Papers

Let $S$ be a smooth complex projective surface endowed with a fibration $f \colon S \to C$ onto a smooth projective curve $C$. We prove that, if $S$ is a Mori dream space (or, more generally, if its pseudo-effective cone is polyhedral) then the Picard number $\rho(S)$ can be effectively computed by counting the irreducible components of the reducible fibres of $f$. A first simple consequence is that, given an elliptic fibration $f \colon S \to \mathbb{P}^1$ with a section and such that $S$ is a Mori dream space, the Mordell-Weil group of the general fibre of $f$ is finite. The main application is a simple criterion for proving that a surface fibred over a curve is not a Mori dream space. We show that certain Horikawa surfaces, Fermat surfaces in $\mathbb{P}^{3}$ of every degree $\ge 4$, particular product-quotient surfaces, and the minimal simply connected numerical Godeaux surface constructed by Craighero and Gattazzo are not Mori dream spaces.

We establish interior $C^{2,\alpha}$ regularity for admissible solutions of the quadratic Hessian equation with positive $C^\alpha$ right-hand side on the full positive branch. The main ingredients are a quantitative large-trace propagation argument and an adaptive Dirichlet comparison method.

De Cock and De Moor conjectured that two matrices built from the solutions of three coupled Lyapunov equations always share their eigenvalues, an identity linking principal angles between model subspaces with the canonical correlations between the past and future of a stochastic process. The conjecture was recorded as Problem 9.1 of Unsolved Problems in Mathematical Systems and Control Theory. We prove it under the natural nonresonance condition, without stability or diagonalizability, and show that the two matrices are similar. The proof exposes a Cauchy-matrix structure created by the rank-one perturbation and reduces the problem to rational interpolation, and the main theorem has also been verified in the Lean~4 proof assistant. The same structure gives a normal form that depends only on the poles and zeros of the model. Off the stable set the eigenvalues become squared cosines of angles in an indefinite metric. We prove an inertia theorem for them and show that reflecting a pole or a zero across the unit circle acts on the angles by a principal pivot transform. For several inputs and outputs, where the conjecture fails, we identify the subspaces whose angles do give the canonical correlations. We also relate our approach to the independent proof of Gillberg and Löfberg.

This paper is devoted to developing a framework for stochastic growth models with environmental risk, in which rare but catastrophic shocks interact with capital accumulation and pollution. Building on the Poisson point process formulation studied in arXiv:2511.13568, we extend the model to disasters driven by a marked Hawkes process, allowing past disasters to temporarily increase the likelihood of subsequent shocks. Our work focuses on a subcritical Markovian Hawkes specification, in which the state space is augmented by the self-excitation component of disaster risk. We establish the well-posedness and nonexplosion of the resulting controlled Hawkes dynamics. Using the Hamilton-Jacobi-Bellman characterization of the corresponding Poisson control problem, we obtain quantitative estimates for the Hawkes excitation process and prove, under a small-excitation scaling, that the Hawkes value function converges to its Poisson counterpart as the magnitude of self-excitation vanishes. This provides a rigorous Poisson approximation of the stochastic control problem and quantifies the effect of self-excitation on optimal growth under environmental disaster risk.

We develop an unobservable M/M/1 queueing model to analyze advance selling and spot selling under uncertainty. Customers are heterogeneous in waiting costs, drawn from a uniform distribution. We disentangle two fundamental sources of uncertainty: customer valuation and service capacity. We derive the respective revenue functions for both advance and spot selling strategies and compare their performance. Our analysis reveals that advance selling is more profitable when customer valuations are uncertain. Conversely, spot selling emerges as the more profitable strategy when uncertainty originates from service capacity. When both sources are present, our numerical experiments suggest that the preferred regime depends on the relative dispersion of the two uncertainties: advance selling tends to be favored when valuation uncertainty is sufficiently pronounced relative to capacity uncertainty, whereas spot selling tends to be favored when capacity uncertainty is more pronounced. These results highlight how the origin and magnitude of uncertainty shape the optimal selling strategy in service queues and offer practical guidance for platforms and service providers deciding between advance and spot selling.

