Papers

For K-unstable toric varieties, Székelyhidi's optimal test-function $\Theta_{P}$ is a mysterious concave function over the moment polytope $P$, which gives the maximal destabilizer for K-stability, and encodes the limiting behavior of the divergent Calabi flow. As balanced norms quantize cscK metrics, we show that $\Theta_{P}$ can be quantized by the maximal destabilizers for Chow-stability, which are given by the shortest GKZ vectors, i.e., the least norm point on the secondary polytope for the set $P\cap k^{-1}\mathbb{Z}^{n}$. Properties and algorithm for general sGKZ vectors are given. Our result may provide a new method to detect the K-unstability of toric varieties.

We prove that every weakly group-theoretical modular fusion category, equipped with its canonical positive spherical structure, has generalized property F: the associated Reshetikhin-Turaev mapping class group representations have finite projective image for all compact oriented surfaces with labeled framed points. This generalizes earlier finite-image results for braid group representations associated with weakly group-theoretical braided fusion categories and for mapping class group representations associated with group-theoretical modular fusion categories and Ising categories. In the weakly group-theoretical setting, it also extends the known finiteness result for symmetric mapping class groups of closed surfaces to the full mapping class groups.

In this short note, we study the behavior of the algebraic entropy of birational automorphism groups in families of projective hyper-Kähler manifolds.

Odd girth and the sum of the largest and smallest adjacency eigenvalues are two common indicators of how close a graph is to being bipartite. This paper investigates the asymptotic behavior of this eigenvalue sum under odd-girth constraints. We study the associated abstract moment problem over finite real lists satisfying the vanishing odd-moment identities imposed by odd girth at least $g$ and the standard second-moment bound for graph spectra. We prove that the supremum of $(\lambda_1+\lambda_n)/n$ over these lists lies between $(1-o(1))(\log g)^2/g^3$ and $(10/7+o(1))(\log g)^2/g^3$ as $g\to\infty$ through the odd integers. The upper bound applies in particular to graphs of odd girth at least $g$. For regular graphs of odd girth $g\geq5$, we obtain a stronger bound $(\lambda_1+\lambda_n)/n \leq 12/((g-1)(g-2)(g-3))$, which is optimal up to a constant factor.

We analyse a divergence form stochastic differential equation featuring a Leray-Lions flux function that satisfies standard Carathéodory, growth, monotonicity, and coercivity conditions. We consider two types of noise: multiplicative Lipschitz continuous noise and a transport noise. Our approach utilises the gradient discretisation method, a generic framework encompassing various numerical schemes such as conforming and nonconforming finite element methods, hybrid mimetic mixed methods, and hybrid high order methods. For both types of noise, we propose a stable generic scheme, prove that scheme solutions converge to a strong solution in the probabilistic sense. For the multiplicative Lipschitz continuous noise, we establish the rate of convergence.

Problem definition: Automated operations research (OR) depends on correctly understanding the problem users intend to solve. In practice, limited user knowledge and vague expressions can obscure that problem, creating semantic uncertainty in the descriptions used for modeling. This uncertainty can lead to mathematical models that depart from users' intentions. We study semantic alignment to identify and correct errors or omissions in these descriptions before mathematical modeling. Methodology/results: We develop AMAO, an ambiguity-aware multi-agent framework for automated OR under semantic uncertainty. Its two-stage alignment process first uses semantic cues and contextual information to identify and correct potential semantic errors, then obtains additional information via human–AI interaction for further revision when the available context is insufficient. Specifically, AMAO integrates OR-guided mixture-of-experts (MoE) tuning and trajectory-based harness evolution to enhance semantic alignment capabilities. In addition, an ambiguity-aware benchmark and its matched multi-turn dialogue extension are proposed. Extensive experiments support that AMAO can recover the intended operational problem from ambiguous descriptions and recognize when available information is insufficient. Managerial implications: AMAO provides managers with an integrated workflow for translating ambiguous operational descriptions into mathematical models and optimization solutions. By combining semantic alignment with targeted user interaction, it supports recovery of the intended problem and improves downstream solution accuracy.

