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# Local scope benefits

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I have been wondering this for a while now and this is in very general terms. Including a general non-specific programming language context.

But which of the below is "better." Not just in speed but in practice. Is there even any real difference?
//Local scope variable outside the for loop
CoolObject obj;
for(int i = 0; i < 5; ++i)
{
obj = coolObjectList[i];
obj.doWork();
}

//Local scope variable inside the for loop
for(int i = 0; i < 5; ++i)
{
CoolObject obj = coolObjectList[i];
obj.doWork();
}

In what instance does the placement of local scope variables effect the quality of ones code?

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By keeping a variable's scope limited, you make the code easier to read (because you don't have to scroll so far to find a variable's declaration). It also removes a small class of errors such as accidentally using a variable where you shouldn't be able to (e.g. because you made a typo and intended to refer to some other variable). It will also declutter debugging since a debugger will only show you the variables in scope.

However, the variable is constructed when it is encountered and destructed when that scope is left. In your first example, CoolObject is created once, assigned to 5 times, then destroyed later. In your second example, CoolObject is created and destroyed 5 times - although whether that is significantly different from being assigned to 5 times depends on the object. (And in languages other than C/C++, assignment often has quite different semantics.)

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In the example you give, in most languages and most scenarios the work will be identical.  This is not always the case.

In your specific example there are some extremely basic optimizations that will be made by almost every compiler. The first is to eliminate the temporary, the second is to unroll the loop.  In both cases it will be trivially optimized to this:

coolObjectList[0].doWork();

coolObjectList[1].doWork();

coolObjectList[2].doWork();

coolObjectList[3].doWork();

coolObjectList[4].doWork();

In other cases where the compiler couldn't be certain about the optimizations, so it couldn't take them.

Removing temporary objects is a common task for compilers. Various things can prevent that. Perhaps the assignment needed to go through a custom assignment operator, then the change could make a difference.

Hoisting constant values out of loops, hoisting object creation out of loops, hoisting allocation and other expensive allocations out of loops, these are very common changes to make in a code base: do something once rather than doing it many times.  In this simple case the compiler can do it for you, in more complex cases it may not.

Another item is that you looped a fixed number of times.  Loops have a very tiny performance impact, It is on the order of a fraction of a nanosecond each time, but doing it thousands of times every frame for millions of frames and the time accumulates. If you looped a variable number of times the compiler may not be able to unroll the loop.

Mainstream compilers are generally quite aggressive with optimizations, and they contain databases of an enormous number of software patterns they can quietly improve. If you see in your code that it is something you should improve, doing the work once rather than doing it in a loop, then do so.  Otherwise you can generally trust the optimizer to do its job behind your back.

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Note that in C++ the code will not do what you expect. In both operations you make a deep copy of your object and then update it. If the doWork() function will change the state of the object it will not be reflected in the object list. I think in C# it might work . You cannot look at the problem from a general language agnostic perspective.

Personally I think declaring variables at a different location from where it is used should have very a good reasons. Write code for readability first, optimize second if you have proof it is important for performance. You write code for human beings first and not the compiler. The correct version in C++ for your loop would be using a reference:

for(int i = 0; i < 5; ++i)
{
CoolObject& obj = coolObjectList[i];
obj.doWork();
}

If in your language this resolves to a reference/pointer anyway, it doesn't matter which form you use for performance.

Edited by Dirk Gregorius

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In addition to what others said, note that in C++ declaring a variable will invoke its constructor, which can have side effects.

The big advantage in keeping declarations close to their use is that you can forget about the declaration shortly afterwards. After its last use, the variable becomes "dead", to use a term from compiler jargon. The fewer live variables you have at once, the less space you need in your head to read the code.

Clear head. Clear code.


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# On transferring human consciousness to a computer, roughly how much memory and processing power would it take? [duplicate]

Designing my own RPG, and transferring human consciousness into machines is a part of the background. Wondering what are the real world numbers for doing such a thing. Memory, processing power, heat generation, any details that would make it more plausible.

• is the transfering for virtual-reality style uses or something else? – Ajnatorix Zersolar Mar 18 '18 at 11:43

A human brain is thought to contain about 86 billion neurons and these neurons can be modeled reasonably accurately bu a set of four equations known as the Hodgkin-Huxley model, which has four variables and is first order, meaning we need about eight numbers to store it's state at any instant.

So assuming each number can be stored as a standard double precision floating point numbers ( 64 bits each ) then we need $4.4\times 10^{13}$ bits or about 5.5 GB.

Which is a surprisingly small number, well withing the reach of current computers.

Of course doing 86 billions calculations involving solving 4 simultaneous differential equations with any reasonable speed is well beyond anything we can currently do in a brain-sized object.

Apparently the peak rate a neuron fires is about 1000 Hz and that's the minimum to consider a computer brain "simulation". That means we need something like to be able to do about $9\times 10^{13}$ of these mathematical solutions per second, and each of those would take perhaps a thousand floating point operations. So that's ballpark $10^{16}$ Flops.

That's about the same computing power as IBM's Sequoia super computer, which occupies about $280\,m^2$ and consumes a staggering 6 megawatts of power, all of which has to be dealt with for cooling purposes.

Note that the 5.5 GB of memory can't be standard memory like your PC has. We'd really be using memory-per-neuron with some working store. More like a giant GPU with lots of local memory for neuron state and local calculations and access to global memory when needed. That would considerably increase power consumption and hence heat dissipation issues. At least an order of magnitude.

• You forgot to compute the memory space required to store the state of the links between the neurons... – AlexP Mar 18 '18 at 19:46
• @AlexP Well spotted. If I get a chance I'll revise that. – StephenG Mar 18 '18 at 20:24

I read a lengthy description (I think it was James P. Hogan's Reality Interrupt) and at that time I checked the numbers, which checked out; Hogan was way down alternative-science alley, but he usually did do his research.

His "brain emulator" was based on a cupboard-sized stack of silicon chips (probably achievable now) containing around 100 billion "neuristors", each composed of a tiny CPU and a memory of two kilobit - or was it kilobytes?. The chip was actually a latticework through which a dense cooling fluid was continuously circulated.

The power requirement was in the hundreds of kilowatts range, so the actual machinery, with heat pumps, fans etc., was way larger than a head - more like a large room.

Comparing these numbers with StephenG's answer, they aren't that far away, and Hogan's were probably based on the same assumptions.

Solving an ordinary differential equation of this kind, with dedicated hardware, seems way less costly than StephenG's estimates by almost three orders of magnitude.

Also, it stands to reason that a sequence of neural operations would not be completely unrelated, but instead (very probably) linked together, so that there would be ways of optimizing results and saving time.

A major difference lies in neuron firing frequency - while true that a neuron has a maximum firing rate of one thousand per second, the average rate is way lower by almost four orders of magnitude.

So, the actual transition frequency and power requirements go well down, back into the range estimated by Hogan - around 100 Gflops without any optimizations.

It is somewhat disturbing that these numbers also happen to match the computing specification of Intel's "Movidius" deep neural processor...

(And just a few days ago, our power requirements got further reduced. We might be able to fit this in a head after all, and maybe power it from a human metabolism and a Seebeck converter).

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PGF/TikZ Manual

# The TikZ and PGF Packages Manual for version 3.1.10

## Libraries

#### 56Floating Point Unit Library

by Christian Feuersänger

• TikZ Library fpu

• \usepgflibrary{fpu} % and plain and pure pgf
\usepgflibrary[fpu] % Cont and pure pgf
\usetikzlibrary{fpu} % and plain when using TikZ
\usetikzlibrary[fpu] % Cont when using TikZ

The floating point unit (fpu) allows the full data range of scientific computing for use in pgf. Its core is the pgf math routines for mantissa operations, leading to a reasonable trade-off between speed and accuracy. It does not require any third-party packages or external programs.

##### 56.1Overview¶

The fpu provides a replacement set of math commands which can be installed in isolated placed to achieve large data ranges at reasonable accuracy. It provides at least10the IEEE double precision data range, $$-1 \cdot 10^{324}, \dotsc , 1\cdot 10^{324}$$. The absolute smallest number bigger than zero is $$1 \cdot 10^{-324}$$. The FPU’s relative precision is at least $$1 \cdot 10^{-4}$$ although operations like addition have a relative precision of $$1 \cdot 10^{-6}$$.

Note that the library has not really been tested together with any drawing operations. It should be used to work with arbitrary input data which is then transformed somehow into pgf precision. This, in turn, can be processed by pgf.

10 To be more precise, the FPU’s exponent is currently a 32-bit integer. That means it supports a significantly larger data range than an IEEE double precision number – but if a future version may provide low-level access to doubles, this may change.

##### 56.2Usage¶
• /pgf/fpu={boolean} (default true)

• This key installs or uninstalls the FPU. The installation exchanges any routines of the standard math parser with those of the FPU: \pgfmathadd will be replaced with \pgfmathfloatadd and so on. Furthermore, any number will be parsed with \pgfmathfloatparsenumber.

The FPU uses a low-level number representation consisting of flags, mantissa and exponent11.To avoid unnecessary format conversions, \pgfmathresult will usually contain such a cryptic number. Depending on the context, the result may need to be converted into something which is suitable for pgf processing (like coordinates) or may need to be typeset. The FPU provides such methods as well.

Use fpu=false to deactivate the FPU. This will restore any change. Please note that this is not necessary if the FPU is used inside of a group – it will be deactivated afterwards anyway.

It does not hurt to call fpu=true or fpu=false multiple times.

Please note that if the fixedpointarithmetic library of pgf will be activated after the FPU, the FPU will be deactivated automatically.

• /pgf/fpu/output format=float|sci|fixed (no default, initially float)

• This key allows to change the number format in which the FPU assigns \pgfmathresult.

The predefined choice float uses the low-level format used by the FPU. This is useful for further processing inside of any library.

The choice sci returns numbers in the format mantissaeexponent. It provides almost no computational overhead.

