Papers

The incorporation of non-orthogonal multiple access (NOMA) and reconfigurable intelligent surface (RIS) provides a new solution to improve spectral efficiency. Unfortunately, the channel estimation error of RIS is noteworthy due to the absent active radio frequency link. Meanwhile, the transceivers at base stations (BS) and users also exist hardware impairments (HWI). The imperfect channel state information (CSI) and HWI will degrade the communication performance such as outage probability or data rate. To alleviate such performance deterioration, we propose a robust design scheme to minimize the BS transmit power when the target rate of users is satisfied in RIS-assisted single-beam and multi-beam NOMA networks with transceiver HWI and imperfect CSI. Further, the perfect successive interference cancellation (SIC) condition in single-beam and multi-beam NOMA networks is investigated, and the optimization problem is proposed based on perfect SIC conditions. To tackle the non-convex problem, we provide an effective algorithm combining Bernstein-type inequality and semi-definite relaxation. Numerical results demonstrate that the proposed scheme outperforms non-robust RIS-assisted NOMA scheme that ignores HWI and imperfect CSI. Moreover, the multi-beam NOMA network obtains higher beam gain compared to the single-beam NOMA network, and thus there is a superior performance in multi-beam NOMA networks.

The limit set of Conway's Game of Life collects the configurations that can still appear at arbitrarily late times: those admitting predecessors of every finite depth. Answering a question of Salo and Törmä (ICALP 2022), we prove that two fixed finite patterns can each occur in the limit set, yet can never be found together in a single configuration of it, at any relative position; equivalently, the spatial translation action on the limit set is not topologically transitive. Our method is to make a persistent marker force a periodic lane to grow in sufficiently deep predecessors, until two perpendicular lanes are forced to intersect and prescribe incompatible values at a common cell. The argument uses two computer-verified local implications.

Online resource allocation is a fundamental problem in revenue management, sponsored search, and platform operations. Most prior work assumes nonincreasing assignment rewards, capturing diminishing returns. We instead study increasing returns, where assigning more customers to the same product can unlock larger value through scale, visibility, or network effects. We consider capacity-limited products assigned to sequentially arriving customers, where the reward of each product depends on the total number of customers assigned to it. We focus on full compatibility, where every product can be assigned to every customer. The number of customers is unknown to the online algorithm. We show that representative classical approaches, such as online greedy algorithm and LP-based independent rounding, can perform arbitrarily bad in this setting, even when products share a common bonus function. For homogeneous bonus functions, we give a simple polynomial-time algorithm that achieves a tight $1/2$-competitive ratio. For arbitrary heterogeneous bonus functions, we show that no constant competitive ratio independent of the number of products is possible: for $m$ products, the optimal competitive can be as small as $\Theta\left({1}/{\sqrt m}\right)$. This shows that heterogeneous delayed rewards can force any online algorithm to guess the realized arrival counts. We then identify a structured heterogeneous regime that restores a constant guarantee. When each bonus function is nonnegative, nondecreasing, and discrete concave, we design an intermediate target repair algorithm with a deterministic pathwise guarantee of $0.2466$. The algorithm repeatedly computes an offline target for a larger demand level and repairs the current allocation toward it using a prefix-robust assignment order. This coordinates buildup while remaining robust to early stopping and yields a distribution-free guarantee.

Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires $\mathrm{U}(1)$ symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site $\mathrm{U}(1)\rtimes\mathbb{Z}_2$ symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite $S=1$ antiferromagnetic Heisenberg chain with both indices equal to $-1$, without assuming infinite-volume uniqueness. The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.

