum parameter forces the search to take into account its movement from the previous iteration. By doing so, the system will tend to avoid local minima or saddle points, and approach the global minimum. We will discuss these terms in greater detail in the next section.

The parameter δ is what makes this algorithm a “back propagation” algorithm. We calculate it as follows:

 * $$\delta_j^l = \frac{dx_j^l}{dt}\sum_{k=1}^r \delta_k^{l+1} w_{kj}^{l+1}$$ (4)

The δ function for each layer depends on the δ from the previous layer. For the special case of the output layer (the highest layer), we use this equation instead:

 * $$\delta_j^l = \frac{dx_j^l}{dt}(x_j^l - y_j)$$ (5)

In this way, the signals propagate backwards through the system from the output layer to the input layer. This is why the algorithm is called the backpropagation algorithm.

Log-Sigmoid Backpropagation
If we use log-sigmoid activation functions for our neurons, the derivatives simplify, and our backpropagation algorithm becomes:

 * $$\delta_j^l = x_j^l(1 - x_j^l) (x_j^l - y_j)$$

For the output layer, and

 * $$\delta_j^l = x_j^l(1 - x_j^l) \sum_{k=1}^r \delta_k^{l+1} w_{kj}^{l+1}$$

for all the hidden inner layers. This property makes the sigmoid function desirable for systems with a limited ability to calculate derivatives.

Learning Rate
The learning rate is a common parameter in many of the learning algorithms, and affects the speed at which the ANN arrives at the minimum solution. In backpropagation, the learning rate is analogous to the step-size parameter from the gradient-descent algorithm. If the step-size is too high, the system will either oscillate about the true solution, or it will diverge completely. If the step-size is too low, the system will take a long time to converge on the final solution.

Momentum Parameter
The momentum parameter is used to prevent the system from converging to a local minimum or saddle point. A high momentum parameter can also help to increase the speed of convergence of the system. However, setting the momentum parameter too high can create a risk of overshooting the minimum, which can cause the system to become unstable. A momentum coefficient that is too low cannot reliably avoid local minima, and also can slow the training of the system.


text:
Artificial Neural Networks/Competitive Learning

Competitive Learning
Competitive learning is a rule based on the idea that only one neuron from a given iteration in a given layer will fire at a time. Weights are adjusted such that only one neuron in a layer, for instance the output layer, fires. Competitive learning is useful for classification of input patterns into a discrete set of output classes. The “winner” of each iteration, element i*, is the element whose total weighted input is the largest. Using this notation, one example of a competitive learning rule can be defined mathematically as:

 * $$w_{ij}[n+1] = w_{ij}[n] + \Delta w_{ij}[n]$$
 * $$\Delta w_{ij} = \left\{ \begin{matrix} \eta (x_i - w_{ij}) & \mbox{ if } i = j \\ 0 & \mbox{ otherwise}\end{matrix}\right.$$

Linear Vector Quantization
In a learning vector quantization (LVQ) machine, the input values are compared to the weight vector of each neuron. Neurons who most closely match the input are known as the best match unit (BMU) of the system. The weight vector of the BMU and those of nearby neurons are adjusted to be closer to the input vector by a certain step size. Neurons become trained to be individual feature detectors, and a combination of feature detectors can be used to identify large classes of features from the input space. The LVQ algorithm is a simplified precursor to more advanced learning algorithms, such as the self-organizing map. LVQ training is a type of competitive learning rule.


text:
Artificial Neural Networks/Hebbian Learning

Hebbian Learning
Hebbian learning is one of the oldest learning algorithms, and is based in large part on the dynamics of biological systems. A synapse between two neurons is strengthened when the neurons on either side of the synapse (input and output) have highly correlated outputs. In essence, when an input neuron fires, if it frequently leads to the firing of the output neuron, the synapse is strengthened. Following the analogy to an artificial system, the tap weight is increased with high correlation between two sequential neurons.

Mathematical Formulation
Mathematically, we can describe Hebbian learning as:

 * $$w_{ij}[n+1] = w_{ij}[n] + \eta x_i[n]x_j[n]$$

Here, η is a learning rate coefficient, and $$x_i[n]$$ and $$x_j[n]$$ are, respectively, the outputs of the ith and jth elements at time step n.

Plausibility
The Hebbian learning algorithm is performed locally, and doesn’t take into account the overall system input-output characteristic. This makes it a plausible theory for biological learning methods, and also makes Hebbian learning processes ideal in VLSI hardware implementations where local signals are easier to obtain.


text:
Artificial Neural Networks/Boltzmann Learning

Boltzmann learning is statistical in nature, and is derived from the field of thermodynamics. It is similar to error-correction learning and is used during supervised training. In this algorithm, the state of each individual neuron, in addition to the system output, are taken into account. In this respect, the Boltzmann learning rule is significantly slower than the error-correction learning rule. Neural networks that use Boltzmann learning are called Boltzmann machines.

