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# American Institute of Mathematical Sciences

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October  2006, 2(4): 399-423. doi: 10.3934/jimo.2006.2.399

## Good programs in the RSS model without concavity of a utility function

 1 Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Received  December 2005 Revised  July 2006 Published  October 2006

In this paper, we obtain results on the good ''choice of techniques'' in the long-run in a model proposed by Robinson, Solow, and Srinivasan. We study this model without an assumption that a utility function which represents the preferences of the planner is concave and establish the existence of good programs.
Citation: Alexander J. Zaslavski. Good programs in the RSS model without concavity of a utility function. Journal of Industrial & Management Optimization, 2006, 2 (4) : 399-423. doi: 10.3934/jimo.2006.2.399
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# Confused on how Method of Manufactured solutions works?

I am new to computational science and I am trying to wrap my head around how MMS works. I am solving the time independent Helmholtz equation as a simple test of the technique

so my starting equation is following with homogeneous boundary conditions on a line segment

$$\Delta u +k^2u = F$$

So I am supposed to take something that is sufficiently differentiable. So, following other people I pick the following for my 1D case

$$\bar{u} = \sin(\pi x)$$

My new source term created from MMS would be

$$G(x) = -\pi^2\sin(\pi x) + k^{2}\sin(\pi x) - F$$

Now here is where I am stuck. I picked a u and now I have a new F. Somehow I am supposed to be able to verify my rate of convergence of my actual problem with this fictitious one. Are the points the same?

• First of all, the non-homogeneous time-independent Helmholtz equation is: $$\nabla^{2} u + k^{2} u = F$$ Then if I assume your fabricated solution of $u = \sin(\pi x)$, I would have this for $F$ as: $$F = (k^{2} - \pi^{2}) \sin(\pi x)$$ Basically, if you have this equation: $$\frac{d^{2} u}{d x^{2}} + k^{2} u = (k^{2} - \pi^{2}) \sin(\pi x)$$ The analytical solution would be: $u = \sin(\pi x)$. – Alone Programmer Apr 23 at 19:46
• In your code to solve this Helmholtz equation, you put the right-hand side as $F = (k^{2}-\pi^{2})\sin(\pi x)$ then compare your numerical result with the analytical solution of $u = \sin(\pi x)$ to find the rate of convergence of your method with respect to mesh size. – Alone Programmer Apr 23 at 20:22
• The point is that you cannot independently choose whatever solution and right hand side you have in the equation. You can only choose one, and then the other is determined. So if you pick a $\bar u$, then it follows from the equation what $F$ needs to be. – Wolfgang Bangerth Apr 23 at 22:29

format: plain


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# C++ Program to Represent Graph Using Incidence List

C++Server Side ProgrammingProgramming

This program represents a graph using incidence list and the time complexity of this algorithm is O(e).

## Algorithm

Begin
Take the input of the number of vertex ‘v’ and edges ‘e’ and also
take the input of ‘e’ pairs of vertexes of the given graph in e[][].
For each edge print the corresponding vertex involved in that connection.
End

## Example Code

#include<iostream>
using namespace std;
int main() {
int i, v, e, j, c;
cout<<"Enter the number of vertexes of the graph: ";
cin>>v;
cout<<"\nEnter the number of edges of the graph: ";
cin>>e;
int edge[e][2];
for(i = 0; i < e; i++) {
cout<<"\nEnter the vertex pair for edge "<<i+1;
cout<<"\nV(1): ";
cin>>edge[i][0];
cout<<"V(2): ";
cin>>edge[i][1];
}
cout<<"\n\nThe incidence list representation for the given graph: ";
for(i = 0; i < e; i++) {
// For each vertex print, its adjacent vertex.
cout<<"\n\tE("<<i+1<<") -> { ";
cout<<"V("<<edge[i][0]<<") , "<<"V("<<edge[i][1]<<")";
cout<<" }";
}
}

## Output

Enter the number of vertexes of the graph: 3
Enter the number of edges of the graph: 4
Enter the vertex pair for edge 1
V(1): 2
V(2): 1
Enter the vertex pair for edge 2
V(1): 1
V(2): 2
Enter the vertex pair for edge 3
V(1): 3
V(2): 2
Enter the vertex pair for edge 4
V(1): 2
V(2): 3
The incidence list representation for the given graph:
E(1) -> { V(2) , V(1) }
E(2) -> { V(1) , V(2) }
E(3) -> { V(3) , V(2) }
E(4) -> { V(2) , V(3) }
Published on 14-Mar-2019 10:34:16


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SYLLABUS  Previous: 3 FINITE ELEMENT METHOD  Up: 3 FINITE ELEMENT METHOD  Next: 3.2 An engineer's formulation

## 3.1 Mathematical background

Slide : [ Weighted residuals - Variational form - Partial integration - Approximation - BC || VIDEO login]

To approximate a set of linear partial differential equations

 (1)

for an unknown that is continuously defined with  derivatives in the volume and is subject to the boundary conditions

 (2)

a mathematician involves first the so-called

Weighted residuals.
Having defined a scalar product and a norm , the calculation essentially amounts to the minimization of a residual vector

 (3)

which is carried out using tools from the variational calculus.

Variational principle.
A quadratic form is constructed for that purpose by choosing a test function in a sub-space that is sufficiently general'' and satisfies the boundary conditions. The linear equation (3.1#eq.1) can then be written as an equivalent variational problem

 (4)

Integration by parts.
If is differentiable at least once in , the regularity required by the forth-coming discretization can often be relaxed by partial integrations. Using for illustration, Leibniz' rule states that

 (5)

Integrating over the volume and using Gauss' divergence theorem

yields a generalized formula for partial integration:

 (6)

For the special case where , this is known as Green's formula

 (7)

The last (surface-) term can sometimes be imposed to zero (or to a finite value) when applying so-called natural boundary conditions.

Numerical approximation.
It turns out that the formulation as a variational problem is general enough that the solution of (3.1#eq.4) remains a converging approximation of (3.1#eq.1) even when the sub-spaces are restricted to finite, a priori non-orthogonal, but still complete sets of functions. The overlap integrals involving non-orthogonal functions are often evaluated with a numerical quadrature that can be easily be handled by the computer.

In general, the discretized solution is expanded in basis functions , which either reflect a property of the solution (e.g. the operator Green's function in the Method of Moments), or which are simple and localized enough so that they yields cheap inner products and sparse linear systems (e.g. the roof-top function for the linear Finite Element Method). Different choices can also be made for the test functions : among the most popular are the Galerkin method, where test and basis functions are both chosen from the same sub-space and the method of collocation, which consists in taking Dirac functions that lead to point-wise evaluations of the integrand on the mesh .