We study the weighted bilinear Hardy inequality for the same-direction product $H_I f H_I g$ on an arbitrary real interval $I$, with $0<q<\infty$, $1\le p_1,p_2\le\infty$, and measurable weights allowed to take the values $0$ and $+\infty$. The characterisation is formulated through directed compact restrictions $J\Subset I$. On each compact interval, endpoint-safe input profiles retain the $p_i=1$ boundary contribution as a Stieltjes atom, while the lower-triangle reduction separates a closed Hardy part from a strict Copson part so that common atoms are counted exactly once. Upper and mixed regimes follow from exact freezing and compact linear Hardy estimates. The lower regimes are obtained by power lifting and a measure-valued Hardy-Copson theorem. After simultaneous regularisation of the weights, the compact optimal constant is equivalent, with exponent-only constants, to an operational local characteristic, and the global constant satisfies $C_I\asymp\sup_{J\Subset I} A_J^{\mathrm{op}}$. In the regular interior range, the resulting conditions recover the classical $A_1,\ldots,A_7$ characterisations.

We give local combinatorial criteria for realizing prescribed ideal triangulations by nondegenerate hyperbolic truncated tetrahedra. A local bichromatic criterion allows valence-7 edges when tetrahedra meeting low-valence edges use at most two quotient-edge classes. Using the extended combinatorial Ricci flow, we obtain parameter-dependent criteria for triangulations of minimum valence 6. Symmetric and pair-min angle estimates yield explicit valence conditions, including mixed conditions on entire edge stars. The proofs use analytic angle inequalities and rigorous interval evaluations. Explicit face pairings realize the criteria and distinguish their scope, while cyclic constructions and finite covers give infinite families. Each criterion yields a unique zero-curvature hyper-ideal metric and exponential convergence from any positive initial length vector.

We consider the convex problem $(P): {\min}_{y\in \mathbb{R}^n}\, f(Ay)+g(y)$, where $A:\mathbb{R}^n\to \mathbb{R}^m$ is a linear operator, $f$ is convex and $(M,\nu)$-Hölder smooth on $\mathbb{R}^m$, and $g$ is proper, closed convex on $\mathbb{R}^n$, and admits a (generalized) linear minimization oracle (LMO). An active-hull (AH) method calls the LMO in each iteration and outputs a point that lies in the convex hull of all the past LMO returns. We show that under the Euclidean geometry (and in the high-dimensional regime), the optimal LMO complexity for the class of AH methods to solve $(P)$ is $O\big(M^{{2}/({1+\nu})}D^2\varepsilon^{-{2}/({1+\nu})}\big)$, where $D$ denotes the diameter of the feasible region and $\varepsilon$ denotes the objective sub-optimality gap. In particular, the optimal LMO complexity is achieved by a single-loop first-order primal-dual splitting method. Since this method requires knowledge of the problem parameters (e.g., $M$, $D$ and $\nu$), we propose two meta parameter-search algorithms that aim to resolve this issue. These algorithms are parameter-free, but come with the price of an additional log-factor in their LMO complexities.

Constrained nonconvex optimization has seen increasing application in modern machine learning, such as safe LLM alignment/finetuning, transfer learning and rank-constrained continual learning. The commonly used approach is Projected SGD which requires projection at every iteration. However, projection onto a functional constraint can cost substantially more than a stochastic first-order update. We study whether this operation can be deferred until the end of nonconvex stochastic optimization, and only do it once. For weakly convex, possibly nonsmooth objectives with regular convex constraints, we give a penalized proximal method that uses $\widetilde O(\epsilon^{-4})$ stochastic subgradients and one projection onto the constraint set. The returned point is exactly feasible and has a small Moreau-envelope stationarity measure; for smooth objectives, the same algorithm controls the projected-gradient mapping. For smooth nonconvex constraints, we assume a global lower bound on the infeasible constraint slope which permits arbitrary, possibly infeasible initialization. An exact-penalty variant attains the same stochastic-oracle order and returns an exactly feasible point within $O(\epsilon)$ of an $\epsilon$-KKT point. All intermediate updates use only projections onto a simple Euclidean ball. These are oracle-complexity guarantees: the cost of the single terminal projection is separate, and no efficient projection algorithm for a general nonconvex set is assumed to follow from regularity.

