Papers

For a positive integer $n$, let $f_n(X)=X^4-nX^3-6X^2+nX+1$ and let $K_n=\mathbb{Q}(\rho_n)$, where $\rho_n$ is a root of $f_n$. We determine all coincidences among these fields: for distinct positive integers $m,n$, $K_m=K_n \Longleftrightarrow \{m,n\}\in\{\{1,103\},\{2,22\},\{4,956\}\}$. Thus the three previously known equalities are the only ones. This extends Hoshi's finite-range classification to all positive integral parameters and, in particular, subsumes the uniqueness results of Pincus and Washington. The proof combines Hoshi's correspondence between equal simplest quartic fields and primitive solutions of a quartic Thue equation with estimates of Lettl–Pethő–Voutier for rational approximations to two of its real roots. A Gaussian-integer identity yields a lower bound for the denominator of the resulting rational approximation; the continued-fraction information and the approximation estimates then exclude every parameter exceeding $1000$.

Let $k$ be a field, let $V=k^n$, and let $G$ be a classical group acting on $V$. For a finite subset $E\subset V$ and a basis $u=(u_1,\dots,u_n)$ of $V$, we study the set of $G$-frames of type $u$ contained in $E$, or, equivalently, the set $T_{E,u}^G:=\{g\in G(k)\ |\ g u_i\in E\text{ for each $i$}\}$. In the cases described below, we prove estimates of the form $|T_{E,u}^G|\ll_n |E|^\alpha$ that are uniform over all fields, with sharp exponents in this uniform setting. For $G=\operatorname{SL}_n$, the uniform sharp exponent is $n-1/n$. For orthogonal groups in dimensions $n=2,3$, with $\operatorname{char}(k)\neq 2$, the uniform sharp exponent is $n/2$. We propose an algebro-geometric Brascamp–Lieb inequality which would lead to the orthogonal exponent $n/2$ in all dimensions. For the related setwise stabilizer $R_E^G$ of $E$ in $G(k)$, where $E\subset V$ is finite and spans $V$, we also prove the sharp characteristic-zero bound $|R_E^G|\ll_n E|^{\operatorname{rank}G}$ for the special linear, orthogonal, and symplectic groups, where $\operatorname{rank}G$ denotes the absolute rank.

This note extends the main result of \cite{IS} 2007 — torsion of the extended Chern–Simons (regulator) classes of the Deligne canonical extension of a flat bundle with unipotent monodromy at infinity — from the case of a smooth irreducible boundary divisor to the case of a boundary divisor $D = D_1\cup D_2$ with two smooth irreducible components meeting transversally along a smooth center $Z=D_1\cap D_2$. Let $X$ be a smooth projective variety defined over $\mathbb{C}$, and $U:=X-D$. Given a flat bundle $(E,\nabla)$ on $U$ with unipotent monodromy around the components of $D$ consider Deligne's canonical extension $(\overline{E},\overline{\nabla})$ on $X$. Then the extended Chern-Simons classes $$ c_p(\overline{E},\overline{\nabla})\in H^{2p-1}(X,\mathbb{C}/\mathbb{Z}) $$ are torsion, for $p\geq 2$. These notes were prepared in 2009-2010, and the preprint \cite[2026]{IS2} treats the full normal crossing case via a different approach.

