Papers

Let $S$ be a closed, oriented surface of genus at least $2$, let $h$ be a smooth Riemannian metric on $S$ with curvature $K\in (-1,0]$, and let $c\in \cT_S$ be a conformal structure on $S$. There exists a unique equivariant isometric immersion of $(S,h)$ in $\HH^3$ such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is $c$. Dually, if $h^*$ is a smooth metric on $S$ with curvature $K^*\in (-\infty, 0)$ and if $c\in \cT_S$, there exists a unique equivariant immersion of $S$ into $\HH^3$ with third fundamental form $h^*$ and such that the pull-back of the conformal class at infinity by the Gauss map is $c$. Equivalently, given $h$ and $c\in \cT_S$, there is a unique pair $(E,u)$ where $E$ is a hyperbolic end with conformal structure at infinity $c$ and $u$ is an isometric embedding of $(S,h)$ in $E$. Given $h^*$ and $c$, there exists a unique pair $(E,u^*)$, where $E$ is a hyperbolic end with conformal structure at infinity $c$, and $u^*:S\to E$ is an embedding inducing the third fundamental form $h^*$. Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.

Communication-centric integrated sensing and communication~(ISAC) directly employs random data-bearing communication signals for both information transmission and environmental sensing. Existing studies generally describe pulse-shaped transmission in continuous time or directly adopt periodic discrete-time sensing models, leaving the digital signal processing~(DSP) operations connecting the physical transceiver and the periodic models unclear. This paper develops a DSP-oriented transceiver framework for single-antenna communication-centric ISAC. A common transmitter is constructed through modulation, cyclic-prefix~(CP) insertion, upsampling, and pulse shaping. For communication reception, the physical transceiver chain is reduced to a symbol-rate equivalent linear channel, which becomes circulant after CP removal and thereby supports frequency-domain equalization. For sensing reception, a high-rate reference waveform is extracted from the transmitted waveform and equivalently represented by circular pulse shaping. CP removal at the receiver then converts physical target delays into circular shifts of this reference waveform, leading to a periodic matched-filtering model and the corresponding range profile. Numerical results validate the developed framework through target range estimation and the symbol error rate performance of pulse-shaped CP-OFDM over frequency-selective Rayleigh fading channels.

Using the duality between the categories of inverse semigroups and ample groupoids, we prove the triviality (global and local) of the central term of the extension in terms of twisted crossed products.

Let $F=(f_1,\dots,f_r)$ be a family of nonzero polynomials on $\mathbb C^n$. More than ten years ago, Budur conjectured that $B_F$, the Bernstein–Sato ideal of $F$, is generated by products of linear polynomials with rational coefficients. With the aid of AI, we construct two counterexamples to Budur's conjecture. The examples indicate that main results in [Wu26] provide the optimal linearity properties that the zero locus of Bernstein-Sato ideals satisfies in general.

Optimal experimental design is often formulated as the maximization of the Expected Information Gain (EIG), which measures the expected reduction in uncertainty about a parameter of interest after observing the data. For nonlinear forward models and nonGaussian priors, computing the EIG typically requires costly nested Monte Carlo estimators or sophisticated density-approximation techniques. In this work, we derive new upper and lower bounds on the EIG using dimensional logarithmic Sobolev inequalities. The proposed bounds are expressed in terms of signal-to-noise ratios (SNR), preserving the simple matrix-based structure of the EIG in the linear-Gaussian setting and depending solely on covariance matrices and Fisher information matrices. We introduce two families of bounds: conservative bounds, designed to retain the entire information content, and incremental bounds, that prioritize early information acquisition for sequential experimental design. When model gradients are available, we further derive computable approximations of the Fisher-information terms that require only a small number of gradient evaluations, yielding substantial computational savings. The tightness of these bounds is controlled by the nonlinearity of the forward model, and in the linear-Gaussian setting, these bounds recover the exact value of the EIG. By eliminating the nested-sampling bottleneck, the proposed framework offers a tractable and theoretically grounded alternative to direct EIG estimation for large-scale Bayesian experimental design, while retaining the interpretability of the SNR-based bounds.

