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# Configuration file description¶

The configuration file is in the format described in the Python configparser documentation as “a basic configuration file parser language which provides a structure similar to what you would find on Microsoft Windows INI files.”

Most settings must be explicitly enumerated in the configuration file. (This makes it less likely that library code changes will result in your project silently using different NEAT settings. There are some defaults, as noted below, and insofar as possible new configuration parameters will default to the existing behavior.)

Note that the Config constructor also requires you to explicitly specify the types that will be used for the NEAT simulation. This, again, is to help avoid silent changes in behavior.

The configuration file is in several sections, of which at least one is required. However, there are no requirements for ordering within these sections, or for ordering of the sections themselves.

## [NEAT] section¶

The NEAT section specifies parameters particular to the generic NEAT algorithm or the experiment itself. This section is always required, and is handled by the Config class itself.

• fitness_criterion
The function used to compute the termination criterion from the set of genome fitnesses. Allowable values are: min, max, and mean
• fitness_threshold
When the fitness computed by fitness_criterion meets or exceeds this threshold, the evolution process will terminate, with a call to any registered reporting class’ found_solution method.

Note

The found_solution method is not called if the maximum number of generations is reached without the above threshold being passed (if attention is being paid to fitness for termination in the first place - see no_fitness_termination below).

• no_fitness_termination

If this evaluates to True, then the fitness_criterion and fitness_threshold are ignored for termination; only valid if termination by a maximum number of generations passed to population.Population.run() is enabled, and the found_solution method is called upon generation number termination. If it evaluates to False, then fitness is used to determine termination. This defaults to “False”.

New in version 0.92.

• pop_size
The number of individuals in each generation.
• reset_on_extinction
If this evaluates to True, when all species simultaneously become extinct due to stagnation, a new random population will be created. If False, a CompleteExtinctionException will be thrown.

## [DefaultStagnation] section¶

The DefaultStagnation section specifies parameters for the builtin DefaultStagnation class. This section is only necessary if you specify this class as the stagnation implementation when creating the Config instance; otherwise you need to include whatever configuration (if any) is required for your particular implementation.

• species_fitness_func
The function used to compute species fitness. This defaults to mean. Allowed values are: max, min, mean, and median

Note

This is not used for calculating species fitness for apportioning reproduction (which always uses mean).

• max_stagnation
Species that have not shown improvement in more than this number of generations will be considered stagnant and removed. This defaults to 15.
• species_elitism
The number of species that will be protected from stagnation; mainly intended to prevent total extinctions caused by all species becoming stagnant before new species arise. For example, a species_elitism setting of 3 will prevent the 3 species with the highest species fitness from being removed for stagnation regardless of the amount of time they have not shown improvement. This defaults to 0.

## [DefaultReproduction] section¶

The DefaultReproduction section specifies parameters for the builtin DefaultReproduction class. This section is only necessary if you specify this class as the reproduction implementation when creating the Config instance; otherwise you need to include whatever configuration (if any) is required for your particular implementation.

• elitism
The number of most-fit individuals in each species that will be preserved as-is from one generation to the next. This defaults to 0.
• survival_threshold
The fraction for each species allowed to reproduce each generation. This defaults to 0.2.
• min_species_size
The minimum number of genomes per species after reproduction. This defaults to 2.

## [DefaultGenome] section¶

The DefaultGenome section specifies parameters for the builtin DefaultGenome class. This section is only necessary if you specify this class as the genome implementation when creating the Config instance; otherwise you need to include whatever configuration (if any) is required for your particular implementation.

• activation_mutate_rate
The probability that mutation will replace the node’s activation function with a randomly-determined member of the activation_options. Valid values are in [0.0, 1.0].
• aggregation_mutate_rate
The probability that mutation will replace the node’s aggregation function with a randomly-determined member of the aggregation_options. Valid values are in [0.0, 1.0].
• bias_init_stdev
The standard deviation of the normal/gaussian distribution, if it is used to select bias values for new nodes.
• bias_init_type

If set to gaussian or normal, then the initialization is to a normal/gaussian distribution. If set to uniform, a uniform distribution from $$\max(bias\_min\_value, (bias\_init\_mean-(bias\_init\_stdev*2)))$$ to $$\min(bias\_max\_value, (bias\_init\_mean+(bias\_init\_stdev*2)))$$. (Note that the standard deviation of a uniform distribution is not range/0.25, as implied by this, but the range divided by a bit over 0.288 (the square root of 12); however, this approximation makes setting the range much easier.) This defaults to “gaussian”.

New in version 0.92.

• bias_max_value
The maximum allowed bias value. Biases above this value will be clamped to this value.
• bias_min_value
The minimum allowed bias value. Biases below this value will be clamped to this value.
• bias_mutate_power
The standard deviation of the zero-centered normal/gaussian distribution from which a bias value mutation is drawn.
• bias_mutate_rate
The probability that mutation will change the bias of a node by adding a random value.
• bias_replace_rate
The probability that mutation will replace the bias of a node with a newly chosen random value (as if it were a new node).
• compatibility_threshold
Individuals whose genomic distance is less than this threshold are considered to be in the same species.

Note

It is currently possible for two homologous nodes or connections to have a higher contribution to the genomic distance than a disjoint or excess node or connection, depending on their attributes and the settings of the above parameters.

The probability that mutation will add a connection between existing nodes. Valid values are in [0.0, 1.0].
• conn_delete_prob
The probability that mutation will delete an existing connection. Valid values are in [0.0, 1.0].
• enabled_default
The default enabled attribute of newly created connections. Valid values are True and False.

Note

“Newly created connections” include ones in newly-created genomes, if those have initial connections (from the setting of the initial_connection variable).

• enabled_mutate_rate
The probability that mutation will replace (50/50 chance of True or False) the enabled status of a connection. Valid values are in [0.0, 1.0].

Adds to the enabled_mutate_rate if the connection is currently enabled.

Adds to the enabled_mutate_rate if the connection is currently not enabled.

New in version 0.92: enabled_rate_to_false_add and enabled_rate_to_true_add

• feed_forward
If this evaluates to True, generated networks will not be allowed to have recurrent connections (they will be feedforward). Otherwise they may be (but are not forced to be) recurrent.
• initial_connection
Specifies the initial connectivity of newly-created genomes. (Note the effects on settings other than unconnected of the enabled_default parameter.) There are seven allowed values:
• unconnected - No connections are initially present. This is the default.
• fs_neat_nohidden - One randomly-chosen input node has one connection to each output node. (This is one version of the FS-NEAT scheme; “FS” stands for “Feature Selection”.)
• fs_neat_hidden - One randomly-chosen input node has one connection to each hidden and output node. (This is another version of the FS-NEAT scheme. If there are no hidden nodes, it is the same as fs_neat_nohidden.)
• full_nodirect - Each input node is connected to all hidden nodes, if there are any, and each hidden node is connected to all output nodes; otherwise, each input node is connected to all output nodes. Genomes with feed_forward set to False will also have recurrent (loopback, in this case) connections from each hidden or output node to itself.
• full_direct - Each input node is connected to all hidden and output nodes, and each hidden node is connected to all output nodes. Genomes with feed_forward set to False will also have recurrent (loopback, in this case) connections from each hidden or output node to itself.
• partial_nodirect # - As for full_nodirect, but each connection has a probability of being present determined by the number (valid values are in [0.0, 1.0]).
• partial_direct # - as for full_direct, but each connection has a probability of being present determined by the number (valid values are in [0.0, 1.0]).

Changed in version 0.92: fs_neat split into fs_neat_nohidden and fs_neat_hidden; full, partial split into full_nodirect, full_direct, partial_nodirect, partial_direct

The probability that mutation will add a new node (essentially replacing an existing connection, the enabled status of which will be set to False). Valid values are in [0.0, 1.0].
• node_delete_prob
The probability that mutation will delete an existing node (and all connections to it). Valid values are in [0.0, 1.0].
• num_hidden
The number of hidden nodes to add to each genome in the initial population.
• num_inputs
The number of input nodes, through which the network receives inputs.
• num_outputs
The number of output nodes, to which the network delivers outputs.
• response_init_stdev
The standard deviation of the normal/gaussian distribution, if it is used to select response multipliers for new nodes.
• response_init_type

If set to gaussian or normal, then the initialization is to a normal/gaussian distribution. If set to uniform, a uniform distribution from $$\max(response\_min\_value, (response\_init\_mean-(response\_init\_stdev*2)))$$ to $$\min(response\_max\_value, (response\_init\_mean+(response\_init\_stdev*2)))$$. (Note that the standard deviation of a uniform distribution is not range/0.25, as implied by this, but the range divided by a bit over 0.288 (the square root of 12); however, this approximation makes setting the range much easier.) This defaults to “gaussian”.

New in version 0.92.

• response_max_value
The maximum allowed response multiplier. Response multipliers above this value will be clamped to this value.
• response_min_value
The minimum allowed response multiplier. Response multipliers below this value will be clamped to this value.
• response_mutate_power
The standard deviation of the zero-centered normal/gaussian distribution from which a response multiplier mutation is drawn.
• response_mutate_rate
The probability that mutation will change the response multiplier of a node by adding a random value.
• response_replace_rate
The probability that mutation will replace the response multiplier of a node with a newly chosen random value (as if it were a new node).
• single_structural_mutation

If this evaluates to True, only one structural mutation (the addition or removal of a node or connection) will be allowed per genome per generation. (If the probabilities for conn_add_prob, conn_delete_prob, node_add_prob, and node_delete_prob add up to over 1, the chances of each are proportional to the appropriate configuration value.) This defaults to “False”.

