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Background: Glenoid bone loss (GBL) is common in patients with shoulder instability and plays a major role in surgical decision-making. While a plethora of GBL estimation methods exist, all of which present specific challenges, recent studies have developed simple linear formulas estimating GBL based on glenoid height. Purpose: To assess the correlation between glenoid height and width, and to develop specific formulas based on age and sex to calculate the native glenoid width in the Lebanese population. Study Design: Cross-sectional study; Level of evidence, 3. Methods: Computed tomography scans for 202 normal shoulders were extracted from our database. The glenoids were reconstructed in 3 dimensions and their width and height were measured. Glenoid width and height were compared between male and female groups. Correlation analysis was also performed on the width, height, age, and body mass index. Formulas estimating glenoid width were developed using regression analysis including all variables significantly influencing the model. Results were then compared with the values calculated using previously published formulas to determine the external validity when using linear formulas to estimate GBL. Results: Significant differences were found between men and women. Regression analysis found that glenoid height and width strongly influenced the model, and that age showed a weak but significant correlation; therefore, the following 2 sex-specific formulas were developed: width (mm) = 6.1 + 0.51 × height + 0.03 × age, and width (mm) = 4.55 + 0.51 × height + 0.03 × age, in men and women, respectively. The values yielded from the formulas developed in this study and the true width significantly differed from those calculated from previous reports. Conclusion: A strong correlation was found between glenoid height and width in a the Lebanese population and demonstrated that glenoid width can be accurately calculated based on the glenoid height and patient's age and sex using the following simplified formulas: width (mm) = 6 + 0.5 × height + 0.03 × age, and width (mm) = 4.5 + 0.5 × height + 0.03 × age, in men and women, respectively. [ABSTRACT FROM AUTHOR]

We prove that continuous sentences preserved by the ultramean construction (a generalization of the ultraproduct construction) are exactly those sentences which are approximated by linear sentences. Continuous sentences preserved by linear elementary equivalence are exactly those sentences which are approximated in the Riesz space generated by linear sentences. Also, characterizations for linear Δn-sentences and positive linear theories will be given. [ABSTRACT FROM AUTHOR]

A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n -ary operations defined on set A for all natural numbers n ≥ 1 and the operations are the so-called superposition operations S m n for natural numbers m , n ≥ 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties. [ABSTRACT FROM AUTHOR]

A term t is linear if no variable occurs more than once in t. An identity s ≈ t is said to be linear if s and t are linear terms. Identities are particular formulas. As for terms superposition operations can be defined for formulas too. We define the arbitrary linear formulas and seek for a condition for the set of all linear formulas to be closed under superposition. This will be used to define the partial superposition operations on the set of linear formulas and a partial many-sorted algebra Formclonelin(τ, τ′). This algebra has similar properties with the partial many-sorted clone of all linear terms. We extend the concept of a hypersubstitution of type τ to the linear hypersubstitutions of type (τ, τ′) for algebraic systems. The extensions of linear hypersubstitutions of type (τ, τ′) send linear formulas to linear formulas, presenting weak endomorphisms of Formclonelin(τ, τ′). [ABSTRACT FROM AUTHOR]

Abstract: and are important variants of the propositional satisfiability problem (SAT). XSAT contains all CNF formulas that can be satisfied by setting exactly one literal in each clause to 1, whereas NAE-SAT requires satisfying truth assignments not setting all literals equally in any clause. Both variants are studied here regarding their computational complexity of linear CNF formulas, whose clauses are allowed to have at most one variable in common. We prove that both variants remain -complete for (monotone) linear formulas, yielding the conclusion that also bicolorability of linear hypergraphs is NP-complete. The reduction used gives rise to the complexity investigations of several monotone linear subclasses of both variants that are parameterized by the size of clauses, or by the number of occurrences of variables. For particular values of those parameters, we are able to show the -completeness of and , though we cannot provide a complete treatment. Finally, we focus on exact linear formulas, where clauses intersect pairwise, and for which SAT is known to be polynomial-time solvable. Here, we prove -completeness of for exact linear formulas. If in addition the clauses are required to be of uniform length, we show that both XSAT and NAE-SAT are decidable in polynomial time, what even holds for their counting versions. [Copyright &y& Elsevier]

We define an ultraproduct of metric structures based on a maximal probability charge and prove a variant of Łoś theorem for linear metric formulas. We also consider iterated ultraproducts (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]