We study the problem of implementing multivariable matrix-valued functions on a quantum computer and propose two quantum algorithms for multivariable matrix eigenvalue transformations acting on tuples of pairwise commuting matrices. The first algorithm is based on multivariable contour integrals and applies to arbitrary holomorphic functions on $\mathbb{C}^n$. The second algorithm is based on high-dimensional Fourier transforms and designed for smooth functions of commuting Hermitian matrices. For both algorithms, we discuss and analyze the complexity of their quantum implementation leveraging quantum singular value transformation, compression gadgets, and linear combination of unitaries. We additionally study their hybrid quantum–classical variants that reduce the number of required ancilla qubits to logarithmic dependence on the number of variables, at the cost of increased query complexity. As an application, we show that multivariable matrix polynomials can be implemented with no additional explicit degree dependence, depending only on global properties of the polynomials.

Pinching-antenna systems (PASS) reconfigure wireless channels by activating radiating elements at selected locations along dielectric waveguides. Port selection is a core issue in improving the performance of multiuser PASS systems. In this paper, we study joint port selection and precoder design for multiuser sum rate maximization, which leads to a large-scale nonconvex mixed-integer problem coupling discrete port selection with continuous precoder design. We propose a discrete diffusion method that does not require optimal or near-optimal port selection solutions as training labels and recasts the resulting optimization problem as sampling from a target distribution. Instead, we learn a potential function from local objective differences between neighboring feasible realizations. The learned potential is incorporated into a Metropolis–Hastings sampling rule to guide a parallel diffusion process while preserving the port selection constraints. Simulation results show that the proposed method approaches exhaustive search within a $0.30\%$ sum rate gap while achieving over $150\times$ speedup, and outperforms state-of-the-art methods including greedy and beam search. In particular, it also achieves an $18.8\times$ speedup over greedy search.

Non-commutative friezes have been introduced by Berenstein and Retakh and studied further by the authors. In this paper we consider non-commutative friezes over normed division rings, like for instance Hamilton's quaternions or more general quaternion algebras. We address the fundamental question in the theory of friezes of whether over a certain subset there are finitely or infinitely many non-commutative friezes (with 1's on the boundary) for any height. As an application of a theorem bounding the norm of quiddity entries we deduce that for every norm-finite subset of a normed division ring there are only finitely many such non-commutative friezes for every height. In particular this result applies to the Lipschitz quaternions and the Hurwitz quaternions of Hamilton's quaternions. We then study more generally non-commutative friezes over Lipschitz subrings of non-split quaternion algebras $(a,b)_{\mathbb{Q}}$. We determine the frieze subrings for all $a,b<0$, and as a consequence we see that all such non-commutative friezes are known if $a\le -4$ and $b\le -4$.

Operator learning of periodic solutions requires deciding how simulation data should be recorded and represented. A natural choice is to integrate long enough for transients to decay and record a window wide enough to contain at least one full period of all trajectories. We find that these conservative choices can make the resulting trajectories difficult to learn, even when the underlying periodic orbits vary regularly with system parameters. Unaligned trajectories generalize poorly even within the training distribution. Phase alignment substantially improves in-distribution generalization, but models trained on a fixed physical-time window still have large errors on trajectories with periods outside the training range. We explain both failures through a common mechanism: frequency differences accumulate over time, so the target phase varies rapidly with the parameters. Predictors that cannot track this variation incur a population MSE floor in both settings; for fixed window prediction, we also derive a per-sample lower bound. We then study one of the simplest representations that escape these floors: learning an aligned, normalized waveform and its period separately. We establish regularity of the decoupled targets under ODE assumptions and show experimentally that this approach avoids both failures in ODE systems and a PDE case study.

