Papers

Advances in computational modeling, neuroimaging, and artificial intelligence are revolutionizing the modeling of neurological disorders for improved diagnostics, prognosis, and treatment planning. Mechanistic models provide valuable scientific insight into the disorders, but in practice they are often simplified with assumptions or computationally expensive and slow to solve. However, while purely data driven approaches provide speed and scalability, they require large, high quality data to train and generally suffer from interpretability and generalization issues. This perspective paper presents a structured overview of hybrid modeling strategies, which combine deep learning models with physics based solvers, and are categorized into parallel, series, and parallel-series architectures. Three main approaches that have been emphasized are residual modeling for missing or incomplete physics, Neural Ordinary Differential Equations (NODEs) for continuous time dynamics approximation, and solver in the loop that accelerates traditional solvers with neural approximations. These hybrid models integrate the governing differential equation based formulations and deep learning to characterize the evolution of neurological disorders, and promise advanced personalized neurological modeling. In addition, the study explores and proposes different hybrid configurations to improve diagnosis accuracy, predict disease progression, and inform treatment strategies across a range of neurological disorders. These capabilities outperform standalone mechanistic or purely data driven approaches, making hybrid modeling a powerful tool, especially in applications involving modeling the progression and treatment responses in neurological conditions such as brain tumors, Alzheimer's disease, and stroke.

Laplace transforms and their numerical inverses arise throughout applied mathematics, physics, finance, and probability theory. Numerical inversion, however, quickly becomes intractable in high dimensions because the number of quadrature evaluations grows exponentially with dimension. We develop a tensor train (TT) formulation of the multidimensional inverse Laplace transform. The method constructs a TT approximation of the transformed function on the complex quadrature grid and then performs the inversion through a sequence of tensor contractions. Under suitable low-rank assumptions, this reduces the computational cost from exponential to polynomial in the dimension, provided that the relevant bond dimensions remain bounded. The method has only a small number of tunable parameters and admits error estimations. We demonstrate its performance in numerical experiments, benchmarked against Monte Carlo estimates and exact references, for multivariate normal-inverse Gaussian, Wishart, and correlated Gamma-type distributions.

In this paper, we introduce a notion of Higgs bundles on the Fargues-Fontaine curve. We establish a version of the BNR correspondence, which relates Higgs bundles to line bundles on suitable curves. We then describe an action of a Picard stack on the moduli stack of Higgs bundles and show that, modulo this action, there is a natural injective map of étale-stacks from the product of $B_{dR}^+$-affine Springer fibers to the Hitchin fiber that induces an equivalence of categories on every geometric point. Finally, we discuss connections with number-theoretic objects.

In this paper, we prove a generalization of a conjecture of Amdeberhan, Andrews, and Ballantine on double Lambert series. Motivated by a question raised by Cui, Kumar, and Singh concerning the existence of a generalization of this conjecture, we establish an identity in which the coefficients are given by the generalized divisor function $\sigma_k(n)$. As a special case, our result includes the original conjecture.

We prove a version of GAGA theorem for a normal complex analytic variety $X$ equipped with an invertible holomorphic contraction $\gamma$ with center in $x$. We show that $X$ admits a natural structure of an affine variety, and any $\gamma$-equivariant complex analytic reflexive coherent sheaf on $X$ admits a natural algebraic structure. We prove a structure theorem for $X_0:=X\backslash x$, showing that it admits a proper action of ${\Bbb C}^*$, and is isomorphic to the space of non-zero vectors in the total space of an ample line bundle over the projective variety $Z:= X_0/{\mathbb C}^*$ equipped with an orbifold structure. We show that the quotient $M:=X_0/\gamma$ admits a holomorphic embedding to a Hopf manifold, and, conversely, any normal subvariety $M$ in a Hopf manifold is obtained this way. We prove a form of structure theorem, showing that any reflexive coherent sheaf on $M$, $\dim M > 2$, admits a filtration such that its associated graded subquotients, tensored with an appropriate line bundle, are obtained as pullbacks of coherent sheaves on the projective variety $Z=X_0/{\mathbb C}^*$. This is used to show that any reflexive coherent sheaf on $M$ is filtrable, that is, admits a filtration with associated graded quotients of rank $\leq 1$.