We study the self-intersections of billiard trajectories in a square with small pockets of size $\varepsilon$ removed from its four corners. In particular, we establish an asymptotic formula for the $r$-th moment of the number of self-intersection points for each positive integer $r$, as well as the distribution of such points.

The boundary of a plane amoeba is always contained in its contour, and equality is a characteristic feature of simple Harnack curves. We show that the converse fails, even under strong smoothness and nondegeneracy assumptions. For every two-dimensional lattice polygon, except unimodular triangles, we construct a smooth Newton-nondegenerate curve with smooth logarithmic critical locus and smooth embedded contour satisfying $\mathcal C(\mathscr A_f)=\partial\mathscr A_f$, although the curve is not Harnack. We also provide explicit primitive and nonprimitive families that are not torus-equivalent to simple Harnack curves. A concrete primitive example is certified by exact elimination and Sturm root counting. These results disprove contour–boundary rigidity and show that the contour as a set does not detect the real structure or the multiplicity of coincident critical sheets, thereby refining the compensation problem proposed by Lang, Shapiro, and Shustin.

The contour of an amoeba records the critical values of the logarithmic map and forms a fundamental interface between complex algebraic geometry and real-analytic geometry. Building on the known semianalyticity of amoeba contours, we prove the stronger statement that the componentwise exponential image of the contour of any algebraic hypersurface in $(\mathbb C^*)^n$ is semialgebraic. For a smooth hypersurface, we establish a natural criterion guaranteeing that the contour is a closed real-analytic hypersurface: the logarithmic Gauss map is transverse to $\mathbb RP^{n-1}$ and the logarithmic map has maximal rank along its critical locus. We then show that neither hypothesis is necessary, since singular or ramified critical parametrizations may still have complete real-analytic images. Finally, we derive intrinsic restrictions imposed by analyticity. In particular, every irreducible contour component of dimension $n-1$ is generated, over a dense open subset of its regular locus, by a real-analytic critical stratum on which the logarithmic map has maximal rank. These results distinguish the analytic geometry of the contour from the singularities of its critical parametrization and answer the question of Lang, Shapiro, and Shustin concerning effective sufficient conditions for analyticity.

Let $\Lambda=\operatorname{diag}(m_1,\ldots,m_d)$ be a diagonal expanding integer matrix with $m_1\ge\cdots\ge m_d\ge2$ and at least two distinct diagonal entries, and let $X\subset\mathcal A^{\mathbb N}$ be a subshift with weak specification over $\mathcal A=\prod_{r=1}^d\{0,1,\ldots,m_r-1\}$. Let $K=R(X)$ be the corresponding self-affine set. Writing $\gamma=\dim_{\mathrm H} K$, we prove a sharp critical-measure dichotomy: \begin{align*} \dim_{\mathrm H} K=\dim_{\mathrm B} K \Longleftrightarrow 0<\mathcal H^\gamma(K)<\infty,\quad \dim_{\mathrm H} K<\dim_{\mathrm B} K \Longleftrightarrow \mathcal H^\gamma(K)=\infty. \end{align*} In the second case, $\mathcal H^\gamma$ is not $\sigma$-finite on $K$. This resolves Peres' 1994 conjecture for planar systems and extends to higher dimensions. For sofic subshifts, we prove the existence of the box dimension and show that dimension equality implies that the critical Hausdorff measure is positive and $\sigma$-finite, although it may be infinite. We also construct two examples with $\dim_{\mathrm H} K<\dim_{\mathrm B} K$: in one example the critical Hausdorff measure is positive and finite, whereas in the other it is infinite but $\sigma$-finite.

We answer Problems 8.1 and 8.3 from \cite{Tianyuan2026}. We first note that the condition attributed there to ordinary Prikry forcing is not correct as written: the appropriate formulation uses finite symmetric difference of the ranges of the generic sequences, rather than eventual equality at the same coordinates. We record answers to the two problems for both formulations. A length-$\omega$ Magidor forcing gives a genuinely Prikry-type example satisfying both conditions. Rigid real-adding forcing gives a broad source of further examples, and a forcing of Jech and Shelah shows that the condition does not carry any large cardinal strength. An $E_0$-invariant Jensen-type forcing of Kanovei and Lyubetsky gives a stronger nontrivial example in which the generic reals in a fixed extension form exactly one full $E_0$-class. Finally, a simple recoding turns the real-forcing previous examples into answers for the version of the problems with corrected condition.

