Papers

A queue layout of a graph consists of a linear order of the vertices and a partition of the edges into queues so that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue number. The planar product structure theorem states that every planar graph is a subgraph of the strong product of a graph of simple treewidth at most $3$, a clique $K_3$, and a path. Such a strong product admits a queue layout with $49$ queues (Wood, 2005), which implies that the queue number of planar graphs is at most $49$. Recently, Bekos, Gronemann, and Raftopoulou (Algorithmica, 2023) investigated how the general approach based on the product structure can be optimized for planar graphs. They claim that by appropriately reordering the three vertices in each bag arising from a tripod, it is possible to reduce the queue number of planar graphs to~$42$. In this note we highlight a gap in their queue layout algorithm: one of the choices required by the algorithm is not guaranteed to exist. Hence the claimed upper bound of $42$ queues is not established by the published proof.

In this paper, we consider a class of multiblock nonconvex nonsmooth optimization problems, which covers many applications such as the analysis of pre-earthquake anomalies and machine learning. To solve this class of problems, we propose the inertial block proximal linearized method with two-phase adaptive momentum (IBPL$^+$-TP). Compared to the current methods, our method possesses three main advantages: (1) it introduces a two-phase adaptive momentum strategy to effectively update the extrapolation parameters, (2) it allows using two different extrapolation points to accelerate the convergence, (3) it allows the extrapolation parameters of these two extrapolation points to be independent of and unconstrained by all other parameters. While maintaining the above advantages, we prove that our method ensures the monotonic convergence of the objective function of this class of problems, and we also prove that the sequence generated by our method globally converges to a critical point, as well as establish the convergence rate of our method. To demonstrate the effectiveness of our method, we apply it to solve two nonconvex and nonsmooth machine learning problems, namely sparse nonnegative matrix factorization with $\ell_0$-constraints and sparse nonnegative CP decomposition with $\ell_0$-constraints. The numerical experimental results on solving these problems show that our method outperforms several state-of-the-art methods.

We develop and analyze a second-order, fully implicit, monolithic time-discretization for the coupled time-dependent Stokes and quasi-static Biot system. The Biot subsystem is formulated in a three-field total-pressure formulation, in which the total pressure is introduced as an additional unknown together with the solid displacement and pore pressure. This reformulation improves robustness in nearly incompressible regimes and suppresses volumetric locking. All variables are discretized in time using the BDF2 scheme, and the coupled problem is solved monolithically without operator splitting. Discrete energy stability is established using the BDF2 $G$-stability identity. The resulting estimates exhibit robustness with respect to the Lamé parameter, Biot–Willis coefficient, and storage coefficient. A consistency–stability argument is then used to derive second-order convergence in time together with optimal-order convergence in space. Numerical experiments confirm the theoretical results and demonstrate second-order temporal accuracy in the corresponding energy norms.

We establish several new series evaluations involving the central $q$-binomial coefficients, with the inspiration coming from earlier work by Vignat and one of the authors on the limiting case at $q\to 1$.

This paper presents a discrete-time passivity-based analysis of first-order momentum-based methods for a class of functions whose gradient has lower and upper sector bounds of $0$ and $L$, respectively. Through a loop transformation, it is shown that momentum-based methods can be represented as a passive controller in negative feedback with an output strictly passive (OSP) system. The weak passivity theorem is then used to derive explicit hyperparameter conditions under which the shifted gradient asymptotically vanishes. Under an additional assumption that requires the existence of a unique stationary point and excludes arbitrarily small gradients far from that point, convergence of the iterates to the global minimizer is established.

We study the enumerative geometry of elliptic curves in toric threefolds. We consider enumerative integer invariants, called well-spaced counts, which can be studied using well-spaced genus-one tropical curves in $\mathbb{R}^3$. By comparing this with the logarithmic degeneration formula, we obtain an explicit relationship between logarithmic virtual invariants and these geometric invariants. The result is a logarithmic analogue of a formula of Getzler–Pandharipande for elliptic curves in $\mathbb{P}^3$. As an application, we show that the virtual logarithmic invariants for $\mathbb{P}^3$ with respect to its toric boundary are strictly less than the ordinary Gromov–Witten invariants once the degree is sufficiently large. Several examples are included.

