Computer Science and Mathematics

Abstract: We determine two finite models motivated by open problems in discrete geometry. Let \(T_4=\{-1,0,1\}^4\). We prove that the maximum Borsuk number of a subset of \(T_4\) is exactly 4: every nondegenerate subset can be partitioned into four sets of strictly smaller diameter, and a four-point equidistant set shows sharpness. Let \(U_5=\{v/\lVert v\rVert:v\in\{-1,0,1\}^5\setminus\{0\}\}\). We also prove that a spherical code in \(U_5\) with pairwise inner products at most \(1/2\) has at most 40 points, with equality only for the \(D_5\) root system inside this fixed library. The first result is certified by exact distance-layer colorings and an exhaustive symmetry-reduced enumeration of 15,056 maximal feasible sets; the second by two exact maximum-clique computations using integer predicates. These are finite-library theorems: they do not settle Borsuk's conjecture in \(R^4\) or the unrestricted kissing numbers in dimensions 5 and 6.

Abstract: For \( q \geq 1 \) and \(gcd(a,q) = 1\) we study the restricted weighted Goldbach sum\[R_{a,q}(N): = \sum_{\substack{p_{1} + p_{2} = N \\ p_{1} \equiv a\,(q)}}^{}(\log p_{1})(logp_{2})\]for even \(N\). We prove an elementary local-obstruction theorem: if \(q\) has an odd prime factor \(\mathcal{l}\), then \(R_{a,q}(N) = O_{q}(logN)\) on a positive-density set of even \(N\), so no asymptotic of size \(\asymp N/\varphi(q)\) can hold uniformly; the restriction \(q = 2^{k}\) used throughout the rest of the paper is therefore forced by the truth of the statement, not by convenience. For \(q = 2^{k}\) we identify the correct main term \(M_{a,q}(N) = 2C_{2}\varphi(q)^{- 1}S(N)N\), prove a qualitative almost-all theorem by a complete major/minor-arc argument, and give a fully self-contained, elementary derivation of an explicit pointwise minor-arc bound (via Vaughan's identity with the balanced parameters \(U = V = X^{2/5}\)) from which an effective almost-all theorem follows, with a threshold that grows like \( (logX)^{(5 + A)/2} \); we further prove that this growth is necessary, in the sense that no threshold constant independent of \(X\) can achieve the same exceptional-set bound once the underlying second moment is genuinely of order \(X^{3}/logX\). We give a sub-exponential exceptional-set bound via the explicit formula and McCurley's zero-free region, a gap theorem for the exceptional set, an additive-energy decay statement, a fully unconditional restricted Chen-type theorem via the Selberg--Chen sieve and the classical Bombieri--Vinogradov theorem, a ternary transfer via prime anchoring, and positivity of the restricted quaternary singular series through explicit local densities. We then prove that the odd part $S(N)$ of the singular series, evaluated along the shifted-prime sequence \(N = p + h\) for a fixed nonzero even \(h\), converges in distribution to an explicit random Euler product \(Y_{h}\) with independent Bernoulli local factors, with every integer moment given by a certified constant, an entire Mellin transform, a non-atomic law of unbounded support, and a superpolynomial upper tail, strictly amplified relative to the analogous statistic on generic even integers; this identifies the amplitude of the restricted Goldbach main term along shifted primes as a rigorously known heavy-tailed statistic rather than an approximately fixed quantity. Every numerical constant used is independently certified in this paper by direct computation with a rigorous tail bound; a previously circulated numerical optimisation purporting to convert the effective almost-all theorem into one with a threshold constant independent of \(X\) is shown, by direct computation of the relevant Chebyshev inequality, to be unachievable, and is not asserted here. We explain, as honestly labelled remarks rather than theorems, why several heuristic routes to unconditional finiteness of the exceptional set do not currently constitute proofs. Every result in this paper is unconditional; no Generalized Riemann Hypothesis, density hypothesis, or other unproved conjecture is used anywhere.