Boundary conditions.
If natural boundary conditions are not already sufficient, essential boundary conditions have to be imposed either by choosing the functional space (e.g. let and overlap in a periodic domain as in the next section) or by replacing one or several equations of the linear system (e.g. use higher order finite differences to preserve the convergence rate as in exercise 3.02).

SYLLABUS  Previous: 3 FINITE ELEMENT METHOD  Up: 3 FINITE ELEMENT METHOD  Next: 3.2 An engineer's formulation


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## How to modify the WooCommerce shop page to tweak the display price without affecting the actual price under the hood?

The woocommerce shop I am working on imports product price as "£ per g", this is correct for the calculation. On the shop page my client would like the price to be displayed as "£ per 100g" so I need to multiply the price by 100. In functions.php I have the following snippet which lets me change what is displayed (as you can see, I have already edited the text to display correctly), does anyone provide help on how to translate this import price to display price?

add_filter( 'woocommerce_get_price_html', 'wb_change_product_html' );     // Change and return $price_html variable using the$  price and weight amount     let a = 10     function wb_change_product_html( $price ) {$  price_html = '' . $price . ' per 100g '; // change weight measurement here return$  price_html;     }          add_filter( 'woocommerce_cart_item_price', 'wb_change_product_price_cart' );     // Change the cart prices with $price variable and weight amount function wb_change_product_price_cart($  price ) {         $price =$  price . ' per g'; // change weight measurement here              return $price; }  I’m sure that there’s a quick fix here! I have also added a screenshot of the import columns so that the price change I’m looking for can be seen. ## WordPress + Vue — Single page app giving me 404s when I use query params I’ve got my rest api powered site working really well with Vue. Except for the issue of WordPress and urls. Site: http://mytely.ca/ Search for a name or click on a name and the url updates. Now refresh the page. Everything still works, in that Vue will take over and do all the requests and set things up. But I’m getting a 404 on every page like that. I don’t really know anything about this whole thing. I tried setting up a redirect plugin but that does what it says — it redirects from http://mytely.ca/&page=1&search=John%20Doe back to http://mytely.ca, which isn’t what I want. I basically want WordPress to ignore the urls completely — just have index.php (and hence Vue) takeover everything. Basically, not 404 and just pass on through with the url. I’m not currently using Vue Router at all in my app — just appending query strings and using popstate. I’m wondering if I actually need to be using Vue Router to handle this. ## jQuery Ajax not loading page with ACF fields I’m using the following jQuery code to load contents from a template part into another on click. Without the ACF PHP tags, it works. However, when I add these tags in <?php the_field('field_name'); ?> it throws a 500 (Internal Server Error) error. Does anyone have a solution for this, please?  var baseUrl = "http://projectname:8888/wp-content/themes/custom-theme"$  (".cross__functional").click(function (e) {         e.preventDefault();         console.log('cross functional Clicked')         $("#keContent").load(baseUrl + "/template-parts/components/filename.php"); });  ## Add post featured image to default WP search results page function snippet I did search a lot for such a snippet, but did not find any. Could anyone assist? ## Page Rank Recommendations? What PR is recommended for good quality backlinks in gsa ser setup? sorry for a basic question but im newbie in backlinking, hope some guru will help me with this Thanks ## Nested loops join algorithm – additional available page buffers The basic version of nested llops join algorithm only uses two main memory page buffers for loading the pages. But if additional page buffers are available the I/O cost can be reduced. Should I chose to use the additional available page buffers to hold more pages of the relation considered by the outer loop or the inner loop? And why? I apologies if this is the wrong place to post this question Thanks in advance, ## Multi-level paging where the inner level page tables are split into pages with entries occupying half the page size A processor uses $$36$$ bit physical address and $$32$$ bit virtual addresses, with a page frame size of $$4$$ Kbytes. Each page table entry is of size $$4$$ bytes. A three level page table is used for virtual to physical address translation, where the virtual address is used as follows: • Bits $$30-31$$ are used to index into the first level page table. • Bits $$21-29$$ are used to index into the 2nd level page table. • Bits $$12-20$$ are used to index into the 3rd level page table. • Bits $$0-11$$ are used as offset within the page. The number of bits required for addressing the next level page table(or page frame) in the page table entry of the first, second and third level page tables are respectively (a) $$\text{20,20,20}$$ (b) $$\text{24,24,24}$$ (c) $$\text{24,24,20}$$ (d) $$\text{25,25,24}$$ I got the answer as (b) as in each page table we are after all required to point to a frame number in the main memory for the base address. But in this site here it says that the answer is (d) and the logic which they use of working in chunks of $$2^{11} B$$ I feel ruins or does not go in with the entire concept of paging. Why the system shall suddenly start storing data in main memory in chucks other than the granularity defined by the page size of frame size. I do not get it. ## Will breaking all the titles on a page boost SEO in anyway? So, I have this suggestion by a client – asking me to break the title on every section into two tags. They want to replace <h*> tag with a combination of <h*> and
 tag. So according to them I should rewrite
Hello World
 as
Hello

World
, but somewhere it can even go like this
Hello
<h*>World</h*>
there
 I’m trying to understand what negative impact it could have? One accessibility concern I know is – that it is causing problems with some Screen Reader tools – as the reading agent only reads a bit of title and stops, then I need to focus on another word and then reads.

SEO Experts

to Maximize the Impact of Web Content

  Browser Render: ## How to calculate the size of the page table of a process? Consider a system with A bytes of physical RAM, and B bytes of virtual address space per process. Pages and frames are C bytes in size. Every page table entry is P bytes in size. Calculate the size of the page table of a process? ## How to display the categories on page using shortcode? I am trying to display the categories name on my page but it’s not displaying. I am using the below code and I added the shortcode gridCategories on my page. I am getting only array function createGridCategories(){$  categories = get_categories( array(     'taxonomy'   => 'category',     'orderby'    => 'name',     'parent'     => 0,     'hide_empty' => 0, // change to 1 to hide categores not having a single post ) ); var_dump($categories); return$  categories; } add_shortcode( 'gridCategories', 'createGridCategories');

I tried this also

$categories = get_the_category(); var_dump($  categories);

I added a shortcode like this in textblock

But still, I am not getting any output on my page. Is there any issue with my code?


text:
# Trustless LETS

## Trustless LETS#

LETS involves several parties that agree to use some form of "local currency", usually pegged to the country's main currency at a 1:1 rate. Assume that our LETS is based in a European country where the currency is Euros, and the exchange is done in "local Euros", which are considered equivalent to national Euros.

