In computability theory, a probabilistic Turing machine is a non-deterministic Turing machine which chooses between the available transitions at each point according to some probability distribution.
In the case of equal probabilities for the transitions, it can be defined as a deterministic Turing machine having an additional "write" instruction where the value of the write is uniformly distributed in the Turing Machine's alphabet (generally, an equal likelihood of writing a '1' or a '0' on to the tape.) Another common reformulation is simply a deterministic Turing machine with an added tape full...
In computability theory, the theory of real computation deals with hypothetical computing machines using infinite-precision real numbers. They are given this name because they operate on the set of real numbers. Within this theory, it is possible to prove interesting statements such as "The complement of the Mandelbrot set is only partially decidable."
These hypothetical computing machines can be viewed as idealised analog computers which operate on real numbers, whereas digital computers are limited to computable numbers. They may be further subdivided into differential and algebraic models (digital...
In physics, the Bekenstein bound is an upper limit on the entropy S, or information I, that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the quantum level. It implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy is finite. In computer science, this implies that there is a maximum information-processing rate (Bremermann's limit...
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Hamkins showed that his infinite time Turing machine has the power to decide some $\Delta_2^1$ sets. I wonder if some modifications of the machine could be made to reach level $\Sigma_1^2$ sets, or, if no modifications on sight, if the power of his machine plus infinitely iterated super jumps fro...
In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for determining an upper bound on the number of positive or negative real roots of a polynomial. It is not a complete criterion, because it does not provide the exact number of positive or negative roots.
The rule is applied by counting the number of sign changes in the sequence formed by the polynomial's coefficients. If a coefficient is zero, that term is simply omitted from the sequence.
== Descartes' rule of signs ==
=== Positive roots ===
The rule states that if the terms of a...
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Let $f=u+iv$ be an entire function where $u,v$ are the real and
imaginary parts of $f$ respectively. If the Jacobian $$ J_a=
\left[ {\begin{array}{cc} u_x(a) & u_y(a) \\ v_x(a) & v_y(a) \\
\end{array} } \right] $$ is symmetric $\forall a\in \mathbb C$, then
(1) $f$ is a p...
(Using the advice from Mathoverflow, I have rephrased and splitted up the question)
(Might be a bit layman because I don't have rigorous math term to describe the concept) Generalize it to mathematical spaces, are there spaces which are sort of like
a. Having multiple $\mathbb{R}^2$ spaces th...
For $A$ and $B$ sets that is different from null,
$4. |A-B| = 3 . |A' \cap B|$
$3. |A| - |B| = 20$
What is the maximal value of $|A\cup B|$?
It seems so confused right now. Can you take a look?