New in version 0.92.

• structural_mutation_surer

If this evaluates to True, then an attempt to add a node to a genome lacking connections will result in adding a connection instead; furthermore, if an attempt to add a connection tries to add a connection that already exists, that connection will be enabled. If this is set to default, then it acts as if it had the same value as single_structural_mutation (above). This defaults to “default”.

New in version 0.92.

• weight_init_stdev
The standard deviation of the normal/gaussian distribution used to select weight values for new connections.
• weight_init_type

If set to gaussian or normal, then the initialization is to a normal/gaussian distribution. If set to uniform, a uniform distribution from $$\max(weight\_min\_value, (weight\_init\_mean-(weight\_init\_stdev*2)))$$ to $$\min(weight\_max\_value, (weight\_init\_mean+(weight\_init\_stdev*2)))$$. (Note that the standard deviation of a uniform distribution is not range/0.25, as implied by this, but the range divided by a bit over 0.288 (the square root of 12); however, this approximation makes setting the range much easier.) This defaults to “gaussian”.

New in version 0.92.

• weight_max_value
The maximum allowed weight value. Weights above this value will be clamped to this value.
• weight_min_value
The minimum allowed weight value. Weights below this value will be clamped to this value.
• weight_mutate_power
The standard deviation of the zero-centered normal/gaussian distribution from which a weight value mutation is drawn.
• weight_mutate_rate
The probability that mutation will change the weight of a connection by adding a random value.
• weight_replace_rate
The probability that mutation will replace the weight of a connection with a newly chosen random value (as if it were a new connection).


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# T-invariant subspace;inner product

You might note that you can generate a $T$-invariant subspace by taking a vector $x\in V$ and taking the basis $\{x, Tx, T^2x, T^3x,\ldots\}$.

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# Homework Help: Cohen Tannoudji QM VOl.1 Chapter 1 Exercise 4 -- Eigenvalue & Fourier Transform of a Function

1. Jan 28, 2015

### Helloaksdoq

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

I did Fourier transform directly to the eigenvalue equation and got

Psi(p)=a*Psi(0)/(p^2/2m-E)

But the rest, I don't even know where to start.
Any opinion guys?

2. Feb 1, 2015

### Goddar

Hi. What you got so far seems right.
Now i can't think of any easy "trick" to show the uniqueness of E in this representation, so i would suggest you work on how to enforce:
E(p)|φE'(p)> = δ(E–E')
I suppose you can look up the integral; given the result you should see:
1 - how E needs to be positive
2 - how E must be = E' (i.e.: E is unique)
That's a start...

Last edited: Feb 1, 2015


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Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an element of $\mathbb{C}$ and thus would be $z \ \stackrel{\mathrm{def}}{=}\ Ae^{s T}$ (but then it would be different to the Laplace Transform...). I don't understand why the Z-Transform is not defined as: $$X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] e^{-\omega n}$$ or something like that.

Z-transform

The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of $$z \ \stackrel{\mathrm{def}}{=}\ e^{s T} \$$ where $T = 1/f_s \$is the sampling period (in units of time e.g., seconds) and $f_s \$is the sampling rate (in samples per second or hertz)

Let $$\Delta_T(t) \ \stackrel{\mathrm{def}}{=}\ \sum_{n=0}^{\infty} \delta(t - n T)$$ be a sampling impulse train (also called a Dirac comb) and \begin{align} x_q(t) & \stackrel{\mathrm{def}}{=}\ x(t) \Delta_T(t) = x(t) \sum_{n=0}^{\infty} \delta(t - n T) \\ & = \sum_{n=0}^{\infty} x(n T) \delta(t - n T) = \sum_{n=0}^{\infty} x[n] \delta(t - n T) \end{align} be the continuous-time representation of the sampled x(t) \ $$x[n] \ \stackrel{\mathrm{def}}{=}\ x(nT) \$$ are the discrete samples of x(t) The Laplace transform of the sampled signal x_q(t) \ is \begin{align} X_q(s) & = \int_{0^-}^\infty x_q(t) e^{-s t} \,dt \\ & = \int_{0^-}^\infty \sum_{n=0}^\infty x[n] \delta(t - n T) e^{-s t} \, dt \\ & = \sum_{n=0}^\infty x[n] \int_{0^-}^\infty \delta(t - n T) e^{-s t} \, dt \\ & = \sum_{n=0}^\infty x[n] e^{-n s T}. \end{align} This is precisely the definition of the unilateral Z-transform of the discrete function $x[n] \$. $$X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}$$ with the substitution of $z \leftarrow e^{s T} \$.

Comparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal: $$X_q(s) = X(z) \Big|_{z=e^{sT}}$$ The similarity between the Z and Laplace transforms is expanded upon in the theory of time scale calculus.

Here: http://en.wikipedia.org/wiki/Z-transform it says that $z \in \mathbb{C}$.

-
The $s$ in the definition is a complex number also, and the exponential over the complexes is surjective (even if not injective). –  guaraqe Jun 15 '12 at 13:00
Thanks for the reply. I looked up the words surjective and injective but I don't really understand what you mean. Are you saying that since $s$ is a complex number, $e^{st}$ can basically represent any complex number? If it can, then why not stick a $z$ in the Laplace transform instead of $e^{st}$? –  user968243 Jun 15 '12 at 14:21
Exactly. The $z$ or the exponential depend on how you are going to use these series. The exponential is used mostly in the solution of differential equations. If you put a $z$ there, you will see many $\log z$ terms that are not necessary. –  guaraqe Jun 18 '12 at 23:32
@JuanSimões: The exponential is not surjective over $\mathbb{C}$, since $e^z\neq 0$. But on $\mathbb{C}\setminus\{0\}$ it is. –  Mårten W Oct 8 '13 at 20:20


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# Diatomic molecule problem

1. Nov 5, 2007

### quantum_prince

We have Heteronuclear diatomic molecules with atomic masses m1 and m2 and an internuclear distance R have rotational energy eigenvalues
EJ = BJ(J + 1) , J = 0, 1, 2, . . . when they are considered as a rigid rotor. The rotational constant is given by
B = h^2/2µR^2 with the reduced mass µ = m1m2/(m1 + m2) .

I need to Describe the frequency spectrum observed when photons are emitted
in a pure rotational transition J + 1 -> J .

also describe a method to estimate the internuclear distance R
from such a rotational spectrum ?.

How to proceed in this regard.

Regards.
QP.


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A Look at the Idris Internals, Part IV - From TT Onward

published on July 19th, 2016

So after looking at the invocation of the elaborator in the last post, I now want to look into how the resulting TT terms are further processed and simplified into the other intermediary languages. The cross-platform compilers paper by Edwin Brady gives a high level description of the later stage intermediate languages of Idris. This post will take a look at where the intermediate languages are located in the code and what functions are invoked to simplify TT further.

Idris's later stage intermediate languages

After converting Idris to TT, there are a number of additional intermediate languages, which get produced in subsequent steps. The idea is that these languages are progressively simpler and that each code generator can pick a lanugage which it is easiest to generate the desired code from. The cross-platform compilers paper by Edwin Brady ilustrates the intermediate languages used in Idris. After Idris and TT, the paper lists the following:

• TT_case in which pattern matching is converted to simple case trees via a pattern match compiler.

• IR_case in which all types are erased, as well as any values which are provably not used at run-time.

• IR_lift in which there are no lambda bindings other than at the top level.

• IR_defunc in which all functions are fully applied. All functions are first-order, by defunctionalisation.

• IR_ANF in which all functions are in Applicative Normal Form (ANF), that is all arguments to functions are trivial. By “trivial”, we mean that evaluation of an argument must terminate immediately, that is arguments are either constant or variables.

Where are the intermediate languages produced?

Looking at the idrisMain function in Idris/REPL.hs again, we first loadInputs (which ends up calling either loadSource or reloading the equivalent from ibc files). As discussed in the last post, the loadSource function from Idris/Parser.hs does a lot of work, namely parsing source files, invoking the elaborator via the elabDecls function from Idris/ElabDecls.hs, simplifying case definitions via the simplifyCasedef function (this converts TT to TT_case), totality checking (first building a size change graph via buildSCG, then checkDeclTotality and verifyTotality), and finally writing out the ibc file.

Next we call the process function with a Compile command. process (also in REPL.hs) calls the compile function in IRTS/Compiler.hs, so this seems to be the main entry point for further processing of the TT code.

compile does some usage analysis to find reachable names, then calls mkDecls on a list of these names. mkDecls has a type of [Name] -> Idris [(Name, LDecl)], so outputs a list of declarations of type LDecl. LDecl is the first post-TT intermediate language, it is defined in IRTS/Lang.hs. If I am not mistaken, this should correspond to IR_case in the above list (TT_case is a special case of TT which is represented in the same data structure as TT). LDecl just has two constructors, LFun and LConstructor. The latter just has a Name, an Int (tag) and another Int (arity), the former has options, names (name and argument names) and an expression of type LExp, which is the actual definition of the function (defined in the same file).

mkLDecls is basically a wrapper around the build function, which creates a (Name, LDecl) pair for each (Name, Def) declaration that it gets, so LDecls seem to correspond directly to Defs. If the provided name is a primitive operation build directly constructs an LFun which contains an LOp with some machine generated names for the arguments, otherwise it calls mkLDecl to do this.

mkLDecl handles the constructors Function, CaseOp, TyDecl DCon and TyDecl TCon explicitly, for everything else it creates an LFun [Inline] name [] LNothing construction, this seems to be a function which does a no-op. For the Function and the CaseOp constructors, the irTerm and irTree seem to be the real work horses, we'll check these out next.

