Search Results (18)

The aim of this study is to prove the existence and uniqueness of fixed point and common fixed point theorems for self-mappings in modular ultrametric spaces. These theorems are proved under varying contractive circumstances and without the property of spherical completeness. As a consequence, the examples of fixed point and common fixed point problems are correctly formulated. As an application, the well-posedness of a common fixed point problem is proved. This study expands on prior research in modular ultrametric space to provide a more comprehensive understanding of such spaces using generalized contraction. Full article

In this paper, we propose clustering methods for use on data described as tropically convex. Our approach is similar to clustering methods used in the Euclidean space, where we identify groupings of similar observations using tropical analogs of K-means and hierarchical clustering in the Euclidean space. We provide results from computational experiments on generic simulated data as well as an application to phylogeny using ultrametrics, demonstrating the efficacy of these methods. Full article

In this paper we study pseudoultrametrics, which are a natural mixture of ultrametrics and pseudometrics. They satisfy a stronger form of the triangle inequality than usual pseudometrics and naturally arise in problems of classification and recognition. The text focuses on the natural partial order on the set of all pseudoultrametrics on a fixed (not necessarily finite) set. In addition to the “way below” relation induced by a partial order, we introduce its version which we call “weakly way below”. It is shown that a pseudoultrametric should satisfy natural conditions closely related to compactness, for the set of all pseudoultrametric weakly way below it to be non-trivial (to consist not only of the zero pseudoultrametric). For non-triviality of the set of all pseudoultrametrics way below a given one, the latter must be compact. On the other hand, each compact pseudoultrametric is the least upper bound of the directed set of all pseudoultrametrics way below it, which are compact as well. Thus it is proved that the set $CPsU\left(X\right)$ of all compact pseudoultrametric on a set X is a continuous poset. This shows that compactness is a crucial requirement for efficiency of approximation in methods of classification by means of ultrapseudometrics. Full article

In this paper, the concept of ultrametric structure is intertwined with the SLAM procedure. A set of pre-existing transformations has been used to create a new simultaneous localization and mapping (SLAM) algorithm. We have developed two new parallel algorithms that implement the time-consuming Boolean transformations of the space dissimilarity matrix. The resulting matrix is an important input to the vector quantization (VQ) step in SLAM processes. These algorithms, written in Compute Unified Device Architecture (CUDA) and Open Multi-Processing (OpenMP) pseudo-codes, make the Boolean transformation computationally feasible on a real-world-size dataset. We expect our newly introduced SLAM algorithm, ultrametric Fast Appearance Based Mapping (FABMAP), to outperform regular FABMAP2 since ultrametric spaces are more clusterable than regular Euclidean spaces. Another scope of the presented research is the development of a novel measure of ultrametricity, along with creation of Ultrametric-PAM clustering algorithm. Since current measures have computational time complexity order, $O\left(n^3\right)$ a new measure with lower time complexity, $O\left(n^2\right)$, has a potential significance. Full article

We propose a model of a quantum $\mathsf{N}$-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of p-adic numbers. As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an $\mathsf{N}$-dimensional p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)—a suitable p-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where $\mathsf{N}=2$, thus providing a description of p-adic qubit states and 2-dimensional SOVMs. The analogies—but also the non-trivial differences—with respect to the qubit states of standard quantum mechanics are then analyzed. Full article

Continuing investigations initiated by the first author, we associate relational structures for metric spaces and investigate their model theoretic properties. In this paper, we consider ultrametric spaces. Among others, we show that any elementary substructure of the relational structure associated with a totally bounded ultrametric space $\mathcal{X}$ is dense in X. Further, we provide an explicit upper bound for a splitting chain of atomic types in ultrametric spaces of a finite spectrum. For ultrametric spaces, these results improve previous ones of the present authors and may have further practical applications in designing similarity detecting algorithms. Full article

This paper is devoted to the foundational problems of dendrogramic holographic theory (DH theory). We used the ontic–epistemic (implicate–explicate order) methodology. The epistemic counterpart is based on the representation of data by dendrograms constructed with hierarchic clustering algorithms. The ontic universe is described as a p-adic tree; it is zero-dimensional, totally disconnected, disordered, and bounded (in p-adic ultrametric spaces). Classical–quantum interrelations lose their sharpness; generally, simple dendrograms are “more quantum” than complex ones. We used the CHSH inequality as a measure of quantum-likeness. We demonstrate that it can be violated by classical experimental data represented by dendrograms. The seed of this violation is neither nonlocality nor a rejection of realism, but the nonergodicity of dendrogramic time series. Generally, the violation of ergodicity is one of the basic features of DH theory. The dendrogramic representation leads to the local realistic model that violates the CHSH inequality. We also considered DH theory for Minkowski geometry and monitored the dependence of CHSH violation and nonergodicity on geometry, as well as a Lorentz transformation of data. Full article