We develop entropy-stable isothermal-wall conditions for the compressible Navier–Stokes equations at continuous and semi-discrete levels. For a stationary, impermeable, no-slip wall at constant positive temperature, adding total-energy density divided by wall temperature to the canonical convex entropy cancels the wall contribution without suppressing heat transfer. This affine modification preserves the entropy Hessian, relative entropy, and volume entropy production. It is proportional to total ballistic-energy density and yields a conditional $L_2$-type bound based on the initial field under the stated assumptions. Two wall formulations use a common discontinuous Galerkin discretization with the summation-by-parts property and simultaneous approximation terms. Both give zero adapted wall contribution without a penalty and a nonpositive contribution with it. The complete-residual formulation penalizes the full wall-data residual and balances fluid energy through its complete numerical wall flux. The Fourier-energy-exact formulation combines an entropy-neutral skew correction with a momentum-projected penalty so that the complete numerical outward energy flux equals the one-sided numerical Fourier heat flux at every wall node and spatial resolution. Pointwise tests verify the local identities. Manufactured-solution studies with entropy-stable interior coupling demonstrate convergence rates consistent with order $p_s+1$ in the discrete $L_2$ norm, where $p_s$ is the solution polynomial degree. Thermal-relaxation calculations verify the semi-discrete energy and adapted-entropy balances. Three-dimensional calculations demonstrate balance closure in a closed domain, convergence of curved-wall traces, and applicability to a complex supersonic separated flow.

Let $H$ be an infinite-dimensional Hilbert space and let $\set{x_n}\inN$ be a frame for $H$. We say that a sequence $\set{y_n}\inN$ is an alternative dual of $\set{x_n}\inN$ if $$x=\sumli\ip{x}{y_n}x_n\quad \text{for all }x\in H,$$ with the convergence of the series in the norm of $H.$ We say that a sequence $\set{z_n}\inN$ is a pseudo-synthesis dual of $\set{x_n}\inN$ if $$x=\sumli\ip{x}{x_n}z_n \quad \text{for all }x\in H,$$ with the convergence of the series in the norm of $H.$ We prove that, at every level of smoothness of the generator, there exist Gabor frames for $L^2(\R^d)$ with infinitely many alternative duals (and pseudo-synthesis duals) such that none of them is a Gabor system

We construct a degeneration of Hitchin's non-standard minitwistor spaces associated with toric gravitational instantons of type A$_{\rm odd}$ to the minitwistor space of a simple hyperbolic orbifold. We show that the corresponding families of minitwistor lines also converge under this degeneration. Consequently, the associated non-standard Einstein–Weyl spaces converge to the hyperbolic orbifold.

We justify the vortex-wave system introduced by Marchioro and Pulvirenti in the 1990s as the vanishing-viscosity limit of the two-dimensional Navier-Stokes equations on every prescribed finite time interval. The initial vorticity consists of a point vortex and a smooth, compactly supported background that vanishes near the vortex. The regular component converges strongly to the background vorticity governed by the vortex-wave system, while the concentrated component is approximated by a Lamb-Oseen vortex centered on the limiting point-vortex trajectory. We obtain quantitative estimates for both components, with constants and a viscosity threshold depending on the chosen time interval, thereby providing a global-in-time justification of the vortex-wave system for this class of initial data.

The sudden removal of a potential barrier from a Bose-Einstein condensate (BEC) gives rise to a quantum version of a hydrodynamic dam break and leads to rich wave dynamics that have similarities to other localized defect problems such as domain wall dynamics in spin systems. Denoting $\Delta n$ as the initial difference in density between the upper $n_{1}$ and lower $n_{0}$ reservoirs on either side of the dam, we use the Gross-Pitaevskii equation to study the quasi-one dimensional case in both perturbative ($\Delta n \ll n_{0}$) and non-perturbative ($\Delta n \sim n_{1}$) regimes. In the perturbative regime a pair of outwardly propagating dispersive wavepackets forms which can be viewed as quantum versions of undular tidal bores that have analytic forms at long times in terms of the integrals of Airy functions with a wavelength that grows as $(\hbar^2 t)^{1/3}$. Airy functions are the universal wave functions that dress structurally stable fold caustics where pairs of rays coalesce, and, indeed, we show that the quantum dam break problem has the same ray structure as the naturally occurring caustic phenomenon of a double rainbow, including Alexander's dark band between the two bows where no light is scattered: we identify the intermediate density plateau in the dam break as an analogous `silent band' where only evanescent sound waves can exist. In the opposite regime of a non-perturbative dam break we show that a self-induced sonic horizon occurs when $\Delta n \geq (8/9)n_{1}$. A discussion of possible experimental schemes for amplification of quantum undulations is included as well as an alternative scheme for dispersive wave generation where a constant flow is imprinted on a BEC in box trap.

