Computer Science and Mathematics

Abstract: Let G be a finite simple graph, let \( m = (m_v)_{v∈V(G)} \) with \( m_v ≥ 1 \), and form \( G^m \) by attaching \( m_v \) pendant vertices at each \( v \). For the weighted adjacency matrix \( A_φ(G^m) \) induced by a real symmetric degree-dependent edge weight φ, a generalized Fiedler-type decomposition gives M − n forced zero eigenvalues and \( χ_{A_φ(G^m)}(λ) = λ^{M−n} det(λ²I_n − λB_φ − R_φ) \).Here \( n = |V(G)| \), \( M = ∑_v m_v, B_φ \) is the core block formed from the core-vertex degrees in the thorny graph, and \( R_φ \) is diagonal with squared pendant-fiber couplings. The decomposition gives an arbitrary-core nullity bound, exact nullity and inertia for nonzero pendant couplings, Cauchy interlacing with \( B_φ \), and a 2n-dimensional compression independent of M. For fixed r ≥ 1, m, and φ, the nonzero weighted spectrum of a uniform attachment to an r-regular core determines the core adjacency spectrum whenever the induced core-edge weight is nonzero. On connected walk-regular cores, let exceptional vertices receive m + q rather than m pendant vertices. For independent exceptional sets S, T with |S| = |T| ≥ 2, all moments below twice the smaller minimum separation agree for every real symmetric φ. If the separations differ, then at the first possible mixed order the moment of the smaller-separation placement minus that of the larger-separation placement is an explicit nonnegative multiple of a sum of squared geodesic counts. If the ordinary–ordinary core-edge weight is nonzero and its square differs from that of the exceptional–ordinary core-edge weight, the spectrum determines the minimum separation. On distance-regular cores, it also determines the closest-pair count. Both the ordinary Sombor and Randić weights satisfy this condition.

Abstract: Every neighborhood assignment on a charming space \( X \) has a countable kernel whose closure in \( X \) is Lindelöf \( \Sigma \). For a neighborhood assignment \( P \) on \( X \) and a set \( A\subseteq X \), write \( P[A]=\bigcup_{a\in A}P(a) \). Consequently, for any two neighborhood assignments \( M \) and \( N \) on \( X \), there is a closed discrete set \( D\subseteq X \) such that \( M[N[D]]=X \). The \( D \)-property also admits an exact one-step reduction to covering a fixed Lindelöf \( \Sigma \) kernel by assigned neighborhoods centered at an ambient closed discrete set. The condition that every neighborhood assignment \( \varphi \) on \( X \) admit a pointwise square refinement, meaning a neighborhood assignment \( V \) on X\( X \) with \( V[V(x)]\subseteq\varphi(x) \) for every \( x\in X \), is exactly the classical \( z \)-space condition formulated in terms of normal neighbornets. For a continuous closed surjection \( p:Z\to X \), a fixed neighborhood assignment \( \varphi \) on \( X \) has a pointwise square refinement on \( X \) if and only if there is a neighborhood assignment \( V \) on \( Z \) such that \( p(V[V(z)])\subseteq\varphi(p(z)) \) for every \( z\in Z \). Every uncountable weakly hereditarily finally covering (\( \mathrm{HFC}_{\mathrm w} \)) subspace of \( [\omega_1]^{\leq\omega} \) has a closed countable kernel for every neighborhood assignment but is not charming. For each Lindelöf \( \Sigma \) kernel \( Y \) of X\( X \), put \( B=\overline Y^{\,X} \) and \( H=\overline{B\setminus Y}^{\,B} \). The Hu--Yin boundary obstruction is then \( B\setminus Y \). The space \( X \) has a closed Lindelöf \( \Sigma \) kernel exactly when there is a Lindelöf \( \Sigma \) kernel \( Y \) such that, with \( B=\overline Y^{\,X} \) and \( H=\overline{B\setminus Y}^{\,B} \), the set \( H\cap Y \) is \( G_\delta \) in \( H \). For any fixed Lindelöf \( \Sigma \) kernel \( Y \), the closed-core reduction can be iterated. At every stage, the current iterate is a \( D \)-space if and only if \( X \) is a \( D \)-space. The iteration reaches the largest closed subspace in which the kernel trace and its complement are both dense, and it stabilizes before the successor of the hereditary Lindelöf number of \( Y \). This bound is sharp. For each infinite cardinal \( \kappa \), every \( \gamma<\kappa^+ \) occurs as the first stabilization stage for a compact space with a dense \( \sigma \)-compact Lindelöf \( \Sigma \) kernel whose cardinality and hereditary Lindelöf number are both \( \kappa \). The general charming-space \( D \)-problem remains open.

