Papers

We study variable-length feedback (VLF) codes over a discrete memoryless channel under average decoding-time and error-probability constraints. In the non-vanishing error probability regime, Polyanskiy, Poor, and Verdú (2011) derive achievability and converse bounds on the logarithm of the maximum achievable codebook size. These bounds establish the $\epsilon$-capacity but leave an order-$\log N$ gap in the second-order expansion, where $N$ is the average decoding time. Yavas and Tan (2025) improve the coefficient of $\log N$ in the achievability bound from $-1$ to $-\frac{C}{C_1}$, where $C$ is the channel capacity and $C_1$ is the largest Kullback–Leibler divergence between two conditional output distributions. We derive a converse with the same coefficient, establishing the second-order fundamental limit for every positive-capacity discrete memoryless channel with finite $C_1$. The result also covers the moderate-deviations and error-exponent regimes, including polynomially decaying error probabilities. The converse uses Rényi entropy and the extrinsic Jensen–Shannon divergence. We also derive necessary properties of asymptotically optimal VLF codes. First-order-optimal codes must have an early-stopping branch, and second-order-optimal codes must additionally exhibit communication and confirmation behavior. Finally, for the binary erasure channel, we determine the exact minimum expected decoding time for every message-set size and admissible error probability.

This article introduces four parallel numerical techniques for computing entries of the inverse of large sparse symmetric systems, all grounded in incomplete $LDL^T$ (ILDL) factorizations: (1) the selected inversion method (SelInv), which applies the $LDL^T$ factorization to recover entries of the matrix inverse within the sparsity pattern of the computed factors; (2) an approximate inversion method based on a truncated Neumann series expansion applied to the inverse of the L factor (NInv), providing an alternative at the cost of reduced accuracy; (3) a Mix approximation that merges the best of both SelInv and NInv; and (4) Mix-SPAI, which applies sparse approximate inverse (SPAI) refinement on the output of the Mix method to improve entry-level accuracy. To further improve accuracy while maintaining a stable sparsity pattern, we additionally employ a low-rank correction based on eigenvector updates, providing an alternative to tightening the drop tolerance. We report the performance of the proposed numerical techniques on a comprehensive collection of sparse matrices from scientific and industrial applications.

The minimal Kolmogorov complexity of a string within Hamming distance r of a given string x is the algorithmic rate-distortion function of x, and Vereshchagin and Vitanyi characterized completely which shapes it can have. This paper is about the opposite extreme. For a binary string x of length n let g_x(r) denote the maximal Kolmogorov complexity of a string within Hamming distance r of x; we study which values, and more generally which functions of r, this quantity can attain. First we characterize, up to an additive error O(log n), the possible values of the triple (C(x),r,g_x(r)): writing r_k for the radius of a Hamming ball of cardinality about 2^k, a triple (k,r,l) is realizable if and only if log V(r_k + r) < l < min{n, k+log V(r)}, where V(a) is the cardinality of a ball of radius a. In particular, for r_k+r > n/2 both bounds collapse to n and only l = n is realizable. The two ends of this interval correspond to the two extreme ways of placing a set of complexity k in the cube: a single Hamming ball, where the lower bound comes from Harper's isoperimetric inequality, and an error-correcting code, which for the intermediate parameters we relax to a family of centers with bounded covering multiplicity, in the spirit of list decoding. Then we turn to the function r -> g_x(r) as a whole: we establish four properties that it always has, and show that the minimal and the maximal functions consistent with these properties are both attained, for every complexity level k. Which intermediate profiles are attainable remains open.

