Normal Quotient Research Articles

A Longitudinal Follow-Up Study of Intellectual Function in Duchenne Muscular Dystrophy over Age: Is It Really Stable?

The aim of the study was to retrospectively evaluate the consistency of longitudinal findings on intellectual functioning in DMD boys and their relationship to behavioral and neuropsychiatric difficulties. The cohort included 70 patients of age 3 to 17 years with at least two assessments using the Wechsler scales. CBCL and clinical observation of behavior were also performed. Changes in total intelligence quotient were interpreted as stable or not stable using the reliable-change method. On the first assessment 43/70 had normal quotients, 18 borderline, 5 mild, and 4 moderate intellectual disability, while 27/70 had no behavioral disorders, 17 had abnormal CBCL, and 26 patients had clear signs of attention deficits despite normal CBCL. The remaining seven were untestable. The mean total intelligence quotient change in the cohort was -2.99 points (SD: 12.29). Stable results on TIQ were found in 63% of the paired assessments. A third of the consecutive cognitive assessments showed a difference of more than 11 points with changes up to 42 points. Boys with no behavioral/attention disorder had smaller changes than those with attention (p = 0.007) and behavioral disorders (p = 0.002). Changes in IQ may occur in Duchenne and are likely to be associated with behavioral or attention deficits.

Classifying edge-biregular maps of negative prime Euler characteristic

An edge-biregular map arises as a smooth normal quotient of a unique index-two subgroup of a full triangle group acting with two edge-orbits. We give a classification of all finite edge-biregular maps on surfaces of negative prime Euler characteristic.

Introducing edge-biregular maps

We introduce the concept of alternate-edge-colourings for maps and study highly symmetric examples of such maps. Edge-biregular maps of type (k, l) occur as smooth normal quotients of a particular index two subgroup of T_{k,l}, the full triangle group describing regular plane (k, l)-tessellations. The resulting colour-preserving automorphism groups can be generated by four involutions. We explore special cases when the usual four generators are not distinct involutions, with constructions relating these maps to fully regular maps. We classify edge-biregular maps when the supporting surface has non-negative Euler characteristic, and edge-biregular maps on arbitrary surfaces when the colour-preserving automorphism group is isomorphic to a dihedral group.

CME Sonography 99/Answers: Kidney Echo Changes

CME Sonography 99/Answers: Kidney Echo Changes Abstract. Normal kidneys have a smooth contour, kidney length 9-12 cm, volume 90-180 ml/1.73 m2 body surface, parenchyma width 13-18 mm and an echogenicity of the cortex which is hypoechoic compared to the liver and spleen, as well as medullary pyramids, which are again hypoechoic compared to the cortex. Digital ultrasound images enable the echogenicity to be measured between 0 (black) and 255 (white). A normal quotient between the echogenicity of the liver and the renal cortex is 1.0-1.4. A normal quotient between the spleen and the renal cortex is> 1.0. The normal quotient between the renal cortex and the renal medulla is> 1.0. In diffuse renal parenchyma diseases, changes in echogenicity are observed. There are kidneys with increased echogenicity of the cortex and a ratio to liver and spleen of <1.0 and kidneys with hyperechoic renal medulla and the ratio between renal cortex and renal medulla <1.0.

CME Sonography 99: Kidney Echo Changes

CME Sonography 99: Kidney Echo Changes Abstract. Normal kidneys have a smooth contour, kidney length 9-12 cm, volume 90-180 ml/1.73 m2 body surface, parenchyma width 13-18 mm and an echogenicity of the cortex which is hypoechoic compared to the liver and spleen, as well as medullary pyramids, which are again hypoechoic compared to the cortex. Digital ultrasound images enable the echogenicity to be measured between 0 (black) and 255 (white). A normal quotient between the echogenicity of the liver and the renal cortex is 1.0-1.4. A normal quotient between the spleen and the renal cortex is >1.0. The normal quotient between the renal cortex and the renal medulla is >1.0. In diffuse renal parenchyma diseases, changes in echogenicity are observed. There are kidneys with increased echogenicity of the cortex and a ratio to liver and spleen of <1.0 and kidneys with hyperechoic renal medulla and the ratio between renal cortex and renal medulla <1.0.

Normal edge-transitive Cayley graphs and Frattini-like subgroups

For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist of those automorphisms of G which leave C invariant. We define an Aut(G;C)-invariant normal subgroup Φ(G;C) of G which has the property that, for any Aut(G;C)-invariant normal set of generators for G, if we remove from it all the elements of Φ(G;C), then the remaining set is still an Aut(G;C)-invariant normal generating set for G. The subgroup Φ(G;C) contains the Frattini subgroup Φ(G) but the inclusion may be proper. The Cayley graph Cay(G,C) is normal edge-transitive if Aut(G;C) acts transitively on the pairs {c,c−1} from C. We show that, for a normal edge-transitive Cayley graph Cay(G,C), its quotient modulo Φ(G;C) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ(G;C) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi.