In this paper, we investigate the Lyapunov exponents and hyperbolicity of random billiard systems under small stochastic perturbations. Our study begins with a review of the necessary theoretical background on Lyapunov exponents for stationary sequences of matrices and Markov processes. We then consider several classical billiard systems and demonstrate that random perturbations can lead to significant changes in their dynamical properties. Specifically, we show that non-circular elliptic and lemon billiards exhibit both ergodicity and hyperbolicity under random perturbations, whereas circular billiards, while ergodic, maintain zero Lyapunov exponents. We also establish conditions under which the largest Lyapunov exponent becomes positive, indicating the emergence of hyperbolic behavior in these systems. These results provide new insights into the stability and chaotic properties of dynamical systems subjected to random noise, with implications for the broader study of smooth dynamical systems.

For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's theta function and Dedekind's eta-function, implying the threefold equality $$\underbrace{\prod_{n\ge1}(1-q^n)\cdot\Biggl(\sum_{n=0}^{\infty}\frac{|M_n^{a,b}(\mathbb{F}_q)|}{|\mathrm{GL}_n(\mathbb{F}_q)|}\Biggr)\Biggr|_{q\mapsto q^{-1}}}_{\text{point count}}\;=\;\underbrace{Z_{a,b}(q)}_{q\text{-series}}\;=\;\underbrace{P_{a,b}(q)}_{\text{theta quotient}}$$ If true, the point count on $X^a=Y^b$ is essentially a modular function on $\Gamma(a+b)$. The conjecture is layered in $a$, with an identity for each $b$. The $a=2$ layer is classical, including identities of Rogers–Ramanujan and Andrews–Gordon. For $a\geq3,$ nothing was known. We prove the $a=3$ layer in full: a new infinite family of Rogers–Ramanujan identities, and a geometric origin for Warnaar's products. AxiomProver verified these new identities in Lean assuming existing literature.

Objective: Emergency department (ED) overcrowding threatens care quality, yet Full Capacity Protocols (FCPs) are typically activated only after crowding thresholds have been exceeded. This study evaluated whether machine learning-guided proactive FCP activation improves ED crowding control compared with conventional reactive activation. Materials and Methods: We developed and validated an integrated machine learning and discrete-event simulation framework representing ED patient flow and capacity management. An embedded machine learning model predicted hourly waiting-room crowding six hours ahead and triggered proactive FCP activation. Using empirically calibrated operational parameters from a large US hospital, we compared proactive and reactive deployment strategies over a 365-day simulation horizon with 10 independent replications. Outcomes included ED length of stay, wait time, percentage of time above the crowding threshold, number of FCP activations, and protocol duration. Results: Proactive deployment produced statistically significant reductions in the percentage of time above the crowding threshold and average protocol duration across all threshold settings. It also reduced crowding variability and extreme congestion events, resulting in smoother ED operations. Although proactive deployment generally improved ED length of stay and wait time, these differences were not statistically significant. Conclusion: Embedding machine learning predictions within an operational simulation enables proactive, prediction-guided capacity management before severe congestion develops. These findings support the feasibility and potential value of proactive FCP deployment as a decision-support strategy for improving ED crowding control and provide evidence to support prospective implementation in clinical practice.

In this paper, We prove that: (A) If $m\ge 4,$ there exists a Platonic geodesic tree having $m$ edges (geodesic segments), which form $m$ platonic angles with measure $120^{o}$ and $(\frac{m(m-1)}{2}-m)$ platonic angles with measure $90^{o}.$ (Existence of a Fermat Coordinate system) (B) If $m=2,$ there exists a Platonic geodesic tree with three vertices and 2 edges, which meet at $120^{\circ}$ or $150^{\circ}$ or $180^{\circ}$ (Minimal spanning geodesic tree). (C) If $m=3,$ there exists a Platonic tree and the Steiner point is identical with the Fermat point, having three edges, which meet at $120^{\circ}$ (Hexagonal coordinate system). (D) If $m$ is even and $m \ge 4,$ there exists a Platonic geodesic tree having $m$ edges , which form platonic angles with only two distinct measures $90^{o}$ and $180^{\circ}.$ (Existence of a Cartesian coordinate system). (E) If $m\ge 4$, there does not exist a Platonic geodesic tree having $m$ edges , which form only platonic angles with measure $120^{o}$ (Non existence of a Fermat coordinate system). (F) If $m\ge 3,$ there does not exist a Platonic geodesic tree having $m$ edges, which form platonic angles with only one measure $150^{o},$ or two distinct measures $120^{\circ}$ and $150^{\circ}$ or two distinct measures $90^{\circ}$ and $150^{\circ}$ (Non existence of a Fermat coordinate system). A classification of platonic geodesic trees with boundary vertex set $V$ and added vertex set $Q$ in the $K-$Space is given by conditions (A), (B), (C), (D), (E), (F), which characterize each vertex in $V\cup Q.$

