Updating GitHub Pages

CBOR/docs/PeterO.Numbers.EInteger.html

@@ -1331,7 +1331,7 @@ <h3>shiftLeft</h3>
     int numberBits);
 </code></pre>
 
-<p>Returns a big integer with the bits shifted to the left by a number of bits. A value of 1 doubles this value, a value of 2 multiplies it by 4, a value of 3 by 8, a value of 4 by 16, and so on.</p>
+<p>Returns a big integer with the bits shifted to the left by a number of bits. A value of 1 doubles this value, a value of 2 multiplies it by 4, a value of 3 &times; by, a value of 4 &times; by, and so on.</p>
 
 <p><b>Parameters:</b></p>
 

CBOR/index.html

@@ -136,7 +136,7 @@ <h2 id="examples">Examples</h2>
  using (var stream = new MemoryStream(byteArray)) {
     // Read the CBOR object from the stream
     var cbor = CBORObject.Read(stream);
-    // The rest of the example follows the one given above.
+    // The rest of the example follows the one given earlier.
  }
 </code></p>
 

Encoding/index.html

@@ -36,7 +36,7 @@ <h2 id="how-to-install">How to Install</h2>
 </pre>
 
 <p>In other Java-based environments, the library can be referred to by its
-group ID (<code>com.github.peteroupc</code>), artifact ID (<code>encoding</code>), and version, as given above.</p>
+group ID (<code>com.github.peteroupc</code>), artifact ID (<code>encoding</code>), and version, as given earlier.</p>
 
 <h2 id="documentation">Documentation</h2>
 

MailLib/docs/PeterO.Mail.NamedAddress.html

@@ -60,7 +60,7 @@ <h3 id="member-summary">Member Summary</h3>
 <p><b>Parameters:</b></p>
 
 <ul>
-  <li><i>address</i>: A text string identifying a single email address or a group of email addresses. Comments, or text within parentheses, can appear. Multiple email addresses are not allowed unless they appear in the group syntax given above. Encoded words under RFC 2047 that appear within comments or display names will be decoded. An RFC 2047 encoded word consists of “=?”, a character encoding name, such as  <code>utf-8</code> , either “?B?” or “?Q?” (in upper or lower case), a series of bytes in the character encoding, further encoded using B or Q encoding, and finally “?=”. B encoding uses Base64, while in Q encoding, spaces are changed to “_”, equals are changed to “=3D”, and most bytes other than the basic digits 0 to 9 (0x30 to 0x39) and the basic letters A/a to Z/z (0x41 to 0x5a, 0x61 to 0x7a) are changed to “=” followed by their 2-digit hexadecimal form. An encoded word’s maximum length is 75 characters. See the third example.</li>
+  <li><i>address</i>: A text string identifying a single email address or a group of email addresses. Comments, or text within parentheses, can appear. Multiple email addresses are not allowed unless they appear in the group syntax given earlier. Encoded words under RFC 2047 that appear within comments or display names will be decoded. An RFC 2047 encoded word consists of “=?”, a character encoding name, such as  <code>utf-8</code> , either “?B?” or “?Q?” (in upper or lower case), a series of bytes in the character encoding, further encoded using B or Q encoding, and finally “?=”. B encoding uses Base64, while in Q encoding, spaces are changed to “_”, equals are changed to “=3D”, and most bytes other than the basic digits 0 to 9 (0x30 to 0x39) and the basic letters A/a to Z/z (0x41 to 0x5a, 0x61 to 0x7a) are changed to “=” followed by their 2-digit hexadecimal form. An encoded word’s maximum length is 75 characters. See the third example.</li>
 </ul>
 
 <p>.</p>

MailLib/index.html

@@ -42,7 +42,7 @@ <h2 id="how-to-install">How to Install</h2>
 </pre>
 
 <p>In other Java-based environments, the library can be referred to by its
-group ID (<code>com.upokecenter</code>), artifact ID (<code>maillib</code>), and version, as given above.</p>
+group ID (<code>com.upokecenter</code>), artifact ID (<code>maillib</code>), and version, as given earlier.</p>
 
 <h2 id="source-code">Source Code</h2>
 <p>Source code is available in the <a href="https://github.com/peteroupc/MailLib">project page</a>.</p>

