Search Results (1,082)

We characterize observability in representation learning through Platonic Projection Structures (PPS), an operator-theoretic framework for analyzing representation accessibility under partial observation. Rather than treating observable outputs as direct reflections of latent representations, PPS models observation as a geometry induced by a self-adjoint positive semidefinite operator acting on a latent Hilbert space. A system is represented as a triple $(\mathcal{H},\Pi ,O),$ where $\mathcal{H}$ denotes a latent representation space, $\Pi ⪰0$ is an observation operator, and $O\left(v\right)=⟨v,\Pi v⟩$ defines an induced scalar observable. The framework characterizes observability through the quotient geometry $\mathcal{H}/ker\left(\Pi \right),$ which represents equivalence classes of latent states that are indistinguishable under observation. From this perspective, observable behavior is governed not by latent representations themselves, but by the geometry induced through the observation operator. We show that both quantum measurement and representation inference under linear observation models can be formulated within this common operator-theoretic structure while differing in the algebraic properties of their observation operators. Within this perspective, quantum measurement serves primarily as a mathematically canonical example of projection-mediated observability. The correspondence developed in PPS is therefore structural rather than physical. Within the same framework, representation transfer and knowledge distillation can be interpreted as approximate preservation of observable geometry through the intertwining condition $\Phi {\Pi }_T\approx {\Pi }_S\Phi .$ PPS further reveals a structural limitation of output-based interpretability: latent components contained in $ker\left(\Pi \right)$ are fundamentally inaccessible from observables generated through the induced observation process. Accordingly, attribution and explanation methods inherit intrinsic constraints imposed by the observation geometry itself. We provide controlled empirical validations demonstrating kernel-invariant observability, projection-induced attribution gaps, and rank-controlled observable geometry in latent representation spaces. Overall, PPS provides a mathematically explicit characterization of observability through operator-induced quotient geometry, offering a unified perspective on representation accessibility, interpretability, and representation transfer. Full article

We introduce the local kernel-based calculus, a unifying framework for local differential and integral operators based on an arbitrary positive continuous kernel function. This framework encompasses conformable, non-conformable, and our newly introduced local Euler-kernel derivatives as special cases. The parameter of the kernel is unrestricted and may take negative values, reflecting its role as a genuine parameter rather than an order of fractional differentiation. Within this general setting, we rigorously prove a complete set of foundational theorems: linearity, the product rule, continuity, Rolle’s theorem, the mean value theorem, and the fundamental theorem of calculus via the associated integral operator. We also derive a new formulation of the chain rule that expresses the chain rule entirely in terms of the kernel-based derivatives. While algebraically equivalent to the classical form, this representation preserves the intuitive structure of the chain rule without reference to the classical derivative. We further establish the Fundamental Theorem of Local Euler Calculus and its generalization, the Fundamental Theorem of Local Kernel-Based Calculus, confirming that the derivative and integral operators are genuine inverses, with the classical fundamental theorem recovered as special cases when the kernel reduces to unity. As an important illustration, we develop the local Euler calculus with the exponential kernel in full detail, providing explicit derivative and integral formulas for elementary functions. This special case demonstrates the simplicity and power of the functional approach. Overall, the local kernel-based calculus provides a solid, self-contained foundation that unifies a wide class of local operators and extends far beyond the traditional setting. Full article

With the development of the socio-economic landscape and the increasing demand for urban power supply, user expectations for power supply reliability have risen significantly. To address this challenge, dual-petal distribution networks, characterized by multiple tie-line structures and inter-regional load transfer paths, have significantly enhanced fault recovery capability and are gradually replacing traditional radial configurations as a key form of modern distribution systems. However, their multi-regional coupling characteristics introduce complex issues such as dynamic changes in load transfer paths and islanded operation, resulting in significant limitations in the accuracy and adaptability of existing reliability assessment methods. To this end, this paper proposes an analytical method for calculating reliability indices of dual-petal distribution networks, considering the characteristics of load transfer zones. First, typical operation modes of dual-petal distribution networks are extracted, and a time-sequential component reliability analysis model is established. Second, a load transfer zone matrix is constructed based on the impact of distribution network faults on load nodes across different regions. Third, based on the fault ride-through capability of distributed generation (DG), a load restoration strategy considering load transfer zone characteristics is formulated, and the DG Island Recovery Matrix (DGIRM) is derived. Finally, by performing algebraic operations among various matrices and reliability parameter vectors, an explicit analytical calculation of reliability indices for dual-petal distribution networks with different DG configurations is achieved. The effectiveness of the proposed method is validated using a typical dual-petal network. The results demonstrate that the proposed method offers high computational efficiency and accuracy, effectively quantifying the impact of DG on the power supply reliability of dual-petal distribution networks, and providing theoretical and methodological support for the reliability assessment and planning of complex distribution systems. Full article

