Papers

We introduce a damped variant of the Shannon Wavelet Inverse Fourier Technique (SWIFT) for pricing European options when the characteristic function of the underlying model is available. The key idea is to apply an exponential damping transformation to the payoff, which enables the direct computation of Fourier coefficients in the frequency domain without introducing an additional physical-domain truncation parameter. We provide a rigorous analysis of the decay of these coefficients by exploiting the singularity structure of the associated Fourier transforms. For light-tailed models, we obtain Gaussian-type decay estimates, while for semi-heavy and heavy-tailed models whose singularities are poles, algebraic branch points, or logarithmic branch points, we derive exponential decay bounds with explicit polynomial prefactors. The resulting sharp bounds make it possible to truncate the Fourier series without relying on the cumulants of the underlying density, which are often unavailable or difficult to compute in practice. We further derive an error decomposition separating projection, truncation, and quadrature errors, and translate the analysis into practical rules for selecting the damping parameter, resolution level, and truncation range. Numerical experiments demonstrate that the proposed approach consistently improves the accuracy of the original SWIFT method while requiring a significantly smaller number of Fourier coefficients and remaining stable in cases where the undamped method deteriorates.

The Fisher–Kolmogorov equation models the spatio-temporal evolution of interacting biological species and is extensively employed in fields such as ecology, population dynamics, and the modelling of neurodegenerative diseases. Under suitable assumptions on the data, the solution $c$ is non-negative, a key feature because $c$ typically denotes a population density or a relative concentration. However, standard discretisation methods often fail to preserve this property, resulting in non-physical oscillations and unstable numerical approximations. In this work, we propose and analyse a numerical method to stabilise the dynamics of the Fisher–Kolmogorov model around the unstable equilibrium $c=0$. The proposed approach combines a discontinuous Galerkin spatial discretisation on general polygonal and polyhedral meshes with the Crank–Nicolson time integration scheme. The main idea is to suitably modify the formulation at the continuous level so that, on the one hand, it is strongly consistent with the original model, and, on the other hand, it ensures stability when moving to the discrete setting. We prove well-posedness of the semi-discrete formulation, derive stability bounds and prove optimal a priori error estimates in a suitable energy norm. The theoretical results are demonstrated through a comprehensive set of numerical examples. Moreover, we consider an application arising in computational neuroscience by simulating the propagation of $\alpha$-synuclein, a key pathogenic protein implicated in Parkinson's disease and other neurodegenerative diseases, demonstrating that the proposed scheme is stable, high-order accurate, and robust in a biologically relevant computational setting.

Wave scattering by a thin, porous circular plate submerged in deep water is investigated. The problem is formulated as a second-kind hypersingular Fredholm integral equation over the unit disk, solved numerically using the Boundary Element Method. The analysis focuses on calculating hydrodynamic forces, specifically added mass (real part) and damping coefficient (imaginary part). Results demonstrate the influence of the porosity parameter G: less porous plates (G real) increase added mass and hydrodynamic force, while more porous plates (G imaginary) reduce these effects but increase the damping coefficient. The proposed formulation is validated, showing excellent agreement with established literature.

Channel knowledge maps (CKMs) have emerged as a promising technique for providing scene-specific and location-dependent propagation knowledge to enable environment-aware wireless network design. This paper investigates the construction and dynamic updating of a particular type of CKM, namely grid-based channel gain maps (CGMs), for large-scale networks using three-dimensional Gaussian splatting (3DGS). First, we formulate a grid-based channel gain model, where each map entry is defined as the locally averaged channel gain over a receiver grid, thereby suppressing phase-sensitive small-scale fluctuations. The resulting channel gain is decomposed into distance-dependent attenuation, path transmittance, and effective scattering contributions. Based on this decomposition, we develop a physics-informed Gaussian-splatting-based channel gain (GS-CG) model, which represents the propagation environment as a set of Gaussian primitives. The proposed model maps Gaussian geometry, opacity, and directional features to propagation-related factors and renders grid-level channel gains through a differentiable process. To accommodate real-time environmental changes, we further propose an incremental learning mechanism that updates a static reference GS-CG representation into a dynamic CGM. Specifically, the reference Gaussian primitives are frozen, while a compact set of tunable Gaussians is introduced to capture newly induced local channel-gain variations from sparse measurements.Numerical results demonstrate that the proposed GS-CG methods accurately reconstruct grid-based CGMs, efficiently adapt to dynamic environmental changes, and achieve a favorable accuracy-complexity tradeoff for fast CGM refinement.