Abstract: Execution of discrete systems inherited its vocabulary from set theory, and with it an abstraction: a set element carries no origin, a relation no history. For mathematics this is a virtue; for execution it is a loss: a binary relation by itself records which pairs belong to it, but in the general non-functional case does not determine which among several available successors is admitted, under what criterion, or against which accumulated history. The consequence is direct: the same value admitted under different conditions is a different event. This paper proposes a specific way past that limitation.This paper develops a calculus in which admission is constitutive. An event enters the record only as a pair of a composite key — the ontology under which it was possible, the criterion under which it was admitted, and its position in the record — and a payload. The record is append-only, and the criterion is evaluated over it before every admission: Execute(Aτ, Cⁿ) = eₙ₊₁ iff τ(Cⁿ, ωτ), with C⁰ = ∅. The three objects this rests on — the record C, the ontology ωτ, and the criterion τ — occupy strictly increasing ranks and are mutually non-derivable, a fact that follows from rank separation together with independent specification.Three results follow under the explicit structural and effectiveness assumptions stated below, none requiring the Axiom of Choice. The sequential dependency of Execute is structurally analogous to dependent choice in that each admission is conditioned on the context produced by prior admissions, but no instance of the Axiom of Dependent Choice is invoked: where continuation is defined, the next admission is determined, for the fixed invocation, by the recorded context and the specified ontology and criterion. Admission requires no choice selector: each ωτ is single-valued where defined and τ is Boolean. Pre-admission addresses additionally provide a definable ordering wherever identification or serialization among independently determined abstractions is required — the instance coordinate is assigned only after admission. Since admission is decided from a finite prefix and, at each finite step, only finitely many active invocation instances contribute candidates, each contributing at most one realization, the admissible executions form a finitely branching tree whose body is closed, compact and Borel — the execution space is measurable, a fact that follows from finite branching, prefix-local admission, and the carrier's explicit encoding (§8), with determinism supplying the definable well-ordering that keeps the argument choice-free rather than supplying measurability on its own. And since convergence requires well-foundedness rather than monotonicity, criteria that consume a bounded resource — budgets, quotas, deadlines — fall inside the theory rather than outside it.The cardinality of that space separates two regimes by branching: countable where T contains no nonempty perfect pruned subtree, and continuum where it does. The countable case splits further by termination: T may be well-founded and terminate, or it may contain one or more infinite branches despite having no perfect subtree. Evolutive change is one possible source of persistent splitting, not a condition equivalent to continuum cardinality. Nondeterminism is therefore not removed but structured — it is the generative capacity of the specification, resolved before commitment and recorded with the coordinates under which it was resolved.Two consequences are of direct engineering interest, established in §7: Corollary 6 establishes that verification replay is deterministic from the recorded execution: the preceding prefix fixes the admission context and the ontology and criterion specifications in force, while the recorded event preserves the realization whose admission is being verified, and Corollary 5 formally identifies the control predicate with the admission predicate within the calculus, so that control and execution collapse into one act rather than two layers, structurally aligned with the Good Regulator principle — which permits evolutive change when a governing abstraction is itself admitted into the record it governs. Five minimal realization conditions are given, substrate-agnostic across architectural families; one independently published implementation is shown to instantiate them, and a further application, from the same research program, illustrates a concrete incompleteness result for external AI safety guardrails in a field currently searching for new formal concepts, following from the same decomposition. The scope is execution of discrete systems: no claim is made about decidability.