The general problem of describing quantum states not only by wave functions and density operators but also by probability distribution functions is discussed in this paper. The qubit-state density-matrix elements expressed in terms of probability distributions are connected with the irreducible representation of group $SU\left(2\right)$. The transform of the probability distributions corresponding to the unitary transform of the qubit state of the spin-$1/2$ system is obtained and expressed in terms of matrix elements of the group $SU\left(2\right)$. The possibility to extend the introduced formalism to other quantum states is suggested. For qubit systems, the notion of density operators algebra and the relation with the probability representation of quantum states are discussed. The Schrödinger equation for qubit states is obtained as an equation for probabilities. Full article

TheLinear Equivalence Signature Scheme (LESS) is a code-based post-quantum candidate in the National Institute of Standards and Technology’s (NIST) standardization process for additional digital signatures. In this paper, we present an area-efficient FPGA accelerator for the Reduced Row Echelon Form (RREF) kernel of LESS, designed for embedded RISC-V SoCs where resource overhead is the primary constraint. Our architecture targets the scheme’s primary computational bottleneck: the linear-algebra core responsible for RREF processing. By implementing an optimized pivot-reuse workflow, our design significantly reduces redundant row-reduction operations across related computations. The accelerator features a matrix-oriented execution engine paired with a streaming control interface to minimize synchronization overhead. Implementation on a Xilinx Artix-7 FPGA shows that despite its compact footprint, the accelerator achieves up to 21× speedup over the embedded software RREF baseline. By prioritizing a minimalist footprint, our design requires only 1.38 to 8.7 KeSlice, depending on the targeted security level. By covering all LESS security levels and providing comparisons with existing post-quantum cryptographic hardware, this work establishes a performance baseline for a signature scheme that has remained largely unexplored in the hardware domain. Full article

Initial value problems of the third order featuring explicit dependence on velocity, denoted as $y^{‴}=f(x,y,y^{\prime })$, emerge regularly across applications such as electromechanical networks, structural mechanics, and robotic trajectory control. Despite their practical prevalence, these differential equations remain insufficiently addressed by standard numerical integration techniques. Orthodox Runge–Kutta–Nyström (RKN) schemes are fundamentally formulated for differential equations lacking the first derivative, specifically $y^{‴}=f(x,y)$. Due to this algorithmic constraint, researchers frequently resort to computationally demanding first-order system reductions or rely upon standard Runge–Kutta methods. The present study resolves this methodological gap by defining an explicit s-stage integration architecture that natively incorporates the first derivative within the internal stage evaluations. Such structural modifications require the deployment of a supplementary coefficient matrix, denoted as D, to formulate the corresponding order theory. The complete set of algebraic order conditions is systematically established up to the seventh order, accompanied by a generic mathematical framework for generating schemes of arbitrary order. Based on this analytical foundation, an embedded $6(4)$ method is constructed. This specific pair achieves strict error tolerances utilizing merely six function evaluations per integration step, representing a substantial operational reduction compared to the eight computations strictly required by equivalent Runge–Kutta pairs. Direct numerical integration of the native third-order system prevents the dimensionality increase from reducing to first-order systems. Performance validation of the numerical solver involves two representative physical benchmarks: a coupled robotic appendage subjected to platform excitation and an electromechanical actuator array regulated by transient control inputs. Both dynamical systems exhibit severe velocity-dependent dissipation mechanisms and nonlinear external forcing. Quantitative numerical evaluations confirm that the constructed 6(4) pair yields higher precision and demands less computational expenditure than prevailing RK and RKN integrators. The analytical and empirical findings establish that derivative-capable Nyström integration algorithms furnish mathematically rigorous and computationally efficient numerical solutions for velocity-coupled third-order dynamics. Full article

