Papers

This paper investigates the inverse problem of determining the spatially dependent damping coefficient in a time-fractional damped wave equation, where the damping term is given by a Caputo derivative of order \(\alpha\in(0,1)\). We first prove the well-posedness of the direct problem in exponentially weighted Sobolev spaces. Then, by means of the Fourier–Laplace transform in time, the nonstationary problem is reduced to a family of stationary elliptic equations with complex frequencies. Based on complex geometrical optics solutions and a suitable integral identity, we estimate the Fourier transform of the difference of two damping coefficients in terms of the difference of their Dirichlet-to-Neumann maps. Combining low-frequency estimates with a high-frequency decay argument, we obtain a conditional logarithmic stability result.

This article resolves some errors in the paper “Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance. Math. Ann. 392, 1051-1097 (2025)”. The errors are in the proof of contraction of a map associated with our equation in two and three dimensions.

Consider the following plaquette model from statistical physics: a lamp lies at every vertex of the triangular lattice and a switch lies at every even vertex of the (bipartite) dual hexagonal lattice. Each switch toggles the three lamps on its face. The energy of a configuration is the number of ON lamps. For the Glauber dynamics associated with the Gibbs measure defined by this Hamiltonian at any inverse temperature $\beta>0$, we show that, in any dimension $d\ge 2$, the infinite volume relaxation time satisfies \[e^{\beta^2/C}/C \le T_{\mathrm{rel}}\le Ce^{e^{C\beta}}\] for some $C>0$. Our result entails that the Gibbs measure is unique. The $e^{\beta^2}$ scaling was conjectured by Newman and Moore in 1999 and matches the behaviour of supercritical rooted kinetically constrained models such as the East model, thus recovering fragile glass phenomenology in the absence of kinetic constraints. More precisely, we show that, on a torus of side length $2^k$, when $\beta\to\infty$ and $k/\beta\to0$, we have $T_{\mathrm{rel}}=e^{2\beta k(1+o(1))}$. Quite surprisingly, however, we also prove that, on non-periodic finite domains of size $n\le e^{\beta/C}$ for large $C>0$, we have the much larger asymptotics $\ln T_{\mathrm{rel}}=\beta n^{\Theta(1)}$. The main ingredients of the proofs are new results in extremal and enumerative combinatorics and rely on renormalisation ideas for the dynamics and its groundstates also known as the Ledrappier subshift. We note consequences of our results to geometric group theory (more precisely to the complexity of the word problem for the Baumslag finitely presented group) and to ergodic theory.

This paper introduces a unified abstract variational and topological framework to characterize the spectral gap and eigenvalue distribution in high-contrast multiscale partial differential equations (PDEs). We rigorously prove that the exact location of the spectral gap is universally determined by the dimension of the local null space associated with the high-contrast inclusions. For systems with infinite-dimensional kernels, this location is strictly determined by the topological Betti numbers. Furthermore, we establish a multiscale Weyl's law via a spectral decoupling theorem, demonstrating that as the contrast approaches infinity, the multiscale spectrum bifurcates into two independent components: the Dirichlet spectrum of the background matrix and the internal Neumann spectrum of the inclusions. Using spectral homogenization theory, we also show that in the limit of vanishing periodicity, this expanding topological eigenspace asymptotically spans the entire spectral space of the macroscopic homogenized operator. These theoretical results are comprehensively verified through numerical experiments on diffusion, elasticity, fourth-order plate, Maxwell, and grad-div operators.

In the study of linear retarded functional differential equations (RFDEs), the classical bilinear form proposed by J. Hale has long served as an indispensable tool for the geometric theory. Nevertheless, its original definition suffers from ambiguities, such as the non-commutativity of matrix products and the precise meaning of the involved double integrals, due to its definition motivated primarily by eigenvalue problems. Here, by establishing the Green and Lagrange identities associated with a given autonomous linear RFDE, we derive a bilinear form $B$, which is none other than the celebrated Hale bilinear form. This not only resolves the aforementioned issues in the original formulation, but also clarifies its structural origin within the theory of differential equations. Furthermore, based on the recent definition of the $M^p$-space via the memory measure, we demonstrate that the bilinear form $B$ uniquely extends to the product space $M^{*q} \times M^p$ by dense extension, and show that the Green and Lagrange identities still hold in the sense of mild solutions. In terms of its potential for extension to non-autonomous systems, neutral functional differential equations, and differential equations with delay and spatial structures, this work opens a new avenue for the adjoint theory of linear functional differential equations.

In his 2009 paper, Thakur posed three conjectural hypotheses for the degrees of the power sums \[ S_d(k)=\sum_{\substack{a\in \mathbb F_q[t] \text{ monic}\\ \deg a=d}} a^{-k} \qquad\text{and}\qquad s_d(k)=-\deg_t S_d(k). \] For prime fields $q=p$, we prove Hypotheses H1 and H2, giving a unique greedy description of the extremal term in Carlitz's formula and establishing the recursion \[ s_d(k)=s_{d-1}(s_1(k))+s_1(k). \] As consequences, the prime-field recursion gives the strict Newton-polygon convexity used in the prime-field Carlitz-Goss Riemann-hypothesis theorem, and it recovers Thakur's nonvanishing theorem for positive multizeta values over $\mathbb F_p[t]$. We also prove Hypothesis H3 for all finite fields $q=p^f$, establishing the monotonicity \[ s_d(k)<s_d(k+1)\qquad (p\nmid k). \] We provide Lean formalizations of the arguments in this paper, generated by AxiomProver.

We investigate the Cauchy problem of three-dimensional compressible Navier-Stokes equations with far-field vacuum. Based on delicate energy estimates and structures of the systems under consideration, we show the global well-posedness and decay rates of strong solutions provided that some coefficient-coupled scaling invariant initial quantity is suitably small. In particular, our smallness conditions are independent of any initial data and known parameters in the systems. Moreover, there is no need to require the compatibility conditions on the initial data. Our results improve previous works.

The paper is concerned with the well-posedness of backward stochastic partial differential equations with jumps, also called backward stochastic partial differential integral equations. We start from the proof for the existence and uniqueness of solution to backward stochastic evolution equation with jump in the Gelfand triple framework. Then the well-posedness of both weak solution and strong solution to backward stochastic partial differential integral equation is obtained with the Gelfand triple replaced by specific Sobolev spaces. Finally, the comparison principle for backward stochastic partial differential integral equation is proved, which has potential applications in financial mathematics.

Exact diagonalization of quantum lattice Hamiltonians returns floating-point eigenvalues whose accuracy is not certified: rounding error, eigensolver behaviour, and ill-conditioning can corrupt a result without warning. We present symveig, a pure NumPy/SciPy package that computes rigorous, machine-checkable enclosures of all eigenvalues of a Hermitian matrix, with an optional symmetry-sector decomposition. For a matrix that commutes with an abelian conserved quantity diagonal in the working basis (for example the total magnetization of a spin model), the package verifies each symmetry sector independently. Because each sector block is much smaller than the full matrix, this yields enclosures that are both tighter (by a factor of $3$-$9$ across system sizes $L = 4$-$12$) and dramatically faster (a wall-clock speedup of up to $130\times$ at $L = 12$) than verifying the full matrix, while never forming or diagonalizing it. Every enclosure half-width is a guaranteed upper bound on the distance from a computed eigenvalue to the nearest true eigenvalue under IEEE~754 round-to-nearest arithmetic, obtained by explicit floating-point error analysis with no heuristic slack. The implementation requires neither INTLAB nor MATLAB, bringing rigorously verified eigenvalue enclosure into the standard scientific-Python stack used in computational physics. We validate the package on $1$D Heisenberg (open and periodic), $J_1$-$J_2$ Heisenberg, and $2$D Heisenberg lattices, confirming that every computed eigenvalue is contained in its enclosure across all tested configurations.

Let $1<q\le \infty$ and $v>-1$. Let $(X,d,\mu)$ be an $s$-Ahlfors-regular quasi-metric measure space. Suppose that $B^{0,v}_q(X)$ is the block space which consists of all functions that admit a decomposition into $q$-blocks supported on balls. In this paper, we study the relationship between the block space $B^{0,v}_q(X)$ and the Orlicz-type space $L(\log^+\!\!L)^{1+v}(X)$. More precisely, we show that the block space $B_q^{0,v}(X)$ is a proper subspace of the Orlicz space $L(\log^+\!\!L)^{1+v}(X)$ for any fixed $1<q\le \infty$ and $v>-1$. Namely, $$B_q^{0,v}(X)\subsetneq L(\log^+\!\!L)^{1+v}(X),$$ which gives a confirmed answer to a longstanding open problem concerning the relationship between block spaces and Orlicz-type spaces on the unit sphere $\mathbb S^{n-1}$. We further show that $L(\log^+\!\!L)^{1+v}(X)$ is the smallest Orlicz-type space containing $B^{0,v}_{q}(X)$. We also introduce a generalized block space $\mathscr B_q^{0,v}(X)$ that depends only on the measure structure and show that this space is equivalent to the Orlicz space $L(\log^+\!\!L)^{1+v}(X)$ when $\mu(X)<\infty$. Finally, we consider two special cases that further clarify the roles of the parameter $q$ and the logarithmic weight.

In this paper, we study sign changes and vanishing for the number of representations of an integer as a sum of triangular numbers plus three-colored sums of generalized pentagonal numbers with an even number of parts minus those with an odd number of parts. We study this by investigating coefficients appearing in the $q$-series expansion of $F(z) = \frac{\eta(z)}{{\eta(2z)}^2 {\eta(3z)}^3}$, where $\eta$ is the Dedekind-eta function. We use the Hardy-Ramanujan-Rademacher circle method to give an asymptotic formula for the coefficients.

In this paper, we investigate the boundedness properties of the Dunkl-type Hausdorff operator on generalized Dunkl-type Morrey spaces and generalized Dunkl-type Campanato spaces. Furthermore, we introduce the fractional Dunkl-type Hausdorff operator and study its boundedness from $L^p(\mathbb{R}, d\mu_\alpha)$ to $L^q(\mathbb{R}, d\mu_\alpha)$. We also establish the boundedness of the fractional Dunkl-type Hausdorff operator on generalized Dunkl-type Morrey spaces under suitable conditions. As a consequence of our main results, we derive several applications. Finally, we evaluate the exact norm of the Dunkl-type Hausdorff operator on Dunkl-type Morrey spaces and Dunkl-type Campanato spaces.

We consider the two species (electrons and positive charged ions) Vlasov-Poisson system linearized around BGK waves. We formulate the problem as an equivalent Vlasov-Ampère system, that we write as a system of Schrödinger type in an appropriate Hilbert space, and with a selfadjoint Vlasov-Ampère operator as a Hamiltonian. We develop a complete stationary scattering theory. We identify the absolutely continuous spectrum and the singular spectrum of the Vlasov-Ampère operator, we construct the generalized Fourier maps, we prove that the wave operators exist, are complete, satisfy Birman's invariance principle, and that the stationary formulae hold. Using these results we prove that the BGK waves are asymptotically stable. We obtain a precise description of the large time behaviour of the solutions to the Vlasov-Ampère system Namely,for large times the phase-space densities of electrons and ions are asymptotic to the phase-space densities of solutions of the unperturbed Vlasov-Ampère system. This implies that they follow the trajectories of solutions to Newton's equations for electrons and ions with the potential of the BGK wave, in the sense that they are transported along these trajectories. Furthermore, we prove that Landau damping holds, that is to say, the electric field tends to zero in pointwise sense for large times.

We construct an infinite-lattice discrete Calderón projection for the Helmholtz equation by convolution with the lattice Green's function (LGF), and apply it to active noise shielding and confinement on Cartesian grids with arbitrary geometry. The LGF fixes the outgoing radiation condition and removes the geometry-dependent auxiliary Helmholtz problem and artificial outer boundary from the projection and control synthesis; its finite numerical tabulation depends only on $(h,k)$ and is reusable across geometries. We prove idempotence, characterize the range as the trace space of interior lattice-Helmholtz solutions, and establish range equivalence with a well-posed Tsynkov-type projection. The two projectors coincide as operators when the auxiliary problem reproduces the exact lattice radiation condition. A capacity-matrix realization yields closed-form shielding and confinement densities supported on the exterior and interior sublayers of a single lattice boundary strip, respectively. For pure-noise shielding, exterior-sublayer measurements suffice under explicit invertibility assumptions; preservation of an unknown wanted interior field requires the full strip trace. Experiments on circular, L-shaped, and star-shaped regions verify machine-precision cancellation for LGF-consistent sources and near-second-order convergence for analytic plane waves and point sources. Conditioning and measurement noise tests quantify the configuration dependence of the reconstruction.

We study the parametric family $p_{k,m,\varepsilon,q}=km(m+1)+\varepsilon +2kq$ with $k,m\in\mathbb{N}^{\ast}$, $\varepsilon \in \{\pm1\}$, $q\in\mathbb{Z}$, which extends the elementary observation $p\equiv\pm1\pmod 6$ for every prime $p>3$. Each prime $p>k+1$ has a canonical triple $(m,\varepsilon,q)$ with $|q|$ minimal, mapping it to a generalised hexagonal base $H^{(k)}_m:=km(m+1)+1$ and a signed minimal offset $q_{\rm min}(k,m,\varepsilon)$. Every quantitative assertion is labelled rigorous, conditional, or heuristic. (Rigorous, structural.) For every prime $\ell\mid 2k$ the family satisfies $p \equiv \varepsilon \pmod\ell$: the degenerate modular axis is fixed exactly, independently of $q$. (Rigorous, constructive.) For the $3$-smooth subfamily $p=3m(m+1)+1$ with $m=2^a3^b-1$, Pocklington–Lehmer certificates are unconditional; two a-priori arithmetic filters remove $\approx 87\%$ of candidates before any large-integer arithmetic, and the algorithm produced an unconditionally certified prime of $29\,998$ decimal digits, independently re-verified in a separate computational environment. (Rigorous, negative.) The earlier-reported spectral correlation between the cumulative functional $Q(r)=\sum q_n$ and the zeros of $\zeta$ is a spurious-regression artefact. Beyond three permutation tests and a bias-free test on $10^8$ primes ($R^2=1.16\times10^{-7}$), we prove unconditionally that the regression amplitude $A_N(\gamma)\to 0$ for every fixed $\gamma$, by reduction to square-root-phase prime exponential sums. The Riemann zeros are not spectral frequencies of $Q(r)$. (Conditional, GRH / Bateman-Horn.) $|q_{\rm min}|=O_k(m\log^2 m)$; $\mathbb{E}[|q|\mid m]=m/4+O((\log m)^2/\sqrt m)$; a modular distribution law. (Conditional, RH.) The per-prime $\zeta$-footprint on the layer occupancy is $\ll p^{-1/4}(\log p)^2$ (Selberg variance) – the mechanism behind the negative result. (Heuristic.) $\mathbb{E}[|q_{\rm min}|]\sim\log m/C_k$ ($R^2=0.984$); the geometric constant $C_0(k)=\langle|q|\rangle/\sqrt p=1/(4\sqrt k)$, stable to ${<}0.02\%$ across $3\le k\le 29$; and, for $\ell\mid 2k$, $q_{\rm min}$ is empirically equidistributed modulo $\ell$ (not a consequence of Bombieri–Vinogradov). This is experimental mathematics with rigorously tracked hypotheses: two genuinely unconditional pillars – constructive (the certified prime) and analytic-negative (the vanishing $\zeta$-signal) – together with a precise law on each axis. No classical question is settled.

We study an open problem of understanding the effects of the minimum component separation on the convergence rates of parameter estimation in finite Gaussian mixtures. We address this by developing a unified geometric framework based on novel Hellinger lower bounds that directly relate discrepancies between mixture densities directly to Wasserstein distances between their underlying mixing measures, with explicit dependence on both the minimum separation and the minimum weight. Our approach combines carefully designed interpolation polynomials with confluent divided difference techniques to construct specialized moment-extraction test functions. When the number of components is known, these bounds uncover a localization phenomenon: the separation complexity is driven strictly by the spatial configuration of mixture components, namely, whether they are concentrated in a single cluster, partitioned into multiple clusters separated by a macroscopic gap, or arranged without any structural constraints. On the other hand, when the number of components becomes unknown and is over-specified, the separation complexity is slightly reduced, while the minimum mixture weight disappears entirely from the convergence rates due to a transition from first-order to second-order Wasserstein geometry. As a consequence, we obtain separation-dependent convergence rates that continuously interpolate between point-wise and uniform estimation regimes, thereby settling the fundamental limits of parameter recovery in finite Gaussian mixtures.

We study refined asymptotics of scalar-flat ALE four-manifolds in the Tian–Viaclovsky setting, namely for self-dual or anti-self-dual metrics and for metrics with harmonic curvature. Starting from the ALE coordinates obtained by Tian–Viaclovsky, we construct preferred coordinates at infinity and identify the homogeneous $|x|^{-2}$ term in the metric expansion. This term splits canonically into a scalar part determined by the ALE ADM mass and an algebraic Weyl tensor at infinity. As an application, we consider scalar-flat Kähler ALE metrics on minimal resolutions $\pi:X\to\mathbb C^2/\Gamma$ of quotient surface singularities. In this case, the leading Weyl tensor at infinity vanishes exactly when the minimal resolution is crepant.

This paper proves weighted discrete Strichartz, or equivalently discrete restriction, estimates for two-dimensional exponential sums with fractional and perturbed polynomial phases. The new results cover three related classes of curves: the fractional Schrödinger-type curve $(n,n^{1+\nu})$, the fully fractional curve $(n^{1+\mu},n^{1+\nu})$ with $0<\mu<\nu$, and bounded perturbations of the polynomial curve $(n,n^k)$. In each case we obtain $\ell^2$-weighted $L^p$ estimates with decoupling-type dependence on the frequency scale, including both low- and high-$p$ regimes. The proofs are based on finite-type decoupling, together with localization and rescaling arguments adapted to curvature degeneracy, anisotropic homogeneity, and bounded perturbations of the phase. The introduction frames these estimates from the viewpoint of decoupling theory, with arithmetic results obtained by efficient congruencing used only as comparison points for the known cases.

We study balanced subfamilies of the middle layer $\binom{[2n]}{n}$ of the Boolean lattice $2^{[2n]}$. A family $\mathcal{F}\subseteq\binom{[2n]}{n}$ is said to be balanced if every element in $[2n]$ appears in the same number of members of $\mathcal{F}$. A balanced subfamily of size 2 is exactly a complementary pair $\{A,[2n]\setminus A\}$, and therefore a family with no balanced subfamily of size $2$ has at most $\frac{1}{2}\binom{2n}{n}$ members. We show that for every $k\geq 1$ and all sufficiently large $n$, this maximum size is compatible with delaying the smallest size of a balanced subfamily until $2k+2$. More precisely, there exists a family $\mathcal{F}\subseteq\binom{[2n]}{n}$ of size $\frac{1}{2}\binom{2n}{n}$ with no balanced subfamilies of sizes $2,4,\ldots,2k$, but with a balanced subfamily of size $2k+2$. The proof is constructive and uses a general lifting from simple games to self-dual selectors in the middle layer. This provides an efficient constructive proof of a recent conjecture of Moss and Pedersen.

We develop a low-regularity Sewing theory for the semigroup coboundary $\hat\delta=\delta-a$ associated with a strongly continuous semigroup $S$. Unlike the ordinary low-regularity Sewing problem, the semigroup setting has an intrinsic algebraic non-uniqueness below the threshold $1$, in the sense that solutions are canonical only modulo semigroup cocycles. Accordingly, the natural target is a quotient space rather than an increment space. We identify this quotient structure and construct the corresponding semigroup Sewing map. The construction uses a frozen terminal-time transform, which rewrites semigroup defects, for each terminal time, as ordinary low-regularity Sewing problems on a frozen simplex. This reduction, however, does not by itself produce a genuine semigroup increment; the main additional step is to prove that the frozen solution classes are compatible as the terminal time varies and hence assemble into a canonical quotient class for $\hat\delta$. This yields canonical classes for $0<\gamma<1$, and at $\gamma=1$ under logarithmic control. We further provide a scale-dependent criterion for selecting genuine representatives, verified for heat semigroups on Sobolev scales through a parabolic Littlewood–Paley tail condition.

The Landau–Lifshitz–Gilbert (LLG) equation is a fundamental model in micromagnetics for the magnetisation dynamics of ferromagnets at temperatures well below the Curie temperature. In chiral magnetic materials, the bulk Dzyaloshinskii–Moriya interaction (DMI) contributes a first-order term to the effective field, induces a natural chiral boundary condition, and plays an important role in the dynamics of skyrmions. In this paper, we analyse a mass-lumped midpoint finite element scheme for the LLG equation with exchange interaction and bulk DMI. The fully discrete scheme preserves the nodal unit-length constraint exactly and satisfies a discrete energy law. Under suitable regularity assumptions on the exact solution, we prove an optimal-order convergence estimate in the energy norm, first order in space and second order in time. To our knowledge, this is the first such a priori error estimate for the mass-lumped midpoint finite element method, even in the exchange-only case. Since the midpoint scheme is nonlinear, its practical implementation requires an algebraic solver that is compatible with the geometric structure of the method. To this end, we propose a structure-preserving fixed-point iteration which preserves the nodal unit-length constraint at every iterate. We further prove that, under an appropriate residual stopping criterion, the resulting inexact scheme retains the same convergence rate as the ideal midpoint scheme, up to the contribution of the solver tolerance. Numerical experiments support the theoretical results and illustrate the robustness and structure-preserving properties of the method.

In this paper, we study the 3-D incompressible fractional stationary MHD equations on the torus $\mathbb{T}^3$. For fractional power indices $\alpha_1,\alpha_2 >0,$ we prove that there exist infinite non-trivial stationary singular solutions of 3-D fractional MHD equations via convex integration. In particular, this result shows the non-uniqueness of weak solutions to the 3-D stationary MHD equations in some sub-critical Besov spaces.

Given a knot $K$ in $S^3$, its knot Floer completely determines the bordered Floer homology of its complement by a work of Lipshitz–Ozsváth–Thurston. Furthermore, the determination is combinatorial: given a model for $CFK(S^3, K)$ there is a method for producing an explicit model for $\widehat{CFD}(S^3 \smallsetminus \nu(K))$. In this paper, we show that a similar formula holds between certain classes of chain endomorphisms of knot Floer chain complex and type D endomorphisms of bordered Floer homology, up to a 1-dimensional ambiguity in the type D side; both the formula and the ambiguity can be computed combinatorially. It follows that, for any concordance from a knot to itself and any satellite pattern, we can combinatorially compute the knot Floer cobordism map of the satellite concordance (up to conjugation) from the knot Floer cobordism map of the given concordance and the bordered Floer homology of the pattern complement.

This paper studies safe distributed consensus for single-integrator multi-agent systems over connected undirected graphs under simultaneous asymmetric actuator constraints and output safety constraints. Each agent is equipped with a continuously differentiable asymmetric actuator dynamics that maps a commanded control signal to the realized plant input while keeping the latter strictly inside a prescribed admissible interval. To address output safety, a barrier-coordinate transformation is introduced over a common time-varying safe interval, and a distributed synchronization law is designed in the transformed coordinates. The resulting controller integrates a graph-based coordination layer with an actuator-side tracking layer, thereby enabling simultaneous enforcement of input admissibility, forward invariance of the safe output set, and asymptotic synchronization. For compact admissible sets of initial conditions, it is shown that the closed-loop solution is complete, all signals remain bounded, the actuator inputs remain strictly within their asymmetric bounds, and the agent outputs remain inside the prescribed safe interval for all time. Moreover, the transformed synchronization errors converge exponentially to zero, and the original agent outputs asymptotically synchronize to a designer-selected admissible trajectory embedded in the common safe interval. Numerical simulations validate the proposed framework and demonstrate safe consensus under both asymmetric actuation bounds and time-varying output constraints.

