Iterate cartesian product (cross product) of sorted lists of positive numbers in order of decreasing product of entries (MVCE of a stub included)

Suppose I have several sorted lists of positive numbers, like so for example:

double[] a1 = new double[]{0.70, 0.20, 0.10}; double[] a2 = new double[]{0.80, 0.10, 0.05, 0.05}; double[] a3 = new double[]{0.60, 0.15, 0.14, 0.10, 0.01}; 

I want to iterate Cartesian product of these arrays in the order of decreasing product of entries, like so:

0000: Combo[product=3.36e-01, vals=[0.70, 0.80, 0.60], indexes=[0, 0, 0]] 0001: Combo[product=9.60e-02, vals=[0.20, 0.80, 0.60], indexes=[1, 0, 0]] 0002: Combo[product=8.40e-02, vals=[0.70, 0.80, 0.15], indexes=[0, 0, 1]] 0003: Combo[product=7.84e-02, vals=[0.70, 0.80, 0.14], indexes=[0, 0, 2]] 0004: Combo[product=5.60e-02, vals=[0.70, 0.80, 0.10], indexes=[0, 0, 3]] 0005: Combo[product=4.80e-02, vals=[0.10, 0.80, 0.60], indexes=[2, 0, 0]] ... 

E.g. in the example above the first entry is obvious (as arrays are sorted) and it is a combination of the first values: [0.70, 0.80, 0.60] with product 0.70*0.80*0.60 = 3.36e-01 and corresponding value indexes in arrays a1, a2, a3 are [0, 0, 0]. Now the second entry is less obvious, should we change 0.70 to 0.20? Or 0.60 to 0.15? Or 0.80 to 0.10? The second should be [0.20, 0.80, 0.60]with product 9.60e-02, indexes [1, 0, 0].

Here is a program in java to generate/print them: (all the logic is in printWholeCartesianProduct() method)
This program generates them in lexicographic order and then sorts the whole set by product.

Question: Is there an easy way to actually generate the combos in correct order in the first place?

The reason for this: I don’t have the lists in the first place, only iterators over some sorted collections of numbers. Possibly veeery long, length not known ahead of time, but it is known that the numbers in each iterator are sorted.

MVCE to play with (same as in link above):

import java.text.DecimalFormat; import java.util.ArrayList; import java.util.Arrays; import java.util.List; import java.util.StringJoiner; import java.util.function.Consumer; import;  public class Main {      public static void main(String[] args) {         List<List<Double>> data = createData();         printWholeCartesianProduct(data);     }      public static List<List<Double>> createData() {         double[] a1 = new double[]{0.70, 0.20, 0.10};         double[] a2 = new double[]{0.80, 0.10, 0.05, 0.05};         double[] a3 = new double[]{0.60, 0.15, 0.14, 0.10, 0.01};         return createData(a1, a2, a3);     }      public static void  printWholeCartesianProduct(List<List<Double>> data) {         final DecimalFormat df = new DecimalFormat("0.00");          // print input data         String matrix =             .map(l ->", ")))             .map(row -> "[" + row + "]")             .collect(Collectors.joining("\n"));         System.out.println("Input data:\n" + matrix);          // collect combos as they are generated         final List<Combo> combos = new ArrayList<>();         Consumer<int[]> callback = indexes -> {             double[] v = new double[indexes.length];             double prod = 1;             for (int i = 0; i < indexes.length; i++) {                 List<Double> col = data.get(i);                 int index = indexes[i];                 v[i] = col.get(index);                 prod *= v[i];             }             combos.add(new Combo(prod, v, indexes.clone()));         };          // generate combos         int[] c = new int[data.size()];         int ptr = c.length - 1;         while (ptr >= 0) {             callback.accept(c);             c[ptr]++; // increment             if (c[ptr] == data.get(ptr).size()) { // carry                 do {                     ptr--;                 } while(ptr >= 0 && c[ptr] == data.get(ptr).size() - 1);                 if (ptr < 0) {                     break;                 }                 c[ptr]++;                 // zero out                 while (++ptr <= c.length - 1) {                     c[ptr] = 0;                 }                 ptr = c.length - 1;             }         }          // cheating - sort after generation and print result         combos.sort((o1, o2) ->, o1.product));         StringBuilder sb = new StringBuilder();         double totalP = 0;         for (int i = 0; i < combos.size(); i++) {             sb.append(String.format("%04d: ", i)).append(combos.get(i)).append("\n");             totalP += combos.get(i).product;         }         System.out.printf("Cartesian product in descending product (total p=%.3e):\n%s", totalP, sb.toString());     }      public static List<Double> asList(double[] a) {         return;     }      public static List<List<Double>> createData(double[]... arrays) {         final List<List<Double>> vals = new ArrayList<>(); -> vals.add(asList(a)));         return vals;     }      static class Combo {         final double product;         final double[] vals;         final int[] indexes;          Combo(double product, double[] vals, int[] indexes) {             this.product = product;             this.vals = vals;             this.indexes = indexes;         }          @Override         public String toString() {             return new StringJoiner(", ", Combo.class.getSimpleName() + "[", "]")                 .add("product=" + String.format("%.2e", product))                 .add("vals=[" + -> String.format("%.2f", v)).collect(                     Collectors.joining(", ")) + "]")                 .add("indexes=" + Arrays.toString(indexes))                 .toString();         }     } } ``` 

Apps loaded in App Catalog do not have the correct App Version or Product ID

I’m working with a client who just set up a new SharePoint 2016 site on their server. They created an App Catalog but when I load in a custom SharePoint App, the App Catalog doesn’t read in the version from the AppManifest.XML and sets the version to nor is the app assigned a Product ID – the Product ID is left blank.

The same App is uploaded with the correct metadata on other sites.

Is there something that needs to be configured so the App Catalog library behaves like an App Catalog?

Thanks in advance

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Is {a^n: n is a product of exactly two primes} regular?

I am struggling to prove the following question.

$ L_1 = \{a^n: n \text{ is a product of exactly two primes}\}$

I feel like the language is not regular but I am having trouble proving it. I tried using pumping lemma but got stuck at the end. Here’s how I did it:

Assume that the language is regular and $ m$ is a constant of Pumping Lemma. Now let $ w = a^M$ where $ M > m$ and $ M$ is a co-prime number. Clearly $ w$ is in the language and $ |w| > m$ .

Now let $ y=a^j$ where $ j$ is between $ 1$ and $ m$ , with $ |xy| \leq m$ and $ |y| \geq 1$ .

This is where I am getting stuck. I feel like we should pump up but I don’t know by “how much”. Also, I feel like I have to know what is the next co-prime number after $ M$ , but can’t figure it out.

Translating print product catalog (with 5000 products) to digital with good UX: how?

I’m currently working at a large B2B company which operates in a very traditional market (ordering happens mainly by phone, e-mail and even fax).

Since many years we are using a product catalog (print) with over 5000 products in it, but since it costs a lot of money to print and send it to our customers, we slowly want to move to digital.

The big question is: how can we translate this print manual to a digital version?

There are many PDF e-publishing tools, but that doesn’t really work with a 500 page catalog.

There are some examples like IKEA (where you can browse through pages online and download a pdf version):

What are best practices here? What is the purpose of an online version? For reading? For buying (should we link products to a webshop)?

Another question: could a digital product catalog replace a printed version when customers are still ordering by phone / e-mail?

Many thanks in advance, looking forward to your reply.