Papers

We prove that any real-analytic Hamiltonian in five or more degrees-of-freedom, with a locally integrable non-degenerate elliptic equilibrium with indefinite quadratic part, can be perturbed within the real analytic category, while preserving the Birkhoff normal form at the equilibrium up to any arbitrary order, so that the equilibrium becomes Lyapunov unstable.

A critical quasilinear chemotaxis model with indirect signal production in space dimensions higher than two is studied under radial symmetry. Exploiting the variational structure of the system, the existence of a sharp mass threshold M\textsubscr{c} > 0 is established: all solutions with mass below M\textsubscr{c} are global and bounded, while those with mass above M\textsubscr{c} and sufficiently negative energy are global but blow up in infinite time. Moreover, the value of the critical mass M\textsubscr{c} is related to the best constant of a variant of a Hardy-Littlewood-Sobolev inequality.

We prove that the values set of \(S=|A|^2\) attained by closed embedded minimal hypersurfaces in \(\mathbb S^{n+1}(1)\) with constant \(S\) and constant \(f_3=\operatorname{tr}(A^3)\) is locally finite, where \(A\) denotes the shape operator. Neither the topology of the hypersurface nor the value of \(f_3\) is fixed.

In dimension 3, following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that when the ambient manifold is hyperbolic, the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that on any closed hyperbolic 3-manifold, the number of Anosov flows modulo isotopy orbit equivalence is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non- hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.

We develop theory for the identification of complex causal queries from non-IID data. We propose a formal description of stochastic systems based on symmetries in data leaving causal mechanisms invariant. These symmetries become the source of statistical repetition for identification and mandate knowledge-transfer from observations to queries. The result is an abstract, simple and very general language for causal reasoning. We give descriptions of models and queries, and the formal infrastructure for their mathematically rigorous identification. The proposed structure enables a number of simplifying principles: A shared description of mechanisms and interventions, a duality of observed and queried structure, and the sub-division of the identification task into three steps - extraction, computation and query-expression. Our approach reproduces and matches standard theoretical results on IID data and transport of experimental and non-experimental data. By design, it additionally applies beyond IID data and captures complex causal queries not expressible via do- or soft-interventions.

Let $4\le n\le7$ and let $(M^n,g)$ be a complete, connected, orientable, noncompact Riemannian manifold of positive scalar curvature. We prove that if the asymptotic quadratic scalar curvature coefficient of $g$ is greater than $(n-1)/n$, then $M$ carries a complete smooth metric whose scalar curvature is at least one. The threshold $(n-1)/n$ and the strict inequality are optimal. This confirms the second part of Gromov's critical rate of decay conjecture in dimensions four through seven.

This paper discusses a bi-objective two-phase transportation problem (BTPTP) in which transportation takes place in two phases. The whole set of source-destination links is partitioned into two disjoint sets namely, Phase-I and Phase-II. The transportation among the source-destination links of Phase-II set is done only after the transportation among the source-destination links of Phase-I set is complete. The transportation among the source-destination links in each phase is done in parallel. The problem discussed in this paper concentrates on minimizing two objectives viz., sum of transportation times and sum of transportation costs of both the phases, simultaneously. Due to conflicting nature of the objectives, a polynomial time iterative algorithm named as BTPTP-Algorithm is developed that finds all of its non-dominated solutions. At each iteration, the BTPTP-Algorithm solves some specific restricted cost minimizing transportation problems and finds corresponding optimal feasible solutions. The restrictions imposed on the source-destination links of Phase-I and Phase-II sets, in each of these problems, depend upon their corresponding Phase-I and Phase-II transportation times obtained at the previous step. The algorithm is developed in such a way that it expels all the dominated solutions and records all the non-dominated solutions of the problem, systematically. A proof of the BTPTP-Algorithm's capability of recording all the non-dominated points is provided. Further, a numerical illustration is given in support of the theory and computational behavior of the BTPTP-Algorithm for various BTPTP instances of different sizes, is provided in terms of CPU time.

This paper addresses a class of integral representations of the form \begin{equation} f(t,x)=g(t,x)+\int_0^t k(t,s)\, p(t-s,x,y)\, \partial_x f(s,y^+)\,ds, \qquad 0 \le t \le T, \end{equation} where $f$ is unknown, $p$ is the transition density of a diffusion process, and $g, k$ are prescribed functions. For an arbitrary diffusion process with sufficiently regular coefficients, we prove that this problem is equivalent to a Volterra integral equation of the second kind. This reduction provides a unified framework for both theoretical analysis and numerical approximation. An example of the implementation in the context of financial mathematics is presented.

The derived set of multiple zeta-star values is the half-line $[1,+\infty)$. In this paper we study the corresponding two-dimensional problem for multiple star polylogarithms on the unit circle. We first prove that every shifted multiple star polylogarithm is the generating function of a completely monotone sequence and hence admits a normalized Hausdorff–Stieltjes representation. Both the shifted and the unshifted functions, as well as their infinite-depth limits, are shown to be univalent on the half-plane $\mathrm{Re}\,z<1$. For finite indices, the representing densities satisfy a strict monotone likelihood-ratio order with respect to the reverse lexicographic order. Together with closure properties of the Hausdorff–Stieltjes class and a limiting quotient argument, this shows that, along the upper semicircle, the argument of each translated curve increases strictly while its modulus decreases strictly; the corresponding lower-semicircle statement follows by conjugation. The same idea gives strict separation of the curves attached to different indices. We then establish a one-to-one correspondence between a natural set of pairs consisting of a point of the unit circle and an infinite index, and the closed half-plane $\mathrm{Re}\,w\geq \frac12$ with the point $1$ removed. Under a binary coordinate on the index set, this correspondence is a homeomorphism. As an application, a subclass of shifted cyclotomic multiple zeta-star values of all levels form a countable dense subset of the open half-plane $\mathrm{Re}\,w>\frac12$. Consequently, the derived set, and every higher derived set, of this cyclotomic family is the closed half-plane $\mathrm{Re}\,w\geq\frac12$.

We prove the Jantzen conjecture for standard modules of graded affine Hecke algebras with equal parameters, real central character in the dominant deformation direction. The Jantzen layers are shown to be semisimple, and the multiplicities are given by the full local intersection cohomology of the corresponding orbit closures. The proof identifies the algebraic standard-to-contragredient map with the costalk-to-stalk map on a contracting transverse slice. Hard Lefschetz theorem determines its elementary divisors, and a residual grading on the specialized convolution algebra proves semisimplicity. An example in type $A$ is included to show that dominance is necessary.

We establish a non-Archimedean Caffarelli-Silvestre extension theory for the Vladimirov-Taibleson operator $D^s$ for $s>0$ and, more generally, for hierarchical Laplacians $L_C$. For every $f\in\mathcal{S}(\mathbb{Q}_p^n)$, we equip the Bruhat-tits tree $\mathcal{T}_{p^n}$ with the weight $w_{v_{k-1}v_k}=p^{k(s-n)}$ under the horocyclic coordinates. We prove that the resulting Dirichlet problem admits a unique bounded weighted-harmonic solution that is continuous on the end compactification. Its boundary traces converge to $f$ uniformly and in $L^2$, while the associated deformed normal derivative converges pointwise and in $L^2$ to $D^s f$. We derive explicit Fourier and Poisson representations of the extension and establish an energy identity between its weighted tree energy and the quadratic form of $D^s$. Motivated by $p$-adic AdS/CFT, we also give an equivalent massive formulation on the unweighted tree, based on a renormalized trace and a scale-corrected normal derivative. Finally, under natural local-finiteness and scaling assumptions, we extend the construction to hierarchical Laplacians $L_C$ on ultrametric spaces. We characterize the canonical edge conductances by requiring the cancellation Green operator on the ultrametric tree $\mathcal{T}_X$ to coincide with $L_C^{-1}$ on $\mathcal{S}_0(X)$. In the corresponding canonical flux class, the extension is unique, its Dirichlet-to-Neumann map is $L_C$, and it satisfies the associated energy identity. The Vladimirov-Taibleson construction is recovered as the homogeneous special case.

