Search Results (37)

The Zeta-Minimizer Theorem establishes that the Riemann zeta function $\zeta \left(s\right)$ and the primes arise variationally as unique minimizers of a phase functional defined on a symmetric measure space $\left(X,\mu ,G\right)$ equipped with helical operators. Three fundamental axioms—strict concave entropy maximization (Axiom 1), spectral Gibbs minima with non-vanishing ground states (Axiom 2), and irreducible bounded oscillations with flux conservation (Axiom 3)—allow for the selection of the non-proper Archimedean conical helix as the sole topology satisfying all constraints. Primes emerge as indivisible minimal cycles in the associated representation graph $\Gamma$ (via Hilbert irreducibility and Maschke’s theorem), while the Euler product is recovered through the spectral Dirichlet mapping of the helical eigenvalues. The partial zeta product, $Z\left(s\right)=\prod _j{\displaystyle \frac{1}{1-p_j^{-s}}},s\in \mathbb{R}\left\{0\right\},$ constitutes the exact grand partition function of any finite subsystem. Numerical inversion of this product directly recovers the mixture frequency $s$ from any experimental compressibility factor $Z_{\mathrm{m}\mathrm{i}\mathrm{x}}$. Mole fractions $x_i\left(s\right)$, interaction parameters $\Delta \left(x_i\right)$, and the Lyapunov spectrum $\lambda \left(x_i\right)$ then follow deductively via the helical transfer matrix and the closed-form linear ODE for $\Delta$. Occupation numbers $N\left(x_i\right)$ attain sharp maxima precisely at Fibonacci ratios $F_r/F_{r+1}$, leading to the molecular prime-ID rule. For twelve representative purely binary (irreducible) systems spanning atomic noble gases, simple diatomics, polar molecules, and an aromatic ring, the residuals satisfy $|Z\left(s\right)-Z_{\mathrm{m}\mathrm{i}\mathrm{x}}|<1.5\times {10}^{-8}$. The resulting $\lambda \left(x_i\right)$ curves accurately reproduce critical points, liquid ranges, and thermodynamic anomalies with zero adjustable parameters. The Riemann Hypothesis follows rigorously as a theorem: the unique fixed point of the duality functor $s\mapsto 1-s$ that preserves the orthogonality condition $\sum {\mathrm{c}\mathrm{o}\mathrm{s}}^2{\theta }^k=1$ is $\mathrm{R}\mathrm{e}\left(s\right)=1/2$, enforced by Axiom 1 concavity and Axiom 3 irreducibility. The framework is fully deductive and parameter-free and extends naturally to arbitrary mixtures and multiplicities through the helical representation graph. It provides a variational unification of analytic number theory, spectral geometry, thermodynamic phase behavior, and the Riemann Hypothesis from first principles. Full article

Using the instanton partition function for five-dimensional $U\left(1\right)$ gauge theory with eight supercharges and a single adjoint massive hypermultiplet on the $\Omega$ background, we give explicit expression for non-perturbative corrections to the topological string theory in the holomorphic limit. It was argued that in this case the theory is compactified on the twisted affine line bundle over $\mathbb{C}\times T^2$. We perform calculations in two ways. First we modify the integration contour by adding poles responsible for non-perturbative physics in accordance with a recent proposal. Then, we compute the genus zero Gopakumar–Vafa invariants for our case and evaluate the non-perturbative corrections to the partition function. We check that both calculations give the same result. Full article

This paper addresses the problem of cooperative coverage control for heterogeneous Autonomous Underwater Vehicle (AUV) swarms operating in complex underwater environments. The objective is to achieve optimal coverage of a target region while simultaneously ensuring collision avoidance—both among AUVs and with static obstacles—and satisfying the inherent dynamic constraints of the AUVs. To this end, we propose a hierarchical control framework that fuses Control Barrier Functions (CBFs) with consensus theory. First, addressing the heterogeneity and limited sensing ranges of the AUVs, a cooperative coverage model based on a modified Voronoi partition is constructed. A nominal controller based on consensus theory is designed to balance the ratio of task workload to individual capability for each AUV. By minimizing a Lyapunov-like function via gradient descent, the swarm achieves self-organized optimal coverage. Second, to guarantee system safety, multiple safety constraints are designed for the AUV double-integrator dynamics, utilizing Zeroing Control Barrier Functions (ZCBFs) and High-Order Control Barrier Functions (HOCBFs). This approach unifies the handling of collision avoidance and velocity limitations. Finally, the nominal coverage controller and safety constraints are integrated into a Quadratic Programming (QP) formulation. This constitutes a safety-critical layer that modifies the control commands in a minimally invasive manner. Theoretical analysis demonstrates the stability of the framework, the forward invariance of the safe set, and the convergence of the coverage task. Simulation experiments verify the effectiveness and robustness of the proposed method in navigating obstacles and efficiently completing heterogeneous cooperative coverage tasks in complex environments. Full article

