User:Scwarebang/sandbox

https://en.wikipedia.org/wiki/User:Scwarebang/Books/uncertain_reals August 27, 1958

The Institute for Risk and Uncertainty (commonly known as the "Risk Institute") is a component of the Faculty of Science and Engineering of the University of Liverpool in the United Kingdom. The Risk Institute conducts internationally recognised research and training in methods and tools to manage the inherent risks and inescapable uncertainties that arise in natural, social and engineered systems across all academic disciplines and fields of endeavour.

The Institute is funded primarily by research and training grants from UKRI, but it collaborates with researchers, academic institutions, government agencies, and industrial concerns from around the world. Collaboration with industry and NGOs is central to its research and training.

The Institute hosts the Centre for Doctoral Training in Quantification and Management of Risk & Uncertainty in Complex Systems & Environments with funding from the EPSRC and ESRC, which is collocated with the new EPSRC Centre for Doctoral Training in Distributed Algorithms.

Mission
The Risk Institute's mission is to create methods to quantify, mitigate and manage risk and uncertainty to help people and organisations create a safer, more secure, and more efficient world.

where natural and engineered systems can exhibit extreme or unfavourable states that can lead to injury or death, financial loss, or just suboptimal performance. Anticipating and preventing these outcomes requires understanding how they arise given fluctuations in environmental conditions, actions by adversaries, imperfect or limited measurements, and incomplete scientific understanding of the underlying physical processes.

The Risk Institute promotes risk analysis and uncertainty quantification as a part of science and engineering that uses knowledge from physics, biology, chemistry, environmental and life sciences, medicine, economics and finance, psychology and social sciences for solving diverse problems. It develops new methods, experimental and numerical tools, products, and technological and service innovations for engineering and mathematical modeling of the natural and social resources.

Research and development
Research and development at the Risk Institute is both inter- and trans-disciplinary, spaning four broad areas:
 * Measurement and uncertainty characterisation to make proper use of the statistical uncertainty in measurements collected in new or fluctuating environments or when imprecision cannot be neglected,
 * Numerical simulation methods to integrate available data and incomplete scientific knowledge about the underlying processes into models to forecast relevant risks,
 * Risk and uncertainty communication to overcome recognised biases afflicting human perception and cognition involving uncertainties, and
 * Decision making under risk and uncertainties to optimise planning and management of complex systems.

History
In response to a diverse series of dramatic disasters in the first decade of the century , the Risk Institute was founded in 2012 by Professor Michael Beer, who was its director until 2015.

Mathematical representation
Let Ω be a set. Often, this set is either the reals or the integers, but it could be any topological space. <> Suppose A &sub; Ω is the set of all solutions in Ω that we are trying to characterize.

Let A, B, and C denote subsets of Ω.

Let ω, θ, and φ denote elements of Ω.

If ω &isin; A, we call ω a solution. If ω &notin; A, we say ω is not a solution. If θ &isin; B, we say θ is implied by B.

If ω &isin; A, we call ω a solution. If ω &notin; A, we say ω is not a solution. If θ &isin; B, we say θ is implied by B.

We say that B &sub; Ω bounds A (and that A is bounded by B) if every element of A is also an element of B. If B bounds A then
 * A &sub; B,
 * A &sube; B, or
 * A = B.

We say that B is an inner bound for A if A bounds B. We say that C constrains A if either
 * C bounds A or
 * A bounds C,

and that we know which of these two statements is true.

Structured bounds
Structured bounds are less resolved than arbitrary sets but often far simpler to work with. Call D a structured bounding space for Ω if every subset of Ω is bounded by some element of D, i.e., if the following conditions hold:
 * D &sube; 2Ω, and
 * A &sube; Ω implies A &sube; B for some B &isin; D.

This definition implies that every element of Ω is covered by some element of D, and also that Ω itself is covered by some element of D which would be called a vacuous bound. Note that the empty set may or may not be an element of D.

The union of the elements in any subset of D is bounded by some element of D?

The bound B &isin; D is said to be a best-possible (relative to D) bound on A &sube; Ω if both
 * A &sube; B, and
 * there exists no C &isin; D such that A &sube; C &sub; B.

It may be further true that B is the unique best-possible bound in the sense that
 * A &sube; C implies that B &sube; C, for all C &isin; D.

Thus, B is the unique best-possible bound on A if B bounds A and it is the tightest bound in D that does so.

Some structured bounding spaces, such as intervals bounding sets of reals, have only unique best-possible bounds. Their elements form a tree-shaped upper lattice.

Examples
If Ω is the real numbers, then 2ℝ then is a trivial structured bounding space (i.e., one with no structure) because no subsets of Ω are missing. The set of traditional intervals
 * { [a, b] : a ≤ b, a &isin; ℝ, b &isin; ℝ }

is not a structured bounding space for ℝ, because there is no interval that bounds the whole of ℝ itself. Likewise all half-bounded intervals of the form [a, &infin;] and [&minus;&infin;, b] are also not bounded by any element of the set of real intervals.

However, a structured bounding space for ℝ can be constructed as
 * { [a, b] : a ≤ b, a &isin; ℝ*, b &isin; ℝ* }

where ℝ* = ℝ ∪ {&minus;&infin;, ∞}.

A finite-precision number is

A significant-digit interval is the set of real numbers that are d-close to a real number e whose

Are significant-digit intervals a structured bounding space for the reals? SCOTT SAYS NO!

Are significant-digit intervals a structured bounding space for the reals? Are significant-digit intervals a structured bounding space for the reals? Are significant-digit intervals ℝ a structured bounding space for the reals? Are significant-digit intervals a structured bounding space for the reals? Are significant-digit intervals a structured bounding space for the reals?

There are various notation systems in use for denoting open, closed and partially open intervals. (a,b) = ]a,b[ = [a+, b-] = [a plus, b minus] (a,b] = ]a,b] = [a+, b] = [a plus, b] [a,b) = [a,b[ = [a, b-]  = [a, b minus] [a,b] = [a,b] = [a, b]  = [a, b]

The system using both parentheses and brackets is conventional, but is often confused with ordered sets. The backward-bracket notation ]a,b[ was introduced by Bourbaki is efficient, but it is easy to misread. Notation should strive to be helpful in addition to being efficient. When something should be noticed, the notation should help readers notice it. Thus, we prefer the plus-minus notation, which cannot be misread and, arguably, is the most intuitive of the three.

The plus-minus notation is more flexible, as it can designate a range which is a halo around a closed interval, for instance [a minus, b plus].

Material on kinds of bounds: socks sleeves mittens gloves

Simultaneous (distributional) confidence bounds versus point-wise confidence bounds

Algebraic structure of bounding and uncertainty calculi See https://en.wikipedia.org/wiki/Magma_(algebra)#Types_of_magma

An algebra A over a set Ω is a non-empty set of sets of Ω that satisfies: (i) Ω∈A, (ii) if a∈A then ac∈A, and (iii) if a,b∈A then a∪b∈A. Thus an algebra is a set containing Ω that is closed under complements and finite unions and intersections.

A σ-algebra is a non-empty set of sets that is closed under complements and countable unions and intersections.

A measurable space (Ω,F) consists of a set Ω and a σ-algebra of subsets of Ω.