Of the two irTree itself is short, handing of its work to irSC, which pattern matches on a variable of type SC, this seems to be the case tree representation of the TT terms generated by simplifyCasedef earlier. irSC is pretty involved and looks like it is handeling a lot of special cases. A better understanding of the case trees is probably necessary to understand what is happening in irSC and why.

irTerm on the other hand handles conversions of TT terms, handling App, P, V, Bind, Proj, Constant, TType, Erased and Impossible constructors. The latter three are mapped directly to LNothing, transformations for P, V, Bind, Proj and Constant seem pretty straight-forward, however, the App case is very involved, it unApplys the application and has about 20 different pattern matches on the P constructor alone.

Nevertheless mkDecls and the functions it calls handle the transformation from TT to LDecl - some complications along these lines should have been expected ;).

Further processing

Returning to the compile function, the inlineAll function from IRTS/LangOpts.hs is invoked on our LDecls, which, for all LDecls basically searches for LApp constructs (so function applications), checks a number of conditions to see that these are inlineable, invents fresh names for the variables and replaces the application with the function body.

Next allocUnique and addTags make some tranforms to maintain unique names (?) and tag constructors(?). In there liftAll does lambda lifting for all LDecls, the work horse here seems to be the lift function in IRTS/Lang.hs. So this innocent looking step does the transformation from IR_case to IR_lift, all the while the data type is still LDecl.

Next up is a call to defunctionalise which turns LDefs into DDefs, this is the representation of IR_defunc where all function applications are fully applied and first order. DDefs is a Ctxt around DDecl, which looks pretty similar to LDecl (some cases have been simplified, I suppose). Next, DDefs are inlineed, as far as I can tell this does not do anything at the moment.

The next call is to simplifyDefs which turns the DDefs into SDefs, where SDefs are definitions in the simplified language defined in IRTS/Simplified.hs, I think this corresponds to IR_ANF, xthe simplest of the intermediate languages.

With this we're pretty much through the intermediate language generation and return a CodegenInfo, which contains the simplified (c, IR_ANF), defunctionalized (defuns IR_defun) and lifted (tagged, IR_lift) representations.


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# nginx配置备忘

Linux

$request_time request processing time in seconds with a milliseconds resolution; time elapsed between the first bytes were read from the client and the log write after the last bytes were sent to the client 服务端从客户端读取到第一个字节，直到发送最后一个字节到客户端所需要的时间，单位是秒，精度是毫秒。 $upstream_response_time

keeps time spent on receiving the response from the upstream server; the time is kept in seconds with millisecond resolution. Times of several responses are separated by commas and colons like addresses in the \$upstream_addr variable.

Syntax:	upstream name { ... }
Default:	—
Context:	http

Defines a group of servers. Servers can listen on different ports. In addition, servers listening on TCP and UNIX-domain sockets can be mixed.

Syntax:	keepalive connections;
Default:	—
Context:	upstream
This directive appeared in version 1.1.4.

Activates the cache for connections to upstream servers.

The connections parameter sets the maximum number of idle keepalive connections to upstream servers that are preserved in the cache of each worker process. When this number is exceeded, the least recently used connections are closed.


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Interpreting Robust Optimization via Adversarial Influence Functions
Robust optimization has been widely used in nowadays data science, especially in adversarial training. However, little research has been done to quantify how robust optimization changes the optimizers and the prediction losses comparing to standard training. In this paper, inspired by the influence function in robust statistics, we introduce the Adversarial Influence Function (AIF) as a tool to investigate the solution produced by robust optimization. The proposed AIF enjoys a closed-form and can be calculated efficiently. To illustrate the usage of AIF, we apply it to study model sensitivity — a quantity defined to capture the change of prediction losses on the natural data after implementing robust optimization. We use AIF to analyze how model complexity and randomized smoothing affect the model sensitivity with respect to specific models. We further derive AIF for kernel regressions, with a particular application to neural tangent kernels, and experimentally demonstrate the effectiveness of the proposed AIF. Lastly, the theories of AIF will be extended to distributional robust optimization.
Authors:
; ; ;
Editors:
Award ID(s):
Publication Date:
NSF-PAR ID:
10217370
Journal Name:
Proceedings of Machine Learning Research
Volume:
119
ISSN:
2640-3498
4. Due to the surge of cloud-assisted AI services, the problem of designing resilient prediction serving systems that can effectively cope with stragglers and minimize response delays has attracted much interest. The common approach for tackling this problem is replication which assigns the same prediction task to multiple workers. This approach, however, is inefficient and incurs significant resource overheads. Hence, a learning-based approach known as parity model (ParM) has been recently proposed which learns models that can generate parities’’ for a group of predictions to reconstruct the predictions of the slow/failed workers. While this learning-based approach is more resource-efficient than replication, it is tailored to the specific model hosted by the cloud and is particularly suitable for a small number of queries (typically less than four) and tolerating very few stragglers (mostly one). Moreover, ParM does not handle Byzantine adversarial workers. We propose a different approach, named Approximate Coded Inference (ApproxIFER), that does not require training any parity models, hence it is agnostic to the model hosted by the cloud and can be readily applied to different data domains and model architectures. Compared with earlier works, ApproxIFER can handle a general number of stragglers and scales significantly better with the numbermore »


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# Isometry of two metric spaces

How to prove that $(\Bbb R^2, d_1)$ and $(\Bbb R^2, d_\infty)$ are isometric?

My approach

Let $f:(\Bbb R^2, d_1)\to(\Bbb R^2, d_\infty)$ be a function defined by $f(x, y) = (x+y,x-y)$. I can easily prove that $f$ is bijective, but I can not prove the isometry of $f$.

• Please write the question properly. What is $d_1$ and $d_{\infty}$? It will be better if you write those in your question, rather than asking people to guess. – Krish Aug 29 '17 at 8:56
• Is it $d_1(x,y)=|x_1-y_1|+|x_2-y_2|$ ? – Marios Gretsas Aug 29 '17 at 9:01
• @Marios yes absolutely correct – Anil Kumar Pandey Aug 29 '17 at 9:03
• The standard notation is $d_n(x, y) = \left(|x_1-y_1|^n + |x_2-y_2|^n\right)^{1/n}$, so that $d_1$ is the taxicab metric (as suggested by Marios), $d_2$ is the Euclidean metric, and $d_\infty = \lim_{n\to \infty}d_n$ (pointwise limit) is the max-metric (sup-metric for infinite dimensions, like polynomial spaces). – Arthur Aug 29 '17 at 9:03
• @JeanMarie: I was not aware of the terminology. Thanks Arthur – Krish Aug 29 '17 at 9:07

Since both metrics are induced by norms and your function is linear, it is enough to check that $f$ preserves norms, i.e. we must check that $$\max \{|x+y|,|x-y|\}=|x|+|y|$$ for all $x,y\in \Bbb R$.

We can assume WLOG that $x$ and $y$ and non-negative, since changing their sign would not change the absolute values in the RHS and would only swap the two terms in the LHS.

Then the RHS would be equal to $\max\{x+y,|x-y|\}$; but since $$|x-y|\leq |x|+|y|=x+y,$$ both sides must then be equal to $x+y$.

• In order to prove that $\max\{|x+y|,|x-y|\}=|x|+|y|$, I would square both sides. The square of the LHS is $\max\{x^2+y^2+2xy,x^2+y^2-2xy\}$ and the square of the RHS is $x^2+y^2+2|xy|$. It is clear that they are equal. – José Carlos Santos Aug 29 '17 at 9:18

The map $f$ maps the square $Q_1$ with vertices $(1,0)$, $(0,1)$, $(-1,0)$, $(0,-1)$ onto the square $Q_\infty\subset({\mathbb R}^2,d_\infty)$ with vertices $(1,1)$, $(1,-1)$, $(-1,-1)$, $(-1,1)$.

Since $Q_1$ is the closed unit ball in $({\mathbb R}^2,d_1)$, and $Q_\infty$ is the closed unit ball in $({\mathbb R}^2,d_\infty)$, it follows from general principles about norms and linear maps that $f$ is an isometry.

Here it is important to keep indices and dimensions straight. Remember that both the $d$'s take in two pairs of numbers as arguments, while $f$ only takes in one pair. Both a basic pair (vector / point) and a pair of pairs is tempting to write as $(x, y)$, so this is a breeding ground for confusion. I will try to keep it straight in my answer, using indexed variables for numbers and unindexed variables for points.

Let $x = (x_1, x_2)$ and $y = (y_1, y_2)$. Then $d_1(x, y) = |x_1-y_1| + |x_2-y_2|$, while $$d_\infty(f(x), f(y)) = \max(|f(x)_1 - f(y)_1|, |f(x)_2 - f(y)_2|)\\ = \max(|(x_1 + x_2) - (y_1+y_2)|, |(x_1-x_2) -(y_1-y_2)|)\\ = \max(|(x_1 -y_1) + (x_2-y_2)|, |(x_1-y_1) - (x_2-y_2)|)$$ which is now of the form $\max(|a+b|, |a-b|)$, which is equal to $|a|+|b|$ (you can, for instance, argue by cases: do $a$ and $b$ have the same sign or not?), and we're done.