By a result of Bogomolov and Zarhin, a K3 surface $X$ over a number field $L$ has ordinary reduction at a density 1 set of primes after passing to a finite extension of $L$. In this paper, we refine this result for non-CM K3 surfaces whose transcendental Hodge structure has endomorphism field $F$ abelian over $\mathbb{Q}$. We further assume a condition on the connected components of the $l$-adic monodromy group of $X$, and, when $F$ is totally real, a parity condition on the rank of the transcendental lattice over $F$. Under these assumptions, we prove that the set of primes of $L$ at which $X$ has $\mu$-ordinary reduction has density $1$. As a corollary, the set of primes of $L$ at which $X$ has ordinary reduction has density $1/[FL:L]$. We include explicit examples of K3 surfaces satisfying these conditions.

We prove polynomial-in-time Sobolev bounds on the approximation error between the twisted discrete nonlinear Schrödinger equation and its magnetic continuum limit. The analysis exploits complete integrability of the continuum flow through its Birkhoff representation to control approximation by a non-integrable Hamiltonian finite-difference lattice. We establish a focusing-defocusing dichotomy in uniform polynomial control: the bounds hold for arbitrary defocusing data and focusing data of sufficiently small mass, while modulational instability and aliasing yield an obstruction for large focusing data. The accompanying algebraic spatial rates are generically optimal in the Baire-category sense. The proof combines Birkhoff coordinates with modified energies for discrete Sobolev growth, while Fourier filtering recovers uniform spacetime estimates below the energy space despite degenerate lattice dispersion.

We give a dynamical characterization of the Chandrasekhar mass for cold white dwarfs governed by the radial Euler–Poisson equations with the exact Chandrasekhar equation of state. This mass is the infimum of masses for which a finite-energy classical solution with a physical vacuum can contract its entire support to a point in finite time. Below it, conservation of energy prevents complete collapse. For every mass in a sufficiently small interval above it, we construct a solution whose support shrinks to the origin and whose density concentrates there with the full mass of the star. The interior approaches a collapsing Goldreich–Weber profile. Near the vacuum boundary, a different rescaling gives a limiting profile that connects the high- and low-density regimes of the pressure law. We determine this profile and show that, after adding the electron rest energy, the conserved energy equals that of the Goldreich–Weber reference solution. The proof combines matched asymptotic expansions with local existence and energy estimates that extend solutions backward from times approaching collapse to a common earlier time.

We prove that every acylindrical action on uniformly locally finite product of trees induces the hyperfinite orbit equivalence relation on the Roller boundary. As a byproduct, we construct an example of a standard Borel space and two commuting bounded-to-one surjective partial Borel maps that generate a universal countable Borel equivalence relation. This contrasts to Shinko-Weilacher-Yu's theorem on hyperfiniteness of bounded-to-one actions of commutative monoids.

Motivated by the structure of various standard statistical arrays, Bailey, Praeger, Rowley and Speed (1983) described the automorphism groups of poset block structures as generalized wreath products. We observe that these are exactly the groups used to define DAG exchangeability in Jung, Lee, Staton and Yang (2021), and we restate the representation theorem of the latter work in this language. As a concrete example, we write out the resulting representation for block matrices whose block rows and block columns are separately exchangeable, and whose rows and columns may also be permuted independently inside each block.

We study a one-dimensional nonlinear stochastic wave equation driven by Gaussian noise that is white in time and rough in space. We prove a bi-parameter local linearization for mixed increments along the two characteristic directions. The proof combines characteristic cancellation with a localized fractional-energy estimate that controls boundary interactions caused by the rough spatial noise. As an application, we establish quadratic-variation limits on arbitrary anisotropic rectangular meshes and construct a consistent estimator of a multiplicative diffusion parameter. Numerical experiments illustrate the finite-sample performance of the estimator.