Boltzmann learning is similar to an error-correction learning rule, in that an error signal is used to train the system in each iteration. However, instead of a direct difference between the result value and the desired value, we take the difference between the probability distributions of the system.


text:
Artificial Neural Networks/ART Learning

ART Learning
Adaptive Resonance Theory (ART) learning algorithms compare the weight vector, known as the prototype, to the current input vector to produce a distance, r. The distance is compared to a specified scalar, the vigilance parameter p. All output nodes start off in the uncommitted state. When a new input sequence is detected that does not resonate with any committed nodes, an uncommitted node is committed, and it’s prototype vector is set to the current input vector.

title: Artificial Neural Networks/ART Learning


text:
Artificial Neural Networks/Self-Organizing Maps

Self-Organizing Maps
Self-organizing maps (SOM), sometimes called Kohonen SOM after their creator, are used with unsupervised learning. SOM are modeled on biological neural networks, where groups of neurons appear to self organize into specific regions with common functionality.

Neuron Regions
Different regions of the SOM network are trained to be detectors for distinct features from the input set. Initial network weights are either set randomly, or are based off the eigenvectors of the input space. The Euclidean distance from each input sample to the weight vector of each neuron is computed, and the neuron whose weight vector is most similar to the input is declared the best match unit (BMU). The update formula is given as:

 * $$w_j[n+1] = w_j[n] + \Theta[j, n] \alpha[n](x[n] - w_j[n])$$

Here, w is the weight vector at time n. α is a monotonically decreasing function that ensures the learning rate will decrease over time. x is the input vector, and Θ[j, n] is a measure of the distance between the BMU and neuron j at iteration n. As can be seen from this algorithm, the amount by which the neuron weight vectors change is based on the distance from the BMU, and the amount of time. Decreasing the potential for change over time helps to reduce volatility during training, and helps to ensure that the network converges.


text:
Artificial Neural Networks/Boltzmann Machines

Boltzmann learning compares the input data distribution P with the output data distribution of the machine, Q [24]. The distance between these distributions is given by the Kullback-Leibler distance:

 * $$w_{ij}[n+1] = w_{ij}[n] - \frac{\partial G}{\partial w_{ij}}$$

Where:

 * $$\frac{\partial G}{\partial w_{ij}} = -\frac{1}{T}[p_{ij} - q_{ij}]$$

Here, pij is the probability that elements i and j will both be on when the system is in its training phase (positive phase), and qij is the probability that both elements i and j will be on during the production phase (negative phase). The probability that element j will be on, pi, is given by:

 * $$p_i = \frac{1}{1 + e^\frac{-\Delta E_i}{T}}$$

T is a scalar constant known as the temperature of the system. Boltzmann learning is very powerful, but the complexity of the algorithm increases exponentially as more neurons are added to the network. To reduce this effect, a restricted Boltzmann machine (RBM) can be used. The hidden nodes in an RBM are not interconnected as they are in regular Boltzmann networks. Once trained on a particular feature set, these RBM can be combined together into larger, more diverse machines.

Because Boltzmann machine weight updates only require looking at the expected distributions of surrounding neurons, it is a plausible model for how actual biological neural networks learn.


text:
Artificial Neural Networks/Activation Functions

Activation Functions
There are a number of common activation functions in use with neural networks. This is not an exhaustive list.

Step Function
A step function is a function like that used by the original Perceptron. The output is a certain value, A1, if the input sum is above a certain threshold and A0 if the input sum is below a certain threshold. The values used by the Perceptron were A1 = 1 and A0 = 0.

These kinds of step activation functions are useful for binary classification schemes. In other words, when we want to classify an input pattern into one of two groups, we can use a binary classifier with a step activation function. Another use for this would be to create a set of small feature identifiers. Each identifier would be a small network that would output a 1 if a particular input feature is present, and a 0 otherwise. Combining multiple feature detectors into a single network would allow a very complicated clustering or classification problem to be solved.

Linear combination
A linear combination is where the weighted sum input of the neuron plus a linearly dependent bias becomes the system output. Specifically:

 * $$y = \zeta + b$$

In these cases, the sign of the output is considered to be equivalent to the 1 or 0 of the step function systems, which enables the two methods be to equivalent if

 * $$\theta = -b$$.

Continuous Log-Sigmoid Function
A log-sigmoid function, also known as a logistic function, is given by the relationship:

 * $$\sigma(t) = \frac{1}{1 + e^{-\beta t}}$$

Where β is a slope parameter. This is called the log-sigmoid because a sigmoid can also be constructed using the hyperbolic tangent function instead of this relation, in which case it would be called a tan-sigmoid. Here, we will refer to the log-sigmoid as simply “sigmoid”. The sigmoid has the property of being similar to the step function, but with the addition of a region of uncertainty. Sigmoid functions in this respect are very similar to the input-output relationships of biological neurons, although not exactly the same. Below is the graph of a sigmoid function.