This paper studies the weak convergence order of structure-preserving stochastic theta methods for a class of index-$1$ stochastic differential algebraic equations with time-dependent singular matrices. The singular matrix is allowed to vary in time but preserves a fixed differential-algebraic splitting, thereby extending the constant singular-matrix setting while retaining the projector structure required for constraint preservation. By exploiting the index-$1$ algebraic-differential decomposition of the exact solution, we establish an abstract weak convergence theorem for constraint-preserving one-step approximations and apply it to the stochastic theta method with $\theta \in (0,1]$. Under global Lipschitz, linear growth, and suitable smoothness assumptions, the considered method is proved to be well posed, to preserve the algebraic constraints at all time levels, and to converge with weak order one. Numerical experiments are finally presented to confirm the structure-preserving property and the theoretical convergence order.

Let $pp (n)$ denote the number of ways to write $n$ as a sum of primes. In this paper, we show that $$ \log pp (n) \sim 2\pi\sqrt{\frac{n}{3\log n}}\;. $$ While sharper estimates are already known, they rely on highly involved and lengthy proofs. In sharp contrast, our approach uses a short, elementary recipe that easily adapts to yield similar asymptotic estimates for several related, extensively studied problems.

This paper is concerned with uniqueness and stability of Lagrange multipliers for constrained optimization problems in abstract spaces. It is well known that validity of the strict Robinson-Zowe-Kurcyusz condition implies the so-called isolated calmness, a one-sided Lipschitz property tailored for set-valued mappings, of some Lagrange multiplier mapping associated with a perturbed version of the original optimization problem, and the latter indeed is enough to guarantee uniqueness of the Lagrange multiplier. The paper studies the isolated calmness of the Lagrange multiplier mapping in detail. Exemplary, it is shown that this condition is sufficient for the Robinson-Zowe-Kurcyusz constraint qualification and, in the presence of additional assumptions, even equivalent to the strict Robinson-Zowe-Kurcyusz condition. Illustrative examples are presented to underline the necessity of postulated assumptions.

Let $a_L(n)$ be the maximum number of regions into which $n$ lollipops divide the plane. Paulsen introduced a second obstruction for this problem, based on pairs of circles meeting at obtuse angle, in addition to the stem-direction obstruction of Cutler-Karlsson-Sloane. We recast Paulsen's argument as a weighted problem for two graphs: a $K_4$-free graph $D$ of non-close stem pairs and a $K_5$-free graph $E$ of non-intriguing circle pairs. For the total number $C$ of pairwise crossings, $$ C\le 4\binom n2+|D|+|E|+|D\cap E|. $$ Paulsen bounds the final term by $|D|$. We keep the overlap term and analyze near-extremal configurations of $D$ and $E$. This closes all of Paulsen's remaining gaps up to $n=17$, and also closes $n=19$: $$ \begin{array}{c} a_L(0),a_L(1),\ldots,a_L(17)\\ =1,2,10,25,45,71,104,142,186,237,294,356,425,500,580,667,761,859, \end{array} $$ and $$ a_L(19)=1076. $$ The same method gives the one-region gaps $$ 964\le a_L(18)\le965,\qquad 1193\le a_L(20)\le1194. $$

Tensor linear systems widely arise from high-dimensional data mining and computing, for instance, natural language processing and machine learning. A class of residual-based tensor Kaczmarz method is proposed for tensor linear equations with t-product. Theoretical analyses prove the convergence and give an upper bound of the convergence rate of the proposed method. Furthermore, an accelerated residual-based Kaczmarz method with heavy ball momentum is developed. Numerical experiments verify the efficiency of the proposed methods and demonstrate that they are faster than the existing tensor Kaczmarz methods.

We prove a sharp forbidden interval for the nontrivial adjacency eigenvalues of trivially perfect graphs. More precisely, we show that if $G$ is a trivially perfect graph, then $\operatorname{Spec}(G)\cap [\sqrt{8}-4,0]\subseteq \{-1,0\}$. Moreover, we prove that the interval is best possible at both endpoints: there are connected trivially perfect graphs with eigenvalues arbitrarily close to $\sqrt{8}-4$ from below, and connected trivially perfect graphs with positive eigenvalues converging to $0$.

Let $\Omega\subseteq \mathbb C^d$ be a bounded symmetric domain, $G$ a finite complex reflection group acting on $\mathbb C^d$, and $\boldsymbol \theta:\Omega\to \boldsymbol \theta(\Omega)$ the associated proper holomorphic map factored by $G.$ In this paper, we investigate commutant lifting and interpolation by Schur functions on the quotient domain $\boldsymbol \theta(\Omega).$ For a given quotient module of the Hardy space $H^2(\boldsymbol\theta(\Omega))$, we obtain equivalent criteria for a contractive module map to admit a Schur-class lift: one in terms of the contractivity of an associated functional on a subspace of $L^1(\partial\boldsymbol\theta(\Omega))$, and another in terms of a geometric distance formula in the same $L^1$-space. Specializing to quotient domains of the polydisc factored by imprimitive finite complex reflection groups, we obtain a commutant lifting criterion formulated in terms of inner functions. Finally, we apply these operator-theoretic results to finite-point Nevanlinna-Pick type interpolation problems on $\boldsymbol \theta(\Omega)$. Since the symmetrized bidisc and the tetrablock arise as quotient domains of suitable bounded symmetric domains, these criteria apply in particular to those domains.

We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques. Building on a dimension-incremental framework, we combine product basis expansions with sparse recovery methods, specifically orthogonal matching pursuit (OMP), to substantially reduce the required sample size compared with a previously considered cubature-based approach. We evaluate the resulting method numerically on several examples, comparing it against both cubature-based sparse approximation and Fourier neural operators in terms of accuracy, runtime, and sample size. The experiments show that our approach considerably reduces the number of required PDE solves relative to its predecessor while maintaining competitive accuracy, particularly when the solution admits a sparse representation in the chosen basis. Furthermore, the recovered sparse index sets yield interpretable insights into the relevant variables and parameter interactions.

We prove an Erdős-Ko-Rado type extremal result for tilings of a $1 \times n$ chessboard by tiles whose lengths belong to a set $\Lambda$. Two tilings are said to intersect if they contain a tile spanning the same set of squares. We prove that if $1\in\Lambda$, then the maximum size of an intersecting family of tilings is attained by the set of all tilings containing a fixed singleton tile at one of its ends. This result generalizes a theorem of Butler, Horn and Tressler, which is equivalent to the case $\Lambda=\{1,2\}$.

In 1983, Henri Cohen and Hendrik Lenstra proposed a conjecture about the distribution of the N-torsion of the class group of a random quadratic field, supported by what was at the time a large amount of computational evidence. The Cohen-Lenstra heuristics, which are still almost entirely unproven, have become one of the central foundational problems in arithmetic statistics. Recent years have seen a rapidly accelerated pace of development in Cohen-Lenstra problems. I will give a tour of these developments, including the work of Wood and her collaborators developing a fully fleshed out roster of generalized Cohen-Lenstra conjectures, with support from topology; Smith's theorems proving the Cohen–Lenstra conjectures for the 2-primary part of the class group, as part of more general theorems about Selmer groups in quadratic twists, leading to a resolution of the minimalist conjecture for elliptic curves; and recent work by Koymans and Pagano in the ell-primary case, expanding on Smith's work and proving Stevenhagen's conjecture on the negative Pell equation.

We consider a periodic scalar reaction-diffusion system over a two-dimensional domain under natural conditions on the reaction function and with initial condition $\theta$. We show that on the global attractor $\mathcal A$ of the resulting dynamical system $(u_\theta(t):t>0)$, a reverse Poincaré inequality holds true, and that as a consequence the map $\theta \mapsto u_\theta(t)$ satisfies an $L^2(\Om)$-Lipschitz stability estimate on $\mathcal A$ for any $t>0$ fixed. We then show that statistical recovery of an initial condition $\theta$ in the attractor $\mathcal A$, as well as prediction of the states $u_\theta$, is possible from discrete measurements of the system at parametric $1/\sqrt N$ convergence rates.

We establish a relativised variational principle for the Feng–Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a recent theorem of Yin, this yields an almost-everywhere equivalence between the Feng–Huang entropy and its combinatorial version on fibers. As an application, we compute the Hausdorff dimension of the intersection of random translates of two Bedford–McMullen carpets. The resulting formula extends the Kenyon–Peres formula from the self-similar to the self-affine setting, and also points to a new problem concerning random matrix products.

In a smoothly bounded domain $\Omega\subset R^n$, $n\le 5$, the manuscript considers the variant of the Keller-Segel system given by \[ \left\{ \begin{array}{l} u_t = D \Delta u - \nabla \cdot (u\nabla v), \\[1mm] v_t = d \Delta v + \nabla \cdot (u\nabla v) - v + u, \end{array} \right. \] which involves an additional contribution $\nabla \cdot (u\nabla v)$ to the chemoattractant evolution, in line with refined modeling literature reflecting an anisotropic correction to the isotropic signal production term $+u$ in the classical Keller-Segel model. It is shown that for arbitrary $D>0$ and $d>0$ and any nonnegative intial data from $W^{1,\infty}(\Omega)\times W^{1, \infty}(\Omega)$, an associated Neumann problem admits a global weak solution $(u,v)$ which, inter alia, satisfies \[ \sup_{t \in (0,\infty)\setminus N} \int_\Omega e^{u^\alpha(\cdot,t)} < \infty \] with some $\alpha>0$ and some null set $N\subset (0,\infty)$.

We construct canonical stable Steenrod squares on modified compactly supported étale cohomology of separated finite-type schemes over rings of $S$-integers in number fields with $2$ invertible, extending Feng's absolute Wu classes to arithmetic bases. For a flat projective morphism $f:X\to B$ of pure relative dimension, with $X$ regular and $B$ such a base, we prove $v_X=\operatorname{Sq}^{-1}(w_{\mathrm{et}}(\tau_f+\mathcal O_X^{\oplus3}))$ in completed mod-$2$ étale cohomology. Here $\tau_f$ is the virtual relative tangent bundle, $w_{\mathrm{et}}$ the total étale Stiefel–Whitney class, and $\operatorname{Sq}^{-1}$ the inverse total Steenrod square. Over finite fields of odd characteristic, the formula holds without the three trivial summands. The proof uses a modified compactly supported relative Wu formula extending Benoist's theorem. Our generalized Hecke theorem gives universal mod-$2$ relations involving Chern classes and the Kummer class of $-1$, governed by an arithmetic deformation of Hirzebruch's $2$-Todd series. These hold modulo an explicit archimedean ideal and become vanishing identities over finite fields, over totally imaginary arithmetic bases, or when $-1$ is a square on $X$. Nonempty real loci force infinitely many nonzero Wu components. Applications include Hecke's theorem on the different away from $2$, a finite-field analog of Atiyah's theorem on theta characteristics, and new higher-dimensional relations. We also revisit Serre's Riemann–Hurwitz theorem for spin bundles, recover the Shusterman–Sawin theorem for smooth branched covers of closed $3$-manifolds and prove a function-field analog of the Lusztig–Milnor–Peterson formula, expressing the difference between mod-$2$ and $2$-adic semicharacteristics using the middle Wu class and its Tate-twisted Bockstein.

It is an open question whether every topological group is precompact or ambitable. This note presents a topological group that is ambitable but does not satisfy previously known sufficient conditions for being so.

This paper employs shortfall risk to measure portfolio risk. Assume stock returns follow polynomial relations with their influence factors. We propose a moment-based distributionally robust optimization (DRO) shortfall risk portfolio model. For piecewise linear loss functions, we show that this DRO model can be transformed into a tractable linear conic optimization problem with nonnegative polynomial cones. A Moment-SOS relaxation algorithm is proposed to solve the transformed problem. Its finite and asymptotic convergence properties are studied. For optimizers computed from our algorithm, we give convenient conditions verify their global optimality for the original DRO problem. Numerical experiments involving real stock market data are given to show the efficiency of our approach.