Successive cancellation perturbation-enhanced (SCP) decoding improves the performance of finite-length polar codes by performing multiple SC decoding attempts with receiver-side perturbations. However, many existing perturbation schemes generate or update subsequent perturbations according to the outcomes of previous decoding attempts, resulting in additional decoding latency. In this paper, we propose an offline variance design (OVD) method for parallel SCP (PSCP) decoding of short- and medium-length polar codes. First, we formulate the exact recovery objective conditioned on ordinary SC failure and classify failed frames by the position of the first genie-aided intrinsic error and the number of subsequent intrinsic errors. We also derive a consistent Gaussian representation of the perturbed channel that preserves min-sum SC hard decisions. Second, we construct a class-based approximation of the recovery objective using Gaussian approximation and backward recursions, accounting for both error correction and new errors introduced by perturbations. We prove that both the exact and analytical objectives are nondecreasing and exhibit diminishing marginal gains as independent branches are added. Third, we develop a greedy algorithm to select variances from a finite candidate set for a given code, signal-to-noise ratio (SNR), and number of perturbation branches. All variances are determined offline, allowing the original SC branch and all perturbation branches to start simultaneously. Simulations for rate-$1/2$ polar codes of lengths $64$, $128$, $256$, and $512$ show that OVD-PSCP achieves lower block error rates (BLERs) than conventional SCP with the same number of perturbation branches. The gains are larger for shorter codes and increase as the number of perturbation branches grows.

We prove that if $G$ is a residually finite group, $\widehat{G}$ is its profinite completion and the map $H_2(G, \mathbb{Z}) $ $ \to H_2(\widehat{G}, \widehat{\mathbb{Z}}),$ induced by the canonical map $G \to \widehat{G}$, is injective, then the non-abelian exterior square $G \wedge G$ is residually finite. We show that if $G$ is a finitely presented centre-by-metabelian group then $\nu(G)$, the non-abelian tensor product $G \otimes G$ and the non-abelian tensor square $G \wedge G$ are residually finite. Furthermore we prove that if $G$ be a finitely presented metabelian group then the weak commutative construction $\X(G)$ is residually finite. We discuss a criterion for the $q$-exterior square $G \wedge^q G$ to be residually finite.

We compute the missing three-color relation in type $H_3$ as an explicit $R$-linear combination of double leaves. The expression has $43\,424$ nonzero polynomial coefficients with scalars in $\mathbb{Z}[\varphi]$, where $\varphi = (1 + \sqrt{5})/2$. We verify the relation using exact arithmetic in the computer algebra system GAP. This completes the diagrammatic presentation of the Hecke category of Elias and Williamson.

Non-negative matrix factorization (NMF) is a singular statistical model whose Bayesian asymptotics are governed by the real log canonical threshold (RLCT). We study the local geometry of the factorization map and derive an upper bound for the RLCT of NMF. Let $H$ be the model inner dimension and $H_0$ the non-negative rank of the true $M\times N$ matrix. Assuming that the true matrix admits a strictly positive factorization of inner dimension $H_0$ in the interior of the parameter domain, we prove, for smooth positive priors, that $\lambda\leq \{(H-H_0)\min(M,N)+H_0(M+N-H_0)\}/2$. This bound strictly improves the previous bound when $H_0\geq3$. The proof uses a local analytic normal form that separates independent linear coordinates from a residual matrix product. When $H=H_0$ also equals the ordinary rank of the true matrix, we obtain the exact value $\lambda=H_0(M+N-H_0)/2$. Under the standard assumptions of singular learning theory, these results bound the leading coefficients of the expected Bayesian generalization error and the Bayesian free energy.

We determine the vanishing cycles of Xiao's genus-two fibrations $X_d\to P^1$, $d=3,4,5$, whose fibers admit degree-$d$ maps to a fixed elliptic curve $E$. We tile the base $P^1$ by triangles and realize the surface $X_d$ as a degree-$d$ branched cover of $P^1\times E$. The covering description gives an explicit algorithm for the vanishing cycles. A vanishing path is recorded by the sides it crosses in the tiling; each crossing changes a labelled picture of the fiber by a local hexagon move, and the vanishing cycle is read from the terminal picture. We obtain the $7$, $13$, and $31$ vanishing cycles as explicit curves in a marked reference fiber. For $d=4,5$, these give new positive factorizations of the identity in the genus-two mapping class group, of types $(6,7)$ and $(12,19)$.