Normal quotients of diameter at most two of finite three-geodesic-transitive graphs

An s-geodesic of a graph is a path of length s such that the first and last vertices are at distance s. We study finite graphs Γ of diameter at least 3 for which some subgroup G of automorphisms is transitive on the set of s-geodesics for each s≤3. If Γ has girth at least 6 then all 3-arcs are 3-geodesics so Γ is 3-arc-transitive, and such graphs have already been studied fruitfully; also graphs of girth 3 with these properties have been investigated successfully. We therefore focus on those of girth 4 or 5. We study their normal quotients ΓN modulo the orbits of a normal subgroup N of G and prove that, provided ΓN has diameter at least 3, then Γ is a cover of ΓN and ΓN,G/N have the same girth and transitivity properties as Γ,G (so if N≠1 we may reduce consideration to a smaller graph in the family).We then focus on the ‘degenerate case’ where ΓN has diameter at most 2. In these cases also, Γ is a cover of ΓN provided N has at least three vertex-orbits. If ΓN is a complete graph Kr (diameter 1), then we prove that Γ is either the complete bipartite graph Kr,r with the edges of a perfect matching removed, or a unique 6-fold-cover of K7. In the remaining case where ΓN has diameter 2, then ΓN is a 2-arc-transitive strongly regular graph. We classify all the 2-arc-transitive strongly regular graphs, and using this classification we describe all their finite (G,3)-geodesic-transitive covers of girth 4 or 5, except for a few difficult cases.

Symmetric graphs of valency five and their basic normal quotients

A graph Γ is symmetric or arc-transitive if its automorphism group Aut(Γ) is transitive on the arc set of the graph, and Γ is basic if Aut(Γ) has no non-trivial normal subgroup N such that the quotient graph ΓN has the same valency as Γ. In this paper, we classify symmetric basic graphs of order 2qpn and valency 5, where q<p are two primes and n is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order 2q with 5|(q−1), the complete graph K6 of order 6, the complete bipartite graph K5,5 of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order kpn for some small integers k and n are classified.

Quotient-Complete Arc-Transitive Latin Square Graphs from Groups

We consider latin square graphs $$\varGamma = \text {LSG}(H)$$ of the Cayley table of a given finite group H. We characterize all pairs $$(\varGamma ,G)$$ , where G is a subgroup of autoparatopisms of the Cayley table of H such that G acts arc-transitively on $$\varGamma $$ and all nontrivial G-normal quotient graphs of $$\varGamma $$ are complete. We show that H must be elementary abelian and determine the number k of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with $$k = 1$$ or $$k = 2$$ .

New structural results on tetravalent half-arc-transitive graphs

Tetravalent graphs admitting a half-arc-transitive group of automorphisms, that is a subgroup of the automorphism group acting transitively on its vertices and its edges but not on its arcs, are investigated. One of the most fruitful approaches for the study of structural properties of such graphs is the well-known concept of alternating cycles and their intersections which was introduced by Marušič 20 years ago.In this paper a new parameter for such graphs, giving a further insight into their structure, is introduced. Various properties of this parameter are given and the parameter is completely determined for the tightly attached examples in which any two non-disjoint alternating cycles meet in half of their vertices. Moreover, the obtained results are used to establish a link between two frameworks for a systematic study of all tetravalent graphs admitting a half-arc-transitive group of automorphisms, namely the one proposed by Marušič and Praeger in 1999, and the much more recent one proposed by Al-bar, Al-kenai, Muthana, Praeger and Spiga which is based on the normal quotients method.New results on the graph of alternating cycles of a tetravalent graph admitting a half-arc-transitive group of automorphisms are obtained. A considerable step towards the complete answer to the question of whether the attachment number necessarily divides the radius in tetravalent half-arc-transitive graphs is made.A number of questions and open problems are posed.

On top locally-[formula omitted]-distance-transitive graphs

A graph with some graph symmetry property is top, if it cannot be viewed as a nontrivial normal quotient of some other graph with the same graph symmetry property. Therefore, a graph being top implies that it has no nontrivial normal multicovers, including normal covers. John Conway proved that every s-arc-transitive graph has a nontrivial s-arc-transitive normal cover, so there is no top s-arc-transitive graph. However, there exist top locally-s-distance-transitive graphs, and complete multipartite graphs are examples of this. In this paper, we give a generic condition for locally-s-distance-transitive graphs to be top. Also, examples and characterizations of graphs that admit this condition are given.

