Papers

This paper presents a real-time control-constrained Differential Dynamic Programming (DDP) framework for underactuated legged robots. To address the limitation of classical DDP in handling control constraints, we propose an Accelerated Projected Gradient (APG)-based control-constrained DDP (ABC-DDP), which efficiently computes constrained solutions and identifies active sets without repeated Karush-Kuhn-Tucker (KKT) inversions. A virtual constraint is introduced to integrate control constraints within a feasibility-driven multiple-shooting framework, enabling stable optimization even from dynamically infeasible initializations. The proposed method supports real-time model predictive control (MPC) with short horizons under strong underactuation. Simulation results demonstrate static two-leg standing under external disturbances, along with diverse dynamic motions including slow catwalk, upright walking, and high-speed running within a unified MPC framework. To the best of our knowledge, this is the first demonstration of static two-leg standing of a quadruped robot achieved using real-time finite-horizon MPC.

Let $\mathcal{C}_{[4,r]}$ be the family of cycles $\{C_4, \dots, C_r\}$. A graph $G$ is said to be $\mathcal{C}_{[4,r]}$-saturated if $G$ does not contain a copy of cycle $C_i$ for $4\le i\le r$, but the addition of any edge $e\notin E(G)$ creates at least one copy of $C_i$ for $4\le i\le r.$ The saturation number $sat(n, \mathcal{C}_{[4,r]})$ is the minimum number of edges in an $n$-vertex $\mathcal{C}_{[4,r]}$-saturated graph. In 2025, Ma determined that $sat(n, \mathcal{C}_{[4,5]})=\lceil \frac{5n}{4} - \frac{3}{2} \rceil$, and conjectured that for any $r \ge 5$, $sat(n, \mathcal{C}_{[4,r]}) = \lceil\frac{5n}{4} - \frac{3}{2} \rceil$ holds for large $n$. In this paper we prove that $sat(n, \mathcal{C}_{[4,r]}) \le \lceil\frac{5n}{4} - \frac{r+1}{4}\rceil$ for $n \ge r+1$, which disproves Ma's conjecture for $r\ge 6.$ For $r=6,$ we determine that $sat(n, \mathcal{C}_{[4,6]})=\lceil\frac{5n}{4}-\frac{7}{4}\rceil.$ Keywords: Saturation graphs; Saturation number; Cycles; Edge minimization

Combining classical group theory and partial differential equation theory, this paper investigates the symmetry group $\operatorname{Sym}(u)$ of the unique solution $u$ to the second-order linear elliptic boundary value problem on an $n$-dimensional bounded domain $\Omega$ $-\sum_{i,j=1}^{n} a_{ij}(x)u_{x_ix_j} + \sum_{i=1}^{n} b_i(x)u_{x_i} + c(x)u = f(x), x\in \Omega$, $u(x) = h(x), x\in \partial \Omega$, The following symmetry groups are defined and characterized respectively: the symmetry group $\operatorname{Sym}(A)$ of the second-order coefficient matrix function $A(x)=(a_{ij}(x))_{n\times n}$; the symmetry group $\operatorname{Sym}(b)$ of the first-order coefficient column vector function $b(x)=(b_{1}(x),b_{2}(x),\cdots,b_{n}(x))^{T}$; the symmetry group $\operatorname{Sym}(c)$ of the zero-order coefficient function $c(x)$; the symmetry group $\operatorname{Sym}(f)$ of the internal source function $f(x)$; and the symmetry group $\operatorname{Sym}(h)$ of the boundary source function $h(x)$. This paper rigorously proves that the common symmetry group $\operatorname{Sym}(A)\cap\operatorname{Sym}(b) \cap\operatorname{Sym}(c) \cap\operatorname{Sym}(f) \cap\operatorname{Sym}(h)$ is a subgroup of $\operatorname{Sym}(u)$. In addition, if the common symmetry group contains several mirror symmetry elements, the original second-order linear elliptic boundary value problem on the entire domain $\Omega$ can be reduced to the corresponding boundary value problem on a certain subdomain. It is strictly proven in this paper that the new boundary condition imposed on the boundary of the subdomain is the homogeneous generalized Neumann boundary condition. The linear finite element method is used to numerically solve the second-order linear elliptic boundary value problem on the subdomain, thereby achieving domain reduction and significantly reducing the computational cost.

