Maximal Rank Research Articles

Representation Matters: One Urology Residency Program's Approach to Increasing Workforce Diversity

ObjectiveTo address the challenge in urology in recruiting physicians from backgrounds racially and ethnically underrepresented in medicine (URiM), we sought to design, implement, and evaluate methods for recruiting URiM candidates to our urology residency program. MethodsWe developed a 3-pronged approach aimed at increasing the number of interviewed applicants, and subsequently number of URiM residents recruited to our program. The 3 facets included: (1) funded visiting student rotation, (2) holistic evaluation of applications, (3) implemented targeted outreach. Statistical analysis of the applicants interviewed and matched into our residency program, as well as traditional metrics used for residency recruitment, were performed from 2015 to 2022. ResultsThe number of URiM interviewees significantly increased from 6.1% in 2015 to its peak, 40%, in 2020. In 2015, there were no URiM residents in our urology residency program. By 2022, the total URiM complement increased to 35%. In evaluating traditional metrics of residency recruitment, there was no significant difference in mean USMLE Step 1 score before compared with after the implementation of our recruitment approach. The maximum rank number reached to fill the urology residency positions also remained relatively stable throughout the study period, with a range from 5 to 38. ConclusionWe demonstrate that the implementation of our innovative and intentional 3-pronged recruitment approach effectively increased the number of URiM interviewees and residents in our residency program. The diversification of our urology workforce depends on the implementation of such efforts, and we encourage urologists to lead the way on such initiatives.

Geometry of solutions to the c-projective metrizability equation

On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a nondegenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions the underlying manifold into strata, in a manner that generalizes the model, a certain Lie group orbit decomposition of \(\mathbb{C}\mathbb{P}^m\). We describe the smooth nature and geometric structure of each strata component, generalizing the geometries of the embedded orbits in the model. This includes a quasi-Kähler metric on the open strata components that becomes singular at the strata boundary. The closed strata inherit almost CR-structures and can be viewed as a c-projective infinity for the given quasi-Kähler metric.

Composite Numerical Methods and Scalable Tile Algorithms

Numerical modeling is an important tool when studying various natural processes and phenomena. Fractional diffusion can be used for modeling many processes in biology, for example in silico experiments in molecular biology and medicine design, protein diffusion within cells, complex media geometry, etc. The problem is usually reduced to a system of linear algebraic equations and in many cases this system has a dense coefficient matrix. Numerically solving such problems with the traditional LU factorization is a computationally expensive endeavour - $O(n^3)$. In this paper we explore the use of a hierarchical compression method based on Hierarchical Semi-Separable compression and ULV-like factorization from the STRUctured Matrices PACKage (STRUMPACK) software library for a flow around airfoils problem discretized with Boundary Element Method and fractional diffusion problem discretized with the Finite Element Method. The HSS based method promises better overall computational complexity of $O(r^2n)$ for problems with suitable structure - low rank off-diagonal blocks. Here $r$ is the maximum rank of the off-diagonal blocks. We present analysis of the performance and accuracy of the HSS based method and compare it with the state of the art direct LU factorization solvers. This paper is based on the PhD thesis Composite Numerical Methods and Scalable Tile Algorithms of the author defended on 17.05.2022 in the Institute of Information and Communication Technologies at the Bulgarian Academy of Sciences.

HYPERBOLIC MANIFOLDS THAT FIBRE ALGEBRAICALLY UP TO DIMENSION 8

AbstractWe construct some cusped finite-volume hyperbolic n-manifolds $M^n$ that fibre algebraically in all the dimensions $5\leq n \leq 8$ . That is, there is a surjective homomorphism $\pi _1(M^n) \to {\mathbb {Z}}$ with finitely generated kernel. The kernel is also finitely presented in the dimensions $n=7, 8$ , and this leads to the first examples of hyperbolic n-manifolds $\widetilde M^n$ whose fundamental group is finitely presented but not of finite type. These n-manifolds $\widetilde M^n$ have infinitely many cusps of maximal rank and, hence, infinite Betti number $b_{n-1}$ . They cover the finite-volume manifold $M^n$ . We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes $P^5, \ldots , P^8$ , and then applying some arguments of Jankiewicz, Norin and Wise [18] and Bestvina and Brady [7]. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to $P^n$ , and the algebra of integral octonions for the crucial dimensions $n=7,8$ .