We establish that finite expressions form a symbolic representation class capable of overcoming the curse of dimensionality for several classes of high-dimensional partial differential equations. For semilinear heat equations, we show how finite expression approximations of the terminal condition and nonlinearity can be propagated through the multilevel Picard framework to produce randomized pointwise approximations with prescribed root-mean-square accuracy. For semilinear Kolmogorov equations with Laplacian diffusion and zero drift, diagonal Black–Scholes equations, and the Laplace Dirichlet problem on a half-space, we construct deterministic finite expression approximations with arbitrarily small spatial $L^p$ error. Under suitable growth and regularity assumptions, the evaluation cost of the resulting approximants is bounded polynomially in the dimension and the reciprocal accuracy. A key ingredient in the nonlinear setting is the construction of finite expressions that approximate the PDE solution while preserving the growth and Lipschitz structures required by the stochastic solution theory. More broadly, our results show that dimension-robust approximation of high-dimensional PDEs is not restricted to conventional neural-network architectures: structured symbolic expressions generated from a fixed dictionary can achieve comparable polynomial-complexity guarantees. This provides a rigorous foundation for finite expression methods as a representation paradigm for high-dimensional scientific computing.

The Hamilton-Jacobi [HJ] analysis and the Faddeev-Jackiw [FJ] quantization for the linearized, non-projectable $\lambda R$ model, extended by the lowest-order term of Blas, Pujolàs, and Sibiryakov and the higher-order term involving the square of the Cotton tensor, were carried out. The cases $\lambda \neq \frac{1}{3}$ and $\lambda = \frac{1}{3}$ are analyzed. Within the Hamilton-Jacobi framework, we obtain the complete set of Hamiltonians, generalized brackets, fundamental differential, characteristic equations, and gauge transformations of the theory. In the Faddeev-Jackiw framework, a symplectic tensor is constructed from which the fundamental brackets are identified. The HJ and FJ brackets coincide, and from the symplectic tensor, the measure for the quantum path integral is calculated. The differences between the cases $\lambda \neq \frac{1}{3}$ and $\lambda = \frac{1}{3}$ at the quantum path integral are discussed.

For a reduced crystallographic root system with Weyl group $W$, Weyl alternants are signed sums of exponentials indexed by $W$. They occur in the numerators of the Weyl character formula and the explicit formula for spherical functions on complex semisimple groups. We establish rootwise bounds that quantify cancellation in these oscillatory sums, with each positive root contributing a factor depending on both the spatial variable and the spectral parameter. More generally, we establish rootwise bounds for all mixed directional derivatives of normalized Weyl alternants. Our proof extends to arbitrary rank the descent strategy used by the second author for pointwise character bounds on $\mathrm{SU}(3)$. The key ingredients include the decomposition of the Weyl group into parabolic double cosets and the relative BGG–Demazure identity. For spherical functions on complex semisimple groups, these estimates recover the recent pointwise bound of Brumley, Marshall, Matz, and Peterson, and they refine the derivative bounds of Cowling and Nevo by controlling arbitrary mixed radial derivatives uniformly across spatial and spectral root hyperplanes. Furthermore, by a standard localization argument relative to the affine Weyl arrangement, we establish periodic alternant estimates for regular integral weights and corresponding rootwise bounds for all mixed radial derivatives of irreducible characters of compact connected semisimple groups.