For four finite discrete random variables, we prove that the negative of the Ingleton expression, divided by the joint entropy, is at most $19/124 \approx 0.15323$, improving the previously reported upper bound $3/19$ by $11/2356$. The proof uses a copy program with four copy steps and six auxiliary variables, certified by exact rational linear-programming certificates. We also show that two published information inequalities, due to Dougherty-Freiling-Zeger and Csirmaz, imply the bound $5/32$ by an explicit nonnegative combination with Shannon inequalities, and that $5/32$ is the exact optimum both of the union of the three published inequality catalogues considered here and of the full copy-lemma extensions of all 213 distinct copy programs occurring in them. The new program is not among these programs, and the bound $19/124$ remains optimal when all of them and all catalogue inequalities are added. Ancillary files provide the parsed catalogues, exact primal and dual certificates, exact extension certificates, and verifiers that use only the Python standard library.

We prove a geometric hierarchy of Fourier layer energies for the weakly nonlinear cubic Schrodinger equation on $\mathbb T_\pi\times\mathbb T_1$, retaining its nontrivial resonant dynamics. For initial Fourier support in a rectangle, form nested rectangles with half-widths $(2j+1)$ times the initial ones; layers are the differences between consecutive rectangles. On $0\le t\le T\delta^{-1}$, the energy in layer $j$, measured by its squared $H^1$ norm, is at most $C_N\delta^{2j}$ for $1\le j\le N$; the energy outside the $N$th rectangle is at most $C_N\delta^{2N+2}$. These bounds hold at every fixed depth $N$ under a uniform $H^s$ bound, $s>2$, with constants depending on $N$. The Sobolev bound holds for every fixed $T$ in the defocusing case. We also obtain layer estimates for support bounded in one coordinate and for geometrically decaying initial Fourier tails. The proof uses a filtered Poincare-Dulac normal form: input layers select which nonresonant interactions are removed, while the same layer weights control the interactions left in the equation. Finite coordinate changes and stability estimates transfer this hierarchy to the exact solution.

We study two-step recurrences with positive step sizes and nonnegative momentum coefficients whose terminal residual is the Chebyshev minimax polynomial on a prescribed spectral interval $[\mu,L]$. These recurrences have the same terminal error on a fixed quadratic, but their first-order responses to stepwise Hessian perturbations can differ. We prove that the sine-Jacobi recurrence of Part I is asymptotically optimal for this response norm within a factor of $1.000004$, uniformly over $0<\mu<L$. Every terminal-exact recurrence satisfies $A_N\ge(2\sqrt{c_{cert}}-o(1))N^{3/2}\epsilon_N^\star/(L-\mu)$ with $c_{cert}>1.14268353624$, whereas the sine-Jacobi recurrence attains the constant $2\sqrt{c_{\sin}}$ with $c_{\sin}<1.14269218$. The proof reduces the gain to the diagonal energy of a Jacobi Green matrix. A centering lift represents this matrix by an exponential kernel; packing its diagonal into a density on the line preserves the energy and controls the change in the Birman-Schwinger spectrum. A calibrated identity gives a lower bound using the ground parameter and one Fredholm determinant. Two further determinants and a trace observation improve the bound through a pointwise inequality on an explicit compact state region. One rational certificate verifies this inequality by interval arithmetic. A second certificate constructs a density satisfying the selected observations and both normalizations with energy below the sine value. Finally, no $C^1$ function of finitely many observations of the specified kinds gives a locally sharp lower bound at the sine density.

We establish the log Sarkisov Program for strongly $\mathbb{Q}$-factorial compact Kähler threefolds.

