Papers

Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann–Poincaré operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min–max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}\lambda_j^+(\Omega)\ge \frac{d-2}{2}.$ It follows that $\lambda_1^+(\Omega)\ge \frac{d-2}{2d},$ with equality if and only if $\Omega$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin–Shtrikman inequality $|\Omega|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partial\Omega}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton's theorem identifies $\Omega$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.

We provide some information about the Drinfeld center $\Z(\Ver_p)$ of the Verlinde category $\Ver_p$. We compute explicitly the graded Cartan matrix, bound quiver and cohomology of $\Z(\Ver_p^+)$, and prove that the category $\mathscr{Z}(\Ver_p^+)$ is wild for every $p\ge 7$. In the special case $p=5$, we show that $\Z(\Ver_5^+)$ has exactly $10$ non-isomorphic indecomposable objects, classify them and describe the Green ring of $\Z(\Ver_5^+)$, and compute the semisimplification of $\Z(\Ver_5^+)$.

Feature selection is a highly relevant task in a data-driven knowledge discovery project. Several techniques have been developed aiming at finding the features that influence most an outcome to predict, including mutual information and, in recent years, the data-based sensitivity analysis. The present research focus on analyzing the advantages and disadvantages of each of these two techniques, by applying both to a bank telemarketing case. Thereafter, a logistic regression model is built on the tuned set of features identified by each of the two techniques as the most influencing set of features on the success of a telemarketing contact, in a total of 13 features for mutual information and 9 features for the data-based sensitivity analysis. The latter performs better for lower values of false positives while the former is slightly better for a higher false positive ratio. Thus, mutual information becomes a better choice if bank managers intend to reduce slightly the cost of contacts without risking losing a high number of successes. Such results show that mutual information, although not recent, is still a valid method for feature selection. On the other side, the data-based sensitivity analysis selection achieved good prediction results with less features.

A generalized monoid of words tilde{Sigma}* is constructed as the quotient of moderate nets of finite words by an asymptotic equivalence relation based on a bidirectional prefix-suffix metric. The main algebraic result is that this quotient is a monoid containing Sigma* faithfully, with a reversal involution, a natural divisibility preorder, failure of cancellation, and nontrivial idempotents. The construction is designed so that any functional depending only on a logarithmic prefix descends to the quotient, yielding a well-defined action of logarithmic-prefix functionals. Finite scalar values arise only after applying an additional renormalization functional, which depends on the chosen mould and window. With respect to the fixed truncation injection, the monoid strictly enlarges the classical set Sigma^{infty}: an explicit oscillating net is exhibited that has no limit in the Cantor space but defines a genuine element of tilde{Sigma}* not coming from a finite or right-infinite word under that injection.

Selecting the optimal neural-operator prediction during deployment is challenging when high-fidelity reference solutions are unavailable. We demonstrate that under a squared Hilbert-space loss, ranking a finite model library depends strictly on the low-dimensional span of candidate differences, allowing us to score all models simultaneously using a single anchor-based linearized response of the governing equation. This shared physical diagnostic accurately recovered over 99.6% of pairwise preferences and 99.0% of optimal checkpoints across diverse Fourier and convolutional operator libraries for fluid, reaction-diffusion, and wave dynamics. Furthermore, the corrected physical proxy frequently outperformed the best individual candidates, and we establish computable sufficient conditions that rigorously certify exact decisions for strongly monotone discretizations. By exploiting the local dynamical response rather than raw defect magnitude, this framework enables the reliable and highly efficient deployment of scientific surrogates without requiring ground-truth data.

An efficient $k$-limited dominating broadcast, or $k$-ELDB, is a $k$-limited broadcast in which every vertex is dominated exactly once. This notion brings together efficient domination and limited broadcast domination in a common framework. For a graph $G$, we write $mcr(G)$ for the smallest integer $k$ for which $G$ admits a $k$-ELDB. For an admissible value $k\ge mcr(G)$, we denote by $\gamma_{ebk}(G)$ the minimum cost of a $k$-ELDB on $G$, and is called the $k$-efficient broadcast domination number of $G$. In this paper, we study these parameters from an algorithmic perspective with complexity analysis. We develop a dynamic programming algorithm for trees which, for fixed $k$, computes $\gamma_{ebk}(T)$ and thereby obtains a polynomial-time procedure for determining $mcr(T) $. In contrast, we prove that, for every fixed integer $k\ge 1$, deciding whether a graph admits a $k$-ELDB is NP-complete for arbitrary graphs. These results place efficient limited broadcast domination in a natural complexity framework, with trees forming a tractable class and arbitrary graphs remaining computationally hard.