Each user in LETS has an account, which contains the LETS balance of that user (in Local Euros). On joining, each user has a balance of zero. The balance is stored in a (possibly decentralized) ledger. An interesting feature of LETS is that a user with zero balance can also "withdraw" money, but only for paying another LETS user. At any time, the sum of LETS balances of all the users is zero.

As an example, Alice, with zero balance, wishes to purchase one litre of milk for 2 Euros from Bob, a member of LETS with zero balance. She transfers 2 Euros from her account to Bob's, making her balance -2 and Bob's +2. Bob can then transfer some or all of his balance to another LETS user in exchange for goods or services.

### Implementation#

Since we desire a trustless LETS, we cannot depend on any trusted group of people to admit users. Note that we will still have a committee to perform tasks such as setting up the LETS parameters (local currency, the maximum number of members, etc.) and consuming any joining fee.

We will only assume a trusted pricing oracle that gives the current rate of euros to ergs identified by some global id (rateTokenID) and a singleton box containing exactly one token with this id. A singleton box is a box containing a singleton token, i.e., a token with only one quantity in existence. This box also contains the rate of ergs to euros at any given period. The rate is updated by spending this box and creating another singleton box with the new rate.

At any instance, our LETS is uniquely defined by a global token box that contains some membership tokens with id letsTokenID. This box defines the LETS parameters such as the location, the currency unit, rateTokenID, etc. The token box is initially started with, say, 10000 membership tokens. Users can spend this box and create their individual LETS boxes as outputs of the transaction, such that each such output has exactly one membership token. The balance membership tokens are put in a newly created token box.

A LETS box represents a LETS member and must be used in every transaction. For simplicity, this article restricts all LETS transactions to involve exactly two members, one being the sender and the other the receiver, such that the sender transfers some positive amount of the LETS currency (local euros) to the receiver. Such a transaction consumes the member's boxes and recreates them as output with the updated balance.

#### Variants#

To prevent spam and DDoS attacks, we require at least a minimum number of ergs (minErgsToJoin) to be locked in the newly created member's box. The ergs will be locked until at least the minWithdrawTime number of blocks has been mined. A box can have a negative LETS balance up to the amount that can be covered by the locked ergs (using the rate at the time of trade).

// a tokenBox stores the membership tokens and has this script
val tokenBox = OUTPUTS(0) // the first output must also be a tokenBox
// first output contains remaining LETS tokens

def isLets(b:Box) = { // returns true if b is a LETS box
// A LETS box must have exactly 1 membership token in tokens(0)
b.tokens(0)._1 == letsTokenID && b.tokens(0)._2 == 1 &&
blake2b256(b.propositionBytes) == memberBoxScriptHash &&
SELF.R4[Long].get == 0 && // start the box with zero LETS balance
b.value >= minErgsToJoin && // the box must contain some minimum ergs
b.R6[Long].get <= HEIGHT // store the creation height in R6
}

// how many lets boxes creared in the tx
val numLetsBoxes = OUTPUTS.filter({(b:Box) => isLets(b)}).size

// In the transaction following is preserved for the token box ...
tokenBox.tokens(0)._1 == SELF.tokens(0)._1 &&                //  token id
tokenBox.tokens(0)._2 == SELF.tokens(0)._2 - numLetsBoxes && //  quantity
tokenBox.propositionBytes == SELF.propositionBytes           //  script

A LETS member's box is protected by the script below, whose hash memberBoxScriptHash is used above. The script requires exactly one (sender, receiver) pair per transaction.

val validRateOracle = CONTEXT.dataInputs(0).tokens(0)._1 == rateTokenID
val rate = CONTEXT.dataInputs(0).R4[Int].get
val inBalance = SELF.R4[Long].get    // LETS balance of current input
val pubKey = SELF.R5[SigmaProp].get  // owner of the current input
val createdAt = SELF.R6[Long].get    // height at which current input was mined

val index = getVar[Int](0).get       // index of the corresponding output
val out = OUTPUTS(index)
val outBalance = out.R4[Long].get    // LETS balance of the output

// A LETS box is one that has the same script as the current box
val isMemberBox = {(b:Box) => b.propositionBytes == SELF.propositionBytes}
val letsInputs = INPUTS.filter(isMemberBox)    // all LETS input boxes
val letsOutputs = OUTPUTS.filter(isMemberBox)  // all LETS output boxes

// The current input belongs to the receiver if its LETS balance increases
// There may be some ergs in receiver's input box. We need to ensure that
// the receiver's output box also contains the same amount of ergs as input
val receiver = outBalance > inBalance && out.value == SELF.value

val getBalance = {(b:Box) => b.R4[Long].get} // returns LETS balance of a box

val letsBalIn = letsInputs.map(getBalance).fold(0L, {(l:Long, r:Long) => l + r})
val letsBalOut = letsOutputs.map(getBalance).fold(0L, {(l:Long, r:Long) => l + r})

// sender box can contain less amount of ergs (sender may withdraw ergs provided
// that any negative LETS balance of sender in out is backed by sufficient ergs)
val correctErgs = out.value >= -outBalance * rate && (
out.value >= SELF.value || SELF.R6[Long].get + minWithdrawTime > HEIGHT
)

// for the receiver, we don't touch the erg balance,
// since a receiver is not actively involved in the transaction

inBalance != outBalance && // some transaction should occur; balance must change
SELF.tokens(0)._1 == letsTokenID && // the current input has the right token
out.tokens(0)._1 == letsTokenID && // corresponding output has the right token
validRateOracle &&          // oracle providing rate has the correct "rate token"
letsBalIn == letsBalOut &&  // total LETS balance is preserved in the transaction
letsInputs.size == 2 && letsOutputs.size == 2 &&  // only two LETS inputs, outputs
out.propositionBytes == SELF.propositionBytes &&  // out is a LETS box ...
out.R5[SigmaProp].get == pubKey &&                // ... with the right pub key
out.R6[Long].get == SELF.R6[Long].get &&          // ... and creation height
(pubKey && correctErgs) // ... or out has correct ergs and tx has signature
)

The transaction spending a box with the above script requires: - The sum of the LETS balance of inputs and outputs is preserved - There are two LETS inputs and two LETS outputs - The public keys (stored in R5) are preserved in the corresponding output - The creation height (stored in R6) be preserved in the corresponding output

Some public key is the receiver if the LETS balance of its output is higher than that of its input.

The last condition requires that either the input and output boxes belong to the receiver (so that the ergs are preserved), or, if they belong to the sender, a signature is provided. The required number of ergs backs the output if its LETS balance is negative. Furthermore, it requires that the sender's ergs balance cannot be reduced until at least the minWithdrawTime number of blocks have been mined after the ergs were locked.

