Searcharxiv⌕ Search

arXiv subjects

Boyan Zlatanov

Publications and source records attributed to Boyan Zlatanov.

10 recordsLinked to original sources
arXiv2026

Context-Free Fixed Points and Complete Classification of Orbits in Picard Iteration for Guarded Power Language Operators

We study the language-theoretic structure of fixed points and finite Picard iterates for guarded q-power language operators. For the general operator, we prove that a context-free seeded language deter- mines a unique context-free fixed point and give an effective construction of a context-free grammar generating this fixed point, independently of the initial language.We then consider a marked single-guard special case in which two new symbols separate the recursive contribution from the the contribution of the seed. In this setting, the initial language can be traced and recovered exactly from every finite Picard iterate by means of a regular slice and fixed-word quotients. This yields injectivity of the finite-time maps and an abstract finite-time class-preservation prin- ciple. As a consequence, we obtain a classification of the finite Picard iterates according to the exact position of the initial language in the Chomsky hierarchy. Thus the exact language-theoretic complexity may persist at every finite stage, while all Picard orbits converge to the same context-free fixed point.

cs.FL↗
arXiv2026

Equilibrium in the Canonical Stackelberg Triopoly via Response Functions and Fixed Point Theory

We analyze a canonical extension of the Stackelberg duopoly to a sequential framework, where each firm strategically anticipates the reactions of all subsequent players. In a triopoly (three-firm) settings, we obtain existence and uniqueness of market equilibrium via a reformulation of the equilibrium conditions that draws on coupled fixed-point theory. Even with linear demand, convergence of myopic best-response dynamics is not guaranteed. A recursive equilibrium formulation enables the analysis of the limiting case as the number of participants grow.

math.FA↗
arXiv2026

Banach Spaces Generated by Finite-Valued Functions: Superreflexive Rigidity, Hankel Operators, and Universality

We investigate Banach spaces generated by uniformly bounded families of functions taking values in a fixed finite set and establish a rigidity principle connecting the cardinality of the generating family with the geometry of its closed linear span. We prove that such a space is superreflexive precisely when the generating family is finite, or equivalently, when the resulting Banach space is finite-dimensional. The main ingredient is a finite-range spreading-model obstruction showing that an infinite family of finite-valued functions cannot generate a superreflexive space under the supremum norm. This principle is applied to Banach spaces generated by the characteristic functions of the left derivatives of formal languages. It yields geometric characterizations of regular languages in terms of finite dimensionality, superreflexivity, and the existence of an equivalent uniformly convex norm. We also obtain a canonical representation of the associated language space as a coordinate-function subspace of a space of continuous functions on a compact shift-orbit closure. The corresponding language Hankel operator is shown to be compact exactly for regular languages. In the nonregular case, we determine its exact distance from both the compact and finite-rank operators and compute all its nontrivial approximation numbers. Finally, we construct a single binary language whose associated Banach space contains an isometric copy of every separable real Banach space. These results reveal a sharp contrast between the geometric rigidity associated with regular languages and the universality that may occur in the nonregular setting.

math.FA↗
arXiv2026

Ultrametric Convergence of Guarded Automata and Applications to Structural Input Validation

We equip language-equivalence classes of deterministic finite automata with a distinguishing-word ultrametric and identify the resulting space isometrically with the regular languages. This space is incomplete, while its metric completion is naturally identified with the complete ultrametric space of all formal languages. Guarded language operators induce contractions on the automaton space, and their Picard iterates converge in the completion to the unique language fixed point, which is represented by a finite automaton exactly when it is regular. Motivated by structural input validation, we use this framework to construct depth-capped deterministic finite automata with certified finite-depth correctness. These automata provide efficient pre-filters for nested input structures, such as parenthesised SQL parameters, while avoiding the backtracking risks of regular-expression engines and the runtime overhead of full context-free parsers. We also outline a practical WAF pipeline combining learned grammar models, finite-state construction, and \(O(1)\)-memory runtime validation.