Abstract: For \(q \geq 1\) and \(\gcd(a,q)=1\), consider the restricted weighted Goldbach sum \[ R_{a,q}(N):=\sum_{\substack{p_1+p_2=N\\p_1\equiv a\pmod q}}(\log p_1)(\log p_2). \] We give a fully rigorous, self-contained treatment of the restricted binary, ternary and quaternary Goldbach problems in this setting. We first prove an elementary but decisive local obstruction: if an odd prime \(\ell\) divides \(q\), then \(R_{a,q}(N)\) collapses to \(O_q(\log N)\) on a positive-density set of even \(N\), so no main term of size \(\asymp N/\varphi(q)\) can hold uniformly; for \(q=2^k\) this obstruction is absent, and we identify the correct local main term \[ M_{a,q}(N)=\frac{C_2}{\varphi(q)}S(N)N, \] Within this scope we prove a qualitative almost-all theorem, with a complete major/minor-arc derivation, showing the exceptional set has density zero. We record a conditional impossibility theorem showing that no \(X\)-independent threshold can upgrade this to an effective almost-all statement once a matching second-moment lower bound is granted, and we isolate, as honestly labelled structural cautions rather than theorems, four classical routes that fail to upgrade the almost-all theorem to unconditional finiteness. We add a fully unconditional restricted Chen-type theorem obtained from the Selberg--Chen sieve and the classical Bombieri--Vinogradov theorem, a ternary prime-anchoring transfer, and positivity of the restricted quaternary singular series through explicit local densities. As a companion study, we then prove that the variable factor \[ S_0(n):=\prod_{\ell\nmid n}\frac{\ell-1}{\ell-2} \] of the Hardy--Littlewood singular series, evaluated along shifted primes \(n=p+h\) for fixed \(h\neq 0\), converges in distribution to an explicit random Euler product \[ Y_h=\prod_{\ell>2,\ell\nmid h}\left(\frac{\ell-1}{\ell-2}\right)B_\ell \] with independent Bernoulli local factors \(P(B_\ell=1)=1/(\ell-1)\); we identify its Mellin transform as an entire function of order one, prove convergence of every integral moment, and establish that the law is non-atomic, has unbounded support, and superpolynomially decaying tails, strictly amplified relative to generic integers. We then connect the two studies: since \(S(N)=S_0(N)\) is exactly the amplitude of the restricted Goldbach main term \(M_{a,q}(N)\), the limit law furnishes a rigorous probabilistic description of how that amplitude fluctuates as \(N\) ranges over the shifted-prime sequence \(N=p+h\), a connection neither source study states. Throughout, every asserted theorem is unconditional and every numerical constant is independently certified via partial Euler products with explicit tail bounds; statements retracted at an earlier stage of this programme after failing independent verification are recorded only as open problems.

Abstract: The numerical simulation of incompressible viscous flows remains a central pillar of modern computational fluid dynamics (CFD). Over the past decades, a wide spectrum of numerical methodologies has been developed, reflecting fundamentally different mathematical formulations and discretization philosophies. Among these, domain-based approaches—including finite difference (FDM), finite volume (FVM), and finite element (FEM) methods—have emerged as versatile and general-purpose frameworks, while boundary element methods (BEM) provide efficient alternatives for selected classes of problems governed by linear physics, particularly in unbounded domains. Meshfree and particle-based methods constitute an alternative paradigm for problems involving large deformation, moving interfaces, free surfaces, and fragmentation. This review provides a unified comparison of FDM, FVM, FEM, BEM, and meshfree approaches, emphasizing their mathematical foundations, treatment of nonlinear operators, stability, consistency, convergence, computational characteristics, and compatibility with turbulence modeling. Particular attention is given to the interaction between discretization and Reynolds-averaged Navier– Stokes(RANS), large-eddy simulation (LES), direct numerical simulation (DNS), and hybrid RANS–LES methodologies, including detached-eddy simulation (DES), delayed detached-eddy simulation (DDES), improved delayed detached-eddy simulation (IDDES), and stress-blended eddy simulation (SBES). Recent developments in highorder discretization, immersed-boundary and hybrid formulations, meshfree methods, and artificial-intelligence-assisted CFD are also discussed. The review demonstrates that no numerical method is universally optimal; rather, methodological suitability is governed by the interplay among flow physics, Reynolds number, geometry, boundary conditions, and computational requirements.

Abstract: Bakshali square root formula is an amazing achievement of ancient Indian Mathematics. Exploring the logic behind the Bakshali formula, an extended Bakshali algorithm in two variables for square root extraction is presented. The algorithm leads to Bakshali-type formulas of high orders of convergence that goes as 2ⁿ, where n=1,2,3, …. . Further, a transformed Bakshali algorithm is presented which ensures fast convergence with just one division operation per iteration. Just in a few iterations square roots can be computed with millions of digits of accuracy. Algorithms with 8-fold and 16-fold convergences are demonstrated.