The codegree Turán density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the problem of what 3-graphs $F$ satisfy $\pi_{\mathrm{co}}(F) = 0$. They introduced layered $3$-graphs and conjectured that a $3$-graph has zero codegree Turán density if and only if it is layered and has zero uniform Turán density. For $k\ge 3$, a $k$-graph is called layered if its vertices can be labelled so that every edge has a unique maximum label and two edges with the same maximum label have the same label multiset. In this paper, we show that every non-layered $k$-graph $F$ on $m$ vertices satisfies \[ \pi_{\mathrm{co}}(F)\ge q_{k,m}^{-q_{k,m}}>0, \quad \text{where}\quad q_{k,m}=\frac{(k-1)^{m+1}-1}{k-2}, \] which implies any $k$-graph with zero $\ell$-degree Turán density is layered, and the case $k=3$ confirms the conjecture of Ding, Lamaison, Liu, Wang, and Yang.

We present a simple geometric proof of Kepler's first law. The core of the argument rests on a kinematic invariant that we call the velocity projection ratio - the ratio between the radial projection of the velocity and its projection onto a fixed direction parallel to the periapsis line. We show that the constancy of this ratio is an immediate consequence of Newton's laws, and that it leads directly to the focus-directrix property of the orbit. By establishing a direct link between planetary dynamics and the classical geometric definition of conics, our proof reveals the conic nature of the orbit as an inevitable consequence of the inverse-square law.

Encoding quantum information with low circuit overhead is a fundamental challenge in fault-tolerant quantum computation. Random circuits provide a natural mechanism for rapidly spreading logical information through simple gates applied in parallel. Brown and Fawzi showed that random Clifford circuits on two-qubit Clifford gates provide such encoders that achieve the quantum Gilbert-Varshamov rate-distance tradeoff with depth $O(\log^3 n)$. We show that the same asymptotic tradeoff is attained in optimal $O(\log n)$ depth under a gate distribution with a more restricted support. For every fixed $\delta>0$ and sufficiently large $n$, if $\frac kn < 1 - H(\frac{d}{n}) - \frac{d}{n}\log_2 3 - \delta$, we can construct random circuits of depth $O(\log n)$ which define, with high probability, an $[n,k]$ stabilizer code of distance at least $d+1$, which matches the $\Omega(\log n)$ light-cone lower bound for linear distance encoders. Our ensemble employs a random matching circuit architecture consisting of $T$ independent permutation-invariant layers. In each layer, the qubits are paired up by a uniformly random perfect matching, and a random independent two-qubit Clifford gate is applied to each pair. The gate distribution need not be uniform over, or even have full support on, the two-qubit Clifford group; rather, we allow for very general distributions on Clifford gates satisfying three regularity conditions. In particular, the construction can be implemented using $n/2$ CNOT gates on randomly matched pairs in each layer, with parallel one-qubit Clifford twirls. These regularity conditions allow us to reduce the second-moment dynamics of our random circuits to a reversible Markov chain on binary support strings. We establish logarithmic hitting-time bounds for this Markov chain and comparisons of its stationary distribution to prove the coding properties of the circuits.

It has been a long standing open problem to find generators of the automorphism group of the complex Fermat quartic K3 surface; in fact, even an explicit number of generators has not been known before. In this paper, we provide the first solution to this problem, presenting $16$ geometric generators of finite order for this group explicitly.

For a divisor on $\overline{M}_{0,n}$, we introduce the polytope of its effective boundary expressions. We establish structural properties of these polytopes under the forgetful maps of $\overline{M}_{0,n}$ forgetting marked points, and give equivalent graph-theoretic descriptions. We compute these polytopes for several families of divisors. For psi-classes and their pullbacks by forgetful maps, we show that the polytopes are unimodular simplices. For the log-canonical class and its modifications by psi-classes, we prove that the nonnegative parts of the corresponding polytopes recover spanning forest polytopes and the subtour elimination (Held–Karp relaxation) polytope of the symmetric traveling salesman problem. As an application, we obtain a Minkowski-like decomposition of the subtour elimination polytope into simplices. Finally, for symmetric level-one $\mathfrak{sl}_p$ conformal block divisors, we show that the defining inequalities are local Turán bounds and the $0/1$-points are balanced Turán graphs. Moreover, for $p=2$ and $p=n/2$, these polytopes recover the perfect matching and fractional perfect matching polytopes.