The spin-one Ising model on the honeycomb lattice has never been solved exactly in spite of its simplicity. Even its exact critical temperature is not known. The exact integer values for the density of states of the spin-one Ising model on the $L\times 2L$ honeycomb lattice are enumerated up to $L=14$. The partition function zeros in the complex temperature plane of the spin-one Ising model on the $L\times 2L$ honeycomb lattice are exactly obtained, using the density of states. The properties of the partition function zeros in the complex temperature plane are related to the behaviors of various thermodynamic functions, in particular, their singular behaviors. The unknown properties of the spin-one Ising model on the honeycomb lattice are investigated, based on its partition function zeros in the complex temperature plane. Full article

The problem of multi-function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node is required to compute with zero error multiple vector-linear functions, where each vector-linear function has distinct inputs generated by multiple source nodes. The computing rate tuple of an admissible code is defined as a tuple consisting of the average number of zero-error computations for each vector-linear function when the network is used once jointly. From the information theoretic point of view, we are interested in characterizing the rate region, which is defined as the closed set of all achievable computing rate tuples. In particular, when the sink node is required to compute a single vector-linear function, the network multi-function computation problem degenerates to the network function computation problem. We prove an outer bound on the rate region by developing the approach of the cut-set strong partition. We also illustrate that the obtained outer bound is tight for a typical model of computing two vector-linear functions over the diamond network. Furthermore, we establish the relationship between the network multi-function computation rate region and the network function computation rate region. Also, we show that the best known outer bound on the rate region for computing an arbitrary vector-linear function over an arbitrary network is a straightforward consequence of our outer bound. Full article

The heat capacity decomposition method, a well-established analytical approach in polymer thermodynamics for elucidating thermal transitions in homogeneous polymers, is extended here to heterogeneous systems. We demonstrate that the decomposition of heat capacity based on partition function zeros allows the identification of transition-like crossovers originating from compact low-energy states, thereby enabling the evaluation of the foldability of HP sequences. The occurrence of significant crossovers between the collapse and folding transitions indicates slow folding behavior, whereas their absence characterizes good folders. This criterion is further validated through kinetic Monte Carlo simulations of two representative sequences. Full article

To address the challenge of frequent extreme events causing power grid failures and traditional post-fault recovery modes struggling to meet the rigid demand for continuous power supply to critical loads, this paper proposes a partition-based two-layer optimization strategy. First, a partitioning index coupling active power flow with reactive voltage sensitivity is constructed, and the Fast Newman algorithm is applied to obtain partitions tailored to extreme events. Second, a two-layer optimization model is established: the upper layer performs network reconfiguration to minimize the total load curtailment, while the lower layer coordinates adjustable resources and power mutual aid between partitions. A simulation verification using the IEEE 39-bus system shows that the proposed method efficiently makes decisions during extreme events, achieving zero interruption for critical loads and reducing the curtailment of non-critical loads to 374.9 MW—a reduction of 54 percent compared to the traditional centralized dispatch model. The partitioning results also exhibit a high modularity of 0.6554 and a low boundary power flow factor of 0.1321, confirming the structural and functional advantages of the proposed approach. The method demonstrates good practical application value in enhancing grid resilience. Full article