Let $(x,y), (u,v) \in \mathbb{R}^2$.

$$D =d_\infty(f(x,y),f(u,v))$$ $$=d_\infty((x+y,x-y),(u+v,u-v))$$ $$= \|x+y-u-v, x-y-u+v||_{\infty}$$ As, $$\|(A,B)\|_{\infty}= \max(A,B) =\frac{|A+B|+|A-B|}{2}$$

we can write $$D = \frac{1}{2}(|(x+y-u-v)+(x-y-u+v)|+|(x+y-u-v)-(x-y-u+v)|)$$

$$= \frac{1}{2}(|2x-2u|+|2y-2v|)$$ $$= |x-u|+|y-v|$$ $$=d_1((x,y),(u,v))$$


text:
• ### Announcements

#### Archived

This topic is now archived and is closed to further replies.

# Texture UVs

## Recommended Posts

Nazrix    307
I''m trying to play with UV''s. From what I understood the U was the x of the picture and the V was the y. So, what I''m trying to do for now is display the upper left corner of the picture on a Quad. I''m using the following code, but it''s not giving the desired effect.

glTexCoord2f(0,0);
glVertex3f(0,0,-10);

glTexCoord2f(.5f,0);
glVertex3f(3,0,-10);

glTexCoord2f(.5f,.5f);
glVertex3f(3,3,-10);

glTexCoord2f(0,.5f);
glVertex3f(0,3,-10);

glEnd();

A CRPG in development... Need help? Well, go FAQ yourself. "Just don''t look at the hole." -- Unspoken_Magi

##### Share on other sites
Brian Jones    122
It looks like you''re using coordinate of the lower left corner of the texture

##### Share on other sites
Well, specify what effect you get, and what effect you would liek to get.

Height Map Editor | Eternal Lands | Fast User Directory

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Sander    1332
In an image editor (0,0) is the top-left corner of your image. In OpenGL (0,0) is the bottom-left corner. so u=x and v=1-y

Sander Maréchal


text:
# Writing. Part 2. Домбровская И.В - 28 стр.

Составители:

Рубрика:

• ## Иностранный язык

27
First and Last Paragraph Techniques.
The first paragraph may:
- state a strong, firm opinion e.g. Vegetarians are definitely healthier than meat-
eaters.
- make reference to a strange scene or situation e.g. Will frogs ever become
household pets?
- be addressed to the reader directly (in less formal articles) e.g. What do you
think ? Have you ever ? If you want to
- start with a quotation e.g. Try not to become a man of success but rather try to
become a man of value. or a rhetorical question (a rhetorical question is a
question which needs no reply) e.g. Do the students know how fortunate they
- start with a problem which needs a solution e.g. Can you explain the problem
of
The last paragraph may:
- state a personal opinion e.g. I believe , In my opinion , It seems that etc
- give the reader something to consider e.g. After all, is it fair to
- summarise the article e.g. In conclusion , To sum up , On the whole etc
- end with a quotation e.g. Just a little more loving and a lot less fighting and
the world would be all right.
Exercise 22
Match the first with the last paragraphs. Which techniques are used for the
first/last paragraphs? Which outlines should be used for each topic?
Suggest a title for each topic.
First paragraphs
1. Physical education has long been part of the school timetable for children;
imagine an educational programme which does not include some form of
physical education.
2. Throughout history people have taken part in dangerous sports. This does
not change the fact that it is unacceptable for athletes to be seriously
injured or even lose their lives for the sake of competition.
3. The advance of technology has meant that most schoolchildren are
familiar with computers, which can be used to teach a wide variety of
subjects. But can they, and should they, replace teachers?
4. As our cities continue to grow, and villages become smaller than ever, it
is true to say that it is far better to live in a busy, crowded metropolis than
in a rural area where the pace of life is considerably slower.
5. What do you think of air travel? In a hi-tech world where air travel has
become commonplace, many people still feel that it is an unnecessary and
dangerous luxury.
6. The problem of teenage alcoholism is one that today's society must deal
with before it is allowed to increase.
                                         27

First and Last Paragraph Techniques.
The first paragraph may:
- state a strong, firm opinion e.g. Vegetarians are definitely healthier than meat-
eaters.
- make reference to a strange scene or situation e.g. Will frogs ever become
household pets?
- be addressed to the reader directly (in less formal articles) e.g. What do you
think …? Have you ever …? If you want to …
- start with a quotation e.g. “Try not to become a man of success but rather try to
become a man of value.” or a rhetorical question (a rhetorical question is a
question which needs no reply) e.g. Do the students know how fortunate they
- start with a problem which needs a solution e.g. Can you explain the problem
of …
The last paragraph may:
- state a personal opinion e.g. I believe …, In my opinion …, It seems that …etc
- give the reader something to consider e.g. After all, is it fair to …
- summarise the article e.g. In conclusion …, To sum up …, On the whole … etc
- end with a quotation e.g. “Just a little more loving and a lot less fighting and
the world would be all right.”

Exercise 22
Match the first with the last paragraphs. Which techniques are used for the
first/last paragraphs? Which outlines should be used for each topic?
Suggest a title for each topic.
First paragraphs
1. Physical education has long been part of the school timetable for children;
imagine an educational programme which does not include some form of
physical education.
2. Throughout history people have taken part in dangerous sports. This does
not change the fact that it is unacceptable for athletes to be seriously
injured or even lose their lives for the sake of competition.
3. The advance of technology has meant that most schoolchildren are
familiar with computers, which can be used to teach a wide variety of
subjects. But can they, and should they, replace teachers?
4. As our cities continue to grow, and villages become smaller than ever, it
is true to say that it is far better to live in a busy, crowded metropolis than
in a rural area where the pace of life is considerably slower.
5. What do you think of air travel? In a hi-tech world where air travel has
become commonplace, many people still feel that it is an unnecessary and
dangerous luxury.

6. The problem of teenage alcoholism is one that today's society must deal
with before it is allowed to increase.


text:
$$\renewcommand\AA{\unicode{x212B}}$$

# SelectCellWithForm v1¶

## Summary¶

Select a conventional cell with a specific form number, corresponding to the UB stored with the sample for this peaks works space.

## Properties¶

Name

Direction

Type

Default

Description

PeaksWorkspace

InOut

IPeaksWorkspace

Mandatory

Input Peaks Workspace

FormNumber

Input

number

Mandatory

Form number for the desired cell

Apply

Input

boolean

False

Update UB and re-index the peaks

Tolerance

Input

number

0.12

Indexing Tolerance

NumIndexed

Output

number

The number of indexed peaks if apply==true.

AverageError

Output

number

The average HKL indexing error if apply==true.

AllowPermutations

Input

boolean

True

Allow permutations of conventional cells

TransformationMatrix

Output

dbl list

The transformation matrix

## Description¶

Given a PeaksWorkspace with a UB matrix corresponding to a Niggli reduced cell, this algorithm will allow the user to select a conventional cell corresponding to a specific form number from the Mighell paper. If the apply flag is not set, the information about the selected cell will just be displayed. If the apply flag is set, the UB matrix associated with the sample in the PeaksWorkspace will be updated to a UB matrix corresponding to the selected cell AND the peaks will be re-indexed using the new UB matrix. The output transformation matrix $$M$$ will change $$UB$$ to $$UBM^{-1}$$ and map each $$(HKL)$$ vector to $$M(HKL)$$. It can be further used by the TransformHKL algorithm. NOTE: The possible conventional cells, together with the corresponding errors in the cell scalars can be seen by running the ShowPossibleCells algorithm, provided the stored UB matrix corresponds to a Niggli reduced cell.

This algorithm is based on the paper: Alan D. Mighell, Lattice Symmetry and Identification—The Fundamental Role of Reduced Cells in Materials Characterization. Journal of research of the National Institute of Standards and Technology 106.6 (2001): 983, available from: nvlpubs.

## Usage¶

Example:

ws=LoadIsawPeaks("TOPAZ_3007.peaks")
FindUBUsingFFT(ws,MinD=8.0,MaxD=13.0)
print("Lattice before SelectCellWithForm:")
lattice = ws.sample().getOrientedLattice()
print(" ".join("{:.9f}".format(x) for x in [lattice.a(), lattice.b(), lattice.c(),
lattice.alpha(), lattice.beta(), lattice.gamma()]))
SelectCellWithForm(PeaksWorkspace='ws', FormNumber=25, Apply=True)
print("\nLattice after SelectCellWithForm:")
lattice = ws.sample().getOrientedLattice()
print(" ".join("{:.9f}".format(x) for x in [lattice.a(), lattice.b(), lattice.c(),
lattice.alpha(), lattice.beta(), lattice.gamma()]))

Output:

Lattice before SelectCellWithForm:
8.605818643 11.935925461 11.941812766 107.429088323 98.752912466 98.951193475

Lattice after SelectCellWithForm:
14.131051152 19.247332564 8.605818643 89.881170675 105.071333770 89.970386662

Categories: AlgorithmIndex | Crystal\Cell


text:
# Nomenclature of This State?

I've read this in a book

It says the state having three parallel spin is called triplet state . But as far as I know it is determined by "2S+1", from this it comes out to be 4 . Then how it is "Triplet (3) " ?

• You are correct in that if S is the total spin then $2S+1$ gives the spin multiplicity. As you note in the example $2(3/2)+1 = 4$, a quartet not triplet. – porphyrin Feb 14 '17 at 11:36


text:
anonymous one year ago A plane intersects a sphere to form a circle as shown in the figure. The distance from the center of the circle to the center of the sphere is 10in. If the area of the circle formed by the intersection is 576 pi inches squared, what is the volume of the sphere?

1. anonymous

|dw:1443066944419:dw|

2. anonymous

@ganeshie8 @mathstudent55 @Zale101 @kohai @dan815 @zepdrix

3. anonymous

I'm totally lost on this one, I keep getting similar problems wrong, the last equaled out to 4000/3 pi in.^3

4. ganeshie8

Hey if psble could you attach the original diagram

5. ganeshie8

I'm finding it a bit hard to relate the word problem to the picture that you have drawn

6. anonymous

the picture i drew is pretty much it.