The central charge occupies a unique role in conformal field theory, simultaneously serving as a measure of degrees of freedom, as the determinant of Casimir energy through modular transformations, and as an obstruction to the naive extension of the Witt algebra. The Virasoro central extension itself is rigid: it fixes c as a label of a given conformal field theory. In this work, we propose that higher categorical coherence—the pentagon and hexagon constraints governing fusion and braiding data, one level above the cocycle responsible for the Virasoro extension—supplies an additional, physically controllable handle. We show that controlled deformations of this higher coherence (higher categorical coherence breakdown, HCCB), implemented consistently through anomaly inflow, shift the effective central charge read out by anomaly-sensitive observables in quantized steps, opening the possibility of treating the measured central charge not as a fixed label but as an experimentally addressable piecewise-quantized quantity. We then focus on the fractional quantum Hall effect (FQHE), where the chiral central charge $c_{-}$ directly governs the quantized thermal Hall conductance. After reviewing the role of edge conformal field theories and current bounds on thermal transport, we propose experimental modifications—such as engineering multi-component edge states, coupling to non-Abelian quasiparticles, or introducing controlled categorical perturbations—that could render higher coherence breakdown detectable as shifts in the effective central charge. Two further elements complete the program. First, we show that within the consistent framework, all route- and bracketing-dependent observables vanish identically (route blindness), so that the pentagon and hexagon interferometers and thermal Y-junction networks we design operate as precision null tests of the modular-functor axioms themselves—the axioms stating that anyonic amplitudes are determined by the topology of a process rather than by the bookkeeping route used to compose it. Second, we show that a quantized remnant of route sensitivity survives in exactly one consistent form: the holonomy of closed cycles of categorical controls, realizing a central-charge pump for which the integer count per cycle is a family invariant beyond any static stacking description. The resulting framework provides both a conceptual reinterpretation of the central charge as a higher obstruction in categorical terms and a concrete experimental route for probing its dynamical behavior. Beyond the quantum Hall setting, these ideas suggest a broader program: anomalies, topological phases, and even string worldsheet central charges may admit reinterpretation through higher coherence. We conclude by outlining a research agenda in which categorical methods yield new experimental observables, potentially transforming the interplay between mathematics, condensed matter physics, and high-energy theory. Full article

Vertically linked commodity markets—global futures, regional spot, and farm-gate prices—transmit information through directed cross-market channels whose strength varies with latent volatility regimes. Standard deep learning forecasters absorb both the directed cross-market dependence and the regime dependence of intrinsic-mode-aligned latent components into shared model weights, with no explicit architectural mechanism that exposes either as an inspectable structure. This paper proposes HVB-RA, a modular framework that combines two such mechanisms with a per-tier Variational Mode Decomposition and bidirectional LSTM backbone: (i) a directed cross-market attention layer in which the upstream-to-downstream topology is supplied from domain knowledge and the time-varying upstream-source attention intensities at the farm-gate tier (the regional-spot tier, with a single upstream key, reduces algebraically to a fixed residual upstream fusion) are extracted from data, and (ii) a regime-informed modal-weighting layer that mixes two trainable softmax weight profiles over IMF-aligned latent components through a filtered Markov-switching state probability fitted in a separate stage. An auxiliary post hoc projection enforces an exact linear constraint defined by long-run sample-mean ratios across tiers; the paper does not claim that these descriptive ratios are cointegrating relations or equilibrium coefficients. The framework is evaluated on three tiers of daily natural-rubber prices spanning 2038 trading days, against three external benchmarks (random walk, ARIMA(2,0,2), and an exogenous-only LSTM) and a contemporary neural hierarchical-interpolation forecaster (NHITS). Root mean squared error is reported per tier-horizon cell; a decision-aware income-smoothing metric quantifies the operational value of $h=5$ farm-gate forecasts under a 5-day selling rule; and a within-method comparison evaluates the marginal contribution of the auxiliary constraint projection. On the present single-regime test window, HVB-RA attains a lower point error than the contemporary NHITS baseline at every tier-horizon cell, while no method—including HVB-RA—improves on the random-walk floor at most cells; the regime-conditional components of the architecture are not identifiable because every calibration and test origin is classified as a high-volatility regime by the trained Markov-switching model. The paper contributes to machine learning and knowledge extraction by demonstrating how time-varying upstream-source attention intensities at the farm-gate tier and regime-dependent latent-component-weight profiles—two forms of latent structure typically absorbed into model weights—can be exposed as explicit, inspectable, and individually testable components of a multi-tier forecasting architecture, and by providing a reproducibility package documenting the conditions under which each component is expected to be identifiable. Full article

In this paper, we investigate the theory of nonexpansive mappings in the framework of fuzzy normed GE-algebras. After recalling the fundamental concepts of GE-algebras and fuzzy GE-norms, we introduce the notion of fuzzy nonexpansive mappings and examine their basic structural properties. We show that the composition of two fuzzy nonexpansive mappings remains fuzzy nonexpansive, and establish that every fuzzy nonexpansive mapping is sequentially continuous with respect to fuzzy convergence. Further, by employing the concept of a fuzzy GE-interpolation family, we define $\alpha$-averaged mappings in fuzzy normed GE-algebras and prove that such averaged operators preserve nonexpansiveness. We also develop a demiclosedness-type principle and provide fixed point equivalence results between a mapping and its associated averaged mapping under suitable assumptions. Finally, we prove that the fixed point set of a fuzzy nonexpansive mapping is sequentially closed. These results extend classical ideas from metric fixed-point theory and Banach space theory to the algebraic setting of fuzzy normed GE-algebras. Full article

