My kids helped unpack groceries last night and left a bag of uncooked pork chops in the original packaging out overnight.  They were still in the shopping bag on the floor of the kitchen.  There was also ground beef that comes in the chub packaging left out.  The chubs were frozen when I bought them. When I put them away in the refrigerator, they were still partially frozen.  I am on a very limited budget and can't afford to replace these items.  Are these items safe to eat?
If I left food out of the refrigerator for some period of time, is it still safe? If I left it out too long, can I salvage it by cooking it more?
Computing the best constant in classical Hardy's inequality Classical Hardy's inequality (cfr. Hardy-Littlewood-Polya Inequalities, Theorem 327)  If $p&gt;1$, $f(x) \ge 0$ and $F(x)=\int_0^xf(y)\, dy$ then   $$\tag{H} \int_0^\infty \left(\frac{F(x)}{x}\right)^p\, dx &lt; C\int_0^\infty (f(x))^p\, dx $$  unless $f \equiv 0$. The best possibile constant is $C=\left(\frac{p}{p-1}\right)^p$.   I would like to prove the statement in italic regarding the best constant. As already noted by Will Jagy , the book suggests stress-testing the inequality with   $$f(x)=\begin{cases} 0 &amp; 0\le x &lt;1 \\ x^{-\alpha} &amp; 1\le x \end{cases}$$   with $1/p&lt; \alpha &lt; 1$, then have $\alpha \to 1/p$. If I do so I get for $C$ the lower bound  $$\operatorname{lim sup}_{\alpha \to 1/p}\frac{\alpha p -1}{(1-\alpha)^p}\int_1^\infty (x^{-\alpha}-x^{-1})^p\, dx\le C$$  but now I find myself in trouble in computing that lim sup. Can someone lend me a hand, please?    UPDATE: A first attempt, based on an idea by Davide Giraudo, unfortunately failed. Davide pointed out that the claim would easily follow from   $$\tag{!!} \left\lvert \int_1^\infty (x^{-\alpha}-x^{-1})^p\, dx - \int_1^\infty x^{-\alpha p }\, dx\right\rvert \to 0\quad \text{as}\ \alpha \to 1/p. $$  But this is false in general: for example if $p=2$ we get   $$\int_1^\infty (x^{-2\alpha} -x^{-2\alpha} + 2x^{-\alpha-1}-x^{-2})\, dx \to \int_1^\infty(2x^{-3/2}-x^{-2})\, dx \ne 0.$$ Algorithm to return all combinations of k elements from n I want to write a function that takes an array of letters as an argument and a number of those letters to select.   Say you provide an array of 8 letters and want to select 3 letters from that. Then you should get:  8! / ((8 - 3)! * 3!) = 56   Arrays (or words) in return consisting of 3 letters each. How do you convert a byte array to a hexadecimal string, and vice versa? Convert stored procedure result to model object Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD.   I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$ are all irreducible. Is there someway to conclude that $\mathbb{Z}[\sqrt{-n}]$ is not a UFD based on this? Thanks. Overheating results in system shutdown How to do a Point to Polygon analysis with multiple variables in QGIS? I have a database of almost 10k crimes, which are divided in to 10 categories and plotted on a map  I also have a database of 12k addresses, which are also plotted on the map.   I aim to perform a multiple regression to determine whether there are correlations between types of crime and house price.  I have created a buffer of 250 meters around each of the addresses, and would like to list a count of each of the crimes which are located within this buffered zone for each of the addresses, however, I can only get QGIS to give me a count of all types of crime, rather than the specifics.   Does anyone know how to do a point to polygon analysis of this type? I am in the middle of breaking the crime databases to analyse each one separately, which is a bit difficult! I am trying to prove that the set of irrational numbers $\mathbb{R} \setminus \mathbb{Q}$ is not an $F_{\sigma}$ set. Here's my attempt:  Assume that indeed $\mathbb{R} \setminus \mathbb{Q}$ is an $F_{\sigma}$ set. Then we may write it as a countable union of closed subsets $C_i$: $$ \mathbb{R} \setminus \mathbb{Q} = \bigcup_{i=1}^{\infty}  \ C_i $$ But $\text{int} ( \mathbb{R} \setminus \mathbb{Q}) = \emptyset$, so in fact each $C_i$ has empty interior as well. But then each $C_i$ is nowhere dense, hence $ \mathbb{R} \setminus \mathbb{Q} = \bigcup_{i=1}^{\infty}  \ C_i$ is thin. But we know $\mathbb{R} \setminus \mathbb{Q}$ is thick, a contradiction.  This seems a bit too simple. I looked this up online, and although I haven't found the solution anywhere, many times there is a hint: Use Baire's Theorem. Have I skipped an important step I should explain further or is Baire's Theorem used implicitly in my proof? Or is my proof wrong? Thanks.  EDIT: Thin and thick might not be the most standard terms so:  Thin = meager = 1st Baire category When should I use exposure compensation? When should I use exposure compensation, rather than ISO, shutter speed or aperture? What does outputting to /dev/null accomplish in bash scripts? Create any number of paragraph fields in a multiple value field on node create form I have a paragraphs field that takes a term reference.  I need to count how many terms there are for a specific vocabulary and then create that many instance of the paragraphs multiple value field for when creating a node of a certain type. Can this be done with Hook_form_alter ?  It is important that these fields are loaded when the node creation form is loaded rather than the user having to click add more  Any help would be greatly appreciated  I have tried this but when saving the node it doesn't save the new fields  function hook_form_alter(&amp;$form, $form_state, $form_id) {    if ($form_id == 'my_form') {    $item = $form['field_features']['widget']['0'];    // get taxonomy terms   $query = \Drupal::entityQuery('taxonomy_term');   $query-&gt;condition('vid', "my_voc");   $tids = $query-&gt;execute();   $terms = \Drupal\taxonomy\Entity\Term::loadMultiple($tids);    $count = count($terms);    $i = '1';   foreach($terms as $term){     $form['field_features_para']['widget'][$i] = $item;     $form['field_features_para']['widget'][$i]['#delta'] = $i;     $form['field_features_para']['widget'][$i]['#weight'] = $i;     $i ++;   }  } } What is "preimage resistance", and how can the lack thereof be exploited? How is this different from collision resistance, and are there any known preimage attacks that would be considered feasible? What's the hardest or most misunderstood aspect of LINQ? Prove that for any square matrix, an invertible matrix B exists, so that BA is triangular I'm given a matrix A, its dimensions are n x n.  I am required to prove that an invertible matrix B exists, such that the product of the matrices BA is triangular.  Any help? Multiple forms in delphi Change Legend Right to Left Reading in QGIS I am a new QGIS user and i have a challenge to convert legend from right to left reading in QGIS like ArcGIS. Does Lin's  assume that the 2 data sets have linear or monotonic tendencies? Or, can I measure the concordance between 2 data sets that have a sinusoidal tendency? Resizing point on zoom out I've added a point on map like this but it is static in size.   How to resize that point on zooming?  Map on zoom out Deserialize JSON object into dynamic object using Json.net How to wait in bash for several subprocesses to finish and return exit code !=0 when any subprocess ends with code !=0?