Compared to the managed LETS, the above system has the following differences:

• No membership record: Unlike the managed LETS, We don't store any membership information here.
• Multiple-boxes: A person can create multiple membership boxes, which is permitted. We only require that any negative balance be backed by the corresponding number of ergs locked.
##### LETS-1: Zero Sum, Collateral#

The above is the basic variant, which we call LETS-1. It has the following features:

• Time-locked Joining-Fee: To prevent spam attacks, a member has to pay a certain minimum fee in ergs at the time of joining. This fee is refundable but only after a predefined number of blocks.
• Zero-Sum: The sum of the LETS balances of all member boxes is zero. Member boxes can have a negative balance as long as it is within a certain limit.
• Collateral: For the sender's output, ergs are used as collateral to cover the negative LETS balance at the current exchange rate.

The following are some variations of LETS-1.

##### LETS-2: Zero Sum, No collateral#

(A slight variation of LETS-1)

• Non-refundable joining fee: Like LETS-1, a joining fee is needed to prevent spam attacks. However, unlike LETS-1, this fee is non-refundable and must be sent to some predefined management committee.
• Zero-Sum: As in LETS-1.
##### LETS-3: Positive-Sum, Collateral#

The above two variants require the total LETS balance to be always zero. Here we consider a positive value for this sum. In particular, this variant has the following properties:

• Time-locked Joining Fee: As in LETS-1.
• Positive Sum: The LETS balance of every member must always be non-negative, which ensures that the sum of the LETS balances of all member boxes is positive. The initial LETS balance is set to a positive value based on the joining fee at the current rate, capped to some maximum value.
• Collateral: Any reduction in the ergs balance of the sender must be accompanied by a reduction of the corresponding LETS balance at the current exchange rate.

We can also allow topping up the LETS balance during a transaction by adding the equivalent amount of ergs.

##### LETS-4: Positive-Sum, No collateral#

This is similar to LETS-3 but with some small variations:

• Non-refundable Joining Fee: As in LETS-2
• Positive-Sum: As in LETS-3

The following table summarizes the variants:

Zero Sum Positive Sum
Collateral LETS-1 LETS-3
No collateral LETS-2 LETS-4

We considered LETS transactions involving a single sender-receiver pair. More advanced models can allow multiple senders and receivers and need not be in pairs.

Crossing the last mile

The need to reform the global financial system has been clear since the last crisis in 2008. Now, COVID-19 has forced our hands. We cannot delay, and the best way to start is from the grassroots.

As lockdowns across the world effectively shut down whole sectors and a large part of the economy, we are entering a new era at a breakneck pace. Central banks are taking unprecedented action in the form of money printing, but, just like last time around, they will almost certainly fail at the 'last mile' (intentionally or otherwise): while their efforts will benefit the financial sector and the wealthy, the funds will not reach those small businesses and individuals who need them the most.

Technology like the Ergo platform can address this problem efficiently, building bridges into and within our local economies without the need for commercial or central banks. A key principle of Ergo, and one stated in the white paper, is that it is for regular people. The developers have spent considerable time building technology that can be applied to real-world use cases. (This is the idea behind 'Smart contracts for the people', for example.)


text:
# Ticks / nanosecond delay

On an STM32F103C8T6 (72MHz) I want to use a delay in range of 100's of nanosecond (to create a LED strip driver, WS2813B).

Since HAL only provides a ms delay, I found at various places code to use the ticks/debug counter as timer.

I have the code below:

Initialization:

volatile unsigned int *DWT_CYCCNT = (volatile unsigned int *)0xE0001004; //address of the register
volatile unsigned int *DWT_CONTROL = (volatile unsigned int *)0xE0001000; //address of the register
volatile unsigned int *SCB_DEMCR = (volatile unsigned int *)0xE000EDFC; //address of the register

unsigned int count = 0;

Within main():

  *SCB_DEMCR = *SCB_DEMCR | 0x01000000;
*DWT_CYCCNT = 0; // reset the counter
*DWT_CONTROL = *DWT_CONTROL | 1 ; // enable the counter

int ticks = 1;

while (1)
{
unsigned int start, current;
start = *DWT_CYCCNT;
do
{
current = *DWT_CYCCNT;
} while((current - start) < ticks);

if (count++ % 1000 == 0)
{
HAL_GPIO_TogglePin(GPIOB, GPIO_PIN_6);
}
}

When I connect a logic analyzer, I see a PWM signal of 0.53 ms per toggle. Since it is toggled every 1,000 counts, this means 0.53 us (530 ns). However, with 1 tick, I would expect something like 1/72,000,000 = 14 ns (plus some overhead for the while loop/counter). (note: the counter for 1,000 I used because my logic analyzer can only handle 200 kHz).

I'm not sure if the addresses in the initialization are correct (found them only for STM32F103VET6).

So my question: What is the mistake in the program above?

(side question: is this method good anyway for a LED driver, or should I send the data by interrupt or DMA?).

Update

The reason I need these kind of delays are for the T0H, T1H, T0L and T1L times which can be found at WS2813B datasheet, page 5.

Data Transfer Time (TH+TL=1.25µs±300ns)

T0H 0-code, High-level time 300ns~450ns

T1H 1-code, High-level time 750ns~1000ns

T0L 0-code, Low-level time 300ns~100us

T1L 1-code, Low-level time 300ns~100us

RES Frame unit, Low-level time 300µs or more

• I can't wrap my head around how the µC and LED strip driver are related, or why you feel that you need 100 ns resolution for something. I'd love to hear what your final goal is. Is your final goal perhaps to make 10 LEDs in series to glow brighter/weaker in 1 second oscillation like some LED Christmas strips? – Harry Svensson Apr 11 '18 at 23:43
• @HarrySvensson I updated my question (see update, last part), that shows I need different delays depending on a 0 or 1. However, I could use e.g. a 400 ns 'fixed' time in case I need a timer. – Michel Keijzers Apr 12 '18 at 2:57
• Have you already tried the Systick handler? I think you can configure this timer and achieve the desired delay. – Abel Tom Apr 12 '18 at 6:21
• @AbelTom I have not tried, I will do in case the SPI solution will not work. – Michel Keijzers Apr 12 '18 at 9:34
• Somewhat unrelated, but in your initialization section those int * should be int *const – Reinderien May 29 '18 at 8:14

Running at 72 MHz one instruction cycle is ~ 14 ns. Instruction timing is roughly 1 instruction per cycle, but memory access, the peripheral bus, branch penalty, and flash timing all get in the way. To get an external waveform that is accurate enough for a single WS2813B is doable in assembler, but for a chain of these LEDs it is a very tough (or impossible) job.

Your best chance is to (mis) use the SPI peripheral, and feed it bytes that are pre-calculated to produce the desired waveform.