Abstract: Deterministic differential equations describe dynamical systems in idealized states, neglecting any random influences. Within biomathematical modeling, incorporating stochasticity requires a clear distinction between environmental (extrinsic) noise and demographic (intrinsic) noise. The latter framework assumes that temporal fluctuations arise strictly from the demographic dynamics of interacting populations rather than environmental variability. The literature thoroughly documents how demographic noise can be modeled and simulated as a stochastic process acting on individual members of a population, which yields discrete stochastic systems. For large population sizes, these discrete processes approximate continuous ones, leading to stochastic differential equations (SODEs). If random effects are omitted, these SODEs simplify back to standard ordinary differential equations (ODEs). Conversely, deducing how demographic noise impacts a natural system previously modeled by ODEs represents a major challenge. In this paper, we present an initial comparison of two distinct methodologies for reconstructing demographic noise by working backward from a deterministic, continuous differential system to its discrete stochastic process: the traditional Allen’s method and the backward approach recently introduced by Carletti and Banerjee (2019).

Abstract: This paper introduces the new three-dimensional (3D) fractional-order Zernike moments obtained by extending the radial component of classical Zernike polynomials to admit fractional exponents. This formulation provides a mathematically consistent extension of integer-order Zernike moments to fractional orders while preserving their orthogonality in the unit ball. A reformulation of the fractional Zernike functions and a recurrence relation are also presented for efficient computation of the fractional-order moments. Rotationally invariant descriptors are further derived to allow orientation-independent shape representation. Numerical experiments demonstrate that the proposed fractional-order moments improve the accuracy of shape reconstruction and representation compared to their integer-order counterparts, highlighting their effectiveness for 3D shape analysis.

Abstract: Freshwater quality is degrading globally, and regulatory monitoring remains largely retrospective, identifying non-compliance only after it occurs. This study identifies whether multi-year, regulatory-relevant compliance breaches can be forecast from sparse monthly monitoring records alone, and whether such forecasts improve on the assumption that next year resembles the current one. Using approximately two decades (2004–2024) of Land, Air, Water Aotearoa (LAWA) data from 497 South Island, New Zealand river sites, a single pooled gradient-boosted (LightGBM) classifier was trained to forecast Escherichia coli worst-band (Band E) non-compliance under the National Policy Statement for Freshwater Management at one-, two-, and three-year horizons, benchmarked against persistence, trend projection, and majority-class baselines under strictly temporal validation. The model discriminated breaches reliably (AUC 0.84–0.85) and exceeded persistence in balanced accuracy at all three horizons. Its principal value was early warning: among currently-compliant sites, it recovered roughly half of subsequent breaches, transitions that persistence cannot detect by construction, yielding a forward watchlist of 116 sites at risk of entering the worst band (Band E) by 2027, concentrated in pastoral catchments. Forecasting from monitoring data alone imposes an honest ceiling; scores are reported as risk rankings. The findings support a shift from reactive to anticipatory freshwater management.

Abstract: We revisit the Hunter–Saxton equation using its classical first-order Lagrangian. We reduce the determining system for all multipliers depending on (t,x,u,u_t,u_x) completely to a single linear equation for Φ(t,ux); the resulting family includes characteristics of both point and generalized variational symmetries. Within the known infinite-dimensional point-symmetry ideal Y_f=f(t)∂_x+f′(t)∂_u, we show that every generator is a Noether divergence symmetry and derive its arbitrary-function family of conserved currents, while the finite generator X1 is excluded from the algebra of Noether point-divergence symmetries. We also correct the symmetry–conservation-law associations and remove trivial current terms. For the scaling generator X₃=t∂_t+x∂_x, the canonical representative of its current has an identically vanishing transformed radial component. We derive alignment and multiplier criteria for this degeneracy; the latter proves that, among associated representatives, it is a property of the conservation-law equivalence class. Thus association guarantees invariance of the transformed component, not that it retains differential content. Finally, we reduce the Lagrangian itself in canonical coordinates for two representative variational symmetries. The reduced Euler–Lagrange equations reproduce the known similarity equations, while their first integrals follow directly from the reduced variational structure. For the scaling reduction, the reduced Noether symmetry is induced by the commuting ideal element Y_t. The resulting analysis separates new structural results from previously known similarity solutions.