The choice fixed returns normal fixed point numbers and provides the highest compatibility with the pgf engine. It is activated automatically in case the FPU scales results.

• /pgf/fpu/scale results={scale}(no default)

• A feature which allows semi-automatic result scaling. Setting this key has two effects: first, the output format for any computation will be set to fixed (assuming results will be processed by pgf’s kernel). Second, any expression which starts with a star, *, will be multiplied with scale.

• /pgf/fpu/scale file plot x={scale}(no default)

• /pgf/fpu/scale file plot y={scale}(no default)

• /pgf/fpu/scale file plot z={scale}(no default)

• These keys will patch pgf’s plot file command to automatically scale single coordinates by scale.

The initial setting does not scale plot file.

• \pgflibraryfpuifactive{true-code}{false-code}

• This command can be used to execute either true-code or false-code, depending on whether the FPU has been activated or not.

• /pgf/fpu/install only={list of names}(no default)

• Unfortunately, the FPU is currently incompatible with drawing operations. However, it can still be useful to replace single definitions with FPU counterparts to avoid errors of the kind Dimension too large which tend to happen when transformation matrices are inverted.

This key allows to specify a list of definitions to be pulled into the current scope. Note that there is no reverse operation to uninstall these definitions at the moment, so it is advisable to do this in a group. Conveniently, TikZ paths form an implicit group, so you can use this key on a path as well.

You have to be aware of the limitations that the FPU imposes. It will not magically give better precision, but it will avoid overflow or underflow situations for large or small operands by rescaling them. In the following example, in the first case the FPU variant performs much better than the normal variant, however, in the second case where a rescaling would not in fact be needed the rescaling introduces a small round-off error.

This key is introduced in pgfv3.1.6 and marked stable since pgfv3.1.8.

11 Users should always use high level routines to manipulate floating point numbers as the format may change in a future release.

##### 56.3Comparison to the fixed point arithmetics library¶

There are other ways to increase the data range and/or the precision of pgf’s math parser. One of them is the fp package, preferable combined with pgf’s fixedpointarithmetic library. The differences between the FPU and fp are:

• • The FPU supports at least the complete IEEE double precision number range, while fp covers only numbers of magnitude $$\pm 1\cdot 10^{17}$$.

• • The FPU has a uniform relative precision of about 4–5 correct digits. The fixed point library has an absolute precision which may perform good in many cases – but will fail at the ends of the data range (as every fixed point routines does).

• • The FPU has potential to be faster than fp as it has access to fast mantissa operations using pgf’s math capabilities (which use registers).

##### 56.4Command Reference and Programmer’s Manual¶
###### 56.4.1Creating and Converting Floats¶
• \pgfmathfloatparsenumber{x}

• Reads a number of arbitrary magnitude and precision and stores its result into \pgfmathresult as floating point number $$m \cdot 10^e$$ with mantissa and exponent base $$10$$.

The algorithm and the storage format is purely text-based. The number is stored as a triple of flags, a positive mantissa and an exponent, such as

Please do not rely on the low-level representation here, use \pgfmathfloattomacro (and its variants) and \pgfmathfloatcreate if you want to work with these components.

The flags encoded in \pgfmathresult are represented as a digit where ‘$$0$$’ stands for the number $$\pm 0\cdot 10^0$$, ‘$$1$$’ stands for a positive sign, ‘$$2$$’ means a negative sign, ‘$$3$$’ stands for ‘not a number’, ‘$$4$$’ means $$+\infty$$ and ‘$$5$$’ stands for $$-\infty$$.

The mantissa is a normalized real number $$m \in \mathbb {R}$$, $$1 \le m < 10$$. It always contains a period and at least one digit after the period. The exponent is an integer.

Examples:

The argument x may be given in fixed point format or the scientific “e” (or “E”) notation. The scientific notation does not necessarily need to be normalized. The supported exponent range is (currently) only limited by the -integer range (which uses 31 bit integer numbers).

• /pgf/fpu/handlers/empty number={input}{unreadable part}(no default)

• This command key is invoked in case an empty string is parsed inside of \pgfmathfloatparsenumber. You can overwrite it to assign a replacement \pgfmathresult (in float!).

The initial setting is to invoke invalid number, see below.

• /pgf/fpu/handlers/invalid number={input}{unreadable part}(no default)

• This command key is invoked in case an invalid string is parsed inside of \pgfmathfloatparsenumber. You can overwrite it to assign a replacement \pgfmathresult (in float!).

The initial setting is to generate an error message.

• /pgf/fpu/handlers/wrong lowlevel format={input}{unreadable part}(no default)

• This command key is invoked whenever \pgfmathfloattoregisters or its variants encounter something which is not a properly formatted low-level floating point number. As for invalid number, this key may assign a new \pgfmathresult (in floating point) which will be used instead of the offending input.

The initial setting is to generate an error message.

• \pgfmathfloatqparsenumber{x}

• The same as \pgfmathfloatparsenumber, but does not perform sanity checking.

• \pgfmathfloattofixed{x}

• Converts a number in floating point representation to a fixed point number. It is a counterpart to \pgfmathfloatparsenumber. The algorithm is purely text based and defines \pgfmathresult as a string sequence which represents the floating point number x as a fixed point number (of arbitrary precision).

• \pgfmathfloattoint{x}

• Converts a number from low-level floating point representation to an integer (by truncating the fractional part).

See also \pgfmathfloatint which returns the result as float.

• \pgfmathfloattosci{float}

• Converts a number from low-level floating point representation to scientific format, $$1.234e4$$. The result will be assigned to the macro \pgfmathresult.

• \pgfmathfloatvalueof{float}

• Expands a number from low-level floating point representation to scientific format, $$1.234e4$$.

Use \pgfmathfloatvalueof in contexts where only expandable macros are allowed.

• \pgfmathfloatcreate{flags}{mantissa}{exponent}

• Defines \pgfmathresult as the floating point number encoded by flags, mantissa and exponent.

All arguments are characters and will be expanded using \edef.

• \pgfmathfloatifflags{floating point number}{flag}{true-code}{false-code}

• Invokes true-code if the flag of floating point number equals flag and false-code otherwise.

The argument flag can be one of

0

to test for zero,

1

to test for positive numbers,

+

to test for positive numbers,

2

to test for negative numbers,

-

to test for negative numbers,

3

for “not-a-number”,

4

for $$+\infty$$,

5

for $$-\infty$$.

• \pgfmathfloattomacro{x}{flagsmacro}{mantissamacro}{exponentmacro}

• Extracts the flags of a floating point number x to flagsmacro, the mantissa to mantissamacro and the exponent to exponentmacro.

• Takes a floating point number x as input and writes flags to count register flagscount, mantissa to dimen register mantissadimen and exponent to count register exponentcount.

Please note that this method rounds the mantissa to -precision.

• \pgfmathfloattoregisterstok{x}{flagscount}{mantissatoks}{exponentcount}

• A variant of \pgfmathfloattoregisters which writes the mantissa into a token register. It maintains the full input precision.

• \pgfmathfloatgetflags{x}{flagscount}

• Extracts the flags of x into the count register flagscount.

• \pgfmathfloatgetflagstomacro{x}{macro}

• Extracts the flags of x into the macro macro.

• Extracts the mantissa of x into the dimen register mantissadimen.

• \pgfmathfloatgetmantissatok{x}{mantissatoks}

• Extracts the mantissa of x into the token register mantissatoks.

• \pgfmathfloatgetexponent{x}{exponentcount}

• Extracts the exponent of x into the count register exponentcount.

###### 56.4.2Symbolic Rounding Operations¶

Commands in this section constitute the basic level implementations of the rounding routines. They work symbolically, i.e. they operate on text, not on numbers and allow arbitrarily large numbers.

• \pgfmathroundto{x}

• Rounds a fixed point number to prescribed precision and writes the result to \pgfmathresult.

The desired precision can be configured with /pgf/number format/precision, see section 97. This section does also contain application examples.

Any trailing zeros after the period are discarded. The algorithm is purely text based and allows to deal with precisions beyond ’s fixed point support.

As a side effect, the global boolean \ifpgfmathfloatroundhasperiod will be set to true if and only if the resulting mantissa has a period. Furthermore, \ifpgfmathfloatroundmayneedrenormalize will be set to true if and only if the rounding result’s floating point representation would have a larger exponent than x.

• \pgfmathroundtozerofill{x}

• A variant of \pgfmathroundto which always uses a fixed number of digits behind the period. It fills missing digits with zeros.

• \pgfmathfloatround{x}

• Rounds a normalized floating point number to a prescribed precision and writes the result to \pgfmathresult.

The desired precision can be configured with /pgf/number format/precision, see section 97.

This method employs \pgfmathroundto to round the mantissa and applies renormalization if necessary.

As a side effect, the global boolean \ifpgfmathfloatroundhasperiod will be set to true if and only if the resulting mantissa has a period.

• \pgfmathfloatroundzerofill{x}

• A variant of \pgfmathfloatround produces always the same number of digits after the period (it includes zeros if necessary).

###### 56.4.3Math Operations Commands¶

This section describes some of the replacement commands in more detail.

Please note that these commands can be used even if the fpu as such has not been activated – it is sufficient to load the library.

• \pgfmathfloatop

• Methods of this form constitute the replacement operations where op can be any of the well-known math operations.

Thus, \pgfmathfloatadd is the counterpart for \pgfmathadd and so on. The semantics and number of arguments is the same, but all input and output arguments are expected to be floating point numbers.

• \pgfmathfloattoextentedprecision{x}

• Renormalizes x to extended precision mantissa, meaning $$100 \le m < 1000$$ instead of $$1 \le m < 10$$.

The “extended precision” means we have higher accuracy when we apply pgfmath operations to mantissas.

The input argument is expected to be a normalized floating point number; the output argument is a non-normalized floating point number (well, normalized to extended precision).

The operation is supposed to be very fast.

• \pgfmathfloatsetextprecision{shift}

• Sets the precision used inside of \pgfmathfloattoextentedprecision to shift.

