Maximal Intersection Research Articles

The spiderweb of error-detecting codes in the genetic information

Nature possesses inherent mechanisms for error detection and correction during the translation of genetic information, as demonstrated by the discovery of a self-complementary circular C3-code called X0 in various organisms such as bacteria, eukaryotes, plasmids, and viruses (Arquès and Michel, 1996; Michel, 2015, 2017). Since then, extensive research has focused on circular codes, which are believed to be remnants of ancient comma-free codes. These codes can be regarded as an additional genetic code specifically optimized for detecting and preserving the proper reading frame in protein-coding sequences. A study by Fimmel et al. in 2014 identified that a total of 216 maximal self-complementary C3-codes can be grouped into 27 equivalence classes with eight codes in each class.In this work, we study how the 27 equivalence classes are related to each other. While the codes in each equivalence class obtained by Fimmel et al. in 2014 are permutations of each other, i.e. one code can be obtained from the other by applying a permutation of the bases, it has not been clear how the equvalence classes are connected. We show that there is an ordering of the equivalence classes such that one gets from one class to the next one by substituting only one pair of codon/anticodon in the corresponding codes, i.e. the corresponding codes have a maximal intersection of 18 codons. To perform this analysis, we define two graphs, G216 and G27, whose vertices are, respectively, all 216 maximal self-complementary C3-codes and 27 equivalence classes. Several properties of the graphs are obtained. Most surprisingly, it turns out that G27 contains Hamiltonian paths of length 27. This fact ultimately leads to a representation of the set of all 216 maximal self-complementary C3-codes as a kind of spider web. Finally, we define dinucleotide cuts of such codes by projecting each codon to its first two bases and show that the paths of lengths 27 in G216 can even be chosen so that all the codes contain a special subset of dinucleotides defined by Rumer’s roots. These observations raise a lot of new questions about the biological function of such structures.

Probabilistic Properties of Zipf Sets and Their Maximal Intersections

In the article, the weak and strong laws of large numbers are obtained for various characteristics of a generalized Zipf set. Based on these results, we investigate the maximal intersection size between a random Zipf set and the elements of a large base of independent random sets of the same type but, eventually, with different parameters. This question is significant for applications.

Probabilistic properties of Zipf sets and their maximal intersections

В работе получены слабые и усиленные законы больших чисел для различных характеристик обобщенного случайного множества Ципфа. На их основе исследован интересный для приложений вопрос о размере максимального пересечения случайного множества Ципфа с элементами большого набора независимых множеств того же типа, но, возможно, с другими параметрами.

Some New Positions of Maximal Volume of Convex Bodies

In this paper, we extend and generalize several previous works on maximal volume positions of convex bodies. First, we analyze the maximal positive-definite image of one convex body inside another, and the resulting decomposition of the identity. We discuss continuity and differentiability of the mapping associating a body with its positive John position. We then introduce the saddle-John position of one body inside another, proving that it shares some of the properties possessed by the position of maximal volume, and explain how this can be used to improve volume ratio estimates. We investigate several examples in detail and compare these positions. Finally, we discuss the maximal intersection position of one body with respect to another, and show the existence of a natural decomposition of identity associated to this position, extending previous work which treated the case when one of the bodies is the Euclidean ball.

On Explicit Representations of Isotropic Measures in John and Löwner Positions

Abstract Given a convex body $K \subseteq {{\mathbb {R}}}^n$ in Löwner position, we study the problem of constructing a non-negative centered isotropic measure supported in the contact points, whose existence is guaranteed by John’s Theorem. The method we propose requires the minimization of a convex function defined in an $\frac {n(n+3)}2$ dimensional vector space. We find a geometric interpretation of the minimizer as $\left . \frac {\partial }{\partial r}(A_r, v_r)\right |_{r=1}$, where $A_r K + v_r$ is a one-parameter family of positions of $K$ that are in some sense related to the maximal intersection position of radius$r$ defined recently by Artstein-Avidan and Katzin.

Combinatorial k-systoles on a punctured torus and a pair of pants

In this paper, $S$ denotes a surface homeomorphic to a punctured torus or a pair of pants. Our interest is the study of \emph{\textbf{combinatorial $k$-systoles}} that is closed curves with self-intersection numbers greater than $k$ and with least combinatorial length. We show that the maximal intersection number $I^c_k$ of combinatorial $k$-systoles of $S$ grows like $k$ and $\underset{k\rightarrow+\infty}{\limsup}(I^c_k-k)=+\infty$. This result, in case of a pair of pants and a punctured torus, is a positive response to the combinatorial version of the Erlandsson - Parlier conjecture, originally formulated for the geometric length.

Maximal Intersections in Finite Groups

For a finite group G, we investigate the behavior of four invariants, text {MaxDim}(G),text {MinDim}(G),text {MaxInt}(G) and text {MinInt}(G), measuring in some way the width and the height of the lattice {mathcal {M}}(G) consisting of the intersections of the maximal subgroups of G.

