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# Biblioteca

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## Travelling At The Speed Of Thought

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# Trainable Weka Segmentation - How to install new classifiers

### Introduction

From version 3.7.2, Weka has a package manager (understanding package not in the java sense but as a bundle of additional functionality, separate from that supplied in the main weka.jar file). Many learning algorithms and tools that were present in earlier versions of Weka have become separate packages from version 3.7.2 and need to be installed via the package manager.

In this tutorial we describe step by step how to add new classifiers to the Trainable Weka Segmentation plugin using the Weka package manager.

### Calling the package manager

We assume you are familiar with the Trainable Weka Segmentation plugin and know why and how to use it, otherwise, please go back to a more basic tutorial.

To get started, from the main plugin GUI click on the Weka button (on the bottom left corner of the window):

This will launch the Weka GUI Chooser and, if it is the first time, a warning dialog will be displayed to warn you about the package manager:

We can just click OK, go the Weka GUI Chooser window and look for the package manager under the menu Tools › Package manager , or by pressing Ctrl\|U :

This will open the Weka Package Manager window:

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## MATH 422 Lecture Note #13 (2018 Spring)Smooth manifolds

Jae Choon Cha
POSTECH $\def\id{\operatorname{id}} \def\rel{\text{ rel }} \def\Z{\mathbb{Z}} \def\Q{\mathbb{Q}} \def\R{\mathbb{R}} \def\C{\mathbb{C}} \def\sm{\smallsetminus} \def\vp{{\vphantom{1}}}$

The remaining part of this semester will focus on differentiable techniques in topology. We begin with the notion of smooth manifolds, which comes with tangent spaces.

### Definition of smooth manifolds

To define smooth manifolds, we need to use smooth maps. Briefly speaking, "smooth" means that all partial derivatives exist. The easiest case to formulate this is when the domain and codomain of the concerned map are open subsets in Euclidean spaces, as in the following definition.

Definition. Suppose $U$ and $V$ are open subsets in $\R^n$ and $\R^m$ respectively. For a map $f\colon U\to V$, let $f_i$ be the $i$th coordinate function ($i=1,\ldots,m$). That is,

$f(x_1,\ldots,x_n) = (f_1(x_1,\ldots,x_n),\ldots,f_m(x_1,\ldots,x_n)).$

We say that $f$ is smooth if all partial derivatives of arbitrary order

$\frac{\partial^k f_i}{\partial x_{j_1}\cdots\partial x_{j_k}}$

exists and are continuous for each $f_i$.

More generally, suppose $X\subset \R^n$ and $Y\subset \R^m$ are just subsets, not necessarily open. In this case, a map $f\colon X\to Y$ is called smooth if the following holds: for every $x\in X$, there is an open neighborhood $W$ of $x$ in $\R^n$ and a smooth map $g\colon U\to \R^m$ such that $g|_ {W\cap X} = f|_ {W\cap X}$. (Here, the smoothness of $g$ is defined as above, since the domain $U$ is an open subset of $\R^n$.)

Note that a composition of smooth maps is smooth.

A map $f\colon X\to Y$ is called a diffeomorphism if $f$ is a smooth map which has a smooth inverse $f^{-1}\colon Y\to X$.

Definition. A subspace $M\subset \R^k$ is a smooth manifold of dimension $n$ if for every $x\in M$, there is a neighborhood $V$ of $x$ in $M$ and a diffeomorphism $f\colon V\to U$ onto an open subset $U$ in $\R^n$.

The above $f\colon V\to U$ is called a coordinate system. Its inverse $f^{-1}\colon U\to V$ is called a parametrization.

Example: $S^n$ is a smooth manifold of dimension $n$.

Recall the definition $S^n=\{x\in \R^{n+1}\mid x_1^2+\cdots+x_n^2=1\}$. Define

$\phi\colon \R^n \to U=\{x=(x_1,\ldots,x_{n+1})\in S^n \mid x_{n+1}>0\}$

by $\phi(x_1,\ldots,x_n) = (x_1,\ldots,x_n,\sqrt{1-(x_1^2+\cdots+x_n^2)})$. This is smooth, and has smooth inverse $\phi^{-1}\colon U\to \R^n$ which is defined by $\phi^{-1}(x_1,\ldots,x_n,x_{n+1}) = (x_1,\ldots,x_n)$. (Verify that this is smooth, using the above definition!) Therefore $\phi$ is a diffeomorphism, and thus $\phi$ is a parametrization. Exchanging the roles of the coordinates $x_i$ and $x_{n+1}$, we obtain more parametrizations, and the images of these $n+1$ parametrizations cover $S^n$. It follows that $S^n$ is a smooth manifold of dimension $n$.

### Derivatives of maps on Euclidean spaces

Recall that the first order partial derivatives of a multi-variable vector-valued function form a linear map, which we call the derivative. More precisely, for a smooth map $f\colon U\to V$ with $U\subset \R^k$ and $V\subset \R^\ell$ open, and for each point $x\in U$, the map $df_x\colon \R^k \to \R^\ell$ defined by

$df_x(h) = \lim_{t\to 0} \frac{f(x+th)-f(x)}{t}$

is a linear map, which is represented by the matrix

$df_x = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_k} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_\ell}{\partial x_1} & \cdots & \frac{\partial f_\ell}{\partial x_k} \end{bmatrix}$
with respect to the standard basis. When it causes no confusion, we will sometimes write $df$ instead of $df_x$.

The following are some basic properties:

1. The chain rule holds: $d(g\circ f)_ x = dg_ {f(x)} \circ df_x$.

2. For the identity map $I\colon U\to U$, $dI_x$ is the identity (matrix) for each $x\in U$.

3. If $f\colon \R^k \to \R^\ell$ is a linear transformation, then $df_x = f$ for all $x\in \R^k$.

The following is an immediate consequence of the above properties (1), (2).

Theorem. Suppose $U\subset \R^k$ and $V\subset \R^\ell$ are open. If $f\colon U\to V$ is a diffeomorphism, then $df_x \colon \R^k \to \R^\ell$ is a linear isomorphism for every $x\in U$. In particular, $k=\ell$.

Proof. Let $f^{-1}\colon V\to U$ be a smooth inverse of $f$. Let $x\in U$ and $y=f(x)$. Then, since $I=f^{-1}f$ and $I=ff^{-1}$, we obtain that $d(f^{-1})_y \circ df_x$ and $df_x \circ d(f^{-1})_y$ are equal to the identity, by applying (1) and (2) above. It follows that both $df_x$ and $d(f^{-1})_y$ are isomorphisms. $\quad\square$

A partial converse of this is the well-known inverse function theorem:

Inverse function theorem. For a smoooth map $f\colon U\to V$ with $U\subset \R^k$, $V\subset \R^\ell$ open, if $df_x$ is nonsingular, then there exist open neighborhoods $U'$ of $x$ in $U$ such that $V'=f(U')$ is an open neighborhood of $f(x)$, and $f|_{U'} \colon U' \to V'$ is a diffeomorphism.

### Tangent spaces and derivatives of smooth maps on manifolds

Note that the above discussion on the derivatives is limited to maps defined on an open neighborhood of the Euclidean space. To generalize this to the case of smooth maps on smooth manifolds, we first define tangent spaces.

Definition. Suppose $M\subset \R^k$ is a smooth manifold of dimension $n$, and $g\colon U\to M$ is a parametrization, where $U\subset \R^n$ is open. Let $x=g(p)$ for some $p\in U$. We call $TM_x = dg_p(\R^n)$ the tangent space of $M$ at $x$.

Note that $TM_x$ is a linear subspace of the vector space $\R^k$ when $M$ is contained in $\R^k$.

We list some basic properties:

1. $TM_x$ is determined by $x$, independent of the choice of the parametrization $g$ used in the definition. Indeed, if $h\colon V\to M$ is another parametrization with $h(q)=x$, then $U'=g^{-1}(g(U)\cap h(V))$ and $V'=h^{-1}(g(U)\cap h(V))$ are open subsets, and $h^{-1}g|_{U'}\colon U'\to V'$ is a diffeomorphism. So $(dh_q)^{-1} dg_p$ is an isomorphism. It follows that $dh_q(\R^n) = dh_q((dh_q)^{-1} dg_p(\R^n)) = dg_p(\R^n)$.

2. $TM_x$ is an $n$-dimensional vector space if $M$ has dimension $n$. To prove this, take a parametrization $g\colon U\to M\subset \R^k$ with $g(p)=x$. Then by definition, $g^{-1}$ is smooth. This implies that there is a smooth map $F\colon W\to \R^n$ which is defined on an open neighborhood $W$ of $x$ in $\R^k$ and satisfies $F|_ {W\cap g(U)} = g^{-1}|_ {W\cap g(U)}$. Let $U' = g^{-1}(g(U)\cap W)$. Then, $F\circ g|_{U'}$ is equal to the identity on $U'$. It follows that $dF_x dg_p = I$, so $dg_p$ is 1-1. Therefore $TM_x = dg_p(\R^n)$ has dimension $n$.

Now, we generalize the notion of derivative to the case of smooth maps defined on manifolds, not only for those on open subsets of Euclidean spaces.

For a smooth map $f\colon M\to N$ with $M\subset \R^k$ and $N\subset \R^\ell$ smooth manifolds and for $x\in M$, define

$df_x \colon TM_x \to TN_{f(x)}$

by $df_x(v) = dF_x(v)$, where $F\colon W\to \R^\ell$ is a smooth map defined on an open neighborhood $W$ of $x$ in $\R^k$ satisfying $F|_ {W\cap M} = f|_{W\cap M}$. Note that such $F$ exists by the definition of the smoothness of $f$.

To show that $df_x$ is well-defined, we first claim that $df_x(v)\in TN_{f(x)}$. To show this, choose parametrizations $g\colon U\to M$ and $h\colon V\to N$ with $U\subset \R^m$ and $V\subset \R^n$ open, $g(p)=x$, $h(q)=f(x)$. We may assume $g(U)\subset W$ and $f(g(U))\subset h(V)$. Then, $Fi=f$, so $Fg=fg=h(h^{-1}fg)$. It follows that $dF_x dg_p = dh_q d(h^{-1}fg)_p$. Applying this to $\R^n$, we have $dF_x(TM_x) \subset TN_{f(x)}$.

Second, we claim that $df_x(v)$ is independent of the choice of $F$. Indeed, it follows from the above: we have $dF_x = dh_q d(h^{-1}fg)_ p (dg_p)^{-1}$ on $TM_x=dg_p(\R^n)$.

The above properties of derivatives generalize to this more general context:

1. Chain rule: if $M\xrightarrow{f} N \xrightarrow{g} P$ are smooth maps of manifolds, then $d(g\circ f)_x = dg_{f(x)}\circ df_x$.

2. For the identity map $I\colon M\to M$, $dI_x$ is the identity on $TM_x$. More generally, if $M\subset N$, then for the inclusion $i\colon M\to N$, $di_x$ is the inclusion $TM_x \to TN_x$.

Theorem. If $f\colon M\to N$ is a diffeomorphism of smooth manifolds, then $df_x\colon TM_x \to TN_{f(x)}$ is an isomorphism for all $x\in M$. Moreover, $M$ and $N$ has the same dimension.

Proof. The first statement is an immediate consequence of the above properties 1 and 2. For the second conclusion, recall that the dimension of a manifold is equal to the dimension of the tangent space. Since $TM_x \cong TN_{f(x)}$, $TM_x$ and $TN_{f(x)}$ have the same dimension, so $M$ and $N$ have the same dimension. $\quad\square$


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Math Planes Intercept equation

# Intercept equation of a plane

The intercept equation is a special case of the cartesian equation. It looks like this:

$\text{E: } \frac{x}a+\frac{y}b+\frac{z}c=1$

The special feature is that the intercepts of the plane can be read directly.