Sigmoid functions are also prized because their derivatives are easy to calculate, which is helpful for calculating the weight updates in certain training algorithms. The derivative when $$\beta=1$$ is given by:

 * $$\frac{d\sigma(t)}{dt} = \sigma(t)[1 - \sigma(t)]$$

When $$\beta\ne1$$, using $$\sigma(\beta,t) = \frac{1}{1 + e^{-\beta t}}$$, the derivative is given by:

 * $$\frac{d\sigma(\beta,t)}{dt} = \beta[\sigma(\beta,t)[1 - \sigma(\beta,t)]]$$

Continuous Tan-Sigmoid Function

 * $$\sigma(t) = \tanh(t) = \frac{e^{t}-e^{-t}}{e^{t}+e^{-t}}$$

Its derivative is:
 * $$\frac{\operatorname{d}\sigma(t)}{\operatorname{d}t} = 1-\tanh^{2}(t) = \operatorname{sech}^{2}(t) = 1-\frac{(e^{t}-e^{-t})^{2}}{(e^{t}+e^{-t})^{2}}$$

ReLU: Rectified Linear Unit
The ReLU activation function is defined as:

$$\sigma(t) = \max(0, t)$$

It can also be expressed piecewise:

$$\sigma(t) = \begin{cases} 0 & \text{if } t < 0\\ t & \text{if } t > 0 \end{cases}$$

Its derivative is:

$$\sigma(t) = \begin{cases} 0 & \text{if } t < 0\\ 1 & \text{if } t > 0 \end{cases} $$

ReLU was first introduced by Kunihiko Fukushima in 1969 for representation learning with images, and is now the most popular activation function for deep learning. It has a very simple formula and derivative, making backpropagation calculations easier to calculate. As the output of the derivative function is always 1 for positive inputs, it suffers less from the gradient vanishing or exploding problem than other activation functions. ReLU does present the unique problem of "dying ReLU", where some nodes may very rarely activate or "die", providing no predictive power to the model. Variants of ReLU including Leaky and Parametric ReLU have been introduced to avoid the dying ReLU problem.

Softmax Function
The softmax activation function is useful predominantly in the output layer of a clustering system. Softmax functions convert a raw value into a posterior probability. This provides a measure of certainty. The softmax activation function is given as:

 * $$y_i = \frac{e^{\zeta_i}}{\sum_{j\in L} e^{\zeta_j} }$$

L is the set of neurons in the output layer.


text:
Artificial Neural Networks/MATLAB Neural Networking Toolbox

MATLAB
MATLAB ® is an ideal tool for working with artificial neural networks for a number of reasons. First, MATLAB is highly efficient in performing vector and matrix calculations. Second, MATLAB comes with a specialized Neural Network Toolbox ® which contains a number of useful tools for working with artificial neural networks.

This book is going to utilize the MATLAB programming environment and the Neural Network Toolbox to do examples and problems throughout the book.

Other Neural Network Software
Even though this book is going to focus on MATLAB for its problems and examples, there are a number of other tools that can be used for constructing, testing, and implementing neural networks. E.g.

 * GNU Octave.
 * pybrain
 * NEST


text:
Artificial Neural Networks/Pattern Recognition

Artificial neural networks are useful for pattern matching applications. Pattern matching consists of the ability to identify the class of input signals or patterns. Pattern matching ANN are typically trained using supervised learning techniques. One application where artificial neural nets have been applied extensively is optical character recognition (OCR). OCR has been a very successful area of research involving artificial neural networks.

An example of a pattern matching neural network is that used by VISA for identifying suspicious transactions and fraudulent purchases. When input symbols do not match an accepted pattern, the system raises a warning flag that indicates a potential problem.


text:
Artificial Neural Networks/Clustering

Similar to pattern matching, clustering is the ability to associate similar input patterns together, based on a measurement of their similarity or dissimilarity. An example of a clustering problem is the Netflix Prize, a competition to improve the Netflix recommendation system. Competitors are attempting to produce software systems that can suggest movies based on movies the viewer has already rated. Essentially, clustering movies based on the predicted like or dislike of the viewer.

title: Artificial Neural Networks/Clustering


text:
Artificial Neural Networks/Feature Detection

Feature detection or “association” networks are trained using non-noisy data, in order to recognize similar patterns in noisy or incomplete data. Correctly detecting features in the presence of noise can be used as an important tool for noise reduction and filtering.

For example, neural nets have been used successfully in a number of medical applications for detection of disease. One particular example is the use of ANN to detect breast cancer in mammography images.


text:
Artificial Neural Networks/Curve Fitting

Curve Fitting
Curve-fitting problems represent an attempt for the neural network to identify and approximate an arbitrary input-output relation. Once the relation has been modeled to the necessary accuracy by the network, it can be used for a variety of tasks, such as series prediction, function approximation, and function optimization.

Function Approximation
Function approximation or modeling is the act of training a neural network using a given set of input-output data (typically through supervised learning) in order to deduce the relationship between the input and the output. After training, such an ANN can be used as a black box with an input-output characteristic approximately equal to the relationship of the training problems. Because of the modular and non-linear nature of artificial neural nets, they are considered to be able to approximate any arbitrary function to an arbitrary degree of accuracy. More accuracy in this case represents a tradeoff with system complexity, and the ability to generalize.


text:
Artificial Neural Networks/Series Prediction

ANN can be trained to match the statistical properties of a particular input signal, and can even be used to predict future values of time series. There are a number of applications of feature prediction that have received significant research attention. Among these are financial prediction, meteorological prediction, and electric load prediction. ANN have shown themselves to be very robust in predicting complicated series, including non-linear non-stationary chaotic systems.

Financial prediction is useful for anticipating events in the economic system, which is considered to be a chaotic system. ANN have been used to predict the performance and failure rate of companies, changes in the exchange rate, and other economic metrics.