We show that every polytope $P\subseteq[0,1]^n$, and more generally every compact convex set, has Chvátal rank at most $12.22n^2+n\log_2 n+2n+4$. This improves the $O(n^2\log n)$ bound of Eisenbrand and Schulz and, together with the $\Omega(n^2)$ lower bound of Rothvoß and Sanità, shows that the maximum Chvátal rank of a polytope in $[0,1]^n$ is $\Theta(n^2)$. More precisely, if $P$ contains an integer point, then for every $c\in\mathbb{Z}^n\setminus\{0\}$ the inequality $cx\le\max\{cy: y\in P\cap\mathbb{Z}^n\}$ is valid for the $k$-th Chvátal closure of $P$ for some $k\le 12.22n^2+2n+2\log_2\|c\|_\infty+4$. Following Eisenbrand and Schulz, we derive this inequality along a chain of coarser and coarser integer vectors, but instead of halving the vector in each step, we round $\tau c$ for a scale $\tau$ chosen freely in each dyadic window $[2^{-t-1},2^{-t}]$. The main new ingredient is a multiscale version of Dirichlet's approximation theorem, proved by an elementary volume argument: for every $c\in\mathbb{R}^n$, the $\ell_1$-distances of $\tau c$ to $\mathbb{Z}^n$, minimized within each dyadic window and summed over all windows, total less than $1.222n^2$, independently of $\|c\|_\infty$.

Andrews and Newman introduced the minimal excludant $\mathrm{mex}(\lambda)$ of an integer partition $\lambda$ and studied the summatory function $\sigma\mathrm{mex}(n)$, and Baruah et al. refined this to the odd- and even-restricted functions $\sigma_o\mathrm{mex}(n)$ and $\sigma_e\mathrm{mex}(n)$. In this paper, we introduce and study the overpartition analogue $\overline{\sigma_o\mathrm{mex}}(n)$, defined as the sum of odd minimal excludants over all overpartitions of $n$. We first derive the exact generating function for $\overline{\sigma_o\mathrm{mex}}(n)$, and relate it to the bivariate generating function of Aricheta and Donato for the overpartition minimal excludant. Using elementary $q$-series arguments, together with a weight-one eta-quotient identity for $\varphi(q)^2$ verified via the Gordon–Hughes–Newman–Ligozat criterion, we establish an infinite family of congruences satisfied by $\overline{\sigma_o\mathrm{mex}}(n)$. Consequently, we obtain that $\overline{\sigma_o\mathrm{mex}}(0)=1$ and, for $n\ge1$, $\overline{\sigma_o\mathrm{mex}}(n) \equiv 0 \pmod{4}$ if and only if $n$ is a perfect square. We further obtain congruences for the partial sums and self-convolution of $\overline{\sigma_o\mathrm{mex}}(n)$; in particular, an infinite family of congruences modulo $8$ for the self-convolution, expressed in terms of the divisor functions $d_1(n)$ and $d_3(n)$. We conclude the paper by establishing a Hardy–Ramanujan-type asymptotic formula for $\overline{\sigma_o\mathrm{mex}}(n)$ via the Wright circle method.

We develop a high-order and positivity-preserving kinetic solver for the multi-species Vlasov-Ampère system that is built within a semi-Lagrangian discontinuous Galerkin (SLDG) framework. The method is designed for efficiency and accuracy across a broad range of conditions, including stiff, multi-scale regimes with strong ion-electron scale separation. Phase space is discretized via high-order discontinuous Galerkin finite elements; time stepping is handled via high-order operator splitting and a series of one-dimensional, unconditionally stable, semi-Lagrangian updates. To improve both accuracy and efficiency, each plasma species is represented on a different phase-space mesh. On every element and for each species, local positivity-preserving limiters are applied to prevent spurious undershoots and rigorously ensure positivity of the cell averages at the next time step. The accuracy and robustness of the resulting method are verified on several standard single- and two-species test cases, including manufactured solutions, two-stream instability, weak and strong Landau damping, ion-acoustic wave propagation, ion-acoustic shock formation, and KEEN wave dynamics.

Motivated by the systematic study of spiral minimal products, we consider harmonic maps obtained by coupling two spherical eigenmaps through a profile curve in $\mathbb S^3$, with a doubly warped product metric on the source. The source warps are independent of the profile magnitudes. This gives two complementary problems: reconstructing a compatible source metric from prescribed profile or source data, and finding closed profiles when the source metric is fixed. For arbitrary prescribed warps, homogenization of the reduced Lagrangian gives a strongly convex conic Finsler metric whose oriented graph geodesics are exactly the harmonic profiles. In the unit-speed inverse problem, the phase momenta reduce reconstruction to a scalar magnitude equation. We obtain global reconstructions from a prescribed magnitude, one source warp, or a monotone volume factor, together with periodic families in which profiles closing on finite covers are dense. In the fixed-metric problem, a change of time gives an autonomous Neumann system. One fixed source metric then carries continuous families of closed profiles and infinitely many closed graph geodesics. For fixed even $p,q\geq 2$, every fixed metric in this Neumann class on $\mathbb S^1\times\mathbb S^p\times\mathbb S^q$ supports harmonic maps into $\mathbb S^{p+q+1}$ of degree $4m$ for every $m\geq 1$. In the matched constant-warp case, these maps reduce to explicit eigenmaps.