This paper considers a class of charged-particle dynamics problems in which the particle is subjected to a magnetic force, with a magnetic flux density inversely proportional to a small parameter $0<\varepsilon\ll 1$, and a nonlinear electric force. The resulting highly oscillatory behavior poses significant challenges for numerical computation. To enhance the performance of exponential integrators (EIs), this paper employs a technique that linearizes the ordinary differential equation through a dimension-raising approach. Based on this technique, a new family of EIs is developed that achieves arbitrarily high order. For short-time simulations on the interval $[0,T]$, it is rigorously proved that the proposed method–which employs auxiliary polynomials of degree $k$ and a time step $\Delta t$–satisfies two distinct error bounds: $O(\varepsilon \Delta t^{k+1})$ and $O(\varepsilon^{k+2})$. The latter bound guarantees that the algorithm stays accurate even when the step size is of order $O(1)$. Furthermore, when a large step size $\varepsilon^{-1}\Delta t$ is used to simulate the long-term dynamics over $[0,\varepsilon^{-1}T]$, the numerical scheme attains a uniform convergence rate of $O(\Delta t^{k+1})$. Several numerical experiments confirm these theoretical results.

Metasurfaces, two-dimensional arrangements of subwavelength nanopillars, provide local control over the phase of light, enabling flat optical functions beyond conventional refractive components. Their inverse design, finding the pillar distribution producing a target response, faces two obstacles: the immense dimensionality of the design space and the prohibitive cost of rigorous electromagnetic simulations, precluding exhaustive exploration at device scale. This thesis addresses the challenge in three stages. The first assesses the reliability of rigorous Maxwell solvers: comparing three implementations of Li's factorization rules for RCWA shows that spectral convergence does not guarantee physical fidelity, as these rules implicitly distort the simulated permittivity, most severely in the plasmonic regime. FDTD, immune to such artifacts, is retained as ground truth throughout. The second stage removes the computational bottleneck: a local phase-approximation model, then fully convolutional surrogates trained on FDTD simulations of large pillar metasurfaces, predict the near field almost instantaneously. Exploiting problem symmetries quadruples the training database, and the surrogates generalize to much larger apertures while remaining differentiable. The third stage benchmarks three strategies under a common FDTD protocol ($R^2$ between realized and target far fields): Gerchberg-Saxton retrieval and Local Model ($R^2\approx0.925$), surrogate-based and heuristic-initialized gradient descent ($R^2\approx0.975$), and a generative framework based on diffusion models and Schrödinger bridges. Hybrid posterior sampling and amplitude constraints restore scale-invariant fidelity on surfaces over 230 times larger than training, with diverse, fabrication-tolerant designs. Database enhancement lifts every approach to $R^2\approx0.97$.

We extend Hamana's theory of injective envelopes, along with several key features of the theory, to the realm of partial C*-dynamical systems. In particular, we show that a partial C*-dynamical system has the ideal intersection property if and only if its injective envelope does. A key ingredient in our arguments is a new kind of unitization of a partial action on a unital C*-algebra $A$ arising from the C*-algebra generated by the orbits of $A$ in its injective envelope~$I(A)$. For an arbitrary unital partial C*-dynamical system, which is known to have an enveloping action, we establish a natural relationship between the injective envelope of the system and the injective envelope of its enveloping action. For an abelian partial C*-dynamical system, we show that our construction coincides with the algebra of continuous functions on the Furstenberg boundary of the corresponding transformation groupoid. It is crucial in our work to consider a notion of generalized unital partial C*-dynamical systems, in which the unital ideals are replaced by unital hereditary subalgebras.

We propose a novel probabilistic formulation for optimal stopping mean field games (OS-MFGs) with randomized strategies. We characterize mean field equilibria through a new class of coupled forward-backward systems, termed coupled reflected forward-backward McKean–Vlasov stochastic differential equations (MKV-RFBSDEs). An equilibrium is represented by a quintuple $(X,Y,Z,A,L)$, where $L$ is an adapted, $[0,1]$-valued, non-increasing càdlàg process representing the randomized stopping strategy. The optimality of randomized stopping strategies is characterized through two novel Skorokhod-type conditions involving $L$. This characterization is new even for classical optimal stopping problems without mean field interactions. We rigorously prove an equivalence between solutions of the MKV-RFBSDE system and OS-MFG equilibria in randomized strategies. We establish the existence of equilibria by applying the Kakutani–Fan–Glicksberg fixed-point theorem to a set-valued best-response correspondence, relying on new stability, compactness, and continuity results for the coupled MKV-RFBSDE system. We also prove uniqueness under suitable conditions. Under alternative monotonicity assumptions, we develop a new order-theoretic approach based on Tarski's fixed-point theorem, yielding the existence of extremal equilibria and constructive schemes for the minimal and maximal solutions. We further show that a mean field equilibrium induces an approximate Nash equilibrium for the associated $N$-player stopping game. Finally, we connect our probabilistic formulation with the analytical approach characterized by a coupled system of constrained partial differential equations.

This paper develops a unified method for the exponential stabilization of first-order linear hyperbolic balance laws under general actuation, including both underactuated boundary and in-domain control. The proposed framework brings together a wide range of underactuated configurations within a single formulation and substantially extends existing results restricted to particular actuation settings. Using cutting and folding transformations, the proposed approach is further applied to networks of hyperbolic balance laws, including configurations with cycles. The control design is developed under a stabilizability condition and a robustness assumption. It is based on an invertible backstepping transformation, which partially decouples the system, followed by a reformulation of the stabilization problem at the level of an Integral Difference Equation (IDE). The gains of the resulting dynamic feedback law are constructed at the IDE level by combining a stable rank-reduction procedure, which reduces the problem to a single-input design, with the solution of an interpolation equation arising from a Corona problem. Numerical simulations are presented for a relevant cycle network that cannot be addressed by existing methods in the literature.

We introduce a weak elastic energy for rectifiable curves on compact orientable smooth Riemannian surfaces without boundary. The energy is defined by relaxation starting from a notion of $p$-rotation of inscribed geodesic polygonals, that is obtained by a local construction in normalized isothermal coordinates. For every exponent $p>1$, the resulting relaxed functional detects precisely the intrinsic second-order Sobolev regularity of the arc-length parameterization of the curve. Furthermore, when the relaxed energy is finite, it agrees with the integral of the $p$-power of the geodesic curvature.

We develop an explicit geometric and algebraic description of shadows in the affine Coxeter complex of type $\tilde{A}_2$, by introducing a coordinate system based on a decomposition of the complex into tunnels and channels. We also provide an algorithm for converting arbitrary reduced words into coordinates. Using this framework, we identify geometric symmetries of shadows and show that they are governed by the underlying channel structure. This allows us to derive explicit, piecewise formulas for the cardinality of shadows in $\tilde{A}_2$, depending on the parity of the coordinates. Furthermore, we establish a simple criterion for shadow membership via a counting function that detects admissible positions within channels. These results provide a concrete and computationally effective description of shadows in affine type $\tilde{A}_2$, bridging the gap between combinatorial definitions and geometric realisations.

Let $P$ be a $p$-group of maximal class of order $p^n$ where $p,n>2$. If $n>p+1$ then the exponent of $P$ is $p^k$, where $k$ is the least integer greater or equal to $(n-1)/(p-1)$. If $n=p+1$ then $P$ has exponent $p^2$. And if $n\leq p$ then $P$ has exponent $p$ or $p^2$, with both possibilities arising. The Schur multiplier of $P$ has exponent at most $p^k$, where $k$ is the least integer greater than or equal to $(n-2)/(p-1)$.

The root lattice $A_n$, equipped with its graph distance (equivalently, one half of the ambient $\ell_1$ metric), is isometric to $\mathbb{Z}^n$ with the asymmetric Manhattan metric. We study two extremal problems in this space – the isodiametric problem, i.e., determining the maximum anticode cardinality, and the (non)existence of linear diameter-perfect codes, i.e., lattice tilings by optimal anticodes – and solve them in dimension $3$. We show that, for every integer $D\ge 0$, the largest cardinality of a diameter-$D$ subset of $A_3$ is $\binom{D+3}{3}+(D+1)\lfloor D^2/4\rfloor$, and this value is attained by the balanced difference of two discrete simplices. We then prove an integrality-refined simplex-packing obstruction: a sublattice of $\mathbb{Z}^n$ of asymmetric Manhattan distance greater than $D$ induces a lattice packing by $(D+1)\Delta_n$ in $\mathbb{R}^n$. Combining this observation with the exact lattice-packing density of the tetrahedron yields a complete classification in dimension $3$: lattice diameter-perfect codes in $A_3$ exist precisely for $D=1$ and $D=2$. We also give the equivalent statement for perfect $B_h$ sets of cardinality four. Finally, we formulate a conjecture regarding optimal anticodes in arbitrary dimension, and restate it as an intersection problem for uniform multisets.

We show that abelian coverings of the projective line of order prime to $p$ are generically $\mu$-ordinary in characteristic $p$. The images of the irreducible components of Hurwitz spaces of abelian coverings of the projective line by the Torelli morphism lie in some Shimura varieties. The stratification by Newton polygons of these varieties is known, and we show that the generic Newton polygon for the Hurwitz space coincides with the generic (or $\mu$-ordinary) Newton polygon of the smallest Shimura variety that contains its image. In order to do this, we compute the generic Newton polygons for $L$-functions associated to multiplicative character sums over the projective line.

We develop the theory of functions of bounded variation and the structure theory of finite-perimeter sets on arbitrary Riemannian manifolds without relying on global curvature bounds or completeness of the manifold. To this end, we build a localization framework that permits a synthesis of techniques from Euclidean geometric measure theory and analysis on metric measure spaces while preserving genuinely Riemannian features, such as polar and normal vector fields, reduced boundaries, and approximate tangent spaces. As a consequence, we recover key results of the Euclidean theory, such as a differentiation theorem for a Riemannian generalization of vector-valued measures and the structure theorems by De Giorgi and Federer, formulated intrinsically on Riemannian manifolds. This makes a large portion of the classical Euclidean $BV$ theory available to the Riemannian setting. We demonstrate this by providing the theoretical background for boundary value problems on domains in manifolds, including trace and Gauss-Green theorems. Finally, we prove an approximation result for finite-perimeter sets in a strict sense that respects a prescribed Dirichlet boundary portion of a given ambient domain and apply this to prove Gamma-convergence of a family of energy functionals for capillarity problems with mixed boundary conditions.

We prove Liouville rigidity theorems for translation-invariant real and complex Hessian equations in the viscosity sense, where the PDE is encoded by an admissible set $\mathcal{A}$. The main structural notion is Liouville admissibility, a recursive geometric condition requiring each quotient set to be either boundary compatible or to fall into a terminal class. Our main theorem states that every bounded, globally $C^{0,\alpha}$ entire viscosity solution of \[ \mathrm{Hess}_{\mathbb F}u\in\partial\mathcal{A} \] is constant if and only if $\mathcal{A}$ is Liouville admissible; thus the Liouville-type property is characterized as a geometric property of the admissible set. A central class of examples arises from polarizations of univariate Gårding polynomials satisfying the monotone root sequence condition, producing mixed elementary-symmetric admissible sets and recovering the standard $k$-Hessian equations as monomial cases. The framework also allows anisotropic constructions, including linear pullbacks and intersections of admissible sets.