• $a$ is the x-intercept
• $b$ is the y-intercept
• $c$ is the z-intercept

### Example

• $\text{E: } \frac{x}4+\frac{y}3+\frac{z}3=1$

$X(4|0|0)$, $Y(0|3|0)$, $Z(0|0|3)$

• $\text{E: } \frac{x}1+\frac{y}6=1$

$X(1|0|0)$, $Y(0|6|0)$, $Z$ does not exist

• $\text{E: } 2x+4y+z=1$

$\text{E: } \frac{x}{\frac12}+\frac{y}{\frac14}+\frac{z}1=1$

$X(\frac12|0|0)$, $Y(0|\frac14|0)$, $Z(0|0|1)$

## Cartesian equation → intercept equation

The intercept equation can be formed using the cartesian equation by deviding by the number on the right side of the equation.

### Example

$\text{E: } 2x-2y+3z=6\quad|:6$

$\text{E: } \frac{x}3+\frac{y}{-3}+\frac{z}{2}=1$


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# Windows 2000 pro Setup and Startup

The Windows 2000 startup process is similar to the startup process for Windows NT 4.0 but is significantly different from other operating systems. In MS-DOS, Windows 95, and Windows 98, the computer loads the IO.sys file followed by Msdos.sys and Command.com. However, in Windows 2000, these files only exist on computers that have multiple-boot configurations with Windows 95, Windows 98, or MS-DOS.

The following list describes the startup sequence on a computer running Windows 2000 Professional:

• Power-on self test (POST)
• Initial startup process
• Operating system selection (if the computer has a multiple-boot configuration)
• Hardware detection
• Hardware configuration selection (if you are using more than one hardware profile)
• Operating system logon process

Note

The system startup sequence only applies to a system that is started from a reboot or a power off state. This sequence does not apply to a system that is turned on from a hibernated state.

This startup sequence applies to all versions of Windows 2000.

Power-on self test (POST)

When a computer is started or reset, it runs the power-on self test (POST), which determines the amount of memory in the computer and checks that required hardware devices, such as the keyboard, are present and functioning. After the POST sequence, adapters that have their own basic input/output system (BIOS) run individual POST routines.

Note

Some adapters (for example, some video adapters) run their POST sequence before the memory and device check, which runs during the initial POST.

Initial startup process (Starting the Computer)

After the computer finishes the POST, the system BIOS attempts to start an operating system. The sequence that the BIOS follows depends on the BIOS configuration. For example, if the BIOS search order is set to A and then C (A, C), the BIOS searches the floppy disk drive for a bootable disk. If drive A contains a bootable floppy disk, the BIOS loads its first sector (the partition boot sector) into memory. If the floppy disk is not bootable, the following error message appears.

Non-system disk or disk error

Replace and press any key when ready

If the BIOS does not find a floppy disk in the drive, it then searches for the active partition on the hard disk drive. If there is no system partition on the first hard disk, the master boot record (MBR) displays one of the following errors:

• Invalid partition table
• Missing operating system

When the active partition of the hard drive is found, the system BIOS reads the MBR and loads it into memory. The MBR then scans the partition table for the system partition information. When the system partition information has been read, it loads sector 0 of the system partition into memory and starts it.

Note

Sector 0 of the system partition can be a utility, a diagnostic program, or a partition boot sector that contains the startup code for the operating system.

The system partition must be on the first physical disk and contain the system startup files listed in Table 6.1.

Table 6.1 Windows 2000 Startup Files

File Name Location
Ntldr Root of startup disk
Boot.ini Root of startup disk
Bootsect.dos Root of startup disk (on multiple-boot systems)
Ntdetect.com Root of startup disk
Hyberfil.sys %systemdrive%
Ntbootdd.sys Root of startup disk (for SCSI and some large IDE drives)
Ntoskrnl.exe %systemroot%/System32
Hal.dll %systemroot%/System32
System key %systemroot%/System32/Config
Device drivers %systemroot%/System32/Drivers
Cdldr Root of startup disk

Note

If the name of the path in Boot.ini uses small computer system interface (SCSI) syntax, the Ntbootdd.sys must be in the root folder of the system partition.

The boot partition can be on the same partition as the system partition, on a different partition, or on a different hard disk.

For the most part, the MBR is independent of the operating system. For example, the MBR can start Windows NT, Windows 95, Windows 98, MS-DOS, Microsoft® Windows® 3.x, and Windows 2000 Professional. However, after boot sector 0 has been loaded into memory, the partition boot sector is dependent on both the operating system and the file system. The Windows 2000 partition boot sector performs the following functions:

Ntldr loads the operating system files from the boot partition. If you have a multiple-boot configuration, you are prompted to choose an operating system.

Ntldr controls the operating system selection and the hardware detection process before passing control to the Windows 2000 kernel. Ntldr must be in the root folder of the startup disk.

Ntldr performs the following steps:

• Sets the processor to run in 32-bit flat memory mode. When you first start a computer, it runs in real mode, which is similar to an 8088 or 8086 CPU. Because Ntldr is a 32-bit program, it must switch the processor to 32-bit mode so that it can continue to load the operating system.
• Starts the NTFS file system or the file allocation table (FAT) 16 or 32 file system. The code to access the appropriate file system is built into Ntldr.
• Reads Boot.ini, which displays the operating selections on the boot loader screen. If you select an operating system other than Windows 2000 Professional, Ntldr loads and passes control to the Bootsect.dos file, and the selected operating system starts. If you select Windows 2000 Professional, Ntldr starts Ntdetect.com, which gathers information about the computer hardware.
• Asks you to choose a hardware profile if you have one or more hardware profiles on your computer.
• Loads and passes the information from Ntdetect.com to Ntoskrnl.exe, starting the startup screen.

Operating system selection (if the computer has a multiple-boot configuration)

Boot.ini contains the list of available operating systems. Each entry includes the path to the boot partition for the operating system, the string to display in the boot loader screen, and optional parameters. Boot.ini supports starting multiple versions of Windows 2000, as well as starting other operating systems, such as Windows 95, Windows 98, Windows NT 4.0, MS-DOS, and OS/2.

When you install Windows 2000, Setup places Boot.ini at the root of the system partition.

The following is a sample Boot.ini file:

[boot loader]

timeout=30

default=multi(0)disk(0)rdisk(0)partition(1)/WINNT

[operating systems]

multi(0)disk(0)rdisk(0)partition(1)/winnt= “Microsoft Windows 2000 Professional” /fastdetect

C:/=“Windows 98”

The information displayed on the boot loader screen is based on the contents of Boot.ini, such as in the following example:

Please select the operating system to start:

Microsoft Windows 2000 Professional

Microsoft Windows 98

Use and to move the highlight to your choice.

Please Enter to choose.

Seconds until highlighted choice will be started automatically: 29

For troubleshooting and advanced startup options for Windows 2000, press F8.

The first operating system listed is the default. In this configuration, if you do not select an operating system before the timer reaches 0, Windows 2000 Professional is started.

Hardware detection

After you select the operating system, Ntdetect.com detects the hardware, gathers a list of the currently installed hardware components, and passes the information to Ntldr.

Ntdetect.com detects the following components:

• Computer ID
• Keyboard
• Communications ports
• Floppy disks
• Mouse or other pointing devices
• Parallel ports

Hardware configuration selection (if you are using more than one hardware profile)

During the Ntdetect phase, if you have multiple hardware configurations, the following information appears:

Hardware Profile/Configuration Recovery Menu

This menu allows you to select a hardware profile

to be used when Windows 2000 is started.

Note

Profiles are only necessary under non–Plug and Play conditions. Plug and Play negates the need for profiles.

The bootstrap loader pauses for a few seconds or until you press Enter. If you have only one hardware profile, Windows 2000 Professional uses the default settings.

Windows 2000 Professional automatically uses the default settings if you do not select a hardware profile. When the default configuration is used, the bootstrap loader uses the registry information that Windows 2000 Professional saved at the completion of the last shutdown. After you select a hardware profile, Ntdetect.com passes control back to Ntldr.

During the next phase, Ntldr loads the kernel and the hardware abstraction layer (HAL) into memory. Next the bootstrap loader loads the registry key HKEY_LOCAL_MACHINE/SYSTEM from the folder %systemroot%/system32/Config/System. Ntldr creates the control set that it uses to initialize the computer. The loader uses the control set identified by the default value unless you choose the Last Known Good Configuration from the Hardware Profiles screen.

The kernel initiates the Windows 2000 Professional screen, and the Starting Up progress bar is displayed. When the status bar completes, Ntoskrnl prepares the network information.

The kernel uses the information that was passed from the boot loader to create the HKEY_LOCAL_MACHINE/HARDWARE key, which contains the hardware data that is collected at system startup. The data includes information about various hardware components on the motherboard and the system interrupts allocated by each device.

The kernel then creates the Clone control set, which is a copy that points to the Current control set. The Clone control set represents the state of the computer during configuration and is not changed or modified.

Operating system logon process

The Windows subsystem automatically starts Winlogon.exe, which starts the Local Security Administration. The Begin Logon dialog box appears. Windows 2000 might still be initializing network device drivers, but you can log on.

The Service Controller searches the registry for services that are configured to load automatically. Autoload services have a Start value of 0x2 in the subkey HKEY_LOCAL_MACHINE/SYSTEM/CurrentControlSet/Services/DriverName.The services that are loaded during this phase are loaded in parallel and so are loaded based on their dependencies. The dependencies are described in the DependOnGroup and DependOnService entries in the subkey HKEY_LOCAL_MACHINE/SYSTEM/CurrentControlSet/Services/DriverName.

Note

Windows 2000 startup is not considered complete until a user successfully logs on to the system. After a user logs on, the Clone control set is copied to the LastKnownGood control set.

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## Model-based vs. Model-free

### Comparison

Model-free methods use the reward prediction error (RPE) to update values:

\delta_t = r_{t+1} + \gamma \, V^\pi(s_{t+1}) - V^\pi(s_t)

\Delta V^\pi(s_t) = \alpha \, \delta_t

Encountered rewards propagate very slowly to all states and actions. If the environment changes (transition probabilities, rewards), they have to relearn everything. After training, selecting an action is very fast.

Model-based RL can learn very fast changes in the transition or reward distributions:

\Delta r(s_t, a_t, s_{t+1}) = \alpha \, (r_{t+1} - r(s_t, a_t, s_{t+1}))

\Delta p(s' | s_t, a_t) = \alpha \, (\mathbb{I}(s_{t+1} = s') - p(s' | s_t, a_t))

But selecting an action requires planning in the tree of possibilities (slow).

Relative advantages of MF and MB methods:

Inference speed Sample complexity Optimality Flexibility
Model-free fast high yes no
Model-based slow low as good as the model yes

A trade-off would be nice… Most MB models in the deep RL literature are hybrid MB/MF models anyway.

### Goal-directed learning vs. habit formation

Two forms of behavior are observed in the animal psychology literature:

1. Goal-directed behavior learns Stimulus \rightarrow Response \rightarrow Outcome associations.
2. Habits are developed by overtraining Stimulus \rightarrow Response associations.

The main difference is that habits are not influenced by outcome devaluation, i.e. when reard slose their value.

The classical theory assigns MF to habits and MB to goal-directed, mostly because their sensitivity to outcome devaluation. The open question is the arbitration mechanism between these two segregated process: who takes control? Recent work suggests both systems are largely overlapping.

