Papers

Let $p$ be a prime number. We prove the sharp Furstenberg set bound in the $p$-adic plane $\mathbb{Q}_p^2$: every $(s,t)$-Furstenberg set $E\subset\mathbb{Q}_p^2$ satisfies $$ \dim_H E\ge \min\left\{s+t,\frac{3s+t}{2},s+1\right\}. $$ This matches the sharp lower bound in the Euclidean plane. We also derive two related consequences: a $p$-adic projection theorem for the maps $\pi_\theta(x,y)=x+\theta y$, together with the corresponding exceptional set estimate giving a $p$-adic analogue of Oberlin's projection question; and a discretized fractal sum-product estimate over $\mathbb{Q}_p$, showing that sufficiently non-concentrated subsets of $\mathbb{Z}_p^\times$ cannot have both small sum set and small product set. The proof follows the projection-theoretic and multiscale machinery developed in the Euclidean works of Orponen-Shmerkin (arXiv:2301.10199) and Ren-Wang (arXiv:2308.08819). The main task is to rebuild this machinery in the non-archimedean setting, and along the way we develop several new $p$-adic inputs needed to overcome the ultrametric features of the problem.

We compute the dimension of the center of the Temperley–Lieb algebra $\operatorname{TL_n}(\delta)$ over a field of characteristic zero for every nonzero value of the parameter $\delta$. The proof uses the cellular filtration by cup number, together with known facts about the representation theory of the Temperley–Lieb algebra, especially the structure of its standard modules and their radicals. Dilation and compression maps compare the induced graded pieces of the center at levels $n$ and $n-2$, giving an upper bound of one for each such piece. A deformation argument gives the matching lower bound, and hence $ \dim Z(\operatorname{TL}_n(\delta))=1+\Bigl\lfloor \frac{n}{2}\Bigr\rfloor$. We also prove that every central element is fixed by the canonical anti-automorphism and by the natural diagram-reflection automorphism. Finally, we give a congruence criterion for the trivial-radical case and record a Gram-matrix computation for leading terms.

Few-shot industrial defect detection remains difficult for standard supervised detectors, which achieve poor performance on boundary-dominated industrial defects. This paper proposes rough path signature-guided geometry augmentation (RPS-GA), a geometry-aware approach in which Canny edge contours are treated as ordered planar paths whose truncated second-order signature responses, especially the antisymmetric Lévy-area term, are aggregated into a spatial map that highlights boundary-related structure through two fusion operators, SIG-AUG and SGAA. The approach is evaluated on NEU-DET and PCB-Defect under a few-shot protocol with 5, 10, 20, or 50 labeled images per class, using an unmodified YOLOv8n detector throughout. Compared with the baseline, RPS-GA delivers large gains when supervision is limited, although the margin shrinks as more labels become available. On NEU-DET, SIG-AUG raises 10-shot mAP@0.5 from 0.341 to 0.583, whereas on PCB-Defect, SGAA improves 10-shot mAP@0.5 from 0.086 to 0.299 and yields usable detection at 5-shot where the baseline fails entirely. These trends are confirmed by multi-seed evaluation across independent random partitions. Overall, the results indicate that second-order path-signature geometry offers a practical way to strengthen few-shot industrial defect detection without meta-learning or detector redesign.