Maximal connected 𝑘-subgroups of maximal rank in connected reductive algebraic 𝑘-groups

Let k k be any field and let G G be a connected reductive algebraic k k -group. Associated to G G is an invariant first studied in the 1960s by Satake [Ann. of Math. (2) 71 (1960), 77–110] and Tits [Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962], [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] that is called the index of G G (a Dynkin diagram along with some additional combinatorial information). Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] showed that the k k -isogeny class of G G is uniquely determined by its index and the k k -isogeny class of its anisotropic kernel G a G_a . For the cases where G G is absolutely simple, all possibilities for the index of G G have been classified in by Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62]. Let H H be a connected reductive k k -subgroup of maximal rank in G G . We introduce an invariant of the G ( k ) G(k) -conjugacy class of H H in G G called the embedding of indices of H ⊂ G H \subset G . This consists of the index of H H and the index of G G along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of k k -subgroups of G G , and observe that the G ( k ) G(k) -conjugacy class of H H in G G is determined by its index-conjugacy class and the G ( k ) G(k) -conjugacy class of H a H_a in G G . We show that the index-conjugacy class of H H in G G is uniquely determined by its embedding of indices. For the cases where G G is absolutely simple of exceptional type and H H is maximal connected in G G , we classify all possibilities for the embedding of indices of H ⊂ G H \subset G . Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when k k has cohomological dimension 1 (resp. k = R k=\mathbb {R} , k k is p \mathfrak {p} -adic).

Equivalence and automorphism groups of two families of maximum scattered linear sets

Linear set in projective spaces over finite fields plays central roles in the study of blocking sets, semifields, rank-metric codes and etc. A linear set with the largest possible cardinality and the maximum rank is called maximum scattered. Despite two decades of study, there are only a few number of known maximum scattered linear sets in projective lines, including the family constructed by Csajb\'ok, Marino, Polverino and Zanella 2018, and the family constructed by Csajb\'ok, Marino, Zullo 2018 (also Marino, Montanucci, and Zullo 2020). This paper aims to solve the equivalence problem of the linear sets in each of these families and to determine their automorphism groups.

Rational curves and maximal rank in multiprojective spaces

Rational curves and maximal rank in multiprojective spaces

Maximally non-integrable almost complex structures: an h-principle and cohomological properties

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an h-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension 2nge 10 (respectively ge 6) admit a almost complex structure whose Nijenhuis tensor has maximal rank everywhere (resp. is nowhere trivial). For closed 4-manifolds, the existence of such structures is characterized in terms of topological invariants. Moreover, we show that the Dolbeault cohomology of non-integrable almost complex structures is often infinite dimensional (even on compact manifolds).

Sample distribution theory using Coarea Formula

Let ( Ω , Σ , p ) be a probability measure space and let X : Ω → R k be a (vector valued) random variable. We suppose that the probability pX induced by X is absolutely continuous with respect to the Lebesgue measure on R k and set fX as its density function. Let ϕ : R k → R n be a C 1-map and let us consider the new random variable Y = ϕ ( X ) : Ω → R n . Setting m : = max { rank ( J ϕ ( x ) ) : x ∈ R k } , we prove that the probability pY induced by Y has a density function fY with respect to the Hausdorff measure H m on ϕ ( R k ) which satisfies f Y ( y ) = ∫ ϕ − 1 ( y ) f X ( x ) 1 J m ϕ ( x ) d H k − m ( x ) , for H m − a . e . y ∈ ϕ ( R k ) . Here J m ϕ is the m-dimensional Jacobian of ϕ . When J ϕ has maximum rank we allow the map ϕ to be only locally Lipschitz. We also consider the case of X having probability concentrated on some m-dimensional sub-manifold E ⊆ R k and provide, besides, several examples including algebra of random variables, order statistics, degenerate normal distributions, Chi-squared and “Student's t” distributions.

An asymptotically optimal construction of almost affinely disjoint subspaces

A family of k-dimensional subspaces of Fqn, which forms a partial spread, is called L-almost affinely disjoint (or briefly [n,k,L]q-AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family. The notion of AADs was introduced and its connections with some problems in coding theory were presented by Liu et al. (2021) in [27]. In particular, Liu et al. investigated the size of maximal such sets and presented a conjecture. They also proved that their conjecture is true for k=1 and k=2 when L is sufficiently large. In this work, we give some constructions of large AADs for n≥3k, hence•improve the lower bound of Liu et al. and•prove that their conjecture is still true for k≥3 when n=3k. Our constructions are based on the usage of some certain maximum rank distance codes together with a modified version of the lifting idea and linkage construction.