Fix positive integers $N,k,n$ with $n\ge k$. We seek the minimum number of members of size at least $n$ in a finite family of finite sets closed under union and containing exactly $N$ distinct sets of size $k$. This problem is a specialization of the Leck–Roberts–Simpson weighted conjecture: assign weight one to sets of size at least $n$ and zero to smaller sets. The predicted minimizer consists of the unions of nonempty subfamilies of the first $N$ $k$-subsets of the natural numbers, ordered by their largest elements and, when these agree, by their increasing lists lexicographically. For an integer $t\ge 1$, call the range \[ \binom{n+t-1}{k}<N\le\binom{n+t}{k} \] the $t$-th strip. We prove the layered conjecture throughout the first strip, and throughout the second strip for $k=3$. For arbitrary $k$, we prove the second strip for families of subsets of an $(n+2)$-element set. For $k,t\ge 3$, we prove the $t$-th strip for families of subsets of an $(n+t)$-element set whenever $n\ge(t+1)(k-1)$. With no restriction on the ground set, we prove it for $k\ge 3$ and $t\ge 2$ whenever $n>\frac{5}{2} k^2t$. For sufficiently large $k$, we obtain a sufficient bound of order $k^2t/\log k$, uniformly in $t\ge2$.

This paper develops an explicit spectral-contraction approach to studying fully coupled nonlinear forward–backward stochastic difference equations on infinite horizon and their applications to stochastic control. The estimates for this fully coupled system are assembled into an explicit two-dimensional nonnegative matrix. Its spectral radius yields an explicit sufficient discount threshold, and a corresponding equivalent weighted product norm is constructed to establish the contraction property. Moreover, for an infinite-horizon control problem with convex control constraints and an accumulated discounted running cost, we derive a Pontryagin-type stochastic maximum principle, its equivalent pointwise normal-cone formulation, and a verification theorem. Finally, a recursive risk-adjusted portfolio example is given to demonstrate the applications of our theoretical results, and a projection-type sufficient optimality condition for this example is derived.

We utilize Lempel-Ziv universal compression for music note generation. We control the algorithm's tendency to over-copy or under-copy training data by manipulating the average sequence length saved in the dictionary.

How does continuum operator structure appear in a finite qubit register? We address this question for analytic functions of diagonal operators represented in a binary basis, focusing on the Jordan–Lee–Preskill (JLP) momentum operator, by developing an operator-level generating-function framework for their Walsh-Pauli spectra. The construction determines the spectrum analytically and reveals exact support and parity selection rules, together with a nontrivial intra-shell hierarchy induced by the binary encoding. This provides a systematic characterization of Walsh compressibility and its relation to Pauli locality. As a controlled application, we consider generalized uncertainty-principle (GUP) kinematics as a tunable nonlinear odd deformation of momentum, showing how higher-order momentum terms redistribute spectral weight into progressively higher Pauli-weight sectors

We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit relations and estimates involving the volume of the manifold and the geometry of its boundary. In the scalar-flat case, we prove global splitting and Ricci-flat rigidity results, including for disconnected boundary, while in the negative scalar curvature case we establish a sharp mean-curvature bound and characterize the equality case by an exponential warped-product structure. The proofs rely essentially on the study of the associated Einstein manifold. In appendix we derive several identities for static manifolds with boundary and discuss the associated Einstein manifold technique in the boundaryless setting.

In this paper, we consider the classical time-optimal elevator problem in the presence of higher-order state constraints. The main contribution is a constructive indirect solution framework based on a Pontryagin Maximum Principle specifically formulated for higher-order constrained systems. First, we derive specialized optimality conditions and analyze the resulting structure of extremal trajectories. Second, we show that the infinite-dimensional optimal control problem can be transformed into a finite-dimensional system of nonlinear algebraic equations involving switching times, boundary-contact times, and multiplier parameters. This provides a computationally efficient procedure for calculating candidate optimal trajectories using standard nonlinear equation solvers. Third, we characterize the geometry of boundary contacts and explain the emergence of contact-chattering phenomena in higher-order constrained systems. In particular, we show that normal extremals cannot evolve along the constraint boundary for a positive amount of time, although sequences of boundary contacts may accumulate indefinitely. The proposed approach is demonstrated on a fourth-order elevator model, and the computed solutions are independently verified using a direct IPOPT-based optimization method.