Let $\mathcal{C}$ be an $[n,k,d]$ linear code over $\F_{q^2}$ with parity-check matrix $H$. The rank $r=\Rank(HH^{\dagger})$ measures how far the Hermitian dual $\mathcal{C}^{\perp_h}$ is from being Hermitian self-orthogonal, and it is the least number of coordinates that must be adjoined to $\mathcal{C}^{\perp_h}$ to make it Hermitian self-orthogonal; equivalently, the Hermitian hull $\mathcal{C}\cap\mathcal{C}^{\perp_h}$ has dimension $(n-k)-r$. In this paper, we develop a structure theory of the Hermitian hull of a cyclic code and apply it to the construction of quantum stabilizer codes. For a cyclic code with defining set $T\subseteq\mathbb{Z}_n$, we show that $r$ is additive over the orbits of the involution $\tau(\nu)=-q\nu$ on the $q^2$-cyclotomic cosets and satisfies a parity constraint. A parameter dictionary exhibits $r$ as a gauge trading one-for-one against the code length, and a complementation duality shows that the complement of $T$ realises the logical-qubit count of the quantum code. The extreme values of $r$ are settled by closed subset-sum criteria, and the $\tau$-invariant defining sets are shown to be the unique locus on which the Hermitian dual is again cyclic. On the distance side, we determine the Hermitian dual distance of an $r$-fold extension of an arbitrary linear code exactly, and we prove that the admissible extension vectors lie in the image of $HH^{\dagger}$, which reduces the search space from $q^{2(n-k)}$ to $q^{2r}$. Applied to cyclic, quasi-cyclic and BCH codes, the construction improves $23$ best-known parameters in Grassl's code tables, one of them provably optimal.

Two families \(\mathcal{A}\subseteq\binom{[n]}{k}\) and \(\mathcal{B}\subseteq\binom{[n]}{\ell}\) are said to be nontrivial cross-\(t\)-intersecting if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\), and $|\bigcap_{A\in \mathcal{A}\cup \mathcal{B}}A|<t$. In this paper, we determine the upper bound on \(|\mathcal{A}||\mathcal{B}|\) of two nontrivial cross-\(2\)-intersecting families \(\mathcal{A}\subseteq\binom{[n]}{k}\) and \(\mathcal{B}\subseteq\binom{[n]}{\ell}\) for any positive integers $n,k,\ell$ with \(k\geq \ell \geq 3\) and \(n \geq 3(k-1)\). Moreover, we characterize the extremal families attaining this bound. This settles the last unsolved case of a recent result by He, Li, Wu and Zhang (J. Combin. Theory Ser. A, 217 (2026) 106095).

The character of a connected $\mathfrak{sl}_n$-crystal is a Schur polynomial; the crystal can be further decomposed into quasicrystals, whose characters are the Gessel quasisymmetric functions. Crystal skeletons are obtained by contracting quasicrystals within crystal graphs. They generalize dual equivalence graphs, and can be used to prove the Schur expansion of a symmetric function when the quasisymmetric expansion is known. In this paper, we show that the crystal skeleton can be tiled further into components which we call quasicrystal skeletons, whose characters are Young quasisymmetric Schur functions. We characterize which edges in the crystal skeleton move between quasicrystal skeleton components. Contracting the quasicrystal skeleton components yields Bruhat order. We illustrate how these tools can be applied to symmetric functions by analyzing the Stanley symmetric functions.

We give a reachable-state operator formulation of one-to-one deferred acceptance with strict, possibly incomplete preference lists. A local update operator is defined for each active proposer, while a scheduler selects which local update is applied. On the states reachable from the canonical empty initial state, three invariants are immediate: the set of proposed edges grows strictly, every proposer visits each acceptable receiver at most once, and every receiver holds its most-preferred proposal received so far. These invariants yield termination after at most |E| proposals and stability of every terminal reachable state. The classical rejection lemma then gives proposer optimality and schedule independence. We separate these trajectory statements from two logically different results: the distributive-lattice structure of the full stable-matching set and one-sided strategy-proofness. A diagnostic table records which proof obligation is lost when a model changes the bipartition, ordinal comparisons, proposal irreversibility, or receiver choice rule. The paper makes no new complexity claim; its purpose is a precise operator-level account of the classical proof architecture and of the limits of that architecture.

In this paper, we establish $\Omega$-results for the logarithmic derivative of the Riemann zeta function on vertical homogeneous progressions on the 1-line and in the critical strip. Compared with Yang's work in 2023, our results show that the discrete case has a similar order of magnitude to the continuous case.

In this paper, we establish an Omega result for partial sums of the Riemann zeta function on the \(1\)-line. Our work relies on the long resonance method and an effective asymptotic formula for smooth numbers. For sufficiently long partial sums, our result approaches the work obtained by Aistleitner, Mahatab and Munsch in 2019.