Constructing reducible Brill–Noether curves II

This paper is the second of a two-part series by the author devoted to the following fundamental problem in the theory of algebraic curves in projective space: Which reducible curves arise as limits of smooth curves of general moduli? Special cases of this question studied by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ), and others have been critical in the resolution of many problems in the theory of algebraic curves over the past half century. In this paper, we give sharp bounds on this problem for space curves, when the nodes are general points and the components are general in moduli. We also systematically study a variant in projective spaces of arbitrary dimension when the nodes are general in a hyperplane. The results given here significantly extend those cases established in the previous paper in this series (Eric Larson, Constructing reducible Brill-Noether curves, To appear in documentamathematica, arxiv:1603.02301), as well as those cases established by Sernesi (Sernesi, Edoardo (1984) On the existence of certain families of curves. Invent Math 75(1): 488 25-57 ), Ballico (Ballico, Edoardo (2012) Embeddings of general curves in projective spaces: the range of the quadrics. Lith Math J 52(2): 134-137 ) , and others. As explained in (Eric Larson, Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980), the reducible curves constructed in this paper also play a critical role in the author’s proof of the maximal rank conjecture in a subsequent paper (Eric Larson Degenerations of curves in projective space and the maximal rank conjecture, arXiv:1809.05980).

Quantitative Ultrasound Assessment of Early Osteoarthritis in Human Articular Cartilage Using a High-Frequency Linear Array Transducer.

Quantitative ultrasound (QUS) assessment of osteoarthritis (OA) using high-frequency, research-grade single-element ultrasound systems has been reported. The objective of this ex vivo study was to assess the performance of QUS in detecting early OA using a high-frequency linear array transducer. Osteochondral plugs (n=26) of human articular cartilage were scanned with ExactVu Micro-Ultrasound using an EV29L side-fire transducer. For comparison, the samples were also imaged with SAM200Ex, a custom 40-MHz scanning acoustic microscope with a single-element, focused transducer. Thirteen QUS parameters were derived from the ultrasound data. Magnetic resonance imaging (MRI) data, with T1 and T2 extracted as the quantitative parameters, were also acquired for comparison. Cartilage degeneration was graded from histology and correlated to all quantitative parameters. A maximum Spearman rank correlation coefficient (ρ) of 0.75 was achieved using a combination of ExactVu QUS parameters, while a maximum ρ of 0.62 was achieved using a combination of parameters from SAM200Ex. A maximum ρ of 0.75 was achieved using the T1 and T2 MRI parameters. This study illustrates the potential of a high-frequency linear array transducer to provide a convenient method for early OA screening with results comparable to those of research-grade single-element ultrasound and MRI.

Designing a Winner–Loser Gap for WTA in Subthreshold. Resolution Performance Revisited

The paper pioneers a thorough mathematical approach for the Lazzaro variant of the W(inner) T(ake) A(ll) maximum rank and amplitude analog selector. Two exact levels of output which split the maximum and determine the resolution, are found for the first time. At the input, a list of currents left( I_{1},I_{2},ldots ,I_{N}right) from a large family {{mathcal {L}}} with smallest relative distance Delta on a left[ 0,I_{M}right] scale is applied. To distinguish the largest current I_{w} (the winner) from the second largest I_{l} (the loser), the paper proposes two decision levels, {overline{D}} and {underline{D}}, for the output voltage list left( U_{1},U_{2},ldots ,U_{N}right) . The upper level {overline{D}} is surpassed only by the U_{w} winner and encodes the winning rank w. All other ranks are placed under the lower level {underline{D}}. Two rigorously treated optimization problems with inequality constraints lead to the identification of two input lists that yield the levels {overline{D}} and {underline{D}} as outputs. They are valid for processing any list in the {mathcal {L}} family. The index left( {overline{D}}-{underline{D}}right) /U_{M}—“the output resolution”—expresses how large the gap between the first and the second component on the left[ 0,U_{M}right] scale is. It exceeds “the input resolution,” i.e., the similar index Delta /I_{M} at the input and the two depend monotonically on each other. Widely commented numerical examples are presented.

A Blueprint for Increasing Ethnic and Racial Diversity in U.S. Residency Training Programs.