This survey provides overview of distributed estimation, filtering, and fault detection techniques in the context of cyber-physical systems (CPS). To establish a strong foundation, we first define essential aspects of linear dynamical systems and related graph theoretic concepts, emphasizing observability conditions that are key for local state estimation. After discussing consensus algorithms, this survey highlights single-time and double-time-scale consensus-based estimation and filtering approaches. We provide a detailed comparative analysis of these methodologies, examining their computational requirements, communication overhead, and performance characteristics in resource-constrained environments. We further explore different diffusion-based estimation techniques and observationally redundant designs to enhance resilience and robustness against failures and adversarial attacks. In addition, we investigate distributed fault detection methods that enable local isolation of faults over large-scale CPS. We present both stateless and stateful detection mechanisms, along with threshold-based techniques that balance detection accuracy and false alarm rates. These approaches are essential for maintaining system integrity and preventing cascading failures in critical infrastructure. This survey concludes with an exploration of diverse real-world applications. We examine implementation challenges and algorithm adaptations in smart grid and power networks, social systems, target tracking and localization, and intelligent transportation systems. By bridging theoretical insights with practical applications, this survey and tutorial provides valuable understanding and research directions in the field of distributed algorithm design for CPS, offering both newcomers and experienced researchers a comprehensive resource for addressing current challenges and future opportunities.

It was conjectured that the nonzero idempotents of the integral quandle ring of a finite latin quandle are trivial. We prove this holds for latin dihedral quandles. Furthermore, we discuss counterexamples against this conjecture.

We prove that every Kobayashi distance isometry between $2$-dimensional Lempert manifolds having finite universal set (with atleast three elements) is holomorphic or anti-holomorphic. It is shown that every such isometry is $C_{loc}^{1,1}$ without any regularity assumption on the isometry.

We develop a representation-independent symmetric-Fock formulation of the nonlinear Boltzmann collision operator for Maxwell molecules. Starting from the standard velocity-space operator, we give a short derivation of Bobylev's Fourier identity and isolate its characteristic structure: two linearly related Fourier arguments followed by multiplication. We then prove, independently of collision geometry and of any coordinate realization, a canonical lift–fusion intertwining theorem: one-particle linear maps lift to the two incoming Fock legs and symmetric-algebra multiplication fuses them into a single outgoing state. Bargmann coordinates make this intrinsic construction especially transparent, but do not define it; Bobylev's collision maps subsequently select the Maxwell vertex from the already graded lift–fusion class. The resulting bilinear map obeys the exact grading \[ \widehat{\mathcal Q}(\mathscr H_p,\mathscr H_q)\subseteq\mathscr H_{p+q}, \] which gives an intrinsic origin for the triangular structures of Maxwell kinetics. Its one-vacuum restriction yields the complete Wang Chang–Uhlenbeck spectrum, while moment adaptation makes the nonequilibrium hierarchy strictly triangular and every finite compatible Fock jet an exact autonomous factor with global exponential relaxation. For the inhomogeneous problem, we extend realization covariance to the nonlinear two-leg vertex and evaluate the moving-frame connection independently through Hermite-function and Fourier realizations. The two routes give the same abstract Fock propagation and the same first Chapman–Enskog source; its traceless level-$2$ and vector level-$3$ sectors are relaxed by the exact Maxwell rates, giving $\Pr=2/3$. This provides an explicit computational demonstration that coordinate realizations may be chosen for economy without entering the structural Fock formulation.