Abstract: This study establishes a systematic geometric characterization of modern Iran's boundaries and associated coastlines within the West Asian continental domain. It provides a reproducible framework for quantifying the scale-dependent complexity of the region's political and maritime frontiers using adaptive-scale methods. For Iran's mainland borders, the analysis yields a box-counting dimension of 1.0714 and a divider dimension of 1.0744. The adjoining regional seashores exhibit higher irregularity; the Caspian Sea coastlines produce fractal dimensions of 1.1120 and 1.1577, while the Persian Gulf's fractal dimensions reach 1.1587 and 1.1742, respectively. Interestingly, all three geographical land and maritime objects occupy broadly intermediate positions within their respective distributions: Iran ranks centrally among its 13 West Asian neighbors, while both regional seas occupy a middle-ground position relative to eighteen published global coastline studies. These findings reveal a hierarchy of boundary intricacy, where marine coastlines consistently demonstrate greater spatial occupancy than terrestrial frontiers. By integrating dual-method estimations with automated scaling, this research offers a substantiated geometric perspective on the Iranian Plateau's position within the broader global cartographic context.

Abstract: Arguments that computation underlies physical time frequently identify the work embodied in a complex state with Bennett's logical depth, and identify that quantity in turn with relativistic proper time. This paper examines the identification and finds it untenable in the form usually presented, for reasons that nevertheless indicate a defensible reformulation. Three difficulties are set out. First, computational irreducibility in its informal form is not a theorem; its rigorous surrogate is the P-completeness of prediction problems, which is a statement about parallel time rather than about sequential duration. Second, logical depth is invariant only up to a machine-dependent time factor and a shift in its significance parameter, and it is model-relative in a way that becomes acute once quantum computation is admitted. Third, and most consequentially, logical depth is a minimum taken over producing programs whereas proper time is a maximum taken over timelike curves, so the two quantities are extremal in opposite directions. We propose causal depth, defined as the height of the causal dependency graph of a process, as the quantity that occupies the position the argument requires. Causal depth is invariant under reordering of causally independent events, is extremal in the same direction as proper time, and corresponds to the longest-chain construction through which causal set theory recovers proper time from discrete causal order. The resulting correspondence is stated as a conjecture with explicit refutation conditions. We argue that the substantive open question concerns the gap between the causal depth of a state and the height of the histories that actually produce it.

Abstract: Fractional logistic maps introduce memory into a canonical route to chaos, but how memory shifts successive period-doubling bifurcation points is not yet fully understood. For the finite-memory Grünwald–Letnikov (GL) fractional logistic map, we formulate equations that determine exact periodic orbits and their period-doubling points, with the number of unknowns independent of memory length. The framework reproduces an independently derived exact fixed-point boundary and identifies a genuine period-4 to period-8 doubling, confirming its applicability beyond the fixed point. Across the first seven computed bifurcation levels, increasing memory length shifts the bifurcation points toward smaller control-parameter values, with stronger shifts at lower fractional orders. The GL memory tail decays more slowly at lower fractional order, helping explain why extending retained memory produces larger bifurcation-point shifts. At fixed memory length, decreasing the fractional order from its classical value first moves the bifurcation points toward smaller control-parameter values, before a turnover carries them toward larger values. As the order approaches zero, the GL memory terms vanish and limiting bifurcation points lie one unit above their classical counterparts, so branch continuity requires a turnover. Together, these results provide a validated basis for computing and interpreting how memory reorganizes the period-doubling route to chaos.

Abstract: Exposing foams stabilized by photoswitchable surfactants to UV light induces changes in surface surfactant concentration, leading to significant alterations in foam behaviour such as the generation of Marangoni flow and change in foam drainage patterns. The occurrence of Marangoni flow can be observed when either all elements of the foam or only their films are exposed to UV light. Conversely, changes in foam drainage occur when a macroscale portion of a foam column is exposed to UV light. To explore these phenomena, numerical models are developed and validated using experimental data. These models simulate the scale and profile of Marangoni flow from foam networks to films as well as the drainage flow within the foam network. Microscale findings demonstrate that Marangoni flow can be controlled by adjusting the intensity and duration of UV light exposure. Macroscopically, the drainage profile in exposed foam regions undergoes significant changes with varying UV intensity. Furthermore, beyond a certain threshold, the foam drainage reverses direction, contrary to gravity. The effect of foam interfacial mobility on the reversed drainage of both interior and exterior foams is analyzed. The findings provide a potential tool to control foam drainage behaviour without the need to modify other variables.