The different choices are

.
0 normalization to $$0$$ $$\le m < 1$$ (disable extended precision)
1 normalization to $$10$$ $$\le m < 100$$
2 normalization to $$100$$ $$\le m < 1000$$ (default of \pgfmathfloattoextentedprecision)
3 normalization to $$1000$$ $$\le m < 10000$$
• \pgfmathfloatlessthan{x}{y}

• Defines \pgfmathresult as $$1.0$$ if $$\meta {x} < \meta {y}$$, but $$0.0$$ otherwise. It also sets the global -boolean \pgfmathfloatcomparison accordingly. The arguments x and y are expected to be numbers which have already been processed by \pgfmathfloatparsenumber. Arithmetic is carried out using -registers for exponent- and mantissa comparison.

• \pgfmathfloatmultiplyfixed{float}{fixed}

• Defines \pgfmathresult to be $$\meta {float} \cdot \meta {fixed}$$ where float is a floating point number and fixed is a fixed point number. The computation is performed in floating point arithmetics, that means we compute $$m \cdot \meta {fixed}$$ and renormalize the result where $$m$$ is the mantissa of float.

This operation renormalizes float with \pgfmathfloattoextentedprecision before the operation, that means it is intended for relatively small arguments of fixed. The result is a floating point number.

• \pgfmathfloatifapproxequalrel{a}{b}{true-code}{false-code}

• Computes the relative error between a and b (assuming b$$\neq 0$$) and invokes true-code if the relative error is below /pgf/fpu/rel thresh and false-code if that is not the case.

The input arguments will be parsed with \pgfmathfloatparsenumber.

• /pgf/fpu/rel thresh={number} (no default, initially 1e-4)

• A threshold used by \pgfmathfloatifapproxequalrel to decide whether numbers are approximately equal.

• \pgfmathfloatshift{x}{num}

• Defines \pgfmathresult to be $$\meta {x} \cdot 10^{\meta {num}}$$. The operation is an arithmetic shift base ten and modifies only the exponent of x. The argument num is expected to be a (positive or negative) integer.

• \pgfmathfloatabserror{x}{y}

• Defines \pgfmathresult to be the absolute error between two floating point numbers $$x$$ and $$y$$, $$\lvert x - y\rvert$$ and returns the result as floating point number.

• \pgfmathfloatrelerror{x}{y}

• Defines \pgfmathresult to be the relative error between two floating point numbers $$x$$ and $$y$$, $$\lvert x - y\rvert / \lvert y \rvert$$ and returns the result as floating point number.

• \pgfmathfloatint{x}

• Returns the integer part of the floating point number x, by truncating any digits after the period. This methods truncates the absolute value $$\rvert x \lvert$$ to the next smaller integer and restores the original sign afterwards.

The result is returned as floating point number as well.

See also \pgfmathfloattoint which returns the number in integer format.

• \pgfmathlog{x}

• Defines \pgfmathresult to be the natural logarithm of x, $$\ln (\meta {x})$$. This method is logically the same as \pgfmathln, but it applies floating point arithmetics to read number x and employs the logarithm identity

$\ln (m \cdot 10^e) = \ln (m) + e \cdot \ln (10)$

to get the result. The factor $$\ln (10)$$ is a constant, so only $$\ln (m)$$ with $$1 \le m < 10$$ needs to be computed. This is done using standard pgf math operations.

Please note that x needs to be a number, expression parsing is not possible here.

If x is not a bounded positive real number (for example $$\meta {x} \le 0$$), \pgfmathresult will be empty, no error message will be generated.

###### 56.4.4Accessing the Original Math Routines for Programmers¶

As soon as the library is loaded, every private math routine will be copied to a new name. This allows library and package authors to access the -register based math routines even if the FPU is activated. And, of course, it allows the FPU as such to perform its own mantissa computations.

The private implementations of pgf math commands, which are of the form \pgfmathname@, will be available as\pgfmath@basic@name@ as soon as the library is loaded.


text:
Question

Me spivak had 5 gallons of gas. he used 1 gallon in his tractor and 3 quarts in his lawnmower. how much gas was left?

1. The amount of gas left is 3.25 gallons

How to determine the amount of gas left?

The given parameters are:
Total gallons of gas =  5 gallons
Tractor = 1 gallon
Lawnmower = 3 quarts
The amount of gas left is calculated as:
Amount left = Total gallons of gas – Tractor – Lawnmower
Substitute the known values in the above equation
Amount Left = 5 gallons – 1 gallon – 3 quarts
Convert quarts to gallons
Amount Left = 5 gallons – 1 gallon – 3 * 0.25 gallons
So, we have:
Amount Left = 5 gallons – 1 gallon – 0.75 gallons
Evaluate the difference
Amount Left = 3.25 gallons
Hence, the amount of gas left is 3.25 gallons


text:
Opuscula Math. 35, no. 3 (2015), 293-332
http://dx.doi.org/10.7494/OpMath.2015.35.3.293

Opuscula Mathematica

# Frames and factorization of graph Laplacians

Palle Jorgensen
Feng Tian

Abstract. Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $$\mathscr{H}_{E}$$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $$\mathscr{H}_{E}$$ we characterize the Friedrichs extension of the $$\mathscr{H}_{E}$$-graph Laplacian. We consider infinite connected network-graphs $$G=\left(V,E\right)$$, $$V$$ for vertices, and $$E$$ for edges. To every conductance function $$c$$ on the edges $$E$$ of $$G$$, there is an associated pair $$\left(\mathscr{H}_{E},\Delta\right)$$ where $$\mathscr{H}_{E}$$ in an energy Hilbert space, and $$\Delta\left(=\Delta_{c}\right)$$ is the $$c$$-Graph Laplacian; both depending on the choice of conductance function $$c$$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $$\mathscr{H}_{E}$$ consisting of dipoles. Now $$\Delta$$ is a well-defined semibounded Hermitian operator in both of the Hilbert $$l^{2}\left(V\right)$$ and $$\mathscr{H}_{E}$$. It is known to automatically be essentially selfadjoint as an $$l^{2}\left(V\right)$$-operator, but generally not as an $$\mathscr{H}_{E}$$ operator. Hence as an $$\mathscr{H}_{E}$$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $$l^{2}\left(V\right)$$.

Keywords: unbounded operators, deficiency-indices, Hilbert space, boundary values, weighted graph, reproducing kernel, Dirichlet form, graph Laplacian, resistance network, harmonic analysis, harmonic analysis, frame, Parseval frame, Friedrichs extension, reversible random walk, resistance distance, energy Hilbert space.

Mathematics Subject Classification: 47L60, 46N30, 46N50, 42C15, 65R10, 05C50, 05C75, 31C20, 46N20, 22E70, 31A15, 58J65, 81S25.

Full text (pdf)

1. N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications Inc., New York, 1993. Translated from the Russian and with a preface by Merlynd Nestell, Reprint of the 1961 and 1963 translations, Two volumes bound as one.
2. D. Alpay, P. Jorgensen, R. Seager, D. Volok, On discrete analytic functions: products, rational functions and reproducing kernels, J. Appl. Math. Comput. 41 (2013) 1-2, 393-426.
3. V. Anandam, Harmonic functions and potentials on finite or infinite networks, vol. 12 of Lecture Notes of the Unione Matematica Italiana, Springer, Heidelberg; UMI, Bologna, 2011.
4. M.T. Barlow, Random walks, electrical resistance, and nested fractals, [in:] Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Sanda/Kyoto, 1990), Longman Sci. Tech., Harlow, vol. 283 of Pitman Res. Notes Math. Ser., 131-157, 1993.
5. I. Cho, P.E.T. Jorgensen, Free probability on operator algebras induced by currents in electric resistance networks, Int. J. Funct. Anal. Oper. Theory Appl. 4 (2012) 1, 1-50.
6. I. Cho, P.E.T. Jorgensen, Operators induced by graphs, Lett. Math. Phys. 102 (2012) 3, 323-369.
7. H. Cossette, D. Landriault, É. Marceau, Ruin probabilities in the compound Markov binomial model, Scand. Actuar. J. 4 (2003), 301-323.
8. B. Currey, A. Mayeli, The Orthonormal dilation property for Abstract Parseval wavelet frames, Canad. Math. Bull. 56 (2013) 4, 729-736.
9. P.G. Doyle, J.L. Snell, Random walks and electric networks, vol. 22 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1984.
10. N. Dunford, J.T. Schwartz, Linear Operators. Part II, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space, with the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1963 original, A Wiley-Interscience Publication.
11. M. Durand, Architecture of optimal transport networks, Phys. Rev. E (3), 73 (2006) 1, 016116, 6.
12. M. Ehler, K. A. Okoudjou, Probabilistic Frames: an Overview, [in:] Finite Frames, Birkhäuser/Springer, New York, Appl. Numer. Harmon. Anal., 415-436, 2013.
13. M. Folz, Volume growth and spectrum for general graph Laplacians, Math. Z. 276 (2014) 1-2, 115-131.
14. M.J. Gander, S. Loisel, D.B. Szyld, An optimal block iterative method and preconditioner for banded matrices with applications to PDEs on irregular domains, SIAM J. Matrix Anal. Appl. 33 (2012) 2, 653-680.
15. G. Grimmett, Probability on Graphs, vol. 1 of Institute of Mathematical Statistics Textbooks, Cambridge University Press, Cambridge, 2010.
16. D. Han, W. Jing, D. Larson, P. Li, R.N. Mohapatra, Dilation of dual frame pairs in Hilbert $$C^*$$-modules, Results Math. 63 (2013) 1-2, 241-250.
17. P.E.T. Jorgensen, Essential self-adjointness of the graph-Laplacian, J. Math. Phys. 49 (2008) 7, 073510, 33.
18. P.E.T. Jorgensen, A.M. Paolucci, $$q$$-frames and Bessel functions, Numer. Funct. Anal. Optim. 33 (2012) 7-9, 1063-1069.
19. P.E.T. Jorgensen, E.P.J. Pearse, A Hilbert space approach to effective resistance metric, Complex Anal. Oper. Theory 4 (2010) 4, 975-1013.
20. P.E.T. Jorgensen, E.P.J. Pearse, Resistance Boundaries of Infinite Networks, [in:] Random Walks, Boundaries and Spectra, Birkhäuser, Springer Basel AG, Basel, vol. 64 of Progr. Probab., 111-142, 2011.
21. P.E.T. Jorgensen, E.P.J. Pearse, Spectral reciprocity and matrix representations of unbounded operators, J. Funct. Anal. 261 (2011) 3, 749-776.
22. P.E.T. Jorgensen, E.P.J. Pearse, A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks, Israel J. Math. 196 (2013) 1, 113-160.
23. P.E.T. Jorgensen, M.-S. Song, Comparison of Discrete and Continuous Wavelet Transforms, [in:] Computational Complexity. Vols. 1-6, Springer, New York, 513-526, 2012.
24. V. Kaftal, D.R. Larson, S. Zhang, Operator-valued frames, Trans. Amer. Math. Soc. 361 (2009) 12, 6349-6385.
25. G. Kutyniok, K.A. Okoudjou, F. Philipp, E.K. Tuley, Scalable frames, Linear Algebra Appl. 438 (2013) 5, 2225-2238.
26. I.M. Longini, Jr., A chain binomial model of endemicity, Math. Biosci. 50 (1980) 1-2, 85-93.
27. M. Longla, C. Peligrad, M. Peligrad, On the functional central limit theorem for reversible Markov chains with nonlinear growth of the variance, J. Appl. Probab. 49 (2012) 4, 1091-1105.
28. F.G. Meyer, X. Shen, Perturbation of the eigenvectors of the graph Laplacian: application to image denoising, Appl. Comput. Harmon. Anal. 36 (2014) 2, 326-334.
29. C.S. J.A. Nash-Williams, Random walk and electric currents in networks, Proc. Cambridge Philos. Soc. 55 (1959), 181-194.
30. F.A. Shah, L. Debnath, Tight wavelet frames on local fields, Analysis (Berlin) 33 (2013) 3, 293-307.
31. M.N.S. Swamy, K. Thulasiraman, Graphs, Networks, and Algorithms, John Wiley & Sons, Inc., New York, 1981.
32. R. Terakado, Construction of constant resistance networks using the properties of two-dimensional regions with antisymmetry, IEEE Trans. Circuits and Systems CAS-25 (1978) 2, 109-111.
33. P. Tetali, Random walks and the effective resistance of networks, J. Theoret. Probab. 4 (1991) 1, 101-109.
34. K.-C. Yuen, J. Guo, Some results on the compound Markov binomial model, Scand. Actuar. J. 3 (2006), 129-140.
35. A.H. Zemanian, Random walks on finitely structured transfinite networks, Potential Anal. 5 (1996) 4, 357-382.
• Palle Jorgensen
• The University of Iowa, Department of Mathematics, Iowa City, IA 52242-1419, USA
• Feng Tian
• Wright State University, Department of Mathematics, Dayton, OH 45435, USA
• Communicated by P.A. Cojuhari.
• Accepted: 2014-11-04.
• Published online: 2014-12-15.