Let $H_n(q)$ be the generic Iwahori–Hecke algebra associated to the symmetric group $\mathfrak{S}_n$ of degree $n$ and let $\mathscr{Y}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\leq i\leq \binom{n}{2}$. We show that $\mathscr{Y}_{n,1}$ and $\mathscr{Y}_{n,2}$ are noncommutative whenever $n\geq 4$, which disproves Doikou's Conjecture on the commutativity of the shuffle elements of $H_n(q)$ in (Nuclear Physics B 1029 (2026): 117532). We also show that the matrix of the operator of the left multiplication by $\mathscr{Y}_{n,i}$ in $H_n(q)$ is $q$-symmetric for all $i$.

For a wide class of Helmholtz scattering problems (namely, those fitting in the black-box framework of Sjöstrand–Zworski) we prove that the error incurred by approximating the radiation condition by a Cartesian perfectly-matched layer (PML) decreases exponentially in the width of the PML, the strength of the PML scaling, and the frequency.

Seasonal reservoir operation requires optimization methods that remain reliable under strongly periodic and uncertain inflow dynamics. We present InFlow, a verification-oriented framework combining periodic stochastic dual dynamic programming (SDDP), entropic robustness, control-space Bellman annealing, and an independently discretized Hamilton-Jacobi-Bellman (HJB) benchmark. The entropic transition operator is interpreted as a KL-penalized worst-case expectation and linked to fixed-radius KL ambiguity sets through Lagrange duality. We establish a uniform soft-Bellman approximation bound and prove the validity of Gibbs-tilted SDDP cuts. The benchmark relies on a normalized seasonal Cox-Ingersoll-Ross inflow process whose positivity condition fails during the driest weeks, requiring explicit treatment of the degenerate boundary. Independent verification shows strong agreement between HJB storage gradients and HiGHS storage-balance duals, with water-value correlations of 0.995 and 0.996 for the standard and exact-cut-only implementations. Refining the HJB grid from 61^2 to 81^2 states changes the annual mean water value by only 0.6% and the peak value by less than 0.1%. Out-of-sample simulations based on 3,000 independent trajectories show that a robustness level of gamma=5 increases nominal mean cost by 4.6% while reducing CVaR90 by 11.2%. Under a dry-season stress scenario with inflows reduced by 40%, the mean-cost difference becomes statistically insignificant whereas CVaR90 improves by 11.5%. A robustness sweep confirms monotonicity of the risk-adjusted value. Finally, a two-reservoir cascade reproduces grid-oracle marginal water values with correlation 0.999 and recovers the predicted upstream-to-downstream value ratio of two. InFlow is intended as a transparent methodological benchmark rather than a calibrated operational reservoir model.

In this paper, we study finite dimensional representations of shifted quantum affine algebras of type A. We give an explicit description of the tensor product of simple evaluation modules of the quantum loop algebra and a one-dimensional representation of the shifted quantum affine algebra under the separation condition. As a consequence, we give the q-characters of some finite dimensional simple modules of the shifted quantum affine algebra.

We study quasilinear symmetrizable partially dissipative hyperbolic systems with non-autonomous relaxation coefficients in $\mathbb{R}^d$ ($d\geq1$). The existence of global strong solutions is established in a critical regularity setting for systems satisfying the so-called Shizuta-Kawashima (SK) and entropy conditions. When the initial data are additionally bounded in a lower-regularity norm, we prove that the corresponding solutions converge to equilibrium at optimal algebraic decay rates. Furthermore, we show that the conservative part of the solution behaves asymptotically as the solution of a non-autonomous parabolic equation. Our results apply to the compressible Euler system with the time-dependent damping coefficient $\frac{K}{(1+t)^{\alpha}}$ ($\alpha<1$, $K>0$ or $\alpha=1$, $K\gg 1$) in the velocity equation. The natural low/high-frequency splitting of the autonomous theory persists in the non-autonomous setting, but with a frequency-threshold that evolves in time. To handle this moving frequency structure, we introduce a new class of hybrid Besov spaces adapted to time-dependent thresholds and derive hypocoercive estimates in each frequency regime. Our results reveal the qualitative and quantitative effects of general time-dependent relaxation coefficients on dissipation and large-time dynamics.