Or switch to the WS2801, which has separate clock and data lines.

update:

The advantage of SPI is that there is some buffering - IIRC you can load one byte while the other is being shifted out, and the hardware will take care of putting the bytes immediately after each other. This takes most of the pain out of the timing.

One of my students actually did this for his end-of-first-year project, but that was on an Arduino Due (also Cortex M3, but at 92 MHz, and with a different SPI peripheral). He first tried to get the timing right uysing tweaked C++, but that was too fragile and could not handle a longer chain.

• Without DMA I think the time to load the new word is prohibitive. – Jan Dorniak Apr 12 '18 at 8:08
• Thanks for this answer, I didn't realize the overhead would be that much. So far I want to use only one chain, but I like the SPI solution. It will cost some more memory (since I have to keep the chain of bits ready beforehand), but even the STM32F103C8T6 (20 KB) has enough to run several strips in case needed in future. – Michel Keijzers Apr 12 '18 at 9:08
• @JanDorniak Why? My plan was (after reading the answer), to create a big array containing all data, than send it via SPI (non DMA). Than wait until data changes, and send it again for the next change. So the words are loaded 'slowly' but send with one command (for sending it via SPI). – Michel Keijzers Apr 12 '18 at 9:11
• Btw, I explicitly selected WS2813 above WS2812 for the fact that if one LED breaks, I can use the remainder LEDs. I would have to check what differences are with WS2801. – Michel Keijzers Apr 12 '18 at 9:12
• @MichelKeijzers Read up the manual. HAL obfuscates a lot of stuff, the data on SPI is actually sent using 8 or 16 bit words and after writing one word the processor must load the next one into the perfieral register which may take too long. Seeing as you seem to have an oscilloscope handy - do it and check the results. – Jan Dorniak Apr 12 '18 at 9:16

You can drive the WS leds using DMA and timer generated PWM signal. Read about the DMAR register and how to use it in the RM. I do not know if it is possible using the HAL library as I do not use it.

Bare register is easy and straightforward.

• Thanks, the problelm is that the PWM signal should change often, but maybe it could work. At least another option to try if other fails). – Michel Keijzers Apr 12 '18 at 10:12
• @MichelKeijzers I use this approach all the time without any problems. I do not have a code to share as I do not use the F1 family anymore - F3 is cheap but much better and modern. – P__J__ Apr 12 '18 at 10:18
• Ok ... I like HAL so far, still it's a huge dive from the Arduino. I would need to check for F3, I know they exist but haven't them encountered at e.g. AliExpress. I have some F4 but intending to use them for something else later. – Michel Keijzers Apr 12 '18 at 10:31
• I found out that HAL sometimes extends a clock pulse (including data), so I cannot use it. Not sure if it is HAL or not. I could check DMAR/RM but doubt it is supported (easily) by HAL. Instead I'm going to use a timer directly, either with DMA or not. – Michel Keijzers Apr 14 '18 at 11:25


text:
Last edited by Gasho
Sunday, July 26, 2020 | History

5 edition of Electronic Structure and Properties of Hydrogen in Metals (Nato Conference Series. VI, Materials Science, V. 6) found in the catalog.

# Electronic Structure and Properties of Hydrogen in Metals (Nato Conference Series. VI, Materials Science, V. 6)

• 389 Want to read
• 33 Currently reading

Written in English

Subjects:
• Physical Properties Of Materials,
• Solid State Physics,
• Science,
• Technology & Industrial Arts,
• Science/Mathematics,
• Metallurgy,
• Science / Physics,
• Physics,
• Congresses,
• Electronic structure,
• Hydrogen content,
• Metals

• Edition Notes

The Physical Object ID Numbers Contributions C.B. Satterthwaite (Editor), P. Jena (Editor) Format Hardcover Number of Pages 698 Open Library OL10323230M ISBN 10 030641130X ISBN 10 9780306411304

Electronic structure of metals A. Pseudopotential perturbation theory B. Pseudopotentials in the perfect lattice C. Electron diffraction by pseudopotentials D. Nearly-free-electron bands and Fermi surfaces E. Scattering by defects F. Screening Mechanical properties of metals A. The band-structure . The electronic structure of metals means they are relatively good conductors of ons in matter can only have fixed rather than variable energy levels, and in a metal the energy levels of the electrons in its electron cloud, at least to some degree, correspond to the energy levels at which electrical conduction can occur.

@article{osti_, title = {Pure metals properties: A scientific and technical handbook}, author = {Buch, A}, abstractNote = {This handbook provides detailed data on the properties of pure metals for general scientific and technical users who are involved in the selection, application, testing, and characterization of metals. The more than quick-reference tables and figures contained. Publication date Note Published through SciTech Connect. 01/01/ "bnl" " conf" "DE" International symposium on the electronic structure and properties of hydrogen in metals, Richmond, VA, USA, 4 Mar

These elements (except hydrogen) are known as alkali metals, and they all have similar chemical properties. The elements in group 2 (the second column) form compounds consisting of one atom of the element and two atoms of hydrogen: These are called alkaline earth metals, with similar properties among members of that group. Optical Properties and Electronic Structure of Metals and Alloys. Proceedings of the International Colloquium held at Paris, Sept by Florin Abeles, editor and a great selection of related books, art and collectibles available now at

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### Electronic Structure and Properties of Hydrogen in Metals (Nato Conference Series. VI, Materials Science, V. 6) Download PDF EPUB FB2

The purpose of the "International Symposium on the Electronic Structure and Properties of Hydrogen in Metals" held in Richmond, Virginia, Marchwas to increase our fundamental under standing of hydrogen in metals. Such knowledge is essential in solving technologically important : \$ NATO International Symposium on the Electronic Structure and Properties of Hydrogen in Metals ( Richmond, Va.).

Electronic structure and properties of hydrogen in metals. New York: Published in cooperation with NATO Scientific Affairs Division [by] Plenum Press, © (OCoLC) Material Type: Conference publication: Document.

Electronic Structure and Properties of Hydrogen in Metals - Ebook written by C.B. Satterthwaite, P. Jena. Read this book using Google Play Books app on your PC, android, iOS devices. Download for offline reading, highlight, bookmark or take notes while you read Electronic Structure and Properties of Hydrogen in Metals.

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(ISBN: ) from Amazon's Book Store. Everyday low prices and free delivery on eligible : Hardcover. The book first discusses the electronic structure of atoms, including hydrogen atom and many-electron atom.

The text also underscores bonding between atoms and electrons in metals. Discussions focus on bonding energies and structures in the solid elements, eigenstates of free-electron gas, and electrical conductivity.