Abstract: The Bakshali formula for square root extraction is an amazing achievement of ancient Indian mathematics. Understanding the logic behind the Bakshali formula as a cycle of approximation, error determination, correction, and new approximation, a Bakshali algorithm for square root extraction in two variables (the approximation X_n and its correction Y_n ) is presented. The recursive algorithm extendable to any depth n leads to Bakshali-type formulas of high orders of convergence that goes as 2ⁿ, where n=1,2,3, …. . The Bakshali algorithm is streamlined by recasting it into a transformed form with three variables (y_n, t_n, z_n). The transformed Bakshali algorithm runs by generating integer sequences with no division in the stages and a single division at the end. It obeys the exact invariant z_n² − N t_n² = y_n², which places every iterate on a Pell conic and gives closed forms for the iterate and its error, and the order of convergence 2ⁿ. The iterates coincide with those of Newton’s method in rational form, but the route does not. The correction term of Bakshali algorihm is such the new residual is exactly the square of the correction. This exact error transfer builds stages that no longer involve N and save at least half a squaring each. For an integer radicand and an integer seed every stage is exact. Three consequences follow: the number of correct digits after each stage is known in advance; the integers never exceed the accuracy they deliver by more than one digit, so no precision schedule is needed; and a rigorous enclosure of the error follows from the algorithm’s own variables alone, at no extra cost. Nested versions with 8-fold and 16-fold convergence are demonstrated. In five 16-fold iterations, square root of a 5 digit number is obtained to more than 2.8 million correct digits, and in seven 8-fold iterations 5.6 million correct digits are obtained.

Abstract: For \(q \ge 1\) and \(\gcd(a,q)=1\) we study the restricted weighted Goldbach sum \(R_{a,q}(N) = \sum_{p_1+p_2=N,\ p_i\equiv a(q)} (\log p_1)(\log p_2)\) for even \(N\), with candidate main term \(M_{a,q}(N) = 2C_2\varphi(q)^{-1}\Xi_0(N)N\), where \(C_2\) is the twin-prime constant and \(\Xi_0(N)\) is the odd part of the Hardy--Littlewood singular series. We first prove an elementary local-obstruction theorem: if \(q\) has an odd prime factor \(\ell\), then \(R_{a,q}(N) = O_q(\log N)\) for every \(N\equiv a\ (\ell)\), a set of positive density, so no asymptotic of size \(\asymp N/\varphi(q)\) can hold; the restriction to \(q=2^k\) in the rest of the paper is therefore forced by the truth of the statement. For \(q=2^k\) we evaluate the major arcs through an exact additive-shift decomposition of the progression and a factorisation of the resulting singular-series coefficients, which yields the main term \(M_{a,q}\) with an explicit coefficient, and, combined with the Siegel--Walfisz theorem and Vaughan's bound on the minor arcs, the mean-square estimate \(\sum_{N\le X}|R_{a,q}(N)-M_{a,q}(N)|^2 \ll_{A,q} X^3(\log X)^{-A}\) for every fixed \(A>0\). This gives an exceptional set \(\#\{N\le X \text{ even}: R_{a,q}(N)=0\} \ll_{A,q} X(\log X)^{-A}\). An elementary transfer principle then shows that the same bound controls \(R_{a,q}\) along any injective map into the even integers of at most linear growth. Consequently, for every residue class ("ray") \(c\) coprime to \(q\), almost all primes \(p_n\equiv c\ (q)\) have \(p_n+p_{n+1}\), \(p_n+p_{n+k}\) and \(p_n+h\) (\(h\) odd) representable with first summand in the class \(a\). For a modulus with an odd prime factor \(\ell\) we prove that \(N_n=p_n+p_{n+1}\equiv a\ (\ell)\) is a condition on the pair of rays of \(p_n\) and \(p_{n+1}\), that it holds infinitely often by a theorem of Shiu, and, for \(q=3\), we verify computationally up to \(6\times10^6\) that \(R_{1,3}(N_n)=0\) exactly when \(N_n\equiv1\ (3)\) and \(N_n-3\) is composite (only the implication from this condition to the vanishing of the sum is proved). We document by exact computation below \(10^8\) the digit-repulsion bias between consecutive primes, the repetition of rays according to the size of the gap, the doubling of the relative variance of \(R_{a,q}\) on the class \(N\equiv2a\ (q)\), and the out-of-sample predictability of the next ray; the ray-gap identity and the symmetry \(p_1\leftrightarrow p_2\) behind this doubling are proved. The same circle-method machinery gives the restricted ternary asymptotic \(W_{a,q}(n)=\Xi_3(n)n^2/(2\varphi(q))+O_{A,q}(n^2(\log n)^{-A})\) and positivity for all large odd \(n\). Finally, the amplitude \(\Xi_0(p+h)\) of the main term along shifted primes converges in distribution to an explicit random Euler product \(Y_h\), with all moments converging to absolutely convergent Euler products, an entire moment generating function with \(\log\mathbb E Y_h^s \le s\log\log s+O(s)\), a doubly exponentially thin upper tail that is sharp on a doubly logarithmic scale, and no atoms. Every theorem is unconditional; computations are labelled with their exact range.

