Computer Science and Mathematics

Abstract: Gradient-based latent space planning faces a fundamental trade-off: maintaining high-entropy representations for robust perception under domain shift often yields ill-conditioned manifolds that destabilize trajectory optimization. DEGP introduces a framework designed to isolate gradient interference between representation learning and control dynamics via strict architectural separation. However, empirical evaluation demonstrates that structural decoupling alone is insufficient for closed-loop planning (0/50 per condition; 0/200 total), uncovering a fundamental information-usability gap driven by contact-conditional predic-tion errors and cost signal suppression. The gap is not closed by retraining the encoder to be more decodable: a trained encoder whose T-pose decodability matches DEGP’s (R2=0.686) still yields a predictor that captures essentially no contact-conditioned movement (CDR = 0.046, versus the frozen random encoder’s 0.060). To delineate these operational limits, we provide analytical proofs establishing guaranteed spectral bounds on the predictor’s Ja-cobian under distribution shifts. Furthermore, we validate DEGP across manipulation tasks, ntroducing a telemetry-based runtime contract that quantifies the operational boundaries where geometric conditioning preserves planning stability.

Abstract: Large language models (LLMs) are increasingly being used to support data analysis and have the potential to make statistical methods more accessible to clinicians and medical researchers with limited programming experience. However, the rapid development of technologies surrounding LLMs has introduced new concepts and workflows that may be unfamiliar to potential users. In particular, tool calling, sandboxing, and data science agents provide different mechanisms through which LLMs can perform statistical analyses, yet the distinctions between these approaches are often unclear. This paper introduces and compares these three developments from the perspective of statistical data analysis. Tool calling allows an LLM to access predefined statistical functions, sandboxing provides a controlled computational environment in which model-generated code can be executed, and data science agents extend these capabilities by combining reasoning, tools, memory, planning, and execution environments to pursue analytical objectives through multi-step workflows. Their application is illustrated using the R programming language together with the ellmer and securer packages. Examples demonstrate how statistical analyses can be performed using predefined functions, code execution within a sandbox, and a simple biostatistics agent. The advantages, limitations, and verification requirements of each approach are discussed, with particular emphasis on transparency, reproducibility, and human oversight. This paper aims to provide clinicians, researchers, and statisticians with an accessible introduction to these emerging technologies and their potential role in LLM-assisted data analysis.

Abstract: The hyperstructural model of redox reactions organises the oxidation states of an element into an H_v-group whose hyperoperation returns the set of thermodynamically accessible products. We recast this model in the language of general topology. To every hyperoperation ⊕ on a nite set H we attach a topology τ_⊕, the reaction topology, generated by the family of hyperproducts {x ⊕ y} as a subbasis. We prove that τ_⊕ is an Alexandrov topology whose minimal neighbourhoods are N(x) = ∩ {x ⊕ y : x ∈ x ⊕ y}; that its specialisation preorder admits the chemical reading y accompanies x in every reaction product containing x''; that τ_⊕ is T₀ exactly when no two species are reactionindistinguishable, and discrete exactly when every species is separated from all others by some reaction; and that the connected components of (H, τ_⊕) are the components of the comparability graph of that preorder. We then show that the hyperoperation is continuous for the Vietoris hyperspace topology whenever it is constant on minimal neighbourhoods, so that every H_v-group with discrete reaction topology is a topological H_v-group, and that the quotient by the fundamental relation β^∗ is discrete if and only if the β^∗-classes are open. The constructions are worked out on the redox data for tin, iron, copper and manganese: the hypergraphs induced by the iron H_v-group yield, respectively, a disconnected space with components {Fe³⁺} and {Fe, Fe²⁺}, a connected T₀ space, a connected non-T₀ space with two Kolmogorov classes, and an indiscrete space, while the copper hypergraph yields a connected, non-discrete T₀ space.

Abstract: Large Language Models (LLMs) are widely used for question answering, summarisation, coding, and research support, yet choosing a suitable model remains difficult because models differ in performance, context capacity, cost, privacy, and deployment options. This paper presents a document-based comparative analysis of five LLMs: GPT-4o, Claude 3.5 Sonnet, Gemini 1.5 Pro, Llama 3.1 405B Instruct, and Mistral Large 2. Using published technical reports, official documentation, model cards, and recognised benchmark studies, the models are examined under a common framework covering response quality, reasoning, coding, language understanding, context-window capacity, speed, computational requirements, cost, privacy, and deployment flexibility. No new experiments are conducted, and results obtained under different evaluation conditions are not treated as directly equivalent. The analysis shows that model selection involves trade-offs: hosted models offer easier access, whereas models with downloadable weights give greater deployment control but require more computing resources. No single model is found to be best for every task, which supports a task-specific approach to selecting LLMs.