For a graph $F$, its extremal number ${\rm ex}(n,F)$ and extremal graphs denote the maximum number of edges in $F$-free graphs and the graphs attaining this maximum, respectively. Let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex graph with ${\rm ex}(n,F)+q$ edges, and let $t_F(n,q)$ denote the corresponding minimum obtained by adding $q$ edges to an extremal graph. Mubayi (2013) conjectured that for every non-$r$-partite stable $r$-uniform hypergraph $F$ with $r\ge 2$, $h_F(n,q)\ge q\cdot t_F(n,1)$ holds for all sufficiently large $n$ and all integers $q\ge 1$. In this paper, we prove that Mubayi's conjecture fails in a stronger sense. For every integer $d\ge 2$, we construct a family of non-$d$-partite stable graphs $F$ such that $h_F(n,q)< q\cdot t_F(n,1)$ for all $1\le q\le O\bigl(n^{4/ 3}\bigr)$. In particular, these examplesis completely refute Mubayi's conjecture. \noindentKey words: extremal graph; supersaturation; chromatic number.

Let \(H\) be a multiplicative group, \(R\) a commutative \(H\)-graded ring, and \(M\) a nonzero finitely generated gr-semisimple \(H\)-graded \(R\)-module. We assume that \(M\) admits a fixed decomposition into powers of gr-simple modules with pairwise distinct annihilators, each repeated block carrying the componentwise grading. For a nonzero homogeneous element \(x\), we compute the colon ideal \((Rx:M)\) and show that it is determined by the multiplicity-one blocks on which \(x\) has a nonzero component. This leads to the criterion \( x\leftrightarrow y \text{ if and only if } \sigma(x)\cap\sigma(y)=\varnothing, \) where \(x\leftrightarrow y\) means that \(x\) and \(y\) are adjacent in \(\Gamma_H(M)\). Thus adjacency is governed by disjointness of multiplicity-one obstruction supports. The corresponding obstruction classes form a weighted disjointness structure: higher multiplicities affect the sizes of the classes, but not the adjacency relation between their supports. We use this description to study graph invariants, reconstruction, and Kneser-type layers, and, in the finite case, to obtain formulas for the obstruction-class sizes and a reduction of the adjacency spectrum.

Let $T$ be a nonzero real symmetric third-order tensor. Denote by $r(T)$ its CP rank, by $s(T)$ its symmetric rank, by $d(T)$ its essential dimension, and put $e(T)=r(T)-d(T)$. We establish two complementary upper bounds for the Comon rank gap $$\Delta(T):= s(T)-r(T)\leq\min\{C_{ec}(e(T)),\ C_{dc}(d(T))\}.$$ The excess-controlled function $C_{ec}(e)$ is zero for $0\leq e\leq3$ and equals $9e-3$ for $e\geq4$. In particular, the equality $C_{ec}(e(T))=0$ for $0\leq e\leq3$ extends Friedland's flattening-rank criterion for real symmetric third-order tensors by two additional rank layers. The dimension-controlled function $C_{dc}(d)$ is zero for $d=0$, equals $(d^2-4d+10)/3$ for $1\leq d\leq18$, and equals $d^2/4+d/4+3/2$ for $d\geq19$. The excess estimate follows from the alternating equations and decomposable-bivector support geometry; a low-wedge descent gives the vanishing range $e\leq3$. The dimension estimate combines a transverse polar-completion bound with a shifted estimate for the maximum real Waring rank. Thus the two bounds measure different features of the same tensor and remain effective in complementary regimes.

