Jetpack Infinite Scroll problem

I can’t seem to get Infinite scrolling working on a custom archive page. It seems to me it’s not picking up the arguments from the add_theme_support call. Here’s the code:

/**     * Add theme support for infinite scroll.     *     * @uses add_theme_support     * @return void     */     function twenty_twelve_infinite_scroll_init() {     add_theme_support( 'infinite-scroll', array(         'container'      => 'catalogue-section',         'render'         => 'my_child_infinite_scroll_render',         'posts_per_page' => get_option( 'posts_per_page' ),         'wrapper' => false,         'type' => 'scroll',     ) );          }      add_action( 'after_setup_theme', 'twenty_twelve_infinite_scroll_init' );      function my_child_infinite_scroll_render() {      while ( have_posts() ) : the_post();          // Format of a catalogue entry is defined in catalogue-entry.php         get_template_part( 'catalogue-entry', get_post_type() );      endwhile; } 

The layout of the page shows entries from the initial custom posts correctly. However, when the second set of posts is called, it is not formatted correctly, as it’s no longer sitting inside the section that is targetted in my CSS to have multiple columns.

See the result here:

I can see that Infinite Scroll is adding a div around the rendered posts even though in the add theme support call I set wrapper to false:

<div class="infinite-wrap infinite-view-2" id="infinite-view-2" data-page-num="2" role="region" aria-label="Page: 2."> 

Also it wasn’t picking up my rendering call, so I had to update the default content.php to get the correct format of the entries. I wonder can anyone see why the add_theme_support parameters for infinite scrolling are not being picked up?

Infinite summation giving wierd results

We are searching in our group for closed forms of derivatives of hypergeometric functions. This leads to expressions like

$ \sum\limits_{m=2}^\infty \frac{z^m\Gamma[m-1/2]H_m}{2m^2\sqrt{\pi}\Gamma[m]}$

where $ H_m$ denotes the m-th harmonic number. Now trying to evaluate this in Mathematica 12.0 ( or 12.1.1) using

Sum[(z^m*Gamma[-(1/2) + m]*HarmonicNumber[m])/(2*m^2*Sqrt[Pi]*Gamma[m]), {m, 2, Infinity}]

returns 0. But in this case we actually know a rather complicated closed form expression for this sum in terms of logs and polylogs which are non-vanishing. Moreover, taking the case z=1, Mathematica 12.0 evaluates the sum correctly, i.e.

Sum[Gamma[-(1/2) + m]*HarmonicNumber[m])/(2*m^2*Sqrt[Pi]*Gamma[m]), {m, 2, Infinity}]

returns $ \frac{7 \sqrt{\pi }-\frac{2 \pi ^{5/2}}{3}}{2 \sqrt{\pi }}$ which is correct and non-zero. Thus the result form the original command seams to be wrong. Are we missing something? Is there a way to prevent these wrong evaluations? We would like to use Mathematica to compute some series with a priori unknown closed forms and that behaviour is worrying us.

Does a spell cast from a Glyph of Warding with a range of Self have infinite effective range?

It is widely agreed that spells with a range of "Self" can be stored in a glyph of warding. The top answer to "What are the targeting range limitations of Glyph of Warding?" states:

That’s it: the spell is cast with all its normal statistics including range.

While the trigger of the glyph can be unlimited in range ("Trigger when I move 12,000 miles away") the spell that is cast is cast from the gylph with all its normal limitations.

Spells with a range of "Self" do not have a numerical range limit on them, and as stated above, there is no range limit on the trigger either. So, using glyph of warding, could I effectively trigger a "Self" range spell from any distance?

I cast fire shield into a glyph of warding in my home with the trigger "When I speak the command word ‘flame on’".

Could I then speak the command word to have fire shield cast on myself when I am 100 miles away in a dungeon, or on another plane?

Every decidable lanugage $L$ has an infinite decidable subset $S \subset L$ such that $L \setminus S$ is infinite

Given an infinite decidable language $ L$ , then if $ S \subset L$ such that $ L \setminus S$ is finite, then $ S$ must be decidable. This is true since given a decider of $ L$ we contruct a decider for $ S$ :

Simulate the decider of $ L$ on the input, if it accepts, go over $ L \setminus S$ and check if it is there, if it is, reject. If it isn’t accept. If the decider of $ L$ rejects – reject.

Another point is if $ S \subset L$ is finite then $ S$ also must be decidable, this is immediate that every finite language is decidable.