One of the main challenges for the future in logistics lies in energy consumption. France has the ambition to reach net zero carbon emissions by 2050. The most ambitious research work envisioning the solutions to do so is ___Futurs énergétiques 2050' (October 2021) by RTE, the company in charge of the French Electricity Transportation Network. The work assesses multiple scenarios involving various energy mixes (wind, solar, nuclear etc.). It concludes that the net carbon neutrality in 2050 can only be made possible through sobriety, despite mobility switch to electricity (i.e. at least a 40% decrease final energy consumption, plummeting from 1600 TWh to 930 TWh in 2050). In the logistics field, vast progress has been made to lower the energy intensity of road freight transport. Some promising solutions and technologies have been introduced, namely the improvement of vehicle performances, the alternatives fuels, as well as the parallel evolution of physical and digital infrastructures. It resulted in a cut from 0.077 tonne of oil equivalent per thousand tonne-km in 1992 to 0.057 in 2012 (Llorca and Jamasb, 2017). But the simultaneous soar in freight flows quantity and traffic prevented a global decrease in energy consumption. This challenge is stepped up in urban logistics, as it involves managing -in already congested cities- supply channels and small vehicles that are amplifying, multiplying, and becoming more and more intricate. It seems that all the solutions and technologies that have been introduced are necessary, but insufficient, to meet the expectations in terms of energy sobriety. Significant gains will come from a radical shift in logistics shared infrastructure and from innovative organizations. In this vein, CargoCap (Stein and Schoesser, 2003) and other projects are exploring new infrastructures potential while horizontal collaborative transport as a new organization of freight systems has been largely studied and its effectiveness and efficiency has been proven in recent literature (Pan et al., 2019). In this work we will focus on the energy effectiveness of a new freight dedicated infrastructure. This work assesses the potential a large scale cooperative dedicated system, involving shared infrastructure, for the distribution of freight in the city. This is conceived as an alternative to scattered road haulage, in order to significantly cut down energy consumption. With the goal of energy sobriety, we set the twofold objective of modelling the energy consumption of such a dedicated freight supply system, and of drawing conclusions regarding the best practices by means of an impact assessment of the main design parameters. A distribution system's energy demand strongly depends on specific choices of implementation, which can involve putting back on the agenda some discontinued practices, as well as taking advantage of new technologies and innovative processes. Keeping that in mind, our methodology is devised to comprehensively assess all the available implementation choices, both in technology and in layout. The method consists in taking a step back to fundamental energy theorems, which parameters can be set to stand for any scenario of interest. The model includes transportation modes characteristics (e.g. aerodynamics, rolling resistance etc.), as well as kinetics repercussions of delivery management, handling, and infrastructure accessibility (e.g. driving cycles, congestion etc.). For our purposes of generality, but not limited to, the urban layout and the main routing steps are simplified as a general dense area accessed via a ring road. It is applied to the specific case of Paris, benefiting from previous studies and surveys, as well as government approved open data, quantifying all aspects of freight supply. As far as limits are concerned, it can be put forward that the model is overly simplified and blind on indirect energy consumption which is considerable in overall energy demand. Our work precisely spots out the most impactful parameters, and the high potential axis for refinement, factoring uncertainty. It also specifies where direct and indirect energy drives align, and, on the contrary, a judicious compromise should be investigated. These findings give the necessary tools to further the research into a refined and expanded model, a methodology and insights for urban freight distribution applied here in large cities.

Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl–Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities, with eigenvalues \(d|\chi_\phi(u)|^2\), turning stability into an explicit worst-direction design problem; write \(\lambda\) for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while \(\mathbb E[\lambda^{-1}]=\infty\), and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly \(d^2\) outcomes in every integer dimension has floors \(\Theta(d^{-3})\) for odd \(d\) and \(\Theta(d^{-5})\) for even \(d\); a finite-field family for \(q=2^m\) obeys the uniform bound \(\lambda\ge4/9\). Our main result treats every prime-power dimension of characteristic \(p\ge5\). A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to \([L_q,U_q]\) with \(U_q/L_q\to1\). Its SIC-normalized minimum tends to one, and \(\lambda(\phi_q)/\Lambda_q^\star\to1\) for the global finite-field WH max–min optimum \(\Lambda_q^\star\), without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert–Schmidt error of canonical linear inversion at \(I/d\), while its lower edge controls local Fisher efficiency and canonical-shadow bounds.

In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space $\mathbb{H}^n$ for $n\geq 3$. We note that when $n\geq 4$, the result was obtained in our previous paper [23]. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [13] for the hyperbolic case.