We ask whether it can be certified algorithmically that a self-modifying program keeps a behavioural property, a safety property in the motivating case, after its next rewrite (preservation) and along its whole evolution (persistence). When the rewrite depends only on behaviour, preservation is a behavioural property and Rice's theorem applies. When the rewrite reads the code, preservation is no longer behavioural; yet, under a uniform disruption condition, the s-m-n reduction that proves Rice's theorem works inside a single class of behaviourally identical programs, and preservation inherits the degree of the halting problem. One step never exceeds the degree of the property, while persistence can climb one level of the arithmetical hierarchy. We then isolate the mechanism shared by rewriting, supervision and system comparison, the elevation operator, and prove a normal form: the preserving set is determined by a single finite trigger and a polarity, and the Rice-Shapiro theorem restricts the polarity to the arithmetical class of the property. Runtime monitors, consistency supervision, conformance to a reference and observational equivalence are instances, and no sound theory covers the preserving systems.

This is the first part of a project concerning the regularity of the free boundary of a function, where the free boundary depends on the gradient of the function. We study the minimizer of the expression \begin{equation*} J(u) : = \int_{B_1} F(\nabla u) \ dx, \end{equation*} where $F(x)$ is a uniformly convex function whose second derivatives might jump at $|x|=1$. This results in an Euler-Lagrange equation that varies over the free boundary, which we define as $$ \Gamma = \partial \{x \in B_1: |\nabla u| >1 \} \cap \partial \{x \in B_1: |\nabla u| <1 \}. $$ We consider two-phase flat points for which the second derivatives of $F$ jump over $\Gamma$. In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer $u$ can be expressed as $$ u(x)= a + \nu \cdot x+ \delta p(x) + \delta \epsilon q(x), $$ where $a \in \mathbb{R}$, $\nu \in \mathbb{R}^n$, $0 < \delta, \epsilon << 1$. Here $p$ is a $C^1$ function, which consists of one polynomial in the upper half ball and another polynomial in the lower half ball. The function $q$ is a rest term with bounded $L^2$ norm. Furthermore, assuming that we have a sequence, $u^j$, of minimizers, we show that $q^j$ converges weakly to a function $q^0$ that satisfies a certain PDE. In addition, we show that $q^0$ is $C^3$ in $B_r^+$ and $B_r^-$, respectively, for $0<r<1$. This is the main result of this paper which is intended to be used to show regularity of the free boundary.

For a finite simple graph $G$ and an integer $k\ge0$, let $w_k(G)$ denote the number of walks of length $k$. We prove the conjecture of Täubig, Weihmann, Kosub, Hemmecke, and Mayr for every finite tree and determine all equality cases. If $T$ has $n\ge1$ vertices, then $n w_{k+1}(T)-2(n-1)w_k(T)\ge0$ for every $k\ge1$; for $n\ge3$, equality holds if and only if $T$ is a star and $k$ is even, whereas for $n=1$ or $n=2$, equality holds for every $k\ge1$. For non-Dynkin trees and even indices, the proof is based on a polynomial positivity cone associated with the adjacency operator of a finite graph and a positive real root of its simply-laced Coxeter system. For finite connected bipartite non-Dynkin graphs, we establish sufficient positivity conditions in terms of Coxeter orbits and inversion sets, and verify these conditions for indicator roots supported on connected induced subtrees. For non-Dynkin trees, this yields the rooted even-index inequality and, after summation, the corresponding global inequality. We also prove that if $G$ is a finite connected bipartite non-Dynkin simple graph, $\varnothing\ne U\subseteq V(G)$, and the subgraph of $G$ induced by $U$ is a tree, then $|U|w_{k+1}(G,U)-2(|U|-1)w_k(G,U)\ge0$ for every $k\ge0$, where $w_k(G,U)$ counts the length-$k$ walks in $G$ whose initial and terminal vertices lie in $U$; the intermediate vertices are unrestricted. The remaining even-index cases for finite Dynkin trees are handled by generating-function recurrences, while the odd-index cases follow from a spectral covariance identity.