When studying the dynamical Mordell-Lang conjecture by using the $p$-adic method, experts have found that return sets of the $p$-adic version of the DML problem can be very sparse. In this article, we investigate the similar phenomenon for a complex analytic version of the DML problem and the DML problem for backward orbits.

In this paper, we focus on the positive reflectionless measures rigidity problem proposed by Xavier Tolsa in [16]. Let $1\leq n<d$, we consider positive reflectionless measures for the $n$-dimensional Riesz transform in $\mathbb R^d$. Under polynomial upper growth and a lower mass bound at large scales about one point, we obtain two flatness results for reflectionless measures. The first concerns globally defined Lipschitz graph measures with finite $\dot H^{1/2}$ energy and bounded, uniformly positive parameter densities whose deviations from a constant belong to $L^2$. The second case concerns a reflectionless measure that coincides with a flat measure outside a compact set. In both cases, the measure is a constant multiple of the $n$-dimensional Hausdorff measure on an affine $n$-plane. For integers $s>n/2$, we further obtain $H^s$ estimates controlling the graph gradient and density perturbation by the normalized Riesz field under smallness of the slope and density perturbation in $L^\infty$. The constants are independent of the supports.

The support recovery problem in mixture of linear classifiers aims to identify the features relevant to the underlying decision rules when data is generated by a mixture of several linear decision rules. In particular, the goal is to recover the support (nonzero coordinates) of $l$ unknown $k$-sparse vectors from sign measurements. Each measurement is generated by selecting one of the $l$ vectors uniformly at random, and returning the sign of its inner product with a chosen measurement vector. In this paper, we propose adaptive and non-adaptive schemes that significantly improve upon prior results by simultaneously reducing the number of measurements and achieving sublinear decoding time. In particular, our adaptive constructions substantially reduce measurements compared to existing approaches, while also lowering decoding complexity from super-quadratic to sublinear in the ambient dimension. We further provide a non-adaptive scheme that improves previous measurement bounds while maintaining efficient decoding. Overall, our approach yields a more efficient trade-off between sample complexity and decoding time for support recovery in mixture models compared to previously known methods.

We give a systematic description of $\Hom$-Lie structures on several classes of Lie algebras and of their relations with transposed Poisson structures. Our starting point is the observation that the left-hand side of the defining identity is an alternating trilinear map; this makes $\mathcal{H}(\LL)$, the set of $\Hom$-Lie structures of $\LL$, a subspace of $\End(\LL)$ and reduces its computation to increasing triples of basis vectors. From this we derive, in every finite dimension $n$, the lower bound $\dim \mathcal{H}(\LL)\geq n \dim \mathcal{C}(\LL)$ and the codimension bound $\binom{n}{3}\dim [\LL,[\LL,\LL]]$, together with a refinement of the latter, and we determine $\mathcal{H}(\LL)$ for nilpotent, oscillator and solvable algebras, for $\sltwo \oplus A_{n-3}$, for the Witt, Virasoro and Heisenberg–Virasoro algebras, for all three-dimensional algebras, for $M_n(\bF)$ and $T_n(\bF)$, for the Virasoro-like algebra and its $q$-analogue, and for the planar Galilean conformal algebra. By a result of Filippov, every $\frac12$-derivation, and more generally every $\delta$-derivation with $\delta\neq0,1$, of a Lie algebra is a $\Hom$-Lie structure; we observe that, more generally, for $\delta\neq 0,1$ the $\delta$-derivations of an algebra of any variety defined by identities of degree at most three (associative, Novikov, left-symmetric, Leibniz, \ldots) are $\Hom$-structures of the corresponding $\Hom$-variety. We show that the inclusion of the $\frac12$-derivations into the $\Hom$-Lie structures is far from being an equality by computing the graded $\Hom$-Lie structures of Witt-, Virasoro- and Schrödinger-type algebras and of their central extensions, for which we give an explicit lifting obstruction. Along the way we correct two multiplication tables from the literature that violate the Jacobi identity.