Abstract: A security assistant that explains log lines has a built in weakness, because many parts of a log line are written by the attacker the assistant is meant to describe, so text placed in a web request can act as an instruction to the language model, which is known as indirect prompt injection. This study measures this on a small local assistant, Phi-4-mini in four bit form, using synthetic web server logs that contain real attack strings and one of four kinds of hidden instruction. The assistant labelled each line as malicious or benign and explained it to a learner. The study compares a plain prompt, a prompt that warns the model that the log is untrusted, a quarantine that removes instruction like text found by a rule based detector, and a locked verdict in which the detector sets the traffic light colour and the model only explains it, and repeats the test with reworded instructions that the detector does not recognise. A fake end of record declaring the line benign hid most attacks from the plain prompt, the warning gave no protection and caused false alarms on harmless lines, and how dangerous each instruction was depended strongly on its exact wording. Quarantine stopped the instructions it was written for but let every reworded one through, while the locked verdict hid no attack under any instruction. Explanations proved far more fragile than labels, and the locked verdict removed almost all false denials of the attack except those caused by a fake outcome. A security assistant should therefore decide its verdict outside the language model and check every explanation against that verdict.

Abstract: Large language model (LLM) agents choose which external tool to call from the tools' documentation, and this description matching fails in three measurable ways: similar tools create ambiguity, truncated or anonymized documentation creates degradation, and untrusted catalogs admit adversarial injection. We present Three-Axis Tool Selection (TATS), which answers each condition with a different signal: a contextual bandit that learns online to separate similar tools, schema features that survive the loss of descriptions, and a success-filtered anomaly penalty (ADAB) that downweights candidates unlike anything that has succeeded before. All three run inside one compact eight-feature selector with no LLM call at inference, evaluated axis by axis on BFCL and MetaTool. Under ambiguity, TATS's advantage over description matching widens from 12.6 to 21.4 points as similar candidates grow from 5 to 50; under name-only documentation, schema features keep 62.1% accuracy where description matching falls to 30.2%; under injection, ADAB cuts the selection rate of a malicious tool from 6.9% to 1.7% at a 3.4-point clean-catalog cost. We state when any feature-only defense cannot stop an attacker who knows the scoring rule, confirm it with a relaxed oracle, and show that an unforgeable registry of trusted tools reduces the same attack to about 1% of queries.

Abstract: When several sources bear on two hypotheses, Royall’s three evidential outcomes (strong for one, strong for the other, weak) leave no room for a fourth state: some sources are strong evidence for one hypothesis and others for the other. We integrate four verdicts (supported, refuted, undetermined, contradicted) in one likelihood-based report and state which error probability of each verdict is bounded in finite samples, which vanishes only asymptotically, and when “contradicted” stops describing a real conflict between source parameters. Spurious contradiction has probability at most \( m/c \) for \( m \) sources and per-source threshold \( c \), under any dependence, for any correctly specified family. With a truth midway between two simple hypotheses the contradiction probability tends to \( 1-2^{1-m} \); a guarded rule keeps it at most \( m\Phi (-\sqrt{2\ln\,c}) \) for marginally Gaussian sources with any common mean and any dependence, and at most \( m(m-1)\Phi (-2\sqrt{\ln\,c}) \) under pairwise joint normality with non-negative covariances. For composite hypotheses an integrated-over-maximised likelihood ratio gives the bound \( m/c \) for heterogeneous parameters within one hypothesis. A preregistered simulation confirmed the bounds. In the Bacillus Calmette–Guérin (BCG) vaccine trials the verdict for “risk ratio at most 0.5” against “at least 1” is “contradicted” under normal, binomial-profile and conditional likelihoods; because these hypotheses leave an indifference zone, this does not establish that the trials’ true effects conflict.