This paper introduces a novel Support Vector Regression (SVR) model that incorporates Ordered Weighted Average (OWA) operators to differently penalize deviations of observations from the {\epsilon}-strip. The penalty is determined based on the position of each deviation in the ordered vector of all deviations. A key contribution of this work is the development of two nonlinear formulations: a continuous formulation for non-decreasing monotone weight vectors and a mixed-integer formulation for general weight vectors. By leveraging dual approaches associated with these formulations, the model accommodates nonlinear kernel functions. To enhance computational efficiency, two heuristic approaches are proposed for deriving suitable regression hyperplanes in reduced computation times, as compared to exact methods. Additionally, a third heuristic approach is developed to handle nonlinear kernel functions effectively. Computational experiments conducted on both real and synthetic datasets demonstrate that the exact formulations yield remarkable regression functions with respect to the following standard metrics: mean absolute errors (MAE) and mean squared errors (MSE). The heuristic approaches are shown to be particularly efficient for larger datasets, striking a balance between solution quality and computational effort.

Let $(R,\m)$ be a one-dimensional Cohen–Macaulay local ring with a canonical fractional ideal $R\subseteq K\subseteq\overline R$, and let $v$ denote its embedding dimension. Put $\sigma(R)=\sup\{j\ge0\mid\tr(K^j)\supseteq\m\}$. By definition, $R$ is nearly Gorenstein if and only if $\sigma(R)>0$. We prove that $R$ is almost Gorenstein if and only if $\sigma(R)\ge v$, equivalently, $\sigma(R)=\infty$. Thus every finite value of $\sigma(R)$ is at most $v-1$. We then determine the structure of rings for which $\sigma(R)$ attains its largest finite value $v-1$, just below the threshold for the almost Gorenstein property. This occurs if and only if $K^{v-1}\cong\m$ and $K^v\not\cong\m$. When these conditions hold, $R$ has type two, with multiplicity $e$ greater than $v$ and $e \equiv 3 \pmod{v-1}$. As an application, we characterize the numerical semigroup rings $R$ with $\sigma(R)=v-1$ as those generated by an arithmetic progression, of type two and multiplicity greater than $v$.

In this paper, we prove that for any fixed base $b \ge 2$, there exists an infinite arithmetic progression of positive integers $k$ that are simultaneously generalized Sierpiński and generalized Riesel numbers in base $b$. More precisely, we construct such an arithmetic progression so that both $k\cdot b^n + 1$ and $k\cdot b^n - 1$ have at least two distinct prime factors for every positive integer $n$. Our approach uses covering systems, following the ideas of Erdős and later constructions of Harrington. The argument is then completed using Zsigmondy's theorem on primitive divisors together with properties of multiplicative orders.

We prove strict Pólya inequalities for every positive Neumann eigenvalue of every Euclidean annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. Explicit polynomial positivity arguments and angular-layer comparisons cover the remaining finite spectral ranges. We also obtain a positive counting surplus uniform in the radius ratio, an explicit subtraction of order $\sqrt{k}/(R^2-r^2)$ for all $k\ge1604$, and a positive square-root correction over the entire spectrum with a universal, non-explicit coefficient. The square-root order is sharp, and the first positive pair and the low-energy count admit optimal constants expressed by the first positive zero of $J_1'$. Neumann inequalities for annular products and congruent annular sectors follow by established transfer principles.

This manuscript develops time-varying coefficients neurodynamic models (TVCNDMs) to solve the inclusion problem (IPs) $0 \in \mathcal{P}(w) + \mathcal{Q}(w)$, where $\mathcal{P}$ is a maximal set-valued and $\mathcal{Q}$ is a single-valued operator. The underlying operators are assumed to fulfill a generalized monotonicity condition that is less restrictive than the classical monotonicity assumption. Within this framework, existence and uniqueness of the model trajectories are established, and fixed-time ($\mathrm{FxT}$) convergence of the proposed TVCNDM to the solution of the IPs, along with explicit estimates of the settling time. Furthermore, the robustness of the proposed TVCNDM is investigated in the presence of bounded external disturbances, demonstrating that the $\mathrm{FxT}$ convergence. Numerical simulations support the theoretical results, illustrating rapid error decay and improved transient behavior achieved through the time-varying design.

We propose a single-call variant of Polyak extragradient, termed SPolyakEG, by recycling the previous extrapolation operator evaluation. Under a delayed critical condition, we establish that SPolyakEG achieves an $O(1/K)$ residual convergence rate for monotone operators and linear convergence under strong monotonicity. Compared with PolyakEG, SPolyakEG reduces the number of operator evaluations from two to one per iteration.