Now we have the last case where $ S$ is infinite and $ L \setminus S$ is infinite. We know that there must be some subsets $ S$ corresponding to this case that are undecidable. This is since there are $ \aleph$ such $ S$ but only $ \aleph_0$ deciders. Denote $ D(L) = \{ S \subset L : |S|= |L \setminus S|=\infty \wedge S \text{ is decidable} \}$

Is it true that for all infinite decidable languages $ L$ we have $ D(L) \neq \phi$ ?

If this is true then as a conclusion we will have for all infinite decidable languages $ L$ a sequence of decidable languages $ L_n$ such that $ L_0=L$ and $ L_{n+1} \subset L_n$ and $ |L_n \setminus L_{n+1}| = \infty$

We will also have a limit-set $ L_\infty = \{ e \in L : \forall n \in \mathbb{N} \text{ } e \in L_n \}$ and can dicuss if it is empty/finite/infinite and decicable or not.

This seems like a nice way to study decidable languages, and curious to know if this direction is indeed interesting and whether there are articles published regarding these questions

Thanks for any help

Show that exist a finite set of clauses F in first-order logic that Res*(F) is infinite

I’m kind of desperate at this point about this question.

A predicate-logic resolution derivation of a clause $ C$ from a set of clauses $ F$ is a sequence of clauses $ C_1,\dots,C_m$ , with $ C_m = C$ such that each $ C_i$ is either a clause of $ F$ (possibly with the variables renamed) or follows by a resolution step from two preceding clauses $ C_j ,C_k$ , with $ j, k < i$ . We write $ \operatorname{Res}^*(F)$ for the set of clauses $ C$ such that there is a derivation of $ C$ from $ F$ .

The question is to give an example of a finite set of clauses $ F$ in first-order logic such that $ \operatorname{Res}^*(F)$ is infinite.

Any help would be appreciated!

Show that infinite decidable sets $A$ and $B$ exist

I am stuck in this problem, so any help is appreciated. The problem asks to show that there exists decidable sets $ A$ and $ B$ such that $ A \leq_{m}^{p} B$ but $ B \not \leq_{m}^{p} A$ , and that $ A$ , $ B$ and $ \bar{A}$ and $ \bar{B}$ are infinite.

Here, $ \leq_{m}^{p}$ refers to many-one polynomial time reducibility…. I have a hunch that this may have something to do with letting $ A$ be a decidable set such that $ A \in EXP$ , but $ B \in P$ , so that the reduction cannot be done in polynomial time.

Can one determinize finite automata over infinite trees?

I’m currently considering deterministic, nondeterministic, universal, and alternating automata over infinite words and trees, with Büchi, co-Büchi, Muller, Rabin, Streett, or parity acceptance conditions.

I know that over words, all these automata accept the same languages, except deterministic and universal Büchi automata, as well as deterministic and nondeterministic co-Büchi automata.

I also know that over trees, nondeterministic and alternating Muller, Rabin, Streett, and parity automata accept the same languages, and strictly more languages than nondeterministic and alternating Büchi automata.

But I hardly know anything about deterministic tree automata, and I’m having a hard time finding more in the literature. And this captures my main question, whether one can determinize nondeterministic automata over infinite trees.

After all, they are definitely used. For example, in reactive LTL synthesis, we often convert the formula to a nondeterministic Büchi word automaton, then to a deterministic parity word automaton, from which we derive a deterministic parity tree automaton.

Also note that I didn’t mention the co-Büchi case when it comes to trees. I just know that, again in reactive LTL synthesis, we can alternatively convert the negated formula to a nondeterministic Büchi word automaton, then to a universal co-Büchi word automaton for the original formula, from which we can derive a universal co-Büchi tree automaton. But that alone doesn’t really say anything about the relation of deterministic parity tree automata and universal co-Büchi tree automata, or does it?

Find an infinite set of strings that are compressable better than in $O(\log n)$ space

The task is to find an infinite set of strings $ a_1,a_2\ldots$ , where $ |a_{i+1}|>|a_i|$ and to find a compression algorithm $ f$ for these strings, such that $ |f(a_i)|=o(\log_2 |a_i|)$ with $ i\to\infty$ .

I have considered a set of strings with minimal entropy: $ a_1=b, a_2=bb$ , etc. The optimal compression algorithm for these strings is $ f : a_i\mapsto |a_i|$ , but this compresses each string only to $ O(\log i)$ space.