The multiplicative cascade is a well-known method for generating random fields that are both genuinely non-Gaussian and scale invariant. Its one-point and scaling statistics, namely the multifractal spectra $\tau(q)$, $D_q$, and $f(\alpha)$, are known exactly; however, its geometric and topological measures are not. In this work, we show that the topological structures of a conservative (microcanonical) random permutation cascade can be precisely predicted. We compute the digital (cubical) Euler characteristic, or genus, of the level-$j$ excursion set of the two-dimensional cascade in closed form, using a transfer operator on the cascade's $b$-ary tree closed by its ultrametric structure and a conservative without-replacement sibling split. The result requires no Monte Carlo and matches simulated realizations with a maximum residual of $\sim\!10^{-4}$ in the Euler density, a controlled discretization artifact of the atomic measure. We tie the genus scaling analytically to the multifractal spectrum, and show that for a geometric ladder of weights the genus is exactly self-similar under the multifractal-width dial. The construction uses the tree-sum cumulant machinery of Greiner et al. (Phys. Rev. E 58, 554, 1998), specialized to topological functionals; the conservative split carries the microcanonical fingerprint $\mathrm{Cov}(\ln w_a,\ln w_b)/\mathrm{Var}(\ln w)=-1/(n-1)$ that distinguishes it from canonical and lognormal cascades. The cascade is the multiplicative counterpart of the Rayleigh–Lévy flight and a controlled benchmark for the morphology of strongly non-Gaussian fields.

Safety verification is a fundamental problem in control theory. Barrier certificates (BCs) provide a powerful formal mechanism, yet deriving BCs is computationally intensive. This paper introduces a generative framework that leverages large language models (LLMs) to synthesize BCs through reasoning. Based on the classical Sum-of-Squares (SOS) approach, we train a domain-specific LLM capable of generating high-quality BC candidates for nonlinear systems. Then, the LLM-generated BCs transform the intractable Bilinear Matrix Inequality (BMI) solving problems into convex Linear Matrix Inequality (LMI) feasibility test, significantly improving efficiency while preserving correctness. Experimental results show that our generative method achieves several orders of magnitude speedup over traditional numerical BC approaches and, perhaps surprisingly, surpasses the state-of-the-art dedicated neural BC model. These findings mark a substantive step toward integrating generative AI with formal safety verification for dynamical systems.

Conway's 99-graph problem asks whether a strongly regular graph with parameters srg(99,14,1,2) exists. We develop two complementary lines of attack. Fixing one vertex, the conditions lambda=1 and mu=2 force its neighbourhood to be a perfect matching and determine every edge between that neighbourhood and the remaining vertices. For (99,14,1,2), the unresolved part is therefore a constrained 12-regular graph on 84 vertices. We encode this reduction in CP-SAT and validate it by recovering the unique srg(9,4,1,2). We also prove by exhaustive enumeration that no circulant graph on Z/99 satisfies more than 68.0% of the CAISc constraints, and we give a validated orbit formulation for prescribed automorphisms. We then study the partial-score search problem. Fourteen human-designed search configurations reached at most 69.43%. Separately, we supplied the scoring function to an evolutionary program-search system. It produced a degree-preserving 4-vertex-switch tabu search whose best verified artifact scores 70.73%. The generated move differs from those used in our own searches and crosses a plateau that was stable under them. A tabu search restricted to graphs with a prescribed symmetry group, first Z/33 and then its subgroup of order 11, reaches a 14-regular graph scoring 76.67%. These results do not resolve the existence problem, but they reduce the exact search space and improve the best verified partial construction found in our experiments.

Rashidi, Shajareh Poursalavati, and Tavakkoli [J. Algebra Comb. Discrete Struct. Appl. 7 (2020), no.2, 183-193, Theorem 3.6] claim $Z(P(n,3)) = 8$ for all $n \geq 12$, but this fails at $n = 12$, where $Z(P(12,3)) = 7$. We give an explicit $7$-vertex zero forcing set for $P(12,3)$ and its forcing sequence, and verify by exhaustive search that no $6$-vertex set forces $P(12,3)$. We prove $Z(P(n,3)) \le 8$ for $n \ge 9$ using an explicit forcing set and rotational symmetry. Exhaustive search yields $Z(P(n,3))$ for $7 \le n \le 20$ and identifies the gap in the published proof: its case analysis fails to exclude $7$-vertex sets. We conjecture $Z(P(n,3)) = 8$ for $n \ge 13$, requiring a matching lower bound for all large $n$.