For coprime $a+b=c$, the parity class in which $a$ and $b$ are odd and $c$ is even admits an exact treatment. Triples are measured by the radical excess $E_{\varepsilon}=\log c-(1+\varepsilon)\log\operatorname{rad}(abc)$, and called transgressive at fixed $\varepsilon>0$ when it is non-negative. The $abc$ conjecture asserts that such triples are finite in number. The excess is written exactly in terms of the defect, the mass of repeated primes the radical discards, and rearranges into a linear threshold in which the smaller summand $s=\min\{a,b\}$ appears explicitly. Either $s$ stays bounded along a subsequence, or the defect must overshoot the threshold and rejoin it at an amplified exponent. For the Mersenne family $\mathcal{V}_m = (1, 2^m-1, 2^m)$ transgression at $\varepsilon=0$ holds precisely when $2^m-1$ fails to be squarefree, on a set of exponents of density $47/210$. The defect obeys the exact law $\Delta_m = \Omega_m + \log G_m$, where $\Omega_m$ collects the Wieferich primes dividing $2^m-1$ and $G_m$ is the largest divisor of $m$ whose prime factors divide $2^m-1$. This divisor is trivial on prime-power exponents and largest on $m_k = \operatorname{lcm}(1,\dots,k)$, where the margin $q(\mathcal{V}_{m_k}) - 1 \geq (1-o(1))\log m_k / (m_k\log 2)$ is the most any Wieferich-free mechanism can give. An $abc$ counterexample family along the Mersenne line would require infinitely many Wieferich primes with exponential order–defect growth. A final section separates two elliptic curves attached to the class: the Frey curve, whose minimal discriminant expresses the total defect with a bounded correction at $2$, and a congruent-number Jacobian, whose Szpiro quotient stays below $3$.

Let $\mathcal{H}$ be a complex Hilbert space and $\rho>0$. We establish the explicit formula \[ \omega_\rho (a\otimes b)=\frac{1}{\rho}\|a\| \|b\|+\left|1-\frac{1}{\rho}\right||\langle a,b\rangle|,\qquad a,b\in\mathcal{H}, \] for the $\rho$-numerical radius of rank-one operators, a unified expression interpolating between the operator norm, the numerical radius, and the spectral radius, which correspond to $\rho=1,2$, and $\rho\to\infty$, respectively. As an application, we derive a parametrized family of Buzano-type inequalities for four vectors. We extract from it an explicit closed-form bound and, under a reality condition which is automatically satisfied in real Hilbert spaces, we determine the best bound in the family in closed form. Examples show that the resulting bounds can be strictly sharper than both the Cauchy–Schwarz and the Buzano inequality, and the classical Buzano inequality is recovered as a boundary case, which yields an alternative proof of it.

Let $f:(0,\infty)\to(0,\infty)$ be a nontrivial normalized operator monotone function and set $s=f'(1)$. We introduce the alternative-mean trace functional $$ \altPhi_f(A,B) :=\Tr(A\nabla_s B) -\Tr\!\left( f(A^{-1}\sharp B)\,A\,f(A^{-1}\sharp B) \right). $$ on the positive definite cone. We prove that $\altPhi_f$ is a quantum divergence in the sense of Bhatia–Gaubert–Jain whose diagonal Hessian induces a positive multiple of the Bures–Wasserstein Riemannian metric. We also establish the sharp comparison $$ s(1-s)d_{\rm BW}(A,B)^2 \le \altPhi_f(A,B) \le (1-s+s^2)d_{\rm BW}(A,B)^2. $$

We prove Kato's main conjecture for modular forms at nonordinary primes for weights in the Fontaine-Laffaille range. The key ingredients in the proof are a reformulation of the conjecture in terms of signed Selmer groups due to Lei-Loeffler-Zerbes, certain $p$-adic families of Rankin-Eisenstein classes arising from the work of Lei-Loeffler-Zerbes and Kings-Loeffler-Zerbes, and the lower bound divisibility in an Iwasawa-Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions obtained in our earlier work \cite{CLW}.

We study a quadrature surface free boundary problem on a smooth compact finite-dimensional Riemannian manifold $(M,g)$. The problem is formulated as a shape optimization problem involving a Dirichlet problem for the Laplace–Beltrami operator, with a geometric condition on the free boundary. Under suitable uniform geometric assumptions on the admissible class, we establish its compactness and prove the stability of the corresponding Dirichlet problems, including strong convergence of the associated states in $H_0^1(M)$. We derive the first-order optimality condition for the associated shape functional. Finally, we establish a Riemannian comparison principle for the second fundamental forms and the mean curvatures of tangent boundaries at a contact point. These results provide a rigorous extension of the quadrature surface free boundary framework from the Euclidean setting to compact Riemannian manifolds.