In a seminal paper, Lewis and Zagier constructed variety of functions satisfying the three-term functional equations. In this article, we consider the first example among them and establish that the function is a Hecke eigen form with respect to the Hecke operators, which acts on periods. We then utilize this result to determine the action of the aforementioned operators on the derivative of the Higher order Herglotz-Zagier function. The action leads to a family of multi-term functional equations satisfied by the function.

We study $p$-adic solubility of generalized Fermat equations $Ax^\ell + By^m + Cz^n = 0$ for positive integers $\ell,m,n$. For all but finitely many primes $p$, the probability of having a $p$-adic solution is described by a rational function in $p$ depending only on $\gcd(p-1,\ell,m)$, $\gcd(p-1,\ell,n)$, and $\gcd(p-1,m,n)$. When $\ell,m,n$ are pairwise coprime, we deduce that the proportion of these equations which are everywhere locally soluble is positive, given by a product of these local probabilities; when $\ell,m,n$ are not pairwise coprime, the proportion is 0%. We then give several detailed examples demonstrating the explicit nature of the results.

Large antenna arrays allow wireless systems to serve more users and achieve higher data rates, but they also make channel feedback expensive: the receiving device must repeatedly report a large complex-valued channel matrix to the base station. Most neural compressors treat this matrix like an image and replace it with a fixed-length code that only a matched neural decoder can interpret. The message therefore does not adapt to channel complexity, and changing the antenna count typically requires retraining. We ask whether a device can instead report only the few dominant propagation paths underlying each channel. We introduce the Gramian Chebyshev Neural Operator (GCNO), a physics-based, variable-rate compressor that identifies a sample-dependent set of path directions. GCNO uses receive-transmit channel structure to locate paths, a first-order Taylor correction to refine directions that fall between grid points, and least squares to recover their complex strengths. It is trained without path labels, and the base station reconstructs the channel analytically from the transmitted path tuples rather than through a learned decoder. Across three ray-traced environments, GCNO achieves better reconstruction accuracy at the same payload - or lower payload at the same accuracy - than neural feedback baselines, and transfers to unseen antenna counts without retraining.

We study Sobolev approximation on bounded domains by linearized shallow neural networks whose inner parameters are prescribed independently of the target function. Our main step is a one-dimensional construction for analytic activations. We prove that quasi-Chebyshev parameter sets with univariate resolution $m$ generate fixed feature spaces attaining the sharp $H^r$-to-$H^s$ approximation order $m^{-(r-s)}$ for a class of analytic activations satisfying a quantitative non-cancellation condition on their Taylor coefficients. Combining this result with the ridge-function lifting theorem in [SIAM J. Math. Anal. 30 (1998), pp. 155-189] and its extension to arbitrary quasi-uniform direction sets established in this work, we construct tensor-product-type parameter sets that attain the sharp rate $$\|f-f_n\|_{L^2(\Omega)}\lesssim n^{-\frac rd}\|f\|_{H^r(\Omega)},\quad f\in H^r(\Omega)$$ for all $r>0$. In contrast to the finite-difference construction in [Neural Comput. 8 (1996), pp. 164-177], whose explicit admissibility condition may require an extremely small parameter scale, the proposed parameter sets remain distributed over fixed intervals and are therefore more amenable to practical computation.