## Successor representations

### Principle of successor representations

Successor representations (SR, ) have been introduced to combine MF and MB properties. Let’s split the definition of the value of a state:

\begin{align} V^\pi(s) &= \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, r_{t+k+1} | s_t =s] \\ &\\ &= \mathbb{E}_{\pi} [\begin{bmatrix} 1 \\ \gamma \\ \gamma^2 \\ \ldots \\ \gamma^\infty \end{bmatrix} \times \begin{bmatrix} \mathbb{I}(s_{t}) \\ \mathbb{I}(s_{t+1}) \\ \mathbb{I}(s_{t+2}) \\ \ldots \\ \mathbb{I}(s_{\infty}) \end{bmatrix} \times \begin{bmatrix} r_{t+1} \\ r_{t+2} \\ r_{t+3} \\ \ldots \\ r_{t+\infty} \end{bmatrix} | s_t =s]\\ \end{align}

where \mathbb{I}(s_{t}) is 1 when the agent is in s_t at time t, 0 otherwise.

The left part corresponds to the transition dynamics: which states will be visited by the policy, discounted by \gamma. The right part corresponds to the immediate reward in each visited state. Couldn’t we learn the transition dynamics and the reward distribution separately in a model-free manner?

SR rewrites the value of a state into an expected discounted future state occupancy M^\pi(s, s') and an expected immediate reward r(s') by summing over all possible states s' of the MDP:

\begin{align} V^\pi(s) &= \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, r_{t+k+1} | s_t =s] \\ &\\ &= \sum_{s' \in \mathcal{S}} \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(s_{t+k}=s') \times r_{t+k+1} | s_t =s]\\ &\\ &\approx \sum_{s' \in \mathcal{S}} \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(s_{t+k}=s') | s_t =s] \times \mathbb{E} [r_{t+1} | s_{t}=s']\\ &\\ &\approx \sum_{s' \in \mathcal{S}} M^\pi(s, s') \times r(s')\\ \end{align}

The underlying assumption is that the world dynamics are independent from the reward function (which does not depend on the policy). This allows to re-use knowledge about world dynamics in other contexts (e.g. a new reward function in the same environment): transfer learning.

What matters is the states that you will visit and how interesting they are, not the order in which you visit them. Knowing that being in the mensa will eventually get you some food is enough to know that being in the mensa is a good state: you do not need to remember which exact sequence of transitions will put food in your mouth.

SR algorithms must estimate two quantities:

• The expected immediate reward received after each state:

r(s) = \mathbb{E} [r_{t+1} | s_t = s]

• The expected discounted future state occupancy (the SR itself):

M^\pi(s, s') = \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(s_{t+k} = s') | s_t = s]

The value of a state s is then computed with:

V^\pi(s) = \sum_{s' \in \mathcal{S}} M(s, s') \times r(s')

what allows to infer the policy (e.g. using an actor-critic architecture). The immediate reward for a state can be estimated very quickly and flexibly after receiving each reward:

\Delta \, r(s_t) = \alpha \, (r_{t+1} - r(s_t))

Imagine a very simple MDP with 4 states and a single deterministic action:

The transition matrix \mathcal{P}^\pi depicts the possible (s, s') transitions:

\mathcal{P}^\pi = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}

The SR matrix M also represents the future transitions discounted by \gamma:

M = \begin{bmatrix} 1 & \gamma & \gamma^2 & \gamma^3 \\ 0 & 1 & \gamma & \gamma^2 \\ 0 & 0 & 1 & \gamma\\ 0 & 0 & 0 & 1 \\ \end{bmatrix}

SR matrix in a Tolman’s maze

Source:

The SR represents whether a state can be reached soon from the current state (b) using the current policy. The SR depends on the policy: * A random agent will map the local neighborhood (c). * A goal-directed agent will have SR representations that follow the optimal path (d).

It is therefore different from the transition matrix, as it depends on behavior and rewards. The exact dynamics are lost compared to MB: it only represents whether a state is reachable, not how.

The SR matrix reflects the proximity between states depending on the transitions and the policy. it does not have to be a spatial relationship.

### Learning successor representations

How can we learn the SR matrix for all pairs of states?

M^\pi(s, s') = \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(s_{t+k} = s') | s_t = s]

We first notice that the SR obeys a recursive Bellman-like equation:

\begin{aligned} M^\pi(s, s') &= \mathbb{I}(s_{t} = s') + \mathbb{E}_{\pi} [\sum_{k=1}^\infty \gamma^k \, \mathbb{I}(s_{t+k} = s') | s_t = s] \\ &= \mathbb{I}(s_{t} = s') + \gamma \, \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(s_{t+k+1} = s') | s_t = s] \\ &= \mathbb{I}(s_{t} = s') + \gamma \, \mathbb{E}_{s_{t+1} \sim \mathcal{P}^\pi(s' | s)} [\mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(s_{t+k} = s') | s_{t+1} = s] ]\\ &= \mathbb{I}(s_{t} = s') + \gamma \, \mathbb{E}_{s_{t+1} \sim \mathcal{P}^\pi(s' | s)} [M^\pi(s_{t+1}, s')]\\ \end{aligned}

This is reminiscent of TDM: the remaining distance to the goal is 0 if I am already at the goal, or gamma the distance from the next state to the goal.

If we know the transition matrix for a fixed policy \pi:

\mathcal{P}^\pi(s, s') = \sum_a \pi(s, a) \, p(s' | s, a)

we can obtain the SR directly with matrix inversion as we did in dynamic programming:

M^\pi = I + \gamma \, \mathcal{P}^\pi \times M^\pi

so that:

M^\pi = (I - \gamma \, \mathcal{P}^\pi)^{-1}

This DP approach is called model-based SR (MB-SR, ) as it necessitates to know the environment dynamics.

If we do not know the transition probabilities, we simply sample a single s_t, s_{t+1} transition:

M^\pi(s_t, s') \approx \mathbb{I}(s_{t} = s') + \gamma \, M^\pi(s_{t+1}, s')

We can define a sensory prediction error (SPE):

\delta^\text{SR}_t = \mathbb{I}(s_{t} = s') + \gamma \, M^\pi(s_{t+1}, s') - M(s_t, s')

that is used to update an estimate of the SR:

\Delta M^\pi(s_t, s') = \alpha \, \delta^\text{SR}_t

This is SR-TD, using a SPE instead of RPE, which learns only from transitions but ignores rewards.

The SPE has to be applied on ALL successor states s' after a transition (s_t, s_{t+1}):

M^\pi(s_t, \mathbf{s'}) = M^\pi(s_t, \mathbf{s'}) + \alpha \, (\mathbb{I}(s_{t}=\mathbf{s'}) + \gamma \, M^\pi(s_{t+1}, \mathbf{s'}) - M(s_t, \mathbf{s'}))

Contrary to the RPE, the SPE is a vector of prediction errors, used to update one row of the SR matrix. The SPE tells how surprising a transition s_t \rightarrow s_{t+1} is for the SR.

Summary

The SR matrix represents the expected discounted future state occupancy:

M^\pi(s, s') = \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(s_{t+k} = s') | s_t = s]

It can be learned using a TD-like SPE from single transitions:

M^\pi(s_t, \mathbf{s'}) = M^\pi(s_t, \mathbf{s'}) + \alpha \, (\mathbb{I}(s_{t}=\mathbf{s'}) + \gamma \, M^\pi(s_{t+1}, \mathbf{s'}) - M(s_t, \mathbf{s'}))

The immediate reward in each state can be learned independently from the policy:

\Delta \, r(s_t) = \alpha \, (r_{t+1} - r(s_t))

The value V^\pi(s) of a state is obtained by summing of all successor states:

V^\pi(s) = \sum_{s' \in \mathcal{S}} M(s, s') \times r(s')

This critic can be used to train an actor \pi_\theta using regular TD learning (e.g. A3C).

Note that it is straightforward to extend the idea of SR to state-action pairs:

M^\pi(s, a, s') = \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(s_{t+k} = s') | s_t = s, a_t = a]

allowing to estimate Q-values:

Q^\pi(s, a) = \sum_{s' \in \mathcal{S}} M(s, a, s') \times r(s')

using SARSA or Q-learning-like SPEs:

\delta^\text{SR}_t = \mathbb{I}(s_{t} = s') + \gamma \, M^\pi(s_{t+1}, a_{t+1}, s') - M(s_t, a_{t}, s')

depending on the choice of the next action a_{t+1} (on- or off-policy).

## Successor features

The SR matrix associates each state to all others (N\times N matrix):

• curse of dimensionality.
• only possible for discrete state spaces.

A better idea is to describe each state s by a feature vector \phi(s) = [\phi_i(s)]_{i=1}^d with less dimensions than the number of states. This feature vector can be constructed (see the lecture on function approximation) or learned by an autoencoder (latent representation).

The successor feature representation (SFR) represents the discounted probability of observing a feature \phi_j after being in s. Instead of predicting when the agent will see a cat after being in the current state s, the SFR predicts when it will see eyes, ears or whiskers independently:

M^\pi_j(s) = M^\pi(s, \phi_j) = \mathbb{E}_{\pi} [\sum_{k=0}^\infty \gamma^k \, \mathbb{I}(\phi_j(s_{t+k})) | s_t = s, a_t = a]

Linear SFR supposes that it can be linearly approximated from the features of the current state:

M^\pi_j(s) = M^\pi(s, \phi_j) = \sum_{i=1}^d m_{i, j} \, \phi_i(s)

The value of a state is now defined as the sum over successor features of their immediate reward discounted by the SFR:

V^\pi(s) = \sum_{j=1}^d M^\pi_j(s) \, r(\phi_j) = \sum_{j=1}^d r(\phi_j) \, \sum_{i=1}^d m_{i, j} \, \phi_i(s)

The SFR matrix M^\pi = [m_{i, j}]_{i, j} associates each feature \phi_i of the current state to all successor features \phi_j: Knowing that I see a kitchen door in the current state, how likely will I see a food outcome in the near future?

Each successor feature \phi_j is associated to an expected immediate reward r(\phi_j): A good state is a state where food features (high r(\phi_j)) are likely to happen soon (high m_{i, j}).

In matrix-vector form:

V^\pi(s) = \mathbf{r}^T \times M^\pi \times \phi(s)

The reward vector \mathbf{r} only depends on the features and can be learned independently from the policy, but can be made context-dependent: Food features can be made more important when the agent is hungry, less when thirsty.

Transfer learning becomes possible in the same environment: * Different goals (searching for food or water, going to place A or B) only require different reward vectors. * The dynamics of the environment are stored in the SFR.

How can we learn the SFR matrix M^\pi?

V^\pi(s) = \mathbf{r}^T \times M^\pi \times \phi(s)

We only need to use the sensory prediction error for a transition between the feature vectors \phi(s_t) and \phi(s_{t+1}):

\delta_t^\text{SFR} = \phi(s_t) + \gamma \, M^\pi \times \phi(s_{t+1}) - M^\pi \times \phi(s_t)

and use it to update the whole matrix:

\Delta M^\pi = \delta_t^\text{SFR} \times \phi(s_t)^T

However, this linear approximation scheme only works for fixed feature representation \phi(s). We need to go deeper…

## Deep Successor Reinforcement Learning

Each state s_t is represented by a D-dimensional (D=512) vector \phi(s_t) = f_\theta(s_t) which is the output of an encoder. A decoder g_{\hat{\theta}} is used to provide a reconstruction loss, so \phi(s_t) is a latent representation of an autoencoder:

\mathcal{L}_\text{reconstruction}(\theta, \hat{\theta}) = \mathbb{E}[(g_{\hat{\theta}}(\phi(s_t)) - s_t)^2]

The immediate reward R(s_t) is linearly predicted from the feature vector \phi(s_t) using a reward vector \mathbf{w}.