We present a rigorous strong-coupling Birman-Schwinger analysis of the three-boson lattice Schroedinger operator on Z^2 at the exceptional quasimomentum K = pi. First, we provide an exact closed-form benchmark for the fiber Fredholm determinant, valid for every quasimomentum K, obtained via a uniform elliptic reduction. Second, we formulate and prove a general order-matching criterion determining whether a leading-order Fredholm determinant asymptotic, with relative error O(1/mu), suffices to fix the constant-order additive energy correction, or if the next-order refinement is required. Applying the criterion to the formal branch z = -2mu + d, we identify an algebraic crossing -2mu + 6 + 8/mu + O(mu^{-2}), but demonstrate that this crossing does not correspond to a true eigenvalue of the full Hamiltonian. The actual ground state obeys the rigorous variational bounds -3mu <= z_1^{pi,s}(mu) <= -3mu + 6, so that z_1^{pi,s}(mu) = -3mu + O(1), the same leading branch as at K = 0; direct finite-volume diagonalisation confirms the refined asymptotic -3mu + 6 + O(mu^{-1}) and spectral gap 2mu - 2 + O(mu^{-1}). We independently confirm that the known K = 0 constant C approximately 3.96458 requires no analogous refinement. Finally, we compare the asymptotic precision levels achieved across recent lattice few-body models and draw a structural parallel with the parity-based classification of topological band insulators at time-reversal-invariant momenta.

We consider the fractional Lane-Emden equation with a nonlocal Neumann condition in a half-space. We establish the existence of non-constant solutions for the critical problem in any dimension. On the other hand, we show that, in dimension $n=1$, the subcritical problem admits only the trivial solution. This result follows from a new Pohozaev-type identity, which we obtain by using suitable decay estimates for the solutions.

We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken–Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss–Green formula for divergence-measure fields.

Let $\eta$ be a fixed positive integer. Let $S$ be a subset of $\mathbb{Z}$, $\star:S\times S\to \mathbb{Z}$ be a binary function, and $\zeta_{\eta}:\{\xi\in \mathbb{Z}:\gcd(\xi,\eta)=1\}\to \{0,1\}$ be a function. For a simple graph $G$ of order $n$, a bijective function $f:V(G)\to S$ (where $|S|=n$) is called an arithmetic cordial labeling modulo $\eta$ under the arithmetic structure $\langle S,\zeta_\eta,\star\rangle$ if the induced function $f_\eta^*:E(G)\to \{0,1\}$, defined by $f_\eta^*(ab)=1$ whenever $\gcd(f(a)\star f(b),\eta)= 1$ and $\zeta_\eta(f(a)\star f(b))=1$; otherwise, $f_\eta^*(ab)=0$, satisfies the condition $|e_{f_\eta^*}(0)-e_{f_\eta^*}(1)|\leq 1$, where $e_{f_\eta^*}(i)$ is the number of edges with label $i$ ($i=0,1$). In this paper, the arithmetic cordial labeling of product graphs, namely, corona, lexicographic, cartesian, tensor, and strong, is explored under the operation of addition.

We consider the three-dimensional defocusing cubic nonlinear Schrödinger equation with variable coefficients, a magnetic potential, and a non-negative localized damping term, \[ i\partial_tu+(\nabla-iA)\cdot G(\nabla-iA)u+ia(x)u=|u|^2u, \qquad t>0,\quad x\in\mathbb R^3. \] No non-trapping condition is imposed on the metric $G$. Instead, the variable-coefficient region is assumed to be contained in the effective damping region. Under a one-centre condition on the tangential magnetic field, we prove global well-posedness for initial data in $H^{1+\varepsilon}$, uniform mass and energy bounds, and show the local energy decay. To obtain scattering, we impose a support condition on the full magnetic field inside the damping region. Under these stronger assumptions, the solution scatters to a free Schrödinger evolution in $H^s$ for every $0\le s<1$. The appendix discusses a separate constant-damping framework for abstract Hamiltonians.

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

Current wireless systems combat inter-symbol interference (ISI) by diagonalizing or sparsifying the channel matrix, yet they remain vulnerable to selective fading. To address this, we propose a universal random precoding (RP) transmission framework based on the universality class. RP leverages random transforms to statistically exploit all subchannels and construct an equivalent channel belonging to the universality class, thereby enhancing diversity gain while maintaining backward compatibility with existing waveforms. Low-complexity implementations include the randomly permuted fast transform (FT-RP) and the interleaved block-sparse fast transform (IBSFT-RP). A cross-domain OAMP/MAMP (CD-OAMP/MAMP) detector is designed for RP systems, which is replica maximum a posteriori (MAP)-optimal according to state evolution (SE). Simulation results on MIMO systems demonstrate that RP with CD-OAMP/MAMP achieves near-RM performance with much lower complexity, with additional benefits of flexible compression ratios for spectral efficiency.