Abstract: Can simple spectral properties of a quantum gate, computed classically from its eigenvalues, predict how well quantum phase estimation (QPE) performs on that gate? We study three spectral features, a normalised Spectral Coherence Metric (SCM), the spectral gap, and a circular phase spread, using ideal Qiskit Aer simulations of 1,756 two-qubit gates (1,000 Haar-random unitaries plus three controlled spectral classes) for m = 3, 4, 5 ancilla qubits and three independent random seeds. All prediction targets are measured from simulated QPE circuits. When QPE is run on single eigenstates, the spectral features have no predictive power for the estimation error (cross-validated R² ≤ 0), because that error is fixed by the position of each eigenphase on the 2^m-point measurement grid. When the target register is prepared in an equal superposition of eigenvectors, the spectral features do predict eigenvalue resolvability, the fraction of eigenphases that appear as separate peaks: for Haar-random gates with m = 4, a random forest reaches R² = 0.633 ± 0.020, and the spectral gap dominates the feature importance (0.88). However, a parameter-light physical baseline that counts gaps larger than a multiple of the grid spacing 2π/2^m matches the learned models in accuracy and exceeds them in balanced accuracy (0.78 versus at most 0.63). We conclude that spectral structure, through the spectral gap, governs resolvability in superposition-input QPE in the way expected from the standard resolution limit, and that machine learning adds little beyond this physical rule.

Abstract: Reconstructing missing values in meteorological time series is essential for reliable renewable-energy assessment, but on-device imputation must respect the power, memory and latency budget of edge hardware. We benchmark 46 PyPOTS imputation models—four naïve baselines, TRMF and 41 neural networks—on two tropical coastal stations (UCC and CUC, Colombian Caribbean) resampled to 15 min and subjected to a PyGrinder missingness scenario, running each model on an NVIDIA Jetson Orin Nano (25 W) and measuring MAE, RMSE, power, memory and time. A hybrid Entropy–CRITIC weighting scheme with TOPSIS and MARCOS ranked the alternatives. The best neural networks cut the mean-imputer error by 40–45% (MAE) and 28–40% (RMSE), yet resource usage spanned two orders of magnitude, uncorrelated with accuracy (|ρ|≤0.15). Because both methods placed 64–76% of the weight on resource criteria, naïve baselines and TRMF topped the station-averaged and UCC rankings, TiDE was the most rank-stable neural network, and SCINet led at CUC under the untuned configuration. TOPSIS and SPOTIS agreed closely (ρ≥0.98); MARCOS agreed moderately with both (ρ≈0.68–0.93). Station-to-station agreement was low for TOPSIS (ρ=0.52)—a confound rather than a station effect: the same masking configuration produced an asymmetric gap structure at the shorter, previously incomplete UCC record and a uniform one at the longer, complete CUC record, so station-level conclusions are not directly comparable. An automated per-model hyperparameter-search protocol, which reorders the CUC top, is provided in an appendix. These results offer a reproducible, hardware-validated framework for edge-constrained imputation model selection.