The electron theory of metals describes how electrons are responsible for the bonding of metals and subsequent physical, chemical and transport properties.

This textbook gives a complete account of electron theory in both periodic and non-periodic metallic systems. The author presents an accessible approach to the theory of electrons, comparing it with experimental results as much as possible.3/5(2).

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13 Magnetism, electronic structure and electron transport properties in magnetic metals Prologue Classiﬁcation of crystalline metals in terms of magnetism Orbital and spin angular momenta of a free atom and of atoms in a solid Localized electron model and spin wave theory Itinerant electron model We report the synthesis of high-entropy-alloy (HEA) nanoparticles (NPs) consisting of five platinum group metals (Ru, Rh, Pd, Ir and Pt) through a facile one-pot polyol process.

We investigated the electronic structure of HEA NPs using hard X-ray photoelectron spectroscopy, which is the first direct observation of Celebrating 10 years of Chemical Science. Hydrogen is also prevalent on Earth in the form of chemical compounds such as hydrocarbons and water.

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Electronic structure. Orbital structure of hydrogen atom, principal quantum number n, number of electrons per orbital In the Bohr model, the hydrogen electron orbits the nucleus. In quantum mechanics, hydrogen electron exists in a spherical probability cloud around the nucleus. The principle quantum number, n, defines what shell the electron is in.

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Replace expensive precious metals. Hydrogen, symbol H, has a Simple Hexagonal structure and Colorless color. Hydrogen is a other nonmetal element. Trivial name of Hydrogen is alkali metals*.

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How do alkali metals differ from alkaline earth metals in atomic structure and general properties? The alkali metals all have a single s electron in their outermost shell. In contrast, the alkaline earth metals have a completed s subshell in their outermost shell. In general, the alkali metals react faster and are more reactive than the corresponding alkaline earth metals in the same period.Electronic Structure is a new multidisciplinary journal covering all theoretical and experimental aspects of electronic structure research, including the development of new methods.

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text:
# Caustics in Tachyon Matter and Other Born-Infeld Scalars

@article{Felder2002CausticsIT,
title={Caustics in Tachyon Matter and Other Born-Infeld Scalars},
author={Gary N. Felder and Lev Kofman and Alexei A. Starobinsky},
journal={arXiv: High Energy Physics - Theory},
year={2002}
}
• Published 5 August 2002
• Physics
• arXiv: High Energy Physics - Theory
We consider scalar Born-Infeld type theories with arbitrary potentials V(T) of a scalar field T. We find that for models with runaway potentials V(T) the generic inhomogeneous solutions after a short transient stage can be very well approximated by the solutions of a Hamilton-Jacobi equation that describes free streaming wave front propagation. The analytic solution for this wave propagation shows the formation of caustics with multi-valued regions beyond them. We verified that these caustics…

## Figures from this paper

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We study the role that tachyon fields may play in cosmology as compared to the well-established use of minimally coupled scalar fields. We first elaborate on a kind of correspondence existing between

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### Compact scalar field solutions in EiBI gravity

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## References

SHOWING 1-10 OF 45 REFERENCES

### Rolling Tachyon

We discuss construction of classical time dependent solutions in open string (field) theory, describing the motion of the tachyon on unstable D-branes. Despite the fact that the string field theory

### Sigma model approach to string theory effective actions with tachyons

Motivated by recent discussions of actions for tachyon and vector fields related to tachyon condensation in open string theory we review and clarify some aspects of their derivation within the sigma

### Accelerated expansion of the universe driven by tachyonic matter

It is an accepted practice in cosmology to invoke a scalar ﬁeld with potential V ( φ ) when observed evolution of the universe cannot be reconciled with theoretical prejudices. Since one

### Problems with Tachyon Inflation

• Physics
• 2002
We consider cosmological consequences of string theory tachyon condensation. We show that it is very difficult to obtain inflation in the simplest versions of this theory. Typically, inflation in

### Solution and Quantization of a Nonlinear Two-dimensional Model for a Born-Infeld Type Field

• Mathematics
• 1966
The Cauchy problem for the nonlinear equation for a scalar field of the Born-lnfeld type[ 11 is solved in the two-dimensional case. The solution is obtained in parametric form. It is shown that the

### On Exact Tachyon Potential in Open String Field Theory

• Mathematics
• 2000
In these notes we revisit the tachyon lagrangian in the open string field theory using background independent approach of Witten from 1992. We claim that the tree level lagrangian (up to second order

### Field Theory of Tachyon Matter

We propose a field theory for describing the tachyon on a brane–antibrane system near the minimum of the potential. This field theory realizes two known properties of the tachyon effective action:

### Some exact results on tachyon condensation in string field theory

• Physics
• 2000
The study of open string tachyon condensation in string field theory can be drastically simplified by making an appropriate choice of coordinates on the space of string fields. We show that a very


text:
The Stack Overflow podcast is back! Listen to an interview with our new CEO.

# Tag Info

Huffman's algorithm is known to be optimal, that is, produce a code which minimizes the average codeword length (with respect to the input distribution). Let us notice now that there is a code in which each codeword has length $\lceil \log n \rceil$, and in particular the average codeword length is $\lceil \log n \rceil$ (with respect to any distribution). ...


text:
$\partial \bar{\partial}$ on a complex manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:08:08Z http://mathoverflow.net/feeds/question/87968 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87968/partial-bar-partial-on-a-complex-manifold $\partial \bar{\partial}$ on a complex manifold william 2012-02-09T06:42:38Z 2012-02-11T06:13:58Z
hallo,

i have the following question: let $M$ be a complex $n-$dimensional manifold and $R \subset M$ be a totally real, compact, $n-$dimensional (real) manifold. let $\alpha$ be a smooth nonnegative $(n,n)-$form on $M$. does there exists a smooth plurisubharmonic function $f : U \rightarrow \mathbb{R}$, where $U$ is a open neigbourhood of $R$ in $M$ such that the equation $(\partial \bar{\partial} f)^{n} = \alpha$ is satiesfied. So i am interested in smooth solution in a neigbourhood of the real manifold of the inhomogenous monge-ampere equation. does there exists such a solution ? if yes, can you give me some reference and if no what can be done to ave such a solution ? hope for answers. thanks in advance.

marco


text:
Is always true that: $v(\cup_{n=1}^{\infty} A_n) \leq \sum\limits_{n=1}^{\infty} v(A_n)$?

Let $v : \mathcal {P}(\mathbb R) \to [0,\infty]$ a set-function such that $\displaystyle{ v(F \cup E) \leq v(E) + v(F) \quad \forall E,F \subset \mathbb R}$.