As nutrient allocation trade-offs occur between reproductive and vegetative development in crops, optimizing their partitioning holds promise for improving agricultural productivity and quality. Herein, we characterize the phenotypic diversity of the fruitfulness trait and identify associated genes in tea plants (Camellia sinensis). Over three consecutive years, we monitored the fruitfulness of an F₁ hybrid population (n = 206) derived from crosses of ‘Emei Wenchun’ and ‘Chuanmu 217’. A marked variation was observed in the yield of individual plants, ranging from complete sterility (zero fruits) to exceptionally high fertility (1612 fruits). Using the high-resolution genetic linkage map and the fruitfulness data, we identified a stable major QTL designated as qFN5. To fine-map the underlying gene(s), artificial pollination experiments were conducted with extreme phenotype individuals (with the highest vs. lowest fruit numbers). Bulked segregant RNA sequencing (BSR-Seq) with ovules collected at two and seven days post-pollination (DPP) identified the genomic intervals that exhibit a high degree of overlap with qFN5. Analysis of expression dynamics combined with functional genomics data revealed a prominent candidate gene, CsETR2 (TGY048509), which encodes an ethylene receptor protein. When CsETR2 was overexpressed in Arabidopsis thaliana, the transgenic lines exhibited significantly decreased reproductive performance relative to the wild-type plants. Relative to the wild type, the transgenic lines exhibited a significant decline in several key traits: the number of effective panicles decreased by 72.5%, the seed setting rate dropped by 67.7%, and the silique length shortened by 38%. These findings demonstrate its role in regulating plant fruitfulness. Furthermore, yeast one-hybrid and dual-luciferase assays verified that CsMYB15 (TGY110225) directly binds to the CsETR2 promoter, thus repressing its transcription. In summary, our findings expand the understanding of genetic regulation underlying fruitfulness in tea plants and provide candidate target loci for breeding. Full article

This paper addresses the controller design problem for linear systems with time-varying delays. By constructing a novel Lyapunov–Krasovskii functional incorporating delay-partitioning techniques, we establish delay-dependent stability criteria for the solvability of the robust stabilization problem. The derived conditions are formulated as linear matrix inequalities (LMIs) that become affine upon fixing a single scalar parameter, thereby facilitating efficient numerical computation. Furthermore, these criteria guarantee that the reachable set of the closed-loop system remains bounded within a prescribed ellipsoid under zero initial conditions. The effectiveness and superiority of the proposed approach are demonstrated through two comparative numerical examples, including a benchmark problem with varying delay. Full article

Hyperparameter optimization (HPO), which is also called hyperparameter tuning, is a vital component of developing machine learning models. These parameters, which regulate the behavior of the machine learning algorithm and cannot be directly learned from the given training data, can significantly affect the performance of the model. In the context of relevance vector machine hyperparameter optimization, we have used zero-mean Gaussian weight priors to derive iterative equations through evidence function maximization. For a general Gaussian weight prior and Bayesian linear regression, we similarly derive iterative reestimation equations for hyperparameters through evidence function maximization. Subsequently, after using relative entropy and Bayesian optimization, the aforementioned non-closed-form reestimation equations can be partitioned into E and M steps, providing a clear mathematical and statistical explanation for the iterative reestimation equations of hyperparameters. The experimental result shows the effectiveness of the EM algorithm of hyperparameter optimization, and the algorithm also has the merit of fast convergence, except that the covariance of the posterior distribution is a singular matrix, which affects the increase in the likelihood. Full article

We exploit the equivalence between the partition function of an adsorbing Dyck walk model and the Asymmetric Simple Exclusion Process (ASEP) normalization to obtain the thermodynamic limit of the locus of the ASEP normalization zeros from a conformal map. We discuss the equivalence between this approach and using an electrostatic analogy to determine the locus, both in the case of the ASEP and the random allocation model. Full article