Let $M$ be a simple matroid and let $B(M)$ be the graded Möbius algebra of its lattice of flats. The ordered-basis weights of flats define an inner product for which the adjoints of atom multiplication become ordinary coordinate derivatives under the basis-polynomial realization. From this, we construct a canonical global lowering operator $D_\beta$ which acts as ordinary differentiation on a canonical “radial” copy of a truncated polynomial algebra. Allowing both $D_\beta$ and the coordinate derivatives to act produces a graded cyclic module with Hilbert series \[ H_{\beta,M}(q)=\sum_{k=0}^r h_k^\beta(M)q^k. \] We give examples of matroids with the same Derksen $\mathcal G$-invariant and the same classical apolar Hilbert series but different $H_\beta$. Hence $H_\beta$ cannot be the restriction to simple matroids of a valuative matroid invariant. We conjecture that $H_\beta$ is log-concave and top-heavy in differential degree. For the generalized theta family containing Larson's counterexample to Whitney log-concavity, we compute the first four coefficients and prove the critical log-concavity inequality. Exact computation verifies both conjectures for all $950$ simple matroids on eight elements.

The main result of this paper concerns the stability of Monge solutions to eikonal equations in metric spaces. Although the subslope used in the standard definition of Monge solutions is, in general, not stable under uniform convergence, we overcome this difficulty by introducing an equivalent nonlocal characterization based on sphere-wise subslope conditions. This approach complements existing methods based on metric viscosity solutions and enables us to establish stability directly under uniform convergence and Gromov-Hausdorff perturbations of the domain and metric, without the use of test functions. The method extends to inhomogeneous equations via the optical metric and to a broader class of Hamilton-Jacobi equations. Our stability argument further enables us to develop Perron's method for Monge solutions in general complete length spaces. This yields a new Perron-type existence result for Hamilton-Jacobi equations in metric spaces, without relying on test functions.

The main purpose of this paper is to provide a simple proof of the isometry theorem for one-parameter persistence modules, in which a matrix representing one morphism of an interleaving is reduced and the pivots determine a matching between barcodes. This approach applies to persistence modules of finite type indexed by the reals, and the more general statement for q-tame modules can then be deduced from it using approximations of q-tame modules. The similarity between this use of matrix reduction and that in the persistent homology algorithm motivates some further development of matrix computations for persistence modules, formalized by a category of barcodes in which the morphisms are equivalence classes of matrices. A method for computing induced maps on persistent homology is provided using matrix operations that fit naturally into persistent homology computations, making functorial persistent homology barcodes computable. A method is also given for computing persistent homology barcodes and morphisms when chains do not necessarily have infinite death times.

The two-dimensional bandwidth minimization problem (2DBMP) seeks an injective embedding of a guest graph into a square grid that minimizes the maximum Manhattan distance over its edges. Heuristic methods can provide strong upper bounds, but these bounds do not by themselves certify optimality. We present an efficient exact SAT-based approach for 2DBMP that incrementally searches for the minimum feasible bandwidth and certifies optimality through satisfiability and unsatisfiability results. On the standard $\lceil\sqrt n\rceil \times \lceil\sqrt n\rceil$ host grid, under a 3600 s time limit, the proposed SAT approach certifies optimal bandwidths for 41 of 43 Regular instances and 42 of 93 Harwell–Boeing instances, achieving substantially broader optimality certification within the 3600 s time limit than a previous exact approach evaluated with a 72-hour time limit. In addition, it certifies three bandwidth values that improve all previously published comparison values considered in this study and establishes all three as optimal. We further evaluate the approach on alternative host geometries, namely $2\times\lceil n/2\rceil$ and $n\times n$ grids, to assess its effectiveness beyond the standard host. Overall, the results demonstrate that the proposed SAT approach provides an effective exact method for the small- and medium-sized benchmark instances considered in this study, with fewer than 400 vertices, while heuristic methods remain important for larger and more challenging instances.

A fundamental problem in the algebraic study of biochemical reaction networks is to characterize degeneracy. Two-dimensional zero-one networks, where each reactant appears with stoichiometric coefficient zero or one, constitute the smallest biologically relevant class capable of exhibiting degeneracy. In this paper, we provide a complete classification: a two-dimensional zero-one network without trivial species is degenerate if and only if it is a consistent subnetwork of a species refinement of one of two prototypical networks, namely the complete paired-exchange network or the catalytic-pair conversion network. Equivalently, in matrix-theoretic terms, degeneracy is determined entirely by the row patterns of the stoichiometric and reactant matrices, and can be verified by purely structural inspection without computation. Remarkably, every such degenerate network has a steady-state system consisting entirely of binomials, so its steady-state variety is toric. These results also bear on absolute concentration robustness: since any network exhibiting this property necessarily contains a degenerate subnetwork, the minimal degenerate networks characterized here serve as fundamental building blocks for constructing such networks.