We study weak openness and related geometric properties of convex combinations of slices of the unit balls of Banach spaces. We prove that, for every $m\geq 2$, properties $\mathrm{P1}^{(m)}$ and $\mathrm{CWO}^{(m)}$ are equivalent, and likewise for their closure variants. Consequently, both $\mathrm{P1}$ and $\overline{\mathrm{P1}}$ are determined by convex combinations of two slices; in particular P1 and CWO are equivalent. We construct infinite-dimensional real Banach spaces with $\mathrm{P2}^{(2)}$ but without P2, and with $\mathrm{P3}^{(2)}$ but without P3. In the latter space, every convex combination of two slices contains two points at distance 2, although the space fails the strong diameter two property. Finally, we give an equivalent norm on the real space $c_0$ for which $\overline{\mathrm{P1}}$ holds, but P1 fails. These answers several questions raised in the literature.

We establish some sharp reverse Hardy-Littlewood-Sobolev (HLS) type inequalities on \(\mathbb{R}^n\) and \(\mathbb{R}_+^n\). Using an operator representation, we overcome the difficulty that the symmetric double-integral structure is unavailable in the half-space setting. On \(\mathbb{R}^n\), for \(1 \le n < \alpha\), \(\frac{n}{\alpha} < t < 1\), and \(0 < q < 1\), there holds for nonnegative \(f \) that \[ \|E_\alpha f \|_{L^{t^\prime}(\mathbb{R}^n)} \ge \mathscr{C}(n,\alpha,q,t) \|f \|_{L^1(\mathbb{R}^n)}^{\gamma} \|f \|_{L^q(\mathbb{R}^n)}^{1-\gamma}, \quad \gamma := \frac{n - q\alpha - \frac{n}{t^\prime}q}{n(1-q)} \] for some $\mathscr{C}(n,\alpha,q,t)>0$ iff \(q>\frac{n}{\alpha}\), where \(E_\alpha\) is the extension operator with Riesz kernel and \(t^\prime\) is the conjugate of \(t\). The sharp constant is achieved when \(\frac{n t^\prime}{n + \alpha t^\prime} \le q < 1\). On \(\mathbb{R}_+^n\), with \(2 \le n < \alpha\), \(\frac{n}{\alpha} < t < 1\), and \(0 < q < 1\), we show for nonnegative \(f \) that \[ \|\widetilde{E}_\alpha f \|_{L^{t^\prime}(\mathbb{R}_+^n)} \ge\widetilde{\mathscr{C}}(n,\alpha,q,t) \|f\|_{L^1(\partial \mathbb{R}_+^n)}^{\widetilde{\gamma}} \|f\|_{L^q(\partial \mathbb{R}_+^n)}^{1-\widetilde{\gamma}}, \quad \widetilde{\gamma} := \frac{(n-1) - q(\alpha-1) - \frac{n}{t^\prime}q}{(n-1)(1-q)}, \] for some $\widetilde{\mathscr{C}}(n,\alpha,q,t)>0$ iff \(q > \frac{n-1}{\alpha-1}\), where \(\widetilde{E}_\alpha\) is the extension operator with Poisson-type kernel. The sharp constant is achieved when \(\frac{t^\prime(n-1)}{n + t^\prime(\alpha-1)} \le q < 1\). We further extend results to \(q\ge1\). The proofs use rearrangement inequalities, the sharp Carlson–Levin inequality, and refined pointwise lower bounds for the Riesz and Poisson-type potentials. Our results unify and extend the classical reverse HLS inequalities, especially on \(\mathbb{R}_+^n\).

Quantum field theory allows local violations of the classical energy conditions, such as regions of negative energy density. The Ford-Roman quantum inequalities quantify how negative these regions can be and for how long. We present a derivation of these inequalities from the second law of thermodynamics, in an operational framework that couples the field to a small thermal detector. The detector acts as a physical probe of the field's energy density and, by the non-negativity of the entanglement entropy between the two systems, one can show that its heat loss is bounded from below by a state-independent quantity.

