Papers

We construct a smooth projective Calabi-Yau threefold with a flat elliptic fibration of Mordell-Weil rank twelve and Hodge numbers $(h^{1,1},h^{2,1})=(25,1)$. The construction starts from the Kummer surfaces associated with the products of a fixed elliptic curve and the level-four modular family. The threefold is birational to an anticanonical quartic in $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(3)^{\oplus2}\oplus\mathcal{O}_{\mathbb{P}^1}(4)\oplus\mathcal{O}_{\mathbb{P}^1})$. We give twelve independent sections and construct a projective crepant resolution of the associated Weierstrass model. Varying the fixed elliptic curve gives a family of quartic resolutions that is locally complete at the original member. The resulting fourfold is birational over $\mathbb{C}$ to a diagonal order-four quotient of a product of four elliptic curves with $j$-invariant $1728$, and every smooth projective model has Kodaira dimension zero. An algebraic correspondence identifies the third cohomology of the original quartic resolution with the tensor product of the first cohomology of the fixed elliptic curve and the transcendental cohomology of the modular K3 surface. The Galois representation on the latter factor is associated with a weight-three CM newform.

A Neumaier graph is a non-complete edge-regular graph containing a regular clique; it is called strictly Neumaier if it is not strongly regular. In this paper we present a construction using finite rings that unifies several known results and yields three new families, each containing infinitely many strictly Neumaier graphs. We characterize when the resulting graphs are strongly regular. The family obtained from norm functions of finite field extensions includes the collinearity graphs of generalized quadrangles of \(*\)-Tits type arising from the classical hyperoval. We further show that, when the cyclotomic numbers involved have order at most seven, four types of rings suffice to obtain, up to isomorphism, all strictly Neumaier graphs arising from our construction. As a consequence of our results, each of these four types has now been used to construct Neumaier graphs.

Day-ahead operation of local energy communities (LECs) is affected by uncertainty in PV generation and electricity demand, which may compromise the feasibility of committed exchanges with the upstream grid. This paper proposes a forecast-residual-based chance-constrained optimization (CCO) framework that characterizes uncertain parameters through their forecasts and associated residual distributions. Under the adopted zero-mean Gaussian assumption, the resulting chance constraints admit an exact deterministic-equivalent reformulation in which uncertainty is embedded through analytical safety margins, preserving the mixed-integer linear structure of the original scheduling problem. The model jointly coordinates PV and BESS operation, low-voltage distribution network constraints, internal energy sharing, and day-ahead grid-exchange commitments. Once the physical schedule is determined, an ex-post allocation stage distributes the available community energy pool among users according to predefined participation coefficients, identifies post-allocation surpluses and deficits, and maximizes their internal matching while preserving the aggregate grid exchanges obtained from the CCO. The framework is evaluated on a reduced 206-node European low-voltage feeder with up to 55 community users and benchmarked against a two-stage stochastic programming formulation. Results show that, for forecast-error levels close to 5%, the proposed CCO achieves operating outcomes comparable to the stochastic benchmark while retaining computational requirements close to the deterministic formulation. The ex-post allocation results further show that internal matching can reduce gross energy exchanges that require settlement with the upstream grid.

We analyze a coupled PDE-ODE model for metastatic cancer invasion involving epithelial-like and mesenchymal-like cancer cells, cancer-associated fibroblasts (CAFs), the transforming growth factor $\beta$, matrix-degrading metalloproteinases, and extracellular matrix reconstruction by CAFs. The model incorporates bidirectional phenotypic transitions between epithelial-like and mesenchymal-like cancer cells, cancer-cell haptotaxis, negative fibroblast haptotaxis, as well as cell-specific proliferation. Under explicit parameter restrictions, we establish global classical solvability for the full model in two dimensions using an energy-based approach, and for a three-dimensional variant with constant phenotypic transition rates using coupled $L^p$-estimates. Numerical simulations illustrate primary tumor invasion and metastatic spread to secondary sites.