Meteorological prediction is a difficult process because current atmospheric models rely on highly recursive sets of differential equations which can be difficult to calculate, and which propagate errors through the successive iterations. Using neural nets for meteorological prediction is a non-trivial task, and there have been many conflicting reports of the efficacy of the technique.


text:
Artificial Neural Networks/Optimization

Optimization ANNs are concerned with the minimization of a particular cost function with respect to certain constraints. ANN are shown to be capable of highly efficient optimization.

title: Artificial Neural Networks/Optimization


text:
User talk:209.174.29.15

-- Herby talk thyme 19:37, 29 October 2007 (UTC)

title: User talk:209.174.29.15


text:
Cookbook:Chicken Fajitas

| Tex-Mex cuisine

Ingredients

 * 1 Tbsp olive oil
 * 1–2 lb (½–1 kg) skinless, boneless chicken breast
 * 1 red bell pepper
 * 1 green bell pepper
 * ½ yellow onion
 * 1 tsp ground cumin (or to taste)
 * 1 Tbsp crushed red pepper (or to taste)

Procedure

 * 1) Cut chicken into roughly 1x1-inch (2.5 x 2.5 cm) pieces.
 * 2) Marinate the chicken with a spoonful of turmeric, salt and black pepper.
 * 3) Cook chicken in olive oil, cumin and red pepper over high heat in a large skillet or wok.
 * 4) While meat is cooking, chop/julienne vegetables peppers and onion).
 * 5) When meat is completely cooked, add vegetables, stirring frequently so that all are evenly cooked.
 * 6) Remove from heat when vegetables are soft.
 * 7) Serve with tortilla shells and toppings such as cheese, sour cream, lettuce, tomato, onions, and/or guacamole.

title: Cookbook:Chicken Fajitas


text: User talk:218.186.8.10

title: User talk:218.186.8.10


text:
User talk:Hayesr

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.– Mike.lifeguard  | talk 21:35, 29 October 2007 (UTC)

title: User talk:Hayesr


text:
User:Xania/monobook.css

table.fundraiser-box {display : none; }

title: User:Xania/monobook.css


text:
User talk:216.160.160.116

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.

Thank you for explaining the op-amp single-formula technique at Circuit Idea/Simple Op-amp Summer Design.

--DavidCary 23:02, 29 October 2007 (UTC)

title: User talk:216.160.160.116


text:
User talk:24.247.90.41

Urbane  (Talk)   (Contributions)  23:15, 29 October 2007 (UTC)

title: User talk:24.247.90.41


text:
User talk:70.49.87.159

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Urbane   (Talk)   (Contributions)  23:18, 29 October 2007 (UTC)

title: User talk:70.49.87.159


text:
User talk:68.154.197.164

Thank you for reverting the vandalism on the 8th Grade Science/Section 2: Scientific Problem Solving page. It is greatly appreciated. Urbane  (Talk)   (Contributions)  23:21, 29 October 2007 (UTC)

title: User talk:68.154.197.164


text: User:Ravichandar84/On the road

title: User:Ravichandar84/On the road


text:
User talk:24.185.38.210

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Urbane   (Talk)   (Contributions)  00:25, 30 October 2007 (UTC)

title: User talk:24.185.38.210


text:
User talk:24.174.195.237

Thanks for experimenting with Wikibooks. Your test worked, and has now been removed. Please use Sandbox for any other tests you want to do, since testing material in modules will normally be removed quickly. Please see the welcome page if you would like to learn more about contributing to our project. Thanks. -- Urbane   (Talk)   (Contributions)  00:28, 30 October 2007 (UTC)

title: User talk:24.174.195.237


text:
Understanding Darcs/Dealing with conflicts

Resolving conflicts
For the moment, the best place to go for dealing with conflicts is the conflicts FAQ on darcs wiki.


text: User talk:Thebfg2007

title: User talk:Thebfg2007


text:
Talk:Piano/Scales

Scales and technique
'Do you ask me how good a player you may become? Then tell me how much you practice the scales.' - Carl Czerny

The statements that "scales aren't so superb for technique" and that "practicing the scales is good for technique - but mainly for playing the scales" are patently false. I am changing the article. Please contact me if you object or have a different viewpoint.

title: Talk:Piano/Scales


text: User talk:24.177.122.184

title: User talk:24.177.122.184


text: User talk:130.184.83.233

title: User talk:130.184.83.233


text:
New Zealand History/World War I

‹ Back to The Dawn of the Twentieth Century

New Zealand's Involvement in World War I
After the United Kingdom declared war on Germany in 1914, New Zealand followed without second thoughts. New Zealand only had a small population of just over one million at the time, and was fairly isolated from the rest of the world, but due to New Zealand's strong ties to Britain it offered its services to the Allied Forces.

New Zealand's first act in the war was to seize German Samoa. New Zealand sent 1,413 men to conduct this action and took control of the territory without much resistance. It was the first German territory to be occupied in the name of King George V in the war.

On the 25th of April 1915, New Zealand sent troops to Gallipoli under the command of British General Alexander Godly, along with Australian soldiers, to help seize Constantinople. Due to a navigational error, however, they landed at the wrong point, and the steep cliffs in the cove they had disembarked at offered the Turkish defenders a significant vantage point. Advancement of the New Zealand-Australian forces was impossible. New Zealand suffered casualties of 2,721 dead and 4,852 wounded in the cove, and eventually it was decided to evacuate. The battle was the first great conflict of New Zealand, and the loss was felt greatly in New Zealand.