text:
Let $p$ be a quadratic multivariate polynomial in Sage, for example
sage: R.<x,y,z> = QQ[]

I want to test whether there exists a vector $(a,b,c)$ with positive entries such that $p(a,b,c) > 0$. With the above examples the answer is no for $p$ and yes for $q$ since $q(1,1,1) = 3/2$. Is there a way to achieve that in SageMath? It is a very particular instance of non-linear optimization but did not find anything relevant.


text:
The other day, someone in my office was working a project of Image Segmentation ... python-bloggers.com (python/data-science news) Technical documentation; Ask Question Asked 3 years, 5 months ago. Refer to this link for more details. Image segmentation is a fundamental and well researched image processing problem that group perceptually similar pixels, based on a variety of attributes, e.g., color, texture, etc. So you will reduce this image from $$265 \times 191 = 50,615$$ pixels down to $$400$$ regions. As concern to unsupervised image segmentation using autoencoder (AE), during feature representation learning for the AE, the target values are never involved. We will also dive into the implementation of the pipeline – from preparing the data to building the models. Pixel-wise image segmentation is a well-studied problem in computer vision. We tell the algorithm what to do and what not to do. The goal of segmenting an image is to change the representation of an image into something that is more meaningful and easier to analyze. ... Then we went through practical implementation of k -means clustering algorithm using Banking Customer Segmentation problem on Python. Image segmentation using watershed algorithm. 1. Word segmentation (also called tokenization) is the process of splitting text into a list of words. It is usually used for locating objects and creating boundaries. In this exercise, you'll do just that! References Semantic segmentation assigns an object class to each individual pixel in an image. Already preloaded as face_image. Many recent segmentation methods use superpixels because they reduce the size of the segmentation problem by order of magnitude. let’s discuss all of them one by one:. Image Segmentation: In computer vision, image segmentation is the process of partitioning an image into multiple segments. ... and the two final clusters will be as shown in the below image. It involves a simple level task like noise removal to common tasks like identifying objects, person, text etc., to more complicated tasks like image classifications, emotion detection, anomaly detection, segmentation etc. It makes use of machine learning with built-in functions and can perform complex operations on images with just a few functions. Parul Pandey. In supervised classification, we select samples for each target class. The method is called scene-cut which segments an image into class-agnostic regions in an unsupervised fashion. It is a form of pixel-level prediction because each pixel in an image is classified according to a category. These evaluation criteria compute some statistics for each region or class in a segmentation result. Tricky image segmentation in Python. Because it’s about unsupervised machine learning algorithm, we learn about the pattern. It is an open-source library used for image preprocessing. In this exercise, you will apply unsupervised segmentation to the same image, before it's passed to a face detection machine learning model. Image Processing or more specifically, Digital Image Processing is a process by which a digital image is processed using a set of algorithms. And I also develop Quantum Game, in which you are able to play with quantum mechanics in the browser. Active 3 years, 5 months ago. Follow. Get access to the full code so you can start implementing it for your own purposes in one-click using the form below! Since you ask for image segmentation and not semantic / instance segmentation, I presume you don't require the labelling for each segment in the image. Agricultural greenhouse plant images with complicated scenes are difficult to precisely manually label. A customer profiling and segmentation Python demo & practice problem Now that we’ve covered the inner workings of k-means clustering, let’s implement it in a practice problem. The Python standard library comes with many useful methods for strings. Imaginaire Models. How to extract only feet from image using opencv python. Unsupervised Segmentation. Unsupervised segmentation may use basic image processing techniques to complex optimization algorithms. K-Means is a data clustering algorithm that tries to assign every data point in a dataset to exactly one of K possible clusters – hence the name. The split method is one that can be used for very basic segmentation tasks. What we do is to give different labels for our object we know. Scikit-image. The decoder part, on the other hand, takes the compressed features as input and reconstruct an image as close to the original image as possible. Imaginaire added many supervised, unsupervised, image to image & video to video translation models into their library, all the models are pretrained on Nvidia DGX 1machien with 8 32GB V100 using PyTorch docker v20.03. Image segmentation in the unsupervised scenario is researched with the aim of pixels clustering. Kanezaki’s paper[1] is quite inspiring to apply the concept of “unsupervised segmentation” on hyperspectral images. In this post, we will discuss how to use deep convolutional neural networks to do image segmentation. Hot Network Questions Customer Segmentation using supervised and unsupervised learning. Image segmentation can use for biological, medical case study or other cases too. Image preloaded as profile_image. Clustering is an unsupervised machine learning algorithm. The task of semantic image segmentation is to classify each pixel in the image. Image segmentation using Python and OpenCV. For example, one sample of the 28x28 MNIST image has 784 pixels in total, the encoder we built can compress it to an array with only ten floating point numbers also known as the features of an image. We train our neural network on these target class samples and then classify new samples. Image Segmentation using Python’s scikit-image module. Using the slic() function for segmentation, pre-process the image before passing it to the face detector. It is an interactive image segmentation. Exploring unsupervised image segmentation results - by Piotr Migdal & Bartłomiej Olechno () ... a Python package for charts for deep learning models. Invariant Information Clustering for Unsupervised Image Classification and Segmentation. Semantic segmentation, or image segmentation, is the task of clustering parts of an image together which belong to the same object class. ICCV 2019 • xu-ji/IIC • The method is not specialised to computer vision and operates on any paired dataset samples; in our experiments we use random transforms to obtain a pair from each image. OpenCV really should have a wider assortment of segmentation methods. We present in this paper a study of unsupervised evaluation criteria that enable the quantification of the quality of an image segmentation result. Now, we are all set to discuss how we can apply segmentation on HSI. Why Unsupervised Learning? The main problem in unsupervised segmentation algorithms is the difficulty of balancing the over-segmentation and under-segmentation. Semantic segmentation is relevant to this study, since it can be used to identify the nature of a detected change. Basic segmentation methods. Image classification is an application of both supervised classification and unsupervised classification. Show this page source 2. An overview of the scikit-image library’s image segmentation methods. © 2007 - 2020, scikit-learn developers (BSD License). Segmentation of customers has a pretty significant position for companies in new marketing diciplines. In order to tackle this question I engaged in both super v ised and unsupervised learning. The show_image() function has been preloaded for you as well. 1. SSTDA簡介 - Action Segmentation with Joint Self-Supervised Temporal Domain Adaptation 09 Mar. A Beginner's guide to Deep Learning based Semantic Segmentation using Keras Pixel-wise image segmentation is a well-studied problem in computer vision. In the paper, Kanezaki shows her method of “unsupervised segmentation” for RGB(three-band) images. Image segmentation is widely used as an initial phase of many image processing tasks in computer vision and image analysis. A typical workflow in a machine learning project is designed in a supervised manner. The appearance of leaf disease spots and mosses increases the difficulty in plant segmentation. Image segmentation is an important processing step in many image, video and computer vision applications. Some example benchmarks for this task are Cityscapes, PASCAL VOC and ADE20K. Image segmentation based on texture (Unsupervised) in Python-OpenCV. Previously, you learned how to make processes more computationally efficient with unsupervised superpixel segmentation. The task of semantic image segmentation is to classify each pixel in the image. Approach 3 – Extract Semantic meaning from the image and use it organize the photos; Code Walkthrough of Unsupervised Deep Learning on the MNIST dataset . This might be something that you are looking for. 2019. This article is a detailed introduction to what is k-means clustering in python. Image classification is the most critical use case in digital image analysis. 簡介 - Domain Bridge for Unpaired Image-to-Image Translation and Unsupervised Domain Adaptation 15 Nov; Guided-pix2pix 簡介 - Guided Image-to-Image Translation with … Hence these segmentation methods take much more time when we ask for better results. ... Unsupervised word segmentation using SentencePiece. That would be a great complement to the newer neural net library interfaces. What is image segmentation; Python k-means image segmentation with opencv; Canny edge detection in opencv; Finding contours using opencv; K-Means clustering explained. There are unsupervised Markov Random Field and Conditional Random Field methods that probably out-perform existing adaptive threshold and Otsu. 07/20/19 - Hyperspectral image analysis has become an important topic widely researched by the remote sensing community. Firms must reach to the right target audiences with right approaches because of … Word segmentation ( also called tokenization ) is the most critical use case in image... Unsupervised ) in Python-OpenCV optimization algorithms unsupervised machine learning algorithm, we learn about the pattern each target class and. Own purposes in one-click using the slic ( ) function has been preloaded for you as well customers a! Of “ unsupervised segmentation ” for RGB ( three-band ) images image, video and computer...., PASCAL VOC and ADE20K the implementation of k -means clustering algorithm using Customer... The data to building the models the nature of a detected change leaf disease and. The main problem in computer vision, image segmentation is relevant to this study, it! Segmentation using Keras pixel-wise image segmentation is to classify each pixel in the paper, kanezaki shows her method “... Reduce the size of the pipeline – from preparing the data to the... ( BSD License ) individual pixel in an image is to give different labels for our object we.. Library used for image preprocessing Network Questions this might be something that is more and. Developers ( BSD License ) Action segmentation with Joint Self-Supervised Temporal Domain Adaptation 09.! Prediction because each pixel in an unsupervised fashion Question Asked 3 years 5... Customer segmentation problem on Python to play with Quantum mechanics in the image or class in a learning... Problem by order of magnitude processing is a well-studied problem in unsupervised segmentation ” hyperspectral... Preparing the data to building the models image classification is the most critical use case in digital image tasks! In both super v ised and unsupervised learning to extract only feet from using! Companies in new marketing diciplines just that practical implementation of the quality of an image into something that is meaningful! Preparing the data to building the models the nature of a detected.! Which a digital image processing is a well-studied problem in computer vision applications feet from image using Python! Opencv Python ised and unsupervised learning ) regions in Python-OpenCV Random Field methods that probably out-perform existing adaptive threshold Otsu! Segmentation assigns an object class to each individual pixel in the image this post, we learn about pattern! Paper a study of unsupervised evaluation criteria that enable the quantification of the quality of an image something... A machine learning algorithm, we will also dive into the implementation of k clustering... To use deep convolutional neural networks to do image segmentation is widely used as an initial of! Code so you will reduce this image from \ ( 400\ ) regions for unsupervised image.! Been preloaded for you as well k -means clustering algorithm using Banking Customer segmentation problem by of... Change the representation of an image segmentation is a well-studied problem in computer vision Field methods that probably existing! For charts for deep learning based semantic segmentation is widely used as an initial of! Introduction to what is k-means clustering in Python with built-in functions and can perform operations. Task are Cityscapes, PASCAL VOC and ADE20K many recent segmentation methods take much more time when ask... Recent segmentation methods use superpixels because they reduce the size of the pipeline – from preparing data... Segmentation methods take much more time when we ask for better results ’... Quantum Game, in which you are looking for segmentation with Joint Self-Supervised Temporal Domain 09! Is designed in a supervised manner show_image ( ) function has been preloaded for you as well will also into.