Sheaves of maximal intersection and multiplicities of stable log maps

A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.

Generalized Tonnetze and Zeitnetze, and the topology of music concepts

The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies – whether orientable, bounded, manifold, etc. – reveal some of the topological character of musical concepts.

Intersections of maximal subgroups in prosolvable groups

Given a positive real number we say that an open subgroup H of a profinite group G is an η-intersection if there exists a family of maximal subgroups such that and . We study the finitely generated prosolvable groups G with the property that there exists η such that every maximal intersection in G is an η-intersection. We prove in particular that this is equivalent to a condition on the centralizers in G of the complemented chief factors of G. We use the notion of η-intersection to address an open question asking whether the number of maximal intersections of index n in a finitely generated prosolvable group G is polynomially bounded. We prove that the answer is positive if the derived subgroup of G is pronilpotent.

Isotropic measures and maximizing ellipsoids: Between John and Loewner

We define a one-parametric family of positions of a centrally symmetric convex body K K which interpolates between the John position and the Loewner position: for r > 0 r>0 , we say that K K is in maximal intersection position of radius r r if Vol n ( K ∩ r B 2 n ) ≥ Vol n ( K ∩ r T B 2 n ) \textrm {Vol}_{n}(K\cap rB_{2}^{n})\geq \textrm {Vol}_{n}(K\cap rTB_{2}^{n}) for all T ∈ S L n T\in \rm {SL}_{n} . We show that under mild conditions on K K , each such position induces a corresponding isotropic measure on the sphere, which is simply the normalized Lebesgue measure on r − 1 K ∩ S n − 1 r^{-1}K\cap S^{n-1} . In particular, for r M r_{M} satisfying r M n κ n = Vol n ( K ) r_{M}^{n}\kappa _{n}=\textrm {Vol}_{n}(K) , the maximal intersection position of radius r M r_{M} is an M M -position, so we get an M M -position with an associated isotropic measure. Lastly, we give an interpretation of John’s theorem on contact points as a limit case of the measures induced from the maximal intersection positions.

On the Approximation of Convex Bodies by Ellipses with Respect to the Symmetric Difference Metric

Given a centrally symmetric convex body \(K \subset \mathbb {R}^d\) and a positive number \(\lambda \), we consider, among all ellipsoids \(E \subset \mathbb {R}^d\) of volume \(\lambda \), those that best approximate K with respect to the symmetric difference metric, or equivalently that maximize the volume of \(E\cap K\): these are the maximal intersection (MI) ellipsoids introduced by Artstein-Avidan and Katzin. The question of uniqueness of MI ellipsoids (under the obviously necessary assumption that \(\lambda \) is between the volumes of the John and the Loewner ellipsoids of K) is open in general. We provide a positive answer to this question in dimension \(d=2\). Therefore we obtain a continuous 1-parameter family of ellipses interpolating between the John and the Loewner ellipses of K. In order to prove uniqueness, we show that the area \(I_K(E)\) of the intersection \(K \cap E\) is a strictly quasiconcave function of the ellipse E, with respect to the natural affine structure on the set of ellipses of area \(\lambda \). The proof relies on smoothening K, putting it in general position, and obtaining uniform estimates for certain derivatives of the function \(I_K(\mathord {\cdot })\). Finally, we provide a characterization of maximal intersection positions, that is, the situation where the MI ellipse of K is the unit disk, under the assumption that the two boundaries are transverse.

Attribute reduction based on max-decision neighborhood rough set model

The neighborhood rough set model only focuses on the consistent samples whose neighborhoods are completely contained in some decision classes, and ignores the divisibility of the boundary samples whose neighborhoods can not be contained in any decision classes. In this paper, we pay close attention to the boundary samples, and enlarge the positive region by adding the samples whose neighborhoods have maximal intersection with some decision classes. Applying the mentioned idea, we introduce a new neighborhood rough set model, named max-decision neighborhood rough set model. An attribute reduction algorithm is designed based on the model. Both theoretical analysis and experimental results show that the proposed algorithm is effective for removing most redundant attributes without loss of classification accuracy.

Wall Crossing Invariants from Spectral Networks

A new construction of BPS monodromies for 4d {mathcal {N}}=2 theories of class {mathcal {S}} is introduced. A novel feature of this construction is its manifest invariance under Kontsevich–Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve C of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For A_1-type theories, the graphs encoding the monodromy are “dessins d’enfants” on C, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg–Witten curve.

BPS graphs: from spectral networks to BPS quivers

We define “BPS graphs” on punctured Riemann surfaces associated with AN −1 theories of class mathcal{S} . BPS graphs provide a bridge between two powerful frameworks for studying the spectrum of BPS states: spectral networks and BPS quivers. They arise from degenerate spectral networks at maximal intersections of walls of marginal stability on the Coulomb branch. While the BPS spectrum is ill-defined at such intersections, a BPS graph captures a useful basis of elementary BPS states. The topology of a BPS graph encodes a BPS quiver, even for higher-rank theories and for theories with certain partial punctures. BPS graphs lead to a geometric realization of the combinatorics of Fock-Goncharov N - triangulations and generalize them in several ways.