This paper adapts a semi-analytical framework the Laplace-Residual Power Series Method (LRPSM) to solve the time-fractional Allen–Cahn equation under the Caputo derivative. While the classical Allen–Cahn model successfully describes phase separation, its fractional counterpart is essential for capturing sub-diffusive memory effects in complex heterogeneous materials. However, the interplay between the non-local fractional temporal operator and the cubic nonlinearity of the bistable double-well potential creates significant computational bottlenecks for conventional time-domain series solvers. The proposed approach projects the governing fractional partial differential equation into the Laplace domain, systematically replacing the computation of iterative fractional derivatives with the algebraic evaluation of asymptotic limits at infinity. Furthermore, the nonlinear cubic interactions are managed through Laplace-space convolution theorems. The structural convergence of this approach is evaluated against multi-scenario one-dimensional phase transitions. Graphical analyses, featuring 2D profile trajectories and 3D spatiotemporal surface mappings, visually illustrate the retarded interfacial propagation driven by fractional memory. Ultimately, this study presents the LRPSM as an applicable, continuous mathematical tool for approximating anomalous diffusion in the specific phase-field dynamics evaluated herein. Full article

We introduce the Hamiltonian pseudorandom function (HPRF), a new symmetric cryptographic primitive in which the function family $\left\{F_k\right\}$ is defined by $F_k\left(q\right)=\nabla S_k\left(q\right)$, the gradient of the generating function of a secret Lagrangian submanifold $L_k$ on the symplectic torus ${\mathbb{T}}^{2n}$. The key k specifies a composition of kicked-rotor maps in the strongly chaotic regime, whose classical Lyapunov exponents grow as $log(K/2)$ per kick. The HPRF is best understood as a seeded one-way function with high min-entropy output: $F_k$ is smooth ($C^{\infty }$), so its raw output is not directly usable as a uniform keystream, but it is computationally hard to invert. We construct three symmetric encryption modes—Mode A (key-dependent coordinate frame), Mode C (Lagrangian keystream), and Mode AC (hybrid)—in which the HPRF supplies the hardness and a key derivation function (HKDF) supplies bit-level uniformity. Standard symmetric composition then yields IND-CPA and IND-CCA2 security. Classical security reduces to the Lagrangian identification problem (LIP), shown as equivalent to the Hamiltonian inversion problem of recovering the kick parameters, which we state as an explicit hardness assumption supported by a precision/sample-complexity obstruction from the positive Lyapunov exponents, by the empirical failure of concrete attacks, and (more heuristically) by topological suggestiveness from the Arnold conjecture and Floer theory. We validate a gradient-fitting attack and an algebraic-structure attack and show that both fail. For quantum security, we propose what we believe is the right framing: that the composed Floquet operator ${\widehat{U}}_K^r$ is a candidate pseudorandom unitary (PRU) in the sense of Ji–Liu–Song. We provide three independent pillars of evidence—Wigner–Dyson spectral statistics, Lyapunov-rate scrambling, and conjectural approximate-design behaviour—and reduce the HPRF quantum security to the PRU conjecture for ${\widehat{U}}_K^r$. We then retire the dynamical-localisation argument of previous drafts as inapplicable at cryptographic parameters; the chaotic-pseudorandomness regime that the operator actually inhabits is, we argue, a stronger foundation than the one that localisation would have provided. A deterministic fixed-point arithmetic core ensures cross-platform bit-exact consistency. A reference implementation validates correctness across all modes, and an NIST SP 800-90B analysis of the output min-entropy fixes the parameter sets. As a foundational proposal, the HPRF is intended for settings that seek a symmetric hardness assumption structurally independent of the algebraic problems underlying current cryptography, for example, as a hedge primitive in defence-in-depth designs, or as a basis for further study of geometry- and chaos-based cryptography, rather than as a drop-in replacement for AES or lattice-based schemes at this stage. Full article