R(s_t) = \phi(s_t)^T \times \mathbf{w}

\mathcal{L}_\text{reward}(\mathbf{w}, \theta) = \mathbb{E}[(r_{t+1} - \phi(s_t)^T \times \mathbf{w})^2]

The reconstruction loss is important, otherwise the latent representation \phi(s_t) would be too reward-oriented and would not generalize. The reward function is learned on a single task, but it can fine-tuned on another task, with all other weights frozen.

For each available action a, a DNN u_\alpha predicts the future feature occupancy M(s, s', a) for the current state:

m_{s_t a} = u_\alpha(s_t, a)

The Q-value of an action is simply the dot product between the SR of an action and the reward vector \mathbf{w}:

Q(s_t, a) = \mathbf{w}^T \times m_{s_t a}

The selected action is \epsilon-greedily selected around the greedy action:

a_t = \text{arg}\max_a Q(s_t, a)

The SR of each action is learned using the Q-learning-like SPE (with fixed \theta and a target network u_{\alpha'}):

\mathcal{L}^\text{SPE}(\alpha) = \mathbb{E}[\sum_a (\phi(s_t) + \gamma \, \max_{a'} u_{\alpha'}(s_{t+1}, a') - u_\alpha(s_t, a))^2]

The compound loss is used to train the complete network end-to-end off-policy using a replay buffer (DQN-like).

\mathcal{L}(\theta, \hat{\theta}, \mathbf{w}, \alpha) = \mathcal{L}_\text{reconstruction}(\theta, \hat{\theta}) + \mathcal{L}_\text{reward}(\mathbf{w}, \theta) + \mathcal{L}^\text{SPE}(\alpha)

The interesting property is that you do not need rewards to learn: * A random agent can be used to learn the encoder and the SR, but \mathbf{w} can be left untouched. * When rewards are introduced (or changed), only \mathbf{w} has to be adapted, while DQN would have to re-learn all Q-values.

This is the principle of latent learning in animal psychology: fooling around in an environment without a goal allows to learn the structure of the world, what can speed up learning when a task is introduced. The SR is a cognitive map of the environment: learning task-unspecific relationships.

The same idea was published by three different groups at the same time (preprint in 2016, conference in 2017): , , . The is from Deepmind, so it tends to be cited more…

Visual Semantic Planning using Deep Successor Representations

See the paper:


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### Home > INT3 > Chapter 11 > Lesson 11.2.1 > Problem11-62

11-62.

The height $h$ (in feet) of object A after $t$ seconds is given by the equation $h = 80t – 16t^{2}$. The height of object B is given by the graph at right.

1. Which object was in the air longer? How long was this object in the air?

Find the $x$-intercepts for object A to determine how long it was in the air.

Object A was in the air for $5$ seconds.

2. Which object traveled higher? How high did it travel?

Average the $x$-intercepts for object A to find the vertex.

3. For what times is the height of object A higher than $64$ feet?

Solve $80t−16t^2>64$

$1

4. In the context of this problem, what domain is reasonable for the function that describes object A?

For what values of $t$ is the ball on or above the ground?


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# What does a dot inside two parenthesis mean on a weather depiction chart?

Earlier today, a friend asked me what this symbol (referring to an airport in Canada) on a weather depiction chart represented. I didn't know. (Full chart available here)

What does a dot inside parenthesis mean on a weather depiction chart?

• What agency produced the chart?
– user2168
May 14 '14 at 5:33
• @Articuno - The Aviation Weather Center. I've edited in a link to the full chart. May 14 '14 at 6:30
• Looks like its a NOAA chart.
– user2168
May 14 '14 at 7:04

From here, $(\cdot)$ means that there is precipitation within sight, reaching the ground, not at the station, but < 5 km from the station.
$)\cdot($ means that there is precipitation within sight, reaching the ground, not at the station, but > 5km from the station.


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Minst Symbol Math - starpl.com

# List of LaTeX symbols LaTeX Wiki Fandom.

HTML Charsets HTML ASCII HTML WIN-1252 HTML ISO-8859 HTML Symbols HTML UTF-8 HTML UTF-8 Latin Basic Latin Supplement Latin Extended A Latin Extended B Modifier Letters Diacritical Marks Greek and Coptic Cyrillic Basic Cyrillic Supplement HTML Symbols. Set symbols of set theory and probability with name and definition: set, subset, union, intersection, element, cardinality, empty set, natural/real/complex number set.

2019-12-17 · Set Symbols. A set is a collection of things, usually numbers. We can list each element or "member" of a set inside curly brackets like this: Common Symbols Used in Set Theory. Symbols save time and space when writing. Here are the most common set symbols. In the examples C. 2016-08-06 · Mathematical Symbols with their Name Click here for more symbol /2016/08/all-mathematical-symbols-name-list.html equals sign not e. When you need to communicate mathematical equations or formulas in your notes, you can do that directly on your Mac. Using OneNote, you can insert fractions, exponentials, logarithms, and other mathematical structures in your notes. Minst ett hygienrum ska vara tillgängligt och användbart för personer med nedsatt rörelseförmåga och utformas så att det lätt kan ordnas plats för medhjälpare. I det tillgängliga och användbara hygienrumet ska också gå att ordna en separat duschplats om en sådan saknas från början. explains how symbols are spaced in math mode, presents a LATEX ASCII and Latin 1 tables, and provides some information about this document itself. The Comprehensive LATEX Symbol List ends with an index of all the symbols in the document and various additional useful terms.

How to look up a symbol or identify a math symbol or character? 17 answers. Wish to design new integral symbol with infinity symbol through it. 16. A special mathematical symbol supremum of directed set 8. biblatex symbol instead numbers. 0. How to get an undirected double quote symbol. 2. List of infinity symbols where used in math, physics and more. Html infinity sign, ascii code, unicode, learn how to type an infinity symbol on mac and win. Download free Math Symbol Regular font, MATHS.TTF Math Symbol Regular FontMonger:Math Symbol Math Symbol MathSymbol. För att markera starten för den matematiska formateringen används taggen $. För att avsluta den matematiska formateringen används taggen$. När du behöver använda symboler som inte är tillgängliga på ett tangentbord eller konstruera avancerade formler behöver du använda dig av specialkommandon.

ALT Codes for Mathematical Symbols Alt Code Symbol Description Alt 48 - 57 0 - 9 zero to nine Alt Codes for Basic Operators; Alt Code Symbol Description Alt 43Plus Sign Alt 45 - Minus Sign Alt 0215 × Multiplication Sign Alt 0247 ÷ Obelus / Division ign. LATEX Mathematical Symbols The more unusual symbols are not deﬁned in base LATEX NFSS and require \usepackageamssymb 1 Greek and Hebrew letters α \alpha κ \kappa ψ \psi z \digamma ∆ \Delta Θ \Theta β \beta λ \lambda ρ \rho ε \varepsilon Γ \Gamma Υ \Upsilon. 9 Math mode accents. Copy and paste symbols with this cool symbol picker tool, which help easily get Facebook symbols, Instagram symbols, Twitter symbols, emoji, emoticon text & text art. Just click on a symbol, an emoji or a text art to copy it to the clipboard. You can also check out the ios app and android app. 2019-12-19 · Your private math tutor, solves any math problem with steps! Equations, integrals, derivatives, limits and much more. Steps require an in-app subscription Symbolab Math Solver app is composed of over one hundred of Symbolab's most powerful calculators: Equation Calculator Integral Calculator Derivative Calculator Limit calculator. MSD = Minsta Symbol avstånd Letar du efter allmän definition av MSD? MSD betyder Minsta Symbol avstånd. Vi är stolta över att lista förkortningen av MSD i den största databasen av förkortningar och akronymer. Följande bild visar en av definitionerna för MSD på engelska: Minsta Symbol avstånd.

Alla symboler som hjärtan, blommor, pilar, objekt och mycket mer! Använd dem på Facebook, Twitter, Instagram eller i dina blogginlägg! This means, for example, that you cannot put one symbol over another. While this is a serious limitation, multi-level formulas are not always needed and even when they are needed, proper math symbols still look better than improvised ASCII approximations. Compare: ∀x, y ∈ A ∪ B; x ≠ y x² − y² ≥ 0. As you begin to see, typesetting math can be tricky at times. However, because LaTeX provides so much control, you can get professional quality mathematics typesetting with relatively little effort once you've had a bit of practice, of course!. It would be possible to keep going and going with math topics because it seems potentially limitless.

1. LaTeX symbols have either names denoted by backslash or special characters. They are organized into seven classes based on their role in a mathematical expression. This is not a comprehensive list. Refer to the external references at the end of this article for more information. Letters are.
2. 2019-12-21 · An online LaTeX editor that's easy to use. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more.
3. In inline math mode the integral/sum/product lower and upper limits are placed right of integral symbol. Similar is for limit expressions. If you want the limits of an integral/sum/product to be specified above and below the symbol in inline math mode, use the \limits command before limits specification.
4. LPS = Minst sannolika Symbol Letar du efter allmän definition av LPS? LPS betyder Minst sannolika Symbol. Vi är stolta över att lista förkortningen av LPS i den största databasen av förkortningar och akronymer. Följande bild visar en av definitionerna för LPS på engelska: Minst sannolika Symbol.

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# Electric Charge

Calculators and formulas for calculating the electric charge

### Electric Charge calculator

This function calculates the relationship between electric charge, current and time.

To perform the calculation, use the radio button to select which value should be calculated. Then enter the required values and click the 'Calculate' button.

Electric Charge calculator

 What should be calculated? Charge Current Time Eingabe Charge C Ah Current MA kA A mA µA Time days hr min sec msec µsec Decimal places 0 1 2 3 4 6 8 10
 Result Charge Current
 Time

## Formulas for electric charge

Electric charge is measured in coulombs (C) or ampere-seconds (As). 1 C = 1 As.
The storage capacity of a battery is referred to as the nominal charge, which is given in ampere hours (Ah). 1 Ah = 3600 C.

#### Electric charge

$$\displaystyle Ladung= Strom · Zeit$$
$$\displaystyle C=I·t[s]$$ $$\displaystyle Ah=I·t[h]$$

#### Electric current

$$\displaystyle I=\frac{C}{t[s]}$$ $$\displaystyle I=\frac{Ah}{t[h]}$$

#### Time

$$\displaystyle t[s]=\frac{C}{I}$$ $$\displaystyle t[h]=\frac{Ah}{I}$$


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# Who will win?

Two cars $$A$$ and $$B$$ are racing along a straight line, car $$A$$ is leading and their relative velocity is directly proportional to the distance between the two cars. When the lead of car $$A$$ is $$x_{1}=10\text{ m}$$, it is running $$10\text{ ms}^{-1}$$ faster than car $$B$$. If the time taken by car $$A$$ to increase its lead to $$x_{2}=20\text{ m}$$ from car $$B$$ is in the form $$t=\ln n$$, find $$n$$.

×


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# CS Theory

## On the tractability of $(k,i)$-coloring (Extended Journal Version)

This paper is primarily about NP-completeness of $(k,i)$-coloring of graph and giving a parameterized algorithm for this problem using feedback vertex set as the parameter.

## On the tractability of $(k,i)$-coloring

This paper is primarily provides a parameterized algorithm for $(k,i)$-coloring problem using feedback vertex set as the parameter.


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# Stability of cyclic isomers of C6H12

I was given the problem to arrange all cyclic isomers of $\ce{C6H12}$ from the lowest to the highest heat of combustion. I found twelve isomers and quickly realised that cyclohexane is the most stable and thus has the lowest heat of combustion, since it has zero strain, and that the only cyclopentane isomer - methylcyclopentane, follows after since it does not have much strain. However, what confuses me is the stability of the other isomers (four cyclobutanes as well as six cyclopropanes). I understand that the cyclobutanes are more stable than the cyclopropanes, but what about the substituents? How do they affect stability?

• Not much. That said, multiple and/or branched substitutes would probably make a dense crowd around the ring and thus increase the strain. One long side chain would not do that. – Ivan Neretin Jul 29 '17 at 11:46
• As a sidenote: I think your count is a bit off. I count at least 6 isomers of cyclic butane. – Martin - マーチン Jul 29 '17 at 15:39
• ethyl-, 1,1-dimethyl, cis + trans 1,2-dimethyl-, cis + trans 1,3-dimethyl- – Martin - マーチン Jul 29 '17 at 18:40


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# why is linear b important

It may show for example how demand changes when price changes or how consumption //-->. Linear regression is used to perform regression analysis. Linear relationships are fairly common in daily life. Minoan pottery circulated widely, and

We can determine what effect the independent variables have on a dependent variable. With positive slope The later, Linear B, represented a subsequent intrusion of Mycenaeans from the mainland. than a letter of an alphabet. Both linear scripts Linear B was deciphered as Greek in 1952 by Michael Ventris. The Archaeological area of Eleusis, â¢ the change in y or the change on the vertical axis versus the change in x or In econometrics, linear regression is an often-used method of generating linear relationships to explain various phenomena. Below are the uses of regression analysis. Second, he analysed statistically the placing and frequency of different symbols within individual words in order to build up an understanding of grammatical structure. Accessed Aug. 10, 2020. Slope measures the rate of change in the dependent variable as the independent In the study of linear algebra, the properties of linear functions are extensively studied and made rigorous. Regression analysis helps in understanding the various data points and the relationship between them. variable changes.

In one variable, a linear function can be written as follows: f(x)=mx+bwhere:m=slopeb=y-intercept\begin{aligned} &f(x) = mx + b \\ &\textbf{where:}\\ &m=\text{slope}\\ &b=\text{y-intercept}\\ \end{aligned}f(x)=mx+bwhere:m=slopeb=y-intercept. Some data describe relationships that are curved (such as polynomial relationships) while still other data cannot be parameterized. ALL RIGHTS RESERVED. This is a typical upward sloping supply curve which says that Regression analysis also helps the company provide maximum efficiency and refine its processes. These include white papers, government data, original reporting, and interviews with industry experts. classes, Linear A and Linear B. Tragically, Michael Ventris, the man who made this critical discovery, was almost immediately killed in a car crash. To determine if a series of numbers is a linear sequence, subtract each number by the number before it.

But before 1952 no-one knew who the Mycenaeans were.

Another important variable will result in a change in y by the amount of b. slope = change in y/change in x = rise/run.
He was rewarded almost immediately by the discovery of slabs of baked clay, some rectangular, some leaf-shaped, bearing two types of inscription of hitherto unknown form. The Linear B Tablets revealed Below is the equation that represents the relation between x and y. When Arthur Evans started digging at Knossos on Crete in 1900, a major aim was to find inscriptions and prove that the ancient Cretans had been literate. You can learn more about the standards we follow in producing accurate, unbiased content in our. The usual growth is 3 inches. Hadoop, Data Science, Statistics & others. supply rises as price rises. The Linear B Tablets interpreted Assume that the independent variable is the size of a house (as measured by square footage) which determines the market price of a home (the dependent variable) when it is multiplied by the slope coefficient of 207.65 and is then added to the constant term $10,500. The Linear B tablets were created and preserved because they contained important economic or financial records. Linear B Tablets The Linear B Tablets revealed When Arthur Evans started digging at Knossos on Crete in 1900, a major aim was to find inscriptions and prove that the ancient Cretans had been literate. We should understand are important variables and unimportant variables before we create a model. The formula we use to calculate speed is as follows: the rate of speed is the distance traveled over time. Third, in a speculative leap, he substituted Greek sounds for Linear B symbols. The variable names may differ. They represent the oldest known Greek dialect, elements of which survived in Homer’s language as a result of a long oral tradition of epic poetry. It is commonly used in extrapolating events from the past to make forecasts for the future. Linear A, the older script, has The same is represented in the below equation. The residual standard deviation describes the difference in standard deviations of observed values versus predicted values in a regression analysis. A linear relationship (or linear association) is a statistical term used to describe a straight-line relationship between two variables. the Aegean world. The independent variable can also be called an exogenous variable. Represented graphically with the distance on the Y-axis and time on the X-axis, a line tracking the distance over those 20 hours would travel straight out from the convergence of the X and Y-axis. With negative and the like. Simply put, algebra is about finding the unknown or putting real life variables into equations and then solving them. The oldest Mycenaean writing dates to about 1450 BC. during the period between 1450 and 1400 B.C., when the currently accepted This article is an extract from the full article published in World Archaeology Issue 34. Learning to detect and predict sequences is useful in pattern recognition, both by visual inspection and technological algorithms. Many scholars still doubted the Mycenaeans were Greek, and viewed Homer’s Iliad and Odyssey as wholly mythological. Some linear relationships between two objects can be called a "proportional relationship." Predictive analysis helps in understanding the relationship between the predictor and outcome variable (i.e. interchange comes from Cretan writing on tablets that Evans and With positive slope the line moves upward when going from left to right. Understanding the data and relationship between them helps businesses to grow and analyze certain trends or patterns. Slope means that a unit change in x, the independent Ancient Greek jewelry blog. Crete's strategical position, â¢ What Does a Linear Relationship Tell You? If a home's square footage is 1,250 then the market value of the home is (1,250 x 207.65) +$10,500 = \$270,062.50.


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### Recent Papers

We study a new generating functional of one-particle irreducible diagrams in quantum field theory, called master functional, which is invariant under the most general perturbative changes of field variables. The functional $\Gamma$ does not transform as a scalar under the transformation law inherited from its very definition, although it does transform as a scalar under an unusual transformation law. The master functional, on the other hand, is the Legendre transform of an improved functional W = ln Z with respect to the sources coupled to both elementary and composite fields. The inclusion of certain improvement terms in W and Z is necessary to make this transform well defined. The master functional behaves as a scalar under the transformation law inherited from its very definition. Moreover, it admits a proper formulation, obtained extending the set of integrated fields to the so-called proper fields, which allows us to work without passing through Z, W or $\Gamma$. In the proper formulation the classical action coincides with the classical limit of the master functional, and correlation functions and renormalization are calculated applying the usual diagrammatic rules to the proper fields. Finally, the most general change of field variables, including the map relating bare and renormalized fields, is a linear redefinition of the proper fields.

PDF

Eur. Phys. J. C 73 (2013) 2385 | DOI: 10.1140/epjc/s10052-013-2385-y

arXiv:1205.3584 [hep-th]

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### Book

14B1 D. Anselmi
Renormalization

PDF

Last update: May 9th 2015, 230 pages

Contents: Preface | 1. Functional integral | 2. Renormalization | 3. Renormalization group | 4. Gauge symmetry | 5. Canonical formalism | 6. Quantum electrodynamics | 7. Non-Abelian gauge field theories | Notation and useful formulas | References

Course on renormalization, taught in Pisa in 2015. (More chapters will be added later.)


text:
Using the Intermediate Value Theorem and a calculator, find an interval of length $0.01$ that contains a root of $x^5-x^2+2x+3=0$, rounding off interval endpoints to the nearest hundredth.

$< x <$

You can earn partial credit on this problem.


text:
# clearRange-methods

0th

Percentile

##### Clearing cell ranges in a workbook

Clears cell ranges in a '>workbook.

Keywords
methods, utilities
##### Usage
# S4 method for workbook,numeric
clearRange(object, sheet, coords)
# S4 method for workbook,character
clearRange(object, sheet, coords)
##### Arguments
object

The '>workbook to use

sheet

The name or index of the worksheet in which to clear cell ranges

coords

Numeric vector of length 4 or numeric matrix with 4 columns where the elements of the vector or rows in the matrix refer to the coordinates of the top-left and bottom-right corners of the ranges to clear. I.e. a vector or each row specifies the coordinates {top row, left column, bottom row, right column}. You may use aref2idx to generate such a matrix.

##### Details

Clearing a cell range means to clear all the cells associated with that range.

'>workbook, clearSheet, clearNamedRegion, clearRangeFromReference, clearSheet

##### Aliases
• clearRange
• clearRange-methods
• clearRange,workbook,numeric-method
• clearRange,workbook,character-method
##### Examples
# NOT RUN {
# mtcars xlsx file from demoFiles subfolder of
# package XLConnect
demoExcelFile <- system.file("demoFiles/mtcars.xlsx",
package = "XLConnect")

# Clear range from top left corner (4,2) ^= B4 to
# bottom right corner (6,4) ^= D6
clearRange(wb, sheet = "mtcars", coords = c(4, 2, 6, 4))

# Clear two ranges in one go ...
mat = matrix(c(5, 1, 6, 4, 5, 7, 7, 9), ncol = 4,
byrow = TRUE)
clearRange(wb, sheet = "mtcars", coords = mat)

# The above is equivalent to ...
clearRange(wb, sheet = "mtcars",
coords = aref2idx(c("A5:D6", "G5:I7")))

# This in turn is the same as ...
clearRangeFromReference(wb, reference = c("mtcars!A5:D6",
"mtcars!G5:I7"))
# }

Documentation reproduced from package XLConnect, version 0.2-15, License: GPL-3

### Community examples

Looks like there are no examples yet.


text:
## Beverton-Holt fitting example

Here are some examples of fitting a Beverton-Holt model (a saturating hyperbolic model of the basic form $$y = x/(1+x)$$, also known as Michaelis-Menten; Monod; Holling type II; … ?) to data. It can be done via any nonlinear/maximum likelihood fitting routine, but with a bit of a trick you can also do it via glm(), which allows it be put automatically into a ggplot-based figure …

### Example 1

The data and question come from Holly Kindsvater: they're from Hocking MD and Reimchen TE. 2009. Salmon species, density, and watershed size predict magnitude of marine enrichment in riparian food webs. Oikos 118:1307-1318 (Table 1). $$X$$ is salmon spawning biomass (kg/m); $$Y$$ is the $$\delta^{15}N$$ index (isotopic signature). Other than that, I don't know anything about the biological context.

load("bevholt_ex.RData")
library(bbmle)