7. anonymous

|dw:1443067700784:dw|

8. ganeshie8

Ok got you

9. anonymous

It wants me to find the volume, I even tried the V=4/3*3.14*r^3

10. ganeshie8

Think of the given diagram as a small spherical fish tank, 10cm represents the center of top surface to the center of fish tank : |dw:1443067804228:dw|

11. ganeshie8

They gave you surface area of the top surface : $$576\pi$$ Say, the radius of top surface is $$r$$ : |dw:1443067992574:dw|

12. anonymous

r=10 ?

13. ganeshie8

$\pi r^2 = 576\pi$ solve $$r$$

14. anonymous

r=24 ?

15. ganeshie8

Yes, the radius of top circular disk surface is $$24$$ : |dw:1443068220493:dw|

16. ganeshie8

Next, see if you can find the radius of sphere : |dw:1443068276532:dw|

17. anonymous

would I substitute 24 as r into the same equation?

18. ganeshie8

do you see a right triangle with sides $$10, 24, R$$ ? you need to setup an equation and find the value of $$R$$

19. anonymous

which equation would I use? A=1/2 bh? sorry....

20. ganeshie8

when you see a right triangle, you should think of "pythagoras theorem"

21. anonymous

10^2+24^2=676

22. ganeshie8

yes take square root

23. anonymous

$\sqrt{676} =26$

24. anonymous

@ganeshie8 What do I do next?

25. ganeshie8

so $$26$$ is the radius of sphere

26. ganeshie8

use volume formula of sphere and finish it off

27. anonymous

V=4/3*3.14*26^3 ?

28. anonymous

I get a huge number 73584.85

29. ganeshie8

Looks good!


text:
+0

# solve for x : log base 9 X^2=9

0
300
2

solve for x : log base 9 X^2=9

Guest Aug 6, 2015

#2
+18838
+10

$$\small\text{ Solve for x : \log_9{(x^2)}=9 }}$$

$$\small{\text{Formula: \boxed{~ \log_b{(b^a)}=a~}}}\\\\ \small{\text{ We have b=9, and a = 9 so \log_9{(9^9)}=9, but 9^9 is x^2 }}\\\\ \small{\text{ \begin{array}{rcl} x^2 &=& 9^9\\ x &=& 9^{\frac{9}{2}}\\ x &=& \left(\sqrt{9} \right)^9\\ x &=& \left(\sqrt{3^2} \right)^9\\ x &=& 3^9\\ \mathbf{x} & \mathbf{=} & \mathbf{19683}\\ \\ \hline \\ \end{array} }}\\$$

heureka  Aug 6, 2015
Sort:

#1
+91505
+10

$$\\log_9x^2=9\\\\ 2log_9x=9\\\\ log_9x=9/2\\\\ 9^{log_9x}=9^{9/2}\\\\ x=9^{9/2}\\\\ x=3^9\\\\ x=19683$$

check:

$${{log}}_{{\mathtt{9}}}{\left({{\mathtt{19\,683}}}^{{\mathtt{2}}}\right)} = {\mathtt{9}}$$

Melody  Aug 6, 2015
#2
+18838
+10

$$\small\text{ Solve for x : \log_9{(x^2)}=9 }}$$

$$\small{\text{Formula: \boxed{~ \log_b{(b^a)}=a~}}}\\\\ \small{\text{ We have b=9, and a = 9 so \log_9{(9^9)}=9, but 9^9 is x^2 }}\\\\ \small{\text{ \begin{array}{rcl} x^2 &=& 9^9\\ x &=& 9^{\frac{9}{2}}\\ x &=& \left(\sqrt{9} \right)^9\\ x &=& \left(\sqrt{3^2} \right)^9\\ x &=& 3^9\\ \mathbf{x} & \mathbf{=} & \mathbf{19683}\\ \\ \hline \\ \end{array} }}\\$$

heureka  Aug 6, 2015

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text:
Show transcribed image text

Get this answer with Chegg Study

Practice with similar questions
Q:
An infinitely long cylindrical conductor of radius a is placed along the z-axis. If the current density is $$J=J_{o}/\rho*a_{z}$$


text:
Publicité ▼

## définition - Molecular_mass

voir la définition de Wikipedia

Wikipedia

# Molecular mass

Assuming hydrogen and oxygen are standard weights in this image (as opposed to deuterium oxide) the molecular mass should be 18.01528 u.

The molecular mass (m) of a substance is the mass of one molecule of that substance, in unified atomic mass unit(s) u[1] (equal to 1/12 the mass of one atom of the isotope carbon-12[2]). This is numerically equivalent to the relative molecular mass (Mr) of a molecule, frequently referred to by the term molecular weight, which is the ratio of the mass of that molecule to 1/12 of the mass of carbon-12 and is a dimensionless number. Thus, it is incorrect to express relative molecular mass (molecular weight) in daltons (Da). The terms molecular weight and molecular mass have been confused on numerous websites, which often state that molecular weight was used in the past as another term for molecular mass.[citation needed]

Molecular mass differs from more common measurements of the mass of chemicals, such as molar mass, by taking into account the isotopic composition of a molecule rather than the average isotopic distribution of many molecules. As a result, molecular mass is a more precise number than molar mass; however it is more accurate to use molar mass on bulk samples. This means that molar mass is appropriate most of the time except when dealing with single molecules.

The concept of molecular mass is important for all molecules, especially for complex molecules like polymers and biopolymers such as proteins and carbohydrates. The determination of their molecular mass is often difficult, and is usually inferred from gel permeation chromatography and mass spectrometry.

## Definition

There are varying interpretations of this definition. Many chemists use molecular mass as a synonym of molar mass,[3] differing only in units (see average molecular mass below). A stricter interpretation does not equate the two, as the mass of a single molecule is not the same as the average of an ensemble. Because a mole of molecules may contain a variety of molecular masses due to natural isotopes, the average mass is usually not identical to the mass of any single molecule. The actual numerical difference can be very small when considering small molecules and the molecular mass of the most common isotopomer in which case the error only matters to physicists and a small subset of highly specialized chemists; however it is always more correct, accurate and consistent to use molar mass in any bulk stoichiometric calculations. The size of this error becomes much larger when considering larger molecules or less abundant isotopomers. The molecular mass of a molecule which happens to contain heavier isotopes than the average molecule in the sample can differ from the molar mass by several mass units. Also the molar mass comes into comparison with the number of particles contained within the structure by multiplying the number of moles by Avogadro's constant: NA=6.022·1023 mol−1.

## Average mass

The average molecular mass (sometimes abbreviated as average mass) is another variation on the use of the term molecular mass. The average molecular mass is the abundance weighted mean (average) of the molecular masses in a sample. This is often closer to what is meant when "molecular mass" and "molar mass" are used synonymlousy and may have derived from shortening of this term. The average molecular mass and the molar mass of a particular substance in a particular sample are in fact numerically identical and may be inter-converted by Avogadro's constant. It should be noted, however, that the molar mass is almost always a computed figure derived from the standard atomic weights, whereas the average molecular mass, in fields that need the term, is often a measured figure specific to a sample. Therefore, they often vary since one is theoretical and the other is experimental. Specific samples may vary significantly from the expected isotopic composition due to real deviations from earth's average isotopic abundances.

## Computation

The molecular mass can be calculated as the sum of the individual isotopic masses (as found in a table of isotopes) of all the atoms in any molecule. This is possible because molecules are created by chemical reactions which, unlike nuclear reactions, have very small binding energies compared to the rest mass of the atoms ($<$ 10−9) and therefore create a negligible mass defect. The use of average atomic masses derived from the standard atomic weights found on a standard periodic table will result in an average molecular mass, whereas the use of isotopic masses will result in a molecular mass consistent with the strict interpretation of the definition, i.e. that of a single molecule. However, any given molecule may contain any given combination of isotopes, so there may be multiple molecular masses for each chemical compound.

## Measurement

The molecular mass can also be measured directly using mass spectrometry. In mass spectrometry, the molecular mass of a small molecule is usually reported as the monoisotopic mass, that is, the mass of the molecule containing only the most common isotope of each element. Note that this also differs subtly from the molecular mass in that the choice of isotopes is defined and thus is a single specific molecular mass of the many possible. The masses used to compute the monoisotopic molecular mass are found on a table of isotopic masses and are not found on a typical periodic table. The average molecular mass is often used for larger molecules since molecules with many atoms are unlikely to be composed exclusively of the most abundant isotope of each element. A theoretical average molecular mass can be calculated using the standard atomic weights found on a typical periodic table, since there is likely to be a statistical distribution of atoms representing the isotopes throughout the molecule. This however may differ from the true average molecular mass of the sample due to natural (or artificial) variations in the isotopic distributions.

The basis for determination of molecular weight according to the Staudinger method (since replaced by the more general Mark-Houwink equation[4]) is the fact that relative viscosity of suspensions depends on volumetric proportion of solid particles.

There are several companies that manufacture accurate instruments for determining absolute molecular weight quickly and easily. These companies include Brookhaven Instruments, Malvern Instruments, and Horiba

## Unit type variation

The molar mass of a substance is the mass of 1 mol (the SI unit for the basis SI quantity amount of substance, having the symbol n) of the substance. This has a numerical value which is the average molecular mass of the molecules in the substance multiplied by Avogadro's constant approximately 6.022×1023. The most common units of molar mass are g/mol because in those units the numerical value equals the average molecular mass in units of u.

