Rank of elements of general rings in connection with unit-regularity

We define the rank of elements of general unital rings, discuss its properties and give several examples to support the definition. In semiprime rings we give a characterization of rank in terms of invertible elements. As an application we prove that every element in the socle of a unital semiprime ring is unit-regular.Comment: 13 page

Compressed zero-divisor graphs of noncommutative rings

We extend the notion of the compressed zero-divisor graph $\varTheta(R)$ to noncommutative rings in a way that still induces a product preserving functor $\varTheta$ from the category of finite unital rings to the category of directed graphs. For a finite field $F$, we investigate the properties of $\varTheta(M_n(F))$, the graph of the matrix ring over $F$, and give a purely graph-theoretic characterization of this graph when $n \neq 3$. For $n \neq 2$ we prove that every graph automorphism of $\varTheta(M_n(F))$ is induced by a ring automorphism of $M_n(F)$. We also show that for finite unital rings $R$ and $S$, where $S$ is semisimple and has no homomorphic image isomorphic to a field, if $\varTheta(R) \cong \varTheta(S)$, then $R \cong S$. In particular, this holds if $S=M_n(F)$ with $n \neq 1$.Comment: 30 page

Final solution to the problem of relating a true copula to an imprecise copula

In this paper we solve in the negative the problem proposed in this journal (I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48-66) whether an order interval defined by an imprecise copula contains a copula. Namely, if $\mathcal{C}$ is a nonempty set of copulas, then $\underline{C} = \inf\{C\}_{C\in\mathcal{C}}$ and $\overline{C}= \sup\{C\}_{C\in\mathcal{C}}$ are quasi-copulas and the pair $(\underline{C},\overline{C})$ is an imprecise copula according to the definition introduced in the cited paper, following the ideas of $p$-boxes. We show that there is an imprecise copula $(A,B)$ in this sense such that there is no copula $C$ whatsoever satisfying $A \leqslant C\leqslant B$. So, it is questionable whether the proposed definition of the imprecise copula is in accordance with the intentions of the initiators. Our methods may be of independent interest: We upgrade the ideas of Dibala et al. (Defects and transformations of quasi-copulas, Kybernetika, 52 (2016), 848-865) where possibly negative volumes of quasi-copulas as defects from being copulas were studied.Comment: 20 pages; added Conclusion, added some clarifications in proofs, added some explanations at the beginning of each section, corrected typos, results remain the sam

The total zero-divisor graph of commutative rings

In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graphs of the ring ${\mathbb Z}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb Z}_m$ and ${\mathbb Z}_n$ are isomorphic if and only if $m=n$

Blokada gornjih dišnih putova u zbrinjavanju otežanog dišnog puta u budnom stanju

Airway anesthesia is pivotal for successful awake intubation provided either topically or by blocks. Airway blocks are considered technically more difficult to perform and carry a higher risk of complications. However, in experienced hands, they can be useful as they provide excellent intubating conditions. For complete upper airway anesthesia, bilateral glossopharyngeal and superior laryngeal nerve blocks with translaryngeal injection are required. Superior laryngeal nerve block and translaryngeal injection can be performed easily, safely and with a high success rate in patients with normal anatomy. In those with difficult landmarks, ultrasound can be of assistance. For the superior laryngeal nerve block, other targets than the nerve itself must be established to make the technique consistently successful, easy to teach, learn and perform. The same applies to the translaryngeal injection, where the use of ultrasound is necessary for correct midline identification. Intraoral glossopharyngeal nerve block is also safe and easy to perform, but associated with long lasting discomfort. Bilateral extraoral peristyloid approach should be discouraged since inadvertent blocks of the closely adjacent vagus nerve cannot be prevented in this location. A safe and easy method of blocking the distal portions of the glossopharyngeal nerve for awake intubation is therefore required.Za intubaciju u budnom stanju potrebna je dobra anestezija gornjeg dišnog puta koju se može postići topičnom anestezijom ili regionalnim blokadama. Blokovi su tehnički zahtjevniji i povezani su s mogućim ozbiljnim komplikacijama, ali u rukama stručnjaka omogućavaju dobru anesteziju i odlične uvjete za intubaciju u budnom stanju. Za kompletnu anesteziju gornjeg dišnog puta potrebna je obostrana blokada glosofaringealnog živca, unutarnjih grana gornjeg laringealnog živca i translaringealna blokada. Blokade gornjeg laringealnog živca i translaringealna injekcija su jednostavne, sigurne i uspješne kod većine bolesnika s normalnom anatomijom. U primjeru teške identifikacije anatomskih točaka može poslužiti ultrazvuk. Kod ultrazvučno vođene blokade gornjeg laringealnog živca potrebno je pronaći druge ciljeve od vizualizacije samog živca, jer je ta tehnika slabo reproducibilna, teška za učenje i izvođenje. U primjeru translaringealne blokade ultrazvuk može poslužiti za točnu identifikaciju medijalne linije. Intraoralni pristup blokadi glosofaringealnog živca jednostavan je za izvođenje i dosta siguran, ali zna nanositi neugodnu bol. Obostrani vanjski, peristiloidni pristup se apsolutno ne preporuča zbog blizine vagalnog živca i posljedične opstrukcije dišnoga puta. Iz toga slijedi da je za namjeru opskrbe dišnoga puta u budnom stanju potrebno pronaći jednostavniju metodu za blokadu glosofaringealnog živca koja će biti locirana na njegovom distalnom dijelu

Categorial properties of compressed zero-divisor graphs of finite commutative rings

We define a compressed zero-divisor graph $\varTheta(K)$ of a finite commutative unital ring $K$, where the compression is performed by means of the associatedness relation. We prove that this is the best possible compression which induces a functor $\varTheta$, and that this functor preserves categorial products (in both directions). We use the structure of $\varTheta(K)$ to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.Comment: 14 page

GPS-derived geoid using artificial neural network and least squares collocation

The geoidal undulations are needed for determining the orthometric heights from the Global Positioning System GPS-derived ellipsoidal heights. There ore several methods for geoidal undulation determination. The paper presents a method employing the Artificial Neural Network (ANN) approximation together with the Least Squares Collocation (LSC). The surface obtained by the ANN approximation is used as a trend surface in the least squares collocation. In numerical examples four surfaces were compared: the global geopotential model (EGM96), the European gravimetric quasigeoid 1997 (EGG97), the surface approximated with minimum curvature splines in tension algorithm and the ANN surface approximation. The effectiveness of the ANN surface approximation depends on the number of control points. If the number of well-distributed control points is sufficiently large, the results are better than those obtained by the minimum curvature algorithm and comparable to those obtained by the EGG97 model