In mathematics, distance and similarity are known as dual concepts. However, the concept of similarity is interpreted as fuzzy similarity or T-equivalence relation, where T is a triangular norm (t-norm in brief), when we discuss a fuzzy environment. Dealing with multi-polarity in practical examples with fuzzy data leadsus to introduce a new concept called m-polar $\mathbf{T}$-equivalence relations based on a finitely multivalued t-norm $\mathbf{T}$, and to study the metric behavior of such relations. First, we study the new operators including the m-polar triangular norm $\mathbf{T}$ and conorm $\mathbf{S}$ as well as m-polar implication $\mathbf{I}$ and m-polar negation $\mathbf{N}$, acting on the Cartesian product of $[0,1]$m-times.Then, using the m-polar negations $\mathbf{N}$, we provide a method to construct a new type of metric spaces, called m-polar $\mathbf{S}$-pseudo-ultrametric, from the m-polar $\mathbf{T}$-equivalences, and reciprocally for constructing m-polar $\mathbf{T}$-equivalences based on the m-polar $\mathbf{S}$-pseudo-ultrametrics. Finally, the link between fuzzy graphs and m-polar $\mathbf{S}$-pseudo-ultrametrics is considered. An algorithm is designed to plot a fuzzy graph based on the m-polar ${\mathbf{S}}_L$-pseudo-ultrametric, where ${\mathbf{S}}_L$ is the m-polar Lukasiewicz t-conorm, and is illustrated by a numerical example which verifies our method. Full article

A tropical ball is a ball defined by the tropical metric over the tropical projective torus. In this paper we show several properties of tropical balls over the tropical projective torus and also over the space of phylogenetic trees with a given set of leaf labels. Then we discuss its application to the K nearest neighbors (KNN) algorithm, a supervised learning method used to classify a high-dimensional vector into given categories by looking at a ball centered at the vector, which contains K vectors in the space. Full article

We propose a method for the unsupervised clustering of hyperspectral images based on spatially regularized spectral clustering with ultrametric path distances. The proposed method efficiently combines data density and spectral-spatial geometry to distinguish between material classes in the data, without the need for training labels. The proposed method is efficient, with quasilinear scaling in the number of data points, and enjoys robust theoretical performance guarantees. Extensive experiments on synthetic and real HSI data demonstrate its strong performance compared to benchmark and state-of-the-art methods. Indeed, the proposed method not only achieves excellent labeling accuracy, but also efficiently estimates the number of clusters. Thus, unlike almost all existing hyperspectral clustering methods, the proposed algorithm is essentially parameter-free. Full article

The pooling layer is at the heart of every convolutional neural network (CNN) contributing to the invariance of data variation. This paper proposes a pooling method based on Zeckendorf’s number series. The maximum pooling layers are replaced with Z pooling layer, which capture texels from input images, convolution layers, etc. It is shown that Z pooling properties are better adapted to segmentation tasks than other pooling functions. The method was evaluated on a traditional image segmentation task and on a dense labeling task carried out with a series of deep learning architectures in which the usual maximum pooling layers were altered to use the proposed pooling mechanism. Not only does it arbitrarily increase the receptive field in a parameterless fashion but it can better tolerate rotations since the pooling layers are independent of the geometric arrangement or sizes of the image regions. Different combinations of pooling operations produce images capable of emphasizing low/high frequencies, extract ultrametric contours, etc. Full article

We relate the definition of an ultrametric space to the topological distance algorithm—an algorithm defined in the context of peer-to-peer network applications. Although (greedy) algorithms for constructing minimum spanning trees such as Prim’s or Kruskal’s algorithm have been known for a long time, they require the complete graph to be specified and the weights of all edges to be known upfront in order to construct a minimum spanning tree. However, if the weights of the underlying graph stem from an ultrametric, the minimum spanning tree can be constructed incrementally and it is not necessary to know the full graph in advance. This is possible, because the join algorithm responsible for joining new nodes on behalf of the topological distance algorithm is independent of the order in which the nodes are added due to the property of an ultrametric. Apart from the mathematical elegance which some readers might find interesting in itself, this provides not only proofs (and clearer ones in the opinion of the author) for optimality theorems (i.e., proof of the minimum spanning tree construction) but a simple proof for the optimality of the reconstruction algorithm omitted in previous publications too. Furthermore, we define a new algorithm by extending the join algorithm to minimize the topological distance and (network) latency together and provide a correctness proof. Full article

