Detachable Pairs in 3-Connected Matroids

<p>The classical tool at the matroid theorist’s disposal when dealing with the common problem of wanting to remove a single element from a 3-connected matroid without losing 3-connectivity is Tutte’s Wheels-and-Whirls Theorem. However, situations arise where one wishes to delete or contract a pair of elements from a 3-connected matroid whilst maintaining 3-connectedness. The goal of this research was to provide a new tool for making such arguments. Let M be a 3-connected matroid. A detachable pair in M is a pair x, y ∈ E(M) such that either M\x, y or M/x, y is 3-connected. Naturally, our aim was to find the necessary conditions on M which guarantee the existence of a detachable pair. Triangles and triads are an obvious barrier to overcome, and can be done so by allowing the use of a Δ − Y exchange. Apart from these matroids with three-element 3-separating sets, the only other class of matroids that fail to contain a detachable pair for which no bound can be placed on the size of the ground set is the class of spikes. In particular, we prove the following result. Let M be a 3-connected matroid with at least thirteen elements. If M is not a spike, then either M contains a detachable pair, or there exists a matroid M′ where M′ is obtained by performing a single Δ − Y exchange on either M or M* such that M′ contains a detachable pair. As well as being an important theorem in its own right, we anticipate that this result will be essential in future attempts to extend Seymour’s Splitter Theorem in a comparable manner; where the goal would be to obtain a detachable pair as well as maintaining a 3-connected minor. As such, much work has been done herein to study the precise configurations that arise in 3-separating subsets which themselves yield no detachable pair.</p>

Detachable Pairs in 3-Connected Matroids

<p>The classical tool at the matroid theorist’s disposal when dealing with the common problem of wanting to remove a single element from a 3-connected matroid without losing 3-connectivity is Tutte’s Wheels-and-Whirls Theorem. However, situations arise where one wishes to delete or contract a pair of elements from a 3-connected matroid whilst maintaining 3-connectedness. The goal of this research was to provide a new tool for making such arguments. Let M be a 3-connected matroid. A detachable pair in M is a pair x, y ∈ E(M) such that either M\x, y or M/x, y is 3-connected. Naturally, our aim was to find the necessary conditions on M which guarantee the existence of a detachable pair. Triangles and triads are an obvious barrier to overcome, and can be done so by allowing the use of a Δ − Y exchange. Apart from these matroids with three-element 3-separating sets, the only other class of matroids that fail to contain a detachable pair for which no bound can be placed on the size of the ground set is the class of spikes. In particular, we prove the following result. Let M be a 3-connected matroid with at least thirteen elements. If M is not a spike, then either M contains a detachable pair, or there exists a matroid M′ where M′ is obtained by performing a single Δ − Y exchange on either M or M* such that M′ contains a detachable pair. As well as being an important theorem in its own right, we anticipate that this result will be essential in future attempts to extend Seymour’s Splitter Theorem in a comparable manner; where the goal would be to obtain a detachable pair as well as maintaining a 3-connected minor. As such, much work has been done herein to study the precise configurations that arise in 3-separating subsets which themselves yield no detachable pair.</p>

Symplectic decomposition from submatrix determinants

An important theorem in Gaussian quantum information tells us that we can diagonalize the covariance matrix of any Gaussian state via a symplectic transformation. While the diagonal form is easy to find, the process for finding the diagonalizing symplectic can be more difficult, and a common, existing method requires taking matrix powers, which can be demanding analytically. Inspired by a recently presented technique for finding the eigenvectors of a Hermitian matrix from certain submatrix eigenvalues, we derive a similar method for finding the diagonalizing symplectic from certain submatrix determinants, which could prove useful in Gaussian quantum information.

Space-bounded OTMs and REG ∞

An important theorem in classical complexity theory is that REG = LOGLOGSPACE, i.e., that languages decidable with double-logarithmic space bound are regular. We consider a transfinite analogue of this theorem. To this end, we introduce deterministic ordinal automata (DOAs) and show that they satisfy many of the basic statements of the theory of deterministic finite automata and regular languages. We then consider languages decidable by an ordinal Turing machine (OTM), introduced by P. Koepke in 2005 and show that if the working space of an OTM is of strictly smaller cardinality than the input length for all sufficiently long inputs, the language so decided is also decidable by a DOA, which is a transfinite analogue of LOGLOGSPACE ⊆ REG; the other direction, however, is easily seen to fail.

Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations

<p style='text-indent:20px;'>We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their <inline-formula><tex-math id="M1">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its <inline-formula><tex-math id="M2">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their <inline-formula><tex-math id="M3">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths.</p>

Hypercongruences in fuzzy AG-hypergroupoids

We introduce the notion of fuzzy Abel-Grassmann’s hypergroupoid, hypercongruence, fuzzy hypercongruence, fuzzy strong hypercongruence, compatible relations in an Abel-Grassmann’s hypergroupoid. This paper is aimed to study fuzzy hyperideals, smallest fuzzy hyperideals, fuzzy equivalence relations, fuzzy compatible fuzzy strong compatible, fuzzy hypercongruences, fuzzy strong hypercongruences, fuzzy regular, fuzzy strong regular relations and fuzzy hypercongruences in Abel-Grassmann’s hypergroupoids. Characterizations of hypercongruences, their corresponding quotient structure, homomorphisms and an important theorem on embedding Abel-Grassmann’s hypergroupoids by means of fuzzy sets. We show that each hypergroupoid is embedded into a poe-hypergroupoid of all fuzzy subsets of an Abel-Grassmann’s hypergroupoid.

On a Theorem of Burgess and Stephenson

A theorem of Burgess and Stephenson asserts that in an exchange ring with central idempotents, every maximal left ideal is also a right ideal. The proof uses sheaf-theoretic techniques. In this paper, we give a short elementary proof of this important theorem.

Some properties of quantum fidelity in infinite-dimensional quantum systems

In this paper, we review the infinite-dimensional version of Uhlmann’s theorem of quantum fidelity. Based on this important theorem, we derive some useful properties of quantum fidelity for infinite-dimensional quantum systems. Next, we study the fidelity in the infinite-dimensional discrete-time quantum filters associated Markov chain, and prove that it is a sub-martingale.

A New Kind of Infinite Series and its Applications on Neural Networks

A new important theorem for function expansion and a new infinite series based on the new expansion is proposed. Unlike constant derivatives at a point by Taylors expansion, the terms are functions associated with derivatives. The expansion generated by a function is not the form of polynomials, which is different from Taylors expansion. Some application examples are given to show that the region of convergence is much larger than that obtained by Taylor's Series. As a kind of application, a new learning algorithm based on the new expansion for training neural networks is also proposed. The weight functions using the new expansion are more suitable for training those patterns obtained by some rational problems.

Illustrating the Central Limit Theorem Through Microsoft Excel Simulations

Using Microsoft Excel, several interactive, computerized learning modules are developed to demonstrate the Central Limit Theorem. These modules are used in the classroom to enhance the comprehension of this theorem. The Central Limit Theorem is a very important theorem in statistics, and yet because it is not intuitively obvious, statistics students often have difficulty accepting it. Nevertheless, understanding this theorem is essential because of its importance in statistical inference.