We study tilings of symmetric and alternating groups by conjugacy classes. For $S_n$, the case where the conjugacy class consists of transpositions was first investigated by Rothaus and Thompson in 1966 and remains open. We prove that this long-standing case is the only unresolved case for symmetric groups: every other nonidentity conjugacy class of $S_n$ fails to tile $S_n$. Our proofs combine representation-theoretic methods with combinatorial arguments concerning products of permutations. We also completely resolve the corresponding problem for alternating groups in a stronger form: for $n\geq5$, no normal subset of $A_n$ that avoids the identity can tile $A_n$. \iffalse We study the problem of whether a conjugacy class tiles the symmetric group or alternating group. For $S_n$, the case where the conjugacy class consists of transpositions was first investigated by Rothaus and Thompson in 1966 and remains open. We prove that this long-standing case is the only unresolved case for symmetric groups: every other nonidentity conjugacy class of $S_n$ fails to tile $S_n$. The proof combines representation-theoretic methods with combinatorial arguments concerning products of permutations. We also completely resolve the corresponding problem for alternating groups in a stronger form: for $n\geq5$, no normal subset of $A_n$ that avoids the identity can tile $A_n$. \fi \iffalse In this paper, we study when a conjugacy class or a normal subset tiles the symmetric or alternating group. The case of transpositions was first investigated by Rothaus and Thompson, and in general, it remains open. We show that every other conjugacy class of a symmetric group does not tile. We also prove that normal subsets in $A_n$ ($n\ge 5)$ not containing the identity element cannot tile.

In global metropolitan areas, public transport benefits from bus systems. Bus design widely applies theoretical models, which typically assume static ridership. However, ridership randomness exists, and lack of attention to it might lead to wrong design decisions. Moving beyond static ridership, we extend the traditional single-line model with stochastic ridership. Further, to optimize the bus system under various ridership randomness, we introduce a hybrid model that combines conventional buses (CBs) and flexible buses (FBs). In both models, we apply a continuous approximation approach and find that: 1) Bus capacity increases with ridership randomness in the extended single-line model; 2) The hybrid model exhibits a binary state: either CBs serve the ridership with rejections under low ridership randomness, or CBs serve the majority while FBs serve the minority with rejections under high ridership randomness. Our findings establish a link between bus system design and ridership randomness, contributing to a more adaptive and efficient public transport framework.

We study strictly stable minimal horizons in three-dimensional time-symmetric charged initial data sets with cosmological constant, allowing charged matter. Along the constant-mean-curvature foliation issuing from the horizon through leaves with positive Jacobi operator, we prove exact first-variation identities for the charged Hawking mass. When the charge density has one sign compatible with the horizon charge, Hawking masses are nondecreasing, and equality between the two boundary values of the fixed-charge mass forces the entire collar to be isometric to an annulus in a source-free constant-curvature Reissner–Nordström–(anti-)de Sitter model. From this we recover the local rigidity results under a local maximality condition. At the minimal endpoint the defect entering these identities computes the fixed-charge Hawking-mass Hessian along the normalized lapse, and its vanishing characterizes the model horizon data.

The Sun's atmosphere is significantly influenced by events that often measure only tens of kilometres in the horizontal direction. They happen in both quiet and active regions and are determined by magneto-hydro-dynamic processes. Their origins often lie in the lowest layers of the solar atmosphere. To improve the understanding of chromospheric and coronal heating, the solar dynamo and its activity cycle, or to improve the prediction of space weather events we must therefore enhance our understanding of such small-scale events. In this work I first describe the process of calibrating spectropolarimetric observations with very high spatial resolution and put together the building blocks for the mathematical proof the MOMFBD image reconstruction algorithm. In the second part I describe the FISS-SP prototype which is designed to fully exploit the resolution power of a 1.6m telescope. This instrument is built to provide suitable data for image reconstruction of spectropolarimetric data. The resulting data can resolve small-scale features down to the diffraction limit. The simultaneous interpretation of a few lines is still the norm for spatially highly resolved solar spectra. Theoretical works have shown that the simultaneous interpretation of multiple solar lines can greatly improve the robustness against noise. Using observations from FISS-SP I advance the theoretical idea into a practical technique. With this I can retrieve a coherent model of the local solar atmosphere, resolving structures down to a horizontal size of about 49~km. Finally, I use those new techniques to interpret an observation of a solar pore. In this last part I describe and analyse small-scale structures, which could not be spectropolarimetrically resolved before. Typical pore-boundary features - namely, dark striations with bright grains moving into the pore - are identified as convective features.

In this paper, we consider the quasi-copulas and copulas with a prescribed sub-track section, i.e., with given values along the curve segment connecting the points (x0,0) and (1,1-x0) for 0<x0<1. By employing the technique based on the W-ordinal sum, we provide methods for constructing quasi-copulas and copulas whose sub-track section coincides with a given one. Then, we further investigate construction methods for symmetric quasi-copulas and copulas with a prescribed sub-track section. Furthermore, we determine the best-possible bounds for the class of quasi-copulas sharing a common sub-track section, and verify that the lower bound obtained is also the best-possible lower bound for the class of copulas with a common sub-track section. Finally, we examine several properties of the bound (quasi-)copulas with a prescribed sub-track section, including their comparison with bound (quasi-)copulas with a given track section or a sub-diagonal section, as well as their statistical characteristic and concordance order.