Clouâtre, Ostermann, and Ransford formulated an abstract version of Crouzeix's conjecture involving a bounded unital homomorphism from a uniform algebra into matrices and a unital antilinear map. They conjectured that contractivity of the associated symmetrised map forces the homomorphism to have norm at most two. We prove this conjecture, in fact without requiring the antilinear map to be contractive and for homomorphisms into the bounded operators on a Hilbert space. The proof combines positivity of real parts, an operator-valued Herglotz theorem, and the perturbation lemma of Lorist and Schwenninger used in the recent proof of Crouzeix's conjecture.

Let X be a spectrally negative Lévy process observed up to an inde- pendent exponential time T with parameter {\gamma} > 0. We study the joint law of the maximum drawdown and the maximum drawup of X on [0, T ]. The path is decomposed according to the two possible orderings of its in- fimum and supremum. Conditional on the values of these extrema and on their ordering, the resulting pre-, intermediate, and post- components are independent. We identify their laws as Doob h-transforms of killed spec- trally negative Lévy processes and express the corresponding distribution functions explicitly in terms of the {\gamma}-scale functions W ({\gamma}) and Z({\gamma}) and their derivatives. Combining the conditional laws with the joint densities of the extrema yields integral representations of the joint distribution and moments.

We consider finite-dimensional nonlinear inverse problems arising from finite element discretizations of elliptic inverse coefficient problems such as the Calderón problem with finitely many measurements and unknowns. Such inverse coefficient problems are notorious for their nonlinearity and ill-posedness, and numerical solvers tend to depend strongly on good initial values. In this work, we develop a new locally convergent algorithm with an explicit residual criterion that ensures convergence to the inverse problem solution, and a globalized variant that is guaranteed to automatically switch to the faster locally convergent algorithm after finitely many global search steps.

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $\eta\in(0,\infty)$ and ResNets $\Psi_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $\eta d^{\eta}\varepsilon^{-\eta}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $\xi>0$, the explicit bound $C_\xi d^{4+\xi}\varepsilon^{-(3+\xi)}$ on the number of parameters.

In this paper, we study spectral properties of multiplicative deformations of non-Hermitian random matrices. We consider matrices of the form $\mathbf{A}\mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. We show that, as $N\to\infty$, the boundary of the complex eigenvalue distribution of $\mathbf{A}\mathbf{B}$ is governed by simple equations involving the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$.

Continuous submodular optimization is an important class of nonconvex problems with global guarantees, for which the submodular box-constrained quadratic programming (BCQP) forms a fundamental subclass. It arises naturally as quadratic subproblems of general twice-differentiable submodular problems and also has direct business and management applications such as pricing. This paper studies the worst-case relaxation gap of semidefinite programming (SDP) relaxations with a finite family of valid linear cuts for submodular BCQP. We show that in dimension $n\geq4$, for any SDP relaxation formulated with a finite family of linear cuts, there always exists a submodular BCQP instance with a strictly positive relaxation gap. By restricting to the normalized BCQP instances that remove the problem's scaling factor, we further quantify this limitation with explicit lower bounds. In general dimension $n\geq4$, when the SDP relaxation is formulated with only Boolean-quadric-polytope (BQP) valid cuts, including the commonly used McCormick and triangle inequalities, we establish an $\Omega(n)$ relaxation gap lower bound. For an arbitrary family of $m$ valid linear cuts, we establish an $\Omega(n/m^2)$ lower bound. These results capture the fundamental limitations of SDP relaxations, especially SDP with BQP valid cuts, and provide guidance on what additional constraints may be needed for more effective convex approximations of submodular BCQP.

We propose NB-HalpernSGD via LM-MultiLRSGA, a two-phase optimizer for multi-objective optimization. The process starts with a competitive optimization phase by applying a limited-memory variant of MultiLRSGA, named LM-MultiLRSGA, designed to reduce the memory footprint of the original optimizer while preserving its rotational correction mechanism. It addresses multi-objective tasks by considering the competitive game associated with the original multi-objective problem, approximating a Nash equilibrium. This point defines a Nash-equilibrium-induced disagreement point: we formulate the Nash bargaining problem associated with the original losses and rewrite its Nash product as a logarithmic minimization surrogate. This minimization problem is solved using HalpernSGD, anchored at the computed competitive reference point. Therefore, the method uses the Nash equilibrium as a principled reference point for the bargaining stage and then moves toward a Pareto-oriented solution of the original multi-objective problem. We validate the proposed optimizer on a PINN-inspired neural model, where it outperforms established multi-objective optimizers, including PCGrad, MultiAdam, and DualConeGD. Finally, while the convergence properties of HalpernSGD have been extensively studied, we discuss the convergence and stability properties of the proposed LM-MultiLRSGA phase.

Over the past 25 years, the Traveling Tournament Problem (TTP) has become one of the most extensively studied optimization problems in sports scheduling. At its core, the TTP seeks to minimize the total travel distance incurred by teams that travel directly between opponents' venues during consecutive away games. The problem originated from the scheduling challenges faced by Major League Baseball, where it was identified as the central computational difficulty. This paper provides a comprehensive overview of the literature on the TTP. We review the principal problem variants and benchmark instances, and summarize the current state of the art in lower bounds, approximation guarantees, and exact and heuristic optimization algorithms. Moreover, we contribute to the continued development of the field by tracking and validating lower and upper bounds, while succeeding the repository originally established by Prof. Michael Trick as part of the RobinX sports timetabling project. Finally, we identify several open questions and outline promising directions for future research.

We show that exponential maps could have complicated measurable behaviours by giving two examples in the exponential family both with singular value escaping to infinity “slowly", one of which is recurrent (in the sense that every positive measure set will return to itself infinitely many times), while the other is non-recurrent. This also complements earlier results of Lyubich, Rees, Urbański-Zdunik, and Hemke. We also give an example of exponential maps with singular value escaping to infinity arbitrarily slow, which is a parametric analog of a result of Rempe.

In this paper, we study the scaling limit of the pinning model in correlated Gaussian environment. The tail probability of the underlying renewal process of the model has a polynomial decay with exponent $\alpha>0$. The covariance of the Gaussian environment $\{\omega_n\}_{n\in\mathbb N}$ is given by $\text{Cov}_{\mathbb P}(\omega_n,\omega_m)\sim |n-m|^{2H-2}$ with $H\in(0,1)$. Assuming $\alpha\in(0,\frac12]$, $H\in(\frac12,1)$ and $\alpha+2H>2$, we show that the partition function of the disordered pinning model, under the appropriate scaling, converges in distribution to the $L^1$-solution of the fractional stochastic heat equation driven by Gaussian noise correlated in time and localized at the origin. In particular, it is known that the solution is not $L^2$-integrable when $\alpha<\frac12$.

For categories, being balanced (meaning that every arrow that is both epic and monic is an isomorphism) can be regarded as a strong tameness property which plays an important role, for example, in algebra and logic. We study the problem of associating a balanced companion category, called a Beth companion, to a locally finitely presentable category (equivalently, the category of models of a finitary essentially algebraic theory). We show that, if it exists, the Beth companion is unique and can be described in terms of saturated objects. Under additional assumptions, we prove that Beth companions can be computed as orthogonality classes, and admit a syntactic presentation via Gabriel–Ulmer duality. Furthermore, we establish conditions for the transfer of properties from a category to its Beth companion. A number of examples of categories that do or do not admit a Beth companion are provided. In particular, we show that the category of commutative rings does not admit a Beth companion, and employ Pierce sheaf representations to describe the Beth companion of other classes of rings.

We use a detection method for invariant Einstein equations to construct and distinguish asymmetric homogeneous Einstein metrics on several families of compact homogeneous spaces. The method retains a geometrically meaningful symmetry-breaking parameter, eliminates the remaining Einstein equations, and then reconstructs positive metrics from the resulting detector. We prove a sharp threshold in a two-factor symplectic family, construct two asymmetric Einstein metrics on each admissible space $\SU(n)^3/\Delta\SO(n)$, obtain a sharp threshold for $\SU(2n)^3/\Delta\Sp(n)$, and construct asymmetric metrics on three exceptional three-factor families. The four-factor case exhibits a further phenomenon. On \[ M_9=\SU(18)^4/\Delta\Sp(9) \] the two symmetry types $3+1$ and $2+2$ are both realized: up to homothety and internal block permutations, the corresponding fixed families contain exactly three asymmetric Einstein metrics, two with three equal horizontal scales and one distinct scale (type $3+1$), and one with two equal pairs of horizontal scales (type $2+2$). They are pairwise non-isometric, Riemannian irreducible, and not naturally reductive. The $2+2$ metric is globally unique in the full positive $2+2$ fixed family up to block interchange.

We study the large-depth behavior of residual networks whose weights are correlated across layers at initialization. Our results confirm and extend a conjecture of Marion et al. [2025], according to which correlated initializations should interpolate continuously between the Brownian stochastic differential equation arising from independent initialization and the ordinary differential equation arising from perfectly correlated initialization. When the initialization is obtained from the application of a feature function to a stationary Gaussian sequence with regularly varying correlation, we prove that there exists a unique critical scaling such that the infinite-depth limit is the solution of a Young differential equation driven by a Hermite process. Hermite processes reduce to the fractional Brownian motion if the feature function generating the initialization has Hermite rank one, which is the case for the identity function, for example. We show that the critical scaling and asymptotic limit are uniquely determined by the decay of correlations together with the Hermite rank of the feature function. Consequently, the correlation structure and Hermite rank of the initialization represent meaningful hyperparameters in the asymptotic regime. By contrast, under finite-variance iid initialization, the asymptotic driver is universally Brownian up to normalization regardless of the choice of distribution. Our proofs rely on a collection of novel results establishing a robust stability theory for Young differential equations in Banach spaces.

Let $M$ be a smooth manifold equipped with a para-Kähler structure. We define an action of the diffeomorphism group $\operatorname{Diff}(M)$ on the space of para-Kähler structures and prove that this action preserves the associated affine structures. Although this result is well known, we provide a proof based on this action, showing in particular that an isometry commutes with the Levi–Civita connection. We then construct several natural liftings of this action to the tangent bundle $TM$, the cotangent bundle $T^*M$, and the Whitney sum $E=TM\oplus T^*M$. In certain cases, these lifted actions induce natural dynamical systems on a distinguished class of $3$-webs.

The most important open problem of symbolic dynamics consists in finding a procedure to decide whether two subshifts of finite type are conjugate or, equivalently, whether two nonnegative integer matrices are strong shift equivalent. While the problem remains hard in general, the literature also contains specific examples of matrices that are not known to be strong shift equivalent: the Baker matrices, and Ashley's eight-by-eight. These specific examples have been open for 30 years. Using computer experiments, we show that the Baker matrices are indeed strong shift equivalent for small values of k, and that Ashley's eight-by-eight matrix is indeed strong shift equivalent to the one-by-one matrix 2