Abstract: Accurate and computationally efficient prediction of radioactive plume dispersion is essential for rapid risk assessment and emergency response in the event of accidental atmospheric releases. This study presents a three-dimensional, mass-conservative numerical framework for simulating mesoscale transport and transformation of radionuclides under varying meteorological conditions. The proposed model is based on the advection–diffusion equation and incorporates key physical processes, including turbulent diffusion, gravitational settling, precipitation washout, radioactive decay, surface absorption, and re-emission. A hybrid numerical scheme is developed that combines a semi-implicit Crank–Nicolson discretization for diffusion terms with a second-order Total Variation Diminishing (TVD) scheme employing a Van Leer limiter for advection. To ensure computational efficiency and stability, the governing equations are solved using an alternating direction implicit (ADI) strategy, leading to tridiagonal systems efficiently handled via the Thomas algorithm. In addition, a modified parameterization of turbulent diffusion coefficients is introduced, extending classical Pasquill–Gifford formulations to better represent mesoscale atmospheric behavior with bounded dispersion characteristics. Computational experiments simulating iodine-131 release under different atmospheric stability classes and meteorological conditions demonstrate that the proposed framework reproduces physically consistent plume structures and maintains strict mass conservation with negligible numerical errors. The results highlight the dominant role of advective transport and precipitation scavenging under moderate wind conditions, as well as strong localization effects under stable atmospheric stratification. The proposed model provides a practical compromise between the simplicity of Gaussian plume approaches and the high computational cost of full-physics atmospheric models, making it suitable for rapid engineering calculations and integration into decision-support systems for radiological emergency management.

Abstract: Scientific inquiry relies on causal explanation, whereby phenomena are understood through the antecedent conditions, interactions and laws producing the current configuration of a system. We introduce a complementary approach termed Future Compatibility (FC), which evaluates a present state not only through its causal history, but also through the set of future states structurally accessible from it. The central premise is that scientific understanding may be enriched by characterizing the opportunities and constraints associated with a given state. Rather than focusing exclusively on origins or on the prediction of a specific outcome, FC quantifies the repertoire of coherent futures available from a present configuration and their stability under perturbation. To explore this approach, we implemented simulations of evolving networked systems undergoing branching developmental pathways and external perturbations. For each system state, a set of novel observables describing future accessibility, entropy, robustness and persistence was computed from admissible trajectories satisfying predefined coherence constraints. These compatibility landscapes exposed organizational properties that were not captured by causal reconstruction alone, providing additional information regarding adaptability, resilience and developmental potential. Potential applications include developmental biology, evolutionary theory, ecology, neuroscience, complex systems research and the assessment of resilience and adaptability in technological, organizational and socioeconomic networks.

Abstract: We develop P versus NP as the feasibility instance of the horizon principle over a finite relational substrate. The algebraic carrier is complete as a totality, yet a bounded internal observer has a finite comprehension horizon. The incompleteness migrates from the undecidability of truth to that horizon inaccessibility. Computation is counting along a representation: the hyperoperation ladder (succession, addition, multiplication, exponentiation) carried by the substrate's four cardinal representation charts. A certificate is a near-representation placing a remote target within the horizon: NP asks that one exists, P that it be forward-found, and P=NP iff finding is as feasible as checking. Geometrically the carrier is a shell with the observer at its pole and a computation a geodesic, so P=NP asks whether a geodesic that provably exists is forward-findable. The reading reproduces, under exact substitutions, proof complexity, automatizability, and one-way functions, and grounds the find-check asymmetry in the substrate's scale/discrete-logarithm map and single time arrow. P versus NP is the computational face of the horizon clause: its uniform certificate is an Ω-hard residue, decided by the complete totality yet below the bounded observer's horizon. One-way functions yield the separation; the descent's one-wayness is that residue. The reading is consistent with the relativization, natural-proofs, and algebrization barriers. Every exact claim is verified in finite-field or cyclotomic arithmetic.