Abstract: In this article we solve the diffusion-advection-reaction model problem with Robin boundary conditions in an open bounded domain Ω∈Rⁿ, with n=1,2,3. For the discretization, we use the mixed-hybrid finite element method of lowest order on a simplicial partition of Ω¯. To prevent the occurrence of spurious unphysical oscillations we numerically evaluate the local bilinear form with a quadrature that is equivalent to introducing an optimal amount of artificial diffusion on each edge of the simplex. The resulting scheme is monotone and flux-conservative and is able to compute accurate solutions even with a corse mesh. The robustness and accuracy of the method is demonstrated on examples in Rⁿ, n=1,2,3, characterized by the presence of steep boundary and internal layers.

Abstract: A fixed-time stabilizer for underactuated Euler–Lagrange systems was established in Part 1 of this study , which certified fixed-time reaching on the actuated manifold and supplied a separate certificate for the internal dynamics. This paper asks what that certificate costs and what it buys. We decompose the total settling time into the reaching time governed by the certificate and an internal transient governed by the plant, and show that the two obey different constraints: the settling bound depends on the reaching gains alone and tightens exactly in inverse proportion to their scale, whereas the unactuated coordinate obeys a rate limit fixed by the coupling authority and is untouched by those gains. We prove that this forces a strictly positive floor on total settling time as the gains grow without bound, and we measure that floor to be far more binding in practice than the proof alone guarantees. On the translational oscillator with rotational actuator, scaling the reaching gains over a range of 13.3 tightens the analytical bound by a factor of 13.3 while the full-state settling time improves by 6.0% and the control energy rises by 53%. The floor is not an artifact of actuator saturation: with the actuation limit raised a hundredfold the same gain range still moves settling by only 11.4%. It is not an artifact of the plant either, since the cart-pole shows the separation more sharply, nor of the reaching-law gain family specifically, since settling varies by only 7.0% across a 49-design two-dimensional sweep over which the bound varies by a factor of 53.3. The floor is likewise unchanged at 1% and 5% settling tolerances, across a sixteenfold range of parametric mismatch including its sign, and under bounded disturbances, which the certified channel absorbs completely while the internal channel does not. A search over all eight controller parameters, a sampled-data implementation with measurement noise and torque quantization, three further cost metrics, and a fourfold finer sweep leave the result and the parity finding intact, and show that the designs that buy a tight certificate are also the ones that chatter and amplify noise. The bound itself is valid throughout, honoured by every design with margins between 62% and 95% once the event it governs is identified correctly. A matched-performance protocol puts the fixed-time law and its linear counterpart at energy parity, with ratios between 0.934 and 1.128. By contrast, increasing the coupling authority eightfold makes the response 7.4 times faster while reducing the energy integral by a factor of 14.6. The resulting design rule is to select gains against the guarantee that must be certified and to treat achievable response speed as a question about coupling authority rather than about feedback gain.

Abstract: For q≥1 and gcd(a,q)=1 we study the restricted weighted Goldbach sum R_(a,q) (N)□(:=) ∑_(p_1+p_2=N, p_1≡a (q))^ ▒( logp_1)(logp_2) for even N, with candidate main term M_(a,q) (N)□(:=) 2C_2 φ(q)^(-1) S(N)N, where C_2 is the twin-prime constant and S(N) is the odd part of the Hardy–Littlewood singular series. We first prove an elementary local-obstruction theorem: if q has an odd prime factor l, then R_(a,q) (N)=O_q (logN) on a positive-density set of even N, so no asymptotic of size ≍N/φ(q) can hold uniformly for such q; the restriction to 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 give a fully self-contained, elementary derivation of an explicit pointwise minor-arc bound for S_χ (α) (via Vaughan’s identity with the balanced parameters U=V=X^(2/5)), from which we deduce an unconditional master second-moment bound ∑_(N≤X)^ ▒| E(N)|^2≤G/(2φ(q)) X^3/logX(1+o(1)), with G the Gallagher–Goldston constant. From this we prove a qualitative almost-all theorem, #{N≤X" even":R_(a,q) (N)=0}=o_q (X), and its effective refinement with an explicit 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 decay once the second moment is genuinely of order X^3/logX. Via the explicit formula and McCurley’s zero-free region we give a sub-exponential exceptional-set bound #E_(a,q) (X)≪_q Xexp(-√logX/R_McC), and show that a single real (Siegel) zero cannot invalidate it, since its contribution to the main term is itself o(N). Every result through Section 8 is unconditional; no Generalized Riemann Hypothesis, Density Hypothesis, or other unproved conjecture is used anywhere. We close by stating precisely why the results proved here do not, by themselves, upgrade “almost all” to “all,” and record this gap, together with a precisely stated open question about power-saving and an account of which classical unconditional routes to finiteness are already known to fail, as open questions. This revision corrects an internal inconsistency in the statement of the explicit minor-arc bound found in the previous version of this paper (the coefficient of the middle term), verified against the bound’s own proof and against its use two results later; the two numerical constants C_2 and G were independently re-verified and are unchanged.