Is always true that: $\displaystyle{ v\left(\bigcup_{n=1}^{\infty} A_n\right) \leq \sum_{n=1}^{\infty} v(A_n)}$ ?

I think this is not but I can't find a counterexample. Any help?

-
Thank you both for your answers! –  passenger May 11 '12 at 19:33
However if $v$ is a measure then it is true. –  leo May 15 '12 at 0:56

Let $\nu\colon \mathcal{P}(\mathbb{R})\to [0,\infty]$ be the function $\nu(A) = 0$ if $A$ is finite and $\nu(A) = \infty$ if $A$ is infinite. Then $\nu(\mathbb{N}) = \infty$ but $\sum_{n\in \mathbb{N}}\nu(\{n\}) = 0$.

-
Heh; we're very close. (-: –  Arturo Magidin May 11 '12 at 19:30
You're not kidding! –  froggie May 11 '12 at 19:31

Define $v$ to be $0$ if the set is bounded, and $\infty$ if the set is unbounded. If $F$ and $E$ are bounded, then so is $F\cup E$, so $v(F\cup E)\leq v(E)+v(F)$ is satisfied.

Now let $A_n = [-n,n]$. Then $\bigcup\limits_{n=1}^{\infty}A_n = \mathbb{R}$ is unbounded, and $v(A_n) = 0$ for all $n$, so $$v\left(\bigcup_{n=1}^{\infty}A_n\right) = \infty \gt 0 = \sum_{n=1}^{\infty}v(A_n).$$

-

The purpose of that question is to show that if $\nu$ subadditive it isn't implies that $\nu$ $\sigma$-additive. The answers above shows that fact.

It's well known that, if $\nu$ is a measure, then:

$$\nu \ \sigma\text{-additive} \ {\Longrightarrow} \ \nu \ \sigma\text{-subadditive} \ {\Longrightarrow} \ \nu \ \text{subadditive}$$

-


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# Polynomials/ galois field question

by JamesGoh
Tags: field, galois, polynomials or, solved
 P: 141 Im reading through a section that deals with polynomials Galois fields and ran into something that Im not quite understanding. Say we have an irreducible polynomial, f(x), which has coefficients from GF(2) and roots $$\beta$$, $$\beta$$$$^{2}$$, $$\beta$$$$^{4}$$, $$\beta$$$$^{8}$$, .........$$\beta$$$$^{2}$$$$^{e}$$$$-1$$ where e is the smallest integer such that $$\beta$$$$^{2}$$$$^{e}$$ = $$\beta$$ given by f(x) = $$\prod$$$$^{e-1}_{i=0}$$ ( X + $$\beta$$$$^{2^i}$$) Note: Beta term is Beta^(2^i) In the section Im reading, they do a test to prove f(x) is irreducible. I will state the test below Say f(x) = a(x).b(x) where a(x) and b(x) are polynomials with coefficients from GF(2) if we sub one of the roots of f(x) in, say $$\beta$$, f($$\beta$$) = 0 which means that either a($$\beta$$) = 0 or b($$\beta$$) = 0, hence a(x) = f(x) or b(x) = f(x). This understanding also says that a(x) or b(x) (depending which one had $$\beta$$ subbed into it) has all the roots of f(x) (A theory in my textbook says that if f($$\beta$$) = 0, f($$\beta$$$$^{2^i}$$)=0 for any i) I get how they arrive at their result, however Im still clueless as to how this proves that f(x) is irreducible. insight is appreciated regards James
 P: 141 Ok I read over the notes again and think I may have the answer Since f(x) = a(x) or f(x) = b(x) when the root is substituted in, it cannot be divided into a smaller polynomial with a non-zero degree. Therefore f(x) must be irreducible. Thoughts, comments, insights ??
 Sci Advisor HW Helper P: 2,020 I'm confused. You start out by saying that f(x) is irreducible, presumably over GF(2). Then you go on to prove that f(x) is irreducible - this time over what? Also, why is this in the Set Theory, Logic, Probability, Statistics forum? It really should be in the algebra forum.
P: 141
Polynomials/ galois field question

 I'm confused. You start out by saying that f(x) is irreducible, presumably over GF(2). Then you go on to prove that f(x) is irreducible - this time over what?
I answered my own question (see f(x)=a(x).b(x) proof), it was just that I didn't read over the notes properly.

Sorry about the confusion

Also, someone pls move this topic to the correct forum. ta

 Related Discussions Calculus & Beyond Homework 1 Calculus & Beyond Homework 3 Linear & Abstract Algebra 11 Linear & Abstract Algebra 1 Linear & Abstract Algebra 1


text:
Chaotic Number Shuffling

You have a deck of cards, consisting of cards with non-negative numbers on them, and you haven't shuffled it yet, and no player trusts the other players to be fair at shuffling. You devise a way to shuffle the deck that is completely deterministic, but chaotic enough that the other players are okay with using it. Here's how you perform a single shuffle:

1. Start with the deck face down. There are a series of M empty piles (where M is the maximum number in the deck) labeled as the 1st, 2nd, 3rd, ... ,Mth pile, and there is also an empty storage pile.

2. While the deck is not empty, do the following: Take the first card off the top of the deck, let N be its number and put the card face down on the top of the storage pile. If there are at least N cards left in the deck, deal the top N cards face down such that the top card goes in the 1st pile, the second card from the top goes in the 2nd pile, etc. until the Nth card gets put in the Nth pile. If there are fewer than N cards in the deck, deal all of them.

3. Construct the new deck by putting the first pile on the storage pile, then the second pile on the new pile, then the third pile on that new pile, then the fourth pile on that one, until all piles are in your deck.

This method is pretty intuitive when doing by hand, but after a few games some players want a computer to do it, due to the deep-seated mistrust within the group and a few lucky draws. They refuse to use other deck shuffling programs that would work better. Write a program that takes as input a list of non-negative integers (in some format you specify) and shuffles it once using this method, outputting the result. You may assume the integers and the length of the list are less than 2^16.

