Computer Science and Mathematics

Abstract: In 1970, Dana Scott proposed his highly influential ''mathematical theory of computation'' to define the relationship between the text of a program and what the program computes (or denotes) --- the ''semantics'' of the program. Scott used a complete lattice based on the ''information ordering'', with the bottom element representing undefined --- a program failing to terminate normally, thus producing no information. The top element, however, was unused. Hence most subsequent applications of denotational semantics have used mathematical structures that avoid top elements. We suggest that the information ordering is relevant not only to semanticists, but also to working programmers, as a basis for determining if a program component or a computation is correct according to their intentions. We also suggest that a return to the use of complete lattices is called for if we wish to broaden formal semantics to allow it to encompass programmer intentions. That is because often those intentions permit more than one runtime behaviour for a given input. For example, several declarative debugging tools allow a programmer to declare that a runtime call is ''inadmissible'', loosely meaning the called program component should never be used with such input. If the input is garbage, the programmer does not care what garbage is output --- all results are acceptable, which can be modeled by the top element of a complete lattice. In this paper we explore the connections between the information ordering, correctness of computations and programs, and debugging. We present a general theory and describe several instances where the intention for what our logic/functional code computes plus what it actually computes can be described by elements in a complete lattice. This extends both the theoretical basis and practical flexibility of declarative debugging and reasoning about partial correctness and gives an attractive mathematical framework that encompasses our intentions, our programs and what they compute.

Abstract: We give a certified finite verification of the literal vertex formulation currently displayed as Erd\H{o}s Problem \#580. Namely, for every $1\le n\le 19$, an $n$-vertex graph having at least $\lceil n/2\rceil$ vertices of degree at least $\lceil n/2\rceil$ contains every tree on at most $\lfloor n/2\rfloor$ vertices. The only order requiring new computer-assisted analysis is $n=18$. An edge-minimal counterexample is reduced to a host partition $V(G)=L\sqcup S$ with $|L|=|S|=9$, degree exactly $9$ on $L$, and $S$ independent. Exact embedding theorems cover $42$ of the $47$ non-isomorphic trees on nine vertices. The five remaining trees reduce, by deleting their leaves, to four rooted cores. For each core we construct a Boolean formula whose models are precisely the reduced hosts avoiding that rooted core. All four formulas are unsatisfiable. The deposited data include complete CNF instances and DRUP refutations. The archived traces were validated by reverse unit propagation, all four formulas are independently solved as unsatisfiable by a second SAT solver, a standalone semantic audit reconstructs the complete formulas---including all $3735$ base clauses and every avoidance clause---without importing the production generator, the complete $47$-tree witness table is generated from the classifier, and regeneration from the published encoder reproduces all four CNFs byte for byte. This is a finite partial result concerning trees on at most $n/2$ vertices; it does not settle the stronger classical formulation asking for trees with at most $n/2$ edges.

Abstract: Gallai’s theorem states that whenever the Euclidean plane is colored with finitely many colors, every finite configuration of points admits a monochromatic homothetic copy. This paper presents a detailed exposition of a proof originally published by Ernst Witt in 1952 and subsequently expanded by Alexander Soifer. Additional intermediate steps are supplied throughout, yielding a self-contained and mechanically verifiable proof. The argument is formulated recursively through finite configurations associated with a double induction and thereby makes explicit the finite structures underlying the theorem.

Abstract: The maximum sphere packing density in $\mathbb{R}^4$ is $\pi^2/16$, achieved uniquely by the $D_4$ root lattice. The proof establishes a local Voronoi cell bound: every packing cell satisfies $\mathrm{vol}(V_c)\ge 8$, via a four-layer argument (shell localisation, root-alignment, radial monotonicity, chamber positivity for all $176$ Weyl-orbit types) with positivity certificates exact over $\mathbb{Q}$.

Abstract: The recently introduced Hyperbolic Sombor index HSO(G) is defined as HSO(G)=∑_uv∈E(G)√d_u²+d_v²/min{d_u,d_v}. In this paper, we investigate the mathematical properties of this novel vertex-degree-based topological index for general graphs. We first establish tight upper and lower bounds on HSO(G) in terms of the variable Euler–Sombor index EU(λ,G) for all λ ≥ −2. Furthermore, we derive new bounds connecting HSO(G) with three other well-known degree-based invariants: the second Zagreb index M₂(G), the Elliptic Sombor index ESO(G), and the Forgotten Sombor index FSO(G).