The least primitive root $g(p)$ modulo a prime $p$ is conjectured by Grosswald to satisfy $g(p)<\sqrt{p}-2$ for any $p>409$. We make progress towards the conjecture by proving $g(p)<\sqrt{p}-2$ when $p > 1.77 \times 10^{54}$.

Accounting for information flow on the path space of trajectories of a nonnegative martingale yields exact variational identities for it, even at arbitrary random times. This recovers the widely used classical concentration inequalities, from Ville to PAC-Bayes, and measures what each one discards. The tail a bound controls is itself a relative entropy, resolved by the chain rule into per-step conditional divergences. The discarded slack has an exact form in each of three geometries: a Gibbs tilt for the Azuma-Hoeffding and PAC-Bayes bounds, the crossing itself for Ville's and for pooled tests, and a dominating certificate for the $L^p$ maximal bound. That certificate's optional-stopping deficit resolves per step into Bregman divergences of the running maximum. On a path-time space, the same identity gains one factor that prices anticipation: an arbitrary random time carries an e-process “peeking penalty.” The partition function can be read as a coalescent–a prefix-sharing probability of independent copies–and geometric mixtures of test martingales gain a pooling benefit for multi-model safe testing.

We show that for Abelian Chern-Simons topological orders, torus Berry matrices determine the all-genus extended TQFT. We identify the topological part of the projective Berry holonomy under metric deformations with the mapping-class-group representation of the Abelian Chern-Simons TQFT and prove that the normalized torus data reconstruct its finite quadratic module (G,q). Recent work showed that (G,q) classifies the extended theory up to symmetric monoidal natural isomorphism, and that the associated Abelian defect extension is determined by the pointed modular category C(G,q) arising from the same finite quadratic-module data. Therefore genus-one Berry data determine not only the all-genus bulk theory but also its associated defect structures without choosing a K-matrix presentation. We also prove that, for normalized character row errors $\delta$<21.96%, nearest-row decoding recovers the Abelian fusion algebra independently of the number of anyons. The result applies to Abelian fractional quantum Hall and spin-liquid phases described by even-lattice Chern-Simons theories.

We study large-sample properties of higher-order Markov chains on a finite alphabet $\Sigma$ when the order $m_n$ is allowed to grow with the sequence length $n$. By embedding the process into a first-order chain on $\Sigma^{m_n}$ and exploiting return-time decompositions, we establish a central limit theorem for additive functionals $\sum_{t}\! g_n(Y_t^{(n)})$ under natural ergodicity and sparsity conditions. The normalization involves the stationary return time to a suitably chosen state and accommodates triangular arrays with $m_n\!\to\!\infty$ and $m_n/n\!\to\!0$. We further illustrate the assumptions in a binary variable length Markov chain (VLMC), deriving explicit lower bounds on stationary masses that yield a concrete growth regime (e.g., $m_n\log m_n/n \to 0$) ensuring the CLT. These results provide asymptotic foundations for inference in sparse/partitioned higher-order models; including VLMCs and sparse Markov models (SMMs) where the effective dimensionality grows with the sample size.

We study $G$-equivariant Hörmander pseudodifferential operators on a noncompact symmetric space $G/K$. We define a notion of a complete symbol function, called the Harish-Chandra symbol function, on the spherical tempered dual of $G$, for operators that satisfy a rapid off-diagonal decay condition on their Schwartz kernels, and we characterize the class of such operators in terms of their Harish-Chandra symbol function.

We present a systematic development of the arithmetic of the class of linear orders under the ordered sum $(LO, +)$ and prove a number of new results. Our approach is based on a Euclidean algorithm for pairs of linear orders that almost additively commute. Among our results: (i.) We generalize and give unified proofs of the main classical theorems for $(LO, +)$, including Lindenbaum's division theorem for $(LO, +)$ and a representation theorem for additively commuting pairs of linear orders due to Aronszajn. (ii.) We solve the following problem, posed by Tarski in 1956: is it true that for every pair of linear orders $A, B$ and quadruple of natural numbers $n, m, k, l \geq 1$, if $nA + mB \cong kB + lA$ then $A + B \cong B + A$? Tarski and Chang showed the answer is yes for certain choices of the coefficients $n, m, k, l$. We show the answer is yes in general. (iii.) We prove the following characterization of the additively commuting pairs in $LO$: $A + B \cong B + A$ if and only if $\omega A$ embeds initially in $\omega B$ and $\omega^* A$ embeds finally in $\omega^* B$, or vice versa. We show this can be viewed as a correctly revised version of a refuted conjecture of Tarski. (iv.) We characterize the commutative semigroups $(S, \oplus)$ that can be represented in $(LO, +)$ and show in particular they are all subsemigroups of naturally totally ordered semigroups in the sense of Clifford.