Abstract: Self-attention is fundamental to Transformer architectures, but its computational and memory requirements grow quadratically with sequence length. This paper presents a structured narrative review of computationally efficient self-attention algorithms. It establishes the mathematical foundations of standard self-attention and examines dense, sparse/local, low-rank projection, landmark/Nyström, kernel/linear, memory-efficient exact, and hierarchical mechanisms. The review compares these approaches in terms of arithmetic complexity, memory requirements, sequence-length scaling, approximation, connectivity, I/O behaviour, implementation overhead, and hardware dependence. It distinguishes algorithmic complexity reduction from implementation-level optimization and clarifies the conditions under which complexity is linear in sequence length. Analytical crossover conditions show how sequence length, representation dimension, mechanism-specific parameters, and implementation overhead influence practical advantage. The review demonstrates that efficient attention cannot be characterized solely as quadratic versus linear complexity and identifies research gaps in multidimensional complexity modelling, approximation-error trade-offs, I/O complexity, long-range dependency preservation, Pareto efficiency, and hardware-aware analysis.

Abstract: Vehicular Ad-hoc Networks (VANETs) present significant challenges that are gradually being addressed by the research community. Vehicle-to-vehicle communication is one area requiring improvement, particularly regarding link connectivity between vehicles. Routing protocols used in these networks can be significantly enhanced through the application of artificial intelligence techniques, such as fuzzy logic controllers. The three types of fuzzy controllers address measurement uncertainty with increasing sophistication, ranging from the simplest (Type-1) to the most advanced (Type-3). VANETs are characterized by mobility patterns that introduce considerable uncertainty in various aspects, such as route lifetime. This article presents three fuzzy decision systems (one of each type) sharing the same rules and input parameters. The objective is to determine whether the complexity of these networks justifies the use of more complex systems. Simulation results show that the system based on Type-3 controllers yields the best performance, confirming their potential to improve routing in VANETs.

Abstract: Can topological entropy predict how accurately approximate orbits can be shadowed by true orbits? We show that it cannot, even for Anosov automorphisms on the same two-dimensional torus with the same flat metric. The optimal local Lipschitz shadowing constant is the infimum of error-amplification factors valid at all sufficiently small scales. We construct one family whose entropy tends to infinity while this constant tends to the optimal lower bound one. We construct another family whose maps are smoothly conjugate and have identical entropy, periodic-point counts, and eigenvalues, while their shadowing constants tend to infinity. Thus neither entropy nor the full periodic-point counting sequence provides a finite upper bound. The proofs compute the toral constant through a linear inverse error operator and distinguish the effects of expansion strength and the angle between stable and unstable directions.

Abstract: Let X be a non-Archimedean Hilbert space over a conjugated non-Archimedean valued field K with Char(K) ̸= 2. Let A : D(A) ⊆ X → X and B : D(B) ⊆ X → X be possibly unbounded self-adjoint linear operators. For x ∈ D(A) with ⟨x, x⟩ = 1, define Δx(A) := ∥Ax − ⟨Ax, x⟩x∥. Then for all x ∈ D(AB) ∩ D(BA) with ⟨x, x⟩ = 1, we show that (1) \( \quad \Delta_x(A)+\Delta_x(B)\geq \max\{\Delta_x(A), \Delta_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \) and (2) \( \quad \Delta_x(A)+\Delta_x(B)\geq \max\{\Delta_x(A), \Delta_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \) Inequality (1) may be regarded as a non-Archimedean analogue of the Heisenberg-Robertson-Schrodinger uncertainty principle, while Inequality (2) serves as a non-Archimedean analogue of the Maccone-Pati uncertainty principle.

Abstract: This paper deepens the understanding of the geometric relationships between a regular plane curve and its stereographic lift onto the unit sphere S2. Central to this framework is the Stereo graphically Arc-Length (SAL) parametrization, a non-linear differential constraint that synchronizes coordinate time identically with the canonical spherical arc length t. Within this domain, we construct three automated dimensional-reduction architectures to classify locally 3D space curves based on the continuous evolution of the parameter t: a tangent indicatrix relaxation through a single flat projection channel, a coordinate projection relaxation utilizing a coupled dual-screen setup linked by a shared coordinate bridge, and an invariant phase-space representation inside the standalone γ-plane mapping fundamental 3D curvature and torsion fields. These local classifications rest on our local classification of plane curves, via the characterization of plane curves with polynomial curvature. Related to this classification scheme is the Conformal Binormal Invariant, an out-of-plane functional evaluated within flat coordinate channels. The framework is illustrated by a detailed example: the conical spiral.