New Zealand forces also helped on the Western Front in France, in which, by the time they were relieved, had advanced three kilometres and taken eight kilometres of enemy front line. 1560 New Zealand men were killed, and 7048 were wounded.

Casualties

Out of the 103,000 men recruited, 16,697 New Zealanders serving in the war had been killed and 41,317 had been wounded; a 58 percent casualty rate. This was one of the highest rates per capita of any country involved in the war. Approximately a further thousand men died after the war, as a result of injuries suffered.

Forward to New Zealand in the Great Depression ›

title: New Zealand History/World War I


text:
Programming AI with Leaf/Development/Leaf and Linux

Leaf and Linux
Leaf has been built on windows XP and uses LispWorks as the Lisp development environment. This provides a readily available environment for anyone wanting to build a Leaf robot.

A number of people have expressed interest in trying to port Leaf to a Linux environment. These pages record the trials & tribulations of people’s attempts to do the port.

A message from one of the Linux crowd
A warning to anyone considering a Linux port. The Leaf project is windows based and is a very successful environment for this project. There are a number of features that have been implemented in windows that are going to be VERY challenging to get working under Linux.

Build a standard Leaf before you try to use linux. Please do not expect the Leaf developers to provide any support for Linux. They are doing an excellent job with windows and this is where they will continue to work.

If you are still keen remember you are on your own but not alone.

Major component
The Leaf environment has several major components that we will need to duplicate to be able to do a port.

1) Speech recognition

2) LISP compiler/interpreter environment

3) Text to Speech

4) Face Animation

5) Micro controller interface software (Nav & control program)

Because of the wide range of environments available in Linux it is important to understand what a person is using when they offer a contribution to the Linux port.

People have asked for recommendations on which distribution to use. The distribution is not as important as the kernel version and the version numbers of any software installed.

Contributors & systems

Name: Robin Hartley Hardware: HP Compaq F551AU Turion 64 X2 (I do NOT recommend purchasing this hardware*) Distro: Gentoo 2007.1 AMD64 bit. Kernel: * For Linux the dual core & support chips has poor linux compatability & the Broadcom WiFi has limited support. For Windows systems the machine comes with Vista basic and has sound compatability issues. HP does not supply drivers and the hardware is NOT compatable with a standard XP pro install.

Speech Recognition
This is likely to be the hardest part. Speech recognition work on Linux appears to have stopped around 2000 when the ViaVoice package was removed from the market.

The most likely candidate is one of the Sphinx versions. Experimentation is under way with both Sphinx 2 & Sphinx 3. So far recognition rates have been about 10%

Until this is greatly improved there is little point in progressing the other modules.

Lisp environment
The LISP package is Steel Bank Common LISP (SBCL)

The development environment is SLIME which uses the EMACS editor

LispWorks has a graphical interface, which is only used to set the options at start up, the McCLIM package has been used to get this functionality. McClim is a free software implementation of the CLIM standard

So far the options selection interface has been ported to Linux.

Text to Speech
There are several TTS engines in the public domain. Festival seems to be the most popular and is the same TTS engine used by Cuanimate.

No work has been done on this yet.

Face animation
X-Face looks a likely candidate for this part of the application.

No work yet.

Micro controller program
This is likely to be a re-write of the Nav & control program. This will be done using GCC compiler.

No work yet.


text:
Talk:ERP Internals/Industries

TIME - Telecommunications, IT, Media and Entertainment FMCG (retail) - Fast Moving and Consuming Goods Banking and insurance (Financial Services) Government Public Services Utilities

Or, "transactions" - "Services" (telco, banking, utilities ...)

Behind "manufacturing" are man times specialized (managed in a different way): Pharmaceuticals Beverages

Each one of them are differently process-managed (and thus ERP supported)

Erik.iglesias.abella@gmail.com

title: Talk:ERP Internals/Industries


text:
User talk:72.1.206.163

-- Herby talk thyme 12:59, 30 October 2007 (UTC)

-- Herby talk thyme 14:32, 30 October 2007 (UTC)

Unsurprisingly
You have been blocked from editing, please contribute constructively when the block ends -- Herby talk thyme 14:49, 30 October 2007 (UTC)

title: User talk:72.1.206.163


text: User talk:209.155.60.232

title: User talk:209.155.60.232


text:
Computer Programming/Hebrew calendar

Programming a Hebrew calendar application:

The intended audience for this summary of the mechanics of the Hebrew calendar is computer programmers who wish to design software that accurately computes dates in the Hebrew calendar. The following details may prove useful for validating such software. Note, however, that published Hebrew calendar algorithms are much simpler than the details listed below, and there is no need to employ tables in computer implementation of Hebrew calendar arithmetic. As usual, tables are useful shortcuts for humans carrying out the calculations manually.