Let \(\mathscr{G}_n\) be the family of simple graphs obtained from \(K_n\) by deleting at most five edges. For a fixed integer \(1\leq k\leq n\), let \(\Phi_k(L(G),x)\) denote the immanantal polynomial of the Laplacian matrix associated with the hook partition \((k,1^{n-k})\). We prove that, for \(n>7\) and \(n\neq 2k-1\), every graph in \(\mathscr{G}_n\) is determined by \(\Phi_k(L(G),x)\) among all simple graphs. We prove that, for \(n>7\) and \(n\neq 2k-1\), every graph in \(\mathscr{G}_n\) is determined by \(\Phi_k(L(G),x)\) among all simple graphs. The proof recovers the order, size, and degree-square sum from the first coefficients, and then separates the remaining candidates by explicit third- and fourth-coefficient comparisons based on the finite classification of complements with at most five edges. The case \(n=2k-1\) is left open because the binomial differences used in these comparisons vanish.

We window a delayed periodic square wave with a decaying signum function and evaluate the product at zero frequency in both the time and frequency domains. The window renders every series that appears absolutely convergent, so the argument requires only elementary integrals and a limit. We obtain new unified derivations for the Mittag-Leffler partial-fraction expansion for the hyperbolic tangent, the Fourier expansion of the first Euler polynomial E1, and the Basel sum, zeta(2). The same calculation, followed by termwise integration, yields the Dirichlet beta value beta(3) = pi^3/32, and the values at s = 3 of two other Dirichlet L-functions. Termwise integration of this sine series produces zeta(4), and an induction continues the process to zeta(2k) for every k greater than or equal to 1.

Let $f(x) = \Gamma(x)^2/(2\Gamma(2x))$ and set $\lambda_\alpha = 4\sin^2\alpha$ for $0 < \alpha < \pi/2$. We establish an elementary parameter identity for a weighted translate of $f$ and derive an explicit formula, valid at every derivative order, for the associated weighted sums $\sum_{k\ge1} \lambda_\alpha^{k-1} f^{(r)}(k)$. The coefficients satisfy an effective recurrence in ordinary zeta values. The unique unweighted specialization $\alpha = \pi/6$ proves Conjecture 4.1 of Zhi-Wei Sun; at the fourth order a depth-two value $\mathrm{Gl}_{4,1}(\pi/3)$ occurs. The construction complements general cyclotomic-multiple-zeta methods for inverse-binomial harmonic sums by supplying a continuous master identity, with concrete specializations at $\alpha = \pi/4$ and $\alpha = \pi/3$.

The notion of circuit refers to a cyclic oriented influence between the elements of a dynamical system. There are two classes of circuit: positive and negative. R. Thomas conjectured that a necessary condition of multi stationarity is the existence of positive circuits. In this paper we use dynamical system tools and planar analysis to find conditions for which the conjecture holds for planar systems.

We study the relationship between nowhere-zero three-bit flows, labeled triangles, and cycle double covers of bridgeless multigraphs. Using the flow theorems of Seymour and Tutte, we realize the three flow values at each vertex as the side differences of a triangle whose sides carry two-element subsets of $\F_2^3$. We express agreement of these local labels across graph edges as a binary system and prove its solvability by an inconsistency certificate, a local tester-parity identity, and a global double count. The resulting compatible labels trace cycle components in which every edge occurs exactly twice, proving that every finite bridgeless multigraph has a cycle double cover, proving the cycle double cover conjecture.