Conversion factor of average molecular mass to molar mass:

molar mass = average molecular mass × ((1/6.022)×10-23g/u)×(6.022×1023/mol)
or
molar mass in g/mol= average molecular mass in u

(Note that these relations are true for theoretical and experimental values, but not between experimental and theoretical values. Molar mass is most often theoretical and average molecular mass is most often experimental)

The average atomic mass of natural hydrogen is 1.00794 u and that of natural oxygen is 15.9994 u; therefore, the molecular mass of natural water with formula H2O is (2 × 1.00794 u) + 15.9994 u = 18.01528 u. Therefore, one mole of water has a mass of 18.01528 grams. However, the exact mass of hydrogen-1 (the most common hydrogen isotope) is 1.00783, and the exact mass of oxygen-16 (the most common oxygen isotope) is 15.9949, so the mass of the most common molecule of water is 18.01056 u. The difference of 0.00472 u or 0.03% comes from the fact that natural water contain traces of water molecules containing, oxygen-17, oxygen-18 or hydrogen-2 (Deuterium) atoms. Although this difference is trivial in bulk chemistry calculations, it can result in complete failure in situations where the behavior of individual molecules matters, such as in mass spectrometry and particle physics (where the mixture of isotopes does not act as an average).

There are also situations where the isotopic distributions are not typical such as with heavy water used in some nuclear reactors which is artificially enriched with Deuterium. In these cases the computed values of molar mass and average molecular mass, which are ultimately derived from the standard atomic weights, will not be the same as the actual molar mass or average molecular mass of the sample. In this case the mass of deuterium is 2.0136 u and the average molecular mass of this water (assuming 100% deuterium enrichment) is (2 × 2.0136 u) + 15.9994 u = 20.0266 u. This is a very large difference of ~11% error from the expected average molecular mass based on the standard atomic weights. Furthermore the most abundant molecular mass is actually slightly less than the average molecular mass since oxygen-16 is still the most common. (2 × 2.0136 u) + 15.9949 u = 20.0221 u. Although this is an extreme artificial example, natural variation in isotopic distributions do occur and are measurable. For example, the atomic weight of lithium as found by isotopic analysis of 39 lithium reagents from several manufacturers varied from 6.939 to 6.996.[5]

## Notes

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AMA529 STATISTICAL INFERENCE
Assignment 2

1. Let X1, . . . , Xn be a random sample from the N(µ, σ2
) distribution, where µ is known, and
σ
2 > 0 is unknown.
(a) Calculate E((X1 − µ)
2
).
(b) Show that n
−1 Pn
i=1(Xi−µ)
2
is the best unbiased estimator of σ
2
. You may use without
proof the fact that Pn
i=1(Xi − µ)
2
is a complete statistic.
2. Let X1, . . . , Xn be a random sample from the Gamma(α, θ) distribution with density
f(x; θ) = θ
α
(α − 1)!x
α−1
e
−θx for x > 0,
where θ > 0 and α is a known positive constant.
(a) Show that Pn
i=1 Xi
is a sufficient statistic for θ.
(b) Find the MLE of 1/θ.
(c) It is known that Pn
i=1 Xi
is a complete statistic. Find the best unbiased estimator of
1/θ.
(d) Find the best unbiased estimator of θ. (Hint: Use the fact that Pn
i=1 Xi ∼ Gamma(nα, θ)
and see the example on p.26 of Chapter 2.)
3. Let X1 and X2 be two independent and identically distributed variables with probability
mass function
f(x; θ) = 1
2
x−θ+1 for x = θ, θ + 1, . . .
where θ ∈ {0, 1, 2, . . .}.
(a) Show that min(X1, X2) is a complete sufficient statistic for θ.
(b) Find P(min(X1, X2) = k) for k = θ, θ + 1, . . . and E(min(X1, X2)).
(c) Find the best unbiased estimator of θ.
4. Let X1, . . . , Xn be a random sample with density
f(x; θ) = θxθ−1
for 0 < x < 1.
Consider testing H0 : θ = 1 versus H1 : θ = 2. Show that the rejection region of the most
powerful test takes the form of R =

x :
Qn
i=1 xi ≥ c

for some constant c.
5. Let X1 and X2 be independent and identically distributed random variables whose probability mass function under H0 and H1 is given by
x 0 1 2
f(x; H0) 0.5 0.3 0.2
f(x; H1) 0.4 0.1 0.5
Use the Neyman–Pearson lemma to find the most powerful test for H0 versus H1 with size
α = 0.24. Compute the type II error probability for this test.

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text:
# The Horizon

That horizon stretches out. You know the one. It lies on the far side of a vast, unknowable plain punctuated by our dreams and fears and fantasies of what might be. The horizon retreats as we tread upon that plain, as we encounter the figures and actions of our passage. We watch the horizon, we wonder about it, we follow our footsteps along an indistinct line that meanders in that direction. Call this line destiny, or fate, or the labyrinth, or whatever you like. It is the path that we take.

This is the landscape of our lives: the path upon the plain, and the horizon, far off, yet always in view. We tend not to think in metaphors like these anymore. Nowadays we talk about choice and chance and a myriad of modern forces that render us into a given shape. Genetic and social and environmental factors have become our means of grappling with the trajectories upon which we find ourselves. And we tend to look forward as we move: the future is ahead of us, we reach toward signposts as we try to find the track.

But our modern approaches, our increasingly quantified measures, our notion that we are, indeed, in the vanguard of human progress: these convictions mask a fundamental truth, which is this: the horizon encircles us.

Ancient peoples knew this. They did not perceive time and progress as lines marching forward but rather as curves, as long trammelled tracks that lead both forward and back. For them, time was a circle. The future was also the past. The path ahead was already festooned with the tracks of previous travelers: gods, heroes, wanderers like themselves. Upon that plain stretching toward the horizon the ancients saw their own histories, their own stories lost and found again. They were, in this sense, never alone.

But we’re different. We’ve made fresh tracks and new paths. We have, for the most part, left behind our ancient beliefs and ancestral values. Typically we find them to be a bit antiquated, not quite suited to the complexities and contradictions of modern life. We choose instead to move forward along new paths, trying to make sense of the things we can do and the trouble we’ve caused: to the earth, to our own cultures, to the welfare of all humanity. Every culture tells an old story about our sacred task of guardianship, and we have not honored that sacred trust as well as we might have.

Every culture also tells a story about the return. In this story, people with vision and purpose find the buried bones of wisdom. In some stories that wisdom is symbolized by a treasure; in others it is a sacred book; sometimes it is a human or a god who returns from the horizon with gifts of learning and transformation. In all such stories, the return is the moment we rediscover ourselves. It is a hinge, a crack between the worlds, a brilliant flash of illumination across a field of rumour and wreckage.

The myth of return is not a fable, or a fantasy, or the wishful thinking of lost cultures. It is, rather, the mark of our human powers. It’s an old myth, one that has been preserved and retold for millennia — because, eventually, it comes to pass. As we move forward, we return to discover and re-affirm who we are.

We are living now in an age of return. We’ve begun to unearth and remember the gifts of the past. Within the energetic clamor of the environmental movement, within bootstrapped communities and reinvigorated cultures, beneath the hum and hustle of educational renewal: we’ve started to understand that in many ways, the ancients have gone before us, have laid tracks that intertwine with our own, have left signposts for us to find. They’ve come back from that far-off horizon to offer clues about our own future.

The age of return might be defined by a single theme: the growth of the power of the individual. And we are seeing this now in many ways: the worldwide increase in the rights of women; the global effort of individual entrepreneurs to reduce disease and poverty; the ability of young people to seize a social cause, create a foundation, make real change. This, after all, is the essence of the myth of return. It is not, at heart, a story of being rescued by the gods. Rather, it is a narrative of individual empowerment and social transformation. In the recent history of humanity — and by recent, I mean roughly the last five thousand years — there have been three other moments of return. They don’t come along very often.

The cultures of the ancients are dead, gone, vanished. They cannot come back, they cannot survive in this modern, global meta-culture that we have constructed. But the signposts they have left for us do remain, and will continue to shape our journey. There are three such signposts. One reminds us to care for the earth. Another reminds us to care for one another. The third embraces the other two, and says this: preserve and nurture the sacred, however we define it and wherever we find it.

We are now, for the first time in many centuries, in a position to change the world in fundamental ways. And we’ve discovered, of course, that we cannot accomplish our aims alone. We need those signposts, that ancient wisdom, the hardscrabble knowledge of the archaic peoples we sought to leave behind. We recognize now that the horizon is behind as well as in front. We are, finally, moving beyond the precocious arrogance of our modernity. As one ancient story says, we are the ones we’ve been waiting for.

Many people will follow the ancient signposts and participate in the fundamental reinvigoration of our age. They won’t be heroes or gods or exalted beings. They will stand at the center of the circular horizon, will forge paths that embody ancient wisdom in a modern guise. They will be our guardians and our guides. They will be you.

© 2017. All rights reserved.

Ross Laird


text:
# DIY gauges??

### Help Support HomeBuiltAirplanes.com:

#### phiatt

##### Member
Hello Guys,

My name is Philip Hiatt, it's been a while(I think?), since I posted on this forum, if ever. I ran across this thread in the past and once again this morning. I revisited an old project I started about 8-10 years ago when I started building my first home built, VP-2. Like most of us I didn't have a lot of money to throw into gauges and flight instruments, So I started designing my own. Being a long time Electronics Hobbyist and doing it for a living I had many resources to pull from. After several moves and having to relocate, I wasn't able to commit any time to it. But here recently, in between jobs I have nothing but time. All in all, the plan is to make something affordable( less cost than a panel of steam gauges), simple and portable(used on different aircraft), and reliable.