Two hypergeometric families of mixed-type multiple orthogonal forms are constructed for rank-one $q\times p$ matrices of weights, with arbitrary $p$ and $q$: Jacobi and Laguerre I systems. Both normalized mixed forms are obtained explicitly for admissible near-diagonal multi-indices. The power-vector components are terminating generalized hypergeometric polynomials, while the hypergeometric-vector components are finite sums of such polynomials. In the Jacobi case, these sums are expressed as finite combinations of terminating Kampé de Fériet polynomials evaluated at $(x,1)$. For the mixed beta–Euler Laguerre I system, a finite triangular system relates the residues at finite poles to the terms generated by the Euler operator. Gamma-quotient Mellin formulas, Meijer $G$-representations, and Rodrigues formulas are derived for the complete mixed forms. On the step-line, the two biorthogonal systems satisfy dual recurrences governed by matrices with $p$ subdiagonals and $q$ superdiagonals. All recurrence coefficients are given by finite Gamma–Pochhammer expressions. Under normality and nonvanishing-pivot assumptions, Christoffel transformations and Gauss–Borel factorization yield bidiagonal factorizations of the recurrence matrices. The lower factors have closed Pochhammer formulas, while the upper factors are expressed through finite Christoffel tau-determinants or, equivalently, cross-ratios of shifted moment minors. Both Christoffel chains close explicitly in the mixed Piñeiro specialization.

The Material Point Method (MPM) has drawn great attention in the numerical modelling of large deformation, geotechnical problems. The popularity of MPM is mainly because its formulation shares significant similarities with the Finite Element Method. In MPM, the iteration points can move independently from the mesh, allowing for the resolution of large deformation problems. However, because of this, the original MPM formulation suffers from the well-known cell-crossing noise and volumetric-locking instabilities, resulting in a strongly oscillated stress field. A novel implicit locking-free B-spline MPM that controls stress oscillations to a negligible level is proposed in this paper. A novel, but very straightforward B-spline shape function implementation procedure, avoids the need for a complex material point searching algorithm, providing seamless transformation from the original MPM to this robust B-spline MPM, aiming at modelling large-strain geotechnical problems. The newly proposed volumetric locking mitigation strategy is also very easy to implement, which facilitates the reproducibility of this research. The proposed method is validated against three numerical studies: granular column collapse experiment, slope failure and footing with large penetration. The proposed numerical method agrees well with experiments reported in the literature and previous numerical studies. Also, these numerical examples show that the proposed method provides a more prominent stress field than other available methodologies.

Nielsen and Soelberg exhibited three torsion-free groups G_1, G_2, G_3 carrying 8-element sets without unique products, and asked whether any of the group rings R[G_i], R a domain, is a domain. We record that for every commutative domain R the answer is affirmative for all three groups: each G_i is virtually nilpotent, so the theorem of Kropholler, Linnell and Moody applies over every field, in every characteristic, and commutative coefficients reduce to the fraction field. Every ingredient is in the literature except the nilpotent structure of the relevant finite-index subgroup of G_2, which is supplied by the certified computational model of our companion paper (arXiv:2607.19687). The question therefore remains open exactly for noncommutative coefficient domains, where the unique-product mechanism – the only known ring-independent one – is precisely what these groups are constructed to lack. As a complement we report machine-checkable, DRAT-certified verifications that F_2[G_i] has no zero divisors with both supports in explicit balls of the defining generating sets, obtained by propositional reasoning alone, independent of the K-theoretic machinery; we state precisely what these certificates do and do not add.