We compute homotopy groups of $n$-fold iterated topological Hochschild homology of $\mathbb{F}_p$ for $n\geq1$ and odd primes. Identifying this as the Loday construction $T^n\otimes\mathbb{F}_p$, we construct a bar spectral sequence computing $\pi_*(T^n\otimes\mathbb{F}_p)$ and prove that this spectral sequence collapses at the $E^2$-page. Finally, we resolve the multiplicative extensions and determine the ring structure.

We construct secure cyclic multi-access coded caching schemes by re-encoding the caches of a secure single-access scheme and reusing its multicast message unchanged. For a library of $N$ files, each of $K$ users reads $L<K$ consecutive shared caches and must recover its requested file while learning no information about the other files. Starting from a parent scheme with a cyclic representation of its cached symbols, the transformation lets each access window recover the corresponding parent-cache symbols and, conditioned on them, learn nothing about the remaining symbols. A deterministic Pascal-window encoder handles symbols whose consecutive appearances span at least $L$ users. A randomized complement-dual encoder handles shorter spans. Over a suitable finite field, these encoders attain the exact minimum per-block storage for every input length in the block-separable, parent-preserving class. Applying the transformation to two secure single-access schemes gives an achievable memory-rate family for $N,K\ge2$ and $1\le L<K$, including the exact rate-one cache size $(N-1)(K-1)/L$. Using converse bounds for arbitrary secure multi-access schemes, we prove approximation guarantees for the achievable family and give a proof of the exact $L=K-1$ tradeoff.

An electromagnetic inversion scheme that integrates stochastic optimization (STO) and plug-and-play (PNP) regularization into contrast-source inversion (CSI), termed STO-PNP-CSI, is developed. Standard CSI solves for the contrast source vector of every transmitter at each iteration, which is expensive in a multi-transmitter configuration. STO instead solves for only one randomly selected contrast source vector per iteration, which reduces the per-iteration cost and can help the inversion escape poor local minima and saddle points. The resulting loss of information, however, increases the ill-posedness of the inversion. To counter this, the Swin-Conv-UNet (SCUNet) denoiser is plugged into the CSI scheme as an implicit regularizer, supplying a learned prior that is stronger than conventional hand-crafted ones and stabilizes the reconstruction. The proposed STO-PNP-CSI is applied to both synthetic and experimental data. The results show that it yields accurate reconstructions at substantially lower computational cost than CSI, including under strong nonlinearity and measurement noise.

Variational quantum algorithms (VQAs) generally rely on classical optimization to train parameterized quantum circuits. This training seeks to minimize an objective function, and its efficiency is central to the practical success of these algorithms. However, globally minimizing such training objectives over the circuit parameters is known to be $\mathsf{NP}$-hard, limiting the prospect of general guarantees for efficient training. In this Letter, we prove that even the weaker task of finding a local minimum of such VQA training objectives is strongly $\mathsf{NP}$-hard, including when the objective admits efficient classical evaluation. Moreover, we show that this hardness persists even for the task of finding a parameter vector within $\ell_p$-distance strictly less than $\pi/2$ of some local minimizer, for every $p\geq 1$. Our central technical result is that approximating a local minimizer of a Hermitian trigonometric polynomial is strongly $\mathsf{NP}$-hard. By explicitly constructing quantum circuits whose training objectives reproduce these hard instances, we obtain a polynomial-time reduction to VQA training. Our results establish a fundamental computational barrier to variational quantum training: even reaching the vicinity of a local minimum remains hard in the worst case.

We give a short and simple proof that an anisotropic generalized quadratic form over a quaternion division algebra with involution of the first kind, in arbitrary characteristic, remains anisotropic after scalar extension to any odd-degree field extension.

