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The celebrated Assouad embedding theorem has been known for over 40 years. It states that for any doubling metric space (with doubling constant $C_0$) there exists an integer $N$, such that for any $\alpha\in (0.5,1)$ there exists a positive constant $C(\alpha,C_0)$ and an injective function $F:X\to \mathbb{R}^N$ such that \[ \forall x,y\in X \quad C^{-1} d(x,y)^{\alpha} \leq \|F(x)-F(y) \| \leq Cd(x,y)^{\alpha} \] In the paper we use the remetrization techniques to extend the said theorem to a broad subclass of semimetric spaces. We also present the limitations of this extension -- in particular, we prove that in any semimetric space which satisfies the claim of the Assouad theorem, the relaxed triangle condition holds as well, which means that there exists $K\geq 1$ such that \[ \forall x,y,z\in X \quad d(x,z)\leq K\left( d(x,y)+d(y,z)\right). \], Comment: 9 pages

In the paper we apply some of the results from the theory of ball spaces in the semimetric spaces. This allowed us to obtain some fixed point theorems which we believe to be unknown to this day. We also show the limitations of the ball space approach to this topic. As a byproduct, we obtain the equivalence of some different notions of completness in semimetric spaces where the distance function is $1$-continuous. In the second part of the article, we generalize Caristi-Kirk results for for $b$-metric spaces. Additionally, we obtain characterization of semicompleteness for $1$-continuous $b$-metric space via fixed point theorem analogous to the result of Suzuki. In the epilogue, we introduce the concept of convergence in ball spaces, based on the idea that balls should resemble closed sets in topological sets. We prove several of its properties, compare it with convergence in semimetric spaces and pose several open questions connected with this notion.

The aim of the paper is to compare different approximation algorithms for the travelling salesperson problem. We pick the most popular and widespread methods known in the literature and contrast them with a novel approach (the polygonal Christofides algorithm) described in our previous work. The paper contains a brief summary of theory behind the algorithms and culminates in a series of numerical simulations (or "experiments"), whose purpose is to determine "the best" approximation algorithm for the travelling salesperson problem., Comment: arXiv admin note: text overlap with arXiv:2105.07275

With the aid of the relaxed polygonal inequality (introduced by Fagin et al.) we strive to extend the applicability of Christofides approximation technique to the scope of all complete finite weighted graphs with positive weights. First section acquaints the Reader with the class of semimetric graphs and proves that every finite graph admits $\gamma$-polygon structure. Sections 2 and 3 establish the necessary notions from the graph and optimization theory to tackle the Traveling Salesperson Problem. In section 4 the minimal spanning tree method is introduced, while section 5 focuses on the analysis of this method through the lens of $\gamma$-polygon graphs. The final section of the paper adjusts the technique of Christofides by obtaining $\frac{3\gamma}{2}-$approximation for the TSP.

In the paper we continue the research of Bors\'{i}k and Dobo\v{s} on functions which allow us to introduce a metric to the product of metric spaces. In this paper we extend their scope on broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving $b$-metric inequality. We provide analogues of the results of Bors\'{i}k and Dobo\v{s}, adjusted to the new, broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.

Following the train of thought from our previous paper we revisit the theorems of Pongsriiam and Termwuttipong by further developing their characterization of certain property-preserving functions using the so-called triangle triplets. We develop more general analogues of disjoint sum lemmas for broader classes of metric-type spaces and we apply these to extend results of Bors\`ik an Dobo\v{s} as well as those obtained by Khemaratchatakumthorn and Pongsriiam. As a byproduct we obtain methods of generating non-trivial and infinite strong $b$-metric spaces which are not metric.

The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the $T_{3 \frac{1}{2}}$ space. Motivated by the generalized Arzel\`a-Ascoli theorem for Tichonoff space $T$ via Wallman compactifiaction $\operatorname{Wall}(T)$, we constuct a measure of non-compactness for the space $C^b(T)$. We also study some of the properties of this object and give another version of Darbo-type theorem, suitable for this particular case.