 * 1) The Hebrew calendar is computed by lunations. One mean lunation is reckoned at 29 days, 12 hours, 44 minutes, 3⅓ seconds, or equivalently 765433 parts = 29 days, 13753 parts, where 1 minute = 18 parts (halakim plural, helek singular).
 * 2) A common year must be either 353, 354, or 355 days; a leap year must be 383, 384, or 385 days. A 353 or 383 day year is called haserah. A 354 or 384 day year is kesidrah. A 355 or 385 day year is shlemah.
 * 3) Leap years follow a 19 year schedule in which years 3, 6, 8, 11, 14, 17, and 19 are leap years. The Hebrew year 5758 (which starts in Gregorian year 1997) is the first year of a cycle.
 * 4) 19 years is the same as 235 lunations.
 * 5) The months are Tishrei, Cheshvan, Kislev, Tevet, Shevat, Adar, Nisan, Iyar, Sivan, Tammuz, Av, and Elul. In a leap year, Adar is replaced by Adar II (also called Adar Sheni or Veadar) and an extra month, Adar I (also called Adar Rishon), is inserted before Adar II.
 * 6) Each month has either 29 or 30 days. A 30 day month is full (מלא pronounced: maleh, maley, or malei), whereas a 29 day month is defective (חסר pronounced: ħaser or khaser).
 * 7) * Nisan, Sivan, Av, Tishrei, and Shevat are always full.
 * 8) * Iyar, Tammuz, Elul, Tevet, and Adar (Adar II in leap years) are always defective.
 * 9) * Adar I, added in leap years before Adar II, is full.
 * 10) * Cheshvan and Kislev vary. There are three possible combinations: both defective, both full, Cheshvan defective and Kislev full.
 * 11) Tishrei 1 (Rosh Hashana) is the day during which a molad (instant of the mean lunar conjunction) occurs unless that conflicts with certain postponements (dehiyyot plural; dehiyyah singular). Note that for calendar computations, the Jewish date begins at 6 pm or six fixed hours before midnight when the date changes in the Gregorian calendar, not at nightfall or sunset when the observed Hebrew date begins.
 * 12) * Postponement A is required whenever Tishrei 10 (Yom Kippur) would fall on a Friday or a Sunday, or if Tishrei 21 (7th day of Sukkot) would fall on a Saturday. This is equivalent to the molad being on Sunday, Wednesday, or Friday. Whenever this happens, Tishrei 1 is delayed by one day.
 * 13) * Postponement B is required whenever the molad occurs at or after noon. When this postponement exists, Tishrei 1 is delayed by one day. If this conflicts with postponement A then Tishrei 1 is delayed an additional day.
 * 14) * Postponement C: If the year is to be a common year and the molad falls on a Tuesday at or after 3:11:20 am (3 hours 204 parts), Tishrei 1 is delayed by two days—if it weren't delayed, the resulting year would be 356 days long.
 * 15) * Postponement D: If the new year follows a leap year and the molad is on a Monday at or after 9:32:43⅓ am (9 hours 589 parts), Tishrei 1 is delayed one day—if it weren't, the preceding year would have only 382 days.
 * 16) Postponements are implemented by adding a day to Kislev of the preceding year, making it full. If Kislev is already full, the day is added to Cheshvan of the preceding year, making it full also. If a delay of two days is called for, both Cheshvan and Kislev of the preceding year become full.
 * 17) A reference epoch in modern times is molad Tishrei for Hebrew year 5758, which is at 22:07:10 on Wednesday, 1 October 1997 (Gregorian), or equivalently midnight-referenced Julian day number 2450723 plus 23889 parts. This epoch also marks the beginning of a cycle. Note: Although the Julian day number begins at noon, it can be reckoned twelve hours earlier for programming purposes, which is what is meant here by the phrase, "midnight-referenced."

Calculation by use of partial weeks
There are a number or approaches that can be taken in calculating Hebrew dates. One that is widely documented uses partial weeks and a table of limits. This method relies on all postponements being defined in terms of a seven-day week. That means that whole weeks between the epoch and the molad of the current year can be eliminated, leaving only a partial week with a few days, hours and parts.

 * A nineteen-year cycle has 235 months of 29d 12h 793p each or 6939d 16h 595p. Eliminating 991 weeks leaves a partial week of 2d 16h 595p or 69715p.

 * A common year has 12 months of 29d 12h 793p each or 354d 8h 876p. Eliminating 50 weeks leaves a partial week of 4d 8h 876p or 113196p.

 * A leap year has 13 months of 29d 12h 793p or 383d 21h 589p. Eliminating 54 weeks leaves a partial week of 5d 21h 589p or 152869p.

Postponement B requiring a delay until the next day (beginning at 6 pm) if a molad occurs at or after noon effectively means that the week begins at noon Saturday for computational purposes.

Calculate the partial week between the molad of the desired Hebrew year and the preceding noon Saturday considering the partial week before molad Tishrei of AM 1 (or the first year of a more recent nineteen-year cycle) and the partial weeks from the intervening cycles and years within the current cycle, eliminating whole weeks via mod 181440, the number of parts in one week.

Thus molad Tishrei AM 1, which is 1d 5h 204p after 6 pm Saturday, is increased by 6 hours to 1d 11h 204p or 38004p. This is 5h 204p after the beginning (6 pm) of the second day of the week. In Western terms, this is 23:11:20 on Sunday (because it is before midnight), 6 October 3761 BCE in the proleptic Julian calendar. This date is midnight-referenced Julian day number 347997. Consulting the Table of Limits below, 1 Tishrei is the second day of the week, equivalent to the tabular Western day of Monday (same daylight period as the Hebrew day), which is 7 October 3761 BCE. This means no postponement was needed (both the molad Tishrei and 1 Tishrei were on the second day of the week).

