Search Results (289)

This study develops a one-dimensional covariant Helmholtz model for a vector-valued wave field transported along a circular helical centerline and represented in the Frenet–Serret frame. For a helix with constant curvature $\kappa >0$ and torsion $\tau \ne 0$, the geometric coupling is described by a constant skew-symmetric connection matrix $\mathsf{\Omega }\in \mathfrak{so}\left(3\right)$. The covariant Helmholtz operator is shown to admit an exact gauge reduction to the flat componentwise Helmholtz operator through $u\left(s\right)=e^{-\mathsf{\Omega }s}y\left(s\right)$. Thus, within the one-dimensional centerline formulation, the helix preserves the operator spectrum while redistributing the observed Frenet components through parallel transport. The closed-form solutions show that a monochromatic input with wavenumber k is decomposed into a carrier and two geometric sidebands governed by the Darboux rotation rate $\lambda =\sqrt{{\kappa }^2+{\tau }^2}$. In the sub-geometric regime $k<\lambda$, the lower algebraic sideband is represented by the positive observable wavenumber $q_{-}=|k-\lambda |$, with associated scale $T_{\mathrm{beat}}=L_{-}=2\pi /q_{-}$. The lossless energy analysis proves conservation of the total averaged energy and its redistribution among the carrier and observable sidebands. A representative helical acoustic-channel design is then examined as a conceptual realization of the centerline model. Monte Carlo perturbations and additive-noise tests show that the predicted sideband locations, lower-sideband scale, and energy partition remain stable under prescribed fabrication tolerances and spectrally identifiable under weak and moderate measurement noise. Full article

Background/Objectives: Institutionalized older adults often present overlapping functional, cognitive, nutritional, intake-related, anthropometric, and biochemical abnormalities. We developed a prespecified weighted multidomain nutritional priority score (VIVA-NIPI) to organize the urgency of comprehensive nutritional review and assessed its internal construct concordance and ranking stability. Methods: Routine-care data from 124 residents in 10 long-term care facilities in Valencia, Spain, collected from June 2022 to April 2024, were combined across functional–cognitive, nutritional-intake/anthropometric, and selected biochemical-indicator domains. Weights were defined by the research team from clinical and methodological judgement rather than derived through a formal expert pairwise-comparison exercise; algebraically generated consistency ratios were not interpreted as validation evidence. The reference constructs overlapped with score components and therefore assessed internal concordance rather than independent clinical validity. Results: The continuous score averaged 24.4 ± 12.2. The partly overlapping k-means phenotype yielded an AUC of 0.918 (95% bootstrap CI 0.870–0.958); AUCs were 0.769 for MNA-SF ≤ 11, 0.602 for intake < 75%, and 0.782 for albumin < 3.5 g/dL. These values do not establish diagnostic or prognostic performance. Sensitivity analyses showed Spearman correlations of 0.991 after removing age, 0.960 after removing MNA-SF, 0.957 for an unweighted score, 0.983 when retaining only albumin and total proteins in the biochemical domain, and 0.901 after removing that do-main. Conclusions: VIVA-NIPI is a preliminary weighted multidomain research score, not a clinically validated triage tool. Independent clinical adjudication or prospective outcomes, external centre-aware validation, formal stakeholder weight elicitation, missing-data modelling, and calibration of actionable thresholds are required before implementation. Full article

Wet cooling towers are widely used for low-energy thermal management and ventilation support; however, high-fidelity simulations are computationally expensive for large design studies. This work develops a physics-based static reduced-order model for a two-dimensional axisymmetric counterflow wet cooling tower derived from computational fluid dynamics (CFD) simulations coupled with a user-defined source-term formulation for heat and mass transfer in the fill region. A design of experiments based on advanced Latin hypercube sampling generated 210 configurations, of which 168 valid simulations were retained. The active inputs included tower diameter, fill height, inlet air mass flow rate, inlet air temperature, inlet humidity ratio, inlet water mass flow rate, and inlet water temperature, while the cooling range and evaporation rate were selected as target outputs. Five surrogate families were compared by cross-validation. Kriging was statistically most accurate, with $R_{CV}^2$ values of 0.9999 and 0.9998 for the cooling range and evaporation rate, respectively. Second-order quadratic polynomial models were selected as the engineering reduced order model (ROM) because they capture non-linear boundary curvatures with accuracy, achieving $R_{CV}^2\ge 0.9989$ and root mean square errors of 0.0426 K and 0.00042 kg/s while preserving an explicit, directly implementable algebraic form. Sensitivity analysis indicated that the inlet water temperature and air mass flow rate are dominant factors within the sampled domain. Full article

