Planar Lattice Subsets with Minimal Vertex Boundary

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $X$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $X$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $X$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $G$ of minimal sets whose vertices are congruence classes of minimal sets of $X$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.

A new construction of mutually unbiased maximally entangled bases in ℂq⊗ℂq

This paper is devoted to constructing mutually unbiased maximally entangled bases (MUMEBs) in [Formula: see text], where [Formula: see text] is a prime power. We prove that [Formula: see text] when [Formula: see text] is even, and [Formula: see text] when [Formula: see text] is odd, where [Formula: see text] is the maximal size of the sets of MUMEBs in [Formula: see text]. This highly raises the lower bounds of [Formula: see text] given in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175. It should de noted that the method used in the paper is completely different from that in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175.

Atrial fibrillation is associated with large beat-to-beat variability in mitral and tricuspid annulus dimensions

Aims Beat-to-beat variability in cycle length is well-known in atrial fibrillation (Afib); whether this also translates to variability in annulus size remains unknown. Defining annulus maximal size in Afib is critical for accurate selection of percutaneous devices given the frequent association with mitral and tricuspid valve diseases. Methods and results Images were obtained from 170 patients undergoing 3D echocardiography [100 (50 sinus rhythm (SR) and 50 Afib) for mitral annulus (MA) and 70 (35 SR and 35 Afib) for tricuspid annulus (TA)]. Images were analysed for differences in annular dynamics with a commercially available software. Number of cardiac cycles analysed was 567 in mitral valve and 346 in tricuspid valve. Median absolute difference in maximal MA area over four to six cycles was 1.8 cm2 (range 0.5–5.2 cm2) in Afib vs. 0.8 cm2 (range 0.1–2.9 cm2) in SR, P < 0.001. Maximal MA area was observed within 30–70% of the R-R interval in 81% of cardiac cycles in SR and in 73% of cycles in Afib. Median absolute difference in maximal TA area over four to six cycles was 1.4 cm2 (range 0.5–3.6 cm2) in Afib vs. 0.7 cm2 (range 0.3–1.7 cm2) in SR, P < 0.001. Maximal TA area was observed within 60–100% of the R-R interval in 81% of cardiac cycles in SR, but only in 49% of cycles in Afib. Conclusion MA and TA reach maximal size within a broad time interval centred around end-systole and end-diastole, respectively, with significant beat-to-beat variability. Afib leads to a larger beat-to-beat variability in both timing of occurrence and values of annulus size than in SR.

Radius of Robust Feasibility for Mixed-Integer Problems

For a mixed-integer linear problem (MIP) with uncertain constraints, the radius of robust feasibility (RRF) determines a value for the maximal size of the uncertainty set such that robust feasibility of the MIP can be guaranteed. The approaches for the RRF in the literature are restricted to continuous optimization problems. We first analyze relations between the RRF of a MIP and its continuous linear (LP) relaxation. In particular, we derive conditions under which a MIP and its LP relaxation have the same RRF. Afterward, we extend the notion of the RRF such that it can be applied to a large variety of optimization problems and uncertainty sets. In contrast to the setting commonly used in the literature, we consider for every constraint a potentially different uncertainty set that is not necessarily full-dimensional. Thus, we generalize the RRF to MIPs and to include safe variables and constraints; that is, where uncertainties do not affect certain variables or constraints. In the extended setting, we again analyze relations between the RRF for a MIP and its LP relaxation. Afterward, we present methods for computing the RRF of LPs and of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF. Summary of Contribution: Robust optimization is an important field of operations research due to its capability of protecting optimization problems from data uncertainties that are usually defined via so-called uncertainty sets. Intensive research has been conducted in developing algorithmically tractable reformulations of the usually semi-infinite robust optimization problems. However, in applications it also important to construct appropriate uncertainty sets (i.e., prohibiting too conservative, intractable, or even infeasible robust optimization problems due to the choice of the uncertainty set). In doing so, it is useful to know the maximal “size” of a given uncertainty set such that a robust feasible solution still exists. In this paper, we study one notion of “size”: the radius of robust feasibility (RRF). We contribute on the theoretical side by generalizing the RRF to MIPs as well as to include “safe” variables and constraints (i.e., where uncertainties do not affect certain variables or constraints). This allows to apply the RRF to many applications since safe variables and constraints exist in most applications. We also provide first methods for computing the RRF of LPs as well as of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF.

A sharp upper bound for the size of Lusztig series

The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group 𝐺 of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group 𝐺 in terms of its Lie rank and defining characteristic. When 𝐺 is specified as G(q) and 𝑞 is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of 𝐺.

Bounding the maximal size of independent generating sets of finite groups

Denote by m(G) the largest size of a minimal generating set of a finite group G. We estimate m(G) in terms of $\sum _{p\in \pi (G)}d_p(G),$ where we are denoting by d p (G) the minimal number of generators of a Sylow p-subgroup of G and by π(G) the set of prime numbers dividing the order of G.

k-stretchability of entanglement, and the duality of k-separability and k-producibility

The notions of k-separability and k-producibility are useful and expressive tools for the characterization of entanglement in multipartite quantum systems, when a more detailed analysis would be infeasible or simply needless. In this work we reveal a partial duality between them, which is valid also for their correlation counterparts. This duality can be seen from a much wider perspective, when we consider the entanglement and correlation properties which are invariant under the permutations of the subsystems. These properties are labeled by Young diagrams, which we endow with a refinement-like partial order, to build up their classification scheme. This general treatment reveals a new property, which we call k-stretchability, being sensitive in a balanced way to both the maximal size of correlated (or entangled) subsystems and the minimal number of subsystems uncorrelated with (or separable from) one another.

The organisation and structure of rhotics in American English rhymes

Language-specific maximal size restrictions on syllables have been defined using frames such as moraic structure. In General American English, a trimoraic syllable template makes largely successful predictions about contexts where tense/lax vowel contrasts are neutralised, but neutralisation preceding a coda rhotic has not been adequately explained. We attribute the apparent special properties of coda /ɹ/ to two characteristics of its representation, informed by our articulatory investigation: sequential coordination of dorsal and coronal subsegmental units and a high blending strength specification, corresponding to high coarticulatory dominance. Characteristics of coda laterals are compared. Our approach employs phonological representations where sequencing is encoded directly among subsegments, and coordination is sensitive to strength. Mora assignment is computed over sequencing of subsegments, predicting that complex segments may be bimoraic. The account brings phonotactics for rhymes with postvocalic liquids into line with the trimoraic template, and supports representing coordination and strength at the subsegmental level.