Cubic space reduction variation of PSPACE-COMPLETE(Theoretical)

i’ve been wondering:

if we change the definition of a PSPACE-COMPLETE definition to the following: A language B will be called PSPACE-COMPLETE if:

  • for each language A in PSPACE: $ A \leq _{CS} B$
  • language B belongs to PSPACE

are there any PSPACE-COMPLETE languages by that definiton?

(note: if a language A can be cubical space reducted to language B, meaning that if there exists a reduction mapping of A to B that can be calculated in a cubical space, we’ll denote $ A \leq _{CS} B$ , meaning that if there’s a determinstic turing machine M, with one tape, that upon input a word w, it uses $ O(|w|^3)$ space and upon completion on its tape written the word f(w)).

it is quite confusing for me, since a pspace-complete is a problem that can be solved using a polynomial space in relation to the input, and that every problem that can be solved in polynomial space be mapped(transformed) to it, but i am really not sure because of the $ O(|w|^3)$ requirement.

would really appreciate your opinion about it

How do you empirically estimate the most popular seat and get an upper bound on total variation?

Say there are $ n$ seats $ \{s_1, …, s_n\}$ in a theater and the theater wants to know which seat is the most popular. They allow $ 1$ person in for $ m$ nights in a row. For all $ m$ nights, they record which seat is occupied.

They are able to calculate probabilities for whether or not a seat will be occupied using empirical estimation: $ P(s_i ~\text{is occuped})= \frac{\# ~\text{of times} ~s_i~ \text{is occupied }}{m}$ . With this, we have an empirical distribution $ \hat{\mathcal{D}}$ which maximizes the likelihood of our observed data drawn from the true distribution $ \mathcal{D}$ . This much I understand! But, I’m totally lost trying to make this more rigorous.

  • What is the upper bound on $ \text{E} ~[d_{TV}(\hat{\mathcal{D}}, \mathcal{D})]$ ? Why? Note: $ d_{TV}(\mathcal{P}, \mathcal{Q})$ is the total variation distance between distributions $ \mathcal{P}$ and $ \mathcal{Q}$ .
  • What does $ m$ need to be such that $ \hat{\mathcal{D}}$ is accurate to some $ \epsilon$ ? Why?
  • How does this generalize if the theater allows $ k$ people in each night (instead of $ 1$ person)?
  • Is empirical estimation the best approach? If not, what is?

If this is too much to ask in a question, let me know. Any reference to a textbook which will help answer these questions will happily be accepted as well.

WordPress query through Products variation stock

I have WooCommerce variable products and a filter which is filtering products based on variation attributes. I have variation attribute depth which has numeric value (from 1 to 20) and the filter is working fine. But I want to display In Stock variations only On the other hand it is displaying all products including out of stock selected depth value (variation). So I want to hide product which will have selected depth value but no stock.

Can I query through Products variation stock ?

Here how it is working right now.

$  args = array( 'post_type' => array('product'),         'meta_query' => array(             array(                 'key' => '_stock_status',                 'value' => 'instock',                 'compare' => '=',             ),         ),          'tax_query' => array(             array(                 'taxonomy' => 'pa_depth',                 'field' => 'slug',                 'terms' => $  wccaf_depth,                 'operator'  => $  wccaf_depth ? 'IN' : 'NOT IN'             )         ),                   ); 

Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach

Let $ P\rightarrow M$ be a principal fibre bundle with structure group $ G$ , $ F$ a manifold and $ \alpha: G\times F\rightarrow F$ a smooth left action.

There is an associated fibre bundle $ E\rightarrow M$ with $ E=P\times_\alpha F=(P\times F)/G$ .

As it is well known, one may either treat sections of the associated fibre bundle “directly”, or consider maps $ \psi:P\rightarrow F$ which satisfy the equivariance property $ \psi(pg)=g^{-1}\cdot\psi(p)$ , where $ \cdot$ denotes the left action. Let us refer to this latter method as the “equivariant bundle approach”.

I am interested in describing the gauge field theories of physics using global language with appropriate rigour. However, most references I know treat this topic using the “direct approach”, and not with the equivariant approach, the chief exception being Gauge Theory and Variational Principles by David Bleecker.

Bleecker’s book however doesn’t go far enough for my present needs.

  • Bleecker only uses linear matter fields, eg. the case where $ F$ is a vector space and $ \alpha$ is a linear representation. Some things are easy to generalize, others appear to be highly nontrivial to me.
  • Bleecker treats only first-order Lagrangians. The connection between a higher-order variational calculus based on the equivariant bundle approach and between the more “standard” one built on the jet manifolds $ J^k(E)$ of the associated bundle is highly unclear to me. Example: If $ \bar\psi:M\rightarrow E$ is a section of an associated bundle, its $ k$ -th order behaviour is represented by the jet prolongation $ j^k\bar\psi:M\rightarrow J^k(E)$ , but if instead I use the equivariant map $ \psi:P\rightarrow F$ , what represents its $ k$ -th order behaviour? I assume it is related to something like $ J^k(P\times F)/G$ , but the specifics are unclear to me.

  • In Bleecker’s approach, connections are $ \mathfrak g$ -valued, $ \text{Ad}$ -equivariant 1-forms on $ P$ , however I am interested in treating them on the same footing as matter fields. Connections however are higher order associated objects in the sense that they are associated to $ J^1P$ . Bleecker absolutely doesn’t treat higher order principal bundles.

In short, I am interested in references that consider gauge theories, gauge natural bundles, including nonlinear and higher-order associated bundles and calculus of variations/Lagrangian field theory from the point of view where fields are fixed space-valued objects defined on the principal bundle (equivariant bundle approach), rather than using associated bundles directly.

Configurable variation is unidentifiable – Magento 2.3.1

I’m getting the Configurable variation is unidentifiable error when trying to import the configurable items to Magento 2.3.1.

I think I have the correct format for the csv, using the defined format,

configurable_variations sku=WAL-WM-03-BLA-UNI,color=Negro,size=UNI|sku=WAL-WM-03-GOL-UNI,color=Dorado,size=UNI|  configurable_variation_labels color=Color,size=Size 

Any ideas?

Decomposition of $Z:\mathbb{R}_+\rightarrow\mathbb{R}$ of local bounded variation.

Let $ Z:\mathbb{R}_+\rightarrow\mathbb{R}$ be cadlag and of local bounded variation with $ Z(0)=0$ and $ V_Z(t)$ denotes the value of the total variation of $ Z$ on $ [0,t]$ for all $ t\in\mathbb{R}_+$ . I want to show that there is a unique decomposition $ Z=Z_1-Z_2$ of monotonically increasing and right-continuous functions $ Z_1,Z_2:\mathbb{R}_+\rightarrow \mathbb{R}_+$ , such that $ Z_1(0)=0=Z_2(0)$ and $ V_Z=Z_1+Z_2$ , more precisely

$ $ Z_1=\frac{V_Z+Z}{2},\quad Z_2=\frac{V_Z+Z}{2}-Z.$ $

I know that $ V_Z-Z$ is monotonically increasing and right-continuous. Therefore it holds also for

$ $ Z_2=\frac{V_Z-Z}{2}$ $

But I do not know how to show monotonicity and right-continuity for $ Z_1$ . Also I do not know how to show uniqueness. I would appreciate any help. Thanks in advance!