We study the zeros of the partition function in the complex temperature plane (Fisher zeros) and in the complex external field plane (Lee–Yang zeros) of a frustrated Ising model with competing nearest-neighbor ($J_1>0$) and next-nearest-neighbor ($J_2<0$) interactions on the honeycomb lattice. We consider the finite-size scaling (FSS) of the leading Fisher and Lee–Yang zeros as determined from a cumulant method and compare it to a traditional scaling analysis based on the logarithmic derivative of the magnetization $\mathit{\partial }ln⟨\left|M\right|⟩/\mathit{\partial }\beta$ and the magnetic susceptibility $\chi$. While for this model both FSS approaches are subject to strong corrections to scaling induced by the frustration, their behavior is rather different, in particular as the ratio $\mathcal{R}=J_2/J_1$ is varied. As a consequence, an analysis of the scaling of partition function zeros turns out to be a useful complement to a more traditional FSS analysis. For the cumulant method, we also study the convergence as a function of cumulant order, providing suggestions for practical implementations. The scaling of the zeros convincingly shows that the system remains in the Ising universality class for $\mathcal{R}$ as low as $-0.22$, where results from traditional FSS using the same simulation data are less conclusive. Hence, the approach provides a valuable additional tool for mapping out the phase diagram of models afflicted by strong corrections to scaling. Full article

This study demonstrated the synthesis of Fe₂MnO₄ modified by citric acid, a biodegradable acid, using a simple co-precipitation method. Characterization was performed using qualitative analysis techniques such as Fourier-transformed infrared spectroscopy, scanning electron microscopy equipped with energy-dispersive X-ray spectroscopy, X-ray diffraction, selected-area electron diffraction, N₂ adsorption–desorption, and zero-point charge. The prepared nanoparticles had a rough and porous surface, and contained oxygenous (-OH, -COOH, etc.) functional groups. The specific surface area and average pore size distribution were 83 m²/g and 5.17 nm, respectively. Net zero charge on the surface of the prepared nanoparticles was observed at pH 7.5. The prepared nanoparticles were used as an adsorbent to remove methylene blue dye from water under various conditions. Using small amounts of the adsorbent (2.0 g/L), even a high concentration of MB dye (60 mg/L) could be reduced by about ~58%. Exothermic, spontaneous, feasible, and monolayer adsorption was identified based on thermodynamics and isotherm analysis. Reusability testing verified the stability of the adsorbent and found that the reused adsorbent performed well for up to three thermal cycles. Comparative analysis revealed that the modified adsorbent outperformed previously reported adsorbents and unmodified Fe₂MnO₄ in terms of its partition coefficient and equilibrium adsorption capacity under different experimental conditions. Full article

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin’s Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner’s 6j symbols, in details. The later are rather well known in the theory of $3D$ indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the Q-operators. We investigate the Baxter equation and Q-operators from two points of view. (3) In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. (4) The real parameter q, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to q (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state. Full article

The first step in comprehending the properties of Au₁₀ clusters is understanding the lowest energy structure at low and high temperatures. Functional materials operate at finite temperatures; however, energy computations employing density functional theory (DFT) methodology are typically carried out at zero temperature, leaving many properties unexplored. This study explored the potential and free energy surface of the neutral Au₁₀ nanocluster at a finite temperature, employing a genetic algorithm coupled with DFT and nanothermodynamics. Furthermore, we computed the thermal population and infrared Boltzmann spectrum at a finite temperature and compared it with the validated experimental data. Moreover, we performed the chemical bonding analysis using the quantum theory of atoms in molecules (QTAIM) approach and the adaptive natural density partitioning method (AdNDP) to shed light on the bonding of Au atoms in the low-energy structures. In the calculations, we take into consideration the relativistic effects through the zero-order regular approximation (ZORA), the dispersion through Grimme’s dispersion with Becke–Johnson damping (D3BJ), and we employed nanothermodynamics to consider temperature contributions. Small Au clusters prefer the planar shape, and the transition from 2D to 3D could take place at atomic clusters consisting of ten atoms, which could be affected by temperature, relativistic effects, and dispersion. We analyzed the energetic ordering of structures calculated using DFT with ZORA and single-point energy calculation employing the DLPNO-CCSD(T) methodology. Our findings indicate that the planar lowest energy structure computed with DFT is not the lowest energy structure computed at the DLPN0-CCSD(T) level of theory. The computed thermal population indicates that the 2D elongated hexagon configuration strongly dominates at a temperature range of 50–800 K. Based on the thermal population, at a temperature of 100 K, the computed IR Boltzmann spectrum agrees with the experimental IR spectrum. The chemical bonding analysis on the lowest energy structure indicates that the cluster bond is due only to the electrons of the 6 s orbital, and the Au d orbitals do not participate in the bonding of this system. Full article