We construct smooth collapsing sequences with sectional curvature at least $-1$ whose limits are topological manifolds with nonzero Kirby–Siebenmann class. For every $n\geq6$, the source and limit are simply connected, have dimensions $n+3$ and $n$, and the metrics have a common positive Ricci lower bound. The limits are homeomorphic to the product of the Chern four-manifold with $S^{n-4}$. We also obtain spin examples and five-dimensional lens-space homotopy types, the latter from finite quotients of one fixed $2$-connected eight-manifold. Finally, we realize aspherical manifolds with no closed PL representatives of their homotopy types as collapsing limits with scalar curvatures tending uniformly to $+\infty$, and establish an obstruction to their approximation with almost nonnegative Ricci curvature.

The exceptional group $F_4(\mathbb{C})$ has two conjugacy classes of involutions: an involution $t$ with centralizer $\mathrm{Spin}(9,\mathbb{C})$, and the principal involution, with centralizer $(\mathrm{SL}(2,\mathbb{C})\times\mathrm{Sp}(6,\mathbb{C}))/\{\pm1\}$. We determine the values of all irreducible characters of $F_4(\mathbb{C})$ at $t$. For the irreducible representation with highest weight $a\omega_1+b\omega_2+c\omega_3+d\omega_4$ (in Bourbaki's labelling), we show that the character vanishes at $t$ if and only if $c$ and $d$ are both odd. In all other cases its value is $\pm 2^{-7}$ times the dimension of an explicit irreducible representation of $\mathrm{Spin}(9,\mathbb{C})$. The proof is a limiting argument in the Weyl character formula, combined with a decomposition over the three cosets of $W(B_4)$ in $W(F_4)$ and the triality action of $W(F_4)=W(D_4)\rtimes S_3$ on the weight lattice modulo the $D_4$-lattice. At the principal involution, Nadimpalli, Pattanayak and Prasad showed that the character value is $\pm 1/2$ times the dimension of an irreducible representation of $\mathrm{SL}(2,\mathbb{C})\times\mathrm{Spin}(7,\mathbb{C})$. Comparing the two results, we find that the characters of $F_4(\mathbb{C})$ vanish at the two involutions for exactly the same highest weights. Together these results determine all irreducible characters of $F_4(\mathbb{C})$ at all involutions. We also explain how our formula at $t$ relates to the generalized Weyl character formula of Gross, Kostant, Ramond and Sternberg.

It is well-known that a random Steiner process converges almost surely to a Euclidean ball. On the other hand, as we show in this article, a planar random shaking process converges almost surely to a triangle (in the Banach-Mazur distance). Asking whether these phenomena are generic or exceptional, we study random iterations of an operation called $\lambda$-Steiner symmetrization, which includes Steiner symmetrization and shaking as special cases. We establish abstract criteria for almost sure convergence and divergence of such processes, obtaining criteria for many other symmetrization operations from convex geometry as a byproduct. Based on a joint characterization of ellipses and triangles, we give a full resolution to our question in the planar case.

We develop a family of compact, dispersion-corrected marker-and-cell (MAC)-based finite-difference discretizations for the elastic Helmholtz equation. While dispersion correction is well established for scalar Helmholtz problems, fewer approaches exist for the elastic problems, and most use non-compact stencils. To this end, we introduce a generalized MAC (GMAC) framework that enables compact discretization of the elastic Helmholtz equation. We prove that even though GMAC is non-symmetric, this discretization produces no numerical dissipation and no shear-mode separation. For certain weights, GMAC has lower polarization error than standard MAC. We also introduce an elastic dispersion correction (EDC) process, extending real-shifted-wavenumber dispersion correction methods from acoustic to elastic Helmholtz problems. We prove and demonstrate numerically that GMAC is second-order accurate. Furthermore, we show that EDC substantially reduces dispersion for standard MAC and can provide additional correction in some instances of GMAC. Dispersion analysis shows that the resulting method outperforms existing compact finite-difference discretizations in dispersion accuracy and is competitive with some non-compact schemes. We give numerical evidence for improved phase alignment in mildly heterogeneous media.