A biharmonic conformal immersion asks whether a hypersurface can become a critical point of the bienergy after a conformal change of the domain metric. We study when this flexibility can produce compact nonconstant-mean-curvature geometry in space forms. For surfaces in three-dimensional space forms, with $U=\lambda^2H$, Ou's system is encoded exactly by a Laplace-type operator acting on a restricted ambient Killing spinor. We classify all monic scalar–chiral factorizations of this operator. In nonzero curvature such a factorization exists locally exactly when the principal curvatures are locally constant; the normalized Euclidean branch is equally rigid, and the remaining real spherical non-CMC branch is excluded by a finite-type Frobenius obstruction. For $m>2$, this scalar rigidity does not imply geometric nonexistence. The conformal tension has independent normal and tangential components. We prove that $\mathbb R\xi\oplus TM$ is the minimal natural $O(m)$-equivariant carrier, derive the corresponding strongly elliptic Jacobi block and its two-grade Clifford realization, and identify off-diagonal hybridization as the escape mechanism. Exact cohomogeneity-one reductions, combined with validated interval and Chebyshev/radii-polynomial arguments, yield embedded proper non-CMC, nowhere-umbilical examples in round spheres, including $$ S^1\times\mathbb S^n\times\mathbb S^n\hookrightarrow\mathbb S^{2n+2} \qquad(n\ge1). $$ Thus the scalar obstruction is sharp with respect to coefficient bundle: the smallest enlargement forced by conformal tension already supports compact non-CMC solutions.

Let $\Omega\subset\mathbb R^2$ be a bounded domain whose boundary $\gamma$ is a smooth simple closed curve, and let $r=|B|^{-1}$ be the Larmor radius of a constant magnetic field. Assume $$ 0<r<\frac{r_0(\gamma)}2, $$ where $r_0(\gamma)$ is the maximal embedded tubular radius of $\gamma$. We prove that if the corresponding magnetic billiard admits a nonconstant first integral polynomial in the velocity, then $\gamma$ is a circle and $\Omega$ is a disk. No convexity assumption is imposed. This eliminates the finite set of possible exceptional strong-field values left by the earlier polynomial nonintegrability theory. The proof uses the two highest reflection modes of the polynomial induced on the space of Larmor centers. A boundary index identity shows that the highest coefficient has degree equal to the true Fourier degree and that all of its roots lie inside the table. After complexification and passage to the normalization of the boundary curve, comparison of the two leading modes at infinity excludes simultaneous poles of the coordinate functions and forces all of these roots to coincide. The reflection identity then implies that the boundary has constant distance from their common root and hence is circular.

Building on a commutative geometric algebra developed for 2D and 3D fields of complex vectors, this paper develops a vector framework for the differential geometry of compact Hermitian manifolds embedded in an ambient three-dimensional field of complex vectors. The underlying algebraic framework is constructed from the commutative geometric product of complex vectors, yielding a unified commutative algebra in both the 2D and 3D settings together with the corresponding vector differential and integral identities. These algebraic results provide the intrinsic foundation for a differential geometric formulation that is developed independently of the classical scalar coordinate approach. Within this framework, compact Hermitian manifolds are described entirely in vector form. Fundamental tensor relations governing the intrinsic and extrinsic geometry of complex manifolds are derived in vector form. The resulting framework establishes a unified algebraic and geometric setting in which the differential geometry of complex manifolds is developed directly from the commutative algebra of complex vectors.

For a graph G and a nonempty set S of vertices, the edge multiset representation of an edge e is the multiset of distances from e to the elements of S, where d(uv,s)=min{d(u,s),d(v,s)}. The edge multiset dimension edim_m(G) is the minimum cardinality of a set whose edge representations are pairwise distinct, and is infinite if no such set exists. A recent survey asked whether edim_m(Q_d) is infinite for every d >= 3. We answer this question negatively and determine the finite-infinite transition completely: edim_m(Q_d) is infinite if and only if 2 <= d <= 5. An exhaustive computation proves edim_m(Q_5) = infinity, extending the known nonexistence results for Q_3 and Q_4. Explicit independently verifiable resolving sets are given for Q_6 through Q_10. For all d >= 11 we prove existence probabilistically: equality of two random edge histograms is a zero-divergence event on a graph of distance levels, and conditioning outside a spanning forest bounds its probability by a product of central-binomial atoms. Certified exact rational computations cover 11 <= d <= 50, and an elementary ten-edge forest estimate handles the tail d >= 51. We also prove the lower bound edim_m(Q_6) >= 6.

Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space $({V},\{p_\alpha\}_{\alpha\in A})$, whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative $L^2$ error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual $V'$, including for non-normable input spaces.

We study the following problem: given an edge-colouring of a $K_n$ with $r$ colours, how many colours does one need to keep (in the worst-case scenario) so that the subgraph formed by those colours is connected? We determine this extremal function up to constant factors in every parameter regime, and show that it has the same asymptotic order as the corresponding number (of colours) needed for ensuring that no vertex is isolated. This gives a complete description of the “spanning” threshold for connectivity in edge-coloured complete graphs.

Two-phase phase-type ($PH_2$) service distributions are widely used in call center, healthcare and manufacturing models: they capture coefficients of variation both above and below unity while remaining parsimonious enough for reliable statistical fitting. We derive explicit closed-form queue-length distributions $p_n = A_1 r_1^n + A_2 r_2^n$ and sojourn-time densities $f_{W_s}(t) = c_1 e^{\alpha_1 t} + c_2 e^{\alpha_2 t}$ for the complete $M/PH_2/1$ family (Erlang-2, hypoexponential-2, hyperexponential-2 and Coxian-2), with every coefficient given as an explicit function of the system parameters and consolidated in ready-to-use reference tables. Because the results are functions rather than numerical values, they support analytical operations that numerical output cannot deliver directly: exact sensitivity derivatives, closed-form threshold optimization, capacity sizing by bisection on an exact cumulative distribution, and $O(1)$ tail-probability evaluation at arbitrary queue length. We further prove that $PH_2$ marks the boundary of algebraic tractability: the three-phase queue $M/E_3/1$ has a universally negative discriminant, forcing complex roots at every traffic intensity, so its distribution admits no representation as a sum of real geometric terms, and for $k \geq 5$ phases the Abel-Ruffini theorem precludes radical solutions. Validation against the matrix-analytic library BuTools confirms the derivations and characterizes where such computation degrades: queue-length evaluation holds machine precision throughout, while sojourn-time evaluation via the matrix exponential loses up to ten significant digits when high traffic intensity and high service variability act jointly. An application calibrated to published surgical time data for 46,322 cases yields an exact affine law for the sensitivity of overflow risk to case mix.

In 1986, Ivić and Tenenbaum introduced arithmetic functions with squarefull kernel, which are also called $s$-functions. Later, Erdős and Ivić gave an asymptotic estimate on the shifted convolution sums of $s$-functions. Recently, Bergelson and Richter studied the orbits along the prime Omega function in a uniquely ergodic topological dynamical system and established a new dynamical generalization of the prime number theorem (PNT). These orbits can be viewed as functions of invariant average under multiplications. In this paper, we show that both $s$-functions and their shifted convolutions are asymptotically uncorrelated to the orbits along the prime Omega function in a uniquely ergodic system. As a consequence, we obtain a refinement of the PNT via the local distribution of $s$-functions. Furthermore, several variants of these results are established as well.

In this work, we consider a structured tensor approximation problem, where only a limited number of lateral slices are observed. The proposed algorithm , called Basis and Manifold prior Tensor Approximation (BMTA), exploits both global and local structures of the evolution of a global tensor. Specifically, BMTA integrates two signal models: (i) a quasi-basis model that captures smooth global variations along a physical trajectory, and (ii) a manifold-guided interpolation model that characterizes local relationships among tensor slices. A low-rank Tucker reconstruction framework is incorporated to efficiently capture the priors, resulting in coefficients for basis function estimation and a tensor optimization. In addition, we provide a theoretical analysis which establishes a non-asymptotic reconstruction error bound that characterizes the effects of sampling complexity, optimization convergence, and model mismatch. Numerical experiments are performed on both synthetic and real-world datasets, including quantum chemistry and spatiotemporal sensing applications.