## Loading required package: stats4

Here's (one parameterization of) the Beverton-Holt function, in which $$a$$ is the asymptote ($$\lim_{x \to \infty} f(x)$$) and $$b$$ is the half-maximum. (Since this model has two parameters and three places you could put them ($$\bullet x/(\bullet + \bullet x)$$), and each parameter could be specified as $$c$$ or $$1/c$$, there are at least 8 ways to parameterize the model … alternatively, one could specify the asymptote, half-maximum, initial slope, or reciprocals of any of them …)

bevholt = function(S, a, b) {
S * a/(b + S)
}

Preliminary view of the data along with initial guess:

par(las = 1, bty = "l")
plot(Y ~ X, data = dat)
maxY <- max(dat$Y) curve(bevholt(x, a = maxY, b = maxY), add = TRUE)  We might already guess that this fit will be fairly poorly determined because both the asymptote and the half-maximum will be large (i.e. the fit may be pretty close to linear over the range of the data …) If we were just approaching this as a MLE/non-linear least squares problem we might run into trouble: nls(Y ~ bevholt(X, a, b), start = list(a = maxY, b = maxY), data = dat)  ## Error: singular gradient  If we use bbmle::mle2 it might be a little more robust, but we still run into convergence failures fit2 <- mle2(Y ~ dnorm(mean = bevholt(X, a, b), sd = exp(logsd)), data = dat, start = list(a = maxY, b = maxY, logsd = 0))  ## Warning: convergence failure: code=1 (iteration limit 'maxit' reached)  Increasing the number of iterations (e.g. control=list(maxit=10000)) or using other optimizers doesn't really help much; we'll see why shortly. coef(fit2)  ## a b logsd ## 12.1243 164.4537 0.7619  We see that as previously hinted both the asymptote $$a$$ and the half-maximum $$b$$ are estimated as being well beyond the range of the data: the best fit is very close to a straight line, which would correspond to making both the asymptote and the half-maximum infinite … par(las = 1, bty = "l") plot(Y ~ X, data = dat) with(as.list(coef(fit2)), curve(bevholt(x, a = a, b = b), lwd = 2, add = TRUE))  We can see a very slight curvature, but this is very close to straight – and we can't really tell whether a straight line would be better, since the parameters would have to be infinite and the optimizer can't really handle that. One alternative would be to fit a straight line and compare likelihoods: fit2L <- mle2(Y ~ dnorm(mean = c + d * X, sd = exp(logsd)), data = dat, start = list(c = 0, d = 0.01, logsd = 0)) fit2L0 <- mle2(Y ~ dnorm(mean = d * X, sd = exp(logsd)), data = dat, start = list(d = 0.01, logsd = 0)) -logLik(fit2)  ## 'log Lik.' 187.7 (df=3)  -logLik(fit2L)  ## 'log Lik.' 183.1 (df=3)  -logLik(fit2L0)  ## 'log Lik.' 186.2 (df=2)  It looks like the linear model is actually better (the negative log-likelihood is about 3 units lower). However, the main reason for the improvement is that the linear model allows the intercept to be non-zero: if we force the intercept to be zero, then the linear fit is only slightly better than the Beverton-Holt (but with one less parameter). We could also have used lm(), of course, but sometimes when comparing different fitting methods it's hard to figure out whether the log-likelihoods are really using the same additive constants or not. We can also reparameterize the model so that at least one of the parameters is 0, rather than infinite, in the limit of a straight line. For example, if $$y=x/(ex+f)$$, then $$e$$ is the reciprocal of the asymptote and $$f$$ is the reciprocal of the initial slope (it will turn out that this is a useful parameterization for other reasons as well): fit3 <- mle2(Y ~ dnorm(mean = X/(e * X + f), sd = exp(logsd)), data = dat, start = list(e = 0.1, f = 1, logsd = 1)) -logLik(fit3)  ## 'log Lik.' 186.1 (df=3)  We get a log-likelihood that is very slightly better than the linear model through the origin, but the reciprocal-asymptote parameter is actually negative: coef(fit3)  ## e f logsd ## -0.02049 22.49721 0.74536  This may or may not make any biological sense (it implies that the function goes through a singularity at $$x=-f/e=1098$$) We can constrain $$e$$ to be $$\geq 0$$: fit4 <- update(fit3, lower = c(a = 0, b = 0, logsd = -Inf), method = "L-BFGS-B")  Here's the comparison – it's worth keeping in mind that none of these curves is likely to be “significantly” different in the non-statistical sense: par(las = 1, bty = "l") plot(Y ~ X, data = dat) with(as.list(coef(fit2)), curve(bevholt(x, a = a, b = b), add = TRUE)) with(as.list(coef(fit2L)), curve(c + d * x, col = 2, add = TRUE)) with(as.list(coef(fit2L0)), curve(d * x, col = 4, add = TRUE)) with(as.list(coef(fit3)), curve(x/(e * x + f), col = 5, add = TRUE)) with(as.list(coef(fit4)), curve(x/(e * x + f), col = 6, add = TRUE))  (the red line is the linear fit that is allowed to have a non-zero intercept). AICtab(fit2, fit2L, fit2L0, fit3, fit4)  ## dAIC df ## fit2L 0.0 3 ## fit2L0 4.1 2 ## fit4 6.0 3 ## fit3 6.0 3 ## fit2 9.2 3  But what if we want confidence intervals? Getting confidence intervals on a nonlinear fit is a bit of a nuisance (we can use the delta method, which may under some circumstances be an unreliable approximation, or bootstrapping or MCMC, which may be nuisances to implement). In principle we can fit the last form of the Beverton-Holt shown above ($$y=x/(ex+f)$$) by using a Gaussian GLM with an inverse link and a response variable of $$1/x$$, i.e. $$1/y \sim \text{Normal}(\mu=e+f\cdot(1/x))$$: g1 <- glm(Y ~ I(1/X), family = gaussian(link = "inverse"), data = dat)  However, this doesn't actually work: coef(g1)  ## (Intercept) I(1/X) ## 1.700 -2.041  It's OK if we start from the results of fit3: coef(g2 <- glm(Y ~ I(1/X), family = gaussian(link = "inverse"), data = dat, start = coef(fit3)[1:2]))  ## (Intercept) I(1/X) ## -0.02112 22.56702  Or even from the starting values that we used for fit3: coef(g2B <- glm(Y ~ I(1/X), family = gaussian(link = "inverse"), data = dat, start = c(0.01, 1)))  ## (Intercept) I(1/X) ## -0.02112 22.56697  The convenience of doing things this way is that now ggplot2 can automatically add the curve to a plot, with error bars: library(ggplot2) theme_set(theme_bw()) p1 <- ggplot(dat, aes(X, Y)) + theme_bw() p1 + geom_point() + geom_smooth(method = "glm", family = gaussian(link = "inverse"), formula = y ~ I(1/x), start = c(0.01, 1))  ## Other possibilities • explore whether the 'bad' glm() fit is a local maximum (e.g. via emdbook::slice())? • see whether glm2::glm2() does better with unspecified starting points? How about nls2, minpack.lm ? glm2 gets the same bad answer as glm, although at least we get a warning message: coef(g3 <- glm2::glm2(Y ~ I(1/X), family = gaussian(link = "inverse"), data = dat))  ## (Intercept) I(1/X) ## 1.700 -2.041  -logLik(g3)  ## 'log Lik.' 204.5 (df=3)  -logLik(g2)  ## 'log Lik.' 186.1 (df=3)  What does the likelihood/SSQ surface look like? ## dnorm with sd profiled out dnorm2 <- function(x, mean, log = FALSE) { ssq <- sum((x - mean)^2) dnorm(x, mean, sd = sqrt(ssq/length(x)), log = log) } ff <- function(e, f, X = dat$X, Y = dat$Y) { mu <- e + f * (1/X) pred <- 1/mu -sum(dnorm2(Y, mean = pred, log = TRUE)) }  This agrees with the previous fit: do.call(ff, as.list(coef(fit3)[1:2]))  ## [1] 186.1  -logLik(fit3)  ## 'log Lik.' 186.1 (df=3)  We can draw a likelihood surface to see what's going wrong with the GLM (or MLE) fits, picking a range that includes both the 'bad' and 'good' fits: library(emdbook) par(las = 1) cc <- curve3d(ff(x, y), xlim = c(-0.5, 2), ylim = c(-5, 25), sys3d = "image", useRaster = TRUE, col = gray((100:1)/100), xlab = "e", ylab = "f") with(cc, contour(x, y, z, add = TRUE)) points(coef(g1)[1], coef(g1)[2], pch = 16, col = 4) points(coef(g2)[1], coef(g2)[2], pch = 16, col = 5) points(0, 0, col = 2, pch = 16) points(0.1, 1, col = 6, pch = 16) tred <- adjustcolor("red", alpha = 0.5) curve(-x, add = TRUE, lwd = 3, col = tred) curve(-114 * x, add = TRUE, lwd = 3, col = tred) legend("topright", c("good fit", "bad fit", "GLM start", "MLE start"), pch = 16, col = c(5, 4, 2, 6))  We know we're in trouble whenever $$1/y = e+(f/x) \approx 0$$ for any value of $$x$$ in the data set, which corresponds to $$f=-ex$$. The range of $$x$$ is {1,114}: the red lines show the curves corresponding to the extreme values, which show exactly the range where we have trouble. If we start inside this zone it's likely that we'll end up somewhere arbitrary. By luck, the starting value for the MLE was just outside this zone … So despite the general cleverness of GLM, it's not necessarily easier/more robust for this case. It is apparently possible to fit a GLM with inequality constraints, supposedly one goes to p. 445 of Venables and Ripley. That page isn't shown in Google Books, and I don't have a copy with me at the moment … but since this is apparently an example of fitting a GLM via direct maximization rather IRLS, it may not be that much better than what I've already shown above using bbmle::mle2. ### Example 2 From Holly again: $$X$$=salmon density (kg/m2), $$Y$$=number of salmon captured by brown bears: dat2 <- setNames(read.csv("gende&quinn_1.csv", header = FALSE), c("X", "Y")) par(las = 1, bty = "l") plot(Y ~ X, data = dat2) curve(x/((1/50) + 0.01 * x), add = TRUE)  Let's say this is a straight line with an nearly infinite asymptote ($$e \approx 0$$) and initial slope approximately 50 ($$f \approx 1/50$$) I tried to use method="L-BFGS-B" again to impose bounds, but it complained, so I used bobyqa from the optimx/minqa packages: library(optimx) library(minqa) d2fit1 <- mle2(Y ~ dnorm2(mean = X/(e * X + f)), data = dat2, start = list(e = 0.01, f = 1/50), optimizer = "optimx", method = "bobyqa", lower = c(e = 0, f = 0))  Check out the likelihood surface (not really necessary; I had some initial confusion over whether $$f$$ was the slope or the reciprocal slope, which had me plotting the likelihood surface to see what was going on …) cc2 <- curve3d(ff(x, y, X = dat2$X, Y = dat2$Y), xlim = c(0, 0.2), ylim = c(0, 0.1), sys3d = "image", useRaster = TRUE, col = gray((100:1)/100), xlab = "e", ylab = "f") with(cc2, contour(x, y, z, add = TRUE)) points(coef(d2fit1)[1], coef(d2fit1)[2], col = 4, pch = 16) points(0.01, 1/50, col = 2, pch = 16)  Check out the fit in base graphics: par(las = 1, bty = "l") plot(Y ~ X, data = dat2) with(as.list(coef(d2fit1)), curve(x/(e * x + f), add = TRUE))  Since we now know where to tell glm() where to start we can get ggplot2 to do it: ggplot(dat2, aes(X, Y)) + theme_bw() + geom_point() + geom_smooth(method = "glm", formula = y ~ I(1/x), family = gaussian(link = "inverse"), start = coef(d2fit1)[1:2])  If we needed to generate the confidence intervals and plot themselves, we would use the approach used by glm()/ggplot(), which is: (1) compute the variances of the predicted values via $$X^T V X$$, where $$X$$ is the model matrix of the predictor variables for the desired prediction and $$V$$ is the variance-covariance matrix of the parameters; (2) compute the confidence intervals of the predictions on the linear predictor scale via Normal approximation (i.e. $$\hat x \pm 1.96 \sigma$$); see an example of this general approach on the GLMM FAQ back-transform the confidence intervals to the raw scale; (4) use geom_ribbon() with no border line (color=NA outside of aes()) and alpha set accordingly to add the ribbon to the plot. Similarly difficult fitting was done by Poulsen et al.: Poulsen, J. R., C. W. Osenberg, C. J. Clark, D. J. Levey, and B. M. Bolker. 2007. “Plants as Reef Fish: Fitting the Functional Form of Seedling Recruitment.” The American Naturalist 170 (2) (August): 167–183. doi:10.1086/518945. JSTOR. If I were trying to implement an automatic (self-starting) algorithm that would succeed on as many noisy/small B-H data sets as possible, I would probably try some combination of • constraining the fit to non-negative parameter values; • choosing the most robust parameterization (I think $$y=X/(eX+f)$$ is not too bad and matches the inverse-link GLM, although it might be good to use $$X/(eX+1/g)$$ instead so that $$g=0$$ corresponded to an infinite initial slope (which would give a flat, non-zero line); • using the most robust optimizer that allows box constraints (BOBYQA is not too bad; there are another couple of options such as an alternate BOBYQA implementation in the nloptwrap package; an AD Model Builder solution would be pretty easy to implement for this simple case too); • finding the best way to get automatic starting values (e.g. maybe try linear regression on a subset of the data, taking the mean of a subset of the data, and a glm2 inverse-link fit, and use the best combination of those approaches). ### Example 3 It gets worse. Even if glm() reaches a plausible (positive-coefficient) answer, the confidence intervals can include implausible values. dat3 <- setNames(read.csv("nagasaka.csv", header = FALSE), c("X", "Y")) g3 <- ggplot(dat3, aes(X, Y)) + geom_point() g3 + geom_smooth(method = "glm", formula = y ~ I(1/x), family = gaussian(link = "inverse"), start = c(0.2, 0.01), fill = "blue", alpha = 0.2) + geom_smooth(method = "loess", color = "red", fill = "red", alpha = 0.2) + coord_cartesian(ylim = c(-4, 10))  (g4 <- glm(Y ~ I(1/X), family = gaussian(link = "inverse"), data = dat3, start = c(0.2, 0.01)))  ## ## Call: glm(formula = Y ~ I(1/X), family = gaussian(link = "inverse"), ## data = dat3, start = c(0.2, 0.01)) ## ## Coefficients: ## (Intercept) I(1/X) ## 0.0907 473.8105 ## ## Degrees of Freedom: 11 Total (i.e. Null); 10 Residual ## Null Deviance: 77.6 ## Residual Deviance: 31.9 AIC: 51.8  pframe <- with(dat3, data.frame(X = seq(min(X), max(X), length = 101))) pred <- predict(g4, newdata = pframe, se.fit = TRUE) minval <- 1e-06 cc <- qnorm(0.975) invlink <- g4$family$linkinv pframe <- with(pred, transform(pframe, Y = invlink(fit), lwr = invlink(fit - cc * se.fit), clwr = invlink(pmax(minval, fit - cc * se.fit)), upr = invlink(fit + cc * se.fit))) g3 + geom_line(data = pframe) + geom_ribbon(data = pframe, aes(ymin = lwr, ymax = upr), colour = NA, alpha = 0.2) + geom_ribbon(data = pframe, aes(ymin = clwr, ymax = upr), fill = "blue", colour = NA, alpha = 0.2) + coord_cartesian(ylim = c(-4, 10))  cc3 <- curve3d(ff(x, y, X = dat3$X, Y = dat3$Y), xlim = c(0, 0.8), ylim = c(100, 2500), sys3d = "image", useRaster = TRUE, n = c(71, 71), col = gray((100:1)/100), xlab = "e", ylab = "f") with(cc3, contour(x, y, z, add = TRUE, levels = min(cc3$z) + qchisq(c(0.9, 0.95,
0.975), 2)/2))
points(coef(g4)[1], coef(g4)[2], col = 4, pch = 16, cex = 2)