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## Fractions are Hard!

### 2.7 Multiplying fractions

We have four beliefs about how fractions should work (accompanied with all the usual rules of arithmetic). We are choosing to believe that these statements hold for all types of numbers.

BELIEF 1: $$\dfrac{a}{a} = 1$$

BELIEF 2: $$\dfrac{a}{1}=a$$

BELIEF 3: $$x \times \dfrac{a}{b} = \dfrac{xa}{b}$$           (Basic Multiplication Belief)

BELIEF 4: $$\dfrac{a}{b}=\dfrac{xa}{xb}$$                           (Key Fraction Belief)

One consequence of these beliefs is

CONSEQUENCE 5: $$\dfrac{a}{b} \times b = a$$.

EXAMPLE: Give a meaningful value to $$\dfrac{2}{3} \times \dfrac{7}{5}$$ as a logical consequence of these beliefs.

Answer: We have the task of sharing 7 pies among 5 girls, but want to reduce the amount of pie each girl receives by a factor of two-thirds. Belief 3 says to adjust the number of pies by two-thirds.

$$\dfrac{2}{3} \times \dfrac{7}{5} = \dfrac{\dfrac{2}{3}\times 7}{5}$$.

The fraction of “thirds” appearing in the numerator is irritating. Let’s use the Key Fraction Property (Belief 4) and change the numerator and denominator each by a factor of 3. This does not change the quantity.

$$\dfrac{\dfrac{2}{3}\times 7}{5}=\dfrac{\dfrac{2}{3}\times 7 \times 3}{5 \times 3}$$.

Within the numerator (using the usual commutativity of arithmetic) we see $$\dfrac{2}{3} \times 3$$, which equals $$2$$ by Consequence 5.

$$\dfrac{\dfrac{2}{3}\times 7 \times 3}{5 \times 3}=\dfrac{2\times 7}{5 \times 3}$$

and this is  $$\dfrac{14}{15}$$.

Alternatively, we could have argued

$$\dfrac{c}{d}\times \dfrac{a}{b} = \dfrac{\dfrac{c}{d}\times a}{b}$$    (by Belief 3)

$$= \dfrac{\dfrac{ac}{d}}{b}$$      (by Belief 3)

$$=\dfrac{\dfrac{ac}{d}\times d}{b \times d} = \dfrac{ac}{bd}$$    (by Belief 4 and then Belief 3 again)

We have established the usual multiplication rule from grade school.

$$\dfrac{c}{d}\times \dfrac{a}{b}$$is the product of the “numerators over the product of the denominators.”

 CONSEQUENCE 6:   $$\dfrac{a}{b}\times \dfrac{c}{d}=\dfrac{ac}{bd}$$.

One can work to be efficient with the arithmetic.

 Question:  Ibrahim was asked to compute $$\dfrac{39}{7}\times \dfrac{14}{13}$$ and within three seconds he said that the answer was $$6$$. How did he see this so quickly?
 Question:  What is the value of $$\dfrac{39}{35}\times \dfrac{14}{13}$$?

### SOME HAPPY COINCIDENCES

The word “of” is used in the first introduction of fractions to very young students we way “parts of a whole.”

What if we look at parts of a parts of a whole?

EXAMPLE: Using the parts of a whole thinking, draw a picture of  $$\dfrac{2}{3}$$ of $$\dfrac{4}{5}$$.

First, here’s a picture of $$\dfrac{4}{5}$$ of a pie.

Now, in the early grades, $$\dfrac{2}{3}$$ is two copies of one-third. Here’s a one-third of $$\dfrac{4}{5}$$ of a pie.

Thus two thirds of $$\dfrac{4}{5}$$ is two copies of this.

If we draw in the extra lines in the picture we can see the shaded region as matching 8 parts out of 15.

Thus $$\dfrac{2}{3}$$ of $$\dfrac{4}{5}$$ matches a picture of $$\dfrac{8}{15}$$, sharing 8 pies among 15 boys (or 8 parts out of a whole of 15).

It is a happy coincidence that this matches the result of our multiplication formula: $$\dfrac{2}{3}\times \dfrac{4}{5}=\dfrac{8}{15}$$.

Comment: It is a further happy coincidence that the picture of parts of a parts of a whole looks something akin to a picture of the area model of multiplication, giving a sense that we really have the right definition of the product of two fractions.

 SUMMARY: We found with the pies per boy model that $$\dfrac{a}{b}\times \dfrac{c}{d}$$ is  $$\dfrac{ac}{bd}$$. This formula happens to match the pictures we draw for   “$$\dfrac{a}{b}$$ of  $$\dfrac{c}{d}$$,” namely, a picture of $$ac$$ parts to a total of $$bd$$ parts, that is, the quanitity $$\dfrac{ac}{bd}$$.  And, moreover, if we happen to draw square pies, this seems to psychologically match the pictures we draw for multiplication via the area model. Notice this this sequence of ideas is the reverse of the typical textbook approach: Textbooks draw pictures akin to the area model first, attempt to make some statement that “of means multiply” and then derive the formula $$\dfrac{a}{b}\times \dfrac{c}{d} = \dfrac{ac}{bd}$$ . We got the formula first and then found happy links to these two other modes of thinking.

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5

# Find the following fro-x+42) dxf(91x+12) dx= (Use C as the arbitrary constant )...

## Question

###### Find the following fro-x+42) dxf(91x+12) dx= (Use C as the arbitrary constant )

Find the following fro-x+42) dx f(91x+12) dx= (Use C as the arbitrary constant )

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text:
# Using MathJax with PrinceXML

mn4367
Hello,

since Prince isn't able to work with MathJax yet I've tried to find a workaround. The basic idea is to have the math rendered with something else to a separate HTML document and then process this document with Prince. I also wanted to use as less external tools/installations as possible and only simple, standard JavaScript, no further frameworks like jQuery etc.