We establish the existence of a three-parameter family of compact quantum metric space structures on cocycle twisted crossed products by discrete groups. We are mainly interested in the case where the acting group has exponential/subexponential growth. We prove that the family is jointly continuous with respect to the parameters when the acting group is exact. We obtain quantitative upper and lower bounds for the associated metric dimensions. In particular, the bounds are helpful to prove the failure of lower semicontinuity of the metric dimension with respect to the quantum Gromov-Hausdorff distance. We also prove invariance of metric dimension under zero quantum Gromov-Hausdorff distance.

We study the evolutionary equation $$\frac{\partial u}{\partial t} = (\mathcal{L}_{\infty})^{\alpha}u \quad \text{in} \quad \Omega,$$ where $(\mathcal{L}_{\infty})^{\alpha}u$ denotes a nonlocal infinity Laplacian acting on the function $u$, $0<\alpha \leq 1$ and $\Omega$ is a bounded open set in $\mathbb{R}^{n+1}$. We prove existence and uniqueness using Perron's method for the Dirichlet problem when $\Omega$ is a cylinder and $0<\alpha<1$.

We establish a sharp criterion for global-in-time existence versus finite-time blow-up of strong solutions to the 3D spherically symmetric compressible Navier-Stokes equations on a solid ball, based solely on the initial sign of effective velocity. The viscosity coefficients are assumed to satisfy the Bresch-Desjardins structure $\mu=\rho^{\alpha}$, $\lambda=(\alpha-1)\rho^{\alpha}$, with the effective velocity given by $v=u+\alpha \rho^{\alpha-2}\rho_r$. Previously, global existence results for strong solutions in higher dimensions were restricted to the case $\alpha\le 1$ with the endpoint $\alpha=1$ corresponding to the viscous Saint-Venant (shallow water) system. In this paper, we extend the global existence theory beyond this threshold to the supercritical regime $\alpha>1$. Specifically, whenever the initial effective velocity is nonnegative on the boundary of the ball, we prove the global-in-time existence of strong solutions for arbitrarily large initial data. The main difficulty lies in deriving a uniform lower bound for the density, which requires handling singular estimates at the center of the ball. To overcome this, we exploit several novel quantities near the center and employ a refined maximum principle to establish, for the first time, the Lipschitz continuity of the velocity at the center. Subsequently, we derive Dini-Gronwall-type inequalities for suitably constructed functions, which close the density lower bound estimate. Conversely, we can construct a family of initial data for which the initial effective velocity is negative on the boundary, such that the corresponding strong solutions blow up in finite time, with vacuum appearing exactly on the boundary at the blow-up time. Our results establish a clean dichotomy between global regularity and singularity formation, governed purely by the initial sign of the effective velocity.

We study when the image of a measure under a random Hölder map attains full packing dimension. Our main tool is a family of packing intermediate dimension profiles $\dim_{P,\theta}^{s}\mu$, indexed by $\theta\in(0,1]$ and $s>0$, which refine the packing dimension profiles of Falconer and Howroyd and reduce to them at $\theta=1$. For a large family of random $\alpha$-Hölder maps $f_\omega:\mathbb{R}^n\to\mathbb{R}^m$, which includes index-$\alpha$ fractional Brownian motion as a particular case, we prove that for every compactly supported Borel probability measure $\mu$ on $\mathbb{R}^n$, $$\dim_P \mu_{f_\omega}=m \quad\text{almost surely} \quad\Longleftrightarrow\quad \alpha m\leq \lim_{\theta\to 0}\dim_{P,\theta}^n \mu.$$ We further study the profiles $\dim_P^s\mu$ themselves, obtaining a quantitative lower bound and a Marstrand-type identity for their limiting behavior as $\theta\to 0$. Finally, we obtain the analogous characterization for analytic sets: $$\dim_P f_\omega(E)=m \quad\text{almost surely} \quad\Longleftrightarrow\quad \alpha m\leq \lim_{\theta\to 0}\dim_{P,\theta}^n E,$$ where $\dim_{P,\theta}^n E=\sup\{\dim_{P,\theta}^n \mu :\, \mu\in \mathcal{M}_c^+(E) \}$.