For a class of nonlinear Fokker-Planck equations for probability densities, we construct distributional solutions by using an approximation scheme with nonlinear Neumann problems in balls. Moreover, we introduce nonlinear functional inequalities to estimate the convergence rates of the distributional solutions to the stationary solution, with respect to the $p$-variance for $p\in [1,2]$ including the relative entropy $(p=1)$ and the variance $(p=2)$. These nonlinear functional inequalities are then established in both non-degenerate and degenerate settings, so that the main results are applied to a number of typical models with various convergence rates.

In 1965, D. H. Lehmer conjectured that the permutations of every multiset admit an imperfect Hamiltonian traversal by adjacent swaps: a walk in the neighbor-swap graph that visits every word, with some words visited twice in order to reach a neighbor and return. The question is posed as an unsolved research problem in Knuth's Art of Computer Programming. Verhoeff (2017) chose the stutter words, in which every domino is a double, as the words to be reached this way, and reformulated the conjecture as the Hamiltonicity of the graph $N(S)$ on the non-stutter words, with two exceptional families — binary signatures with an odd multiplicity, and the permutations of $(2k,1,1)$ — that admit a Hamiltonian path but no cycle. This article proves the reformulated conjecture, and with it Lehmer's conjecture. The key structure is a partition of the words into hypercubes: the swaps inside dominoes turn each class of words with the same domino contents into a hypercube, and the stutters are exactly the $0$-dimensional classes. When every multiplicity is even, Hamiltonian cycles of the hypercubes are glued along a spanning tree, with no finite check. The case of exactly one odd multiplicity reduces to the all-even case and to a theorem of Stachowiak (1992), the one inherited Hamiltonicity input, which also settles two or more odd multiplicities. The only finite ingredients are two explicit cycles, of 28 and 84 words. Every construction is implemented in Python and checked against brute-force graphs, and the proof is formalized in Lean 4 over Mathlib.

Task-oriented communication conveys information that is necessary for downstream tasks. For binary decision tasks, this paradigm is information-theoretically formalized by Boolean function computation (BFC) via channels, where the receiver aims to determine the value of a function unknown to the transmitter. In this paper, we devise a practical code construction for the BFC problem based on a Reed-Solomon code. For noiseless binary channels, we derive finite-blocklength worst-case error bounds. By defining a rate function that captures how supported message length scales with channel uses, we characterize the rate-reliability tradeoff for different Boolean function families. With respect to this rate function, the proposed construction achieves an asymptotic computation rate of $1/2$. We further extend this construction to noisy channels by packing multiple BFC tasks into a single block and concatenating them with a conventional channel code. The corresponding finite-blocklength guarantees are expressed in terms of effective channel uses per function evaluation. With a channel code rate $R_c$, this construction achieves an asymptotic computation rate of $R_c/2$, yielding $C/2$ when capacity-achieving channel codes are employed. Numerical results illustrate the derived bounds and demonstrate substantial performance gains over conventional transmission. As examples, the proposed scheme achieves SNR coding gains of approximately $3.4$ and $6.4$ dB at an average task error probability of $10^{-5}$ for the exact-weight and rank-test tasks, respectively.

In this paper, we study boundary-driven behavior of the BGK model and the Boltzmann equation in a half-space with vacuum initial data. For any $1\leq\beta\leq2$ and $\alpha>0$, we construct a unique local-in-time BGK solution whose inflow boundary data have an arbitrarily small norm with the tangential exponential weight $e^{\alpha|v_h|^\beta}$, but whose norm with weight $e^{\alpha'|v_h|^\beta}$ is infinite for every $0<\alpha'\leq\alpha$ and every positive time of existence, where $v=(v_h,v_z)\in\mathbb{R}^n$ and $v_h\in\mathbb{R}^{n-1}$ denotes the tangential velocity. Thus, this BGK solution instantaneously leaves the exponentially weighted spaces associated with the inflow data, even when the exponential weight is weakened. The construction relies on sharp estimates for the density, bulk velocity, and temperature on a singular scale together with a fixed-point argument in a time-dependent anisotropic space. For comparison, we establish local existence and uniqueness for the cutoff Boltzmann equation under isotropic exponential weights $e^{\alpha|v|^\beta}$. In the Appendix, we further clarify the role of these different weights by showing that the macroscopic stability required for the BGK construction can fail even for Maxwellian inflow data and that it can be recovered by suitably weakening the normal velocity decay of the inflow data. These observations highlight a structural difference between local Maxwellian relaxation and the Boltzmann collision operator in the presence of boundary inflow and vacuum initial data. This work extends the whole-space analysis of D. Lee, S. Park, and S.-B. Yun \cite{LPY2024} to the half-space setting.

