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I constructed a projected gradient descent (ascent) algorithm with backtracking line search based on the book "Convex optimization," written by Stephen Boyd and Lieven Vandenberghe.
The problem what I consider and the pseudocode to solve it is presented as follows:
\begin{array}{cl}
\text{maximize}
In optimization, the line search strategy is one of two basic iterative approaches to find a local minimum
x
∗
{\displaystyle \mathbf {x} ^{*}}
of an objective function
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
. The other approach is trust region.
The line search approach first finds a...
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I am trying to prove it is true for all $k \in \mathbb N \,\backslash\,{\{1}\}$ that:
$$\Biggl\lfloor \ln \left( \frac{1}{k^{n}}
\right) \Biggr\rfloor-\Biggl\lfloor \ln \left( \frac{1}{k^{n+1}}
\right) \Biggr\rfloor=\cases{m_{{i}}&$n \in\mathfrak{B} \left( \alpha_{{k}} \right) $\cr m_{...
In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples
of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926.
Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number.
Beatty sequences can also be used to generate Sturmian words.
== Definition ==
A positive irrational number...
1
Proposal: Make division-ring a synonym of division-algebras.
I don't think there is much difference. We can argue whether a division algebra is always associative or not. I wrote the tag wiki with octonions in mind, and phrased it "inclusively". A division-ring is surely always associative, but d...
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
== Definitions ==
Formally, we start with a non-zero algebra D over a field. We call D a division algebra if for any element a in D and any non-zero element b in D there exists precisely one element x in D with a = bx and precisely one element y in D such that a = yb.
For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has...
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. A division ring is a type of noncommutative ring under the looser definition where noncommutative ring refers to rings which are not necessarily commutative.
Division rings differ from fields only in that their multiplication is not required...
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons, commonly called bits) obtained about one random variable through observing the other random variable. The concept of mutual information is intricately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.
Not limited to real-valued...