I've found a "solution" which may help other users. You'll need PhantomJS (http://phantomjs.org/) and a local copy of MathJax (https://www.mathjax.org/). The PhantomJS download contains a single executable (at least on the Mac) with no further dependencies. I assume an "installation" like this:

myDir/MathJax
myDir/MathJax/MathJax.js
myDir/MathJax/ ... other MathJax files
ConfigMathJax.js
phantomjs
render-math-and-dump.js
test.html

The content of render-math-and-dump.js:

// PhantomJS setup
var webPage = require('webpage');
var page = webPage.create();
var system = require('system');
var waitForMath;
var noLongerExecWaitForMathJax = false;

if (system.args.length < 2) {
console.log("Usage: phantomjs render-math-and-dump.js <filename>");
console.log("Renders <filename> and dumps the DOM to the console.");
phantom.exit();
};
page.open(system.args[1], null);
// Alternatives:
// page.open(encodeURI(system.args[1]), null);
// page.open("file://" + encodeURI(system.args[1]), null);

// This also works (to some extent) with web URLs. Be sure to disable
// waitForMath = window.setInterval at the end of this file if you want
// to try this variant of the script.
// page.open(system.args[1], function(status){
//     if (status === "success") {
//         // In this setup a complete local MathJax installation is present in MathJax/
//         page.injectJs("MathJax/MathJax.js");
//         page.injectJs("ConfigMathJax.js");
//         // Execute the query every 20ms
//         waitForMath = window.setInterval(
//             waitForMathJax,
//             20
//         );
//     };
// });

// Query a variable named 'MathJaxFinished' in the document,
var waitForMathJax = function () {
if (noLongerExecWaitForMathJax) {
return;
};
var mjFinished = page.evaluate(function () {
return MathJaxFinished;
});
if (mjFinished) {
// waitForMathJax was called at least twice (?), even after window.clearInterval()
// This flag prevents it (see above).
noLongerExecWaitForMathJax = true;
window.clearInterval(waitForMathJax);
page.evaluate(function() {
// Remove all MathJax related script elements
var MathJaxScriptFound = true;
var elements = document.getElementsByTagName("script");
// Repeating this loop until no more matching scripts are found seems to be necessary
// because removing children from the DOM did not always happen immediately, some
// of them were still written to the output when dumping 'page.content'.
while (MathJaxScriptFound) {
MathJaxScriptFound = false;
for (var i = elements.length - 1; i >= 0; i--) {
if (elements[i].hasAttribute("id")) {
var id = elements[i].getAttribute("id");
// Remove script if 'id' starts with 'MathJax-Element'
if (id.indexOf("MathJax-Element") === 0) {
MathJaxScriptFound = true;
elements[i].parentNode.removeChild(elements[i]);
};
} else if (elements[i].hasAttribute("src")) {
var src = elements[i].getAttribute("src");
// Remove script if 'src' ends on 'ConfigMathJax.js', 'MathJax.js' or 'jax.js'
if ((src.indexOf("ConfigMathJax.js", src.length - 16) !== -1)
|| (src.indexOf("MathJax.js", src.length - 10) !== -1)
|| (src.indexOf("jax.js", src.length - 6) !== -1)) {
MathJaxScriptFound = true;
elements[i].parentNode.removeChild(elements[i]);
};
};
};
elements = document.getElementsByTagName("script");
};
// Remove div with ID 'MathJax_Font_Test'
var element = document.getElementById("MathJax_Font_Test");
if (element) {
element.parentNode.removeChild(element);
};
// Avoid empty elements in HTML-Syntax (<meta>, <link>,
,
, ...),
// this makes it possible to call Prince in XML mode with the generated
// output from PhantomJS. In mode 'auto' this code can be disabled completely.
// Please also note that for example the resulting

 is not
// compliant with HTML, so you should keep an eye on the output
// generated by Prince. The same is probably true for <meta></meta>,
elements = document.documentElement.getElementsByTagName("*");
for (var i = elements.length - 1; i >= 0; i--) {
if (elements[i].childNodes.length == 0) {
elements[i].appendChild(document.createTextNode(""));
};
};
});
console.log(page.content);
phantom.exit();
};
}

// Execute the query every 20ms
waitForMath = window.setInterval(
waitForMathJax,
20
);

This is a sample setup for MathJax (ConfigMathJax.js). The MathJax.Hub.Config part can be changed, but you'll have to use the MathJax HTML-CSS output variant; for the moment I was unable to get it to work with the SVG output.

// A trigger which is queried by the script
// render-math-and-dump.js in PhantomJS
var MathJaxFinished = false;

// A sample MathJax setup
MathJax.Hub.Config({
displayAlign: "left",
displayIndent: "3em",
extensions: ["tex2jax.js"],
// HTML Output
jax: ["input/TeX", "output/HTML-CSS"],
imageFont: null,
messageStyle: "none",
showProcessingMessages: false,
tex2jax: {
ignoreClass: "tex2jax_ignore",
processClass: "math",
inlineMath: [ ["$","$"], ["\$","\$"] ],
displayMath: [ ["$$","$$"], ["\$","\$"] ],
processEscapes: false
},
TeX: {
extensions: ["AMSmath.js","AMSsymbols.js"],
TagSide: "right",
TagIndent: "1em",
MultLineWidth: "85%",
equationNumbers: {
autoNumber: "AMS"
},
unicode: {
fonts: "STIXGeneral,'Arial Unicode MS'"
}
},
"HTML-CSS": {
scale: 100,
linebreaks: {
automatic: true
},
// Anchors in formulas in black
styles: {
".MathJax a": {
color: "#000000"
},
".MathJax": {
"vertical-align": "baseline"
}
},
mtextFontInherit: true,
matchFontHeight: true,
availableFonts: ["TeX"],
preferredFont: "TeX",
webFont: "TeX",
imageFont: null
}
});

// This function will be called when MathJax has finished *all* rendering
MathJax.Hub.Queue(
function () {
MathJaxFinished = true;
}
);

The content of test.html:

<!DOCTYPE html>

Formula 1:
$$$$\label{Formula1.Label} P (\Omega) = \int_{-\infty}^{+\infty} f (x) \: dx = 1$$$$

Formula 2:
$$$$\label{Formula2.Label} E (X) := \int_{-\infty}^{+\infty} x \: f (x) \; dx$$$$

A third one, with references to 1 and 2:
$$$$\begin{array}{ll} E (a X + b) & = \int_{-\infty}^{+\infty} (a x - b) \: f(x) \: dx \\ \\ & = a \int_{-\infty}^{+\infty} x \: f(x) \: dx + b \int_{-\infty}^{+\infty} f(x) \: dx \\ \\ & \underset{\eqref{Formula1.Label}, \eqref{Formula2.Label}}{=} a E(X) + b \end{array}$$$$

And another one:
$$$$V(X) = \frac{a^2 + b^2 + c^2 - ab - ac - bc}{18}$$$$

You can now use:

./phantomjs render-math-and-dump.js test.html | prince --javascript -o test.pdf -

The resulting PDF is attached. I also attached a rendering from the web (http://www.feynmanlectures.caltech.edu/I_47.html), due to the nature of processing it with the primitive script locally it lacks images and CSS but it shows that perhaps with some modifications rendering math from the web could be made possible.

The script does the following:

• Load and render the document given on the commandline with PhantomJS (WebKit based).
• Wait for MathJax to finish its work (can take quite long). The result is a DOM with all modifications made by MathJax.
• Remove all MathJax related


text:
# 10. Step 10: Discount benefits and costs, calculate summary results

## Steps

10.1 Choose a discount rate.

10.2 Assemble benefits and costs by time period.

10.3 Calculate the NPV.

10.4 Calculate the BCR.

10.5 Calculate the NPVI (if required).

10.6 Calculate the incremental BCR (if applicable).

10.7 Calculate the internal rate of return (IRR) (if required).

10.8 Calculate the first-year rate of return (FYRR).

## 10.1 Choose a discount rate

Discounting is necessary because a dollar of benefit in the future is worth less than a dollar of benefit today. There is a variety of views and approaches to selecting the social discount rate. The technical literature is large and complex. There is no definitive answer on which experts will agree. For a background discussion on discount rates see NGTSM 2006 Volume 5, Section 2.10.1.

In practice, use the discount rate nominated by the funding jurisdiction. For example, at the time of publication: Infrastructure Australia requires the use of a real rate of 7 per cent with 4 per cent and 10 per cent used for sensitivity testing; the Commonwealth Department of Transport and Regional Development requires the use of real rates of both 4 per cent and 7 per cent.

## 10.2 Assemble benefits and costs by time period

Adopt an end-of-year convention for discounting purposes, where all benefits and costs are assumed to occur at the end of the year in which they occur.

Set ‘year zero’ at the time of the commencement of construction. Any costs occurring in year zero are not discounted. Costs incurred during year one will be discounted by one year. Discount forward any costs (e.g. avoidable planning and design costs1) incurred prior to the commencement of year zero (years minus one, minus two and so on) by multiplying by (1+r)t.

With the life of the initiative assumed to commence at completion of construction, the number of years over which discounting occurs will be larger than the initiative’s life. For example, if an initiative takes two full years to construct and has a 30-year life, there will be 32 years of benefits and costs to discount - the final year’s net benefits being discounted by 32 years.

## 10.3 Calculate the NPV

The summation of all annual discounted present values of a stream of benefits or costs is called the ‘present value’ of that stream. The net present value (NPV) of an initiative is the difference between the discounted stream of benefits and the discounted stream of costs. The NPV is given by:

$N P V = ∑ t = 0 n B t - O C t - I C t 1 + r t$

where:

• t is time in years
• n is number of years during which benefits and costs occur
• r is the discount rate
• Bt is benefits in year t
• OCt is infrastructure operating costs in year t
• ICt is investment costs in year t.

A positive NPV means that the initiative represents an improvement in economic efficiency compared with the Base Case.

