On some unary algebras and their subalgebra lattices

We first define lattices, called normal, which are uniquely represented by directed graphs. Secondly, we describe all unary algebras (called normal, too) such that their subalgebra lattices are normal. Next, we characterize pairs (A,L) such that the subalgebra lattice of A is isomorphic to L, where A is a normal unary algebra and L is a normal lattice. Further, we describe pairs of normal unary algebras with isomorphic subalgebra lattices. We use these results in the second part of the paper to find necessary and sufficient conditions for pairs of lattices to be isomorphic to a pair of the weak and strong subalgebra lattices of one normal unary algebra

Optimization of Free Space Optical Wireless Network for Cellular Backhauling

With densification of nodes in cellular networks, free space optic (FSO) connections are becoming an appealing low cost and high rate alternative to copper and fiber as the backhaul solution for wireless communication systems. To ensure a reliable cellular backhaul, provisions for redundant, disjoint paths between the nodes must be made in the design phase. This paper aims at finding a cost-effective solution to upgrade the cellular backhaul with pre-deployed optical fibers using FSO links and mirror components. Since the quality of the FSO links depends on several factors, such as transmission distance, power, and weather conditions, we adopt an elaborate formulation to calculate link reliability. We present a novel integer linear programming model to approach optimal FSO backhaul design, guaranteeing $K$-disjoint paths connecting each node pair. Next, we derive a column generation method to a path-oriented mathematical formulation. Applying the method in a sequential manner enables high computational scalability. We use realistic scenarios to demonstrate our approaches efficiently provide optimal or near-optimal solutions, and thereby allow for accurately dealing with the trade-off between cost and reliability

Subcubic algorithm for (Unweighted) Unrooted Tree Edit Distance

The tree edit distance problem is a natural generalization of the classic string edit distance problem. Given two ordered, edge-labeled trees $T_1$ and $T_2$, the edit distance between $T_1$ and $T_2$ is defined as the minimum total cost of operations that transform $T_1$ into $T_2$. In one operation, we can contract an edge, split a vertex into two or change the label of an edge. For the weighted version of the problem, where the cost of each operation depends on the type of the operation and the label on the edge involved, $\mathcal{O}(n^3)$ time algorithms are known for both rooted and unrooted trees. The existence of a truly subcubic $\mathcal{O}(n^{3-\epsilon})$ time algorithm is unlikely, as it would imply a truly subcubic algorithm for the APSP problem. However, recently Mao (FOCS'21) showed that if we assume that each operation has a unit cost, then the tree edit distance between two rooted trees can be computed in truly subcubic time. In this paper, we show how to adapt Mao's algorithm to make it work for unrooted trees and we show an $\widetilde{\mathcal{O}}(n^{(7\omega + 15)/(2\omega + 6)}) \leq \mathcal{O}(n^{2.9417})$ time algorithm for the unweighted tree edit distance between two unrooted trees, where $\omega \leq 2.373$ is the matrix multiplication exponent. It is the first known subcubic algorithm for unrooted trees. The main idea behind our algorithm is the fact that to compute the tree edit distance between two unrooted trees, it is enough to compute the tree edit distance between an arbitrary rooting of the first tree and every rooting of the second tree.Comment: 20 page

Some properties of the weak subalgebra lattice of a partial algebra of a fixed type

summary:We investigate, using results from [[p3]], when a given lattice is isomorphic to the weak subalgebra lattice of a partial algebra of a fixed type. First, we reduce this problem to the question when hyperedges of a hypergraph can be directed to a form of directed hypergraph of a fixed type. Secondly, we show that it is enough to consider some special hypergraphs. Finally, translating these results onto the lattice language, we obtain necessary conditions for our algebraic problem, and also, we completely characterize the weak subalgebra lattice for algebras of some types

Elastic Multi-resource Network Slicing: Can Protection Lead to Improved Performance?

In order to meet the performance/privacy requirements of future data-intensive mobile applications, e.g., self-driving cars, mobile data analytics, and AR/VR, service providers are expected to draw on shared storage/computation/connectivity resources at the network "edge". To be cost-effective, a key functional requirement for such infrastructure is enabling the sharing of heterogeneous resources amongst tenants/service providers supporting spatially varying and dynamic user demands. This paper proposes a resource allocation criterion, namely, Share Constrained Slicing (SCS), for slices allocated predefined shares of the network's resources, which extends the traditional alpha-fairness criterion, by striking a balance among inter- and intra-slice fairness vs. overall efficiency. We show that SCS has several desirable properties including slice-level protection, envyfreeness, and load driven elasticity. In practice, mobile users' dynamics could make the cost of implementing SCS high, so we discuss the feasibility of using a simpler (dynamically) weighted max-min as a surrogate resource allocation scheme. For a setting with stochastic loads and elastic user requirements, we establish a sufficient condition for the stability of the associated coupled network system. Finally, and perhaps surprisingly, we show via extensive simulations that while SCS (and/or the surrogate weighted max-min allocation) provides inter-slice protection, they can achieve improved job delay and/or perceived throughput, as compared to other weighted max-min based allocation schemes whose intra-slice weight allocation is not share-constrained, e.g., traditional max-min or discriminatory processor sharing