We propose a log-Gaussian scale-space limiter for hybrid continuum–ballistic gas dynamics. The method defines complementary continuum and ballistic weights as Gaussian cumulative probabilities in logarithmic Knudsen-number space and blends Navier–Stokes–Fourier and half-range Maxwellian kinetic fluxes. The method is formulated as a lightweight hybrid closure intended for future implementation in finite-volume, discrete Boltzmann, or gas-kinetic solvers. Reduced one-dimensional closure/profile comparisons against DVM/BGK Fourier and Couette data show that log-Gaussian weighting of NSF and jump/slip-corrected branches improves the tested macroscopic profiles relative to NSF. The same six DVM/BGK profiles are used both as reference profiles and to calibrate K0 and sigma; therefore, the approximately 40 percent reduction in combined mean profile error is an in-sample calibration result for the reduced profile model rather than independent validation or numerical validation of the proposed finite-volume face flux. Additional diagnostics assess non-equilibrium moments, internal parameter robustness, and shock-layer activation.

We establish a unified framework for exact dimensional reductions in network dynamical systems using Lie-Scheffers theory. For network dynamical systems with mean-field Lie-Scheffers structure, we prove that networks of $n$ nodes with local dimension $d$ can be exactly reduced from $ n d $ dimensions to a fixed macroscopic system of dimension $ m d $, where $m$ is the number of fundamental solutions required by the nodal dynamics. Crucially, the superposition principle resulting from the Lie-algebraic structure allows the mean-field coupling to be expressed explicitly in terms of the macroscopic variables, yielding a closed self-consistent system independent of network size. This reduction collapses the high-dimensional network flow onto invariant manifolds parameterized by $ \gamma = d(n-m) $ independent constants of motion. Our framework rigorously explains known reductions and provides a systematic method to discover new ones. We illustrate the theory with ensembles of Riccati equations (encompassing the Kuramoto model and Theta neuron model), quasi-linear ODEs, and generalized Bernoulli equations, explicitly deriving the macroscopic flows and conserved quantities for each case.

We show that the Benjamini–Hochberg procedure can fail to control the false discovery rate (FDR) at its nominal level for correlated two-sided Gaussian $p$-values. We construct a factor model for which, at level $\alpha=0.01$, a rigorous interval-arithmetic certificate proves $FDR>0.0104$ for all sufficiently large numbers of hypotheses. This disproves a conjecture widely believed to be true for twenty years. Monte Carlo experiments are consistent with the theoretical result. The proof was obtained by GPT-5.6 Pro and carefully checked by the author.

Homotopy type theory (HoTT) is a type theory that allows for synthetic reasoning about $\infty$-groupoids. Several proof assistants (such as Rocq and Agda) implement variants of HoTT. Directed type theory is a type theory for synthetic reasoning about $\infty$-categories, where morphisms (or paths) of dimension 1 are not necessarily invertible. Among the proposals for directed type theory, the most developed is Riehl and Shulman's simplicial type theory (RSTT), based on simplicial shapes such as directed intervals and triangles. We present Rzk, a proof assistant implementing (a refinement of) RSTT for synthetic reasoning about $\infty$-categories. Specifically, the type theory implemented by Rzk is a computational variant of RSTT adjusted to make type checking practical. We define a translation from RSTT to Rzk and prove that it is sensible: every RSTT proof translates to an Rzk proof (faithfulness), and Rzk proves nothing new about RSTT types (conservativity). We also give a tutorial introduction to proving in Rzk, and describe its implementation, including the type-checking algorithm and the automated prover for the logic of shapes.

We revisit the classical topic in dynamic optimization: sequential vs simultaneous methods for solving DAE-constrained optimization problems, with a particular focus on how graphics processing unit (GPU) computing changes their effectiveness. Sequential methods offer key advantages through adaptive time stepping at the differential-algebraic equation (DAE) solver level, which is especially effective for handling stiff systems. However, long-time-horizon simulations remain a computational bottleneck, as time integration is inherently sequential and limits parallelization within the optimization algorithm. In contrast, simultaneous approaches are well-suited for parallel computing. They address these limitations by exploiting the highly repetitive structure of discretized DAE systems at the function evaluation level and leveraging sparse linear algebra routines that enable elimination tree-level parallelism. Although simultaneous methods typically lack adaptive time stepping, this limitation can often be mitigated by choosing a sufficiently fine initial mesh or iteratively adjusting mesh coarseness in an outer loop. In this work, we revisit the capabilities of the simultaneous approach in a GPU computing environment and assess its performance against a sequential method baseline. We employ a simultaneous approach based on orthogonal collocation within an open-source modeling framework and apply it to parameter estimation benchmarks from systems biology. We evaluate both GPU and CPU solvers on the simultaneous formulation. Our results show that, although less reliable, the simultaneous approach achieves up to 5.4x speedup compared to the sequential baseline among the largest instances where both methods solve successfully. The advantage of the simultaneous method with respect to problem size is more pronounced on GPUs than on CPUs.