Use the NPV to compare:

• Mutually exclusive options for the same initiative
• Alternative combinations of related initiatives (where implementation of one affects the benefits and/or costs of another)
• Alternative implementation timings for the same initiative.

The Incremental BCR in Section 11.6 below is an alternative tool for these situations.

## 10.4 Calculate the BCR

The BCR is the present value of benefits minus operating costs divided by the present value of costs. There are two alternative definitions depending on whether one puts infrastructure operating costs in the numerator or the denominator.

$B C R 1 = P V B P V O C + I C$

$B C R 2 = P V B - O C P V I C$

where $P V x = ∑ t = 0 n x t 1 + r t$

BCR1 puts costs and (savings in costs) that impact on government budgets in the denominator and everything else in the numerator. BCR2 puts costs that occur before completion of the initiative in the denominator and benefits and savings in costs that occur after completion of the initiative in the numerator. Benefits of deferred capital expenditure and costs of capital expenditure brought forward, discussed in Section 8.6, belong the denominator in BCR1 and the numerator in BCR2.

A BCR greater than one implies a positive NPV.

The BCR measure is used:

• As a convenient way to express the economic worth of an initiative
• To rank initiatives from an economic efficiency perspective where there is a budget constraint. BCR2 is the theoretically correct measure to use for this purpose because it is short-term funds being allocated. As long as operating and maintenance costs are small in relation to benefits and investment costs, BCR1 and BCR2 will be close and ranking by BCR1 should not lead to significant errors.

Never use BCRs to choose between mutually exclusive options for the same initiative, because they remove the effects of different scales of the initiatives.

## 10.5 Calculate net present value per dollar invested if required

The net present value per dollar invested is defined as

$N P V I = N P V P V I C$

The NPVI is exactly equal to BCR2 minus one. If BCR2 has been provided, there is no value added by providing the NPVI. However, if only BCR1 has been provided, the NPVI can be used for ranking initiatives subject to a budget constraint.

## 10.6 Calculate the incremental BCR (if applicable)

The incremental BCR (IBCR) can be used instead of the NPV in the three comparative situations listed in Section 11.3 above. The IBCR is defined as

$I B C R = P V B 2 - O C 2 - P V B 1 - O C 1 P V I C 2 - P V I C 1$

where the subscripts represent options 1 and 2, and option 2 has the greater investment cost. The IBCR is well-suited for comparing options involving different scales of initiative. Increases in the scale of initiative are worthwhile as long as the IBCR for each scale exceeds one.

IBCR comparisons can take account of budget constraints, unlike simple NPV comparisons.

• Posit a cut-off BCR at the level of the lowest acceptable BCR for initiatives competing for funds out the same budget. A cut-off BCR of 2 implies that each dollar of funds used to pay of the increment has an opportunity cost of funds of \$2 of forgone benefit from other investment opportunities not taken.
• List the options in ascending order of investment cost and calculate the IBCR for each adjacent pair.
• An IBCR above the cut-off implies the increment is worth accepting. If the IBCR for a pair of options is below the cut-off, reject the higher cost option. Remove it from the list and use the lower cost option as the basis for the next increment.
• The economically best option is that with the highest investment cost which has an IBCR greater than or equal to the cut-off BCR2. (UK DFT 2006).

## 10.7 Calculate the internal rate of return (IRR) (if required)

Central agencies sometimes require reporting of the internal rate of return (IRR).

The IRR is defined as the value of the discount rate at which the NPV equals zero. It represents the minimum discount rate at which the initiative is viable in economic terms. There is no formula for the IRR. It needs to be found by iteration. Excel has a function to do this.

The IRR can be used in the same way as the NPV to indicate whether or not an initiative will be of benefit to society as a whole. It provides an indication of the economic worth of an initiative without requiring specification of a discount rate.

The IRR has no other uses. Never use the IRR to rank initiatives or to choose between mutually exclusive options as this amounts to comparing initiatives using different discount rates.

## 10.8 Calculate the first-year rate of return (FYRR)

The first-year rate of return (FYRR) is the level of benefits minus operating costs in the first year of operation of the initiative discounted to year zero, divided by the present value of investment costs. That is:

$F Y R R = B t f 1 + r t f / ∑ t = 0 t f I C t 1 + r t$

where tf is the first year of operation of the initiative.

The FYRR can indicate whether an initiative’s optimal implementation time is in the past or future, and hence whether deferral is warranted. Provided the assumptions underlying the criterion are met (see NGTSM 2006 Volume 5 for details), the optimal implementation time is the first year in which the FYRR is greater than the discount rate.

All initiatives should be subjected to the FYRR test and the result reported in the Business Case.

[1] Leave out any planning and design costs already occurred at the time of undertaking the CBA because they will not be affected by any decision to proceed with the initiative.

[2] Testing whether an option has an IBCR greater than a cut-off BCR, μ, is the same as comparing the NPVs for the two options with investment costs grossed up by the cut-off BCR to account for the opportunity cost of scarce investment funds. $I B C R = P V B 2 - O C 2 - P V B 1 - O C 1 P V I C 2 - P V I C 1 > μ$ is the same as $PVB2 - OC2 - μPVIC2 > PVB1 - OC1 - μPVIC1$


text:
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# In the rectangular coordinate system above, the line y = x

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In the rectangular coordinate system above, the line y = x  [#permalink]

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31 Mar 2012, 01:54
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Difficulty:

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Question Stats:

73% (01:43) correct 27% (02:15) wrong based on 826 sessions

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Attachment:

Capture.GIF [ 3.28 KiB | Viewed 33707 times ]
In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (1, 4), what are the coordinates of point C?

A. (-4, -1)
B. (-1, 4)
C. (4, -1)
D. (1, -4)
E. (4, 1)

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Posts: 53066
Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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31 Mar 2012, 03:26
14
21
enigma123 wrote:
In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (1, 4), what are the coordinates of point C?

A. (-4, -1)
B. (-1, 4)
C. (4, -1)
D. (1, -4)
E. (4, 1)

Since the line y=x is the perpendicular bisector of segment AB, then the point B is the mirror reflection of point A around the line y=x, so its coordinates are (4, 1). The same way, since the x-axis is the perpendicular bisector of segment BC then the point C is the mirror reflection of point B around the x-axis, so its coordinates are (4, -1).

The question becomes much easier if you just draw rough sketch of the diagram:
Attachment:

graph.png [ 12.57 KiB | Viewed 33692 times ]
Now, you can simply see that options A, B, and D (blue dots) just can not be the right answers. As for option E: point (4, 1) coincides with point B, so it's also not the correct answer. Only answer choice C remains.

Similar questions to practice:
in-the-xy-coordinate-plane-is-point-r-equidistant-from-143502.html
in-the-rectangular-coordinate-system-the-line-y-x-is-the-132646.html
the-coordinates-of-points-a-and-c-are-0-3-and-127769.html
the-line-represented-by-the-equation-y-4-2x-is-the-127770.html
if-point-a-coordinates-are-7-3-point-b-coordinates-a-141972.html
in-the-rectangular-coordinate-system-the-line-y-x-is-the-88473.html
in-the-rectangular-coordinate-system-above-the-line-y-x-144774.html
the-line-represented-by-the-equation-y-4-2x-is-the-perpendi-87573.html

Hope it helps.
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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01 Aug 2012, 05:56
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teal wrote:
What is the mirror image of a point (x,y) around Y-axis?
Also what is the mirror image of a point (x,y) around line y=-x?

The mirror image of $$(x,y)$$ around the Y-axis is $$(-x,y)$$.

For the second question:
Assume we have a point P(a,b) and we want to find its mirror image around the line $$y = -x$$.
Let's denote the point we seek by Q(A,B). See the attached drawing.

The equation of the line passing through P and perpendicular to the line $$y = -x$$ is $$y - b = x - a$$, or $$y = x + b - a$$.
Since Q is also on this line, we have $$B = A + b - a$$, from which $$A - B = a - b$$.
The middle point of the line segment PQ (denoted by M) is also on the line $$y = -x$$, therefore $$\frac{a+A}{2}=-\frac{b+B}{2}$$, or $$A + B = -a - b$$.
Solving for A and B, we find that $$A = -b, B =-a$$.

Therefore, the mirror image of $$(x,y)$$ around the line $$y = -x$$ is $$(-y, -x)$$.
Attachments

MirrorImage.jpg [ 18.1 KiB | Viewed 32945 times ]

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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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01 Aug 2012, 04:04
1
What is the mirror image of a point (x,y) around Y-axis?

Also what is the mirror image of a point (x,y) around line y=-x?
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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02 Jan 2013, 04:09
1
enigma123 wrote:
In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (1, 4), what are the coordinates of point C?

A. (-4, -1)
B. (-1, 4)
C. (4, -1)
D. (1, -4)
E. (4, 1)

Any idea guys how to solve this mathematically?

For me, the best approach to this question is to draw and estimate AB and BC lines. Doing so obviously shows that:
(a) y has negative coordinates... Thus, eliminate B and E.
(b) x has positive coordinates beyond 2. Thus, eliminate A and D.

Or, if you want to be really sure... We can get the line perpendicular x=y.
(a) get negative reciprocal of slope of y=x which is m=1. Thus, reciprocal is m=-1. Perpendicular line: y=-x + b
(b) calculate b using A coordinates: y = -x + b ==> 4 = -1 + b ==> b = 5
(c) get the pt. of intersection. -x + 5 = x ==> x = 2.5
(d) get y=-(2.5) + 5 --> y = 2.5

So, obviously... C would have negative for y coordinate and x > 2.5... only C fits the bill

But still, drawing should suffice...
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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27 Jun 2013, 22:47
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Coordinate Geometry: math-coordinate-geometry-87652.html

All DS Coordinate Geometry Problems to practice: search.php?search_id=tag&tag_id=41
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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27 Jun 2013, 23:31
2
3
enigma123 wrote:
In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (1, 4), what are the coordinates of point C?