We analyse a non-residual variational multiscale finite element formulation with dynamic subscales for incompressible generalised Newtonian Navier–Stokes flows. The apparent viscosity is assumed to be bounded and Lipschitz continuous with respect to the shear rate, covering several regularised rheological laws. The method separates, through orthogonal projections, the unresolved pressure-divergence, convective, and pressure-gradient contributions. For the linearised semi-discrete problem, we prove well-posedness and an unconditional stability estimate in an anisotropic VMS norm. The estimate controls viscous dissipation, subscale energies, discrete divergence, and a coupled acceleration-convection-pressure balance. For the non-linear formulation, a fixed-point argument yields existence and optimal-order a priori bounds under suitable regularity and smallness assumptions. The analysis also provides weak-in-time control of the discrete pressure-convection balance. A key point is that the dynamic pressure-related subscale provides the time-derivative contribution needed to control the projected pressure-gradient error, a mechanism unavailable in the corresponding quasi-static setting.

The method of ellipcenters (ME) is a recent technique developed for unconstrained minimization. Its iteration relies only on first order information and consists of building a suitable two-dimensional ellipse that tries to capture intrinsic ill-conditioning of the problem. The center of the ellipse is taken as the next iterate, which justifies the name of the method. ME was already shown to converge linearly when the objective function is smooth and strongly convex. The special case when the objective is quadratic was studied in the first paper on ME and is also the subject of our work here. In this paper, we propose two variants of ME: RelaxME which adds in ME an additional relaxation step at each iteration and MomME which embeds momentum in ME. We prove linear convergence of RelaxME and MomME and, in particular, we show convergence of RelaxME and MomME (and also of ME) in one iteration when the matrix of the quadratic form has only two distinct eigenvalues. Finally, we provide the results of numerical experiments which compare on five optimization problems (and different combinations of parameters defining these problems) ME, RelaxME, and MomME, with 3 other optimizers: the conjugate gradient method [10], Barzilai and Borwein gradient method with long step [2], and the gradient method with adaptive spectral step length [7]. On most instances, MomME provides the smallest number of iterations and conjugate gradient and MomME provide the smallest CPU times and similar CPU times. This opens optimistic possibilities for ME with momentum in broader settings.

We study degree-three rational sphere maps in two complex variables. After a standard normalization, the denominator of such a map takes the form \[ g_\sigma(z)=1+\sigma_1 z_1^2+\sigma_2 z_2^2, \qquad \sigma_1,\sigma_2\geq 0. \] A basic question is: which pairs $(\sigma_1,\sigma_2)$ can actually occur as the denominator of a degree-three rational sphere map? The first main result of the paper gives a complete answer: such a denominator occurs if and only if \[ 0\leq \sigma_1,\sigma_2<1, \qquad \sqrt{1-\sigma_1^2}+\sqrt{1-\sigma_2^2}>1. \] Our approach converts the sphere-mapping condition into a finite-dimensional Gram-matrix positivity problem. Furthermore, for each admissible parameter $ \sigma=(\sigma_1,\sigma_2), $ we determine all possible minimal target dimensions in which the corresponding denominator $g_\sigma$ can be realized. We also give a Gram-matrix normal form for maps with a fixed denominator and compute, for each admissible $\sigma$, the dimension of the moduli space of equivalence classes of rational sphere maps realizing $g_\sigma$. Finally, we extend the Gram-matrix method to arbitrary source dimension and obtain a general sufficient condition for the existence of degree-three rational sphere maps.

We survey the theory of vocabulary growth founded in the setting of stochastic processes. In particular, we model the expected number of types through Bernstein functions and Hausdorff sequences. These classes of mathematical objects, defined by alternating signs of their derivatives or differences, can be related to continuous-time Poisson point processes and discrete-time IID processes, respectively. Building on previous accounts of the vocabulary growth, we integrate the broader theories of Bernstein functions and Hausdorff sequences and connect them with recently developed hapax rate models. In particular, we prove that the logistic hapax rate model has a non-negative spectrum and hence it defines a Bernstein function, thereby solving an earlier posed problem. We also analyze the limitations of the Bernstein–Hausdorff theory of the vocabulary growth by considering its generalizations under stationary and Weibull renewal processes.