We study a Riesz gas with interaction parameter $s \in (d-3,d)$ in an arbitrary spatial dimension $d$, confined to a half-space by a hard wall. We prove that the equilibrium measure exhibits a dichotomy according to the interaction range. For $s \in (d-2,d)$, corresponding to the weakly long-ranged regime, the equilibrium measure always retains a non-trivial bulk component and thus is never completely confined to the wall. In contrast, for $s \in (d-3,d-2]$, corresponding to the strongly long-ranged regime, a confinement transition occurs: there exists a critical wall position beyond which the equilibrium measure is supported entirely on the wall. Our theorem generalises recent results established for the Coulomb gas, corresponding to the case $s=d-2$.

This paper presents a data-driven stable manifold (DD-SM) method, which integrates Koopman operator representation learning with the geometric stable manifold approach to Hamilton-Jacobi-Bellman (HJB) equations, enabling end-to-end optimal feedback control synthesis from raw trajectory data without prior knowledge of system dynamics. We construct an augmented control system under a unified symmetric subspace decomposition (SSD) and extended dynamic mode decomposition (EDMD) framework for joint approximation of the drift field, control matrix and their spatial derivatives, and derive probabilistic finite-sample error bounds for invariant and non-invariant dictionary spaces to yield a provably accurate approximate characteristic system of HJB equation. Via Lyapunov-Perron operator and ODE perturbation analysis, we prove the data-driven stable manifold achieves monotonically decreasing semi-global error with growing training data. We further establish closed-loop exponential stability and quantify the optimality gap, both tightenable by refining model accuracy. An efficient algorithm pipeline with adaptive data generation and deep neural approximation is developed, outputting control signals within 1 millisecond. Experiments on a modified van der Pol oscillator verify the effectiveness of our method.

We construct the first countably infinite family of new universal phantom categories on a smooth rational surface, namely the blow-up of the complex projective plane at ten points in general position. Moreover, these phantom categories are pairwise non-equivalent, are not equivalent to any previously known phantom category on a smooth rational surface, and arise as the orthogonal complements of non-full exceptional collections of line bundles of maximal length. As an application, we show that all of these phantom categories admit bounded $t$-structures.

Partial path interception (PPI) asks for a minimum-cost vertex set whose removal from a directed acyclic graph leaves at most a prescribed number of designated terminals able to reach a protected sink. We study how costs, depth, and fan-out affect its complexity. Unit-cost PPI is NP-complete at depth two with exactly two paths per terminal. Unique-path and individual-coverage instances admit polynomial-time algorithms, and bounded treewidth yields fixed-parameter tractability. Parameterized by the terminal threshold, weighted PPI is in XP and W-one-hard already at depth two, with no subexponential-time algorithm in that threshold under ETH. At depth two, undeletable terminals with bounded fan-out admit an FPT algorithm parameterized by the terminal threshold plus the fan-out bound. For unit costs, a dual-gammoid rank identity characterizes the optimum. Unit-cost depth-two instances with intermediate in-degree at most two are solvable in polynomial time for every terminal threshold, whereas depth-three instances are W-one-hard even with terminal out-degree one. On general directed acyclic graphs, minimum cuts and submodular optimization give a randomized approximation whose factor is the smaller of the coverage demand and roughly the square root of the number of initially exposed terminals, up to logarithmic factors. Reductions from smallest specified-edge subgraph, densest subgraph, and minimum union transfer conditional approximation lower bounds. PPI also gives an exact formulation of proof-valid caching under conjunctive derivation semantics: under independent premise erasures, the recovery probability equals one minus the erasure rate raised to the number of exposed premises. We extend this formulation to shared workloads, derive a sorting-based optimum under exact module routing, and give an AND-OR reliability envelope and a separate coded payload-recovery benchmark.