Abstract: We study the Shannon entropy increment generated by the self-power escort transformation on a finite probability simplex, in which each coordinate is raised to its own value and the resulting weights are normalized. With the continuous boundary convention 0⁰ = 1, we prove that entropy increases strictly at every nonuniform probability vector, in every dimension. The proof establishes the stronger fact that the entropy increment is strictly Schur-convex in the interior. Its main ingredient is a monotonicity estimate for the coordinate derivatives, based on an aggregation inequality for self-powers and a uniform two-point weight bound. A facewise argument extends strict entropy increase to all boundary points. We also prove binary output majorization and construct interior counterexamples to output majorization in every dimension at least five. Finally, we determine the sharp quadratic entropy-production coefficient near the uniform distribution. All map-specific ingredients are derived here; the entropy theorem does not assume a separate convergence or contraction theorem for the transformation.

Abstract: Large Language Models (LLMs) have advanced rapidly through increases in model scale, data, and computation, while their underlying architectures have also undergone substantial evolution. Beyond the canonical Transformer architecture, recent designs seek greater efficiency, scalability, and flexibility by rethinking how sequence information is processed, represented, and transformed. We survey efficient attention and state management, modern recurrent and other linear-complexity sequence models, Mixture-of-Experts architectures, and advances in activation functions, normalization, residual connections, positional encoding, and tokenization. We also cover multi-token prediction and adaptive and recurrent depth. Across these areas, we highlight key design principles and trade-offs among representational power, optimization stability, and computational cost. Complementing this component-wise treatment, we take a system-level perspective. We trace representative open-weight LLM families to identify recurring patterns in how architectural choices are combined, and synthesize these developments along four dimensions—processing structure, transformation dynamics, state management, and output commitment—characterizing how recent designs preserve, refine, or relax the assumptions of the canonical decoder. Together, this survey aims to provide a systematic overview of LLM architectural evolution and to chart pathways toward next-generation designs.

Abstract: Games on knot and link diagrams provide a combinatorial setting in which topological invariants can be used to certify terminal outcomes and to guide winning strategies. Separately, bitopological game theory studies situations in which two topologies are relevant to the same underlying set or function space. This paper formulates a bitopological extension of finite knot-diagram games. A bitopological knot game consists of a finite diagram-state space, two topologies, player-dependent admissibility neighborhoods subordinate to those topologies, a turn rule, a terminal payoff, and a knot or link invariant used to evaluate terminal states. We prove a strategic quotient theorem: when a quotient map preserves terminal payoffs and the sets of successor classes available to each player, the quotient game has exactly the same minimax value and winning regions. We then give a two-topology version and explain how invariant-valued terminal payoffs can reduce a diagram game to a finite invariant game. A finite example shows that changing one player's admissibility system can change the value even when the terminal invariant is unchanged. Applications to the Knotting--Unknotting Game and the KnotLink Game are discussed as extensions, not as claims that the published winning-strategy results for those games have been reproved or superseded. The main contribution is a precise framework and a theorem that can be applied once the admissibility systems and quotient hypotheses are verified for a concrete family of knot or link shadows.

Abstract: We develop a groupoid-based framework for gerbes on orbifolds equipped with two compatible topologies. The smooth topology determines the underlying orbifold, whereas a second topology records additional topological structure on the levels of its presenting groupoid. We define paired banded gerbes, formulate their Dixmier–Douady invariants, and establish classification and Morita-invariance results under explicit descent and coefficient hypotheses. In particular, we prove that decorated pairs of circle-banded gerbes are classified by a product of two groupoid-cohomology groups. We then formulate a relative classification for compatible pairs using a homotopy fibre product of gerbe-classification groupoids. We prove that both constructions are invariant under Morita equivalences that preserve the two topological sites and their coefficient systems. A finite-quotient example demonstrates why orbifold isotropy cannot in general be discarded in favor of the coarse orbit space. The example also provides a test that any proposed bitopological Dixmier–Douady theory must satisfy. The standard orbifold-gerbe theory is distinguished throughout from the additional paired structure introduced here. The classification statements are conditional on explicitly stated descent hypotheses; no automatic equivalence between arbitrary pairs of topologies is assumed.