Alternatively, the molad of a more recent Hebrew year may be selected as the epoch if it is the first year of a nineteen-year cycle, such as 5758 (used in rule 9), which is 303 nineteen-year cycles after molad Tishrei AM 1. Thus molad Tishrei 5758 is (38004 + 303×69715) mod 181440 = 114609 parts after noon Saturday, or 4d 10h 129p, which is 4h 129p after the beginning (6 pm) of the fifth day of the week. In Western terms, this is before midnight, which yields the date and time indicated in rule 9. Consulting the Table of Limits, 1 Tishrei is the fifth day of the week, or tabular Thursday 2 October 1997 (Gregorian), again no postponement was needed.

By applying the postponements to the moladot Tishrei at the beginning and end of any Hebrew year, a table of four gates (Hebrew: arba'ah sha'arim), which is also a table of limits, can be developed which uniquely identifies which of the fourteen types the year is (the day of the week of 1 Tishrei, the number of days in Cheshvan and Kislev, and whether common or leap (embolismic)). "Four gates" refers to the four allowable days of the week with which the year can begin. The first table of four gates was developed by Saadiah Gaon (892–942). In the following table, the years of a nineteen-year cycle are listed in the top row, organized into four groups: a common year after a leap year but before a common year (LCC, 1 4 9 12 15), a common year between two leap years (LCL, 7 18), a common year after a common year but before a leap year (CCL, 2 5 10 13 16), or a leap year between two common years (CLC, 3 6 8 11 14 17 19). The week since noon Saturday on the left is partitioned by a set of limits between which the molad Tishrei of the Hebrew year can be found. The resulting type of year in the body of the table indicates the day of the Hebrew week of 1 Tishrei (2, 3, 5, or 7), the four gates, and whether the year is deficient (−1), regular (0), or abundant (+1).

title: Computer Programming/Hebrew calendar


text:
User:Kaisershatner

See my page at WP. Kaisershatner 13:24, 30 October 2007 (UTC)

title: User:Kaisershatner


text:
Artificial Neural Networks/Competitive Models

Competitive Networks
Competitive networks are networks where neurons compete with one another. The weight vector is treated as a "prototype", and is matched against the input vector. The "winner" of each training session is the neuron whose weight vector is most similar to the input vector.

An example of a competitive network is shown below.


text:
AP Biology/LABORATORY 11. Animal Behavior

This is a lab performed by AP Biology students that is used to understand the movement or activity of an organism in response to a stimulus (kinesis) and the direction of the greatest rate of increase of a function under observation and its magnitude is the slope of the graph in a particularly meaningful direction (gradients). The gradiated factor is chosen by the student and is observed with relevance to pill bugs (or meal worms).


text:
User talk:Frdayeen

Please Use my English wikipedia talk page or  Bangla wikipedia talk page to leave me a message

title: User talk:Frdayeen


text:
Template:Calculus/Prerequisite

Prerequisite

title: Template:Calculus/Prerequisite


text: Template:Engineering Tables/Use

title: Template:Engineering Tables/Use


text:
User talk:Kaisershatner

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.– Mike.lifeguard  | talk 15:50, 30 October 2007 (UTC)
 * In the future, please request imports of Wikipedia content at WB:RFI so this can be done in compliance with the GFDL. I will history-merge add the edit history later. – Mike.lifeguard  | talk 15:50, 30 October 2007 (UTC)

title: User talk:Kaisershatner


text:
The Voynich Manuscript/Claude Martin

Claude Martin claims that the Voynich manuscript would contain a phoney text, based on numeric sequences transcribed into anagrams. For example, a line forged on sequence 345678 would be retranscribed into 643875. This « fill in text method » would be used to produce text having some semantic similarities with a real text.

For more details check his site: http://www.ms-voynich.com/introduction.html

title: The Voynich Manuscript/Claude Martin


text:
Talk:Animal Care/Cat

As funny as it may sound, it has nothing to do with wikipedia or -books for that matter.
 * I removed the section describing different methods foe cats to get high on weed.

-Markku

I does have to do, the sell toys with hallucinogens so that cats soothe themselves, it keeps the quiet.--Afa86 (talk) 00:55, 18 October 2009 (UTC)

). Some editing could help. Also merge/link with: http://en.wikipedia.org/wiki/Cat_play_and_toys
 * I agree with the merge. This is a very informative wiki (cat play/toys

 * I agree also, as a chapter.--Afa86 (talk) 00:53, 18 October 2009 (UTC)

meds and cats
can cats have tylenol? My cat has a cut on the side of his face, I put antiseptic gel on it after I cleaned it out!

