In this paper we introduce stable systems of inclusions, which feature chosen arrows A ↪ B to capture the notion that A is a subobject of B, and proposes them as an alternative context to stable systems of monics to discuss partiality. A category C equipped with such a system $\mathscr{I}$, called an i-category, is shown to give rise to an associated category ∂(C,$\mathscr{I}$) of partial maps, which is a split restriction category whose restriction monics are inclusions. This association is the object part of a 2-equivalence between such inclusively split restriction categories and i-categories. $\mathscr{I}$ determines a stable system of monics $\mathscr{I}$+ on C, and, conversely, a stable system of monics $\mathscr{M}$ on C yields an i-category (C[$\mathscr{M}$],$\mathscr{M}$+), giving a 2-adjunction between i-categories and m-categories. The category of partial maps Par(C,$\mathscr{M}$) is isomorphic to the full subcategory of ∂(C[$\mathscr{M}$],$\mathscr{M}$+) comprising the objects of C, and ∂(C,$\mathscr{I}$) ≅ Par(C,$\mathscr{I}$+).

Continuous lattices were characterised by Martín Escardó as precisely those objects that are Kan-injective with respect to a certain class of morphisms. In this paper we study Kan-injectivity in general categories enriched in posets. As an example, ω-CPO's are precisely the posets that are Kan-injective with respect to the embeddings ω ↪ ω + 1 and 0 ↪ 1.

For every class $\mathcal{H}$ of morphisms, we study the subcategory of all objects that are Kan-injective with respect to $\mathcal{H}$ and all morphisms preserving Kan extensions. For categories such as Top0 and Pos, we prove that whenever $\mathcal{H}$ is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock–Zöberlein monad. However, this does not generalise to proper classes, and we present a class of continuous mappings in Top0 for which Kan-injectivity does not yield a monadic category.

We developed an efficient semisupervised feedforward neural network clustering model with one epoch training and data dimensionality reduction ability to solve the problems of low training speed, accuracy, and high memory complexity of clustering. During training, a codebook of nonrandom weights is learned through input data directly. A standard weight vector is extracted from the codebook, and the exclusive threshold of each input instance is calculated based on the standard weight vector. The input instances are clustered based on their exclusive thresholds. The model assigns a class label to each input instance through the training set. The class label of each unlabeled input instance is predicted by considering a linear activation function and the exclusive threshold. Finally, the number of clusters and the density of each cluster are updated. The accuracy of the proposed model was measured through the number of clusters and the quantity of correctly classified nodes, which was 99.85%, 100%, and 99.91% of the Breast Cancer, Iris, and Spam data sets from the University of California at Irvine Machine Learning Repository, respectively, and the superior F measure results between 98.29% and 100% accuracies for the breast cancer data set from the University of Malaya Medical Center to predict the survival time.

The minimum roman dominating problem (denoted by γR(G),the weight of minimum roman dominating function of graph G) is a variant of the verywell known minimum dominating set problem (denoted by γ(G), thecardinality of minimum dominating set of graph G). Both problems remain NP-Complete when restrictedto P5-free graph class [A.A. Bertossi,Inf. Process. Lett. 19 (1984) 37–40; E.J. Cockayne,et al. Discret. Math. 278 (2004) 11–22]. In this paper westudy both problems restricted to some subclasses of P5-free graphs.We describe robust algorithms that solve both problems restricted to (P5,(s,t)-net)-free graphsin polynomial time. This result generalizes previous works for both problems, and improvesexisting algorithms when restricted to certain families such as (P5,bull)-freegraphs. It turns out that the same approach also serves to solve problems for generalgraphs in polynomial time whenever γ(G) and γR(G)are fixed (more efficiently than naive algorithms). Moreover, the algorithms described areextremely simple which makes them useful for practical purposes, and as we show in thelast section it allows to simplify algorithms for significant classes such ascographs.

We describe the C2k+1-free graphs on n vertices with maximum number of edges. The extremal graphs are unique for n ∉ {3k − 1, 3k, 4k − 2, 4k − 1}. The value of ex(n, C2k+1) can be read out from the works of Bondy [3], Woodall [14], and Bollobás [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new.

We obtain that the bound for n0(C2k+1) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turán graph.

We describe an online database of number fields which accompanies this paper. The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.

A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain, F1 ⊂ F2. Erdős extended this theorem to determine the largest family without a k-chain, F1 ⊂ F2 ⊂ . . . ⊂ Fk. Erdős and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds.

In 1966, Kleitman resolved this question for 2-chains, showing that the number of such chains is minimized by taking sets as close to the middle level as possible. Moreover, he conjectured the extremal families were the same for k-chains, for all k. In this paper, making the first progress on this problem, we verify Kleitman's conjecture for the families whose size is at most the size of the k + 1 middle levels. We also characterize all extremal configurations.

This paper presents a new approach to accurately track a moving vehicle with a multiview setup of red–green–blue depth (RGBD) cameras. We first propose a correction method to eliminate a shift, which occurs in depth sensors when they become worn. This issue could not be otherwise corrected with the ordinary calibration procedure. Next, we present a sensor-wise filtering system to correct for an unknown vehicle motion. A data fusion algorithm is then used to optimally merge the sensor-wise estimated trajectories. We implement most parts of our solution in the graphic processor. Hence, the whole system is able to operate at up to 25 frames per second with a configuration of five cameras. Test results show the accuracy we achieved and the robustness of our solution to overcome uncertainties in the measurements and the modelling.

This special issue comprises selected papers which were presented at the workshop on dependently typed programming (DTP 10) in Edinburgh in July 2010 – affiliated with Federated Logic conferences (FLOC 10). Earlier workshops on dependently typed programming took place in Nottingham in 2008 (DTP 2008) and there also has been a Dagstuhl seminar (04381) on this subject in 2004. After DTP 2010, in 2011 a workshop on dependently typed programming (DTP 11) took place in Nijmegen affiliated with Interactive Theorem Proving 2011 (ITP 11). In September 2011 there also was a DTP workshop in Shonan, Japan.