Let $(G_n)_{n\ge n_0}$ be a family of graphs on $[n]$ whose edge ideals $I_n\subseteq R_n=k[x_1,\dots,x_n]$ form an $\mathrm{Inc}(\mathbb{N})$-invariant chain, $I_{n+r}=\mathrm{Inc}(\mathbb{N})_{n,n+r}(I_n)$ for $n\ge n_0,\ r\ge0$. We determine which pairs $(n,r)$ make $R_{n+r}/I_{n+r}$ Cohen–Macaulay, for five classical families: line graphs $L_n$, complements of line graphs $L_n^c$, complete graphs $K_n$, cyclic graphs $C_n$, and complements of cyclic graphs $C_n^c$. For line graphs we give the generators of $I_{n+r}$, the height and Krull dimension of $R_{n+r}/I_{n+r}$, and a complete classification: for $n\ge5$, $R_{n+r}/I_{n+r}$ is Cohen–Macaulay if and only if $r=n-4$. For complements of line graphs, complete graphs, and complements of cyclic graphs, we show the chain is Cohen–Macaulay unconditionally, for every $r\ge0$; the first and third arise from the same underlying phenomenon, in which the $\mathrm{Inc}(\mathbb{N})$-invariant chain reproduces the original graph itself at each step. For cyclic graphs we prove $\mathrm{Inc}(\mathbb{N})_{n,n+r}(I(C_n))=I(K_{n+r})$ once $r\ge n-3$, giving Cohen–Macaulayness in this range, and conjecture – with supporting computational and structural evidence – a complete classification: Cohen–Macaulayness holds if and only if $r\ge\lfloor(n-4)/2\rfloor$ and $r\ne n-4$.

We prove the local Hadamard well-posedness of the higher-order nonlinear Schrödinger equation with negative dispersion coefficient on the half-line under nonzero Dirichlet and Neumann boundary data. The paper uncovers a striking feature of this model: unlike the positive-dispersion case studied in (Alk{\i}n, Mantzavinos, Ozsari, 2024), well-posedness requires two boundary conditions rather than one. This reflects a structural limitation in the underlying global relation, which allows elimination of at most one unknown boundary trace. Using the unified transform of Fokas, we derive an explicit solution formula for the associated forced linear problem and establish sharp Sobolev and Strichartz estimates on the half-line. These linear estimates, together with corresponding Cauchy problem bounds, yield local well-posedness in both high- and low-regularity Sobolev spaces, as well as local Lipschitz continuity of the solution map.

We consider the Schrödinger equation for a charged particle in a randomly fluctuating external magnetic field and establish a singular perturbation limit in which the solution converges to a deterministic Schrödinger equation with Gaussian fluctuations. We propose a stochastic Schrödinger equation incorporating the behavior of both the deterministic average and the fluctuations and show that it has the same limiting behavior as the original model, while providing a simpler way to study the distribution of relevant observables.

Portfolio replication, or the construction of a tradable basket of assets to match the risk-return profile of a target benchmark, is fundamentally an ill-posed inverse problem. When restricted to a subset of available assets, classical variance-minimizing models often yield unstable, over-leveraged portfolios highly vulnerable to market shocks. We propose a unified, two-stage methodology rooted in information theory to achieve robust portfolio replication. First, we model the constituent asset returns against target factors, estimating parameters within data-driven empirical bounds via an entropy minimization principle. Second, using the same entropic approach, we determine the optimal weight replication. In both cases we use an entropy function of the Fermi-Dirac type defined directly on sets of constraints of the inverse problem. We validate this Entropic Factor Model (EFM) against standard Ordinary Least Squares (OLS) across five numerical experiments, including standard equity tracking, multi-asset synthesis, and severe stress-test scenarios. Empirical results demonstrate that the EFM consistently outperforms OLS in terms of annualized turnover and net-of-fees returns. Crucially, during the COVID-19 market crash and under severe idiosyncratic data corruption, the entropic framework acts as a probabilistic “circuit breaker", defensively reducing capital allocation to compromised assets and providing a highly robust, risk-averse solution for generalized portfolio replication.