Abstract: Wave equations with Caputo fractional damping model power-law attenuation in rocks, but their numerical solution is costly because of the non-local memory, and stability proofs for nonlinear variants are error-prone. We study a one-dimensional wave equation with Caputo damping of order α∈(0,1) and a cubic nonlinearity, discretised by Legendre–Gauss–Lobatto (LGL) collocation in space and a linearly implicit three-level scheme with the L1 formula in time. The memory is compressed by a sum-of-exponentials (SOE) approximation with positive weights, which reduces the cost of the history term from O(N²_t) to O(N_eN_t) with N_e growing logarithmically. By a discrete energy argument that does not use a sign of the discrete memory form, we prove stability under a time-step restriction independent of the polynomial degree and an error bound of order τ^2-α+N^1-m+ε_SOE; the energy argument is machine-checked in Lean 4. Experiments confirm the temporal rates, spectral accuracy in space and a speed-up of the memory evaluation of up to 60 times. In a seismic application with the damping calibrated from a prescribed quality factor, the quality factor recovered from simulated seismograms agrees with the dispersion relation to within 0.5%. An extension to Caputo’s constitutive model reproduces a nearly constant quality factor.

Abstract: High-dimensional biological systems can be represented in fewer dimensions to achieve simpler descriptions of complex cellular and physiological data. However, compact organization is frequently interpreted as evidence of reduced underlying dimensionality. We introduce an approach to separate geometrical compression from loss of local mathematical degrees of freedom within a biological solution space. We built a synthetic system of interacting regulatory, metabolic and signaling states governed by nonlinear constraints and examined its admissible solutions across changes in a control parameter. Multistart root finding identified the possible states, while intersolution distances quantified their geometrical compression. In parallel, Jacobian singular values and condition numbers assessed local rank and sensitivity, with nearest-neighbor geometry, participation ratios and residual-gradient dynamics providing complementary information on solution organization and stability. Our analysis revealed progressive geometrical convergence of distinct solutions accompanied by increasing numerical sensitivity, while the Jacobian preserved full rank throughout the investigated range. This suggests that compression in state space can occur without a corresponding reduction in local mathematical degrees of freedom, providing a criterion for separating apparent low-dimensional organization from genuine local dimensional loss. Potential applications include identifying misleading low-dimensional representations, assessing whether compressed biological data preserve their original structure and detecting regions where model solutions become increasingly sensitive to perturbations.

Abstract: Healthcare schools must place students in a limited number of internship wards. Manual planning is slow and may be difficult to inspect or explain, and internship assignment must satisfy department rotation, institution uniqueness, ward capacity, internship round, and student preference rules. This study compares a capacity-constrained regret heuristic with mixed-integer linear programming (MILP). The heuristic builds assignments from the options available at each decision. The MILP reduces the total weighted cost for the full cohort. Both methods were implemented in JavaScript within a standalone browser application. Method-specific records were also exported for a separate large language model explanation step. The evaluation used 133 students, three internship departments, and 101 healthcare institutions in Austria. Each method produced 399 valid placements. The MILP reduced the total weighted cost by 3.5% compared with the heuristic, while the heuristic was about 106 times faster in the recorded run. In one anonymized explanation case, both explanations passed four constraint checks and cited evidence from their source records. The heuristic explanation received higher scores for alternative coverage, uncertainty, and depth. The results show a trade-off between cohort cost, runtime, and the detail available for explaining an individual assignment.