The figure shows the nominal 90%, 95% and 97.5% LRT confidence intervals – as we will see, these are incorrect (too conservative) because part of the area gets cut off at the zero boundaries, so we have to include more area in the feasible region.

Sample from a quasi-Bayesian posterior density (sample automatically rescales its weights to sum to 1.0):

set.seed(101)
ss <- sample(seq_along(cc3$z), prob = exp(-cc3$z), size = 1000, replace = TRUE)

## Warning: Walker's alias method used: results are different from R < 2.2.0

ind2vals <- function(k) {
i <- k%/%ncol(cc3$z) + 1 j <- k%%ncol(cc3$z) + 1
c(cc3$x[i], cc3$y[j])
}
samp <- t(sapply(ss, ind2vals))

image(cc3, xlim = c(0, 0.8), ylim = c(100, 2500), useRaster = TRUE, col = gray((100:1)/100),
xlab = "e", ylab = "f")
with(cc3, contour(x, y, z, level = min(cc3$z) + qchisq(c(0.9, 0.95, 0.975), 2)/2, add = TRUE)) points(samp[, 1], samp[, 2], col = adjustcolor("red", alpha = 0.2), pch = 16, cex = 1.5) points(coef(g4)[1], coef(g4)[2], col = 4, pch = 16, cex = 2)  pframe2 <- pframe allpred <- apply(samp, 1, function(cc) { 1/(cc[1] + cc[2]/pframe2$X)
})
pframe2[, c("lwr", "upr")] <- setNames(t(apply(allpred, 1, quantile, c(0.025,
0.975))), c("lwr", "upr"))
g3 + geom_line(data = pframe2) + geom_ribbon(data = pframe2, aes(ymin = lwr,
ymax = upr), colour = NA, alpha = 0.2) + coord_cartesian(ylim = c(-4, 10))

Alternatively we could do this by real MCMC, either with BUGS or with MCMCpack

### Example 4

dat4 <- na.omit(setNames(read.csv("Hocking_Reynolds_d15N_blueberry.csv", header = FALSE),
c("X", "Y")))
p4 <- qplot(X, Y, data = dat4) + geom_smooth(method = "glm", family = gaussian(link = "inverse"))
coef(glm(Y ~ X, data = dat4, family = gaussian(link = "inverse")))

## (Intercept)           X
##     0.22934    -0.04653

d4fit1 <- mle2(Y ~ dnorm2(mean = X/(e * X + f)), data = dat4, start = list(e = 0.01,
f = 1/50), optimizer = "optimx", method = "bobyqa", lower = c(e = 1e-06,
f = 1e-06))
pframe <- with(dat4, data.frame(X = seq(min(X), max(X), length = 101)))
pframe$Y <- with(c(as.list(coef(d4fit1)), pframe), X/(e * X + f)) p4 + geom_line(data = pframe, colour = "red")  Since geom_smooth(method="glm",...) doesn't work here, we'd have to do something else to get the confidence intervals cc4 <- curve3d(ff(x, y, X = dat4$X, Y = dat4$Y), xlim = c(0.1, 0.3), ylim = c(1e-06, 0.03), sys3d = "image", useRaster = TRUE, n = c(71, 71), col = gray((100:1)/100), xlab = "e", ylab = "f") with(cc4, contour(x, y, z, add = TRUE, levels = min(cc4$z) + qchisq(c(0.9, 0.95,
0.975), 2)/2, labels = c(0.9, 0.95, 0.975)))

et cetera …


text:
### Calendar Capers

Choose any three by three square of dates on a calendar page...

### Card Trick 2

Can you explain how this card trick works?

### Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

# Quick Times

##### Age 11 to 14 Challenge Level:

$$\begin{eqnarray} 32\times 38 &=& 30\times 40 + 2\times 8 \\ 34\times36 &=& 30\times40 + 4\times6\\ 56\times54 &=& 50\times60 + 6\times4\\ 73\times77 &=& 70\times80 + 3\times7\\ \end{eqnarray}$$

Chloe and Nina (Y7) St James Middle School, Bury St Edmunds found these examples:

$$\begin{eqnarray} 81\times89 &=& 80\times90 + 1\times9 = 7209 \\ 72\times78 &=& 70\times80 + 2\times8 = 5616.\\ \end{eqnarray}$$

HOW WE WORKED IT OUT: The units on the first two numbers have to add up to 10 to make this work and the second pair of numbers on the right hand side also have to add up to 10. The tens in the first product on the right hand side have to be 10 apart.

Suzanne Abbott and Nisha Doshi, (Y10) The Mount School, York proved that the method always works. Here is their solution:

Yes these are all correct. Here are some more examples

$$\begin{eqnarray} 51\times59 &=& 50\times60 + 1\times9 \\ 45\times45 &=& 40\times50 + 5\times5\\ \end{eqnarray}$$

It always seems to work. Using algebra, we can try to generalise this

$$\begin{eqnarray} (10a + b)(10a + c) &=& 10a \times10(a + 1) + bc \\ 100a^2 +10ac +10ab + bc &=& 100a^2 + 100a + bc\\ 10ac + 10ab &=& 100a\\ 10a(b + c) &=& 100a\\ so b + c &=& 10\\ \end{eqnarray}$$

This fits with the examples we were given and the ones we have chosen.

This proof depends on the argument being reversible. Set out slightly differently it shows that the method works if and only if $b + c = 10$ and it is perfectly rigorous.

$$\begin{eqnarray} (10a + b)(10a + c) &=& 100a^2 + 10ab + 10ac + bc \\ &=& 100a^2 + 100a + bc + 10ab + 10ac -100a \\ &=& 100a^2 + 100a + bc + 10a(b + c - 10)\\ &=& 10a\times 10(a + 1) + bc \\ \end{eqnarray}$$


text:
# Proving Crazy Vector Space with addition and multiplication

Let $V$ be a vector space. Let $f : V \rightarrow V$ be a bijection. Define two new operations $+_f$ and $\cdot_f$ as follows. If $v$ and $w$ are two vectors in $V$, $v +_f w$ is defined to be the vector $f^{-1}(f(v) + f(w))$ where $f^{-1}$ is the inverse function of $f$. If $a$ is a scalar and $v$ is a vector in $V$, $a \cdot_f v$ is defined to be the vector $f^{-1}(af(v))$. Prove that $V$ together with the new addition of vectors, $+_f$, and the new multiplication of vectors by scalars, $\cdot_f$, is also vector space.

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What have you tried? Your problem is to prove that $V$ is a vector space under the new operation. What would you need to show in order to have proved this? – MJD Mar 26 '12 at 17:42
Google "transport of structure". – Bill Dubuque Mar 26 '12 at 17:48
Noone seems to figure this out. Bummer..! – user28873 Apr 11 '12 at 22:07
Actually Bill's comment is a nearly complete answer. Furthermore this is really nothing more than a simple exercise in verification that the definition of "vector space" holds with the new operations. – Asaf Karagila Apr 11 '12 at 22:48

Replace your problem by the following: Let $X$ be a set and $V$ be a vector space, and let $f:X\to V$ be a bijection. Define operations $\oplus$ and $\otimes$ on $X$ as follows: $\ldots$
It will turn out that you have just transported the vector space structure of $V$ back to $X$, or what is the same thing: The vectors $v\in V$ have got new names $x:=f^{-1}(v)\in X$, but otherwise the operations remain unchanged.


text:
Near Perfect Mood

## latex

0.0.1 • Public • Published

# node-latex

Simple LaTeX wrapper for node.js

# Installation

First, you need to install latex. On any Debian based system, you can do this with the following command:

sudo apt-get install texlive

On OS X, you will need to install MacPorts first. Once that is set up, you can then do:

sudo port install texlive

For Windows, you can try using cygwin though I have not tested this.

Once you have a working version of latex, you can install node-latex using the following command:

npm install latex

# Usage

Here is an example of how to use the library in one line:

require("latex")([
"\\documentclass{article}",
"\\begin{document}",
"hello world",
"\\end{document}"
]).pipe(process.stdout)

This will spit out a formatted PDF article to stdout that says "hello world". The result of calling the function is returned as a stream and can be processed using other tools. If you want to convert the result into an image or pdf, you can use graphics magic.