Abstract: The maximum sphere packing density in $\mathbb{R}^7$ is $\pi^3/105$, achieved uniquely by the $E_7$ root lattice. The proof establishes a local Voronoi bound: every packing cell satisfies $\vol(V_c)\ge16$, via four layers (shell localisation, root-alignment, radial monotonicity, chamber positivity for all $115$ Weyl-orbit types) with positivity certificates exact over $\mathbb{Q}$.

Abstract: Quantum resource estimates for the elliptic-curve discrete logarithm problem (ECDLP) now shape cryptographic migration planning, blockchain security analysis, and fault-tolerant architecture design. Recent work has moved in two complementary directions: Babbush et al. give improved secp256k1 resource estimates supported by zero-knowledge attestation while withholding sensitive circuit details, whereas Luo et al. publish an explicit reversible modular-inversion construction based on the extended Euclidean algorithm, reducing the logical-qubit footprint of prime-field ECDLP and identifying gate count, depth, and architecture-aware implementation as natural optimization targets. This note proposes a third disclosure model: verifiable resource certificates for public reversible arithmetic blocks. A certificate records a circuit commitment, gate basis, resource counts, input-output specification, deterministic test generation, correctness transcript, and optional proof artifact. We specialize the framework to modular inversion blocks |x⟩|0⟩ → |x⟩|x⁻¹ mod p⟩, for \(x\in\mathbb F_p^\times\), which are central to affine-coordinate quantum ECDLP implementations. We prove a basic soundness bound for hash-derived randomized testing and outline a prototype verifier. The goal is not a new quantum attack, but reproducible, comparable, and independently auditable quantum-ECDLP arithmetic claims.

Abstract: We give a source-scoped proof of Conjecture 19 from Written on the Wall II, as stated in the cited WOWII source materials, for finite nontrivial connected simple graphs. For such a graph G, let b(G) denote the maximum order of an induced bipartite subgraph, let ecc(v) denote the eccentricity of a vertex v, and let lambda(G) denote the largest independence number of an open neighborhood. We show that b(G) is at least the floor of the sum of lambda(G) and the average vertex eccentricity. The strict-average case uses the WOWII 13 diameter-local independence bound, for which we include both a paper proof and a formally checked Lean proof. The new mathematical ingredient for WOWII 19 is the equality-case witness, which constructs an induced bipartite subgraph from a diametral geodesic and a maximum independent neighborhood. A supplementary Lean artifact verifies both the WOWII 19 statement and the WOWII 13 bound used in the strict branch. We make no global literature-priority assertion beyond resolving the conjecture in the cited WOWII presentation.

Abstract: A continuous graph, or metric graph, is obtained from a finite graph by replacing each edge by a unit interval and using the induced shortest-path metric. Recent work of Grigoriev and Faulkner formulated continuous analogues of several classical graph-optimization problems, including independent set, vertex cover, chromatic number, and treewidth. In this paper we solve exactly the distance-two independent-set problem for continuous complete multipartite graphs. Let \[ G=K_{n_1,\ldots,n_k},\qquad N=\sum_{i=1}^k n_i,\qquad M=\max_{1\le i\le k} n_i, \] where $k\ge 2$ and $n_i\ge 1$. We prove that \[ \alpha_1(\Gamma(G))=\max\left\{M,\floor{N/2}\right\}, \] where $\alpha_1$ is the maximum cardinality of a set of points in $\Gamma(G)$ whose pairwise metric distances are at least two. The formula shows that the optimum is governed by two support mechanisms: selecting vertices from a largest part, or using interior points supported on the edges of a maximum matching; choosing the midpoints gives a canonical optimal realization of the matching construction. We also give an exact constructive algorithm, EMCIA, for computing the value and producing a labelled optimal configuration. The value is computed in $O(k)$ time; a labelled optimal configuration is produced in $O(N\log k)$ time using a priority queue, or in $O(N)$ time after bucket sorting when the part sizes are bounded. Reproducible numerical illustrations show the transition between the largest-part and matching-supported regimes and include a quantitative balance-entropy diagnostic for the input part-size distribution.