People who identify as African Americans, Latinos, or from indigenous backgrounds, are dramatically underrepresented in the U.S. physician workforce. It is critical for academic health centers to recognize racial and ethnic diversity at the residency level and implement changes to enhance diversity among trainees. The Office of Graduate Medical Education (GME) at the University of Pennsylvania Health System (UPHS) developed a multipronged approach to enhance diversity and inclusion (D&I) among residency trainees. The approach included the development of an underrepresented in medicine (UIM) professional network; UIM-focused visiting clerkship programs; holistic review implementation by selection committees; and targeted outreach to UIM candidates, overseen by an associate designated institutional official for UIM Affairs. The authors reported demographic data on residency applicants invited for interviews and matching for all programs at UPHS from 2014-2015 (baseline) to 2020-2021. They also reported data on maximum ranking number programs reached to fill their positions and the average United States Medical License Examination (USMLE) Step 1 scores of matched candidates. Finally, they discussed the implications for leaders who wish to enhance D&I at academic health centers. During the baseline year (2014-2015), UIMs represented 12.1% of interviewees and 8.7% of all matched candidates into UPHS residency programs. Over the successive 6 years after incremental implementation of the approach, UIM representation steadily increased. In 2020-2021, UIMs represented 23.2% of interviewees and 26.4% of matched candidates. Programs' maximum rank number to fill and USMLE Step 1 scores of matched candidates remained relatively unchanged. The UPHS Office of GME incorporated a purposeful approach to enhance the D&I of its residents. Across 6 years of implementation, UIM representation among resident matches tripled while quantitative program and candidate metrics remained unchanged. Similar efforts should be given further consideration for implementation and evaluation nationwide.

Torsion free endotrivial modules for finite groups of Lie type

In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $\ell$-subgroups in any finite group of Lie type, for any prime $\ell$, which may be of independent interest.

On the classification of nilpotent Lie algebras of maximal rank

On the classification of nilpotent Lie algebras of maximal rank

Scattered linear sets in a finite projective line and translation planes

In [15], Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in PG(1,qt). In this paper a similar construction of a translation plane Af obtained from any scattered linearized polynomial f(x) in Fqt[x] is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by Uf the Fq-subspace of Fqt2 associated with f(x). If f(x) and f′(x) are scattered, then Af and Af′ are isomorphic if and only if Uf and Uf′ belong to the same orbit under the action of ΓL(2,qt). This gives rise to the same number of distinct translation planes as the number of inequivalent scattered linearized polynomials. In particular, for any scattered linear set L of maximum rank in PG(1,qt) there are cΓ(L) pairwise non-isomorphic translation planes, where cΓ(L) denotes the ΓL-class of L, as defined in [5] by Csajbók, Marino and Polverino. A result by Jha and Johnson [8] allows to describe the automorphism groups of the planes obtained from the linear sets not of pseudoregulus type defined in [15].

An Analytic and Computational Condition for the Finite Degree-of-Freedom of Linkages, and Its Relation to Lie Group Methods

Abstract The finite degree of freedom (DOF) of a mechanism is determined by the number of independent loop constraints. In this paper, a method to determine the maximal number of loop closure constraints (which is independent of a specific configuration) of multiloop linkages is introduced and applied to calculate the finite DOF. It rests on an algebraic condition on the joint screws, which stems from the analytic condition that minors of certain rank and their higher derivatives vanish. A computational algorithm is presented to determine the maximal rank of the constraint Jacobian in an arbitrary (possibly singular) reference configuration. This algorithm involves screw products of constant screw vectors only. Unlike the Lie group methods for estimating the DOF of so-called exceptional linkages, this method does not rely on partitioning kinematic loops into partial kinematic chains, and it is applicable to multiloop linkages. The DOF computed with this method is at least as accurate as the DOF computed with the Lie group methods. It gives the correct DOF for any (possibly overconstrained) linkage where the constraint Jacobian has maximal rank in regular configurations. The so determined maximal rank has further significance for classifying linkages as being exceptional or paradoxical but also for detecting singularities and shaky linkages.

Linearized maximum rank correlation estimation

Summary We propose a linearized maximum rank correlation estimator for the single-index model. Unlike the existing maximum rank correlation and other rank-based methods, the proposed estimator has a closed-form expression, making it appealing in theory and computation. The proposed estimator is robust to outliers in the response and its construction does not need knowledge of the unknown link function or the error distribution. Under mild conditions, it is shown to be consistent and asymptotically normal when the predictors satisfy the linearity of the expectation assumption. A more general class of estimators is also studied. Inference procedures based on the plug-in rule or random weighting resampling are employed for variance estimation. The proposed method can be easily modified to accommodate censored data. It can also be extended to deal with high-dimensional data combined with a penalty function. Extensive simulation studies provide strong evidence that the proposed method works well in various practical situations. Its application is illustrated with the Beijing PM 2.5 dataset.

Extending two families of maximum rank distance codes

In this paper, we properly extend the family of rank-metric codes recently found by Longobardi and Zanella (2021) and by Longobardi, Marino, Trombetti and Zhou (2021). These codes are Fq2t-linear of dimension 2 in the space of linearized polynomials over Fq2t, where t is any integer greater than 2, and we prove that they are maximum rank distance codes. For t≥5, we determine their equivalence classes and these codes turn out to be inequivalent to any other construction known so far, and hence they are really new.