We study the continuity equation with reaction $\partial_t\mu + \operatorname{div}(v\mu) = w\mu$ on $[0,T]\times\mathbb{R}^d$, driven by Borel velocity and reaction fields $v$ and $w$. We provide the first version of a Lagrangian superposition principle for the equation in its natural generality, assuming only finite quadratic energy without imposing boundedness conditions on the reaction term neither any form of compactness for the support of the measure. More precisely, every solution $\mu \in \mathcal{C}([0,T];\mathscr{M}_+(\mathbb{R}^d))$ with finite quadratic energy is represented by a probability measure $\eta$ on absolutely continuous curves in the geometric cone over $\mathbb{R}^d$, concentrated on the solutions of the characteristic system $x' = v(x)$, $k' = w(x)\,k$, through the $2$-homogeneous marginal $h^2_t(\eta)=\mu_t$. We also establish the converse implication. The proof passes through a regularization of the triple $(v,w,\mu)$ which produces fields satisfying only local bounds; the core of the argument is therefore a representation theory under such local assumptions, whose key tool, of independent interest, is a two-time representation formula relating $\mu_s$ and $\mu_t$ along the flow of $v$ for arbitrary times $s,t$.

We study locally finite-dimensional ideals in crossed products of totally disconnected spaces by free actions of the integers and in uniform Roe algebras of exact discrete groups. In the first case, we present a dynamical description of the largest locally finite-dimensional ideal, which turns out to coincide with the intersection of all maximal ideals. In the latter case, we provide a coarse geometric characterization of the locally finite-dimensional compact ideals. Moreover, we show that for crossed products of totally disconnected spaces by free actions of exact groups, the largest type I-ideal is locally finite-dimensional. In the case of uniform Roe algebras, we provide coarse geometric conditions for compact ideals guaranteeing that the ideal is type I and admits an embedding of a UHF-algebra, respectively.

The increasing vulnerability of Global Navigation Satellite Systems has motivated renewed interest in Doppler-based navigation using low Earth orbit satellites, which can determine position, velocity, clock bias, and clock drift from carrier Doppler measurements. However, the geometric dilution of precision (GDOP) in this eight-state problem behaves fundamentally differently from the GDOP in pseudorange-based navigation. In particular, volume-based satellite selection, effective for pseudorange-based GDOP minimization, has been empirically found to perform poorly for Doppler GDOP. This paper establishes a geometric foundation for Doppler GDOP characterization. A closed-form geometric parameterization of the Doppler measurement Jacobian in terms of elevation, azimuth, inclination, and altitude ratio is derived. It is shown that the clock bias sensitivity depends on elevation, altitude, and the angle between the satellite velocity vector and the line of sight. This sensitivity produces a correlation with the clock drift that no geometric arrangement can remove. A Schur complement decomposition of the eight-state information matrix is performed, yielding an exact GDOP inflation formula, governed by a collinearity coefficient that measures the alignment between the clock bias column and a seven-state subspace. It is proven that the geometric coupling that renders clock bias observable is the same coupling that inflates estimation covariance, and that satellite altitude diversity reduces the collinearity by separating satellites at equal elevation onto distinct sensitivity bands. An example employing a two-shell satellite configuration illustrates the analysis.

We introduce a family of discrete hyperbolic secant distributions on $\mathbb{Z}$, whose normalizing constants arise from series values calculated by Ramanujan and are expressed in terms of Gauß' constant $G=\varpi/\pi$, where $\varpi=\Gamma^2(1/4)/(2\sqrt{2\pi})$ is the lemniscate constant. Using the elliptic lambda-star function $\lambda^*$, we construct scaled versions of these distributions parametrized by $\sqrt{r}$ and $1/\sqrt{r}$ for $r \in \mathbb{N}$. For the first such distribution, we compute the moments up to the eighth degree in closed form via Poisson summation, exploiting that the hyperbolic secant is a fixed point of the $-2\pi i$-Fourier transform. As a byproduct, we obtain closed-form values for series of odd powers of the hyperbolic secant up to the ninth degree, e.\,g.\ $\sum_{k \in \mathbb{Z}} \operatorname{sech}^3(\pi k) = (G^3+G)/\sqrt{2}$, and reinterpret several classical Ramanujan series probabilistically. Finally, we apply our results to evaluate a contour integral, on the critical line, of the product of Dirichlet's beta, gamma, and Riemann zeta functions.