#### choppergirl

##### Banned
A bunch of steam gauges, even rusty old things bought off of ebay from WW2, would end up costing me more than my entire airplane build and weight a ton. I'm leaning more to a Nexus 7 2013 tablet with a hood from the sun, running some android pilot software, connected via bluetooth to a sensor package (for airspeed mainly). Poor man's 3G/GPS/acelerometer/attitude equipped glass panel, with maps, in an ultralight ;-)

But... not sold on the idea. While I'm flying I don't want to be fiddling with a touchscreen or complex screen that just auto-rotated on me... or rebooting a tablet.... I'd rather have the analog gauges tho... they are all just so darn expensive... new... or even used.

If I could, I'd have all my gauges read out in 7 segment LEDs. No dials, no bar graphs, no needles.. against an analog circular dial where you are squinting and your brain is trying to calculate what exact number that line between two numbers exactly is... just different large color numbers showing different critical information at a glance in big number LEDs. Different colors for different gauge....

Its just so... straight forward... brain associates color with the gauge, so in a pinch you just look for whatever colored number is air speed, altitude, cht, altimiter, tach, or fuel, or compass... and if any get in a danger range, have them... flash... it makes so much sense to me, I don't know why no one hasn't done it before..

Last edited:

#### VAPORTRAIL

##### Well-Known Member
Choppergirl,

Welcome Back!

Just One concern, How do you plan on getting this thing up to 88MPH?

#### Attachments

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# 5 What's New in 2020

## Origin 2020 New Feature Highlights

 Visit www.originlab.com For Origin 2020 Feature Highlights, including a Highlights video.
• Mini Toolbars for Faster Graph Customization
• Context-sensitive popup controls for fast graph edits
• Fast editing of page, layer, plot, axis and label properties
• Faster Import of Text Data
• Multi-threaded import for CSV Connector and ASCII files (Single ASCII, Multiple ASCII, Import Wizard, CSV)
• Fast Scatter Plots for Large Datasets
• Default scatter plot drawing speed has been significantly improved for large datasets
• New templates: Density Dots and Color Dots (color-mapped to a third column)
• Drag out a rectangle with the Data Highlighter to highlight a region of interest
• Copy and Paste Tables as HTML to Other Applications
• A new Copy Table (HTML, EMF) shortcut menu option is added to result tables and Analysis Report Sheets
• The pasted table can be further formatted and edited in supporting applications
• Doughnut Charts and Improved Pie Charts
• Updated Pie charts with several new templates (e.g. "Pie of Pie")
• All-new Doughnut Charts with several template variations
• Sankey Diagrams
• All-new plot type for charting flows
• Excel-like Absolute Cell References for Cell Formula
• You can now use the character in cell formulas when you need to make absolute cell references • Improved Dendrogram • The dendrogram produced by such analyses as Hierarchical Cluster Analysis is much improved for 2020 • Options to color nodes individually, to break out nodes as separate graphs or to draw circular dendrograms • Other Graph Types • Kite Diagram • Pyramid Plot • Before-After Plot • Organize Workbooks in Object Manager • Use the Object Manager to manipulate worksheets in the book ## New Features by Category ### Ease of Use • Context-sensitive Mini Toolbars for faster graph customization • Click on a data plot, graph layer, axis, etc. to display popup controls for typical graph customizations • Includes support for error bars, box charts, histograms, pie charts, contour plots and heat maps • Plot Details: Compact Popup for Font and Line Format • Associated Font and Line format controls are grouped together behind a single popup, to conserve screen space • Plot Menu Improvements: • New Plot menu category for Multi-Panel/Axis plots for easier access • List Mode in Template Library • Toggle between details and preview modes of graph templates • Object Manager for Workbook windows • List all sheets in the active window • Clicking on a sheet in Object Manager will activate the corresponding sheet in the workbook. • Right-click on the window in Object Manager to initiate common operations: Delete, Insert, Add, Move, Copy, Rename, etc. • Simplified plot selection • Previously, a single-click on a plot group would select the entire group, the second click selected the plot and a third click selected a special point • In Origin 2020, a single-click will select a plot, CTRL + click will select a single point and SHIFT + click will select all plots in the group. To revert to the previous behavior, set @GSM=0 • Copy and Paste tables to other applications • Copy a block of cells or results table from an Origin worksheet, including column labels, as an HTML table or EMF image and preserve table structure and special formatting when pasting to other applications such as MS Word • If the target application supports editing, table design and contents can be modified (EMF is useful for Adobe Illustrator as it doesn't support pasting of HTML) • Indicate selected graph in Merge Graph Windows dialog box • Highlighting a graph in the Graphs list highlights the corresponding plot in the Preview window • Add a "Find Apps" menu entry for "Fitting", "Peaks and Baseline" and "Statistics" menus • App Center opens to show only Apps relevant to Fitting, etc. • Added Hotkeys for Worksheet • CTRL + SHIFT + END extends selection to the last filled cell in the sheet • CTRL + SHIFT + HOME extends selection to the cell in row 1, column 1 • SHIFT + arrow key extends the selection to additional columns or rows • Selecting Plot Selects Corresponding Worksheet Data • View: Show Plot Selection in Workbook menu entry is checked by default. Clear check mark to prevent worksheet data selection. The default behavior can be changed by setting LabTalk system variable @PS • Load Template dialog cue for sorting headers • File: New from Template now shows a clearer indicator that the columns in the dialog can be sorted • Change Parameters context menu • You can now right-click directly in the hierarchical result sheet to Change Parameters -- no need to scroll to the top of the sheet and click the lock icon ### New Graph Types  To see new graph samples for Origin 2020, press F11 and in Search, enter 2020. • Pie Chart Improvements • Many pie chart variations (including "doughnut" and "pie of pie") • Set wedge total to < 100% to draw a portion of the pie • Sort wedges by size or by column values • Set pie slice thickness by column values • Combine exploded wedges • plus, label and leader line improvements • Sankey Diagrams • Used to chart flows, with arrow size proportional to flow component • Full control over display of links, nodes, colors (single, gradient), transparency, curvature, etc. • Alluvial Diagrams • Alluvial Diagrams are also used to chart flows but are more similar to Origin's Parallel Sets plots than to Sankey Diagrams • Kite Diagrams • Used to compare trends and relative occurrence among multiple categories. Most commonly used to plot transect data in ecological studies. • Additional layers can be added as supporting plots (e.g. line graphs of topographic profile) • Density Dots and Color Dots • Basic 2D scatter plots for large datasets • Fast-rendering dot plot for viewing data distributions • Color can be mapped to a column of values • Support for Window View mode • Population Pyramid • Composed of two opposing histograms, this chart type is used to display population by age group • Before-and-After Graph • A simple line+symbol plot used to make paired comparisons ### Graph Customization • Faster Drawing Speed • Improved scatter plot drawing speed with Speed Mode off (especially with square symbols + color-mapping where it is 5 - 10 times faster) • Data Highlight Speed Improvement • Much faster response times when dragging out a region of points with the Data Highlighter tool • Offset Data Points in Box Charts • Optionally, alternate the offset in paired plots • Text object Properties dialog box improvements • SuperSubscript button and Align Columns check box (for aligning legends) • Add Arrowhead Pointer to Custom Legend Symbols • Improvement to Skip Points and Speed Mode • Skip Points and Speed Mode now keep last data point, by default • Set @SMEP=0 to roll back to the old default of allowing skipping of last point • Rug Marks in Trellis Plots • Rug marks for individual panels • Plot: 3D: 3D Scatter + Z Error • If there is a Z column immediately following the plotted Z, the second Z column values will plot as error bars in the Z direction • Windrose Plot Additions • Support for uneven bin sizes • Show Calms Circle in center of plot • Paddle Spoke Style option added (Spacing tab of Plot Details) • Data Highlighting for Box Charts • In Origin 2019b, highlighting corresponding worksheet data only worked for basic graphs such as line, scatter, column/bar. This is now supported when selecting individual boxes in a box chart • Color Scale Enhancements • Control scale range from the Color Scale Control dialog • Formula to control tick labels in color scale object • Min- and max-only labels at inside or outside ends of scale • Improved Tooltips for Countour Plot and Heat Map • Display X, Y, Z values on hover. Contour plots support display of interpolated values ### Import & Export • Multi-threaded ASCII Import • Fast, multi-threaded import added for Import ASCII, Import CSV, and the CSV Data Connector. Two system variables, @IAMT and @IAS, are added to modify multi-threading behaviors • CSV Connector Improvements • Import first 50 lines as preview option • Partial Import: Special syntax allows high degree of specificity • Data Block Marker: Enter one or more characters that define the start and end of a data block (e.g. "[Data]" or ">>>>>Start<<<<<") • Excel Connector Improvements • Specify location of Long Names, Units and Comments (From and To) in the Excel file when importing • Drag and Drop Support for Data Connector • When you drag and drop a file type for which there is a Data Connector, a dialog opens to allow you to specify whether to import by Connector (Data: Connect to File) or to use the Import from File (Data: Import from File) methods • Origin Internal SQL-like Query Support • A Query button has been added to the Origin Connector. Use the query function to import just the data that you need • Data Connector-like Support for Database • Faster data import • Support Exclude Imported Data when saving as OPJU/OGWU • New system variable (@IPDB) to control protection of imported range ### Worksheet • Support Excel-like "" Absolute Cell References
• Origin adopts the Excel cell-reference "\$" syntax for more straightforward absolute cell references. This should make constructing cell formulas easier, particularly for those who have used MS Excel
• Worksheet Short Name, Long Name Improvements
• System Variable @SSL is added to control sheet display name
• Plot Setup, Layer Contents dialogs to use same display name, by default
• Support Copy of Label Rows with Subrange Selection
• You can now make a subrange selection in the worksheet, right-click and opt for Copy (including label rows)
• sum() Function Supports Row-wise Calculation