We develop a geometric way to uniformly carry out Hörmander's $L^2$-estimates for Hermitian-Yang-Mills connections with bounded energy on Kähler surfaces. We then show the Uhlenbeck convergence of smooth HYM connections on Kähler surfaces can be realized as algebraic convergence inside an analytic flat family. As applications, we prove a continuity theorem for smoothable families and a boundedness result for $\mu$-semistable sheaves with fixed topology. We also discuss potential applications for the moduli problem.

Trisecting the dihedral angles of an $n$-simplex defines its Morley simplex. We study the original simplices for which this simplex is regular. A criterion in terms of the Gram matrix of the facet normals reduces the problem to a matrix equation. The derivative of the Morley map at the regular simplex has two explicit eigenvalues, both nonzero for $n\ge3$; thus the regular simplex is an isolated solution. We conjecture that in dimensions four and five a regular Morley simplex forces two hyperplane reflections interchanging disjoint pairs of vertices. We prove that this conclusion fails in every dimension $6\le n\le200$ and in every dimension $n=\binom k2-1$ with $k\ge9$: in these dimensions there are simplices with regular Morley simplex and no hyperplane reflection symmetry. The examples for $8\le n\le200$ have dihedral symmetry of order $2(n+1)$, while the infinite family is based on the Johnson scheme. In dimension four we give exact constructions of two nonregular examples, defined by irreducible polynomials of degrees $18$ and $8$ with Galois groups $S_{18}$ and $S_8$. Neither example is expressible by radicals. The computer assisted existence proofs use exact rational arithmetic and intervals with outward rounding.

We introduce the crystal groupoid of a compact semisimple Lie group $K$; it is the Deaconu-Renault groupoid of a higher-rank analogue of a subshift of finite type on an inverse limit of Kashiwara crystals. We show that its Steinberg algebra and its $C^*$-algebra realise the crystal limits of the polynomial function algebra of $K$ and its $C^*$-envelope, respectively, in the sense of Matassa and the second author. We show that the orbit space of the crystal groupoid is in bijection with the Weyl group of $K$, with topology determined by the Bruhat order on the Weyl group. We also define a crystal groupoid for flag manifolds and related Poisson homogeneous spaces, and prove the analogous structural results.

We prove a uniform $L^\infty$-estimate for the parabolic Monge–Ampère equation under uniform Hörmander-Tian type estimates with respect to the initial and prescribed measures. We apply this estimate to rescaled Kähler–Ricci flows on a family of polarized Calabi–Yau manifolds over the punctured disk. An interpolation argument gives uniform Hörmander-Tian type estimates along the flows. We also establish uniform diameter bounds and Gromov–Hausdorff precompactness for this family of flows. In particular, we obtain an alternative proof of the diameter estimate by Li-Tosatti.

We consider low-budget active learning, which consists of selecting a limited number of points, the coreset, such that a model can be trained to high accuracy on the selection only. This problem is particularly relevant in contexts where labeling requires costly expert intervention, as in medical applications. We leverage features extracted from a pretrained self-supervised model to represent the data, and perform coreset selection directly in this feature space. In this paper, we use entropic optimal transport, specifically the Sinkhorn divergence, as the coreset selection criterion, which first allows us to get dimension-free sample complexity results, and second admits computationally efficient gradient evaluations. This opens the way to using gradient-based algorithms to rapidly compute solution candidates, further improved by a swap-based local search, with guarantees on the solution quality. Experiments on image benchmarks and medical datasets show that our method outperforms state-of-the-art heuristics in low-budget settings.

In this work, we study bilevel optimization problems where all variables are integer, all constraints and the leader objective function are linear, and the follower objective function is non-convex quadratic. Relying on bilevel-free sets derived from improving directions, we develop a disjunctive cut approach to exclude bilevel-infeasible solutions within a branch-and-cut algorithm. We show that our disjunctive cuts can be obtained by solving a cut generating linear program. Furthermore, we discuss conditions that allow the number of disjuncts in the cut generating linear program to be reduced, and we propose several strategies to identify improving directions and generate disjunctive cuts efficiently. We evaluate various aspects of the proposed branch-and-cut algorithm on both convex instances from the literature that fit our setting and new non-convex instances and compare the performance of our best approach with existing state-of-the-art approaches, which we significantly outperform.