The pentagram map sends a polygon to the intersections of consecutive short diagonals. We study its shape dynamics after translating and positively rescaling each iterate to restore unsigned area and barycenter. For polygons whose dynamics is generated, after passing to a finite iterate and cyclic relabelling, by one projectivity, we prove a spectral dichotomy. A dominant real projective line yields flattening and unbounded diameter, whereas a dominant real eigenvalue with a subdominant non-real pair produces asymptotic motion on concentric homothetic ellipses. We apply this framework to pentagons and hexagons via Glick's operator and to Poncelet polygons via the Darboux–Schwartz projectivity. We obtain spectral diagrams and a Jacobi-function formula for the Poncelet return spectrum. Consequently, we recover Schwartz's long-and-thin result for strictly convex non-projectively-regular pentagons, prove a line-or-ellipse dichotomy for convex hexagons under explicit nondegeneracy assumptions, and establish flattening for strictly convex non-projectively-regular Poncelet polygons.

Let $p$ be a prime number. A $D^{\Delta}$-pair is a pair consisting of a finite $p$-group and a $p'$-automorphism of the group. In this paper, we introduce diagonal $D^\Delta$-pair bisets and a category whose objects are $D^\Delta$-pairs and whose morphism groups are Grothendieck groups of diagonal $D^\Delta$-pair bisets. Our main result shows that, over suitable coefficient rings, the category of diagonal $p$-permutation functors is equivalent to the category of linear functors on a natural quotient of this new category. In this way, diagonal $p$-permutation functors can be studied through a category built only from finite $p$-groups and their automorphisms of $p'$-order.

Jacobi permutations, invented by Viennot in the context of the Jacobi elliptic functions, are counted by the Euler numbers. Recently, Henke, Hoffman, Stephens, Yuan, and Zhuang studied refined enumerations of Jacobi permutations and proposed three conjectures concerning the distribution of several statistics on Jacobi permutations. In this paper, we prove these conjectures by establishing explicit bijections involving increasing even trees, increasing binary trees, alternating permutations, and André permutations. One highlight of our results is a bijection between Jacobi permutations and André I permutations that transforms the pair of statistics $(\Ascbot, \last)$ to the pair of statistics $(\Desbot, \first)$. Here the statistic $\Ascbot$ (resp., $\Desbot$) denotes the set of ascent bottoms (resp., descent bottoms) of permutations, and the statistic $\first$ (resp., $\last$) denotes the first (resp., $\last$) letter of permutations. Furthermore, we investigate pairs of statistics on André permutations and simsun permutations that are equidistributed with the pair $(\asc, \last)$ on Jacobi permutations, where $\asc$ denotes the number of ascents of permutations. Finally, we obtain a closed-form formula for the trivariate exponential generating function of Jacobi permutations with respect to the number of ascents and the numbers of letters smaller and larger than the last letter.

Width theory identifies economical spaces for compact PDE solution families, but does not provide a stable adaptive selection rule. We introduce Schur-Riesz refinement, which compares ordinary h/p refinement with operator-informed functions on a common variational scale. Projection removes content already represented by the incumbent, Riesz bounds test coefficient stability, and exact Schur gain ranks the surviving directions. New non-regression, bulk-contraction, and mixed near-oracle results provide finite-run error, dimension, and work certificates for coercive Galerkin and noncoercive minimum-residual formulations. We applied Schur-Riesz refinement across coercive and noncoercive PDEs. At matched dimension, automatic modes reduce held-out error by $50.2\%$ for 2D heterogeneous Helmholtz and $30.9\%$ for 2D Darcy. At matched Darcy error, the selected space uses $38.2\%$ fewer coordinates, reducing deployment memory by $38.9\%$ and online time by $30.1\%$. The additional offline construction is reusable and can be amortized across multiple right-hand sides for a fixed operator. On locked 3D heterogeneous Helmholtz tests, the method reduces mean error by $25.1$-$47.3\%$ against matched spectral-polynomial spaces and meets target errors with 24-96 coordinates, versus 1,331-6,859 for native MFEM hp-refinement. In an adaptive audit, 48 coordinates attain mean error $.1092$, compared with $.0967$ using 878 polynomial coordinates, while making online solves $6.8\times$ faster. Thus, the experiments demonstrate that Schur-Riesz refinement can construct smaller stable spaces with lower deployment memory and repeated-solve cost, while preserving a certified incumbent when richer functions do not help.