Abstract: Let r(n) denote the least radius such that n congruent disks cover the unit disk D. We establish three results. First, we prove that the natural linear-programming relaxation of the covering problem coincides with the fractional covering number of the disk, and that its integrality gap converges exactly to the Kershner constant 2π/√27; consequently no bound derived from this relaxation can asymptotically exceed 90.94% of the optimum, which identifies angular information as the missing ingredient for further progress. Second, we derive from this relaxation and its integer strengthening the first systematic table of certified lower bounds for r(n) in the range 3 ≤ n ≤ 21, improving the elementary area and boundary-arc bounds by 7 to 15 percent; the bounds are exact for n = 3, 4, 7 and come with dual certificates verifiable through a single one-dimensional supremum. Third, we introduce an exhaustive branch-and-bound search over configuration space formulated in polar coordinates, in which the covering condition becomes a one-dimensional interval-covering problem on each concentric circle. The method yields rigorous enclosures of r(n) for 3 ≤ n ≤ 8 accurate to nine significant digits at a cost of between one and 26 917 nodes, independently confirming results previously obtained by hand and thereby validating the search against known ground truth. A self-contained reproduction script requiring only NumPy regenerates all enclosures in approximately twenty seconds.

Abstract: In this paper, we consider the properties of the following objects: plafal and geo-space (a general overview). As an application of the created theory, the proof of the equality of complexity classes P and NP will be given. The geo-plafal is a kernel (computational template) of the proof; Khomchenko’ scientific school (the constructive theory of serendipity approximations), Stepanets’ scientific school and the Bogolyubov principle of the decay of correlations for an infinite three-dimensional systems is a shell of the proof.

Abstract: High-frequency trading (HFT) data exhibit complex characteristics, including nonlinearity, non-stationarity, high noise, and long-term temporal dependencies. Traditional quantitative models and basic deep learning methods face three fundamental challenges when processing such data—insufficient feature extraction, temporal information decay, and overfitting—which collectively limit prediction accuracy and robustness. This paper proposes a hybrid LSTM-Seq2Seq deep learning model, TED-Net (Triple Encoder-Decoder Network), which integrates a Multi-head Self-Attention (MSA) mechanism and a Stacked Autoencoder (SAE). The model innovatively constructs a “triple encoding-collaborative decoding” architecture: the SAE serves as a deep feature encoder for feature denoising and reconstruction; the LSTM acts as a temporal encoder capturing long- and short-term dependencies; the MSA functions as a correlation encoder mining global dependencies from multiple representation subspaces; and the collaborative decoder adaptively integrates the triple encoding information through gated fusion and attention mechanisms. Ablation experiments validate the contribution and influence of each module. Comparative experiments on million-level minute-frequency datasets covering 15 stocks demonstrate that TED-Net significantly outperforms mainstream models such as Transformer and TCN, with an average RMSE reduction of 16.2% and an average R² of 0.94. An empirical analysis using CATL stock as an example shows that the model achieves an RMSE of 2.2292 and an R² of 0.97, verifying its adaptability and accuracy for high-frequency trading prediction under volatile market conditions. This study contributes to the growing field of computational finance by demonstrating how the integration of deep learning architectures can enhance predictive modeling in high-frequency financial markets.

Abstract: While polynomial interpolation discretizes fractional operators up to order \(k-\alpha\) (\(k=4,5,6,7\)), we construct a \(4-\alpha\) order approximation for the Caputo derivative directly in the frequency domain using the asymptotic expansion of the polylogarithm function. We determine the scheme's weights by applying the method of undetermined coefficients to a polynomial correction of the generating function \(\operatorname{Li}_{1+\alpha}(e^{-z})\), which cancels low-order truncation errors. To address the continuous solution's initial weak singularity near \(t=0\)—a persistent issue that degrades theoretical \(\mathcal{O}(\tau^{4-\alpha})\) accuracy to first order—we introduce a starting correction that modifies the right-hand side of the discrete equations. Although this modification applies solely to the first three time steps, Laplace transform analysis establishes the uniform recovery of the optimal \(\mathcal{O}(\tau^{4-\alpha})\) convergence rate for any fixed \(t>0\). Numerical tests on both homogeneous and inhomogeneous equations confirm this result.

Abstract: A geometry-aware and stiffness-robust boundary element framework is developed for time-fractional partial differential equations governed by hereditary diffusion, reaction, and transport mechanisms. The formulation couples dual-reciprocity boundary reduction with an Adams predictor and an implicit Moulton correction in which the stiff diffusive boundary element operator is retained at the new time level. This construction moves stability assessment from an isolated scalar test equation to the spectrum, numerical range, and resolvent behavior of the assembled spatial operator, so that geometry, boundary conditions, collocation nonnormality, and stiffness enter the analysis directly. A graded start-up strategy addresses the loss of temporal regularity associated with fractional memory, while compressed history evaluation provides a scalable route for long-time simulations. The theoretical development establishes modal and resolvent-based stability conditions, consistency and convergence properties, and a practical post-assembly stability certificate. Verification on smooth curved and nonconvex domains demonstrates accurate boundary-reduced spatial approximation, robust temporal convergence, resistance to diffusion-driven stiffness, and reliable evolution in the presence of corner singularities. The resulting framework provides a unified and extensible computational architecture for fractional diffusion, reaction transport, thermal memory, and related multiphysics models in geometries for which global spectral discretizations are restrictive.