We study the topological dynamical notion of asymptotic expansivity for linear operators. We also introduce two new notions, namely strong asymptotic expansivity and super asymptotic expansivity. We show that all three notions are equivalent to hyperbolicity for linear operators on finite dimensional spaces. In case of infinite-dimension, we construct examples to show that asymptotic expansivity is weaker than strong asymptotic expansivity which is further weaker than super asymptotic expansivity. More interestingly, we prove that super asymptotic expansivity is equivalent to the notion of hyperbolicity. Then, we prove that the super asymptotically expansive linear operators is a dense class in strong operator topology. Finally, we show elegant ways to construct variants of asymptotically expansive linear operators.

Counterfactual generation seeks to sample outcomes under a hypothetical intervention or decision using observational data collected under the factual assignment mechanism. We develop a flow-matching approach that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes drawn from a fitted conditional outcome model. To enable finite-step generation, we leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation. Our main theoretical contribution is a coupling-sensitive KL bound for constant-step Euler discretization: the error is controlled by moments of the source–target displacement under the chosen coupling, rather than by global uniform regularity of the velocity field, and has near-linear dependence on the ambient dimension. We also establish finite-sample non-parametric guarantees for the learned velocity and score fields when both the conditional outcome model and the source-target coupling are estimated from data. These bounds separate approximation, coupling-replacement, nuisance-estimation, generalization, and Monte Carlo errors and, combined with the sampler analysis, yield an end-to-end guarantee for counterfactual generation. Experiments on synthetic and semi-synthetic image benchmarks support the coupling-dependent theory and show that, at finite discretization budgets, the stochastic sampler can outperform the corresponding deterministic ODE sampler.

We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in $\mathcal{O}\left(\log (n/\varepsilon)\right)$ time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order $1$. We further provide $\mathcal{O}(\text{polylog}(n/\varepsilon))$ depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.

In this work, we study the pricing of American options under stochastic local volatility (SLV) models extended by including stochastic correlation driven by an additional stochastic process. We generalize the class of SLV models by incorporating a flexible stochastic correlation structure. To price options within these extended models, we derive the corresponding reflected forward-backward stochastic differential equations (RBSDEs) and employ data-driven numerical methods to solve them for both pricing and hedging purposes. The RBSDE framework enables the modelling of the future evolution of the option price. Furthermore, we conduct a convergence analysis of the proposed numerical method and present numerical experiments that illustrate the performance of the extended models, as well as the accuracy and efficiency of the RBSDE-based approach.

We consider stochastic games with independent controlled chains and unknown transition kernels, where players observe only their local states and realized payoffs. We develop a fully online, decentralized, and uncoordinated mirror-descent algorithm that operates in the dual space of occupancy measures for approximating stationary Nash equilibrium (NE) policies. The algorithm uses a single transition/reward sample at every primitive time step, relies only on local information, and requires neither coverage of the joint state space nor synchronized episodes. Under uniform-ergodicity and finite-coverage assumptions, we show that, with high probability, the time-averaged fixed-comparator regret decays at the canonical $O(T^{-1/2})$ rate, up to logarithmic factors and polynomial dependence on the game parameters. In particular, the complexity depends on the cover times of the individual local state spaces rather than the product state space, avoiding exponential dependence on the number of players and the sizes of the joint state and action spaces. The resulting finite-time regret bound further yields an approximate coarse-correlated-equilibrium guarantee, which is natural for arbitrary reward functions since computing a stationary $\epsilon$-NE is PPAD-hard in this setting. Under an additional global variational-stability condition, we show that the same fully online algorithm converges asymptotically in the last iterate to a stationary $\epsilon$-NE. Our results provide a fully online and scalable learning framework for stochastic games with unknown independent chains. The algorithm can also be viewed as a primal-dual framework for Markov games that exploits the independence and local structure of the players' controlled transition chains.

We study central algebraic cohomology of Galois gerbes over local fields of characteristic zero and number fields, with coefficients in a connected reductive group and an algebraic central subgroup, not necessarily finite or connected. For gerbes admitting a comparison morphism from Kaletha's rigid gerbe, we obtain fiber product formulas for abelianized cohomology in terms of algebraic fundamental groups of finite central quotients and homomorphisms from the band to the central subgroup. Such comparisons exist whenever the band is of finite type. Over number fields, independently of the comparison hypothesis, the full cohomology set is recovered from its abelianization and its real localizations through a cartesian square. Over non-Archimedean local fields and totally imaginary number fields, abelianization is bijective. We also specialize the formulas to Kottwitz's local and global gerbes.

