Narrow Results

The notions of inner-star and star-inner generalized inverses are introduced on the set of all regular elements in a ∗-ring ${ℛ}$. We thus extend the concept of inner-star and star-inner complex matrices. We study properties of these hybrid generalized inverses on the set ${{ℛ}}^{†}$ of all Moore–Penrose invertible elements in ${ℛ}$ and thus generalize some known results. Partial orders that are induced by inner-star and star-inner inverses are introduced on ${{ℛ}}^{†}$, their properties are examined, and their characterizations are presented.

In this paper, we prove the existence of best proximity point and coupled best proximity point on metric spaces with partial order for weak proximal contraction mappings such that these critical points satisfy some constraint inequalities.

The elements of a finite partial order P can be identified with the maximal indecomposable two-sided ideals of its incidence algebra , and then for two such ideals, I ≺ J ⇔ IJ ≠ 0. This offers one way to recover a poset from its incidence algebra. In the course of proving the above, we classify all of the two-sided ideals of .

Entanglement measures quantify the amount of quantum entanglement that is contained in quantum states. Typically, different entanglement measures do not have to be partially ordered. The presence of a definite partial order between two entanglement measures for all quantum states, however, allows for meaningful conceptualization of sensitivity to entanglement, which will be greater for the entanglement measure that produces the larger numerical values. Here, we have investigated the partial order between the normalized versions of four entanglement measures based on Schmidt decomposition of bipartite pure quantum states, namely, concurrence, tangle, entanglement robustness and Schmidt number. We have shown that among those four measures, the concurrence and the Schmidt number have the highest and the lowest sensitivity to quantum entanglement, respectively. Further, we have demonstrated how these measures could be used to track the dynamics of quantum entanglement in a simple quantum toy model composed of two qutrits. Lastly, we have employed state-dependent entanglement statistics to compute measurable correlations between the outcomes of quantum observables in agreement with the uncertainty principle. The presented results could be helpful in quantum applications that require monitoring of the available quantum resources for sharp identification of temporal points of maximal entanglement or system separability.

Let K be a prime knot in S³ and G(K) = π₁(S³ - K) the knot group. We write K₁ ≥ K₂ if there exists a surjective homomorphism from G(K₁) onto G(K₂). In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then 640, 800 should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial.

A partial order on prime knots can be defined by declaring $J\ge K$, if there exists an epimorphism from the knot group of $J$ onto the knot group of $K$. Suppose that $J$ is a 2-bridge knot that is strictly greater than $m$ distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of $J$ in terms of $m$. Using this bound, we answer a question of Suzuki regarding the 2-bridge epimorphism number $\text{EK}(n)$ which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number $n$. We establish our results using techniques associated with parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K\ge J$ if there exists an epimorphism $f:{\pi }_1(S^3-K)\to {\pi }_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K>J$, then $\gamma (K)\ge 3\gamma (J)-4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K>J$, then $g(K)\ge 3g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.

Covering rough set theory is an important generalization of traditional rough set theory. So far, the studies on covering generalized rough sets mainly focus on constructing different types of approximation operations. However, little attention has been paid to uncertainty measurement in covering cases. In this paper, a new kind of partial order is proposed for coverings which is used to evaluate the uncertainty measures. Consequently, we study uncertain measures like roughness measure, accuracy measure, entropy and granularity for covering rough set models which are defined by neighborhoods and friends. Some theoretical results are obtained and investigated.

The elements of the truth value algebra of type-2 fuzzy sets are the mappings of the unit interval into itself, with operations given by various convolutions of the pointwise operations. This algebra can be specialized and generalized in various interesting ways. First, we consider the more general case of all mappings of a bounded chain with an involution into a complete chain, and delimit some of the properties of the resulting algebra. These include two binary operations each of which give a partial order on the elements of that algebra. These partial orders and their intersection are the principal objects of interest. We specialize this situation in two cases: (1) all mappings of the unit interval into itself, the original version of the truth value algebra of type-2 fuzzy sets introduced by Zadeh, and (2) all mappings of a finite chain into another finite chain. Again, each of these two cases yields two partial orders on the elements of the resulting algebras, and in each case, our principal interest is in these partial orders and their intersection.

In this paper, an order induced by implications on a bounded lattice under some more lenient conditions than the ones given former studies is defined and some of its properties are discussed. By giving an order based on uninorms on a bounded lattice, the relationships between such generated orders are investigated.

Consider a multistate system with partially ordered state space E, which is divided into a set C of working states and a set D of failure states. Let X(t) be the state of the system at time t and suppose {X(t)} is a stochastically monotone Markov chain on E. Let T be the failure time, i.e., the hitting time of the set D. We derive upper and lower bounds for the reliability of the system, defined as P_m(T > t) where m is the state of perfect system performance.

Introduced by Artzner et al. (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini et al. (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results, we study different notions of time consistency and we give examples of vector-valued risk measure processes.

A generalization of Muraki's notion of monotonic independence onto the case of partially ordered index set is given: algebras indexed by chains are monotonically independent, and algebras indexed by non-comparable elements are boolean independent. Examples of central limit theorem are shown in two cases. For the integral-points lattices ℕ^d the moments of the limit measure are related to the combinatorics of the finite heap-ordered labelled rooted trees (if d = 2). For the integral-points lattice ℕ × ℤ^d in Minkowski spacetime the limit measure is given by the recurrence of it's moments, which, for the case d = 1 is related to the inverse error function. Various formulas for computing mixed moments are shown to be related to the boolean-monotonic non-crossing pair partitions.

We give an algorithm for deciding whether an embedding of a finite partial order into the enumeration degrees of the -sets can always be extended to an embedding of a finite partial order .

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

In an earlier paper we defined a relation ≤_Δ between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤_Δ is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤_Δ is always a partial order.

We generalize the notion of the left-sharp and the right-sharp partial orders to ${𝒢}({ℛ})$ where ${ℛ}$ is a ring with identity and ${𝒢}({ℛ})$ the subset of elements in ${ℛ}$ which have the group inverse. We connect these orders to well-known sharp and minus partial orders. Properties of one-sided sharp partial orders in ${𝒢}({ℛ})$ are studied and some known results are generalized.

In this paper, we introduce the concept of Baer $(p,q)$-sets. Using this notion, we define Rickart, Baer, quasi-Baer and $\pi$-Baer $(S,R)$-bimodules, respectively. We show how these conditions relate to each other. We also develop new properties of the minus binary relation, $\le$-, we extend the relation $\le$- to $(S,R)$-bimodules and use it to characterize the aforementioned Rickart, Baer, quasi-Baer, and $\pi$-Baer $(S,R)$-bimodules. Moreover, we specify subsets ${𝒦}$ of the power set of a $(S,R)$-bimodule for which $\le$- determines a partial order and for which $\le$- is a lattice. We analyze the relation $\le$- by examining the associated Baer $(p,q)$-sets. Finally, we apply our results to $C^{\ast }$-modules. Examples are provided to illustrate and delimit our results.

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from $M_n(ℂ)$, the set of all $n\times n$ matrices over $ℂ$, to the set ${{ℛ}}^{\text{ⓓ}}$ of all core-EP invertible elements in an arbitrary unital ring ${ℛ}$ with involution. A new partial order is introduced on ${{ℛ}}^{\text{ⓓ}}$ by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.