This work introduces a proximal stochastic subgradient method for minimizing the sum of an expected cost, whose integrand is potentially nonsmooth and nonconvex, and a lower semicontinuous, prox-bounded function. We target a broad class of integrands obeying a nonsmooth, localized variant of the descent lemma in the decision variable, a structural assumption that simultaneously covers smooth losses with Lipschitz gradient and differences of such losses with convex functions. At each iteration the expected cost is replaced by a sample average that is progressively refined, and the proximal-subgradient stepsize is selected by an Armijo-type line search enforcing a sufficient-decrease property up to stochastic errors induced by the sample-based approximation. This framework accommodates substantially more general problem formulations than existing methods, in particular, it requires neither (weak) convexity of the regularizer nor a uniform bound on the variance of the stochastic oracle, and our analysis yields convergence guarantees that are new even in the smooth setting. Specifically, we establish almost sure convergence of the sequence of function values and stationarity of every accumulation point of the trajectories under the relaxed requirement that the sample-size sequence be merely nondecreasing and unbounded, with no prescribed growth rate. Leveraging the Kurdyka-Lojasiewicz (KL) property, we further upgrade this subsequential guarantee to convergence of the whole trajectory to a single stationary point. Finally, for exponential-type KL desingularizing functions and polynomially growing sample sizes, we derive explicit polynomial convergence rates, up to a logarithmic factor, for both the function values and the iterates.

We study the moduli of sourceless framed Beltrami-Vekua equations under recombinations of the unknown, scalings, and orientation-preserving changes of variables. On every bounded simply connected domain, such an equation reduces to $w_{\bar z}=B\bar w$ on the unit disk, with residual symmetries given exactly by zero-free holomorphic gauges and Möbius transformations. When $B$ is zero-free, the equation is completely classified by two data modulo Möbius: the hyperbolic mass density $\vartheta=\tfrac14(1-|z|^2)^2|B|^2$ and the phase-curvature current $K=\Delta\arg B$, whose hyperbolic density at $C^2$ regularity is $\kappa=\Delta_{\mathrm{hyp}}\arg B$. We determine the exact range of these invariants: every positive Hölder density and every phase current arising as the Laplacian of a Hölder phase occur. For fields with zeros, the classification extends to the phase-integrable sector through the triple $(\vartheta,d\eta,d{\star}\eta)$, where $\eta=\operatorname{Im}(dB/B)$. On the tame sector, $d\eta$ is the atomic charge measure, while $d{\star}\eta$ carries the remaining phase curvature. Thus the pseudo-analytic mass and charge are numerical projections of a larger infinite-dimensional moduli space. Explicit equal-mass, equal-charge, inequivalent equations are exhibited.

Let $\varphi:\Sigma^2\looparrowright\mathbb{RP}^3$ be a closed immersed surface, with no orientability or, equivalently, two-sidedness assumptions. We establish the sharp quantitative estimate $$ \lambda_2(\Delta+|\sigma|^2+2)\leq2-\frac{2}{\operatorname{Area}(\Sigma)}\int_\Sigma H^2d\Sigma+\frac{4\pi\chi(\Sigma)}{\operatorname{Area}(\Sigma)}. $$ Our main contribution is the analysis of the borderline case, which rules out equality when $\chi(\Sigma)\leq0$. More precisely, $$ \lambda_2(\Delta+|\sigma|^2+2)<2\quad\text{whenever}\quad\chi(\Sigma)\leq0. $$ The proof combines the canonical Veronese embedding of $\mathbb{RP}^3$ into $\mathbb{S}^8$, the conformal test function method, an Obata-type rigidity argument, and the classification of closed flat minimal surfaces in $\mathbb{RP}^3$.

Sequential prediction naturally induces an innovation sequence consisting of the prediction errors produced by a causal predictor. We use the collection of all such innovation sequences to define the innovation spectrum of an individual binary sequence and, from it, a new reducibility based on sequential information extraction. We show that this reducibility refines truth-table reducibility while exhibiting a fundamentally different geometry. The degree structure decomposes into two canonical regions: a truth-table spine, whose induced order is isomorphic to the truth-table degrees, and a complementary reservoir-immune region, consisting of sequences from which no infinite computable predictable reservoir can be extracted. We establish bridge constructions connecting the two regions, prove that reservoir immunity is preserved under sequential innovation, and show that the Martin–Löf-random degrees form a proper downward-closed substructure inside the reservoir-immune region. These results reveal a new geometric organization of individual sequences based on causal predictability rather than classical oracle computation.