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

We study finite four-valent graphs \(\Gamma \) admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on \(\Gamma \), and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph \(\Gamma \) and we classify all examples. We show there are five infinite families of such pairs \((\Gamma ,G)\) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.

A normal quotient analysis for some families of oriented four-valent graphs

We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marusic and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected ‘cross-overs’ between these graph families when we formed normal quotients. We determine which of these oriented graphs are ‘basic’, in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.

Determinants of stem form: Application to Tectona grandis (Linn. F) stands

ABSTRACTStem form information can enhance rapid estimation of stem volume. Studies on important determinants of stem form in pure stands are scanty. This study aimed at identifying determinants of stem form using Tectona grandis stands in Omo Forest Reserve, Nigeria. Twenty 0.04 ha square plots were randomly selected across four stand ages. Crown diameter, total and merchantable heights (MH), diameters at breast height (DBH), at 5.27 m tree height and at midpoint from breast height to tip of tree were measured in each plot. Measured trees were classified into four canopy layers (CL)—dominant, codominant, intermediate, and suppressed. Merchantable volume, basal area (BA), form factor (Ff), normal form quotient (Qn), Girard form class (QG), absolute form quotient (Qa), ratio of diameter at merchantable limit to diameter at the base (Dt/Db), and tree slenderness coefficient were derived from the data. Analysis of variance, correlation, and regression analyses were used to analyze the data. Significant differences were observed in stem form and tree size variables under different canopy layers. The DBH, BA, MHT, and CL were identified as important variables associated with stem form. Inclusion of Qa improved most the predictive ability of the selected stem volume equation.

Finite Edge-Transitive Oriented Graphs of Valency Four: a Global Approach

We develop a new framework for analysing finite connected, oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of `basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restrictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

Locally triangular graphs and normal quotients of the n-cube

For an integer $n\geq 2$, the triangular graph has vertex set the $2$-subsets of $\{1,\ldots,n\}$ and edge set the pairs of $2$-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every $2$-path lies in a unique quadrangle. We refine this result and provide a characterisation of connected locally triangular graphs as halved graphs of normal quotients of $n$-cubes. To do so, we study a parameter that generalises the concept of minimum distance for a binary linear code to arbitrary automorphism groups of the $n$-cube.

Semiregular Automorphisms of Cubic Vertex-Transitive Graphs and the Abelian Normal Quotient Method

We characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.

Pathway Evidence of How Musical Perception Predicts Word-Level Reading Ability in Children with Reading Difficulties

ObjectiveTo investigate whether specific domains of musical perception (temporal and melodic domains) predict the word-level reading skills of eight- to ten-year-old children (n = 235) with reading difficulties, normal quotient of intelligence, and no previous exposure to music education classes. MethodA general-specific solution of the Montreal Battery of Evaluation of Amusia (MBEA), which underlies a musical perception construct and is constituted by three latent factors (the general, temporal, and the melodic domain), was regressed on word-level reading skills (rate of correct isolated words/non-words read per minute). ResultsGeneral and melodic latent domains predicted word-level reading skills.

On normal 2-geodesic transitive Cayley graphs

We investigate connected normal 2-geodesic transitive Cayley graphs Cay(T,S). We first prove that if Cay(T,S) is neither cyclic nor K4[2], then ?a??{1}⊆?S for all a?S. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2-arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.

A Novel Method of Handling Tolerances for Analog Circuit Fault Diagnosis Based on Normal Quotient Distribution

Abstract While the Slope Fault Model method can solve the soft-fault diagnosis problem in linear analog circuit effectively, the challenging tolerance problem is still unsolved. In this paper, a proposed Normal Quotient Distribution approach was combined with the Slope Fault Model to handle the tolerances problem in soft-fault diagnosis for analog circuit. Firstly, the principle of the Slope Fault Model is presented, and the huge computation of traditional Slope Fault Characteristic set was reduced greatly by the elimination of superfluous features. Several typical tolerance handling methods on the ground of the Slope Fault Model were compared. Then, the approximating distribution function of the Slope Fault Characteristic was deduced and sufficient conditions were given to improve the approximation accuracy. The monotonous and continuous mapping between Normal Quotient Distribution and standard normal distribution was proved. Thus the estimation formulas about the ranges of the Slope Fault Characteristic were deduced. After that, a new test-nodes selection algorithm based on the reduced Slope Fault Characteristic ranges set was designed. Finally, two numerical experiments were done to illustrate the proposed approach and demonstrate its effectiveness.