We study codimension-one holomorphic foliations admitting invariant singular real-analytic Levi-flat varieties. Our main problem is to determine when the local meromorphic first integrals furnished by the Levi-flat geometry can be continued coherently and globalized. We introduce a pencil-singularity condition and a projective compatibility condition for local meromorphic parameters. In the projective plane these yield a characterization of foliations admitting a primitive rational first integral, while in higher-dimensional projective space they give a rational integrability criterion under a generic local-product hypothesis along the dicritical locus. We also obtain semi-global and Stein versions of these results. In a different direction, we study foliations admitting infinitely many invariant Levi-flat varieties and prove a Darboux-type alternative: either the foliation admits a rational first integral, or infinitely many Levi-flats contain a common invariant algebraic curve whose virtual holonomy is abelian. Under additional assumptions on the singularities, this leads to the existence of a closed rational, and in the absence of saddle-nodes a closed logarithmic, defining \(1\)-form.

We construct the Conway-Wales lattice from a self-dual even code $C$ over $\mathbb{Z}_4$ whose binary layers interweave the Hamming codes $\mathcal{H}_7$ and $\mathcal{H}_8$. Explicit signed symmetries lift every automorphism of the binary residue, and the full signed code group is the coordinate-frame stabilizer of the Gaussian lattice. An integral reflection in an embedded scaled self-dual sublattice supplies an additional symmetry, from which the classical frame exchange is recovered. Counting intrinsically defined frames determines the full Gaussian isometry group, of order $583\,704\,576\,000$. Its quotient by the four Gaussian scalars is the Rudvalis simple group, and the derived subgroup of a cross stabilizer is the Tits simple group of order $17\,971\,200$. The construction, automorphism-group determination, and both simplicity proofs are carried out within the lattice, without assuming the known group orders or using a recognition or classification theorem.

The information-theoretic construct of synergy refers to the statistical structure that is contained in three or more variables, but not in any subset of them. Here we leverage recent advances in multivariate information theory to show that synergy is far more prevalent in complex systems than previously thought. In particular, we show that many-body systems tend to become strongly dominated by synergy as the number of subcomponents grows. At the same time, our results also reveal that commonly used tools for statistical modelling severely underestimate synergistic structures. These findings imply that synergy constitutes a prevalent, yet often invisible, informational component of complex systems. With a certain poetic licence, we interpret these results as suggesting that synergy is the dark matter of statistics - interdependencies that we know exist, but standard instruments fail to detect.

Let $G=\langle a,b\rangle$ be a finite group with distinct nonidentity generators, and suppose that $H=\langle ba^{-1}\rangle$ is normal. We determine exactly which complementary generator choices on the nonterminal cosets of $H$ extend to two arc-disjoint Hamiltonian paths with both terminal vertices in $H$. The associated return residue must avoid an explicit set of at most six residues. When $[G:H]\ge3$, any two nonterminal cosets may be left free while all other choices are prescribed; two is best possible. Every such Cayley digraph has two arc-disjoint Hamiltonian paths; this includes all two-generator Cayley digraphs of finite abelian groups. The criterion follows from a complete distance formula for the Hamiltonian cut spectra of $\mathrm{Cay}(\mathbb{Z}_q;\{D,D+1\})$. Their unique nondegenerate exception comes from the odd central coefficient $(q-1)/2$. The geometric ingredient is a confinement theorem: two sector-filled integer sequences satisfying a length inequality, with overlapping ranges and distance at least two, have one sequence contained in a single gap of the other. Sector filling bounds alternation, and the length inequality excludes interlacing. A self-contained proof of the primitive-ray cycle profile follows the intersection geometry of Curran and Witte.

The quaternary Preparata codes of Hammons, Kumar, Calderbank, Sloane and Solé are $\mathbb{Z}_4$-linear codes of length $2^m$ with $4^{2^m-m-1}$ codewords. They are usually studied through their binary Gray images, that is, in the Lee metric. We consider them instead as codes over a 4-letter alphabet with the Hamming metric. We show that for odd $m$ they have covering radius 2 and minimum distance 4, so they are quasi-perfect. For even $m$ the same construction has covering radius 3. The proof is a short computation with Teichmüller representatives in the Galois ring $\mathrm{GR}(4,m)$, and it also gives the coset weight distribution. The covering density of these codes tends to $9/8$. The best previously known infinite families of quaternary codes with covering radius 2 have covering density about 1.5. Consequently $K_4(n,2)\le 4^{n-m-1}$ for all $n\ge 2^m$ and odd $m$, which improves the best available bounds for a range of lengths above each $2^m$. For $m=3$ the code is the octacode, and it gives $K_4(8,2)\le 256$; the previous bound was 352, from 2005, and the best lower bound is 251. We also give a quaternary code showing $K_4(6,2)\le 48$, where the previous bound was 52, together with consequences for larger alphabets.