Abstract: The increasing reliance on cloud storage across sectors has made the protection of sensitive data a critical concern especially as advances in quantum computing threaten to break traditional public-key cryptographic systems. This growing vulnerability raises urgent questions about how to secure digital assets in a post-quantum future. To address this challenge, this study aims to design, implement, and evaluate a hybrid encryption framework that combines symmetric encryption for data confidentiality with quantum-resistant public-key algorithms for key management and digital signatures. The objective was to develop a secure and practical solution that can be readily adopted using open-source tools. The system was built in a Linux-based virtual environment using an S3-compatible object storage platform, integrated with quantum-safe cryptographic libraries, and automated through scripting. Performance testing was conducted using files ranging from 100 MB to 1 GB, with metrics such as encryption and decryption time, CPU usage, upload/download latency, and internet throughput analyzed. The results showed that while quantum-safe operations added some overhead, even large files could be securely processed in under two minutes with stable system performance and no data corruption. These findings confirm that hybrid encryption models can be deployed today without specialized hardware, offering a forward-compatible and robust approach to cloud security. This work contributes a tested and reproducible framework for early adoption of quantum-safe cryptography in real-world cloud environments.

Abstract: Small open language models that run on an ordinary office computer are an attractive basis for a security awareness assistant, and platforms such as SentinelSphere ship one in a compressed quantised form, that stores the numbers inside the model with fewer bits. This study asks how much cybersecurity knowledge this compression costs, and whether the cost is larger in Greek than in English. A sample of questions from the CyberMetric benchmark was translated into Greek with an open translation model that was not itself tested, the translation was checked by translating it back, and the same questions were put in both languages to Phi-4-mini at three levels of compression, to the general model Qwen2.5-3B and to the Greek adapted model Krikri-8B, all on a two core processor. Phi-4-mini answers Greek questions clearly worse than English ones even with light compression. The four bit compression that platforms ship costs almost nothing in English but a clear share of accuracy in Greek, and the strongest compression brings Greek close to guessing while English stays usable. The general model is the most accurate in English but among the weakest in Greek, where it needs about four times as many text pieces per question, while the Greek adapted model keeps the gap smallest at the price of a larger and slower file. Files made without the calibration step that relies on mostly English text, including the file the platform publishes, lose the same Greek accuracy at four bits. A compression level found acceptable in English should therefore not be assumed acceptable in Greek.

Abstract: Large Language Model (LLM)-based multi-agent systems have demonstrated strong capabilities in complex task automation. However, these systems suffer from a critical vulnerability: hallucination cascade, where erroneous information generated by one agent propagates through the system and is amplified by downstream agents. We propose Dynamic Consensus Auditing (DCA), a novel framework that introduces a lightweight audit agent to monitor inter-agent message flows. DCA combines Retrieval-Augmented Generation (RAG) with uncertainty quantification based on semantic entropy to detect potentially hallucinated content. When high-uncertainty messages are detected, DCA triggers a peer review mechanism where multiple agents vote on the validity of the information. Experiments on HotpotQA, TriviaQA, and MuSiQue benchmarks demonstrate that DCA reduces error propagation rates by 42.5% (relative reduction) on average compared to baseline multi-agent systems, with a token overhead of 15.3% (averaged across datasets). Our results suggest that, in the context of multi-hop question answering with three-agent pipelines, consensus-aware verification provides a practical balance between reliability and computational efficiency in multi-agent collaboration.

Abstract: Remote patient monitoring and telemedicine demand sensor fusion architectures integrating heterogeneous multi-modal streams, such as wearable sensors and continuous telemetry, into timely clinical decisions outside the hospital. We present MedROAD V3, a real-time, closed-loop FHIR R4 streaming platform treating sensor fusion as a first-class architectural framework, validated in a cardiac ICU. Five layers implement the pipeline: FHIR event sourcing from OpenEMR’s FHIR R4 server; Apache Kafka brokering; Faust stream processing assembling a 86-element vector over five-minute tumbling windows; an XGBoost- LSTM-Transformer ensemble combined by logistic regression meta-learning with SHAP attribution; and bidirectional FHIR write-back to the patient chart. On 1,161 held-out patients the calibrated ensemble reaches AUROC 0.774 with an Expected Calibration Error of 0.002. Restricting the feature set to what a three-device home kit measures costs 0.031 AUROC. Inferences exceeding the calibrated threshold trigger a structured three-sentence clinical narrative from the Claude API. The inference path measures 3.1 ms at the median, leaving end-to-end latency dominated by FHIR transport. Because the FHIR R4 Observation resource is device-agnostic, the identical pipeline ingests wearable streams without modification, positioning MedROAD V3 as a general sensor fusion framework for both hospital and telemedicine use.