The Kahn–Lovász theorem gives a sharp upper bound on the number of perfect matchings in a graph in terms of its degree sequence, extending the classical Brégman–Minc inequality for bipartite graphs. In this paper, we establish an asymptotically sharp extension of the Kahn–Lovász theorem to $F$-factors for every Hamiltonian graph $F$. As a consequence, we asymptotically determine the maximum number of $F$-factors in an $n$-vertex $m$-edge graph, yielding an $F$-factor analogue of Kruskal–Katona-type theorems. We also prove a multigraph analogue of the Kahn–Lovász theorem. Combining this with our results for Hamiltonian graphs, we obtain an asymptotically sharp Kruskal–Katona-type bound for a further class of connected graphs $F$, including those containing two vertex-disjoint cycles of equal length whose union spans $V(F)$.

We consider a reference wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region. We give two-point formulas for approximate phase recovering of the radiation solution from the intensity of the total solution for the case of spherical reference wave. We show that these formulas can be used, in particular, for approximate phase recovering from holographic data on a single plane. By these formulas, we continue previous studies with plane wave reference.

Transformational electric vertical take-off and landing (eVTOL) vehicles have gained significant attention over the past decade due to their efficient wing-borne cruise capabilities and reduced reliance on ground-based infrastructure. However, control system design for these vehicles remains challenging, as they must operate across multiple flight phases, each with distinct dominant dynamics. If left unaddressed, this complexity would significantly increase pilot workload, thus motivating the development of pilot control systems for multi-phase flight operations. The Simplified Vehicle Operations concept presents a promising strategy for reducing pilot workload. This study presents the design and implementation of a novel pilot control concept for eVTOL aircraft, validated through a tandem tilt-wing aircraft simulation on a full-motion simulator equipped with an active, force-feedback side stick. The system provides pilots with tactile feedback during specific flight phases, supporting intuitive control. The proposed approach enables seamless transitions and multi-phase flight maneuvers by leveraging the available degrees of freedom. Furthermore, an optimal-control-based methodology is proposed as a metric to evaluate command-filter-induced performance penalties and inceptor activities. The results show that the proposed command filter does not significantly increase the mission duration compared to the closed-loop system, while the active side stick helps reduce inceptor activity.

Variable active power injections arising from device behavior or compromised dispatch complicate voltage regulation in electric power networks with distributed energy resources (DERs). An operator can limit the resulting voltage deviations by remotely selecting DER reactive power parameters before observing the active power injections. IEEE Standard 1547-2018 specifies constant power factor as one such control mode, coupling each device's reactive power to its realized active power. Using a linear voltage model, we formulate the operator's decision as a robust minimax problem in which the operator minimizes the largest feasible aggregate voltage deviation. We solve this problem by expressing the power factor decisions through continuous reactive to active power ratios and exactly decomposing the payoff according to the signs of the voltage deviations. When every feasible voltage residual remains on its initial side of nominal, the resulting ratios cancel each injection's contribution and yield a closed form minimax strategy. We identify realistic DER ratings for which this strategy applies and quantify the regulation capacity lost under restricted power factor ranges. Numerical tests check the cancellation computation, solve the complete minimax problem directly at a representative DER rating, and compare the linear voltage predictions with nonlinear AC power flow.

Sparse pursuit after dictionary learning can yield a precise atom support even when its physical interpretation is not justified by the calibration data, especially for highly coherent dictionaries where alternative calibration-compatible dictionaries may assign different physical meanings to the same selected support. We develop resolution-aware physical-support inference that jointly accounts for uncertainty in the learned dictionary and in the representation of a deployment signal. Our cross-dictionary confidence correspondence retains calibration-compatible dictionaries and deployment-compatible sparse representations, then projects the surviving explanations onto physical-support space. For local coherent-atom classes with separation scale s, once the deployment data resolve the coherent-block explanation and its atom support, the minimax physical resolution from N calibration signals satisfies $\delta_{\mathrm{opt}}(N,s)\asymp\min\{s,\frac{1}{\sqrt{N}s^2}\}$, with relative resolution governed by the orientation-information scale $Ns^6$. Deployment replication improves physical localization only when orientation changes cannot be absorbed by adjusting the active coefficients. For computation, we introduce active endpoint bracketing (AEB), an adaptive finite-bank procedure that evaluates only candidates that can still affect the physical report and otherwise safely coarsens or abstains. Finite-bank experiments, including a four-region synthetic application, show that a point-valued plug-in selector can be physically overprecise, whereas AEB avoids unsupported refinement with fewer candidate evaluations.