Let $\mathcal{A}$ be a ∗-algebra with the unit I and a nontrivial projection P. Assume that $\mathcal{A}$ satisfies the condition that $X\mathcal{A}P=0$ implies $X=0$ and $X\mathcal{A}(I-P)=0$ implies $X=0$. Let $\eta$ be a nonzero complex number. For any $A,B\in \mathcal{A}$, the left $\eta$-Jordan ∗-product is defined by $A{◊}_{\eta }B=A^{\ast }B+\eta B^{\ast }A$. In this study, it is shown that if a map $\phi :\mathcal{A}\to \mathcal{A}$ satisfies $\phi (A_1{◊}_{\eta }{A}_2{◊}_{\eta }\cdots {◊}_{\eta }{A}_n)={\sum }_{k=1}^n{A}_1{◊}_{\eta }\cdots {◊}_{\eta }{A}_{k-1}{◊}_{\eta }\phi (A_k){◊}_{\eta }{A}_{k+1}{◊}_{\eta }\cdots {◊}_{\eta }{A}_n(n\ge 3)$ for all $A_1,A_2,\cdots ,A_n\in \mathcal{A}$, then $\phi$ is an additive ∗-derivation and $\phi (\eta A)=\eta \phi (A)$ for all $A\in \mathcal{A}.$ Full article

This paper proposes a construction of algebraically stabilizing blocks for quantized and finite-precision neural networks. The approach is based on linear transformations defined by integer-valued matrices satisfying a condition of the form $W^k=I+\mu D$, which specifies algebraically controlled $k$-step behavior and modular periodicity in the integer-valued setting. The proposed property is invariant under conjugation, allowing the stabilizing construction to be transferred across equivalent linear representations. The resulting module can be integrated locally into existing neural architectures without requiring the algebraic structure to be imposed globally. The theoretical results concern algebraic and modular stabilization of the linear block and do not constitute a general guarantee of classical spectral or asymptotic stability in real-valued space. The approach is evaluated experimentally under multi-component harmonic, impulsive, and noisy inputs in both floating-point and INT8-quantized settings. Across several experimental configurations, the proposed block reduces output energy, component-wise variation, selected amplitude-related measures, and finite-precision deviation relative to floating-point reference trajectories. Norm-matched control experiments further suggest that the observed effects are not attributable solely to a reduction in operator magnitude, but may also reflect structural properties of the algebraically constructed operator. The proposed construction is particularly relevant to neural systems operating under limited numerical precision, including FPGA-, ASIC-, and edge-oriented implementations. It provides a structural approach for incorporating formally specified algebraic properties into the design of neural-network architectures. Full article

This paper presents a theoretical algebraic block transformation based on bi-periodic Pell matrices and elliptic-curve point-valued blocks. The construction uses block-dependent scalar key matrices $K_i={\tilde{P}}_{m+i-1}(a,b)\in G{L}_2\left({\mathbb{F}}_q\right)$ to transform plaintext blocks whose entries lie in a selected elliptic-curve subgroup $G=⟨G_0⟩\subseteq E\left({\mathbb{F}}_p\right)$ of prime order $q$. The encryption operation is defined by the elliptic-curve scalar matrix action $M_i=B_i\star K_i,$ where the entries of $K_i$ act as scalar coefficients modulo $q$, and point additions are performed in $G$. The main purpose of the construction is to formalize the interaction between recursive Pell-type matrix sequences over ${\mathbb{F}}_q$ and elliptic-curve point-valued block transformations. The algebraic correctness follows from the invertibility of $K_i$ over ${\mathbb{F}}_q$ and the compatibility of the $\star$-action with the matrix multiplication. The deterministic version is not claimed to provide semantic security or IND-CPA security without additional randomization, nonce-dependent masking, and formal security proofs. Full article