Motivated by Igusa's relation between relative cyclic homology and the logarithm of the graded Cartan determinant, we isolate the homological step behind the passage from logarithmic determinants to traces of powers. For a bounded chain complex $(C_*,d)$ and a degree-zero graded endomorphism $f$, not assumed to commute with $d$, we construct the universal quotient on which $f$ becomes a chain map. The kernel of this reflection iterates to a canonical functorial minimal filtration, whose Rees module gives a flat degeneration to an uncurved associated graded. For finite-dimensional chain groups all power supertraces of $f$ are represented by Lefschetz traces on this associated graded, so the resulting logarithmic-determinant and zeta series admit a canonical homological representation.

We characterize compactly supported functions $f$ in Dunkl–BMO spaces as those that can be written as \begin{equation*} f=g+\widetilde{S}_{\mu,\mathcal{P}}, \end{equation*} the sum of some bounded function $g$ and the generalized balayage $\widetilde{S}_{\mu,\mathcal{P}}$ of a Carleson measure on $\mathbb{R}^{N}\times(0,\infty)$ associated with the Dunkl–Poisson semigroup. As a direct application of this representation of Dunkl–BMO functions, we establish a new characterization of the Hardy space in the Dunkl setting via a principal value integral related to $\widetilde{S}_{\mu,\mathcal{P}}$. Moreover, analogous conclusions hold for the Dunkl–heat semigroup as well.

We present a foundational reinterpretation of quantum dynamics based on the standard von Neumann formalism, positing a structural logical bipartition between a classical observational environment and a quantum observed system. Against universalist frameworks, we argue that theoretical consistency requires distinct propositional logics: Boolean logic governs the environment, while non-Boolean logic applies to the observed system. Within this framework, an ideal classical clock is represented as an open system whose energy spectrum is uniformly discretized. The dynamics follow from the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation under the constraint that the Hamiltonian remains observable. Although the clock's temporal evolution is driven by dissipative coupling to the measurement apparatus, it admits a deterministic solution corresponding to information-preserving cyclic dynamics. This deterministic regime may emerge as a logical attractor, stabilized by the Boolean structure inherent to the measurement interaction. Ultimately, $\hbar$ is derived as a fundamental scaling parameter from the classical clock's action operator, revealing that the quantum of action reflects the discrete logical structure underlying classical temporality. Furthermore, this framework establishes an exact mapping with 't Hooft's cellular automaton formalism, offering an alternative ontological interpretation that not require hidden variables and superdeterministic assumptions. The resulting distinction between quantum dynamics (as potential recontextualization) and classical dynamics (as temporal actualization) provides a novel logical approach to the measurement problem.

It is shown that the discriminants of boundary singularities introduced by V.~I.~Arnold naturally arise in integral geometry. They represent the irregularity sets of the volume function. This function maps an affine hyperplane to the volume of the part of a domain in the Euclidean space cut off by this hyperplane. For isolated boundary singularities, the role of the boundary is shown to be related to the hyperplane at infinity in the projective space. Versions of the Milnor fiber bundle adapted to the present problem are defined and studied. Analogs of the local even Petrovsky class are introduced and computed. It is proved that the volume function is locally finite-sheeted only if these classes are trivial. Such components of the complements to the discriminants (quasilacunas) are listed for simple boundary singularities — those having no moduli in the classification. The obtained results establish new obstructions to the algebraic integrability of domains in a Euclidean space.

We prove the existence of infinite time bubble tower solutions for energy critical $k$-equivariant Schrödinger maps $\mathbb{R}^2\to \mathbb{S}^2$ with $k\geq 3$. More precisely, for any integers $k\geq 3$ and $N\geq 1$, we construct a $k$-equivariant solution that is global in one time direction and decomposes asymptotically into a superposition of $N$ bubbles with scales $\lambda_j(t)\sim|t|^{-\beta_j}$, where $\beta_j=\frac{1}{2}\left(\left(\frac{k}{k-2}\right)^{N-j}-1\right)$. Our approach is a backward construction combined with modulation analysis and high order energy estimates. We use Darboux coordinates adapted to the multi bubble profile to express the remainder in a complex scalar form. A key ingredient is the construction of a correction that cancels the leading effect of the nonlinear defect.