Abstract: This paper introduces a ladder-based traversal framework for representing repeated symbolic structures within symbolic and numerical sequences. Repeated structures are interpreted as ordered traversal paths between hierarchical rung states, for example 1→11→111, permitting repetition to be analysed as directed traversal behaviour rather than solely as static substring occurrence. The framework defines traversal states using traversal direction, rung progression between adjacent repetition orders, traversal-gap structure, and climb behaviour. Valid traversal is restricted to transitions between neighbouring rung states only. The model additionally permits chained traversal paths spanning multiple repetition orders, overlapping embeddings in which symbolic occurrences participate in multiple traversals, and recursive overlap between rung states. Unlike conventional repetition frameworks that primarily characterise repeated substrings through periodicity or maximal interval structure, the proposed framework provides a ladder-based and geometric interpretation of symbolic repetition in which repeated structures may be analysed as interconnected traversal systems rather than purely linear strings. Potential applications include symbolic sequence fingerprinting, traversal-density analysis, overlap statistics, traversal entropy measures, and Monte-Carlo comparison of traversal behaviour within random and non-random symbolic systems.

Abstract: A graph G=(V,E) together with a positive real-valued weight-function w:V→R+ or w:E→R+ is called a weighted graph and is denoted by (G;w). In this paper, we introduce the concepts of connectivity, edge-connectivity, and restricted edge-connectivity for a weighted graph (G;w), and we prove general bounds analogous to those in the unweighted case. Furthermore, we study the connectivity and edge-connectivity of the line graph and the P2-path graph of a weighted graph, establishing upper and lower bounds for each of these parameters.

Abstract: A descent sequence is a word π = π₁π₂ · · · πn of nonnegative integers satisfying π₁ = 0 and π_i ≤ 1 + des(π₁π₂ · · · π_i-1) for i = 2, 3, . . . , n, where des(π₁π₂ · · · π_m) denotes the number of indices j such that π_j > π_j+1. In this work, we study descent sequences subject to the additional restriction of avoiding a given pattern of length four. We analyze seven distinct avoidance classes and provide enumerative results for each of them. Our approach is based on the construction of generating trees with one or two labels, from which we derive succession rules and corresponding systems of recurrence relations. These recurrences are then used to compute explicit generating functions for the number of descent sequences of length n avoiding either 0001, 0010, 0011, 0012, 0021, 0110, 0112, 0123, or 0132.

Abstract: This work explores how graph theory concepts and neural networks can assist in strategic planning of metro network expansions using publicly available city data with the São Paulo Metropolitan Region. The methodology consolidated information from multiple public sources, developed a formula to estimate passenger demand based on catchment areas, applied Random Forest to identify the most relevant demographic features, and implemented a GraphSAGE model for demand prediction, extracting predictive capability from the topology of the system as well as socioeconomic features and Origin-Destination trips. The model achieved an R² of 0.874 ± 0.042 with minimal overfitting, outperforming the Random Forest approach. It can be applied to predict demand for future projects in a way that is accurate, inexpensive and computationally efficient, while also not requiring any rail system specific information besides topology. In this project, it was used to analyze multiple real projects and proposals for the São Paulo Metropolitan Region. Analysis revealed that employment, residences, and destinations where people go to eat represent 65% of the predictive capacity in the city.

Abstract: This paper investigates the oscillation properties of solutions to a second-order nonlinear difference equation in noncanonical form with bounded and unbounded neutral terms. By employing the monotonicity of the neutral term together with a linearization technique, we establish new conditions that guarantee all solutions of the equation oscillate. Our results are applicable to various nonlinear forms of the equation, and, notably, the oscillation of all solutions is ensured through a single condition. Consequently, the proposed oscillation criteria are straightforward to apply and distinct from existing results on nonlinear difference equations. Four examples are presented to demonstrate the novelty and significance of the main findings.