cs.FL↗
arXiv2026

Perturbation Method in Musielak-Orlicz Sequence Spaces

We generalize an abstract variational principle in Banach spaces, introduced by Topalova \& Zlateva, by showing that the set $\mathbb{P}_0$ of perturbations for which a perturbed lower semi-continuous function $f$ is WPMC (Well Posed Modulus Compact) not only contains a dense $G_δ$ subset, but is also a complement to a $σ$-porous subset in a specifically defined positive cone. Moreover, if the space is a Musielak-Orlicz sequence space satisfying $\ell_Φ\cong h_Φ$, then the notion WPMC is replaced by the stronger notion of Tikhonov well posedness, which is proved to be equivalent to the single-valuedness and upper semi-continuity of the multivalued mapping assigning a parameter to the solution set. We give several applications. The first one is that the Musielak-Orlicz sequence spaces have the Radon-Nikodym property and, therefore, are dentable by proving the validity of Stegall's variational principle. As a consequence we obtain that the duals of Musielak-Orlicz sequence spaces are $w^*$-Asplund. We establish also a sufficient condition for Musielak-Orlicz and Nakano sequence spaces to be Asplund spaces. The next applications are for determining the type of the smoothness of certain Musielak-Orlicz, Nakano, and weighted Orlicz sequence spaces. We illustrate by an example that it is possible to consider an Orlicz function without the $Δ_2$ condition, by a particular choice of the weighted sequence $\{w_n\}_{n=1}^\infty$ to get $\ell_M(w)\cong h_M(w)$ and to be able to apply the main result.

math.FA↗
arXiv2023

On the UC and UC* properties and the existence of best proximity points in metric spaces

We investigate the connections between UC and UC* properties for ordered pairs of subsets (A,B) in metric spaces, which are involved in the study of existence and uniqueness of best proximity points. We show that the $UC^{*}$ property is included into the UC property. We introduce some new notions: bounded UC (BUC) property and uniformly convex set about a function. We prove that these new notions are generalizations of the $UC$ property and that both of them are sufficient for to ensure existence and uniqueness of best proximity points. We show that these two new notions are different from a uniform convexity and even from a strict convexity. If we consider the underlying space to be a Banach space we find a sufficient condition which ensures that from the UC property it follows the uniform convexity of the underlying Banach space. We illustrate the new notions with examples. We present an example of a cyclic contraction T in a space, which is not even strictly convex and the ordered pair (A,B) has not the UC property, but has the $BUC$ property and thus there is a unique best proximity point of T in A.

math.FA↗
arXiv2020

Application of Coupled Fixed (or Best Proximity) Points in Market Equilibrium in Oligopoly Markets

We present a possible kind of generalization of the notion of ordered pairs of cyclic maps and coupled fixed points and its application in modelling of equilibrium in oligopoly markets. We have obtained sufficient conditions for the existence and uniqueness of fixed (or best proximity) points in complete metric spaces (uniformly convex Banach spaces). We get an error estimates of the fixed (or best proximity), provided that we have used sequences of successive iterations. We illustrate one possible application of the results by building a pragmatic model on competition in oligopoly markets. To achieve this goal, we use an approach based on studying the response functions of each market participant, thus making it possible to address both Cournot and Bertrand industrial structures with unified formal method. In contrast to the restrictive theoretical constructs of duopoly equilibrium, our study is able to account for real-world limitations like minimal sustainable production levels and exclusive access to certain resources. We prove and demonstrate that by using carefully constructed response functions it is possible to build and calibrate a model that reflects different competitive strategies used in extremely concentrated markets. The response functions approach makes it also possible to take into consideration different barriers to entry. By fitting to the response functions rather than the profit maximization of the payoff functions problem we alter the classical optimization problem to a problem of coupled fixed points, which has the benefit that considering corner optimum, corner equilibria and convexity condition of the payoff function can be skipped.

math.OC↗
arXiv2014

An Etude on One Sharygin's Problem

By the methods of the synthetic geometry we investigate properties of objects generated from a complete quadrangle and a line, which lies in its plane. We start with a problem from the book of Sharygin "Problems in Plane Geometry". We generalize this problem with the help of Pappus, Desargues and Pascal's Theorems and we discover new concurrent lines, collinear points, and conic sections.

math.HO↗