Here are some worked-out examples (layout is deck|storage|1st|2nd|3rd|...|Mth, cards go from bottom to top):

9,8,7,6,5,4,3,2,1,0|||||||||||
9,8,7,6,5,4,3,2,1|0||||||||||
9,8,7,6,5,4,3|0,1|2|||||||||
9,8,7|0,1,3|2,4|5|6|||||||
|0,1,3,7|2,4,8|5,9|6|||||||
0,1,3,7,2,4,8,5,9,6||||||||||| (starting another iteration)
0,1,3|6|9|5|8|4|2|7||||
|6,3|9,1|5,0|8|4|2|7||||
6,3,9,1,5,0,8,4,2,7|||||||||||

Here is a long string of successive shuffles:

[9,8,7,6,5,4,3,2,1,0]->[0,1,3,7,2,4,8,5,9,6]->[6,3,9,1,5,0,8,4,2,7]->
[7,3,2,6,4,8,0,5,1,9]->[9,1,5,0,8,4,6,2,3,7]->[7,1,3,9,2,6,4,8,0,5]->
[5,9,0,3,8,1,4,7,6,2]->[2,4,9,6,1,5,7,8,3,0]->[0,3,1,9,8,6,4,7,2,5]->
[5,9,2,1,7,3,4,0,6,8]->[8,5,6,0,4,3,7,1,2,9]->[9,2,1,7,3,4,0,6,5,8]

Another few:

[1,0,1,0]->[0,1,1,0]->[0,1,0,1]->[1,1,0,0]->[0,0,1,1]->[1,0,0,1]->[1,0,1,0]
[1,1,1,0]->[0,1,1,1]->[1,1,1,0]
[1,0,1,1]->[1,0,1,1]
[4,3,2,1]->[1,3,2,4]->[4,2,3,1]->[1,2,3,4]->[4,3,2,1]
[1,2,1,2,1,2]->[2,1,1,1,2,2]->[2,1,2,2,1,1]->[1,2,2,1,2,1]->[1,1,2,2,2,1]

Perl 5, 81 bytes

A naive approach (just shuffling per the specs). Doubtless someone will find a shorter way to do it (even in Perl 5, I mean).

{while(@_){for(1..($.=pop)){push@{$a[$_]},pop}push@{$a[0]},$.}$,=$/;say@$_ for@a}

This is a subroutine, but what it prints (not returns) is the shuffled deck. It requires -M5.01 (which is free).

$perl -M5.01 -e'sub{while(@_){for(1..($.=pop)){push@{$a[$_]},pop}push@{$a[0]},$.}$,=$/;say@\$_ for@a}->(9,8,7,6,5,4,3,2,1,0)'
0
1
3
7
2
4
8
5
9
6

It also adds extra newlines in various places (depending on the deck), but there's… well, nothing technically in the specs forbidding that. :-)

Ruby, 91 bytes

There is probably a better way to create an array of arrays. There's probably a shorter way to do all the shuffling, too. We'll see what future answers and comments bring to the table and all golfing suggestions are welcome.

->d{z=(1..d.size).map{[]};([d[-1]+1,d.size].min.times{|i|z[i]<<d.pop})while d[0];z.flatten}

Ungolfing:

def shuffle(deck)
# an array of arrays
z = (1..deck.size).map{[]}

# while there are still cards in the deck
while deck[0]
# if we're almost out of cards, use how many are left
# else, put in deck[-1] + 1 in the piles (that is, including deck[-1])
card = [deck[-1] + 1, deck.size].min
card.times do |i|
z[i] << deck.pop
end
end

# flatten all of those arrays done and return
return z.flatten
end


text:
SAT Math : How to find the surface area of a pyramid

Example Questions

Example Question #1 : How To Find The Surface Area Of A Pyramid

The Pyramid of Giza has a height of 480 feet. If the length of each side of the base is approximately 756 feet, what is its total surface area? Round to the nearest tenth.

Explanation:

If the length of one side is 756 ft, then multiply to find the area of the base.

Once you've found the area of the base, use the height of the pyramid and half of the side length of the base to determine the length of the side from the apex to the ground using the Pythagorean Theorem.

Using the side length of the base and the height of each of the triangles that form the pyramid, calculate the area of each triangle, then multiply by 4.

Add the surface area of the base to the surface area of the four triangles.


text:
# If Sets Would DWIM

Whenever I work with Sets in Raku they often fail to DWIM. This is a short exploration to see if DWIMminess can be improved.

I recently revisited a script I wrote a while ago that made use of the (-) Set difference operator. The code had a bug lurking there in plain sight because the following code doesn't do what my intuition wants.

  my @allowed = <m c i p l o t>;
my @chars = 'impolitic'.comb;

my @remainder = @allowed (-) @chars;

if +@remainder == 0 {
say 'pangram';
} else {
say "unused: [{@remainder.join(' ')}]";
}
unused: []

The cause of the bug is that (-) produces a Set and assignment to @remainder creates an Array of 1 item. Always. But inconvenently, when it is an empty set, it stringifies to an empty string which just helps to cover for the lurking bug.

my @items = <a b c d e> (-) <a b c d e>;
say @items.raku;
say +@items;
[Set.new()]
1

The fix is relatively simple. Just don't assign to an array. Use a scalar container instead:

my $items = <a b c d e> (-) <a b d>; say$items.raku;
say +\$items;
Set.new("e","c")
2

Or even an associative container works out just fine:

my %items = <a b c d e> (-) <a b d>;
say %items.raku;
say +%items;
{:c(Bool::True), :e(Bool::True)}
2

Or explicity take the list of keys before assignment:

my @items = (<a b c d e> (-) <a b d>).keys;
say @items.raku;
say +@items;
["e", "c"]
2

Great. It works. Just don't use array containers for Setty things. Except that doesn't stop my intuition stumbling into this mistake every now and then. The same class of bug has cropped up in my code on several occasions because it's just so easy to make the mistake. Raku doesn't tell me that I have done something wrong, because maybe it's intentional. But importantly, Raku doesn't manage to DWIM.

The other approach I could take is to get into the habit of adding type information. That does enable Raku to tell me when I fall into this trap.

my Str @a = <a b c d e> (-) <a b d> ;
Type check failed in assignment to @a; expected Str but got Set (Set.new("e","c"))
in sub  at EVAL_0 line 3
in block <unit> at EVAL_0 line 5
in block <unit> at -e line 1

That's a clear example where adding type information helps the Raku compiler to help me avoid introducing this kind of bug.

### Experiment – Custom Array store for Set

I started to dig into the core setting to see what could be done. I was pleasantly surprised to find that I could add to the multi dispatch for Array.STORE to include the semantics I am looking for.

use MONKEY;

augment class Array {
multi method STORE(Array:D: Set \item --> Array:D) {
self.STORE(item.keys)
}
}

my @a = <a b c d e> (-) <a b d>;
say @a.raku;
say +@a;
["c", "e"]
2

It seems prudent to share this to see if there are any gotchas or downsides to my little DWIM that I haven't considered. One possible downside is that you'd need to use , to force a Set into an array if that's what you need to do.

my @a = <a b c d e> (-) <a b d> , ;
say @a.raku;
[Set.new("e","c")]

### What Next

I hope this generates a discussion about this and other cases where our intuition and Raku's behaviour don't quite line up. Maybe there are other related language edges that could be smoothed off to remove this kind of hazard.