The growth rate of Av(1324) is the last unknown Stanley-Wilf limit of a length-four pattern. Since 2020 the best rigorous lower bound has been the 10.271012 of Bevan, Brignall, Elvey Price and Pantone; we raise it to 10.412263. Their construction lays 1324-avoiders along an infinite staircase of two-cell blocks alternating with single connecting cells, and forbids a 1324 by a local condition on how the points of a block interleave with the components of the cell beside it. Certain points of a block, its leaves, may be exempted from that condition. They exempt them on one axis only, and report that they could not count the possibilities when both are exempted. This paper exempts them on both axes and counts the result. The second relaxation is their own argument read in the other axis, and the Harris correlation inequality bounds the joint count below by the product of the two counts taken separately, at the cost of one factor of the component generating function.

The Viterbo isomorphism relates the symplectic cohomology of a cotangent bundle to the homology of the free loop space of its base. We lift this to a relation of modules over (1) the complex bordism spectrum MU and (2) the sphere spectrum $\mathbb{S}$. In particular, by a result of Porcelli and the present author [BP26], it is not the case that, in general, the $\mathbb{S}$-level statement recovers the MU-level statement after base-change – even in the case the base is spin.

We introduce the honeycomb hierarchy, a representation-theoretic framework that gives new asymptotic upper bounds on $R_2(\delta)$. Its first level is the two-row hyperoctahedral representation graph associated with type $S^{(n-k,k)}$. Retaining every two-row irreducible and every coordinate box-transfer channel, together with a moving-projection theorem, yields an explicit four-parameter exponent $\kappa_{\mathrm{HC}}$. The earlier whole-cube exponent $\kappa_H$ is a boundary restriction of this optimization, whereas the fully optimized second MRRW exponent $M_2$ is an exact symmetric slice. The prior best curve is the combined $\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$, which uses a constant-weight branch $\kappa_{\mathrm{CW}}$. Replacing only the whole-cube branch by the honeycomb bound gives $\kappa_{\mathrm{best}}=\min\{\kappa_{\mathrm{CW}}, \kappa_{\mathrm{HC}}\}$. We prove, on $0<\delta<1/2$, \[ R_2(\delta)\le \kappa_{\mathrm{best}}(\delta) \le \kappa_{\mathrm{bin}}(\delta) \le R_{\mathrm{2MQC}}(\delta)<M_2(\delta),\\[-1mm] \kappa_{\mathrm{best}}(\delta) \le \min\{\kappa_{\mathrm{CW}}(\delta), \kappa_{\mathrm{bal}}(\delta)\} <R_{\mathrm{2MQC}}(\delta), \qquad \kappa_H(\delta)=R_{\mathrm{MQC}}(\delta). \] The hierarchy has two further directions. Increasing the representation depth replaces scalar by matrix-valued transfers on the hive. Increasing the anchor depth localizes it in a stable-set hierarchy. The resulting bounds are monotone in both directions and eventually recover $A_2(n,d)$. A complementary Horn–channel hierarchy gives matrix optimizations whose $2\times2$ level is $\kappa_{\mathrm{HC}}$ and whose $3\times3$ level is a stronger bound. Already at low levels, they can be used to improve the strongest previous general bounds, while the honeycomb framework provides a route towards tighter bounds.

We propose computationally efficient tests for equality of mean vectors of two or more high-dimensional populations. Central to our approach is an equivalence between equality of means and a zero population logistic regression parameter. We establish this equivalence for independently distributed observations without imposing common distributional assumptions across populations. Our procedure uses logistic Lasso to screen informative variables and an unpenalized logistic refit for inference in the reduced dimension, yielding asymptotically correct size and consistency. For a specified two-sample Gaussian submodel and sparse discriminative class, the test also attains the minimax separation rate. The framework extends to multiple populations through multi-class logistic regression. Simulations demonstrate accurate size control, strong power, and favorable computational scaling compared with existing tests under unbalanced designs and variance heterogeneity. Applications to gene-expression data with more than twenty-two thousand variables illustrate the practical scalability of the proposed procedures.