Timely distribution of medical items (e.g., whole blood and vaccines) across regional healthcare networks often requires high-volume delivery operations from central facilities (e.g., blood banks) to intermediate regional hospitals, followed by rapid last-mile deliveries to points of injury. Traditional ground-based delivery systems often suffer from limited responsiveness (e.g., traffic congestion) and operational inefficiencies (e.g., blood waste). Leveraging aerial-drone-based delivery systems offers a promising solution for the fast and efficient delivery of time-sensitive medical items across regional healthcare networks. Therefore, we study a hierarchical hub-and-spoke drone-based network for delivering time-sensitive medical items with distinct release and due times to fixed and mobile delivery destinations. We consider a heterogeneous fleet of drones with distinct characteristics (e.g., cost, battery capacity, and speed) and different multi-trip delivery modes. We propose an efficient mixed-integer programming model for location/allocation of mobile delivery destinations, as well as routing and scheduling drones to minimize the total investment and operational costs of the drone delivery network while maintaining the delivery due times. We develop a customized exact solution method integrating problem-specific reformulations and dynamic cutting planes, as well as a fast heuristic algorithm by leveraging a simplified problem variant. Results based on a real-life case study of whole blood delivery data from Pendleton, Oregon, United States, and actual drone flight test data demonstrate that our exact and heuristic solution methods are 31 and 3,375 times faster, respectively, than the Gurobi solver. Results also show that allowing drones to perform multiple trips is 142.8% more cost-efficient than single trips.

Given an ideal $\mathcal{I}$ on $\omega$, a subgroup $H$ of the unit circle $\mathbb{T}$ is said to be $\mathcal{I}$-characterized if there exists an integer sequence $a=(a_n: n \in \omega)$ such that $$ H=\mathsf{H}_{a}(\mathcal{I}):= \left\{x\in\mathbb{T}:\mathcal I\text{-}\lim_{n\to \infty} a_nx=0\right\}. $$ We also consider the corresponding $\mathcal{I}^\star$-version. We provide upper bounds for the topological complexities of those subgroups in terms of the complexity of $\mathcal{I}$. Moreover, we prove that Rudin–Keisler and Rudin–Blass reductions between ideals induce inclusions between the corresponding families of characterized subgroups. As a consequence, every characterized subgroup, and in particular every countable subgroup of $\mathbb{T}$, is $\mathcal{I}$-characterized for every meager ideal $\mathcal{I}$. We also show that if the image of $(a_n: n \in \omega)$ contains arbitrarily large intervals, then every subgroup of $\mathbb{T}$ can be written as $\mathsf{H}_{a}(\mathcal{J})$ for some ideal $\mathcal{J}=\mathcal{J}_{H,a}$. We analyze the descriptive complexity and $P$-properties of these ideals. Finally, we study when the equality $\mathsf H_{a}(\mathcal{I})=\mathbb{T}$ forces $\mathrm{supp}(a)\in\mathcal{I}$. We prove this for a class of ideals satisfying a Katetov-type condition involving $\mathcal{ED}$, including nowhere tall ideals as well as the ideals $\mathsf{nwd}$ and $\mathsf{null}$. We also obtain non-inclusion results between families of $\mathcal{I}$-characterized subgroups: for instance, we show that if the ideal $\mathcal{I}$ is tall and translation invariant then the subgroup $\mathsf{H}_{(2^n)}(\mathcal{I})$ cannot be characterized. We use our results to answer several open problems posed in the literature.