In this paper, we introduce $\vartheta$-fuzzy quasi-k-norms with respect to a continuous t-norm $\vartheta$. Also, we investigate fuzzy operators defined on the product of a ∗-fuzzy measure space and an algebraic group, taking values in a $\vartheta$-fuzzy quasi-k-normed space. Furthermore, by employing a fuzzy controller associated with the ct-norm $\vartheta$, we give sufficient conditions for the approximation of a given operator by a homomorphic fuzzy operator. Full article

The System for Teaching and Assessment using a Computer Algebra Kernel (STACK) is an open source, computer algebra-based online assessment system for teaching and learning mathematics at university. Although the popularity is increasing around the world, its problem generation needs a complex procedure such as algebraic scripting, dynamic randomization, and grading logic, which poses a substantial workload. In this paper, we propose a mathematics problem generation tool using Generative AI for STACK. It adopts a Retrieval-Augmented Generation (RAG) framework to guide the AI to produce pedagogically aligned problems across Depth of Knowledge (DoK) levels, while a Computer Algebra System (CAS) validates mathematical precision. The output is rendered into an XML template and is imported into the STACK system. For evaluation, we measured the success rate of generating 90 problem files for STACK by the proposal and compared the completion time with their manual generation. Learning Object Review Instrument (LORI) was also evaluated for user satisfactions. The results showed that the success rate was 79% while the time was reduced by 35.71%. Furthermore, the LORI evaluations demonstrated a feasibility score of 82.1%, confirming the potential to mitigate teacher workload. Full article

Neural metric learning is often assessed by retrieval accuracy, but a learned dissimilarity can rank examples well while failing to have norm-like algebraic structure. This paper studies a precise finite question: within a Euclidean-anchored residual family of neural dissimilarities, can one reduce sampled defects of homogeneity, subadditivity, and dyadic reconstruction on latent differences without destroying retrieval performance? The construction is inspired by asymptotically metrically normable (AMN) vector spaces, but its claims are finite, sampled, and latent: it does not prove global AMN rigidity or certify a metric on the input space. The framework is motivated by the observation that many learned similarities have the form $K=\exp (-E/\tau )$ and therefore encode an unbounded distance-like quantity or squared distance-like quantity behind a bounded affinity. The AMN-relevant object is this cost, not the bounded kernel value. We formalize bounded-perturbation stability of the large-scale specific energy $E(nv,0)/n$, the conversion of subadditivity into multiplicative affinity consistency, and the quotient interpretation in which directions of zero large-scale cost are collapsed. The mathematical development then introduces finite dyadic diagnostics, learned-gauge and convex-unit-ball interpretations, finite norm-envelope witnesses, dyadic stability bounds, and refinement towers of witness norms. The empirical part reports full official Fashion-MNIST experiments with supervised-contrastive and proxy-anchor-style Euclidean baselines, post hoc audits for shrinkage, residual flexibility, off-training scales, and latent extrapolation, and a ten-seed full-query/full-gallery UCI Human Activity Recognition benchmark. The results show that Euclidean objectives can be stronger for Recall@1, whereas AMN-inspired residual regularization substantially reduces finite norm-like defects inside the residual family. The contribution is therefore a finite diagnostic and regularization framework for learned latent dissimilarities, not a state-of-the-art retrieval objective. Full article

We study support-determined constraints for eigenvectors of nonnegative symmetric tensors whose support may contain repeated indices. Such tensors are naturally encoded by uniform multi-hypergraphs, where each multiedge is represented by an exponent vector $\alpha \in {\mathbb{N}}_0^n$ with $\left|\alpha \right|=k$. Replacing the ordinary vertex-edge incidence matrix by the exponent-incidence matrix, we show that every nonzero-eigenvalue H-eigenvector satisfies linear incidence constraints in the transformed coordinates $y_i=x_i^k$. These constraints are invariant under positive scalar edge weights and reduce to the usual support-incidence constraints for ordinary squarefree hypergraphs. When the exponent-incidence matrix has a nontrivial left kernel, these relations provide nonzero support-level constraints in the transformed coordinates. We also describe the positive branch of the resulting constraint variety and prove a positive-weight realization criterion: a positive vector x can be realized as an eigenvector of some strictly positive edge-weighting of a fixed multi-hypergraph if and only if $x^{\left[k\right]}$ lies in the strictly positive coefficient cone generated by the exponent-incidence columns. Thus, the exponent-incidence constraint variety provides necessary algebraic constraints in general, while the strictly positive coefficient cone gives the exact positive-weight realization region on the positive branch. 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