In a recent article of the authors, we proved a result called the Isomorphism Theorem for holomorphic maps from an irreducible Shimura varieties of rank $\ge 2$. The proof of the Isomorphism Theorem uses in essential ways Kähler geometry, function theory of several complex variables, harmonic analysis and ergodic theory. Here we will focus on a slight variation of the Isomorphism Theorem where the target is uniformized by a simply connected complete Kähler-Einstein manifold $(M,h_M)$ which is moreover assumed to be Carathéodory hyperbolic (i.e., the infinitesimal complex Finsler pseudometric $\kappa_M$ induced from the space of bounded holomorphic maps into the Poincaré disk is a complex Finsler metric) and $\Gamma' \subset {\rm Aut}(M)$ is a torsion-free discrete subgroup such that the quotient manifold $Y_{\Gamma'} := M/\Gamma'$ is of finite volume with respect to the quotient Kähler-Einstein metric. In this setting, we will explain the essential roles played by Kähler geometry, harmonic analysis and ergodic theory in the proof of the Isomorphism Theorem.

We study the spectral behavior of the Fourier blocks of the elastic Neumann-Poincaré operator on a torus. For the full elastic Neumann-Poincaré operator, it is known that eigenvalues accumulate at three points, namely zero and a symmetric pair of nonzero points determined by the Lamé parameters. It remains unclear whether this spectral structure persists within each individual Fourier block. We prove that, for every fixed Fourier mode, each of these three points is approached by infinitely many discrete eigenvalues from both sides. Moreover, we establish precise one-sided eigenvalue counting asymptotics with explicit and strictly positive leading coefficients. Consequently, the essential spectrum of every Fourier block is exactly given by these three points. The proof combines a rotating-frame Fourier decomposition and the Plemelj symmetrization principle to obtain a self-adjoint realization of each block. A cubic polynomial transformation removes the order-zero principal part and reduces the problem to compact pseudodifferential operators of order $-1$. The eigenvalue counting asymptotics are then determined by the principal symbols of these reduced operators.

Let $S_k^\sigma(N)$ denote the space of cusp forms of level $N$, weight $k$, and Atkin-Lehner sign pattern $\sigma$, and $S_k^{\mathrm{new},\sigma}$ denote its new subspace. In this paper, we study the asymptotic behavior of the coefficients of the $m$-th Hecke polynomial over $S_k^\sigma(N)$ and $S_k^{\mathrm{new},\sigma}(N)$. In particular, we show that in certain settings, all but finitely many of these coefficients take a particular sign. We also study settings in which the coefficients do not tend to any particular sign.

This tutorial introduces the algebraic foundations underlying the weight structure of polar codes. Using a monomial-based polynomial formalism, we explain how polar and Reed-Muller codes can be viewed as decreasing monomial codes, enabling systematic characterization and enumeration of low-weight codewords. Its goal is to provide an accessible introduction to affine automorphisms, orbit-based descriptions of minimum and low-weight codewords, and their role in weight enumeration. Through illustrative examples and high-level overviews of recursive and coset-based techniques, the paper aims to prepare readers for deeper engagement with the recent technical literature on the weight distribution of polar codes.