We prove new necessary and sufficient conditions for uniform asymptotic stability under arbitrary switching of two-dimensional switched homogeneous systems with a finite number of subsystems using a worst-case switching analysis. The novelty of our approach is in its explicit nature, which allows us to then study in detail the codimension-one bifurcations of stability of the origin in switched linear systems and further conclude new local and global stability results for certain classes of nonlinear switched systems. In particular, we formulate an analogue of Lyapunov's indirect method for $\mathcal{C}^{1}$ switched nonlinear systems and derive a new method for determining the existence of a bounded basin of attraction for a class of switched nonlinear systems.

Many recently proposed Calderbank-Shor-Steane (CSS) quantum low-density parity-check (QLDPC) codes have sparse decoding graphs, enabling syndrome-based belief propagation (BP) decoding at low complexity. Their construction, however, often results in properties that impair BP performance, such as short cycles and degeneracy. In this work, we propose a general framework for introducing auxiliary variable nodes (AVNs) and auxiliary check nodes (ACNs) into the decoding graph of CSS codes, compatible with the standard stabilizer measurement framework. This provides an additional degree of freedom in the design of the decoding graph itself and can be used to tackle the aforementioned shortcomings. We show that recently proposed techniques, 4-cycle removal and subcode ensemble decoding, can be interpreted as instances of this framework. For 4-cycle removal, we find that the gains depend strongly on the BP iteration count and check-node message scaling. Building on this framework, we further propose a graph-derived subcode ensemble decoder and demonstrate under circuit-level noise that it substantially reduces the per-round logical error rate compared with BP on the corresponding 4-cycle-free decoding graph.

Given $p$-maps $f_1, \cdots, f_p : X \to M,$ $p \geq 2,$ from an arbitrary topological space to an orientable closed connected $m$-manifold, in this paper we define a graded homomorphism $\Lambda_{f_1 \cdots f_p}: H(X) \to H(M^{p-1})$ of degree $-m(p-1)$ called by Lefschetz homomorphism. If the Lefschetz homomorphism is nontrivial then there is a point $x \in X$ such that $f_1(x) = \cdots = f_p(x).$ The Lefschetz homomorphism $\Lambda_{f_1 \cdots f_p}$ can be represented as a Knill-like trace.

For integers $m\ge2$ and $r\ge1$, put $d=m+r-1$ and define the shifted-binomial coordinates $B_{m,r,k}$ by \[ \binom{mn}{d} = \sum_{k=1}^{d+1}B_{m,r,k}\binom{n+k-1}{d}. \] The main purpose of this paper is to identify these coordinates with a specific residue-class decimation of a multinomial coefficient sequence. We prove \[ B_{m,r,k} = [x^{mk-1}](1+x+\cdots+x^{m-1})^{m+r}. \] This identity converts the alternating finite-difference formula for the coordinates into a positive coefficient formula, and yields their exact support. We then prove that the nonzero coefficient vector is palindromic if and only if $r\equiv2\pmod m$. This symmetry criterion is derived here directly within the multinomial-decimation framework. The principal arithmetic results are an unconditional divisibility law and an exact formula for the greatest common divisor of each nonzero row. We prove \[ \frac{m}{\gcd(m,r-1)}\mid B_{m,r,k}, \] and, more precisely, determine \[ \gcd_{1\le k\le K_{m,r}} B_{m,r,k} \] as an explicit product of prime powers determined by the $p$-adic valuations of $m$ and $m+r-1$. Finally, a roots-of-unity filter gives the exact row sum \[ \sum_k B_{m,r,k}=m^{m+r-1}. \] Thus the shifted-binomial coordinates form a positive, arithmetically structured decimation of an $m$-nomial coefficient sequence.

The purpose of this note is to elaborate on the apparent connection between Wronskians and Hessians. More generally, to a given subspace of homogeneous bivariate forms over the complex numbers, we associate two determinantal polynomials called the $W$-polynomial and the $\hat{W}$-polynomial. We give expansion and factorization formulas for these polynomials, and study their behavior under change of coordinates and duality. As an application, we give another proof of Iarrobino's theorem on the strong Lefschetz property for standard graded Artinian Gorenstein algebras in codimension two.