Supervised training of finite-element (FE) surrogate models requires reference solutions, and each reference solution is obtained by solving the system that the surrogate is intended to replace. The assembled discrete potential energy provides a training signal that requires no reference solution. This note records, with proofs, the identities that make this signal exact for linear elastostatics: the difference between the energy of a prediction and the energy of the reference solution equals one half of the squared stiffness-norm error, and the gradient of the energy equals the stiffness-weighted error. Label-free discrete-energy minimisation and supervised regression in the stiffness norm therefore have the same unique minimiser and identical gradients at every point. Around this central result, the note states a conditioning lemma that bounds the displacement error by the energy gap, a modewise contraction identity that explains why the Euclidean displacement error is an unsuitable primary metric, the Chebyshev bound that governs conjugate-gradient post-processing of surrogate predictions, and a conditional latent-separation proposition for joint-embedding predictive architecture (JEPA) pretraining on a shared stiffness operator, with an explicit numerical counterexample that delimits its scope. Every claim with numeric content is implemented as an executable falsification check; the checks were executed twice, on synthetic test problems and on a probe set of 16 instances from the validation split of a pre-registered experimental run, and every inequality holds, with the measured tightness reported. A closing section explains why the construction does not extend to elastodynamics through direct minimisation of the action functional, and which time-discrete formulation restores exactness.

Sequential clinical decision-making often involves more than maximizing average efficacy. Clinicians may need to simultaneously optimize clinically relevant tails of the outcome distribution, control treatment-related risk, and choose among multiple treatment options. Existing quantile dynamic treatment regime (DTR) methods capture distributional features of treatment outcomes but remain largely restricted to efficacy-only objectives and binary treatments. To address these limitations, we propose Risk-Aware Quantile Dynamic Treatment Regimes (RQDTR), a unified framework that optimizes a prespecified quantile of the cumulative potential outcome while explicitly incorporating treatment-related risk. We also develop an angle-based formulation for jointly learning decision rules across multiple treatment categories. Our framework includes three interpretable subclasses: efficacy-only quantile learning, constraint-based learning with population-level risk control, and utility-based learning through a composite benefit-risk utility. Theoretically, we establish identification and oracle equivalence, Fisher consistency of the smoothed surrogate, consistency of the estimated regime, and finite-sample performance error rates. Extensive simulation studies and applications to All of Us major depressive disorder and MIMIC-III sepsis data demonstrate that RQDTR improves tail-oriented efficacy and achieves more favorable benefit-risk trade-offs than existing quantile DTR methods.

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. We prove and widely generalize a conjecture of A.\,I.~Nazarov \cite{Naz21}: for $s\in(1,\frac 32)$ the quadratic form $Q^{\rm SP}_s[u]$ of the spectral fractional Dirichlet Laplacian strictly increases under the map $u\mapsto|u|$ provided $u\in\tilde H^s(\Omega)$ changes sign in $\Omega$.

We establish conditions and give proofs on how an error-correcting code can attain infinite diversity in a time-entanglement quantum key distribution (TE-QKD) reconciliation. The shocking result, never encountered in the literature on coding and communication theory, is that a decoder exhibits an infinite diversity order while the channel has finite diversity and the code has a relatively short finite length. This paper studies the diversity order of coded TE-QKD reconciliation, defined by the asymptotic slope of the error probability at high signal-to-noise ratio. For bounded-distance algebraic decoding, we derive a necessary and sufficient condition in terms of the number of photons per codeword and the decoding radius. For soft-decision decoding, we introduce the maximal finite diversity (MFD) property and prove that infinite diversity is achieved if and only if the code is MFD deficient. The infinite diversity in TE-QKD has no counterpart in classical fading channels, where decoding can only multiply a finite diversity order by a finite factor. Examples of short codes based on Golay, Reed-Solomon, Bose-Chaudhuri-Hocquenghem (BCH), and Reed-Muller codes validate the analysis and illustrate how the TE-QKD system parameters and the relatively short code parameters affect the achievable diversity for both hard and soft information reconciliation.