Oblivious Transfer (OT) extension, in particular, the construction of Ishai, Kilian, Nissim, and Petrank (CRYPTO 2003) requires a hash function H that is correlation-robust(CR). All practical instantiations model H as a random oracle or an ideal cipher, leaving CR with no quantifiable reduction to a structural property of the deployed hash. It is natural to ask whether the most nonlinear balanced Boolean functions available on an odd number of variables, the near-bent functions of the Maiorana–McFarland (MM) class, furnish an algebraic, standard-model CR candidate. We prove that they do not, and we identify precisely why. First, we keep a correct spectral fact: a balanced $H:{\{0,1\}}^n\to \{0,1\}$ is $\epsilon$-CR if and only if ${\max }_{\Delta \ne 0}\left|{\mathcal{A}}_f(\Delta )\right|\le 4\epsilon ·2^n$, reducing CR to an autocorrelation bound. Against this criterion we establish three obstructions: (i) The MM-doubling family ${\mathcal{NB}}_k$ on $n=2k+1$ variables has autocorrelation supported only on the directions $(a,0,1)$, where it equals $2^{k+1}{W}_a$ with ${\sum }_{a\ne 0}{W}_a^2=2^{2k}$; hence $\epsilon \ge \frac{1}{4}{(2^k-1)}^{-1/2}$, a factor $\ge 2^{k/2}$ above the value one would need, and an exhaustive search over all balanced members for $k\le 2$ returns the maximal $\epsilon =\frac{1}{4}$ in every case. (ii) Near-bentness controls the Walsh maximum (nonlinearity), not autocorrelation: every near-bent function satisfies ${\sum }_{\Delta \ne 0}{\mathcal{A}}_f{(\Delta )}^2=2^{2n}$, so ${\max }_{\Delta \ne 0}\left|{\mathcal{A}}_f(\Delta )\right|\ge 2^n{(2^n-1)}^{-1/2}$ and no near-bent function is even approximately CR. (iii) A deterministic $\mathbf{H}:{\{0,1\}}^{\kappa }\to {\{0,1\}}^{\mathcal{\ell }}$ admits the support bound $\mathbf{SD}\left((\mathbf{H}\left(x\right),\mathbf{H}(x\oplus \Delta )),(U_{\mathcal{\ell }},U_{\mathcal{\ell }})\right)\ge 1-2^{\kappa -2\mathcal{\ell }}$, so statistical multi-output CR is impossible for $\mathcal{\ell }>\kappa /2$ and in particular at the IKNP regime $\mathcal{\ell }\approx \kappa$. Together, these results close the near-bent route to standard-model CR and clarify which design objective (low absolute indicator, not high nonlinearity) and which parameter regime ($\mathcal{\ell }\le \kappa /2$) a viable algebraic candidate would have to target. Full article