We present a mathematical model of disease (say a virus) spread that takes into account the hierarchic structure of social clusters in a population. It describes the dependence of epidemic’s dynamics on the strength of barriers between clusters. These barriers are established by authorities as preventative measures; partially they are based on existing socio-economic conditions. We applied the theory of random walk on the energy landscapes represented by ultrametric spaces (having tree-like geometry). This is a part of statistical physics with applications to spin glasses and protein dynamics. To move from one social cluster (valley) to another, a virus (its carrier) should cross a social barrier between them. The magnitude of a barrier depends on the number of social hierarchy levels composing this barrier. Infection spreads rather easily inside a social cluster (say a working collective), but jumps to other clusters are constrained by social barriers. The model implies the power law, $1-t^{-a},$ for approaching herd immunity, where the parameter a is proportional to inverse of one-step barrier $\Delta .$ We consider linearly increasing barriers (with respect to hierarchy), i.e., the m-step barrier ${\Delta }_m=m\Delta .$ We also introduce a quantity characterizing the process of infection distribution from one level of social hierarchy to the nearest lower levels, spreading entropy $\mathcal{E}.$ The parameter a is proportional to $\mathcal{E}.$ Full article

Phylogenetic relatedness is a key diversity measure for the analysis and understanding of how species and communities evolve across time and space. Understanding the nonrandom loss of species with respect to phylogeny is also essential for better-informed conservation decisions. However, several factors are known to influence phylogenetic reconstruction and, ultimately, phylogenetic diversity metrics. In this study, we empirically tested how some of these factors (topological constraint, taxon sampling, genetic markers and calibration) affect phylogenetic resolution and uncertainty. We built a densely sampled, species-level phylogenetic tree for spiders, combining Sanger sequencing of species from local communities of two biogeographical regions (Iberian Peninsula and Macaronesia) with a taxon-rich backbone matrix of Genbank sequences and a topological constraint derived from recent phylogenomic studies. The resulting tree constitutes the most complete spider phylogeny to date, both in terms of terminals and background information, and may serve as a standard reference for the analysis of phylogenetic diversity patterns at the community level. We then used this tree to investigate how partial data affect phylogenetic reconstruction, phylogenetic diversity estimates and their rankings, and, ultimately, the ecological processes inferred for each community. We found that the incorporation of a single slowly evolving marker (28S) to the DNA barcode sequences from local communities, had the highest impact on tree topology, closely followed by the use of a backbone matrix. The increase in missing data resulting from combining partial sequences from local communities only had a moderate impact on the resulting trees, similar to the difference observed when using topological constraints. Our study further revealed substantial differences in both the phylogenetic structure and diversity rankings of the analyzed communities estimated from the different phylogenetic treatments, especially when using non-ultrametric trees (phylograms) instead of time-stamped trees (chronograms). Finally, we provide some recommendations on reconstructing phylogenetic trees to infer phylogenetic diversity within ecological studies. Full article

P-adic numbers serve as the simplest ultrametric model for the tree-like structures arising in various physical and biological phenomena. Recently p-adic dynamical equations started to be applied to geophysics, to model propagation of fluids (oil, water, and oil-in-water and water-in-oil emulsion) in capillary networks in porous random media. In particular, a p-adic analog of the Navier–Stokes equation was derived starting with a system of differential equations respecting the hierarchic structure of a capillary tree. In this paper, using the Schauder fixed point theorem together with the wavelet functions, we extend the study of the solvability of a p-adic field analog of the Navier–Stokes equation derived from a system of hierarchic equations for fluid flow in a capillary network in porous medium. This equation describes propagation of fluid’s flow through Geo-conduits, consisting of the mixture of fractures (as well as fracture’s corridors) and capillary networks, detected by seismic as joint wave/mass conducts. Furthermore, applying the Adomian decomposition method we formulate the solution of the p-adic analog of the Navier–Stokes equation in term of series in general form. This solution may help researchers to come closer and find more facts, taking into consideration the scaling, hierarchies, and formal derivations, imprinted from the analogous aspects of the real world phenomena. Full article