Let $\Sigma=(G,\sigma)$ be a connected signed graph of order $n$ and size $m$, and let $s^{+}(\Sigma)$ and $s^{-}(\Sigma)$ denote the sums of the squares of its positive and negative adjacency eigenvalues, respectively. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan states that every connected graph $G$ of order $n$ satisfies \[ \min\{s^{+}(G),s^{-}(G)\}\ge n-1. \] Liu and Ning~\cite{LiuNing2023} published a wide-ranging paper entitled “Unsolved Problems in spectral graph theory", and this conjectures were placed first in their list of such problems. We prove that every signature $\sigma$ of a connected graph $G$ satisfies the sharp bound \[ s^{+}(\Sigma)\le 2m-n+1. \] For the all-positive signing this gives $s^{+}(G)\le 2m-n+1$, whereas for the all-negative signing it gives $s^{-}(G)\le 2m-n+1$. Since $s^{+}(G)+s^{-}(G)=2m$, these two special cases imply the square-energy conjecture; the present theorem is stronger in scope because the same bound holds for every signing of $G$. Applying the theorem to the negation $-\Sigma$ also yields \[ s^{+}(\Sigma)\ge n-1. \] Both bounds are sharp. The proof is based on a doubly nonnegative matrix inequality. We also shorten the proof of that inequality by replacing its final case distinction with a fixed convex combination.

We classify the entrywise functions that preserve positive semidefiniteness on discrete Green matrices \(G(p,q)=(p_{\min(i,j)}q_{\max(i,j)})\) with positive parameters, without requiring the resulting matrix to retain Green structure. For matrices of all orders, the preservers are the zero function and the functions \(f(t)=\int_{[0,\infty)}t^\alpha\,d\mu(\alpha)\), where \(\mu\) is a nonzero finite positive measure and the integral is finite for every \(t>0\). Requiring the resulting matrix to be totally nonnegative reduces the nonzero preservers to \(f(t)=ct^\alpha\), where \(c>0\) and \(\alpha\ge0\). These power functions also preserve positive semidefinite Green structure, while strict Green structure is preserved precisely when \(\alpha>0\). No regularity assumption is needed for these classifications. We also characterize continuously differentiable functions that are entrywise Loewner monotone on every fixed-\(q\) Green family: this holds precisely when \(f'\) is a positive mixture of nonnegative real powers, with the zero measure allowed.

We propose a novel, robust, and second-order-accurate parameterized penalty-projection ensemble algorithm for incompressible Navier–Stokes flow problems. The resulting linearized algorithm, based on the second-order Backward Differentiation Formula (BDF-2), is computationally efficient because it shares the same coefficient matrix across all realizations for each subproblem at every time step. To enhance robustness in convection-dominated flows, the scheme incorporates Ensemble Eddy Viscosity (EEV) regularization. In addition, it is equipped with grad-div stabilization parameter $\gamma$, which controls the splitting error; under the assumptions of the analysis, the splitting error decreases and vanishes asymptotically as $\gamma\to\infty$. We establish the stability of the proposed scheme and rigorously prove its optimal convergence by demonstrating that, as $\gamma\to\infty$, the scheme converges to an equivalent coupled formulation. We further validate the method through a series of numerical experiments designed to verify the theoretically predicted convergence rates and evaluate its performance on benchmark convection-dominated problems. The numerical results are in excellent agreement with the theoretical analysis and confirm the effectiveness of the proposed scheme.

We give a finite symbolic characterization of the strong topological Rokhlin property in terms of globally realizable finite pattern systems. We use this characterization to prove finite-index ascent: if $H\leq G$ has finite index, $G$ is finitely generated, and $H$ has STRP, then $G$ has STRP. Consequently every finitely generated virtually free group has STRP, whereas among countable locally virtually free groups STRP holds exactly for the finitely generated ones. In the compact coding space $\operatorname{NSub}(F_\omega)$ of countable groups, the STRP locus belongs to ${\boldsymbol \Pi}^0_4$, is ${\boldsymbol \Sigma}^0_3$-hard; the locus of finitely generated virtually free groups is ${\boldsymbol \Sigma}^0_3$-complete. For quotients $F_\omega/N$ with $N$ recursively enumerable, every projectively isolated subshift has decidable finite tuple-language and contains an $N$-recursive configuration. This yields effective SFT obstructions to STRP, including subgroup and direct-product obstructions, and implies that $\mathrm{SL}_n(\mathbb Q)$ does not have \STRP for any $n\geq2$.

In this paper, we establish an almost Kähler analogue of the Cheeger–Gromoll splitting theorem for complete almost Kähler manifolds with nonnegative Ricci curvature. As applications, we use the splitting to obtain Goldberg-type integrability results and establish a relation between symplectic non-hyperbolicity and nonnegative Ricci curvature via a theorem of Bangert.