Let $\Omega \subset \Qp^d$ be a compact open set. Such a set, without lose of generality, admits a representation \(\Omega = \bigsqcup_{c \in C} (c + p^n\Zp^d)\), where $C \subset (\Z/p^n\Z)^d$ and $n \in\N$. We prove that $\Omega$ tiles $\Qp^d$ by translation if and only if $C$ tiles $(\Z/p^n\Z)^d$ by translation. Moreover, $\Omega$ is a spectral set in $\Qp^d$ if and only if $C$ is a spectral set in $(\Z/p^n\Z)^d$.

In this paper, we consider the space $\mathcal{G}raph(\mathcal{C})$ of all closed graphs on the Cantor set $\mathcal{C}$. Equipping this space with Hausdorff metric and employing combinatorial methods, we see that the residual isomorphism classes in the space of single-valued continuous maps on the Cantor set are no longer a $G_\delta$ set but still dense in $\mathcal{G}raph(\mathcal{C})$, and we prove that the set consisting of elements with dense conjugacy classes is a dense $G_\delta$ subset of $\mathcal{G}raph(\mathcal{C})$. Finally, we prove that the set consisting of elements with zero topological entropy is a dense $G_\delta$ set in $\mathcal{G}raph(\mathcal{C}).$

We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of $\mathrm{det}_n$ is always $4$, this gives the first explicit family showing that the dependence on the degree in bounds for strength in terms of Birch rank is unavoidable. Viewing $\mathrm{det}_n$ as an $n$-linear form in its columns, we also prove that its partition rank is at least the largest prime not exceeding $n$. Consequently, \[ n-n^{0.525}\le \operatorname{prk}(\mathrm{det}_n)\le n \] for all sufficiently large $n$, and hence the partition rank of the determinant is $n-o(n)$. The proof introduces an intersection-theoretic method for lower-bounding strength: a short strength decomposition produces a nowhere-vanishing section of a split vector bundle on the complement of the determinantal hypersurface, while a nonzero top Chern class in the Chow ring of $\mathrm{PGL}_n$ obstructs such a section.

A distributed storage system with graph-based replication consists of a collection of databases and the files they contain. The databases (or servers) are represented as the vertices of a graph, while each file is stored in a distinct pair of servers and is represented by an edge of this graph. Private information retrieval (G-PIR) on such a graph-based storage system involves a client which seeks to retrieve a desired file via a query-response protocol, without leaking the identity of the desired file index to any database. The goal of G-PIR is to maximize the rate (reciprocal of the total normalized download) under the privacy constraint. Prior work on G-PIR has involved perfect information-theoretic privacy (i.e., null leakage). However, if the privacy constraint is relaxed, then PIR protocols could be designed that have even higher rates. We term such protocols as Graph-based Weak Private Information Retrieval (G-WPIR) protocols and initiate their formal study in this work. We propose a G-WPIR scheme for arbitrary graphs, and identify the trade-offs it achieves between rate and privacy, under two well known leakage metrics: mutual information leakage and maximal leakage. Our protocol employs minimal subpacketization (representing a file-size constraint) and employs a simple probabilistic query realization to obtain the smooth trade-off. We extend this protocol with some modifications to two special classes of graphs, the complete graphs and the complete bipartite graphs, and identify the corresponding rate-privacy trade-offs achieved.

This paper empirically implements a Solow-type growth model driven by $\alpha$-stable Lévy shocks with time-varying capital elasticity. We extend the framework with an $\alpha$-stable Lévy process, thereby capturing three stylized facts of severe macroeconomic fluctuations: heavy-tailed distributions, jump discontinuities, and infinite variance. We derive the stationary distribution of the capital deviation process, obtain its conditional characteristic function in closed form, and provide an integral representation that explicitly reveals a dual mean-reversion structure separating investment gestation lags from endogenous feedback. We design an estimation strategy based solely on well-defined objective functions that respects the probabilistic properties of Lévy-driven data and circumvents the non-existence of variance. We apply the framework to Argentine quarterly data from 2004 to 2023, with time-varying capital elasticity calibrated from Penn World Table labor shares. Our estimates show that the Lévy specification delivers structural parameters substantially closer to external PWT benchmarks than the Gaussian Ornstein-Uhlenbeck counterpart and substantially improves crisis-period tracking without sacrificing performance in tranquil periods. Cross-country evidence from Colombia and the United States confirms that the quarterly capital adjustment speed $\eta \approx 0.05$ exhibits striking stability across vastly different volatility regimes. Robustness checks across tail index specifications demonstrate that the Lévy framework consistently outperforms the Ornstein-Uhlenbeck benchmark for a broad range of empirically relevant tail indices. These findings establish the Lévy specification as a robust generalization of the Gaussian benchmark, offering a more credible tool for forecasting and structural parameter estimation in both emerging and advanced economies.

Let $D$ be an oriented graph (a digraph with no directed 2-cycles) with maximum degree $\Delta\ge 1$, equipped with nonnegative arc weights of total weight $w(D)$, and let $\mathrm{fas}_w(D)$ denote the minimum weight of a feedback arc set of $D$. Alon (2002) proved $\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{1}{16\sqrt{2\Delta}})w(D)$. We determine the optimal constant: \[\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{\sqrt{2}}{6\sqrt{\Delta}})w(D).\] In fact, we show a stronger result: $\mathrm{fas}_w(D)\le\frac{1}{2}w(D)-\frac{\sqrt{2}}{12}\sum_v w_2(v)$, where $w_2(v)$ is the $\ell_2$-norm of the weights of the arcs incident with $v$. Both bounds are attained by the unit-weight directed triangle, so the constant $\sqrt{2}/6$ is best possible (already among unweighted oriented graphs). The proof combines the vertex-peeling scheme of Berger and Shor with a continuous random-ordering analysis: realizing the random order by independent uniform labels renders the expected local imbalance at each vertex exactly an integrated Khintchine-type functional, and the theorem reduces to the sharp evaluation \[\inf_{\|a\|_2=1}\int_0^1 \mathbb{E}|\sum_j a_j B_j(q)|\,dq = \frac{\sqrt{2}}{6},\] where the $B_j(q)$ are i.i.d. Bernoulli$(q)$ random variables, which we prove via Fourier analysis. The proof also yields a randomized, near-linear-time algorithm attaining the bounds in expectation.

Minimal and constant mean curvature (CMC) surfaces in three-dimensional space forms can be constructed with the Dorfmeister–Pedit–Wu (DPW) method. We introduce CmcMesh, a Python implementation covering the whole pipeline from holomorphic potential to rendered surfaces, using Hoffman and Hoffman's Mesh algorithm. We implement the Weierstrass representation for minimal surfaces in $\mathbb{R}^3$, Bryant's representation for CMC-1 surfaces in $\mathbb{H}^3$, and the DPW method for CMC surfaces in $\mathbb{R}^3$ and for minimal and CMC surfaces in $\mathbb{S}^3$. The package is modular, making it easy to add new DPW variants without modifying existing code.

This paper concerns Lax pairs for circularly symmetric harmonic, Fock-Darwin-type and quartic anharmonic oscillators in two dimensions. Although the 2d isotropic harmonic oscillator is bi-Hamiltonian, its recursion operator does not lead to a Lax pair, nor do we obtain such a pair by taking a limit of the harmonic Calogero model. On the other hand, we show that this superintegrable harmonic oscillator admits a $4 \times 4$ block-form Lax pair with spectral parameter giving two conserved mode energies in involution and a corresponding dynamical $r$-matrix. Interestingly, we also find $2 \times 2$ Lax pairs with spectral parameter that give all three independent conserved quantities satisfying a nonabelian Poisson algebra, thereby providing a simple example of a Lax pair whose conserved quantities are not all in involution. Next, we construct a family of $su(2)$ Lax pairs and $r$-matrices for the quadratic+quartic isotropic anharmonic oscillator. This is then extended to an isotropic oscillator with a rotational energy, which may be viewed as the Fock-Darwin oscillator with a quartic potential. With a change of variables, these Lax pairs and $r$-matrices also apply to the Rajeev-Ranken model, although its noncanonical Poisson structure is distinct from that of the anharmonic oscillator.