The double-directional (DD) wireless channel model is important for realistic system design since it provides complete propagation information. While stochastic and deterministic channel models are widely adopted, and existing machine learning (ML) solutions mostly aim to align future channel realizations, these solutions are often limited to short time spans that may not be statistically significant. Moreover, because the number of multi-path components (MPCs) varies with spatial and temporal variation of the receiver (RX) and/or interacting objects (IOs), typical ML solutions that require fixed, predefined input and output shapes fall short. To curb these limitations, we propose a statistics-aided ML solution that relies on a fixed subset of MPCs selection. More specifically, we first select top-$M$ MPCs, where $M\in\mathbb{Z}^+$ is much smaller than the total number of MPCs, and construct learnable graphs to train our proposed hybrid TimesNet-TimeFilter (TNTF) model. We then use a channel statistics-aided training method to generate future top-M DD channel realizations such that the statistics calculated from these realizations matches closely with those of the actual statistics from the complete time-varying DD channel realizations. We validate the proposed solution using extensive simulations on both synthetic stochastic channel model (SCM)-based and deterministic ray-tracing-based datasets, and demonstrate its effectiveness relative to state-of-the-art baselines.

We prove a conditional Schoen-type Harnack inequality for positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u=g(u),\qquad 1<p<n, $$ under a global critical Sobolev growth assumption and the monotonicity condition that $s^{-(p^*-1)}g(s)$ is nonincreasing. The result is conditional on two inputs, the classification of bounded positive entire blow-up limits as Aubin–Talenti $p$-bubbles and a preliminary singular-rate upper control on the normalized necks. Under these two hypotheses, solutions in $B_{3R}$ satisfy $$ \Big(\sup_{B_R}u\Big)\Big(\inf_{B_{2R}}u\Big)^{p-1} \le C R^{p-n}. $$ The main point is a Pohozaev-neck argument which upgrades the preliminary singular decay rate $|x|^{-(n-p)/p}$ to the sharp $p$-harmonic fundamental rate $|x|^{-(n-p)/(p-1)}$. The argument replaces the Kelvin-transform and moving-sphere methods available in the conformally invariant semilinear case $p=2$, but unavailable for the general $p$-Laplacian.

Uncertainty quantification in neural networks prediction is a main issue for usual applications. Our approach seeks at reducing computation costs by directly evaluating uncertainty using PDE's information on the asymptotic variance, rather than the deep ensemble method which may be seen as a Monte Carlo estimation of the prediction, requiring the training of multiple networks. We thus study the law of the limiting process describing the random fluctuations around the mean-field limit of wide two-layer neural networks trained by stochastic gradient descent in a weak-noise regime. Building on a recent trajectorial central limit theorem, in which this limit is characterized as the weak solution of a linear stochastic evolution equation, we identify its law explicitly. More precisely, we show that it is a centered Gaussian process in the dual of a weighted Sobolev space, and we derive a closed covariance representation for the finite-dimensional distributions obtained by testing it against smooth functions. This covariance is expressed through the solution of a backward transport equation with a nonlocal source term, whose coefficients are driven by the mean-field trajectory. As a consequence, by testing against the activation function at a fixed input, we obtain an expression for the limiting variance of the corresponding network-output fluctuations. We illustrate this result numerically on a one-dimensional regression example.

We consider the inverse medium scattering problem for the Helmholtz equation in two dimensions, i.e., the task to recover a compactly supported penetrable two-dimensional scatterer from full knowledge of the associated far field data or, equivalently, the far field operator. Although this problem is uniquely solvable, it is severely ill-posed and nonlinear. In the regime of weak scattering, the Born approximation yields a linearized relation between the contrast and the far field data, thus overcoming the second difficulty. This linear setting allows to build on recent work on linearized electrical impedance tomography, which relies on triangular Zernike decompositions, to derive an explicit reconstruction formula that expresses the expansion coefficients of the contrast in terms of those of the far field data. By choosing the expansion functions appropriately, the resulting system matrix decouples into separate (infinite) triangular systems for the spatial angular frequencies in the contrast. Consequently, each of these systems can be solved independently by performing forward substitutions. Our numerical experiments indicate that this approach, combined with an adequate regularization method, remains effective even when applied to full nonlinear far field data beyond the Born regime.