Numbers/api/com.upokecenter.numbers.EInteger.html

@@ -2543,7 +2543,7 @@ <h3 id="shiftright">ShiftRight</h3>
 </ul>
 
 <h3 id="shiftleft">ShiftLeft</h3>
-<pre>public EInteger ShiftLeft(EInteger eshift) Returns an arbitrary-precision integer with the bits shifted to the left by  a number of bits given as an arbitrary-precision integer. A value of 1  doubles this value, a value of 2 multiplies it by 4, a value of 3 by 8, a  value of 4 by 16, and so on.
+<pre>public EInteger ShiftLeft(EInteger eshift) Returns an arbitrary-precision integer with the bits shifted to the left by  a number of bits given as an arbitrary-precision integer. A value of 1  doubles this value, a value of 2 multiplies it by 4, a value of 3 &amp;times; by, a  value of 4 &amp;times; by, and so on.
 </pre>
 
 <p><strong>Parameters:</strong></p>
@@ -2566,7 +2566,7 @@ <h3 id="shiftleft">ShiftLeft</h3>
 </ul>
 
 <h3 id="shiftleft-1">ShiftLeft</h3>
-<pre>public EInteger ShiftLeft(int numberBits) Returns an arbitrary-precision integer with the bits shifted to the left by  a number of bits. A value of 1 doubles this value, a value of 2 multiplies  it by 4, a value of 3 by 8, a value of 4 by 16, and so on.
+<pre>public EInteger ShiftLeft(int numberBits) Returns an arbitrary-precision integer with the bits shifted to the left by  a number of bits. A value of 1 doubles this value, a value of 2 multiplies  it by 4, a value of 3 &amp;times; by, a value of 4 &amp;times; by, and so on.
 </pre>
 
 <p><strong>Parameters:</strong></p>

Numbers/docs/PeterO.Numbers.EInteger.html

@@ -3312,7 +3312,7 @@ <h3 id="member-summary">Member Summary</h3>
     int bitCount);
 </pre>
 
-<p>Returns an arbitrary-precision integer with the bits shifted to the left by a number of bits. A value of 1 doubles this value, a value of 2 multiplies it by 4, a value of 3 by 8, a value of 4 by 16, and so on.</p>
+<p>Returns an arbitrary-precision integer with the bits shifted to the left by a number of bits. A value of 1 doubles this value, a value of 2 multiplies it by 4, a value of 3 × by, a value of 4 × by, and so on.</p>
 
 <p><b>Parameters:</b></p>
 
@@ -3978,7 +3978,7 @@ <h3 id="member-summary">Member Summary</h3>
     int numberBits);
 </pre>
 
-<p>Returns an arbitrary-precision integer with the bits shifted to the left by a number of bits. A value of 1 doubles this value, a value of 2 multiplies it by 4, a value of 3 by 8, a value of 4 by 16, and so on.</p>
+<p>Returns an arbitrary-precision integer with the bits shifted to the left by a number of bits. A value of 1 doubles this value, a value of 2 multiplies it by 4, a value of 3 × by, a value of 4 × by, and so on.</p>
 
 <p><b>Parameters:</b></p>
 
@@ -3997,7 +3997,7 @@ <h3 id="member-summary">Member Summary</h3>
     PeterO.Numbers.EInteger eshift);
 </pre>
 
-<p>Returns an arbitrary-precision integer with the bits shifted to the left by a number of bits given as an arbitrary-precision integer. A value of 1 doubles this value, a value of 2 multiplies it by 4, a value of 3 by 8, a value of 4 by 16, and so on.</p>
+<p>Returns an arbitrary-precision integer with the bits shifted to the left by a number of bits given as an arbitrary-precision integer. A value of 1 doubles this value, a value of 2 multiplies it by 4, a value of 3 × by, a value of 4 × by, and so on.</p>
 
