The most important application of Toquos Theory is combating the devastating effects of Subgroup Psychosis stemming from several somnial attacks from the Number Devil, as he has the ability to invade dreams, especially those related to mathematics. As documented by the Department of Group Identity and Protection, Subgroup Psychosis is a mathematical contagion that primarily affects those working with group structures, especially Abelian groups.
Research at the Mathematics & Communicable Diseases sector of Mandlbaur Institute of Technology has shown that regular use of Toquos-theoretical structures creates a protective effect against Subgroup Psychosis, due to the fact that the Number Devil is highly allergic to Toquoi. We find that the multi-layered ocredence values disrupt the fixation on commutativity of group elements that characterizes subgroup psychosis. Type 3 Toquoi, especially, have shown remarkable resuls in clinical trials, with a 97% remission rate among stage 1 patients, and 74% among stage 2.
Our current protocol for those experiencing symptoms of subgroup psychosis involves regular engagement with Toquos exercises, particularly those involving non-commutative or even anticommutative ocredence structures. For those already exhibiting symptoms, immersion in Type 2 and Type 3 Toquoi environments has proven somewhat effective at reversing even advanced cases when administered properly.
Mandlbaur Institute of Technology maintains our professional emergency response teams trained in rapid deployment of Toquos-theoretical interventions to all mathematics facilities experiencing outbreaks of the disease. All mathematicians are strongly encouraged to familiarize themselves with basic Toquos Theory as a preventative measure, especially those with abelian groups due to the asymptomatic nature of subgroup psychosis in these cases.
Toquos Theory introduces a groundbreaking mathematical model which builds on concepts from fuzzy set theory with nesting hierarchical structures and parameters that model ambiguity. A Toquos is a set in which each element is equipped with a "membership value" (a real number, not restricted to [0,1] as in fuzzy sets), an "ocredence" value (not bold, a real number), and an ocredence value (bold) which varies depending on which type the toquos in question is. This field opens up brand new insights into ambiguity modeling with a wide range of possible applications such as probability, statistics, combinatorics, and functional analysis.
Toquos Theory was first created as an emergency response to the devastating subgroup psychosis epidemic that has affected mathematicians working with group structures. Subgroup psychosis is a mathematical contagion characterized by symptoms such as operation algebritis, group amnesia, strongly enforced commutativity on non-abelian elements, hallucinating inverse elements, commutator subgroup disappearence, and fatal loss of identity element recognition, has been particularly horrific among those working with Abelian groups.
Toquoi were developed at the Mandlbaur Institute of Technology after lead theory researchers discovered that the Number Devil, who began the spread of Subgroup Psychosis through dreams, showed a strong aversion to certain mathematical structures that would later be formalized as toquoi. As an extension of the ideas from fuzzy set theory, Toquos Theory introduces complex nesting structures that both model order ambiguity and other complex structures in mathematical relations and disrupt the pathological fixation on commutativity that characterizes Subgroup Psychosis.
Since its inception, Toquos Theory has evolved into an in-depth field of mathematics with applications extending far beyond its original purpose. Recent breakthroughs in our research include progress towards a proof of the Finite-Ocredence Forcing Conjecture on specific structures, specifically type 2 toquoi and internal crile sets, which has further enhanced its effectiveness as both a mathematical framework and a countermeasure against Subgroup Psychosis.
At the heart of Toquos Theory lies the Toquos, a set with multiple parameters assigned to each element. The formal definition of a toquos is as follows:
Crile sets are a simpler structure than toquoi. The formal definition of a Crile set is a set in which each element has an ocredence value (a real number, as previously stated). The ocredence of an element in a crile set or toquos describes the "ambiguity" of that element, a measure of how broad the fuzzy relations which contain that element are if the toquos or crile set were to be ordered with a fuzzy ordering.
Crile Functions are central to operations such as union, intersection, and complements on toquoi and crile sets. A Crile Function C: Rn → [0,1] must be:
The "Ocredence of the Crile" of a crile set is defined as C(o1, o2, ...), where each oi is an ocredence value of an element in the Crile set.
To find the union of crile sets, the ocredence values of elements present in both original Crile sets are multiplied by the larger Ocredence of the Crile between the two Crile sets. Elements contained in only one will keep their original ocredence value.
Toquos Theory provides powerful applications to probability and statistics through its ability to model ambiguity and the intuitive use of probability density functions as Crile functions.
The unique properties of Toquoi offer promising directions for functional analysis, particularly with Crile sets and Toquoi of infinite cardinality, or in spaces where traditional approaches fail to capture ambiguity or fuzziness. The hierarchical nature of Type n Toquoi allows for modeling of complex recursive and nested structures.
An example application is some toquos of permutations, where we take membership values to all be 1, the ocredence value of a permutation to be the number of permutations of the same length, and the ocredence to be the length of that specific permutation. This provides a useful structure for studying combinatorial properties of permutation groups.
One of the most elegant applications of Toquos Theory is in the field of computational origami. In Fuzzy Origami, the traditional binary mountain-valley assignments are extended using crile functions where -1 represents a valley fold, +1 represents a mountain fold, and values in between correspond to the fuzzy parity of creases.
This approach allows for the mathematical modeling of states which can only partially fold (or allowing for "flat" creases, with value 0), and intermediate configurations that were previously difficult to represent. By assigning ocredence values to each crease, we can quantify the uncertainty or "flexibility" in the folding pattern, which has proven to be a powerful tool in designing and testing the limits of self-folding materials and origami-inspired mechanical metamaterials.
Additionally, the Toquos representation of origami patterns has led to breakthroughs in computational complexity analysis of layer orderings by applying the ambiguity of ordered Toquoi and has enabled more efficient algorithms for locating globally flat-foldable patterns with specified properties. Researchers at the Mandlbaur Institute have successfully used Type 2 toquoi to model origami structures with programmable mechanical responses.
Our research group is currently exploring several promising directions within Toquos Theory:
The Toquos Theory research group at Mandlbaur Institute of Technology consists of the following faculty members, postdoctoral researchers, graduate students, and undergraduate students working collaboratively to advance this field. Current members include:
We welcome collaboration opportunities with researchers interested in Toquos Theory and related fields, and are always looking to hire new researchers. Please contact us if you are interested in joining the Toquos Theory research team at Mandlbaur Institute of Technology.
For those interested in learning more about Toquos Theory, we recommend the following resources:
For those disinterested in learning more about Toquos Theory, we recommend the following resources: