Papers

This paper investigates cost-sensitive online window size selection for portfolio management under changing market conditions. Specifically, we propose a two-level framework that constructs portfolios using candidate window sizes and dynamically aggregates them through online learning. By treating candidate window sizes as “experts,” we dynamically update their aggregation weights using turnover-inclusive losses. Moreover, we derive finite-horizon cost-sensitive tracking-regret bounds that account for turnover of the aggregated portfolio, with static regret as a special case. Under bounded losses and cost rates, suitably tuned Fixed Share achieves asymptotically no tracking regret for sublinear switching budgets, with Hedge covering the static case.

This paper is concerned with a no-flux initial-boundary value problem for the degenerate volume-filling chemotaxis system with source terms, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v) + f(x, u, v), \quad v_t = \Delta v + g(u,v), \quad x\in \Omega, \ t>0 \end{align*} in a smoothly bounded domain $\Omega \subset \mathbb{R}^N$ $(N \in \mathbb{N})$. It is shown that when $D$, $h$, $f$ and $g$ satisfy suitable assumptions involving $D(1,\cdot)=h(0,\cdot)=h(1,\cdot)=f(\cdot,0,\cdot) = f(\cdot,1,\cdot) = 0$, for nonnegative initial data $u_0$ and $v_0$ with $u_0\le 1$, there exists a global weak solution $(u, v)$ with $u\le 1$. In addition, uniqueness of global weak solutions is established when $D(r,s) = D(r)$ for all $r\in[0,1]$ and $s\in[0,\infty)$, and $D$, $h$, $f$, $g$ and $v_0$ are supposed to satisfy additional conditions.

We identify the optimal constant and describe all maximizers for the Fourier endpoint extension inequality associated with the moment curve over finite fields. This confirms a conjecture proposed in the authors' earlier work. The proof proceeds by showing that the problem is equivalent to a purely probabilistic extremal statement: among all probability distributions on a finite set, the uniform distribution uniquely maximizes the expected number of distinct rearrangements of an i.i.d. sample.

Let $G= C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $\rr(G)=r$ denote its rank. The Davenport constant $\DD(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\DD^*(G)=1+\sum_{i=1}^r(n_i-1)$ is its classical lower bound. A long-standing conjecture asserts that $\DD(G)\le\DD^*(G)+\rr(G)-1$. In this paper, we disprove this conjecture. More strongly, we prove that $\sup_{\rr(G)=r}\bigl(\DD(G)-\DD^*(G)\bigr)=\infty$ for every fixed $r\ge5$. Thus the classical lower bound does not approximate the Davenport constant within an additive error depending only on the rank. More precisely, our construction gives $\delta(C_m^{r-4}\oplus C_{3m}^3\oplus C_{6m})\ge m$ for all integers $r\ge5$ and $m\ge2$. Consequently, the least rank $r_0$ for unbounded Davenport excess $\DD(G)-\DD^*(G)$ satisfies $3\le r_0\le5$. Our result also disproves the Narkiewicz–Śliwa conjecture of 1982 \cite{NS82} on the Narkiewicz constant, arising in algebraic number theory from the quantitative study of algebraic integers with unique factorization. The same amplification of the Davenport excess yields counterexamples to Girard's conjecture \cite[Conjecture 1.2]{Girard08} on the cross numbers of long zero-sum-free sequences. As a further main result, we establish the uniform upper bound $$\DD(G)\le\frac{16}{5}\rr(G)\exp(G)$$ for every nontrivial finite abelian group $G$. The classical estimate of van Emde Boas and Kruyswijk \cite{vEBK69} gives $\DD(G)\le\exp(G)\left(1+\log\frac{|G|}{\exp(G)}\right) \le\exp(G)\bigl(1+(\rr(G)-1)\log\exp(G)\bigr)$. Our bound removes the logarithmic factor $\log\exp(G)$ from this classical estimate, and replaces it with the absolute constant $16/5$.

The degrees of the iterates of a projective endomorphism grow in two layers: an exponential rate measured by the dynamical degrees, and a polynomial correction carried by the peripheral Jordan blocks. Log-concavity constrains the first layer; we show that the Hodge index theorem already constrains the second, in every codimension and in arbitrary characteristic. Let $f$ be a surjective endomorphism of a normal projective $d$-fold over an algebraically closed field, with dynamical degrees $\lambda_i$, so that $\textrm{deg}_i(f^n)\asymp\lambda_i^n n^{\mu_i}$ for a unique integer $\mu_i\ge0$. We prove that these exponents are coupled across three adjacent codimensions: strict log-concavity of $(\lambda_i)_i$ at $i$ forces $\mu_i=0$, while equality gives \[ 0\ \le\ 2\mu_i-\mu_{i-1}-\mu_{i+1}\ \le\ 4 . \] Consequently, $\mu_i\le2i(d-i)$, and this is sharp for every $i$. For a zero-entropy automorphism, we prove that every mixed orbit function indexed by any weak composition $\gamma$ of $d$ is a multivariate quasipolynomial of even total degree at most $d^2-\sum_j\gamma_j^2$. For a zero-entropy holomorphic automorphism $g$ of a compact Kähler $d$-fold $Y$, the same method gives \[\left\|(g^n)^*|_{H^{p,q}(Y,\mathbb{C})}\right\| =O\bigl(n^{p(d-p)+q(d-q)}\bigr).\] Powers of elliptic curves attain all exponent bounds as well as the constant $4$.

Asymptotic cohomology was recently introduced to study stability problems in group theory. This paper studies vanishing and Hausdorffness questions around asymptotic cohomology. Our first main result is that, in a fixed degree and for a fixed asymptotic coefficient module, vanishing of asymptotic cohomology is equivalent to vanishing of reduced asymptotic cohomology. In the special case of diagonal coefficient modules, we also obtain a full characterisation of vanishing in terms of bounded cohomology. Our second main result is that asymptotic cohomology vanishes in degree 1 for uniformly convex asymptotic Banach modules. From this, we can deduce Hausdorffness of degree-2 bounded cohomology under the same uniform convexity assumption.

This paper shows that one internal stabilization is enough to resolve all the self-referential tubes on an embedding. As an application, this paper gives a complete isotopy classification for embedded surfaces with a common geometric dual in orientable $S^2$-bundle over a surface, provided that the embedded surfaces have greater genus than the base surface of the bundle.

Let $H$ be a pointed Hopf algebra of square-free dimension over an algebraically closed field of characteristic $p>0$. We prove that either $H$ is a group algebra or $\dim H/|\G(H)|=p$, and that in the latter case $H$ belongs to exactly one of two explicit families of rank-one pointed Hopf algebras. We develop a truncated $q$-Weyl freeness principle for finite-dimensional Nichols algebras of quandle type. If $V=\bigoplus_{x\in X}\K e_x$, $X'\subsetneq X$ is a nonempty subquandle, $V'=\bigoplus_{x\in X'}\K e_x$, and $s\in X\setminus X'$, then $\mathcal B(V)\simeq\K[e_s]/(e_s^{m_s})\otimes C_{s,X'}\otimes\mathcal B(V')$ for some graded vector space $C_{s,X'}$, where $m_s$ is the nilpotency order of $e_s$; in particular, $(m_s)_z\,\mathcal H_{\mathcal B(V')}(z)\mid\mathcal H_{\mathcal B(V)}(z)$. In the non-group case, this yields a $p^2$-divisibility obstruction that rules out noncentral support for the infinitesimal braiding. Together with a graded-dual argument, the resulting rank-one reduction forces the diagram of $H$ to have dimension $p$.

The Brinkman penalisation method embeds boundary-value problems on complex domains into a simple computational box by modeling the solid region as a strongly dissipative medium, avoiding body-fitted mesh generation. We show that multi-symplectic Hamiltonian PDEs regularised by Brinkman-type penalisation retain a multi-conformal symplectic structure under a compatibility condition linking the symplectic matrix and the penalisation projection. This yields an exact local conservation law, under which the multi-symplectic two-form is conserved in the fluid region and decays exponentially inside the solid. The linear wave equation with Brinkman friction and Maxwell's equations with artificial Ohmic conductivity satisfy this condition, with explicit modified Hamiltonian densities. Building on this, we propose (i) structure-preserving numerical integrators via Strang splitting that satisfy a discrete conformal conservation law, and (ii) conformal symplectic neural operators that interleave exact dissipative flows with learnable multi-symplectic evolution operators, allowing geometry-dependent operator learning. Numerical experiments on wave and electromagnetic scattering demonstrate that our methods reproduce correct local energy budgets and avoid unphysical energy drift, providing a principled framework for physics-consistent scientific machine learning on complex domains.

In the present work, we study the properties of the sparse entropic quadrature method, a recently proposed method for the closure of systems of moment equations in rarefied gas dynamics. We prove the properties of the method regarding its well-posedness, hyperbolicity, and stability for first-order spatial convection schemes. We apply the method to two model rarefied gas dynamics problems and compare the results with those given by a discrete velocity method and show that it is capable of reproducing rarefied gas effects whilst having significantly fewer degrees of freedom than the discrete velocity method.

The two-component Becker-Döring model is a two-dimensional coagulation-fragmentation equation that describes the dynamics of clusters built from two different types of monomers growing and shrinking through monomer interaction. We show the existence and mass conservation of solutions and give a partial result regarding uniqueness. Furthermore, we prove under the assumption of detailed balance the entropy equation, followed by a detailed study of the equilibrium states and relative entropies. We can determine the minimum of all relative entropies under sequences of fixed Type I and Type II masses and fully describe the minimising sequences in the weak* topology.

In this paper, we prove global well-posedness and scattering below the energy space for the defocusing nonlinear Schrödinger equation with intercritical power nonlinearities in spatial dimensions $d\geq3$, improving the regularity threshold of Vişan and Zhang [Differ. Integr. Equations, 22(2009), 99-124].

We study smooth fibred two-knots whose fundamental quandle is the prism quandle of order fifty-six. The candidates arise from framed circle surgery along a section of a mapping torus, using the two possible normal-framing classes. We prove that both ambient homotopy four-spheres are diffeomorphic to the standard four-sphere. The candidates have the same knot group and diffeomorphic exteriors, but they are inequivalent even as topological pairs. Every smooth fibred realization of this quandle in a homotopy four-sphere is equivalent, up to knot orientation, to exactly one of these two knots. Standardness follows from a Whitehead-link model for the orbit orbifold together with circle-action recognition. Inequivalence is detected by the third cyclic branched covers, on which a spin-bordism secondary invariant takes different values. Thus the parameter-seven prism quandle admits smooth fibred realizations in the standard four-sphere.

A fixed coarse-grained description need not uniquely determine the underlying microscopic rules. Variations in these rules can alter collective behaviour while leaving the retained observables unchanged. Here we develop a method to identify and use such variations for collective-state design. We construct families of microscopic models with exactly matched coarse observables and use the response derivative within each family to identify independent first-order output changes. In a nonreciprocal active mixture, the largest first-order change in interfacial growth rate under a pointwise kernel budget also determines the leading minimax error of predictions based only on the matched data. Near a Hopf bifurcation, this response map gives an initial turning-kernel design, which is refined by jointly solving for the periodic kinetic state and the kernel. Fixed-kernel integrations realize prescribed oscillation powers and frequencies while preserving the selected angular relaxation rates. In a reaction-diffusion family, cubic amplitude equations guide the design of stripe and hexagon amplitudes while the full linear operator remains fixed. Additional constraints reduce the available response even when its rank is unchanged, whereas the matched angular moments retain their exact relaxation rates. These results provide a constructive route to collective-state design within a prescribed coarse-grained description.

Adaptive algorithms increasingly make decisions while reshaping the dynamics that generate their future data. We establish a shrinking-tube concentration bound for projected stochastic approximation driven by an adaptive Markov chain. The bound guarantees, with high probability, that every iterate after a chosen time remains within a tolerance around the target that tightens over time. The probability of any exit after the chosen time admits a polynomially decaying upper bound, and a matching lower bound shows that its polynomial exponent cannot be improved in general under finite second moments. The result therefore identifies a sharp tradeoff between how quickly the tolerance shrinks and how rapidly the probability of any future exit decreases. We also extend the analysis to recursions with additional martingale-difference noise and predictable bias, showing how growth in the martingale-difference noise scale slows the decay of the exit-probability bound while predictable bias restricts the admissible tube shrinkage. The proof combines backward kernel replacement, a finite-time mean-squared-error bound, and a blockwise maximal first-exit argument. We apply the theory to inventory learning with stockout-dependent demand and fixed stockout costs, and quantify how numerical gradient accuracy affects the all-future reliability of the resulting policies.

We relate the theory of $q$-rational and $q$-real numbers introduced by Morier-Genoud and Ovsienko to the classical theory of Kleinian groups and their Teichmüller spaces. This provides a geometric point of view on several recent results about realisations of $q$-rationals for particular values of $ q \in \mathbb{C} $. As an application we resolve a conjecture of Bapat, Becker, and Licata on the topology of the $q$-deformed Farey tessellation.

We establish that the ellipticity of the minimal surface integrand is the sharp threshold for $\mathrm{W}^{1,1}$-regularity of solutions to the Dirichlet problem on $\mathrm{BV}$. Here, we particularly deal with the autonomous case of convex linear growth functionals.

We formulate a generalized Saito criterion for complex projective plane curves, building on a recent generalized Saito criterion for plus-one generated arrangements of hyperplanes due to Junyan Chu.

We study the stability of the log–log blow-up regime for the focusing mass-critical nonlinear Schrödinger equation under small $H^s$ perturbations. Previously, stability was established for every $0<s<1$ in dimension two by Colliander and Raphaël [Math. Ann. (2009)] and subsequently extended to dimensions $d\geq3$ under the restriction $s>1/(1+\min\{1,4/d\})$ by Sun and the fourth author [J. Math. Pures Appl. (2020)]. We remove this restriction and establish stability throughout the full subcritical range $0<s<1$.

For independent bond percolation on $\mathbb{Z}^d$ with parameter $p$, let $p_c^b(d)$ be the critical probability. We prove that for each $d \geq 9$, there is $\epsilon_d>0$ such that for each $p \in (p_c^b(d), p_c^b(d)+\epsilon_d)$, the complement of the infinite open cluster stochastically dominates a supercritical site percolation on $\mathbb{Z}^d$. This improves the previous results by Grimmett, Holroyd and Kozma 2014, and Bock, Damron, Newman and Sidoravicius 2020. Numerical estimates of $p_c^b(d)$ and $p_c^s(d)$ (the site critical probability) suggest that a similar stochastic domination result holds for all $d \geq 4$. For the Ising model on $\mathbb{Z}^d$ with inverse temperature $\beta$, let $\beta_c(d)$ be the critical inverse temperature. We prove that for each $d \geq 8$, there is $\epsilon_d>0$ such that for each $\beta \in [0,\beta_c(d)+\epsilon_d)$, the $+$ spins under the minus phase stochastically dominate a supercritical site percolation on $\mathbb{Z}^d$. This improves the previous result of Aizenman, Bricmont and Lebowitz 1987. Numerical estimates of $\beta_c(d)$ and $p_c^s(d)$ suggest that a similar stochastic domination result holds for all $d \geq 5$.

We propose a discrete-time feedback linearization control approach to solve nonsmooth linearly constrained composite optimization problems. By using proximal operators, we construct a dynamical system whose equilibria correspond to the stationary points of the optimization problem. Interpreting the Lagrange multipliers as control inputs, we employ feedback linearization control to steer the obtained dynamical system to a stable equilibrium. We analyze the convergence of the resulting closed-loop system both in the strongly convex setting and under the Polyak-Lojasiewicz condition. Finally, we illustrate the applicability of the approach through numerical experiments.

In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. In 1977, Nagasaka asked whether this exceptional set must have logarithmic capacity zero. We prove that there exist a transcendental entire function and a regular local inverse for which the exceptional set contains a compact set of positive logarithmic capacity. In fact, for every $0<s<1$, such a function and inverse can be chosen so that this compact set has positive $s$-dimensional Hausdorff measure.

We present various ways of determining generalized weight polynomials of a matroid $M$, and we recall how one can find the generalized weight spectra of a linear code, given these polynomials, for the matroid determined by any generator matrix of the code. A main goal is to give coding theorists different ways to determine these polynomials. We describe how one can find the generalized weight polynomials of any matroid $M$, directly from its lattice of flats, and we also show how one can find them from the Poincare series (in this case polynomials) of the associated Orlik-Solomon algebras of matroids arising as contractions of the flats of $M$. This opens for using information about broken circuits of the matroids to determine generalized weight polynomials. We also recall the well-known connection between the Orlik-Solomon algebra of a matroid, and Whitney numbers obtained by order homology, and use it to show how one can describe weight polynomials in terms of Whitney numbers from order homology of the matroid and its contraction of flats. We recall briefly a well-known relation between the Orlik-Solomon algebra of a matroid $M$, which is representable over the complex numbers, and de Rham homology numbers obtained from complements of intersections of hyperplanes in a hyperplane arrangement corresponding to $M$. We also describe a way to find the generalized weight polynomials of a matroid in terms of polynomials defined in connection with its lattice of cyclic flats. In a simple running example we show how one can calculate the generalized weight polynomials in different ways, including both the methods presented in this paper, and selected methods developed in earlier papers. We also include a less simple example with the projective Reed-Muller code $PR_3(2,2).$

In 2012, Iwaniec, Kovalev and Onninen proposed an upper Nitsche–Grötzsch type estimate for harmonic $K$-quasiconformal homeomorphisms between circular annuli: in normalized form, every such map $h:A(1,s)\to A(1,S)$ should satisfy \[ S \leq \frac{K+1}{2}s-\frac{K-1}{2s}. \] The radial and spiral-radial one-mode models explain why this estimate is natural. We show that this one-mode evidence does not extend to the unrestricted non-radial class. For every $1<s<S$, we construct a non-radial harmonic orientation-preserving diffeomorphism $h:A(1,s)\to A(1,S)$ with \[ \norm{\omega_h}_{\infty}<\frac{S-s}{S-s^{-1}}, \] strictly below the conjectural threshold. Thus, the dilatation lower bound predicted by the conjecture fails, even for smooth harmonic diffeomorphisms with prescribed circular boundary components. The construction uses a small high-frequency reparametrization of the outer boundary: it lowers the first Fourier-mode dilatation by order $t^2$, while the compensating high modes are exponentially damped at the inner boundary. We also record structured regimes in which the one-mode estimate survives, including an inner-boundary anti-conformal energy condition, a Fourier leakage criterion, and a low-frequency spectral stability result. In the minimal-surface interpretation, the upper radial model is helicoidal and vertical-periodic rather than single-valued catenoidal; accordingly, the counterexamples give vertical-periodic minimal annuli with slope below the expected helicoidal threshold, and lead to a non-radial extremal problem for harmonic annulus diffeomorphisms.

In this paper, we consider the existence and stability of waves progressing through a finite-depth two-dimensional body of water that carry a vortex in their bulk. The waves are acted upon by gravity, and sit below a region of air at constant pressure, with the air–water interface being a free boundary along which capillary effects are incorporated. We consider two classes of vortices: point vortices, where formally the vorticity is a Dirac $\delta$, and hollow vortices, where the vortex core is a region of constant pressure outside the fluid domain and about which there is a nonzero circulation. First, we prove that steady small-amplitude wave-borne point vortices exist for any subcritical Froude number and supercritical Bond number, and then show they are conditionally orbitally stable. Second, through a vortex desingularization argument, we construct capillary-gravity wave-borne hollow vortices. Notably, the wave speed is $O(1)$ for both these families.

Enhancing the robustness of deployed networks against failures and disruptions is critical for reliable operation. This requires deciding which new links to install and how strongly to weight them under limited resources. We study this problem through the Kirchhoff index, or total effective resistance, a spectral measure of global connectivity. The resulting augmentation problem couples discrete candidate-edge selection with continuous weight allocation under heterogeneous per-unit deployment costs, a total budget, and an exact-cardinality constraint. For a fixed weighted base graph, this yields a mixed-integer formulation and a semidefinite relaxation whose optimum lower-bounds the mixed-integer optimum. We cast the relaxation as a cone program and solve it numerically using a homogeneous self-dual embedding and first-order operator splitting. Feasible discrete designs are recovered through rounding-and-repair procedures and assessed by a posteriori gap estimates relative to the numerical semidefinite program (SDP) benchmark. As a scalable alternative, we develop an exact-$k$, budget-feasible greedy heuristic built on rank-one Laplacian updates and biharmonic-distance caching, and interpret its progress through a Bellman value-to-go benchmark with a conservative spectral lower bound on the local policy ratio. Experiments on synthetic and real infrastructure networks across graph sizes, budgets, weight distributions, and cost regimes show that, under fixed budgets, distance-proportional costs limit the achievable resistance reduction and shift installed conductance toward shorter links relative to uniform per-unit costs.

Let $T_k(E)$ denote the set of congruence classes of ordered $k$-tuples of pairwise distinct points of $E$. Let $p\equiv3\pmod4$ be prime. For $E\subset\mathbb{F}_p^2$ with $3\leq|E|\leq p^{3/4}$, we prove that $|T_3(E)|\gg|E|^{11/6}$; for $4\leq|E|\leq p^{3/4}$, we prove that $|T_4(E)|\gg|E|^3/\log|E|$. For every fixed $k\geq5$ and $k\leq|E|\leq p^{3(k-2)/(3k-4)}$, we prove that $|T_k(E)|\gg_k|E|^{k-1}$. The proofs proceed by bounding the moments of the overlap function of rigid motions. The main inputs are an exact identity involving the distance energy and an incidence bound obtained by viewing rigid motions as lines over $\mathbb{F}_p(i)$. The bound for $k=4$ is sharp up to a logarithmic factor, while the bounds for $k\geq5$ are sharp up to constants.

In this article, we consider a wave equation on a bounded three-dimensional domain, featuring a degenerate nonlocal damping mechanism that depends on the system's energy and a source term with subquintic growth. This class of models is motivated by applications in optical physics. We establish global existence in the Shatah–Strichartz sense by combining Galerkin approximations with space-time Strichartz estimates on bounded domains. A central difficulty in the long-time analysis is the interplay between three-dimensional wave dynamics, nonlocal damping, and forcing terms that are not locally Lipschitz continuous. We prove the existence of a compact global attractor and derive an upper bound for its Kolmogorov $\varepsilon$-entropy. In the non-degenerate damping case, assuming that the domain has a smooth, strictly convex boundary, we exploit the refined Strichartz estimates of Ivanovici et al.~\cite{Ivanovici-Lascar-Lebeau-Planchon-2023}, which yield, in particular, $u\in L^3_tL^{18}_x$. Under the further assumption $f\in C^2(\mathbb R)$, this improved integrability allows us to establish quasi-stability of the associated dynamical system. Consequently, the global attractor has finite fractal dimension and higher regularity, and a generalized exponential attractor exists.

We study a two-parameter family of quadratic BSDE systems of the form given by Jackson (2023) in his open questions on non-Markovian solvability. Fan, Hu, and Tang (2025) established well-posedness for every bounded terminal datum when $1/\alpha+1/\beta=1$. For every $\alpha,\beta>0$ and $M>0$, we establish existence and uniqueness in $\mathcal{S}^{\infty}\times\mathrm{BMO}$ for a dense $G_\delta$ subset of the space of terminal data satisfying $|\xi_i|\le M$ a.s., $i=1,2$, equipped with the topology of convergence in probability. We also obtain well-posedness for every terminal datum taking at most two values. Furthermore, we show that every bounded terminal problem can be reduced to one with a three-valued terminal datum while preserving its solution set. However, for every positive parameter pair with $1/\alpha+1/\beta\ne1$, we construct a three-valued terminal datum admitting at least two distinct solutions in $\mathcal{S}^{\infty}\times\mathrm{BMO}$. In particular, uniqueness can fail in Jackson's stochastic-game example $\alpha=\beta=1$.

We introduce equivariant persistent analogues of Lusternik-Schnirelmann category, topological complexity, cup length, and zero-divisors cup length for persistent spaces with group actions, and establish their stability. In particular, we investigate their behavior under equivariant interleavings and obtain bounds for the corresponding erosion distances. For Vietoris–Rips filtrations of compact metric spaces equipped with compatible group actions, we relate the erosion distances between the resulting equivariant persistent invariants to appropriate equivariant versions of the Gromov-Hausdorff distance.