Physics-informed neural networks (PINNs) often face ill-conditioned objectives that limit high-accuracy training. Dense quasi-Newton methods improve local conditioning but require expensive optimizer state, while Kronecker-factored methods such as SOAP scale to larger networks but rely on periodic basis updates. We introduce \method, which augments SOAP-style preconditioning with a scalar secant-energy correction adapted to Kronecker geometry and an adaptive basis update followed by variance-state downscaling. We characterize the directional secant matching induced by the scalar correction and give a bound on variance-state mismatch across basis changes. Across eight PDE benchmarks, \method attains the lowest final residual on six, including Burgers and Boussinesq, while SOAP-family baselines perform better on Gray-Scott and Ginzburg-Landau. On Boussinesq, \method reaches a residual of $10^{-5}$ in 4.1 hours with 9.2 GB peak VRAM, while Adam does not reach this target within 14 hours. Three-seed $L^2$ and $H^1$ errors on four representative PDEs support the link between lower residuals and improved solution accuracy. These results position \method as a scalable option for stiff, high-accuracy physics-informed training, rather than a uniform replacement for existing optimizers.

We compute the arithmetic rank of nullcone ideals arising from natural actions of the special linear, orthogonal, and symplectic groups on direct sums of copies of their standard and dual representations. Over an infinite field of characteristic different from two, we show that the arithmetic rank equals the Krull dimension of the invariant ring, and compute this dimension. In positive characteristic, the required lower bounds are not detected by local cohomology, and are obtained using étale cohomology.

Bayart, Pellegrino, and Rueda obtained diagonal absolute summability exponents for continuous \(m\)-linear operators from \((\ell_1)^m\) into \(\ell_p\). Their optimality was known for \(2\le p\le\infty\), whereas the case \(1\le p<2\) remained open. We settle the latter case by means of finite Walsh test operators, and consequently obtain the optimal exponent for every \(1\le p\le\infty\). The same argument, combined with the corresponding interpolation and embedding results, also yields optimal Kwapień-type statements for several related range spaces.

We characterize exact nonuniqueness in the planar interior Radon problem for arbitrary pairs of open convex sets. There exists a nonzero smooth function compactly supported in the first set whose integral over every line meeting the second set vanishes if and only if an ellipse contains the closure of the second set and is compactly contained in the first set. The same criterion holds for a nonzero $L^1$ function whose essential support is compactly contained in the first set and whose Radon transform vanishes almost everywhere on those lines. For concentric open squares, the criterion gives the sharp threshold $1/\sqrt2$ for the inner-to-outer half-side ratio. This yields counterexamples to Conjecture 1.2 of Boman (2021) and to Theorem 40.1 of Boman (2025).

We show that Partizan Arc Kayles (PArcK), a generalization of Domineering to graphs, is PSPACE-complete via a reduction from Positive CNF and with recently-discovered techniques for creating PArcK positions with high temperature. The reduction uses only red and blue edges.

The orbit problem is the problem of whether, given $x, y\in \mathbb{Q}^n$ and an $n\times n$ matrix $A$, there exists an $i\in\mathbb{N}$ such that $A^ix = y$. In 1980, Kannan and Lipton proved that the orbit problem is decidable. We show that a generalization of the orbit problem, where the field is $\mathbb{Q}(X)$ for $X$ a countable set of transcendentals, is also decidable. We define the orbit power problem for an arbitrary group $G$ to be the problem of when, given $x, y\in G$, there exists an $n\in\mathbb{Z}$ such that $x^n = y$. We then use the main result to show that the power orbit problem is decidable for an assortment of groups, including the braid groups and $\operatorname{Aut}(F_2)$.

After recalling basic formulas concerning modular forms of half-integral weight, Gegenbauer polynomials, and Rankin–Cohen brackets, we give four versions of the Shimura lift and several applications to Hilbert modular forms associated to real quadratic fields. We claim no originality, but wanted to collect all the formulas in a single paper.