 <p><b>Parameters:</b></p>
 

bernapprox.html

@@ -1,4 +1,4 @@
-<!DOCTYPE html><html xmlns:dc="http://purl.org/dc/terms/" itemscope itemtype="http://schema.org/Article"><head><meta http-equiv=Content-Type content="text/html; charset=utf-8"><meta name="citation_pdf_url" content="https://peteroupc.github.io/bernapprox.pdf"><meta name="citation_url" content="https://peteroupc.github.io/bernapprox.html"><meta name="citation_date" content="2024/08/22"><meta name="citation_online_date" content="2024/08/22"><meta name="og:type" content="article"><meta name="og:url" content="https://peteroupc.github.io/bernapprox.html"><meta name="og:site_name" content="peteroupc.github.io"><meta name="author" content="Peter Occil"/><meta name="citation_author" content="Peter Occil"/><meta name="viewport" content="width=device-width"><link rel=stylesheet type="text/css" href="/style.css">
+<!DOCTYPE html><html xmlns:dc="http://purl.org/dc/terms/" itemscope itemtype="http://schema.org/Article"><head><meta http-equiv=Content-Type content="text/html; charset=utf-8"><meta name="citation_pdf_url" content="https://peteroupc.github.io/bernapprox.pdf"><meta name="citation_url" content="https://peteroupc.github.io/bernapprox.html"><meta name="citation_date" content="2024/12/07"><meta name="citation_online_date" content="2024/12/07"><meta name="og:type" content="article"><meta name="og:url" content="https://peteroupc.github.io/bernapprox.html"><meta name="og:site_name" content="peteroupc.github.io"><meta name="author" content="Peter Occil"/><meta name="citation_author" content="Peter Occil"/><meta name="viewport" content="width=device-width"><link rel=stylesheet type="text/css" href="/style.css">
             <script type="text/x-mathjax-config"> MathJax.Hub.Config({"HTML-CSS": { availableFonts: ["STIX","TeX"], linebreaks: { automatic:true }, preferredFont: "TeX" },
                     tex2jax: { displayMath: [ ["$$","$$"], ["\\[", "\\]"] ], inlineMath: [ ["$", "$"], ["\\\\(","\\\\)"] ], processEscapes: true } });
             </script><script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-AMS_HTML-full"></script></head><body>  <div class="header">
@@ -157,7 +157,7 @@ <h3 id="approximations-on-the-closed-unit-interval">Approximations on the Closed
   <li>each of the polynomial’s Bernstein coefficients is not less than 0 or greater than 1 (assuming none of $f$’s values is less than 0 or greater than 1).</li>
 </ol>
 
-<p>For some of the polynomials given above, a degree $n$ can be found so that the degree-$n$ polynomial is within $\epsilon$ of $f$, if $f$ is continuous and meets other conditions.  In general, to find the degree $n$, solve the error bound’s equation for $n$ and round the solution up to the nearest integer.  See the table below, where:</p>
+<p>For some of the polynomials given earlier, a degree $n$ can be found so that the degree-$n$ polynomial is within $\epsilon$ of $f$, if $f$ is continuous and meets other conditions.  In general, to find the degree $n$, solve the error bound’s equation for $n$ and round the solution up to the nearest integer.  See the table below, where:</p>
 
 <ul>
   <li>$M_r$ is not less than the maximum of the absolute value of $f$’s $r$-th derivative.</li>
@@ -329,7 +329,7 @@ <h3 id="taylor-polynomials-for-smooth-functions">Taylor Polynomials for “Smoot
   <li>$f(0)$ is known as well as $f^{(1)}(0), …, f^{(n)}(0)$.</li>
 </ol>
 
-<p>Then the $n$-th <em>Taylor polynomial</em> centered at 0, given below, is within $\epsilon$ of $f$:</p>
+<p>Then the $n$-th <em>Taylor polynomial</em> centered at 0, given later, is within $\epsilon$ of $f$:</p>
 
 \[P(\lambda) = a_0 \lambda^0 + a_1 \lambda^1 + ... + a_n \lambda^n,\]
 
@@ -879,8 +879,8 @@ <h3 id="chebyshev-interpolants">Chebyshev Interpolants</h3>
 <hr />
 
