Papers

In this paper we derive a framework for topological expansions of shape functionals with respect to multiple holes based on the expansion of one hole. Our strategy is similar to deriving a Taylor expansion of order two of functions in $\mathbf R^d$ by applying $d$-times a Taylor expansion in $\mathbf R$. For shape functionals defined on subsets of $\mathbf R^2$ we relate the topological expansion with multiple holes to a Taylor expansion with respect to the ball volumes where usual derivatives are replaced by topological state derivatives. The first topological state derivative was introduced in a previous paper and relates the first topological derivative of cost functionals. The second topological state derivative at two distinct points is introduced, similar to a second directional derivatives of functions defined on Banach spaces, as the topological state derivative of the topological state derivative. The second topological state derivative at the same point is defined as the "far-away" component of the compound layer expansion of the state variable. In dimension three the obtained topological expansion still involves first and second topological state derivatives, but it does not follow the same pattern of a Taylor expansion in the ball volumes. As we will show this dimension dependent behaviour is rather of geometrical nature and already appears in cost functionals without PDE constraints.

We show that for a generic real or complex-valued compactly supported potential, the corresponding Schroedinger operator achieves maximal resonance density, in the sense that its integrated resonance counting function achieves the optimal asymptotic upper bound. For odd dimensions this follows from results of Dinh-Vu once we adapt an argument of Christiansen Hislop. The proof for even dimensions constitutes the bulk of the paper, and we prove several new results on resonances which have analogues in the odd dimensional case. This includes a sharp upper bound on the integrated resonance counting function for any compactly support potential, a proof that the characteristic function of a ball has resonance counting function which achieves the optimal upper bound, and an even-dimensional analogue of the result of Dinh-Vu on asymptotics of the resonance counting functions for complements of pluripolar subsets of analytic families of potentials. We use the characterization of resonances as zeros of certain Fredholm determinant functions related to the scattering matrix, allowing us to apply techniques and results from the theories of one and several complex variables. Our proof that the characteristic function of a ball has counting function achieving the optimal upper bound uses the uniform asymptotics of Bessel functions and follows ideas of Zworski, Christiansen-Hislop, and Dinh-Vu.

In this article, we focus on a MHD model that takes into account the Coriolis force. In this case, the strength of the rotation is measured by the Rossby number $\epsilon$ (positive). Under the assumption that the rotation is strong (that is, when $\epsilon$ goes to zero) and the kinematic viscosity goes to zero as $\epsilon$^$\alpha$ ($\alpha$ positive), we first establish the global convergence of weak solutions, and then prove the global existence and convergence of strong solutions. In particular, the Strichartz estimates used to obtain explicit convergence rates depend on the above parameter $\alpha$ and lead us to identify an admissible interval for $\alpha$.

Arithmetic terms are finite fixed compositions of additions, subtractions, multiplications, divisions with remainder and exponentiations, containing variables interpreted as natural numbers. They build a well-defined notion of closed formula. It is known that every Kalmar elementary function can be expressed as an arithmetic term. In this paper, one studies the power of expression of the arithmetic terms. By interpreting Hilbert's Tenth Problem in arithmetic terms, it is shown that it is undecidable whether one-variable arithmetic term takes the value $0$, or whether two such terms take or not the same values. An algorithm constructs the arithmetic term representing an arbitrary function, which has been defined by a recurrence rule. This construction has various applications. Functions with chaotic behavior, like the Logistic Map, can be expressed as arithmetic terms. Finally, we construct a Turing complete arithmetic term and we express a Turing universal function by an arithmetic term. A somewhat unexpected application: there is a wise arithmetic term. It gets (the code of) a sentence, (the code of) a formalized theory and a bound $B$, and after performing a constant number of operations, it outputs (the code of) a proof of the sentence using the given theory if such a proof does exist and its length is less than $B$. Otherwise, it outputs $0$.

The superamalgamation property is a strengthening of the amalgamation property and has found many applications in algebraic logic, more recently, also in model theory. Here we introduce an even stronger notion we call the hyperamalgamation property. We show that lattices, semilattices, Boolean algebras and dual Heyting algebras have the hyperamalgamation property. As a first application, we obtain the amalgamation property for semilattices, Boolean algebras and Heyting algebras with further operators.

We investigate a generalized stochastic fractional neuronal model combining fractional dynamics with correlated stochastic inputs. The proposed framework is described by a fractional differential equation driven by a latent stochastic process with stationary increments and mean-reverting structure. This formulation allows the inclusion of both short-range and long-range dependence structures and naturally produces non-exponential relaxation phenomena. The main goal is the development of a feasible parameter estimation procedure based on discrete observations of the neuronal state process. We propose a two-step methodology. First, the parameters governing the fractional dynamics are estimated by exploiting the asymptotic behavior of Mittag-Leffler functions near the origin. Subsequently, the latent stochastic input is reconstructed through fractional differentiation techniques, allowing the estimation of the parameters governing the hidden noise dynamics. We derive quantitative error bounds for the estimators and analyze the reconstruction error of the latent process under suitable regularity assumptions on the driving noise. In particular, the interplay between the order of the fractional derivative and the Hölder regularity of the noise process naturally emerges in the stability analysis of the reconstruction procedure. Finally, simulation studies illustrate the applicability of the proposed methodology and highlight the influence of memory effects and noise regularity on the quality of statistical inference. The results support the relevance of fractional stochastic analysis for the modeling and inference of neuronal systems with memory and correlated inputs.

This letter presents a generalized technique for cancelling the parasitic inductance of capacitors using a lattice network. The Z-matrix-based equivalent transformation shows that the conventional vertically symmetric inductance condition can be relaxed for cancellation of the dominant high-frequency inductive component. A vertically asymmetric lattice network is then proposed, enabling this component to be cancelled by adjusting only one inductance. The method is further extended to common-mode noise reduction using a common-mode choke, and both concepts are experimentally verified.

Ballantine, Beck, Merca, and Sagan conjectured four identities, collectively Conjecture 19, relating the image of the map pre_k on integer partitions to four OEIS sequences. We prove parts (i) and (iii) unconditionally, prove part (iv) unconditionally using the injectivity of pre_2 on partitions of n (Conjecture 1 of the same paper, proved by Li in arXiv:2508.00971), and show that this injectivity is in fact equivalent to part (iv). For part (ii) we prove the partition-theoretic half unconditionally and reduce the remaining content to a 2006 conjecture of Dean Hickerson on the OEIS concerning Huffman coding. We also correct a sign error in the published statement of part (iii): the correct identity is chi(ImP_3(n)) = A213213(n) - 1, not 1 + A213213(n) as stated.

The inverse spectral problem asks to what extent the Laplace–Beltrami spectrum determines the geometry of a Riemannian manifold. We study a natural weakening, called near isospectrality, in which the spectra of two compact manifolds agree outside a finite set, counted with multiplicity. We prove that for compact quotients of a fixed simply connected symmetric space of nonpositive sectional curvature, near isospectrality already forces full isospectrality. We then extend this rigidity to a broad collection of compact quotients of irreducible symmetric spaces of noncompact type. In this larger setting, near isospectrality determines enough heat invariants to identify the universal cover within the class under consideration, and the fixed-cover rigidity result then implies full isospectrality. Thus, within the class studied here, eventual agreement of the Laplace spectrum already forces complete spectral agreement.

We construct a conformal product structure on $S^3$ using the Reeb foliation. This is the first example of a conformal product structure on a compact simply connected manifold which is not built on a global product manifold.

This paper investigates inverse problems for Linear-Quadratic-Gaussian (LQG) Mean Field Games (MFGs) based entirely on the observation of mean-covariance trajectories. We address three sequential challenges: identifying the optimal control for observed initializations, determining the control for arbitrary initializations, and recovering consistent cost parameters. After establishing the existence and uniqueness of the forward Nash equilibrium under mild hypotheses, we analyze the injectivity of the parameter-to-trajectory mapping, demonstrating that it is inherently non-injective and providing sufficient conditions for parameter equivalence. We prove that while the optimal control is locally identifiable for observed initializations under minimal assumptions, global identifiability requires a deeper structural recovery of the game's costs. To bridge this gap, we propose a constructive semidefinite programming method to infer cost parameters that are strictly consistent with the observed population dynamics. Numerical experiments illustrate this method.

We construct an explicit class of non-separable Gabor windows on $L^2(\R^d)$ by applying chirp deformations to tensor-product dual pairs on separable lattices. Starting from one-dimensional dual Gabor frames, we first obtain separable higher-dimensional dual pairs by tensorization. We then transport these systems through the unitary chirp operator $U_C f(x)=e^{\pi i x^T Cx}f(x)$ and the associated phase-space shear, obtaining Gabor systems on lower block-triangular lattices of the form $ \Lambda_{A,B,C}=\{(Ak,CAk+B\ell):k,\ell\in\Z^d\}. $ {The construction is governed by a single unitary conjugacy identity for the mixed Gabor reconstruction operator. As consequences, compact support, smoothness, frame bounds, exact duality, canonical duals, and approximate-duality errors are transported without loss.} {We also record a covariance-level diagnostic, depending on a chosen STFT reference window, which describes the second-order time–frequency tilt induced by the chirp. This diagnostic is separated from the unchanged frame stability constants and is not used here to claim improved sparsity, denoising, conditioning, or computational complexity.}

We study a hyperbolic approximation ("hyperbolization") of the Cahn-Hilliard (CH) equation, originally proposed by Dhaouadi, Dumbser, and Gavrilyuk (2025, DOI: 10.1098/rspa.2024.0606) and study its convergence towards the CH model in a relaxation limit both via formal asymptotic expansions and, for a slightly modified approximation, via the relative energy framework. Moreover, we develop energy-stable semidiscretizations of the CH equation and of this hyperbolization using upwind summation-by-parts operators in space. Subsequently, we combine them with (additive) implicit-explicit (IMEX) Runge-Kutta methods based on a convex-concave splitting. We show that the resulting method is asymptotic preserving, i.e., it converges in the limit of the relaxation parameter to a stable discretization of the original CH equation. The choice of the necessary parameters is guided by the a priori error estimate based on the relative energy framework.

We present recent advances in applying and analyzing multilevel stochastic gradient descent algorithms to risk-averse, three-dimensional PDE-constrained optimization problems. The algorithm uses adaptive multilevel Monte Carlo gradient estimates, provides parallel scalability as well as improved convergence rates and computational complexity compared to standard batched stochastic gradient descent methods. We study the method in computationally demanding settings using three-dimensional elliptic diffusion problems and large risk-aversion parameters.

In this paper, a new notion, essential contractibility of Banach algebras, is introduced and some of its properties are examined. The main result is to investigate the essential contractibility of (symmetric abstract) Segal algebras.

We classify the long-time decay rate in dispersive estimates for the Schrödinger and wave equations on non-trapping asymptotically conic manifolds and exact metric cones in terms of the intensity of geometric focusing. Letting $X_0$ be a metric cone, one of our main results demonstrates that each multiplicity of conjugate points within distance $\pi$ on $Y=\partial X_0$ leads to a $|t|^{1/2}$-loss in the long-time decay order and a half-order shift in the regularity index in the dispersive estimate for the Schrödinger equation. Unexpectedly, conjugate point pairs on $Y$ at distance $\pi$ do not cause loss when the Legendre submanifold carrying the wave propagation satisfies a natural admissible condition that we propose. In sum, we give a robust framework for proving dispersive estimates that is stable under geometric perturbations and also accommodates perturbations by potentials.

The Material Point Method provides an effective approach for modelling the large deformations that often arise from contact interactions between rigid structures and surrounding continua. However, solving these problems requires accurate representation of the continuum-structure interface, which necessitates high resolution background mesh and material point discretisations. This requirement, combined with evolving continuum-structure interfaces and the fact that most Material Point Method implementations are dependent on structured meshes, can result in large numerical systems and long run times especially when modelling problems in three-dimensions. Motivated by this issue, this paper provides the first octree-based implicit Material Point Method for efficient solution of large deformation continuum-structure interaction problems. The octree background mesh provides a natural way to automatically adapt both the computational mesh and the material point discretisation based on the position of the interaction between the structure and continuum. The new approach is demonstrated on a number of large deformation benchmark and continuum-structure interaction problems, where up to a 5.5-times speed up and a consequent 21-times CO2 saving is achieved when running on a HPC compared to results obtained using a conforming mesh.

These pedagogical notes provide an introduction to nonlinear filtering for diffusion processes. The objective of nonlinear filtering is to estimate an unobserved stochastic state from partial and noisy observations, and thereby characterize the conditional distribution of the state given the observation history. After introducing the state-observation model, we develop the main tools of the theory, including Markov semigroups, infinitesimal generators, and change-of-measure methods. This leads naturally to the Kallianpur-Striebel formula, the Zakai equation for the unnormalized filter, and the Kushner-Stratonovich equation for the normalized filter. The presentation follows, to a large extent, the classical exposition of Bain and Crisan's "Fundamentals of Stochastic Filtering", while placing particular emphasis on intuition, motivation, and the interpretation of the main concepts and equations. Many technical arguments are revisited, with additional details and comments intended to facilitate a first reading of the subject. These notes are not intended to be an exhaustive account of stochastic filtering. Their aim is instead to provide a guided tour of some of its central ideas, methods, and equations.

This paper studies O'Neil-type convolution inequalities in local and generalized Morrey spaces on the n-dimensional torus. We extend classical Minkowski and Young inequalities to the Morrey setting and obtain boundedness results for convolution operators. The main result is a generalization of O'Neil's inequality from Lorentz spaces to Morrey-type spaces via dyadic decompositions. We derive norm estimates for convolution operators between Morrey spaces with different parameters and characterize the admissible range of exponents ensuring boundedness. The proof relies on interpolation theory for Morrey spaces, Köthe duality, and real interpolation methods. We also establish embedding relations between local and generalized Morrey spaces. These results extend classical convolution inequalities and unify Lorentz and Morrey frameworks, with applications to partial differential equations and harmonic analysis.

We solve a boundary value problem arising from Maxwell's equations in the quasi-static approximation, which governs the time evolution of the so-called weighting potential $V_{\mathrm{w}}(t,x)$ in resistive solid-state particle detectors. The model reduces to the third-order time-dependent PDE $$ \varepsilon\,\partial_t \Delta V_{\mathrm{w}}(t,x) + \mathrm{div}\,\big(\sigma(x)\nabla V_{\mathrm{w}}(t,x)\big)=0 \quad\text{in } [0,T]\times\Omega,$$ supplemented with mixed Neumann-Dirichlet boundary conditions, possibly degenerate. Our analysis is based on the decomposition of the weighting potential into a static and a dynamic component. The static part solves a uniformly elliptic mixed boundary value problem, while the dynamic part satisfies a degenerate parabolic Cauchy problem. We also establish interior regularity results.

We prove that the Cartan distribution $C^r_\pi$ on the jet bundle $J^r\pi$, for $\pi:E\rightarrow Q$, is an $N^r_\pi$-contact distribution, where $N^r_\pi$ is the corank of $C^r_\pi$. In other words, $C^r_\pi$ has non-degenerate distributional curvature, and admits, locally, a complementary distribution spanned by $N^r_\pi$ commuting infinitesimal symmetries. Consequently, there exists around every point of $J^r\pi$ an open $U\subset J^r\pi$ and an $R^{N^r_\pi}$-valued one-form $\eta\in\Omega^1(U,R^{N^r_\pi})$ such that $\ker\eta= C^r_\pi|_U$ and $\ker\eta\oplus\ker d\eta=TU$. This recovers the canonical structure of $J^r\pi$, algebraic Spencer contractions, and other mathematical entities in a simple unifying manner while providing a local Hamiltonian structure and other advantages. We will call a manifold endowed with a k-contact distribution a k-contact manifold. New types of polarisations for k-contact distributions lead to our main recognition theorem, which shows that a polarised k-contact distribution is locally equivalent to a Cartan distribution precisely when its polarisation is of jet type. This characterises finite-order jet geometry as polarised $N_\pi^r$-contact geometry of jet type. Moreover, the highest-order vertical polarisation, holonomic submanifolds, characteristics for Lie symmetries and their brackets, and other structures are reconstructed and generalised via k-contact geometry. Solutions of PDEs become polarised Legendrian submanifolds, jet prolongations are polarised k-contact Legendrian prolongations, etc. Our k-contact formalism gives a common language for constructions that are awkward in a single jet presentation, and extends jet geometry to new problems. Our techniques provide new general reduction methods for PDEs retrieving standard and non-local reductions as particular cases, and are applied to PDEs.

Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we find that every topological \(MV\)-algebra is a Mal'tsev space introduced by Mal'tsev in 1954. Hence, applying the theorem of Reznichenko and Uspenskij on pseudocompact Mal'tsev spaces, we show that the product of arbitrary family of pseudocompact topological \(MV\)-algebras are pseudocompact. We also prove that every $\sigma$-compact topological \(MV\)-algebra is ccc. Secondly, we obtain that the Stone-Čech compactification of a pseudocompact topological \(MV\)-algebra carries a natural compact topological \(MV\)-algebra structure extending the original one. Finally, we prove that: let \(I\) be a closed ideal in a pseudocompact topological \(MV\)-algebra \(A\) and \(\iota_1:A\hookrightarrow\bet A\) is the naturally injective; then \(\cl_{\beta A}\iota_1(I)\) is a closed ideal of \(\beta A\) and \( \beta A/\cl_{\beta A}\iota_1(I)\cong \beta(A/I)\).

We construct an explicit sign-changing Poisson kernel for a uniformly elliptic divergence form operator in the unit disk. The coefficient matrix is non-symmetric, and its skew-symmetric part has a jump discontinuity across a fixed diameter, with the size of the jump determined by $k\in\mathbb R$. For each $1<p<\infty$, the associated $L^p$ Dirichlet problem undergoes a sharp transition as $k$ crosses a $p$-dependent threshold. In the unique solvability regime, the solution operator preserves positivity and is represented by a nonnegative Poisson kernel. Beyond the threshold, the homogeneous problem admits a nontrivial solution, and for real-valued boundary data, every bounded linear selection from this nonunique solution family is represented by a sign-changing kernel. Thus, even in bounded domains, uniform ellipticity and the maximum principle in the energy class do not prevent Poisson kernels from changing sign. Using a Riemann-Hilbert formulation, we obtain explicit solution formulas and show that the factorization index governs both nonuniqueness and sign change. To our knowledge, this is the first explicit example of this counterintuitive phenomenon in a bounded domain.

We prove that the asymptotics of the Hermitian-Yang-Mills flow on a slope semistable holomorphic vector bundle over a compact Kähler manifold are determined algebro-geometrically, via the iterated filtration defined by Haiden-Katzarkov-Kontsevich-Pandit. This proves a conjecture of Haiden-Katzarkov-Kontsevich-Pandit in this setting. Moreover, we prove a non-linear analogue, relating the asymptotics of the Calabi flow near a cscK manifold to the iterated balancing filtration of the deformation space. In both settings, we reduce the infinite-dimensional flow to a finite-dimensional flow, following the foundational work of Chen-Sun. In finite dimensions, we prove that the asymptotics of the moment map flow are determined by the iterated balancing filtration, proving a conjecture of Ibáñez Núñez.

We study partial orders induced by constrained variants of finite graph homomorphisms: monomorphisms, embeddings, full homomorphisms, vertex-surjective, edge-surjective and surjective homomorphisms, and locally injective, locally surjective and locally bijective homomorphisms. For each order we ask for analogues of the standard structural properties of the graph homomorphism order: canonical cores, past- or future-finiteness, universality, gaps and finite dualities. The comparison shows which phenomena are specific to ordinary homomorphisms and which are consequences of simpler order-theoretic mechanisms. We identify cores for full and surjective homomorphisms, relate full-homomorphism cores to point-determining graphs, characterize gaps in the full homomorphism order, and give finite obstruction bounds for several one-sided finite orders. We also analyze locally constrained homomorphisms on connected graphs. In particular, locally injective homomorphisms have all connected graphs as cores, admit infinite-chain density under natural degree-refinement assumptions, have explicit gap witnesses, and are universal already on finite connected bipartite subcubic cactus graphs. The paper reorganizes and extends several earlier arguments into a single framework for constrained homomorphism orders.

Let \(\mathbbm{k}\) be a finite field of sufficiently large odd characteristic, let \(F\) be the Frobenius map, and let \(G\) be a symplectic or orthogonal group defined over \(\mathbbm{k}\). For an irreducible representation \(\pi\) of \(G^F\), its rational wavefront set \(\ScO(\pi)\) is the set of \(F\)-rational nilpotent orbits \(\mathcal O\) such that \(\pi\) occurs in the generalized Gelfand–Graev representation \(\Gamma_{\mathcal O}\). We prove that all maximal rational orbits in \(\ScO(\pi)\) lie in a single \(F\)-stable nilpotent orbit, and that the whole rational wavefront set is obtained from its maximal rational orbits by rational closure. We also give an explicit description of these maximal rational orbits in terms of Lusztig parameters and compute the dimensions of the corresponding generalized Whittaker models. For unipotent representations, we prove that the maximal rational wavefront set is a single fiber of Lusztig's canonical quotient. For special unipotent representations, this fiber identifies the kernel of the canonical quotient map, giving a finite-field representation-theoretic realization of the canonical quotient. We also describe the effect of Alvis–Curtis duality on maximal rational wavefront sets.

We consider the asymptotic behaviour of topological indices of random caterpillar graphs, such as the Zagreb, $\alpha$-Randić, Wiener, Gini and level indices, as the size of the graph grows. Our simple, general framework, which is based on a multivariate central limit theorem, the continuous mapping theorem and Slutsky's theorem, can also be used to cover other topological indices. We derive limit laws for these indices and provide exact and asymptotic results on their means and variances. The $\alpha$-Randić index exhibits a discontinuity in its limit distribution at $\alpha=\frac{1}{2}$.

A right group is a semigroup $(S,\cdot)$ in which, for every $a,b\in S$, there is a unique $x\in S$ such that $a\cdot x=b$. In this article, we develop the theory of heaps starting not from groups, but from right groups. We thus get a natural definition of right heap. It is even possible to develop part of the theory starting from a left quasigroup, which is the non-associative analogue of a right group. Our motivation for this study is the investigation of left non-degenerate set-theoretic solutions of the Yang–Baxter equation. Thus, we are led to an analogue of the skew left trusses introduced by T.~Brzeziński.

We show that the logarithmic Sobolev inequality of the Boolean cube is stable under random monotone censoring. More precisely, if $A_n\subseteq \{0,1\}^n$ is chosen uniformly among all monotone subsets, then the logarithmic Sobolev constant of the censored walk on $A_n$ is of order $n$ with high probability. As a consequence, several analytic and probabilistic properties of the Boolean cube persist for a typical monotone subset: the censored semigroup is hypercontractive, the uniform measure on $A_n$ satisfies Gaussian concentration for Lipschitz observables, and the associated walk mixes in time $O(n\log n)$. The latter proves a conjectured mixing bound of Ding and Mossel for almost all monotone sets. The result is genuinely typical rather than universal. We construct monotone sets of density bounded away from zero whose logarithmic Sobolev constant is of order $n^2$. To prove the result, we establish a sharp logarithmic Sobolev inequality for Hamming caps and combine it with a harmonic extension argument transferring this inequality to monotone sets lying between nearby caps, together with a structural theorem of Korshunov on random monotone sets.

Over any number field with sufficiently many roots of unity, we construct homogeneous spaces of $\mathbf{SL}_n$ with finite geometric stabilizers of nilpotency class $2$, whose unramified algebraic Brauer group consists only of constant classes, and which has no rational point but has local points in every completion of the ground field. This provides the first known examples of failures of the Hasse principle for homogeneous spaces of connected linear algebraic groups that are detected only by transcendental Brauer classes. The construction builds on the failure of a global–local principle for cup-product decomposition, based on a previous idea of Borovoi and Kunyavskii.