We determine weak $k$-metric dimensions for several families of Hamming graphs. For $n\ge3$, $m>n$, and $3\le k\le2n$, a cyclic construction proves Conjecture 6.1 of Fernández, Klavžar, Kuziak, Muñoz-Márquez and Yero (2026) on $K_n\square K_m$. For these rectangular products, we also prove that $\operatorname{wdim}_2(K_n\square K_m)=m$ exactly when $m\ge2n-2$. For hypercubes $Q_d$ with $d\ge2$, we show that consecutive parameters $2s-1$ and $2s$ have identical weak resolving sets. Near the maximum parameter, a reduction to restricted-distance binary codes determines $\operatorname{wdim}_{2^d-t}(Q_d)$ for every feasible deficit $0\le t\le15$. For $L\ge1$ and $t\in\{2L,2L+1\}$, we prove the stabilization formula $\operatorname{wdim}_{2^d-t}(Q_d)=2^d-L$ for $d\ge2^L+1$, and show that this threshold is sharp. For each fixed $q\ge3$ and deficit $t$, we also determine $\operatorname{wdim}_{2q^{d-1}-t}(K_q^{\square d})$ in all sufficiently large dimensions, with an explicit sufficient condition.

This paper is the fourth in a series devoted to the finite basis problem for $4$-element additively idempotent semirings. These algebras are divided into five types according to their additive reducts; we study the largest of these types, namely those whose additive reducts are chains. Up to isomorphism, there are $386$ such algebras, denoted by $S_{(4, k)}$, $481 \leq k \leq 866$. We show that all but three of them, namely $S_{(4,545)}$, $S_{(4,634)}$, and $S_{(4,710)}$, are finitely based. This completes the classification, with respect to the finite basis property, of all $4$-element additively idempotent semirings with chain additive reducts.

For a graph $H$, the anti-Ramsey number $\operatorname{ar}(n,H)$ is the maximum number of colors in an edge-coloring of $K_n$ containing no rainbow copy of $H$, where a copy is rainbow if its edges have pairwise distinct colors. Let $s,t$ be nonnegative integers with $s+t\ge2$, and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ odd cycles of fixed lengths at least $5$, all sharing exactly one common vertex and otherwise pairwise vertex-disjoint. Liu et al. (2024) determined $\operatorname{ar}(n,H_{s,0})$ for $s\ge3$ and $n\ge50s^2$. In this paper, we determine the exact value of $\operatorname{ar}(n,H_{s,t})$ for every fixed $H_{s,t}$ with $t\ge1$ and all sufficiently large $n$.

We investigate the scattering of elastic waves by anisotropic inhomogeneous media at a fixed wave number. We prove that, in two dimensions, every penetrable obstacle with piecewise $C^{1}$ boundary and corner singularities, and, in higher dimensions, every obstacle with edge singularities, scatters every incident wave satisfying suitable assumptions. This work extends our previous result [J. Math. Anal. Appl. 2026, doi:10.1016/j.jmaa.2026.130630] from scalar equations to systems of equations, a generalization that requires a more delicate blowup analysis and more involved computations. Roughly speaking, our results establish the principle that corners always scatter for general anisotropic media with variable coefficients.

A wet foam is a cluster of air chambers whose interfaces are coated by a single liquid chamber of prescribed volume. In the corresponding isoperimetric problem the liquid chamber may collapse onto films of zero thickness, so that minimizers need not exist. We identify the relaxed energy accounting for such collapsed films and prove that limits of minimizing sequences are its minimizers. The resulting compactness and energy representation theorems lead to the regularity of these generalized minimizers, including an equilibrium law for the pressures across collapsed films. Finally, as the liquid volume vanishes, the minimal energies converge to those of the classical dry foam.