 <ol>
-  <li>Compute the required degree $n$ as given above, with error tolerance $\epsilon/2$.</li>
-  <li>Compute the values $c_k$ as given above, which relate to $f$’s Chebyshev interpolant of degree $n$.  There will be $n$ plus one of these values, labeled $c_0, …, c_n$.</li>
+  <li>Compute the required degree $n$ as given earlier, with error tolerance $\epsilon/2$.</li>
+  <li>Compute the values $c_k$ as given earlier, which relate to $f$’s Chebyshev interpolant of degree $n$.  There will be $n$ plus one of these values, labeled $c_0, …, c_n$.</li>
   <li>Compute the (<em>n</em>+1)×(<em>n</em>+1) matrix $M$ described in Theorem 1 of Rababah (2003)<sup id="fnref:51" role="doc-noteref"><a href="#fn:51" class="footnote" rel="footnote">51</a></sup>.</li>
   <li>Multiply the matrix by the transposed vector of values $(c_0, …, c_n)$ to get the polynomial’s Bernstein coefficients $b_0, …, b_n$.  (Transposing means turning columns to rows and vice versa.)</li>
   <li>(Rounding.) For each $i$, replace the Bernstein coefficient $b_i$ with $\text{floor}(b_i / (\epsilon/2) + 1/2) \cdot (\epsilon/2)$.</li>

bernapprox.md

@@ -93,7 +93,7 @@ The goal is now to find a polynomial of degree $n$, written in Bernstein form, s
 1. the polynomial is within $\epsilon$ of $f(\lambda)$, and
 2. each of the polynomial's Bernstein coefficients is not less than 0 or greater than 1 (assuming none of $f$'s values is less than 0 or greater than 1).
 
-For some of the polynomials given above, a degree $n$ can be found so that the degree-$n$ polynomial is within $\epsilon$ of $f$, if $f$ is continuous and meets other conditions.  In general, to find the degree $n$, solve the error bound's equation for $n$ and round the solution up to the nearest integer.  See the table below, where:
+For some of the polynomials given earlier, a degree $n$ can be found so that the degree-$n$ polynomial is within $\epsilon$ of $f$, if $f$ is continuous and meets other conditions.  In general, to find the degree $n$, solve the error bound's equation for $n$ and round the solution up to the nearest integer.  See the table below, where:
 
 - $M_r$ is not less than the maximum of the absolute value of $f$'s $r$-th derivative.
 - $H_r$ is not less than $f$'s $r$-th derivative's Hölder constant (for the given Hölder exponent _α_).
@@ -160,7 +160,7 @@ Let $n\ge 0$ be an integer, and let $f^{(i)}$ be the $i$-th derivative of $f(\la
 4. $f$'s $(n+1)$-th derivative is continuous and satisfies $\epsilon\ge M_{n+1}/((n+1)!)$, and
 5. $f(0)$ is known as well as $f^{(1)}(0), ..., f^{(n)}(0)$.
 
-Then the $n$-th _Taylor polynomial_ centered at 0, given below, is within $\epsilon$ of $f$:
+Then the $n$-th _Taylor polynomial_ centered at 0, given later, is within $\epsilon$ of $f$:
 
 $$P(\lambda) = a_0 \lambda^0 + a_1 \lambda^1 + ... + a_n \lambda^n,$$
 
@@ -689,8 +689,8 @@ The following is a method that employs _Chebyshev interpolants_ to compute the B
 
 -------------
 
-1. Compute the required degree $n$ as given above, with error tolerance $\epsilon/2$.
-2. Compute the values $c_k$ as given above, which relate to $f$'s Chebyshev interpolant of degree $n$.  There will be $n$ plus one of these values, labeled $c_0, ..., c_n$.
+1. Compute the required degree $n$ as given earlier, with error tolerance $\epsilon/2$.
+2. Compute the values $c_k$ as given earlier, which relate to $f$'s Chebyshev interpolant of degree $n$.  There will be $n$ plus one of these values, labeled $c_0, ..., c_n$.
 3. Compute the (_n_+1)×(_n_+1) matrix $M$ described in Theorem 1 of Rababah (2003)[^51].
 4. Multiply the matrix by the transposed vector of values $(c_0, ..., c_n)$ to get the polynomial's Bernstein coefficients $b_0, ..., b_n$.  (Transposing means turning columns to rows and vice versa.)
 5. (Rounding.) For each $i$, replace the Bernstein coefficient $b_i$ with $\text{floor}(b_i / (\epsilon/2) + 1/2) \cdot (\epsilon/2)$.