We extend the quantum modulated energy developed in [17] in order to de- rive the magnetized pressureless Euler-Poisson equation as a semiclassical and mean field semiclassical limit from the magnetized Schrödinger-Poisson and von-Neumann equations, respectively. Local well-posedness of the underlying monokinetic PDE is also addressed. In both limits, the magnetic field is external and may be spatially non-uniform. Our results fall in the broader scope of semiclassical and quantum mean field limits for magnetized quantum dynamics.

We prove the nonexistence of eight circulant weighing matrices from the remaining table of orders at most $200$ and weights at most $100$. The proofs combine contraction, character evaluation on the kernel of a contraction, multiplier methods, and exact finite computations. For $CW(105,36)$, the contracted matrix is unique up to equivalence. Applying a nonprincipal character of the $C_3$ kernel gives an element over the Eisenstein integers; reduction modulo $1-\omega$ gives a word in a ternary cyclic code of length $35$, and exact enumeration rules out every required Eisenstein-unit lift. For $CW(140,36)$, the real-valued character $Y\mapsto-1$ of the $C_4$ kernel is incompatible with the same contracted class. For weight $64$, the faithful character $Y\mapsto i$ of a $C_4$ kernel first gives an element of $\mathbb{Z}[i][C_m]$; a generalized multiplier then forces constancy on multiplication-by-$2$ orbits, and exact correlation calculations eliminate orders $140$, $180$, and $196$. The three weight-$49$ cases are settled by the ordinary prime-power multiplier, with contraction where needed. Consequently none of $CW(105,36)$, $CW(140,36)$, $CW(116,49)$, $CW(120,49)$, $CW(192,49)$, $CW(140,64)$, $CW(180,64)$, and $CW(196,64)$ exists.

It is well known that, for each classical pattern $\tau$ of length 3, the number of $\tau$-avoiding permutations of length $n$ is the $n$th Catalan number, and numerous bijections between different length-three avoidance classes have been constructed and studied. In this paper, we refine this classical problem by studying Wilf equivalence among permutations that simultaneously avoid a classical pattern of length three and a flat partially ordered pattern. Partially ordered patterns (POPs) provide a flexible framework for encoding families of classical permutation patterns. For $\ell\geq 3$ and $1\leq x\leq\ell$, let $P_{\ell,x}$ be the length-$\ell$ POP in which the entry at position $x$ is required to be smaller than all the other entries, while no relations are imposed among the remaining entries. Such POPs are called flat POPs. We classify the Wilf equivalences among all pairs $(\tau,P_{\ell,x})$, where $\tau$ is a classical pattern of length three. For every $\ell\geq4$, the resulting $6\ell$ pairs form exactly $2\ell-1$ Wilf equivalence classes, while the exceptional case $\ell=3$ gives four classes. Our proofs combine the derivation of explicit formulas and recurrence relations with the construction of bijections. Moreover, we introduce novel prime-divisor arguments to distinguish the remaining candidate classes, reducing the problem to showing that a certain Diophantine equation has no solutions for $\ell\ge 3{,}274$, where the bound $3{,}274$ is not claimed to be sharp. Finally, by extending our work on POPs, we resolve a conjecture of Qiu and Remmel concerning the distribution of quadrant marked mesh patterns on 132-avoiding permutations and correct an error in their paper that is crucial to the proof.

The primary aim of this paper is to show a linear topological isomorphism between (elements of a wide class {of}) symmetric spaces over the hyperfinite $II_1$ factor $\mathcal{R}$ and certain symmetric operator space over the hyperfinite $II_\infty $ factor $\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. Precisely, we show that for any symmetric function space $E(0,1)$ (in the sense of Lindenstrauss and Tzafriri) such that both $E(0,1)$ and its Köthe dual have the Kruglov property, the symmetric operator space $E(\mathcal{R})$ is isomorphic to some symmetric space $Z_E^2(\mathcal{R}\bar{\otimes}\mathcal{L}(H))$. This result establishes a noncommutative version of a well-known result due to Johnson, Maurey, Schechtman and Tzafriri, and answers the noncommutative version of a question due to Mityagin.