Abstract: We prove, to our knowledge, the first unconditional cumulative closure estimates for the Arnold-Mukherjee-Pouly (AMP) refinement algorithm and the original face-marked tetrahedral algorithm of Bänsch on arbitrary conforming initial tetrahedral meshes. Let \( \mathcal T_0,...,\mathcal T_L \) be an adaptive mesh sequence generated by either algorithm, with \( \mathcal{M}_\ell\subseteq\mathcal{T}_\ell \) denoting the marking set at step \( \ell \). Then \( \#\mathcal{T}_L-\#\mathcal{T}_0\le C_{\mathrm{clos}}(\mathcal{T}_0)\sum_{\ell=0}^{L-1}\#\mathcal{M}_\ell \). The proof organizes conformity refinements into a causal forest and combines a uniform horizontal-propagation estimate with a weighted packing argument to obtain an explicit closure constant. For the original Bänsch algorithm, every history-dependent resolution of the initial two-edge ambiguity is represented by an AMP history associated with one of finitely many initial resolutions. The estimate therefore holds uniformly for arbitrary deterministic or nondeterministic choices. This resolves a long-standing complexity question for the Bänsch-AMP family.

Abstract: This study considers the simulation of hydrological processes in the Mzymta River (Russian Federation) using the Regularized Physics-Informed Neural Networks (PINNs). The Mzymta is the longest river in the Russian Federation that flows into the Black Sea. To formulate the boundary value problem, data on river topography (bottom profile) and hydrological measurements (time series of water level and discharge) from the gauging stations near the settlements of Rosa Khutor, Gorki-Gorod, Krasnaya Polyana, and Kazachiy Brod were collected. This study considers two modeling scales: the full-scale problem and a local-scale formulation for a short river segment, characterized by bed slope flattening, the development of a backwater--drawdown profile, and the occurrence of a hydraulic jump. The Saint-Venant system of equations was considered as a mathematical model. To solve this problem, we implemented PINNs. Two problem statements were considered: stationary and non-stationary. The full-scale simulation revealed that at the entire river scale, the flow rapidly establishes a normal depth profile, with all local perturbations being smoothed out by friction and thus indistinguishable at this resolution. In the unsteady case, the flow profile undergoes perturbations that propagate downstream in response to the time-varying sinusoidal inlet discharge. These perturbations propagate downstream at a finite speed, which is determined by the characteristic velocities of the shallow-water system. The local-scale formulation allowed us to capture the deviation of the flow profile from the normal depth and to study the backwater--drawdown problem, which involves the formation of a hydraulic jump both in stationary and non-stationary formulations. The formation of a hydraulic jump required the introduction of regularization into the loss function, since in the vicinity of the jump, the solution ceases to be smooth and experiences a discontinuity. To capture the discontinuity, we applied a method of relaxing the momentum equation term near the jump region. To enhance the sharpness of the jump, we additionally introduced a term that minimizes the width of the jump region. To properly account for all terms in the loss function, adaptive weighting of the coefficients was employed. Verification of the PINNs was performed by comparing the results against the fourth-order Runge–Kutta method, for both smooth solutions and those involving a hydraulic jump.

Abstract: Karlin, Klein, and Oveis Gharan introduced a randomized better-than-\(3/2\) approximation algorithm for metric TSP [KKO21] and subsequently established the corresponding improvement in the integrality gap of the subtour-elimination LP [KKO22], with an explicit constant \(\varepsilon>1.00000\cdot10^{-36}\). Gurvits, Klein, and Leake subsequently improved the certified saving to \(2.18000\cdot10^{-34}\) [GKL24]. We further obtain a randomized polynomial-time $(3/2-\varepsilon)$-approximation for every fixed \(\varepsilon<\varepsilon_\star\), where \(\varepsilon_\star>2.05522\cdot10^{-30}\), and consequently the subtour-elimination LP has integrality gap at most \(3/2-\varepsilon_\star\).