Abstract: We introduce a generalized fixed-point framework based on \(d\)-\(\psi\)-\(\varepsilon\)-contractions and weak \(d\)-\(\psi\)-\(\varepsilon\)-contractions, where the contraction metric and the metric in which the space is complete may be different and the comparison function belongs to the class \(\Psi_0(\varepsilon)\). This setting extends several classical fixed-point principles and yields existence, uniqueness, localization, and convergence results for Picard iterations. The theory is applied to nonlinear integral equations and to their quadrature approximations. By introducing suitable invariant sets and a \(\psi\)-\(\varepsilon\)-max inequality, existence and uniqueness results are obtained under assumptions that are substantially different from classical Lipschitz-type conditions. The developed framework is further extended to quadrature integral equations generated by numerical integration formulas. Sufficient conditions are established for the existence and uniqueness of solutions as well as for the convergence of the associated Picard sequences. The theoretical results guarantee convergence of the discrete iterations under appropriate assumptions, thereby providing a direct connection between fixed-point theory and numerical computation. Several examples, including Chandrasekhar-type integral equations and nonlinear weighted integral equations with singular kernels, are presented to illustrate the applicability and effectiveness of the proposed approach. Numerical experiments confirm the theoretical findings and demonstrate the accuracy of the resulting quadrature-Picard schemes.

Abstract: Fracture of ceramic femoral heads in total hip arthroplasty is a rare but catastrophic complication requiring urgent revision surgery. Most finite element studies are limited to static loading and do not capture the dynamic behavior of ceramic components under impact conditions generated during stumbling or falling. In the present study, a parametric explicit dynamic analysis was performed using LS-DYNA (version 960) to determine the impact fracture thresholds of alumina (Al₂O₃) and yttria-stabilized zirconia (ZrO₂, Y-TZP) femoral heads. An axisymmetric finite element model of a 32 mm ceramic femoral head articulating with a ceramic liner within a Ti-6Al-4V acetabular shell was developed. Ceramic behavior was described using the Johnson–Holmquist JH-2 damage constitutive model. The viscoelastic bone stock response was represented by a Winkler foundation of discrete spring-dashpot elements (stiffness 50–500 N/mm, damping 0–1.0 N·ms/mm). Impact velocity was varied from 0.01 to 0.45 mm/ms, consistent with velocities recorded by instrumented implant telemetry during stumbling. Fracture was identified by three concurrent criteria: effective plastic strain, maximum principal stress, and inflection of the internal energy–time curve. For Al₂O₃, the critical fracture velocity was 0.08 mm/ms under rigid fixation and 0.05 mm/ms with a viscoelastic foundation. The ZrO₂ femoral head did not fracture at any velocity tested; at V ≥ 0.20 mm/ms, the Ti-6Al-4V neck underwent plastic deformation as a competing failure mode while the ceramic head remained intact. Foundation stiffness and damping had no influence on fracture outcome across the clinically relevant range, indicating inertia-dominated fracture mechanics. These results provide quantitative fracture thresholds to support comparative evaluation of ceramic hip implant components.

Abstract: In the single-depot multi-traveling salesman problem, traditional depot location methods often overlook task balance among traveling salesmen, leading to excessive load on certain units and compromising overall operational efficiency. To address this issue, this paper proposes an optimized depot location method based on clustering and multi-task balancing. The core contribution lies in the design of a multi-weight adaptive depot optimization method. This approach clusters city nodes into multiple groups through cluster analysis and dynamically synthesizes direction vectors using information such as the number of samples within each cluster and the convex perimeter. It iteratively optimizes depot locations, minimizing the total path length while enhancing load balance across all traveling salesman routes. Additionally, a “divide-and-conquer” strategy decomposes the complex MTSP into multiple parallel TSP subproblems, which are then efficiently solved using Or-Tools. A comprehensive evaluation framework is introduced, incorporating Total-Sum distance, Min-Max distance, Distance-balancedness, Cluster separability, Robustness, and Running time. Experimental results on the TSPLIB standard dataset demonstrate that the proposed method exhibits significant advantages over various traditional clustering algorithms in both route optimization and route balancing, validating its effectiveness and practicality. The method's robust performance provides a reliable solution for real-world applications such as logistics distribution, further highlighting its practical value.