A metric graph with a mechanical Hamiltonian on each edge does not, by itself, define a deterministic classical motion through a branching vertex: conservation of energy fixes only the outgoing speed, not the outgoing edge-end. We study the deterministic problem obtained after this missing vertex datum is supplied. On each edge $e$, with coordinate $q\in[0,\ell_e]$, the Hamiltonian is $H_e(q,p)=p^2/2+V_e(q)$, where $V$is continuous on the graph and $C^2$ on every edge. At each vertex we prescribe an energy-preserving Borel isomorphism from incoming to outgoing nonzero boundary covectors. The resulting phase space is the measurable quotient that identifies each incoming boundary covector with its prescribed outgoing one. After excluding the finitely many energy levels $V(v)$, the edgewise Hamilton equations and the vertex laws concatenate to a global one-parameter group of bimeasurable transformations. The group preserves energy and the quotient measure induced by the edgewise Liouville measures $dq\,dp$. The proof uses no smooth symplectic structure on the quotient; the invariance follows from ordinary edgewise Liouville invariance, a uniform no-Zeno estimate on compact regular energy windows, and preservation of the transverse Liouville flux $r\,dr$ by the speedwise vertex permutations. If the vertex laws are compatible with momentum reversal, then the quotient dynamics is reversible. On regular energy surfaces satisfying the usual regular-value condition, the induced time-parametrization measure is invariant as well.

This article develops a rigorous Lagrangian formulation of variational calculus for minimal surfaces, using extensively the concept of pullback covariant derivative. It is shown, in particular, using a geometric argument that all tangential variations vanish. First and second normal variations are then derived.

We investigate the strong identity condition (SIC) for almost-canonically seminormed rings, a class of topological graded rings that includes enveloping algebras of vertex operator algebras. This condition was introduced in the algebro-geometric theory of conformal blocks, where it governs the smoothing of nodal curves. To understand the representation-theoretic meaning of SIC, we develop the representation theory of almost-canonically seminormed rings, including Zhu-type algebras, induced modules, rationality conditions, tensor product compatibility, and an end formula for the mode transition algebra. Our main result characterizes the strong identity condition in terms of orthogonal expansions, projectivity of canonical modules, and Morita-type equivalences induced by Zhu-type algebras. As an application, we show that for vertex operator algebras of CFT type, the smoothing property is equivalent to the Zhu algebra inducing a Morita-type equivalence with the category of admissible modules. Consequently, the strong identity condition identifies the precise representation-theoretic obstruction to extending algebraic smoothing beyond the semisimple setting. We further illustrate the theory through explicit examples, including the Weyl algebra and several irrational vertex operator algebras where the strong identity condition fails.

We prove strong existence and pathwise uniqueness for two stochastic models of a seeker steering toward a target, written in polar coordinates. In both, the angular drift carries a $\frac1{r}$-type singularity which belongs to the supercritical regime in $\mathbb{R}^2$. Standard results for SDEs with singular drift therefore do not apply, and we give a new proof of strong well-posedness based on a pathwise argument. The two models arise from sailboat navigation and proportional navigation. We study the limiting regime in which the stopping radius around the target tends to zero and prove that, despite the singularity at the origin, each system admits a unique strong solution up to the hitting time of the target. These results provide an example of strong well-posedness in a regime where the general theory does not apply.

We develop a concept for the numerical treatment of shape optimization problems based on the combination of phase field and sharp interface methods. On the one hand, phase field methods are very well suited to numerically determine the shape, size and topology of a sought domain, but on the other hand they have problems to sharpen out domains where they e.g. should develop corners. However, this is the strength of a sharp-interface approach developed in our group, which provides shape updates in the Lipschitz topology. This leads to a two-stage process that first determines an optimized shape using the phase field method. The resulting domain is the starting solution for the sharp interface shape optimization method. Both methods are discretized with the finite element method. The starting mesh for the sharp method is constructed from the finite element mesh of the optimal phase field solution using its properly post processed zero-level set. We describe this construction process in detail and investigate the performance of our method on a selection of test problems from the literature and from applications.