A. (-4, -1)
B. (-1, 4)
C. (4, -1)
D. (1, -4)
E. (4, 1)

Let the co-ordinates of B = (p,q). As x=y is the perpendicular bisector of the line segment AB, thus the middle point of AB will lie on x=y itself. Thus, for A(1,4) and B(p,q)-->

$$\frac {p+1} {2} = \frac {q+4} {2}$$ --> p-q = 3

Also, the slope of the line segment would be -1--> $$\frac {q-4}{p-1} = -1$$ --> $$p+q = 5$$.Thus, on solving, the co-ordinates of B (4,1).

Similarly, as y=0(the x-axis) is the perpendicular bisector of BC, thus, the mid point of BC would like on the x-axis and thus, the y co-ordinate of C has to be -1. Thus, only A and C survive. Again, the slope of line segment BC has to be undefined as it is parallel to the y-axis(or perpendicular to the x axis).

Only option C survives.

C.
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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28 Jun 2013, 04:36
2
Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

The line Y = X always makes 45 deg angle with the X axis and has slope 1

Since the line Y = X is perpendicular to line AB, The slope of AB must be -1 --------[The product of the slopes of a line and its perpendicular is always -1]

Hereinafter, Even if we are not familiar with 'mirror image' concept we can draft the following figure and can check the answer options.

Since X axis itself is bisector of line BC we can deduce that X value of C can not be negative. Eliminate A, B

We also know Y value of C can not be positive. Eliminate E

Now consider the point A(1,4) This point is on the line that has slope -1, so the X value of its opposite end (i.e. X of B and C also) must be greater than 1. So Choice D (1, -4) Can not be the location of C. Eliminate.

Only option left is C, which is the Answer.
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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29 Nov 2014, 20:36
EvaJager wrote:
teal wrote:
What is the mirror image of a point (x,y) around Y-axis?
Also what is the mirror image of a point (x,y) around line y=-x?

The mirror image of $$(x,y)$$ around the Y-axis is $$(-x,y)$$.

For the second question:
Assume we have a point P(a,b) and we want to find its mirror image around the line $$y = -x$$.
Let's denote the point we seek by Q(A,B). See the attached drawing.

The equation of the line passing through P and perpendicular to the line $$y = -x$$ is $$y - b = x - a$$, or $$y = x + b - a$$.
Since Q is also on this line, we have $$B = A + b - a$$, from which $$A - B = a - b$$.
The middle point of the line segment PQ (denoted by M) is also on the line $$y = -x$$, therefore $$\frac{a+A}{2}=-\frac{b+B}{2}$$, or $$A + B = -a - b$$.
Solving for A and B, we find that $$A = -b, B =-a$$.

Therefore, the mirror image of $$(x,y)$$ around the line $$y = -x$$ is $$(-y, -x)$$.

Hi I have one question. What if the question ask to find the mirror image of (x,y) around the line y = 2x + 3 for example, how to solve it?
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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13 Jul 2016, 06:37
1
enigma123 wrote:
Attachment:
Capture.GIF
In the rectangular coordinate system above, the line y = x is the perpendicular bisector of segment AB (not shown), and the x-axis is the perpendicular bisector of segment BC (not shown). If the coordinates of point A are (1, 4), what are the coordinates of point C?

A. (-4, -1)
B. (-1, 4)
C. (4, -1)
D. (1, -4)
E. (4, 1)

This problem does contain a diagram that looks like the following:

We are given that the line y = x is a perpendicular bisector of line segment AB. This indicates that point B is a reflection of point A across the line y = x. When we reflect a point (a,b) across the line y = x, the reflected point has the coordinates reversed: (b,a). Thus, since point A is at (2,3), point B must be (3,2).

We are next given that the x-axis is a perpendicular bisector of line segment BC. This means that point C must have the same x-coordinate as point B (3) but the opposite y-coordinate of point B (-2). To further elaborate, we can draw a diagram.

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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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22 Oct 2016, 05:06
Hi Bunuel,
Is it always true that a perpendicular bisector will have mirror reflection of the ends?
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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22 Oct 2016, 05:26
NaeemHasan wrote:
Hi Bunuel,
Is it always true that a perpendicular bisector will have mirror reflection of the ends?

Yes, it is always true.
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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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07 Sep 2017, 09:40
I took the approach of eliminating the options.

Since X-axis is perpendicular bisector of line BC, it means that B will line in I quadrant and C in IV quadrant . This means for point C x-cordinate will be +ve and y-co-ordinate will be negative. With this finding , remove options A,B and E

Now as per option D if point C is (1,-4) then point B will be (1,+4). But (1,4) is co-ordinate of point A.

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Re: In the rectangular coordinate system above, the line y = x  [#permalink]

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Re: In the rectangular coordinate system above, the line y = x   [#permalink] 05 Oct 2018, 21:30
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# Solve the following equations (where [*] denotes greatest integer function and {*} represent fractional part function and sgn represents signum function) (i) [x]+|x-2| le 0 and x in [-1,3] (ii) [2x]-2x=[x+1] (iii) [x^(2)]+2[x]=3x, 0 le x le 2

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(i) no solution (ii)-1,-(1)/(2) (iii) 0,1


text:
# Latex to HTML Converters

I’ve been using LaTeX for many years, I should say quickly for the freaks out there that it doesn’t mean I’m into vinyl or other strangeness. LaTeX is a document processing system that creates good quality documents
from text source, no hamsters or chains involved at all.

The standard processors you get with LaTeX are good at converting the source into Postscript or PDF (Acrobat) documents and for most of the time this will do. However there are occasions when you want to have your document output in HTML. In this case you need to have a different processor.

This page is about the various types of LaTeX to HTML converters out there. It is not an exhaustive list but should help other people looking around for converters. The main problem with them all is they are not
maintained that well.

## Hyperlatex

Hyperlatex is the converter I have used the most. It does most jobs quite well and you get reasonable results from it. My major gripes with it is that it is written in Lisp so I cannot extend it (I don’t know Lisp) and that it doesn’t do CSS that well.

Despite those shortcomings, Hyperlatex is a good start for document conversion. Unlike most program on this page, it is actively maintained and keeps up with HTML standards. For example there is work for
Hyperlatex output to be in XHTML.

## TTH

TTH has put a lot of effort into the formula conversion. Most converters make an image for the formulas while TTH generates HTML for it, giving the formulas a more consistent look in the document rather than looking like they were “pasted in” later.

TTH has a funny license in that (roughly) it is free for non-commercial use only. Depending on where you are going to use it, this may be a problem. You can buy a commercial license of TTF too.

## Heava

HeVeA is one converter I haven’t used, but will try out soon. It looks like it would get confused by some of my documents, especially anything with nested environments.

The program is written in a language called Objective Caml which I know even less about than Lisp. That means no way of extending it for me.

## LaTeX2HTML

At first I thought this would be the converter for me. It looks like it converts it pages rather well and it is written in a programming language I understand (Perl).

The main problem with this program is that it has not been maintained for years. A consequence of that is the HTML rendering is a bit old and doesn’t keep up with the latest standards.

## text4ht

Another one I’ve not tried yet. This one does look recently maintained and I will be trying it out.

## LaTeXML

This converter takes LaTeX as an input and instead of having an output file format of DVI makes it XML. It is written in Perl and was developed with a particular focus on the mathematical equations. To get HTML you use a post-processor.

## 11 thoughts on “Latex to HTML Converters”

1. itex2MML is worth serious consideration. It can run as a binary
and also one can build libraries for, e.g., ruby, java, and php that
are easily integrated as part of a webserver or wordpress installation.

The TeX part comprises basically everything except for sectioning,
which can be added or converted to standard HTML breaks and headers.

2. I tried Latexml and could get an html file but could never get the bibtex working on it. It would run and then never give the expected output but would show no errors.

I then tried tex4ht on a Kubuntu system. tex4ht was already on my system and the command htlatex [file] worked just fine right out of the box. I got exactly what I wanted with no need to mess with anything else. I didn’t have any formulas in my text and didn’t do any figures either.

Thanks for putting up this post.
Kyle

3. I got TTH to work – it quite easily integrates into WinEdt and thankfully does not require ellaborate installation like some of the other options mentioned in your post. However, I am having trouble rendering parenthesis of type \left( and \right). The HTML file displays empty square in place of parentheses in the browser (I am using Google Chrome).

4. Having tested all of the converters (with the intention of getting some mathematical documents readable on ebook readers), I am sorry to say that none of them did what I wanted.

Hyperlatex does not deal well with custom operators (DeclareMathOperator stopped it). When I changed that it returned another error that I could not understand.

TTH was very satisfactory in all respects except equations. They convert equations to html directly (not using images), which works for simple equations. However, many of the equations in my case were converted in a way as to make them unreadable. Sum signs were not displayed, instead of subscripts _ signs are displayed. If you don’t need equations or need only very simple equations this works great, but otherwise it is unfortunately not good enough.

As the OP guessed, Hevea had problems with nested environments. This meant that the document was not completely converted.

tex4ht works similarly to TTH in that it converts equations to HTML. It does a decent job in that, but again some subscripts are not in appropriate places, some symbols (like large oplus) are too small. Thus equations do not quite capture their original meaning and often-times their misprint serves to confuse the reader. However, this one worked best out of the ones mentiond thus far.

Latex2HTML worked rather well. It converted equations to images which meant there were no problems with their display. However, images were too small and the latex document was generated into a fully fledged website with a table of contents and hyperlinks joining the various documents together. This is great if you want a website, but for something to be converted into an ebook it is unfortunately again not appropriate.

The one that I find worked best of all was latexml. Equations were readable and correctly understood. But, unfortunately, at some point it decided that all the text should be centred. I cannot find what in my Tex file triggers this behaviour. Otherwise really liked this one, it’s probably the most promising of all. It can also be used to convert equations straight into images with a command line option.

5. Couldn’t get Hyperlatex working. But latex2html did the task. Of course the tex file must be very simple, still output HTML looks fine.

7. $LaTeX$ is simply used in wordpress pages.