title: Talk:Animal Care/Cat


text:
User:Whiteknight/Basic Prereq

 category :/Prerequisite

title: User:Whiteknight/Basic Prereq


text:
Radiation Oncology/Waldenstroms

Waldenstrom's Macroglobulinemia

 * B-cell proliferative disorder with bone marrow lymphoplasmacytic infiltration and production of monoclonal IgM; considered to be a low-grade lymphoma
 * Symptoms are usually due to blood hyperviscosity, anemia, and hepatosplenomegaly
 * Asymptomatic patients are typically observed
 * Alkylating agents (chlorambucil), nucleoside analogues, and rituximab (eg CHOP-R) are reasonable first line approaches
 * Symptomatic support as necessary
 * For RT treatment of splenomegaly, please see the Spleen chapter
 * Historically, hemibody and total body irradiation was used, but it has fallen out of favor with development of more modern chemotherapies

 * Italy; 2002 PMID 12187029 -- "Advanced waldenström's macroglobulinemia: a case of possible cure after systemic chemotherapy, splenic radiation and splenectomy." (Cavanna L, Acta Haematol. 2002;108(2):97-101.)
 * Case report. 41M with WM, massive splenomegaly. Treated with plasmaparesis, chemo (BCNU, cyclophosphamide, vincristine, melphalan, prednisone) x16 cycles, and splenic RT 36 Gy
 * Outcome: Persistent mild splenomegaly and small IgM band. He then underwent splenectomy, showing persistent disease. 2 months later, complete response, and has remained disease-free x18 years
 * Conclusion: Splenectomy should be considered; spleen RT 36 Gy with chemo didn't seem sufficient to fully eradicate the disease

 * Japan; 1997 PMID 9322902 -- "Durable remission after splenectomy for Waldenström's macroglobulinemia with massive splenomegaly in leukemic phase." (Takemori N, Leuk Lymphoma. 1997 Jul;26(3-4):387-93.)
 * Case report. Patient with WM and massive splenomegaly, refractory to chemo. Splenic RT effectively reduced serum IgM, spleen size, and lymphocyte counts, however, only transient
 * He then underwent splenectomy, and has been in remission for 6 years
 * Conclusion: Recommend splenectomy as part of management

 * 1977 PMID 405054 -- "Report of a case of Waldenström macroglobulinemia treated and controlled by radiotherapy." (Shehata WM, Blood. 1977 Jun;49(6):1023-4.)
 * Case report

 * Review; 2007 PMID 17634838 -- "Treatment of Waldenstrom's Macroglobulinemia." (Dimopoulos MA, Curr Treat Options Oncol. 2007 Apr;8(2):144-53.)

title: Radiation Oncology/Waldenstroms


text:
Template:Prerequisite

Use this template to specify which books, or which chapters of books, are required prior reading.

Related templates:
 * corequisite
 * Signals and Systems/Prerequisite
 * ay prerequisite and Adventist Adventurer Awards/Prerequisite

To suggest that someone read book A first, then book B, Wikibookians often
 * put this prerequisite template on the main page of book B, linking to book A
 * Somewhere near the end of book A, mention a link to book B, perhaps in a "Where to Go From Here" section ( 1), or a "Further Reading" section of the "Resources" appendix ( 2 ).

fr:Modèle:Lecture nécessaire

title: Template:Prerequisite


text:
GtkRadiant/Sky

Sooner or later, you will want to add some kind of sky to your map. Whether its an indoor map with windows, or an outdoor map, most better maps have a sky. So let's first classify a sky in the id tech 3 engine. A sky, or skybox as most mappers call it is a texture which looks the same from all perspectives.

This means it doesn't move when you move in the map like all other textures do, hence giving an illusion of something far away. First off, the myth that to make a, you will have to make a big box around your map, is wrong. it is enough to put the sky shader just where you want the sky to be, so if the sky is outside a window, just put a single brush with a sky on one side (the one facing the window) outside the window. This is enough to give the illusion of a sky on the other side of the window.

Usually, there are already a bunch of premade skies with shaders included with your game. These can be found in the texture folder called skies. Here, make sure you texture your skybox or sky brushes with the shaders, not the actual textures themselves. The shaders have a white frame around them, so you will be able to identify them in the texture browser.

title: GtkRadiant/Sky


text:
User talk:Bob Hegwood

Come introduce yourself at the new users page. If you have any questions, you can ask there or contact me personally.

Selden 17:16, 30 October 2007 (UTC)

p.s. Bob, The above is WikiBook's standard "welcome" template. I thought it might give you a chuckle. :)

s.

title: User talk:Bob Hegwood


text:
Cookbook:Tortellini in Sweet Potato Sauce

Tortellini with sweet potato sauce is a potluck staple.

Ingredients

 * 1 large sweet potato
 * 1 package (16 oz) frozen tortellini
 * 1 tbsp butter
 * 2 garlic cloves, minced
 * 1 cup milk

Procedure

 * 1) Bake or microwave sweet potato in its skin until tender. Let cool.
 * 2) Cook tortellini according to package directions, drain, and return to the pot.
 * 3) Meanwhile, heat butter in a small skillet. Add garlic and sauté over low heat until golden (about 1 minute), then remove from heat.
 * 4) Peel sweet potato and mash well in a small mixing bowl. Add garlic and butter, plus enough milk to make a thick sauce. Season as desired with salt and pepper.
 * 5) Combine sauce with pasta in the pot. Heat just until sauce is heated through.

Notes, tips, and variations

 * Spinach ravioli can be used as well.