In Markov chain Monte Carlo sampling, light-tailed target distributions present something of a poisoned chalice: their light tails offer good confinement, and tend to imply good mixing properties for natural continuous-time dynamics, but the steepness of their tail decay means that they often fall outside of the scope of modern quantitative convergence theory. For usual gradient-based samplers, this reflects a genuine instability issue, whereby Metropolis acceptance rates can degrade badly. In this work, we study the gradient-free random-walk Metropolis sampler, and show that for a wide range of light-tailed targets, the acceptance probability remains stable for reasonable choices of proposal variance, from which effective and favourable mixing time estimates can be deduced. The analysis relies on a simple relationship between the first and second derivatives of the log-density of the target distribution.

In this study, we present an analytical approach based on the extended Nikiforov-Uvarov method to solve the generalized Klein-Gordon oscillator in the presence of a uniform magnetic field within the Som-Raychaudhuri space-time. Exact eigenstate solutions are obtained for two distinct potential models, namely the linear potential and the Cornell potential. The corresponding energy eigenvalues are determined, and the associated eigenfunctions are analyzed and illustrated graphically.

In this paper, we show that the codimension of the singular set of a stable stationary harmonic map to a round $k$-sphere is at least $k+1$ when $k$ is between 3 and 6, and is at least 7 when $k$ is at least 7. The result is sharp in the sense that there exist energy minimizing 0-homogeneous maps in the aforementioned critical dimensions. We also establish non-trivial index bounds for the non-constant harmonic maps from $n$-spheres to $k$-spheres, provided that $n$ is less than $k$.

We construct an explicit finite chain of non-aspherical group presentation complexes $K(\mathcal X)\subset K(\mathcal Y)\subset K(\mathcal Z)$ in which $K(\mathcal Y)$ is Cockcroft, both inclusion-induced maps on $\pi_2$ are zero, and the pair $(K(\mathcal Y),K(\mathcal X))$ has the identity property. This answers a question of Hitchman. The construction is given directly from a short balanced presentation of the binary icosahedral group. As the second result, we construct an integral Lie ring presentation in which a subpresentation has a nontrivial identity among relations, whereas the presentation itself has none.

Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step, and the underlying group structure of $\mathbb{G}$. Our result provides a noncommutative generalization of the dimension-free Euclidean theorem of Ouyang, Spector, and Stockdale arXiv:2608.18068, with the same universal constant. Our proof relies upon a fractional obstacle problem adapted to stratified Lie groups by using the functional calculus of $\mathcal L$ instead of the Fourier transform. As a consequence, by interpolation we obtain uniform $L^p$ bounds for the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ for $p \in (1,2]$. By contrast, for every $p > 2$, we construct a sequence of stratified Lie groups with fixed horizontal dimension $5$ and steps tending to infinity for which the $L^p$ norms of the horizontal Riesz transforms diverge.

We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in the characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H^\infty$-calculus of angle zero, but fails stochastic maximal $L^p$-regularity (SMR$_p$) for every $p\in[2,\infty)$. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q>2$, the Laplacian on $L^q(\mathbb R^d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space.

We solve John Milnor's problem: among all convex regions of a given area on the sphere, the round disk requires the greatest distortion in cartographic projections onto the plane.

A unit-area translation surface is called maximal if it maximizes the length of the shortest saddle connection among all surfaces in the same stratum. We investigate whether a non-maximal translation surface can be continuously deformed into a maximal one while the systole increases strictly monotonically. Although such a deformation does not exist in general due to the existence of local but not global maxima of the systole function, we prove that it exists for square-tiled surfaces and translation surfaces obtained from regular hexagons and regular octagons. In each case, we explicitly construct a continuous deformation to a maximal translation surface along which the systole is strictly increasing. Finally, the preceding construction yields another family of translation surfaces admitting continuous systole-increasing deformations to maximal surfaces.

A scalar field alone does not determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural. Realification illustrates this point: an orthogonal complex structure J^2 = -I selects the physical real operators and the balanced composite. Quaternionic quantum mechanics has an analogous exact representation on a doubled complex space selected by an antiunitary symplectic structure Theta^2 = -I. Within that sector, a Choi fixed-point condition characterizes when a complex channel admits quaternionic Kraus operators, while exact correction of a finite-dimensional right-quaternionic code is characterized by compression coefficients in the real center and admits an explicit recovery. For octonions, nonassociativity precludes a unique continuation; para-linear, categorical, sectorial, Jordan, Clifford-envelope, and Moufang models are compared by the operational structures they retain and the additional data required to define composites, channels, or recovery. Across these cases, an exact ambient representation does not erase the complex or symplectic structure that selects the physical theory.