nt.number theory – Continued Fractions, Chebyshev and non-homogenous approximation

In Khinchin’s book, “Continued Fractions,” he considers the question, given an irrational, $alpha$, and a real number, $beta$, how to find integral $x$ and $y$ such that
$$alpha x – y approx beta$$
to a given level of accuracy.

He then says that Chebyshev “obtained the first basic results connected with it, and has been the subject of continued intensive study, especially by the Soviet arithmetic school.” Khinchin goes on to prove a few interesting theorems concerning this question, but then moves on to other topics.

Where can I find more about work that has been done on this problem?

number theory – Continued Fraction Factorization Algorithm please explain with Code

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How can this continued fraction be assessed and integrated?

Following the answer from my previous question, I would like to apply the same answer to this new continued fraction, where the order starts at n = 1 and not n = 0

f(x_,y_)=1/(1+x y+ContinuedFractionK((n(n+1)I (x/B))-(n(n + 2))/(4(n + 1)^2-1)x^2 A,1,{n,1,Infinity}))

And then I want to integrate it in relation to A.

Thank you very much.

Wolfram Cloud – Can Mathematica evaluate a continued fraction with the Gauss $ K $ operator?

I don't know much about Mathematica and this is a rather unusual notation.

How can I tell Mathematica to rate this? $$ 1 + underset {i = 1} { overset { infty} {K}} frac {(- 1) ^ {i-1}} {i + 1} $$

Where $$ underset {i = 1} { overset { infty} {K}} frac {a_i} {b_i} = cfrac {a_ {1}} {b_ {1} + cfrac {a_ {2} } {b_ {2} + cfrac {a_ {3}} {b_ {3} + cfrac {a_ {4}} {b_ {4} + ddots {}}}} $$

Do I have to convert to another notation first? Or can I create a custom notation like this? Thanks for your time.

P.S. I work in my browser in Wolfram Cloud Open Access if that is important.

Continued fractions – How do you calculate this function?

Please, I want to calculate this function, which contains an infinite continuous fraction

f(y_)=1/(1 + ContinuedFractionK(-(((n + 1) (n + 3))/((2 n + 3) (2 n + 5))) y^2 A, 1, {n, 0, Infinity}))

But no results, Mathematica returns the same continued fraction.

function

Please, is there a way to construct this infinite fraction and evaluate f (y)?

Many thanks.

wp query – How can the layout of post-loop results be varied and continued?

How can I pause at certain intervals when a post loop is output in a certain markup, use a different markup for a number of the next posts, and then continue the loop at the appropriate point in the standard layout?

Specially …

To change the layout on the left to the layout on the right, in which the flow of postcards is interrupted and changed to introduce visual variations …

Enter the image description here

Note that the existing post loop for the regular entire map grid (example on the left) is output as follows …

if ( $posts->have_posts() ) { ?>

    
    
have_posts() ) { $posts->the_post(); // Regular card echo '
'; set_query_var('bgcolor', 'bg-white'); set_query_var('use_logo', true); set_query_var('use_byline', true); set_query_var('use_source', false); get_template_part('pieces/cards/post', multi); echo '
'; } // end while wp_reset_query(); ?>
Nothing here.

The question is not just about changing the postcard CSS within the equal Grid (that's a challenge I've already mastered). Rather, I want to break out of the grid and reuse the alternative function block over the fold, which you see in purple.

This block is rendered with my function display_showcase_features()what as argument a WP_Query Object contains five posts and then outputs the corresponding surcharge using an alternative container grid and an alternative type of postcard.

What I have to achieve is …

  • At Post 13 in $ posts
  • break the divs
  • Submit a WP_Query object with the next five posts to display_showcase_features(), which prints the block with five posts in an alternative layout
  • Restore the opening divs of the row
  • Post 18 continued in the standard layout
  • the same on a regular basis.

I tried the following …

if ( $posts->have_posts() ) { ?>

  
  
have_posts() ) { $posts->the_post(); $i++; /** * ======================================================= * VARY POST CARDS TO BREAK UP LIST * ======================================================= */ if ($i == 13 or $i == 26) { ?>
'; echo '
'; ?> '; set_query_var('bgcolor', 'bg-white'); set_query_var('use_logo', true); set_query_var('use_byline', true); set_query_var('use_source', false); get_template_part('pieces/cards/post', multi); echo '
'; } } // end while wp_reset_query(); ?>
Nothing here.

The markup interruption works fine. It's the WordPress stuff I need to learn more about …

1. How do I get one WP_Query Object with the next five posts?

In the experiment above I send to display_showcase_features() just $showcase_features, which contains the same five posts that were used in the block above the fold. But that's really only for a test.

Instead, I think I need the next five actual posts in the standard layout loop below. Do I have to do a new query for an area here? Will that be expensive in the database?

2. How then do I elegantly continue to output the standard layout loop in the right place?

That is, in the above visual example on the right, Post 13 is actually NOT rendered with the code attempt immediately above it, because I am showing the showcase block here instead. The loop should resume five posts after pausing, provided display_showcase_features() has issued five posts.

3. How do I do this on a regular basis?

The pattern of the intended layout break can be predicted. ie. 12 posts in standard layout, followed by sending five to display_showcase_features(), then another 12.

Though I'm afraid that pagination could ruin things if, for example, it leaves my divs open.