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In this paper, we describe a physical problem, based on electromagnetic fields, whose topological constraints are two-dimensional versions of Kirchhoff’s laws, involving 2-simplicial complexes embedded in $${\mathfrak {R}}^3$$ rather than graphs. The topological constraints are on flux and mmf vectors, which we show satisfy a ‘generalized Tellegen’s Theorem,’ i.e., that they constitute complementary orthogonal vector spaces analogous to voltage and current spaces for ordinary graph-based electrical circuits. We show that the problem of solving this two-dimensional electrical circuit reduces to that of solving an ordinary resistive electrical circuit that can be regarded as dual to it. We present a linear time algorithm for constructing, the above mentioned, dual electrical network. This algorithm is based on the construction of a triangle adjacency graph of the original embedded 2-simplicial complex. This graph contains the information of the order in which we encounter the triangles incident at an edge, when we rotate say clockwise with respect to the orientation of the edge. The triangle adjacency graph is processed through a ‘sliding’ algorithm which simulates sliding on the surfaces of the triangles, moving from one triangle to another which shares an edge with it but which also is adjacent with respect to the embedding of the complex in $${\mathfrak {R}}^3$$. The connectedness information provided by this algorithm is used to construct, in linear time on the size of the 2-simplicial complex, its dual graph.

Purpose To determine the association between Prostate Imaging Reporting and Data System (PI-RADS) version 2 scores and prostate cancer (PCa) in a cohort of patients undergoing biopsy of transition zone (TZ) lesions. Materials and Methods A total of 634 TZ lesions in 457 patients were identified from a prospectively maintained database of consecutive patients undergoing prostate magnetic resonance imaging. Prostate lesions were retrospectively categorized with the PI-RADS version 2 system by two readers in consensus who were blinded to histopathologic findings. The proportion of cancer detection for all PCa and for clinically important PCa (Gleason score ≥3+4) for each PI-RADS version 2 category was determined. The performance of PI-RADS version 2 in cancer detection was evaluated. Results For PI-RADS category 2 lesions, the overall proportion of cancers was 4% (one of 25), without any clinically important cancer. For PI-RADS category 3, 4, and 5 lesions, the overall proportion of cancers was 22.2% (78 of 352), 39.1% (43 of 110), and 87.8% (129 of 147), respectively, and the proportion of clinically important cancers was 11.1% (39 of 352), 29.1% (32 of 110), and 77.6% (114 of 147), respectively. Higher PI-RADS version 2 scores were associated with increasing likelihood of the presence of clinically important PCa (P.001). Differences were found in the percentage of cancers in the PI-RADS category between PI-RADS 3 and those upgraded to PI-RADS 4 based on diffusion-weighted imaging for clinically important cancers (proportion for clinically important cancers for PI-RADS 3 and PI-RADS 3+1 were 11.1% [39 of 352] and 30.8% [28 of 91], respectively; P.001). Conclusion Higher PI-RADS version 2 scores are associated with a higher proportion of clinically important cancers in the TZ. PI-RADS category 2 lesions rarely yield PCa, and their presence does not justify targeted biopsy.

One of the fundamental open problems in control theory is that of the stabilization of a linear time invariant dynamical system through static output feedback. We are given a linear dynamical system defined through \begin{align*} \mydot{w} &= Aw + Bu y &= Cw . \end{align*} The problem is to find, if it exists, a feedback $u=Ky$ such that the matrix $A+BKC$ has all its eigenvalues in the complex left half plane and, if such a feedback does not exist, to prove that it does not. Substantial progress has not been made on the computational aspect of the solution to this problem. In this paper we consider instead `which annihilating polynomials can a matrix of the form $A+BKC$ possess?'. We give a simple solution to this problem when the system has either a single input or a single output. For the multi input - multi output case, we use these ideas to characterize the annihilating polynomials when $K$ has rank one, and suggest possible computational solutions for general $K.$ We also present some numerical evidence for the plausibility of this approach for the general case as well as for the problem of shifting the eigenvalues of the system., 24 pages