The dynamic mode decomposition (DMD), along with its variant for nonlinear systems, the extended DMD (EDMD), are powerful tools for the extraction of meaningful spatio-temporal characteristics of (non)linear dynamical systems from measurement data. Despite some efforts to handle the identification task when dealing with real-world data, the decomposition based methods face a critical challenge: real-world data has two inherent sources of uncertainty, the process and measurement noise. Subspace identification methods, are robust tools able to provide accurate state-space models for multi-variable linear systems directly from input-output data. Combining these two methods, we introduce the p-q quasi-norm Subspace Extended Dynamic Mode Decomposition (pqSEDMD). An algorithm that uses our previous improvements to the EDMD by the use of a p-q-quasi-norm reduction on an orthogonal polynomial basis, the pqEDMD algorithm, along with subspace identification methods. The result is a robust approximation of nonlinear systems in a linear function space, combining the strengths of the two methodologies. Throughout the paper we will use the Duffing oscillator as a benchmark problem to show the effectiveness of the algorithm and illustrate many important aspects related to the development.

We present self-contained martingale proofs of Talagrand's convolution conjecture in the Gaussian and Boolean settings with explicit dimension-free constants. The argument is based on the analysis of the stopped density martingale and a weighted Itô isometry estimate. The Gaussian case was settled by Eldan and Lee, and Lehec; the Boolean case was proved up to a $\log\log$ factor by Chen and recently settled by Lu, Guo and Fang.

Over-the-air computation (AirComp) aggregates distributed data through the wireless multiple-access channel, but coherent implementations require channel state information (CSI), phase alignment, and power control, whereas non-coherent energy methods remain affected by fading. Signal superposition at the receive antenna is linear but requires coherence, and energy superposition is linear only in expectation over fading. We introduce occupancy-domain computation (ODC), whose observable is neither a received amplitude nor an energy: the sum is carried by the silence of the shared resources. With exponential Bernoulli activation, the individual silence probabilities multiply, and the server recovers the sum from the idle fraction using binary activity decisions alone, so that once an activation is detected the amplitude that produced it does not enter the estimate. We characterize the maximum-likelihood estimator, optimal load, and a scale-integrated Fisher-information bound for non-adaptive operation over unknown dynamic ranges. We then introduce balanced occupancy computation (BOC), where each device forms a data-dependent quota of random burst placements. This removes the random placement-count fluctuation of Bernoulli activation; under ideal detection, the leading-order asymptotic root-mean-square error of BOC is no larger than that of Bernoulli ODC at any load and approaches $1/\sqrt{2M}$, where $M$ is the number of resource elements, as the number of devices becomes small relative to $M$. We further analyze unknown-scale operation, finite-frame deviations, and heterogeneous detection misses. Simulations validate the theory and compare ODC/BOC with affine non-coherent energy aggregation and REED.

Recently, Jankowski introduced the Serre criteria for superschemes to define the super versions of normality and reducedness (called generic fermionic regularity and reducedness). We show that for a flat, proper morphism $f:X\rightarrow Y$, where $X$ is a superscheme and $Y$ is a Dedekind scheme of finite type over a field of characteristic $0$, the set of $y\in Y$ for which the fibers $X_y$ are normal, GFRR, regular, and Cohen-Macaulay is open in $Y$. In particular, if one has a degeneration over $\mathbb{A}_{\mathbb{C}}^{1|0}$, this shows that these properties can be lifted from the special fiber to the generic one.

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard–Milanič–Štorgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.