We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations $x \mapsto x+\alpha$ on $\mathbb{T}$ in terms of the irrationality exponent $\mu(\alpha)$. For the binary coding determined by an interval $[0,\beta)$, let $d_N(x,y)$ be the edit distance between the codings of the orbits of the points $x$ and $y$ up to time $N-1$. For every $\alpha \notin \mathbb{Q}$, almost every $\beta \in (0,1)$, and almost every $x,y \in \mathbb{T}$, we determine the edit-distance exponents $\displaystyle (*) \; \; \limsup_{N\to\infty}\frac{\log d_N(x,y)}{\log N} = \frac{\rho(\alpha)}{\rho(\alpha)+1} \; \;$ and $\; \; \displaystyle\liminf_{N\to\infty}\frac{\log d_N(x,y)}{\log N} = \min\Bigl\{\frac{1}{2},\frac{1}{\rho(\alpha)}\Bigr\}$, where $ \rho(\alpha):=\mu(\alpha) - 1$. We find a surprising transition at the golden mean $\varphi= \frac{1+\sqrt{5}}{2}$: If $ \rho(\alpha) \le \varphi$, then both asymptotics in $(*)$ hold for all $\beta$. However, for every finite $\rho>\varphi$, there is an uncountable set of $\alpha$ with $\rho(\alpha)=\rho$ for which both edit-distance exponents are strictly smaller than the corresponding values in $(*)$ for uncountably many $\beta$. We also derive consequences for aperiodic circle homeomorphisms and Sturmian sequences. For rotations of $\mathbb{T}^d$ coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is $d/(d+1)$.

We revisit the Sinkhorn-Knopp (SK) algorithm for the matrix scaling problem. Despite extensive literature on the global convergence of SK and its variants, its local linear convergence behavior remains less understood. We address this gap by providing the first nonasymptotic local analysis of SK that matches the rate obtained from existing asymptotic Jacobian-based arguments. We show that under certain connectivity conditions, SK is a polynomial-time algorithm for doubly stochastic matrix scaling. With the developed tools, we showcase the local suboptimality of SK and provide accelerated variants. Finally, for dense matrices, we improve the complexity of existing first-order matrix scaling algorithms from $O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$ to $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

Power functions with Niho exponents have attracted considerable attention due to their important applications in sequence design, coding theory, and cryptography. This paper investigates the differential properties of Niho type power functions of the form $F(x)=x^{s(2^m-1)+1}$ over $\mathbb{F}_{2^{2m}}$ with $2\leq s\leq 2^m$. We first establish a general characterization of the differential spectrum of $F(x)$ having at most three nonzero values via its Walsh spectrum. Focusing subsequently on the case $s=(2^k+1)^{-1} \pmod{2^m+1}$ where $\gcd(k,m)=e$, we employ a refined analysis of the number of solutions to certain equations over finite fields. Specifically, it is proved that $F(x)$ is locally differentially $2^e$-uniform when $\gcd(2^k-1,2^m+1)=2^e+1$ and locally differentially $(2^{2e}-2^e)$-uniform when $\gcd(2^k-1,2^m+1)=1$, and their differential spectra are completely determined. These results completely characterize the differential properties of this family and yield new infinite families of locally differentially $4$-uniform power functions.

We use the best linear approximation (BLA) to characterize nonlinearities in IM-DD links. The orthogonal nonlinear distortion is shown to be non-Gaussian. The BLA is used to approximate the optimum modulation depth for different equalizer structures.

This paper introduces a novel classification approach for representation-finite artin algebras in terms of the maximal length of their indecomposable modules of finite length, which we call the representation bound. To this end, we develop methods to compute almost split sequences and establish lower bounds for the lengths of the Auslander-Reiten translates of certain modules over artin algebras. We apply these techniques and results to explicitly classify artin algebras of representation bound $n$ for each positive integer $n \le 4.$

Let $X$ be a finite Euclidean Ramsey set. We prove that adjoining any point outside the affine hull of $X$ gives another Euclidean Ramsey set, answering a conjecture of Ivan, Leader, and Walters. We first give an elementary product proof under the additional assumptions that the orthogonal projection of the new point lies in $conv(X)$ and that its distance from $aff(X)$ is sufficiently large. We then prove the general case by combining the product theorem for $E$-Ramsey configurations and Kříž's orbit-gluing theorem with a cyclic construction. No transitivity assumption on $X$ is needed.