Background/Objectives: Pes planus and pes cavus are common foot deformities, which may lead to pain, functional limitations, and impairment of foot biomechanics. While calcaneal pitch, talar declination, and Meary angles, commonly used in diagnosis, provide objective information, their lack of a gold standard and the observer’s dependence on manual measurements limit their reliability. Therefore, in this study, these angles obtained from weight-bearing lateral foot radiographs were evaluated according to literature references, and the aim was to determine the model that provides the most accurate prediction in the diagnosis of pes planus using machine learning algorithms. It should be emphasized that, because the diagnostic labels were derived from literature-based thresholds of these same angles, the machine-learning task addressed here is the automated reproduction and standardization of expert, angle-threshold-based classification, rather than an independent clinical diagnosis from raw images. Methods: This retrospective study was conducted using weight-bearing lateral foot radiographs of 697 male patients obtained from the archives of public hospitals in Gaziantep. Calcaneal pitch, Meary angle, and talar declination angles were evaluated in both feet, and the data were labeled as normal, pes planus, and pes cavus. The dataset, consisting of a total of 1394 feet, was divided into training and test groups and analyzed using Random Forest, XGBoost, Logistic Regression, Support Vector Machine (SVM), and K-Nearest Neighbors (KNN) algorithms; the diagnostic performance of the models was compared using measures such as accuracy, F1 score, sensitivity, and specificity. Results: A total of 1394 feet from 697 male patients (mean age 24.8 ± 5.57 years) were analyzed using five machine learning algorithms with calcaneal pitch angle (CPA), Meary angle (MA), and talar declination angle (TDA) as reference labels. Ensemble-based methods showed superior performance, with XGBoost achieving perfect classification (Accuracy = 1.000) under all three labels for the left foot and 0.996–1.000 for the right foot, while Random Forest reached 0.986–1.000 across all experiments. Logistic Regression and SVM yielded moderate accuracies (0.905–0.973), whereas KNN consistently performed the weakest (0.905–0.964), particularly in the pes cavus subgroup. The near-perfect accuracy obtained when the labeling angle was itself included among the predictors reflects, at least in part, the algebraic reconstruction of the threshold rule from a same-source variable rather than genuine diagnostic generalization; results should therefore be interpreted with this in mind. Conclusions: This study demonstrates that machine learning, particularly ensemble methods such as XGBoost and Random Forest, provides high accuracy and consistency in diagnosing foot arch deformities based on radiographic angle measurements. Traditional models, such as Logistic Regression, still hold value in terms of clinical interpretability despite their lower performance. The findings suggest that machine learning-based approaches can offer objective, rapid, and reliable decision support tools for diagnosing pes planus and pes cavus, but external validation studies are necessary for clinical generalizability. Full article

The volumetric mass transfer coefficient ($k_{L}a$) governs the design, operation, and scale-up of aerobic bioprocesses, yet its dependence on reactor geometry, impeller design, operating conditions, and fluid properties limits prediction by empirical correlations. Machine learning (ML) improves accuracy but faces two barriers in bioprocess practice: selecting the best model among many candidates requires expertise, and small, highly multicollinear data make models chosen based on test error alone prone to overfitting. Using a browser-based, no-code platform, we trained 14 regression algorithms under an identical pipeline on a published $k_{L}a$ dataset, and introduced a composite objective, the generalization-penalized error (GPE), which is the test RMSE plus the absolute train–test RMSE gap. Minimizing GPE rather than test RMSE expanded the top statistically equivalent group to include not only boosting ensembles but also simpler, interpretable models, indicating that black-box models hold no clear advantage once train–test consistency is assessed. Sensitivity analysis showed that tree models produce discontinuous responses, whereas algebraic learning via elastic net (ALVEN) yields smooth surfaces. Shapley additive explanations (SHAP) and an ontology graph, interpreted by a retrieval-augmented language-model agent, identified rotational speed and gas flow rate as dominant, reproducing the established mass transfer mechanism. The framework offers a reproducible, interpretable, expertise-light route to bioprocess model selection. Full article

We review the structure of pseudoscalar mesons within an algebraic model formulated in the light-front framework. The approach provides a unified description of leading-twist parton distribution amplitudes, light-front wave functions, generalized parton distributions, parton distribution functions, elastic electromagnetic form factors, charge radii, and impact-parameter space distributions, all obtained from the same underlying Bethe–Salpeter wave-function representation. The analysis covers light mesons $(\pi ,K)$, the mixed $\eta$–${\eta }^{\prime }$ system, heavy–light states $(D,D_s,B,B_s,B_c)$, and heavy quarkonia $({\eta }_c,{\eta }_b)$, thereby enabling a systematic study of quark-mass effects, flavor-symmetry breaking, and the transition from emergent hadronic mass to heavy-quark dynamics. Where available, results are compared with experimental measurements, functional methods such as lattice-QCD calculations and Dyson–Schwinger Equation formalism, and other phenomenological approaches. The algebraic model thus offers a transparent, symmetry-preserving, and analytically tractable framework for connecting the longitudinal, transverse-momentum, and spatial structure of pseudoscalar mesons across all quark-mass regimes. Full article