In this paper, we study the control co-design (CCD) synthesis problem for a class of systems with parabolic partial differential equation (PDE) dynamics. We first derive a sufficient stability condition for the PDE. By spatially discretizing the PDE and using the sufficient stability condition, we propose a computationally tractable approximate CCD problem. We solve the approximate CCD problem using a gradient-based method. Finally, we justify our proposed approach through an example.

For an odd prime $p$, we consider an action $\alpha$ of the semigroup $\mathbb{N}^{2}$ on the algebra $C(\mathbb{Z}_p)$ induced by the multiplication on the compact topological ring of $p$-adic integers $\mathbb{Z}_p$. We then identify all primitive ideals of the partial-isometric crossed product $C(\mathbb{Z}_{p})\rtimes_{\alpha}^{\textrm{piso}} \mathbb{N}^{2}$ and describe the hull-kernel closures of the subsets of its primitive ideal space.

Let $k$ be a field of characteristic $2$. We exhibit an explicit polynomial endomorphism $F: \mathbb{A}_k^3\to\mathbb{A}_k^3$ whose Jacobian determinant is identically $1$, whose induced extension of rational function fields has degree $3$, and which is nevertheless noninjective. Since $2\neq 3$, this gives a counterexample to the usual Adjamagbo, or separable, formulation of the Jacobian conjecture in characteristic $2$. Stabilization yields analogous counterexamples in every dimension $n\geq 3$

We prove that, among all measurable sets $\Omega\subset\mathbb{C}$ of prescribed Lebesgue measure, there exists a set maximizing the sum of the first $K$ eigenvalues ($K\geq 1$) of the associated Toeplitz operator on the Fock space. In the Fock setting, the case $K=1$ is well known, the optimal sets being balls of prescribed measure, whereas for $K>1$ the existence of optimal sets appears to be new (maximizers are not known explicitly, and the optimality of balls remains conjectural). Moreover, under mild assumptions, our proof extends to localization operators associated with abstract wavelet transforms. In this broader setting, the result is new even for $K=1$. As an application, we prove the existence of optimal sets for the Donoho–Stark concentration problem and its generalization to orthonormal systems.

AI product launches require service-capacity decisions before sufficient product-specific workload histories are available. Early service observations update beliefs about remaining workload, but workload does not physically deplete capacity as inventory demand does. Preserving rapid-response capability may also require costly capacity-access and operational arrangements. We study how a firm should jointly decide current capacity deployment and future response readiness. We formulate a finite-horizon dynamic program with workload beliefs, serviceable capacity, and an absorbing response-channel state. A preservation-deployment decomposition solves one deployment problem for each continuation choice and compares the resulting optimized values. This structure identifies when response readiness should be preserved and how much capacity should be deployed. We operationalize the framework through the Cold-Start Belief Actionability Policy (CBAP), which estimates side-specific values using workload scenarios drawn from predictive beliefs. We also establish bounds on its decision and lifecycle performance losses. Numerical experiments and a BurstGPT-based evaluation show that CBAP reduces lifecycle cost relative to benchmark policies by avoiding premature deployment and unnecessary readiness spending. The results distinguish forecast informativeness from operational actionability and show that current deployment and future response readiness are separate decision margins. Zero deployment may represent Standby rather than Exit, while positive deployment need not imply preserving the response channel.

We study the fractional-part sums $\sum_{k=1}^{n}\{F_n/F_k\}$, where $F_n$ is the $n$th Fibonacci number. Their asymptotic behavior depends on the parity of $n$. For odd $n$, the remainder is expressed in terms of the Gauss circle error term. For even $n$, it is expressed in terms of the Dirichlet divisor error term. Thus determining the optimal remainder exponent for the odd Fibonacci sums is equivalent to the Gauss circle problem, while the corresponding question for the even sums is equivalent to the Dirichlet divisor problem. We also prove analogous formulas for a family of second-order recurrences, including the Lucas sequence, for which the roles of the two parities are exchanged.