We study the wetting transition for a two-dimensional Solid-On-Solid (SOS) interface above a hard wall in the presence of quenched disorder. The interface is represented by a nonnegative integer-valued height function $\phi$, with Hamiltonian $\mathcal H^\omega(\phi)=\beta\sum_{x\sim y}|\phi(x)-\phi(y)|-\sum_x(h+\alpha\omega_x-\lambda(\alpha))\mathbf 1_{\{\phi(x)=0\}}$, where $(\omega_x)_{x\in\mathbb Z^2}$ is an i.i.d. centered field, $\alpha\ge0$ is the disorder strength, and $\lambda(\alpha)=\log\mathbb E[e^{\alpha\omega_0}]$. This normalization is chosen so that the annealed model coincides with the corresponding homogeneous wetting model. As the wall attraction $h$ increases, the interface undergoes a transition from a delocalized phase, in which contacts with the wall have vanishing density, to a localized phase with a positive density of contacts. For sufficiently large $\beta$, the homogeneous wetting point and its sharp near-critical free-energy behavior are known. We prove that, for every fixed $\alpha\ge0$, the quenched critical point coincides with the homogeneous wetting point, $h_c(\beta,\alpha)=h_w(\beta)=-\log(1-e^{-4\beta})$. Moreover, writing $u=h-h_w(\beta)$, we show that the quenched excess free energy satisfies $\overline F_\beta(\alpha,u)=\overline F_{\mathrm{homo}}(\beta,u)+o(u^3)$ as $u\downarrow0$, where $\overline F_{\mathrm{homo}}(\beta,u)$ denotes the leading homogeneous wetting asymptotic and satisfies $\overline F_{\mathrm{homo}}(\beta,u)\asymp u^3$. Thus, at the level of both the critical point and the leading near-critical free-energy asymptotics, the quenched model has the same behavior as the homogeneous SOS wetting model. This contrasts with the disordered SOS pinning model, in which quenched disorder leaves the critical point unchanged but modifies the leading critical behavior of the free energy.

We establish the coefficient sign patterns in the Third Borwein conjecture and the modulus-three Cubic Borwein conjecture. The analytic arguments apply for $n\ge1750$ and $n\ge500$, respectively. They combine exact dissections and positive coefficient identities near the boundary with saddle point estimates that preserve cancellation between primitive-root contributions. Paired Gaussian estimates remove the leading odd error, and explicit remainder bounds cover the complementary contours. The remaining finite intervals are checked by exact integer arithmetic. The Third finite verification through $1749$ is author-confirmed; the Cubic verification through $500$ is supported by two complete integer implementations and additional coefficient crosschecks.

Without needing a $t$-structure, we establish natural long exact homology sequences on finite extension windows with $\ell+1$ layers of extension-closed proper abelian subcategories, assuming vanishing of negative self-extensions in degrees $1,\ldots,\ell+1$, and prove uniform sharpness of this bound. In the Krull-Schmidt setting, we classify intermediate subcategories stable under taking pure homology layers by compatible sequences.

We generalize Serre's conditions for coherent sheaves to triangulated categories equipped with a $t$-structure and duality, which we call Verdier-type categories. We present examples of such categories and their associated Serre conditions, then focus on constructible sheaves, where we refer to these conditions as geometric Serre conditions. We investigate their relationship with perversity and illustrate the theory through a range of examples. Finally, we prove a geometric version of the Auslander-Reiten conjecture.

The Berge conjecture [Proc. London Math. Soc., 1979] asserts that every bridgeless cubic graph can be covered by five perfect matchings. The 5-cycle double cover conjecture, proposed independently by Preissmann (1981) and Celmins (1984), asserts that every bridgeless graph admits five even subgraphs such that every edge belongs to exactly two of them. Hakobyan and Mkrtchyan [Ars Math. Contemp., 2019] proved that the 5-cycle double cover conjecture holds if and only if every bridgeless claw-free cubic graph can be covered by four perfect matchings. In this paper, we prove that the Berge conjecture holds for every bridgeless claw-free cubic graph $G$ and that $G$ has four perfect matchings covering at least $\frac{299}{315}|E(G)|$ edges.