Examples:

• Sum(A:D) // to sum row-wise from column A to D
• Sum(A, C:D, F:H) // to sum row-wise columns A, C to D and F to H
• Show Column Index and Data Size in Column List View
• Appears as tooltip on hover
• Filter Automatically Run on Data Import
• Previously, filters needed to be reapplied manually after data import
• Improved Performance of Large Data + Filter
• Interface is more responsive now when working with large datasets and filtered data

### Analysis

• Improvements to Normalize Tool
• Treat multiple datasets as a single dataset for normalization.
• Support Simple LabTalk Expressions in Insert Straight Line and other X-Functions
• Support LabTalk expression for At Value (e.g. -log(0.05)).
• Support info variables and column values for as input (e.g. page.info.user.variables.noofpoints)
• Simple Curve Math Supports Custom Formula
• Simple Curve math (Analysis: Mathematics: Simple Curve Math) supports custom formula for input data (y1) and reference data (y2) (e.g. y1*5-y2^2)
• Batch Processing Export of Multi-page file
• Export MS Word and PDF as a single, multi-page file
• Support Include Filtered (Hidden) Rows in Analysis
• By default, when there is a data filter on a column that is input for an analysis operation, filtered (hidden) rows are ignored. The analysis lock context menu now has Ignore Hidden Rows = No so that you can include hidden rows
• Open Axis DB from Peak Analysis Preview
• Double-clicking on the axis in PA Preview opens the Axis DB to set axis scale range
• Copy All Open Tables
• Click a table heading in the Analysis Report Sheet to Copy All Open Tables. All expanded tables are copied.
• Copy Table as HTML or EMF
• Result tables pasted to graphs can be opened by double-clicking on them. Right-click on the open table and Copy Table (HTML, EMF), then paste (or paste special) as to the target application as an editable table.
• Kernel Density: Faster Algorithm
• Previously, large 2D Kernel Density plots could take hours to draw. A new Density Method Binned Approximate Estimation method when used with an Interpolate Density Points check box, reduces the time to just seconds

### Programming

• New LabTalk Options for Gadget Output
• gd.Output(); now has the following options:
0 -- normal
1 -- last
2 -- all curves
3 -- all layers
4 -- entire curve // Quick Peaks Gadget only
• Read/Write to Polygon/Polyline objects via LabTalk
• Read and set object properties (e.g. point count and coordinates of each)
• NAG Library is Upgraded to Version Mark 26.1
• Python Update
• Embedded version updated to 3.7.2

### Apps for 2020

The following Apps are new for Origin 2020:

• 2D Peak Analysis
• Import NMR Data
• LeCroy Connector
• Marginal Abatement Cost Curve
• Neural Network Fitting
• Piper Diagram
• Soil and Rock Classification Diagrams
• Video Data Explorer
• Word Connector


text:
1. kropot72

power (watts) = voltage (volts) * current (amps)

2. anonymous

Kropot72 Is correct, Good luck!:D

3. UnkleRhaukus

if you wanted to include resistance as a variable (and assuming ohms law is valid) $\large{\boxed{P=IV=I^2R=\frac {V^2}R}}\qquad\qquad\qquad \scriptsize{ V=IR}\qquad I=\frac{V}{R}$


text:
nLab S5 modal logic

The modal logic obtained from S4 modal logic by adding the axiom that there is the implication

$\lozenge p \to \Box \lozenge p \,.$

In the Kripke frame semantics for modal logic, the accessibility relation for worlds is an equivalence relation.

References

Last revised on December 1, 2014 at 12:24:17. See the history of this page for a list of all contributions to it.


text:
Hello , Let f be a quadratic form of rank n over $\mathbb{Z}/2\mathbb{Z}$. Then:

(i) If n is odd, we have

$f \simeq x_2 + (b_1 {y_1}^2 + {y_1} {z_1} + d_1 {z_1}^2)+(b_2 {y_2}^2 + {y_2} {z_2} + d_2 {z_2}^2)+ \cdots +(b_l {y_l}^2 + {y_l} {z_l} + d_l {z_l}^2)$

(ii) If n is even, we have $f \simeq (b_1 {y_1}^2 + {y_1} {z_1} + d_1 {z_1}^2)+(b_2 {y_2}^2 + {y_2} {z_2} + d_2 {z_2}^2)+ \cdots +(b_l {y_l}^2 + {y_l} {z_l} + d_l {z_l}^2)$

for some $l \in \mathbb{N} , b_l,d_l \in \mathbb{Z}/2\mathbb{Z}$

Question : is this theorem valid for quadratic forms over $\mathbb{Z}/{2^r}\mathbb{Z} ,r>1$.?

if not , what one have in this case ?

Best regards.

-
Get Rational Quadratic Forms by J.W.S. Cassels. Canonical forms in $\mathbb Z_2$ are discussed on pages 117-118, giving some (not necessarily exclusive though). Then, at the top of page 120, mention is made of articles that complete the job, B. W. Jones 1944, Gordon Pall 1945, finally G. L. Watson 1976. –  Will Jagy Feb 17 at 18:48


text: While many of the ideas of calculus had been developed earlier in Greece, China, India, Iraq, Persia, and Japan, the use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce its basic principles. To simplify things, we will consider the relationship between demand and price of the commodity only, assuming that all other factors remain constant. The mass of an object of known density, the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus. One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum (a Riemann sum) of the approximate distance traveled in each interval. Functions y = f(x) is a function of x if and only if, for each x in the domain of f(x), that is the values of x for which f(x) is defined, there is exactly one value of f(x). Discrete Green's Theorem, which gives the relationship between a double integral of a function around a simple closed rectangular curve C and a linear combination of the antiderivative's values at corner points along the edge of the curve, allows fast calculation of sums of values in rectangular domains. A smaller value for Δx will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as Δx approaches zero. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. For example, travelling a steady 50 mph for 3 hours results in a total distance of 150 miles. [11] However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".[10]. e-commerce is of this type, and most experts predict that B2B e-commerce will continue to grow faster than the B2C segment. The Bachelor of Commerce degree is designed to provide students with a wide range of managerial skills, while building competence in a particular area of business (see aside). The derivative is defined as something which is based on some other thing. Derivatives give an exact meaning to the notion of change in output with respect to change in input. A motivating example is the distances traveled in a given time. Graphical analysis cannot tell us easily exactly at what level of output, profits will be maximum, for it takes time to draw a graph and conclude from it. Course Objectives: Students will learn the concepts of di erential and integral calculus in speci c contexts with emphasis on applications to economics. Example 4 A plant produces and sells semiconductor devices. ( In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue. Hence, it could be said that commerce falls within the scope of economics but economics extends beyond commerce. You can learn how to control a system by studying calculus. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. y Grading: Your grade will weighted as follows: , an elongated S (the S stands for "sum"). The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. For instance, if f(x) = x2 is the squaring function, then f′(x) = 2x is its derivative (the doubling function g from above). For other uses, see. For example: In this usage, the dx in the denominator is read as "with respect to x". Calculus for Business, Economics, Life Sciences and Social Sciences, 11th Edition: This is the textbook I'm using in 2009. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today. Cost of a commodity depends upon a number of factors. x Concept: Application of Calculus in Commerce and Economics in … Calculus Applications of the Derivative Applications of the Derivative . [citation needed] A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). ML Aggarwal Applications of Calculus in Commerce and Economics ISC Class-12 Understanding APC Maths Solutions Chapter-1 of Section-C. It is made up of two interconnected topics, differential calculus and integral calculus. The maxima and minima of revenue functions indicate the maximum and minimum revenue earned. were taken to be infinitesimal, and the derivative Thus, the field of economics is riddled with mathematical equations and applications. But it's not a course in physics, biology, economics, or business. Applications of Derivatives There are various applications of derivatives not only in maths and real life but also in other fields like science, engineering, physics, etc. ADVERTISEMENTS: The process of optimisation often requires us to determine the maximum or minimum value of a function. 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# Show that 1/x is NOT uniformly continuous

I have to prove that the function $f(x)=\frac{1}{x}$ on $(0,\infty)$ is not uniformly continuous (for the definition of uniform continuity see here).

I negated the definition of uniform continuity getting:

$$\exists \epsilon>0 \ \text{s.t.} \ \forall \delta>0 \ \text{I can always find} \ x,y \in(0,\infty) \ \text{s.t.} \mid x-y\mid<\delta \ \text{but} \ \mid f(x)-f(y)\mid \geq \epsilon.$$

So I chose $\epsilon=1$. Set $y=\frac{x}{2}$. Then I have to find an $x$ such that this holds: $\mid x-y\mid = \frac{x}{2} <\delta$ & $\mid f(x)-f(y)\mid = \frac{1}{x} \geq 1$, which is equivalent to $x<\min [2\delta,1]$.

Is this argument valid? My concern is if I'm allowed to choose $x$ depending on $\delta$. Thanks!

It is correct. You can choose x depending on $\delta$ because $x$ is quantified before $\delta$.
Alternatively, we have $$1/n - 1/(n+1) \to 0$$ but
$$f(1/n) -f(1/(n+1)) = n - (n+1) \to -1$$
And by the equivalent sequence characterisation of uniform continuity, we conclude that $f$ cannot be uniformly continuous.