Large language models (LLMs) built upon the Transformer architecture have achieved remarkable performance in natural language understanding, text generation and logical reasoning, while their internal working mechanisms remain poorly interpreted. This paper establishes a systematic mathematical analysis framework tailored for decoder-only Transformer LLMs, based on linear algebra, tensor analysis, probability theory, information theory, optimization dynamics and geometric deep learning. We conduct rigorous mathematical modeling and theoretical deduction on core modules including word embedding, position encoding, self-attention, feed-forward networks, training optimization and generalization reasoning, and explore the mathematical nature of semantic representation, contextual correlation, knowledge storage and logical inference within models. In this paper, we strictly distinguish between classic established Transformer theories and our original mathematical derivations and conclusions. Distinct from existing fragmented theoretical studies, this work presents six targeted novel contributions beyond conventional Transformer theories: (1) we construct the first full-process unified mathematical framework covering all core modules and the entire lifecycle of Transformer-based LLMs; (2) we provide strict mathematical proof to verify that single-head self-attention is essentially a kernel weighted average operation in reproducing kernel Hilbert space and derive the low-rank and sparse properties of attention weights; (3) we establish a high-dimensional non-convex optimization dynamics model for pre-training and mathematically prove that model training converges to flat local minima; (4) we derive a tighter upper bound of generalization error and quantify the quantitative relationship among model parameters, sequence length, training data scale and generalization performance; (5) we characterize the latent space as a low-curvature smooth Riemannian manifold and model logical reasoning as geometric transformation on this manifold; (6) we design multi-group controlled experiments on mainstream datasets to quantitatively validate all above theoretical conclusions. This paper further summarizes the inherent mathematical limitations of current Transformer LLMs and proposes feasible theoretical optimization paths, referring to state-of-the-art research published from 2021 to 2026. The outcomes of this research can provide solid mathematical theoretical support for improving model interpretability, optimizing network structures and boosting practical performance, and facilitate the transition of LLM research from empirical engineering practice to theory-driven development. Full article

Edge artificial intelligence (Edge AI) requires real-time inference under stringent constraints on computation, memory, energy, and connectivity. Although training can be offloaded to servers, efficient, high-capacity inference and rapid on-device adaptation remain central challenges. Cloud-based inference offers substantial computational power but depends on connectivity, latency, privacy, and reliability conditions that edge deployments cannot always guarantee. Classical model-compression methods—including quantization, pruning, distillation, and neural architecture search—have extended the feasibility of on-device inference, yet they leave largely unchanged the fundamental cost of the linear-algebraic, sampling, and optimization primitives that dominate modern deep learning. Quantum computing has therefore been proposed as a complementary accelerator for selected AI workloads, with theoretical advantages in linear systems, singular value decomposition, sampling, kernel evaluation, and optimization. This review surveys the emerging field of quantum-accelerated AI for edge systems under a hybrid architectural premise: edge devices remain classical, while quantum processors operate as remote, cloud, MEC, or near-edge accelerators. We synthesize advances across quantum learning models, hybrid optimization methods, hardware and deployment architectures, and quantum-inspired approaches suitable for constrained devices. We also assess the practical barriers that currently separate asymptotic quantum advantage from deployable edge intelligence, including data loading, measurement overhead, noise, latency, and benchmarking gaps. Finally, we outline a staged research roadmap from near-term hybrid workflows to fault-tolerant and integrated quantum-edge architectures. Full article

We develop a categorical and operadic semantics for the diagrammatic proof system underlying the Greimas semiotic square. Building on a prior proof-theoretic formulation, we extract a typed signature of diagrammatic inference rules and construct the corresponding free coloured operad ${\mathcal{O}}_{\Sigma }$, whose elements correspond to proof trees. This establishes a precise correspondence between diagrammatic derivations and operadic terms, making explicit the compositional structure implicit in the original system. We then interpret these terms as wiring diagrams in a symmetric monoidal setting, yielding a graphical semantics in which intermediate semantic configurations are represented as flows through a network of operations. Within this framework, the construction of the semiotic square is realised as a single composite operation $\Omega$, obtained by operadic substitution of generators corresponding to negation, implication, and meta-term formation. Finally, we consider semantic interpretations of this structure as algebras of ${\mathcal{O}}_{\Sigma }$, yielding a category $\mathbf{Alg}\left({\mathcal{O}}_{\Sigma }\right)$, whose morphisms capture structure-preserving translations between interpretations. This provides a formal account of the extensibility of the square across domains such as seme-level analysis, modality, and narratology, and recasts it as a compositional semantic schema rather than a static relational diagram. Full article

Numerous mathematical models, such as elasticity, nuclear reactor dynamics, and heat conduction in memory materials, use systems of fractional integro-differential equations, or FIDEs. In this research, the numerical solution of linear and nonlinear systems of FIDEs is obtained by means of an operational matrix approach based on Hermite wavelets. Using the operational matrix of the Hermite wavelets, the FIDE is transformed into an algebraic equation, and the solution of that algebraic equation is obtained. Three numerical examples are provided to show the accuracy and consistency of the suggested technique. Full article