## require("latex")(doc[, options])

The only exported function from node-latex is a function that takes as input a raw LaTeX document and returns a stream representing the document state. The type of doc must be one of the following:

• A string
• A Buffer
• An array of strings and/or Buffers

In addition, you can also specify the following additional parameters via the options struct:

• command: An optional override for the latex command. By default calls latex.
• format: Either "pdf" or "dvi". By default returns a pdf.

The function returns a readable Stream object representing a LaTeX encoded document in either PDF or DVI format. If there were errors in the syntax of the document, they will be raised as errors on this Stream object.

# Credits

(c) 2013 Mikola Lysenko. MIT License

## Keywords

none

### install

npm i latex

50

0.0.1

none

github.com

github


text:
# American Institute of Mathematical Sciences

November  2019, 12(7): 2127-2141. doi: 10.3934/dcdss.2019137

## On periodic solutions in the Whitney's inverted pendulum problem

 Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Received  November 2017 Revised  April 2018 Published  December 2018

Fund Project: This research is partially supported by the Polish National Science Center under Grant No. 2014/14/A/ST1/00453.

In the book "What is Mathematics?" Richard Courant and Herbert Robbins presented a solution of a Whitney's problem of an inverted pendulum on a railway carriage moving on a straight line. Since the appearance of the book in 1941 the solution was contested by several distinguished mathematicians. The first formal proof based on the idea of Courant and Robbins was published by Ivan Polekhin in 2014. Polekhin also proved a theorem on the existence of a periodic solution of the problem provided the movement of the carriage on the line is periodic. In the present paper we slightly improve the Polekhin's theorem by lowering the regularity class of the motion and we prove a theorem on the existence of a periodic solution if the carriage moves periodically on the plane.

Citation: Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137
##### References:
 [1] V. I. Arnol'd, What is Mathematics?, (Russian), MCNMO, Moscow, 2002. Google Scholar [2] V. I. Arnol'd, Mathematical Understanding of Nature, (Russian), MCNMO, Moscow, 2009. English translation: V. I. Arnold, Mathematical Understanding of Nature, Amer. Math. Soc., Providence, R. I., 2014. doi: 10.1090/mbk/085.  Google Scholar [3] B. E. Blank, Book review "What is Mathematics? An Elementary Approach to Ideas and Methods", Notices Amer. Math. Soc., 48 (2001), 1325-1329.   Google Scholar [4] S. V. Bolotin and V. V. Kozlov, Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney's inverted pendulum problem,, Izv. Math., 79 (2015), 894-901.  doi: 10.4213/im8413.  Google Scholar [5] A. Broman, A mechanical problem by H. Whitney, Nordisk Matematisk Tidskrift, 6 (1958), 78-82.   Google Scholar [6] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.1090/S0002-9947-1992-1042285-7.  Google Scholar [7] L. Consolini and M. Tosques, On the existence of small periodic solutions for 2-dimensional inverted pendulum on a cart, SIAM J. Appl. Math., 68 (2007), 486-502.  doi: 10.1137/070683404.  Google Scholar [8] L. Consolini and M. Tosques, On the exact tracking of the spherical inverted pendulum via a homotopy method, Systems Control Lett., 58 (2009), 1-6.  doi: 10.1016/j.sysconle.2008.06.010.  Google Scholar [9] L. Consolini and M. Tosques, A continuation theorem on periodic solutions of regular nonlinear systems and its application to the exact tracking problem for the inverted spherical pendulum,, Nonlinear Anal., 74 (2011), 9-26.  doi: 10.1016/j.na.2010.08.002.  Google Scholar [10] R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1941.   Google Scholar [11] R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, 2nd edition revised by Ⅰ. Stewart, Oxford University Press, 1996.  Google Scholar [12] C. Davis, Christopher Zeeman Medal Award lecture, The London Math. Soc. Newsletter, 384 (2009), 33-34.   Google Scholar [13] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [14] A. Dold, Lectures on Algebraic Topology, 2nd edition, Springer-Verlag, Berlin, Heidelberg and New York, 1980.  Google Scholar [15] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., 568, Springer-Verlag, Berlin, Heidelberg and New York, 1977.  Google Scholar [16] L. Gillman, Review of "What is Mathematics?" by Richard Courant and Herbert Robbins, revised by Ian Stewart, Amer. Math. Monthly, 105 (1998), 485-488.  doi: 10.2307/3109832.  Google Scholar [17] J. E. Littlewood, A Mathematician's Miscellany, Methuen & Co., London, 1953.  Google Scholar [18] J. R. Newman (ed.), The World of Mathematics, Vol. Ⅳ., Allen & Unwin, London, 1960. Google Scholar [19] I. Polekhin, Examples of topological approach to the problem of inverted pendulum with moving pivot point, (Russian), Nelin. Dinam., 10 (2014), 465-472.   Google Scholar [20] I. Polekhin, Periodic and falling-free motion of an inverted spherical pendulum with a moving pivot point, preprint, arXiv: 1411.1585. Google Scholar [21] T. Poston, Au courant with differential equations, Manifold, 18 (1976), 6-9.   Google Scholar [22] R. Srzednicki, On rest points of dynamical systems, Fund. Math., 126 (1985), 69-81.  doi: 10.4064/fm-126-1-69-81.  Google Scholar [23] R. Srzednicki, Periodic and constant solutions via topological principle of Ważewski, Univ. Iagel. Acta Math., 26 (1987), 183-190.   Google Scholar [24] R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal., 22 (1994), 707-737.  doi: 10.1016/0362-546X(94)90223-2.  Google Scholar [25] R. Srzednicki, Ważewski method and Conley index, in Handbook of Ordinary Differential Equations, Vol Ⅰ., (eds. A. Cañada, P. Drábek and A. Fonda), Elsevier/North Holland, Amsterdam, (2004), 591-684.  Google Scholar [26] I. Stewart, Gem, Set and Math, Blackwell Ltd., London, 1989.  Google Scholar [27] F. Zanolin, Bound sets, periodic solutions and flow-invariance for ordinary differential equations in $\mathbb R^n$: Some remarks, Rend. Ist. Mat. Univ. Trieste, 19 (1987), 76-92.   Google Scholar [28] O. Zubelevich, Bounded solutions to the system of second order ODEs and the Whitney pendulum, Appl. Math. (Warsaw), 42 (2015), 159-165.  doi: 10.4064/am42-2-3.  Google Scholar

show all references

##### References:
 [1] V. I. Arnol'd, What is Mathematics?, (Russian), MCNMO, Moscow, 2002. Google Scholar [2] V. I. Arnol'd, Mathematical Understanding of Nature, (Russian), MCNMO, Moscow, 2009. English translation: V. I. Arnold, Mathematical Understanding of Nature, Amer. Math. Soc., Providence, R. I., 2014. doi: 10.1090/mbk/085.  Google Scholar [3] B. E. Blank, Book review "What is Mathematics? An Elementary Approach to Ideas and Methods", Notices Amer. Math. Soc., 48 (2001), 1325-1329.   Google Scholar [4] S. V. Bolotin and V. V. Kozlov, Calculus of variations in the large, existence of trajectories in a domain with boundary, and Whitney's inverted pendulum problem,, Izv. Math., 79 (2015), 894-901.  doi: 10.4213/im8413.  Google Scholar [5] A. Broman, A mechanical problem by H. Whitney, Nordisk Matematisk Tidskrift, 6 (1958), 78-82.   Google Scholar [6] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.1090/S0002-9947-1992-1042285-7.  Google Scholar [7] L. Consolini and M. Tosques, On the existence of small periodic solutions for 2-dimensional inverted pendulum on a cart, SIAM J. Appl. Math., 68 (2007), 486-502.  doi: 10.1137/070683404.  Google Scholar [8] L. Consolini and M. Tosques, On the exact tracking of the spherical inverted pendulum via a homotopy method, Systems Control Lett., 58 (2009), 1-6.  doi: 10.1016/j.sysconle.2008.06.010.  Google Scholar [9] L. Consolini and M. Tosques, A continuation theorem on periodic solutions of regular nonlinear systems and its application to the exact tracking problem for the inverted spherical pendulum,, Nonlinear Anal., 74 (2011), 9-26.  doi: 10.1016/j.na.2010.08.002.  Google Scholar [10] R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 1941.   Google Scholar [11] R. Courant and H. Robbins, What is Mathematics? An Elementary Approach to Ideas and Methods, 2nd edition revised by Ⅰ. Stewart, Oxford University Press, 1996.  Google Scholar [12] C. Davis, Christopher Zeeman Medal Award lecture, The London Math. Soc. Newsletter, 384 (2009), 33-34.   Google Scholar [13] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [14] A. Dold, Lectures on Algebraic Topology, 2nd edition, Springer-Verlag, Berlin, Heidelberg and New York, 1980.  Google Scholar [15] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., 568, Springer-Verlag, Berlin, Heidelberg and New York, 1977.  Google Scholar [16] L. Gillman, Review of "What is Mathematics?" by Richard Courant and Herbert Robbins, revised by Ian Stewart, Amer. Math. Monthly, 105 (1998), 485-488.  doi: 10.2307/3109832.  Google Scholar [17] J. E. Littlewood, A Mathematician's Miscellany, Methuen & Co., London, 1953.  Google Scholar [18] J. R. Newman (ed.), The World of Mathematics, Vol. Ⅳ., Allen & Unwin, London, 1960. Google Scholar [19] I. Polekhin, Examples of topological approach to the problem of inverted pendulum with moving pivot point, (Russian), Nelin. Dinam., 10 (2014), 465-472.   Google Scholar [20] I. Polekhin, Periodic and falling-free motion of an inverted spherical pendulum with a moving pivot point, preprint, arXiv: 1411.1585. Google Scholar [21] T. Poston, Au courant with differential equations, Manifold, 18 (1976), 6-9.   Google Scholar [22] R. Srzednicki, On rest points of dynamical systems, Fund. Math., 126 (1985), 69-81.  doi: 10.4064/fm-126-1-69-81.  Google Scholar [23] R. Srzednicki, Periodic and constant solutions via topological principle of Ważewski, Univ. Iagel. Acta Math., 26 (1987), 183-190.   Google Scholar [24] R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal., 22 (1994), 707-737.  doi: 10.1016/0362-546X(94)90223-2.  Google Scholar [25] R. Srzednicki, Ważewski method and Conley index, in Handbook of Ordinary Differential Equations, Vol Ⅰ., (eds. A. Cañada, P. Drábek and A. Fonda), Elsevier/North Holland, Amsterdam, (2004), 591-684.  Google Scholar [26] I. Stewart, Gem, Set and Math, Blackwell Ltd., London, 1989.  Google Scholar [27] F. Zanolin, Bound sets, periodic solutions and flow-invariance for ordinary differential equations in $\mathbb R^n$: Some remarks, Rend. Ist. Mat. Univ. Trieste, 19 (1987), 76-92.   Google Scholar [28] O. Zubelevich, Bounded solutions to the system of second order ODEs and the Whitney pendulum, Appl. Math. (Warsaw), 42 (2015), 159-165.  doi: 10.4064/am42-2-3.  Google Scholar
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text:
Question #67709

Sep 8, 2017

The mass of the compound is irrelevant in this problem, probably meant to confuse us.

$6.78 g \cdot \frac{C {O}_{2}}{44.0 g} \approx 0.154 m o l$
$1.94 g \cdot \frac{{H}_{2} O}{18.0 g} \cdot \frac{2 H}{{H}_{2} O} \approx 0.216 m o l$
$0.432 g \cdot \frac{{N}_{2}}{28.0 g} \cdot \frac{2 N}{{N}_{2}} \approx 0.0309 m o l$

Divide every number that's larger than the smallest amount of moles and we get...

${C}_{5} {H}_{7} N$