We develop a finite-band method for the reduction of families of indefinite integrals. The starting point is the case in which the logarithmic derivative $K'(x)/K(x)$ is a rational function. This leads to adapted polynomial and Laurent-type bases and to upper triangular band matrices. The framework includes generalized real Schwarz–Christoffel integrals and more general hyperexponential weights. We then extend the construction to the case in which the logarithmic derivative $K'(x)/K(x)$ is algebraic over the field of real rational functions. The associated finite-dimensional differential module gives rise naturally to block-band systems. Finally, we show that functions satisfying linear differential equations with real polynomial coefficients give rise to finite recurrences for their moments. The resulting reduction is effective: for a prescribed target integral it is enough to solve a finite triangular band system, and only the reduction data associated with the target are required. We also discuss arithmetic complexity and give explicit examples illustrating the scalar and block reductions. The algorithms are implemented in the accompanying Mathematica package BandedReduction.

We study the Ising spin glass with independent, identically distributed couplings on the diamond hierarchical lattice with $n$ branches, of effective dimension $1+\log_2 n$; $n=4$ is the Migdal-Kadanoff lattice of $\mathbb{Z}^3$. For $n\ge4$ and every coupling law with a bounded density we prove that at sufficiently low temperature $T>0$, uniformly in the level of the lattice, the poles are ordered, every spin has Edwards-Anderson parameter $1-O(T)$ when the poles are fixed, and the overlap of two free-boundary replicas has second moment $1-O(T)$. For standard Gaussian couplings, $n=3,4,5,7,8$ and every $T\le T^*(n)=4/15,\,6/7,\,6/5,\,7/4,\,2$ ($0.89$ to $0.98$ times numerical estimates of $T_c$), a computer-assisted proof gives pole order and, as the level tends to infinity, site-overlap order and the blue-cluster statements below; likewise for every larger $n$ at these temperatures. Previously, spin-glass order on these lattices with frustrated independent couplings had been proved only for more than $10^{18}$ branches (Collet and Eckmann, 1984). The key step is growth of the effective coupling under the exact renormalization map: for $n\ge4$, by a one-step contraction, via Rogozin's inequality, of a density bound on an event of high probability; near $T_c$, by interval-arithmetic bounds in the peakedness order. A transfer theorem following the ancestor-chain argument of Collet and Eckmann turns this growth into bulk statements. In the Chayes-Machta-Redner representation the same input gives a blue cluster that contains both poles with probability tending to one and whose expected density is bounded away from zero, while the largest other blue cluster has density tending to zero in probability and in $L^1$; hence blue-cluster density imbalance. To our knowledge this is the first proof of such an imbalance for a spin glass on a graph other than the complete graph.

Power systems are often represented by high-dimensional dynamic models, making nonlinear control design computationally expensive and limiting its practical application. Oscillation analysis and damping control therefore commonly rely on linearized models, which do not capture the nonlinear oscillatory behavior induced by large disturbances. This paper proposes an isostable-based model reduction method for nonlinear modal analysis of power systems. The method constructs an invariant manifold associated with a selected oscillatory mode and uses isostable coordinates to describe the mode with only two real state variables. The isostable formulation gives linear autonomous modal dynamics, with nonlinear state reconstruction and input response computed on the modal manifold. An optimal damping control problem is then formulated directly in the reduced coordinates. Case studies on the IEEE 39-bus system demonstrate the accuracy of the proposed model in describing nonlinear oscillations and show improved damping compared with control designed using a linear reduced model.