Abstract: We present a self-contained exposition of the Lyons carabiner, focusing on the triple obstruction mechanism arising in Section~\ref{sec:triple}. Three independent numerical obstructions---phantom coupling $\pi$, holonomy defect $\rho$, and braid deficit $\beta$---together form the triple $(\pi,\rho,\beta)=(3,4,9)$, which uniquely recovers the MDS code $[6,4,3]_5$ over $\mathbb{F}_5$. This identification is not a postulate but a provable consequence of the weight distribution and the complement involution. We then develop the Pontryagin--Heegner bridge: phantom weights are ``silent frequencies'' in the Pontryagin dual of $\mathbb{R}_+^\times$, and their vacuum generates the $20$-dimensional \emph{inverse Heegner space} $\mathcal{H}_{20}$. This yields a candidate $44$-dimensional lattice $\Lambda_{44} = \Lambda_{24} \oplus \mathcal{H}_{20}$ with Lyons group symmetries. The phantom resolution cascade $\mathrm{Ly}\to\mathrm{HS}\to\mathrm{Ru}$ predicts a chain of lattice dimensions $44\to 34\to 24$, terminating at the Leech lattice $\Lambda_{24}$, with the Rudvalis level providing independent numerical evidence through striking orbit coincidences. All numerical claims are verified by machine-checked computation. ( The Lean~4 formalization is available as the \texttt{HatsuYakitori} library; the key files are \texttt{MachineConstants.lean}, \texttt{LyonsCarabiner.lean}, and \texttt{RudvalisCarabiner.lean}. Remaining \texttt{sorry}s are listed explicitly in Section~\ref{sec:conclusion}.

Abstract: A broom graph is a linear graph with some pendant vertices attached to one of its ends. Using the formula for the commute time of the random walk between two vertices of the graph, which is given in terms of the effective resistance between the vertices, we find closed-form formulas for the hitting time index of some families of broom graphs, extending results found in the literature.

Abstract: We ask for finite field versions of following three: (1) Grothendieck Inequality, (2) Johnson-Lindenstrauss Flattening Lemma, (3) Bourgain-Tzafriri Restricted Invertibility Theorem.

Abstract: We study a pharmaceutical scheduling problem with hybrid batch-continuous manufacturing process in a distributed supply chain. The supply chain consists of heterogeneous plants and one distribution center. Each plant adopts an unrelated permutation flow shop layout consisting of a hybrid batch-continuous production line. Each pharmaceutical order is split and produced in multi-production sites located in various regions. The pharmaceutical medicines manufactured by the production sites are directly shipped to a distribution center To minimize the makespan, we formulate the addressed scheduling problem as a mathematical model. To solve this model, we propose four metaheuristics variants by applying two population-based metaheuristics to two distinct solution structures. We compare the proposed metaheuristics to evaluate their performance in the numerical experiments. Additionally, we present managerial insights through sensitivity analysis.

Abstract: Along with some known and less known results, we discuss new insights relating combinatorics of words and the ordering of the rationals from a dynamical systems point of view, somehow continuing along the path started in [BI]. We obtain in particular a set of results that structure and enrich the correspondence between the Stern-Brocot (SB) ordering of rational numbers and the corresponding ordering of Farey-Christoffel (FC) words, a class of words that, since their appearance in literature at the end of the 18th century, have revealed numerous relationships with other fields of mathematics. Among the results obtained here is the construction of substitution rules that act on the FC words in a parallel way to the maps on the positive reals that generate the permuted SB tree both vertically and horizontally. A complete correspondence is obtained between the vertical and horizontal motions on the SB tree and the geodesic motions along scattering geodesics and the horocyclic motion along Ford circles in the upper half-plane, respectively.

Abstract: A finite hypergraph generalizes an ordinary graph by permitting a hyperedge to connect any nonempty subset of vertices, thereby representing genuine multiway interactions. Extending this idea, a finite SuperHyperGraph is obtained through an iterated powerset construction, so that set-valued objects formed at one level may function as vertices or edge endpoints at the next, providing a natural framework for hierarchical and multilayer relational structures. In contrast, a line graph transforms each edge of a graph into a vertex, with two such new vertices adjacent precisely when the corresponding original edges share an endpoint. In this paper, we introduce the notion of a MultiLine Graph, in which multiple edges can be assigned to a vertex, and then develop its higher-order extensions, namely the MultiLine HyperGraph and the MultiLine Super HyperGraph. We further investigate their fundamental properties and structural characteristics.