An asymmetric multi-item auction with quantity discounts applied to Internet service procurement in Buenos Aires public schools

This article studies a multi-item auction characterized by asymmetric bidders and quantity discounts. We report a practical application of this type of auction in the procurement of Internet services to the 709 public schools of Buenos Aires. The asymmetry in this application is due to firms’ existing technology infrastructures, which affect their ability to provide the service in certain areas of the city. A single round first-price sealed-bid auction, it required each participating firm to bid a supply curve specifying a price on predetermined graduated quantity intervals and to identify the individual schools it would supply. The maximal intersections of the sets of schools each participant has bid on define regions we call competition units. A single unit price must be quoted for all schools supplied within the same quantity interval, so that firms cannot bid a high price where competition is weak and a lower one where it is strong. Quantity discounts are allowed so that the bids can reflect returns-to-scale of the suppliers and the auctioneer may benefit of awarding bundles of units instead of separate units. The winner determination problem in this auction poses a challenge to the auctioneer. We present an exponential formulation and a polynomial formulation for this problem, both based on integer linear programming. The polynomial formulation proves to find the optimal set of bids in a matter of seconds. Results of the real-world implementation are reported.

Performance of Semiautomatic Assessment of Carotid Artery Stenosis on CT Angiography: Clarification of Differences with Manual Assessment

Semiautomated methods for ICA stenosis measurements have the potential to reduce interobserver variability and to speed up its analysis. In this study, we estimate the precision and accuracy of a semiautomated measurement for carotid artery stenosis degree and identify and explain differences compared with the manual method. In this retrospective study involving 90 patients, 2 observers determined the stenosis degree twice, with both the semiautomated and the manual method. Intra- and interobserver correlations were calculated for both methods. The accuracy was estimated by comparing average semiautomated with manual measurements. The semiautomated stenosis calculations were performed using either the minimal or maximal intersection at the reference site. Individual cases with large differences in measurement were retrospectively inspected by 3 observers. Intra- (R = 0.93, 0.96) and interobserver (R = 0.98) correlations for the semiautomated method were excellent and exceeded the manual performance correlations (R = 0.87, 0.86). The semiautomated measurements correlated well with the manual measurements (R = 0.87), with high specificity of 96% and lower sensitivity of 63%. Large differences were caused by misinterpretations of the semiautomated method associated with calcified plaques, resulting in overestimations of the minimal diameter, underestimation of stenosis degree, and incorrect centerlines. The effect of using the minimal diameter at the reference position resulted in a small, but significant, underestimation of the stenosis degree by the semiautomated method. The semiautomated method showed an excellent reproducibility and good correlation with manual measurements with a high specificity and lower sensitivity for detecting a significant stenosis. Erroneous semiautomatic stenosis measurements were associated with the presence of calcium.

Maximal intersections of three real quadrics

We consider real algebraic varieties that are intersections of three real quadrics. For brevity they are referred to as real triquadrics. We construct triquadrics that are -varieties and calculate the cohomology groups of the real parts of such triquadrics with coefficients in the field of two elements using relations between triquadrics and plane curves.

On the number of components of a three-dimensional maximal intersection of three real quadrics

We consider non-singular intersections of three real five-dimensional quadrics. For brevity they are referred to as real three-dimensional triquadrics. We prove the existence of real three-dimensional -triquadrics with components, where is any integer in the range .

SU‐E‐T‐772: Volume Matching for Brachytherapy Planning

Purpose: To perform brachytherapy treatment plan by comparing it with existing data base from earlier treatments of similar type. Method: When brachytherapy target to be treated is closely related in shape and volume to the target treated in the past for which we had well developed plan based on sources energy, activity and spatial distribution we can assume that plan of essentially the same quality will be achieved when the same distribution of sources that was used in the past will be imposed over the new, identical to one in the past treatment volume. The planning strategy under these circumstances relies on matching the volume under consideration with the volume of the target from earlier treatment. Characteristic scalar parameters of volumes will be compared first to filter the database (these will include volume, orientation of principal axes and moments of the characteristic set of the compared volumetric domains). When these parameters match closely between considered volume and one of stored in library volumes they are suspect of close similarity. The next stage of comparisons matches volumes by translating them in 3D space so that the maximal intersection between them is achieved. When closeness criterion is satisfied for this matching the distribution of sources is moved from the old plan to the new target completing the planning process. Results: The program for the algorithm has been written and sequence of plans have been computed for dose distributions obtained by sources placement based on domain comparison planning described above. Conclusion: The excellent agreement between set of plans based on geometrical planning approach and traditional planning optimization approach has been discovered. The domain comparison planning leads to enviable dose distribution in brachytherapy treatment and it is fast enough to be applicable to real time treatments.