A Light Signalling Approach to Node Grouping for Massive MIMO IoT Networks

Massive MIMO is a promising technology to connect very large numbers of energy constrained nodes, as it offers both extensive spatial multiplexing and large array gain. A challenge resides in partitioning the many nodes in groups that can communicate simultaneously such that the mutual interference is minimized. We here propose node partitioning strategies that do not require full channel state information, but rather are based on nodes' respective directional channel properties. In our considered scenarios, these typically have a time constant that is far larger than the coherence time of the channel. We developed both an optimal and an approximation algorithm to partition users based on directional channel properties, and evaluated them numerically. Our results show that both algorithms, despite using only these directional channel properties, achieve similar performance in terms of the minimum signal-to-interference-plus-noise ratio for any user, compared with a reference method using full channel knowledge. In particular, we demonstrate that grouping nodes with related directional properties is to be avoided. We hence realise a simple partitioning method requiring minimal information to be collected from the nodes, and where this information typically remains stable over a long term, thus promoting their autonomy and energy efficiency

Pojedynek – tradycyjny zwyczaj rozprawy honorowej z bronią w ręku czy też forma samodzielnego wymierzania sobie sprawiedliwości

The article explains the notion of duel from the law’s point of view, as well as tradition and honor. It includes the historical development of this phenomenon. It presents the essence of this kind of crime, on the basis of penal acts, i.e. Austrian Penal Act from 1852 year, German Penal Code from 1871 year, Russian Penal Code from 1903 year, Polish Penal Code from 1932 year. The text explains the general definition of duel, it mentions duel’s part and defines the object of legal protection and subject of crime. It indicates the results and participants of dueling battles. Problem of American’s duel has been presented . This thesis shows principles of gentleman’s procedures in daily life on the basis of honorable code. The article allows to understand morale of departed epoch, in legal and custom aspect.Artykuł nie zawiera streszczenia w języku polskim

Piękna i niepiękna przestrzeń dzieciństwa – czy można ją okiełznać w języku? Próby uchwycenia dziecięcego bycia w świecie we współczesnej literaturze dla najmłodszych

The paper analyses the language of child protagonists of contemporary literary works to investigate the probable ways and the extent to which it most likely attempts to get closer to the truth about the being in the world of a young person. Is it possible to define the framework of the linguistic cognitive function in search of a comprehensive as well as a holistic picture of the world and its sense-making interpretation? Is the need to establish the language of children protagonists’ identity with the language that describes unprecedented, unusual, difficult, or even extreme situations from the perspective of an adult literary “Me” self-justifiable enough, and if so – to what extent? This hermeneutic reflection is inseparably inspired by the thought and – at the same time – the concern for truth in the literary vision of the world offered to children, which escapes the infantilising model of early childhood education pedagogy.W artykule analizie poddany został język dziecięcych bohaterów współczesnych utworów literackich w celu zbadania, czy lub w jak dużym stopniu i jakimi sposobami próbuje zbliżyć się do prawdy o byciu w świecie młodego człowieka. Czy można określić ramy językowej funkcji poznawczej w poszukiwaniu całościowego obrazu świata i jego usensowniającej wykładni? W jakiej mierze i czy w ogóle zasadne jest ustalanie jego (języka dziecięcych bohaterów) tożsamości z językiem opowiadającym o sytuacjach wyjątkowych, trudnych, lub nawet ekstremalnych z perspektywy „ja” literackiego dorosłych? Ów hermeneutyzujący namysł inspirowany jest myślą i jednocześnie troską o prawdę w literackiej wizji świata proponowanej dzieciom, która zdecydowanie wymyka się „infantylizującemu modelowi” pedagogiki wczesnej edukacji. Sym-bolicznym punktem wyjścia niniejszych rozważań stał się ujęty w werbalno-graficznym kodzie komunikacyjnym obraz świata ukazanego z perspektywy żydowskiej dziewczynki – mieszkanki Domu Sierot Janusza Korczaka – bohaterki książki Iwony Chmielewskiej Pamiętnik Blumki