A classical theorem of Sheehan in 1977 states that every hamiltonian graph $G$ of order $n$ satisfying $e(G)>\left\lfloor \frac{n^2}{4}\right\rfloor+1$ contains at least two cycles of every length $\ell$, $3\le \ell\le n$. In the same paper, Sheehan recorded a conjecture of Hilton, which strengthens this conclusion by asserting that such a graph contains at least $n-\ell+2$ cycles of length $\ell$ for each $3\le \ell\le n$. We prove Hilton's conjecture for all hamiltonian graphs of order at least $440$.

For every $d \geq 4$, we construct a family of $(d-1)$-dimensional flag simplicial spheres $\mathcal K_n$ whose graphs contain independent sets of size asymptotically equal to the number of vertices. More precisely, we prove that for sufficiently large $n$, $$ \alpha(G(\mathcal K_n)) \geq f_0(\mathcal K_n) - \frac{C\,f_0(\mathcal K_n)}{\left(\log f_0(\mathcal K_n)\right)^{\lfloor d/2 \rfloor-1}},$$ where $C = C(d) > 0$. This disproves a recent conjecture of Chudnovsky and Nevo.

An equivalent version of the Báez-Duarte criterion \cite{baez} for the Riemann Hypothesis (RH) by Bagchi states that the RH holds true if and only if the function $E(s) = 1/s$ belongs to the closed linear span of $G_k(s) = (k^{-s} - k^{-1})\zeta(s)/s,\, k \geq 2$ in the Hardy space \( H^2(\mathbb{C}_{1/2}) \), where $\mathbb{C}_{\alpha}$ denotes the half-plane $\mathrm{Re}(s)>\alpha$. We first show that if $E$ belongs to the closure of span$(G_k)_{k\geq 2}$ in \( H^2(\mathbb{C}_{\alpha}) \) for $\alpha>1/2$, then $\zeta$ is zero-free in $\mathbb{C}_\alpha$. We then use this as the basis for a numerical analysis of the sequence \[ s_n = \left\| \sum_{k=2}^{n} \mu(k) G_k - E \right\|^2_\alpha, \] for $1/2\leq \alpha \leq 1$, where $\left\|.\right\|_\alpha$ is the norm in $H^2(\mathbb{C}_{\alpha})$ and $\mu$ the Möbius function.

We consider the class of acyclic connected directed graphs with $N\geq 1$. In this paper we find the optimal upper bound for the number of paths amongst acyclic, connected graphs with $N$ edges. We prove that it is in fact optimal by finding an acyclic, connected graph with $N$ edges that realizes this bound. We then adapt these methods to find an optimal bound for Leavitt path algebras over a finite, acyclic, connected graph with $N$ edges.

In this paper, we introduce a new idea to study modules of logarithmic differential forms of hyperplane arrangements, which we call residue ideals. We first establish basic properties of these ideals, including their radicals and primary decompositions, and obtain applications for freeness of restrictions of arrangements. Then we apply these ideals to the study of modules of logarithmic differential $1$-forms for graphic arrangements. We give an explicit generating set for these modules and find a new connection to cover ideals of graphs studied in combinatorial commutative algebra. As a consequence we establish several new connections between arrangement theory and Stanley–Reisner theory.

Functional regressors complicate inference in linear regression problems so that the bootstrap can play a useful role in quantifying uncertainty and calibrating intervals. The best bootstrap in practice, though, can depend on factors in the data as well as computational considerations and existing bootstraps can have limitations: residual bootstrap is computationally fast and simple but may fail when the errors are heterogeneous, while paired bootstrap applies more generally in functional linear regression at a cost of much higher computation. To bridge this gap, we develop a wild bootstrap method for functional linear regression, which is akin to a modified version of residual bootstrap but designed to have a wide scope of application like paired bootstrap, including to heteroscedastic errors. Its theoretical consistency is established and numerical studies suggest that wild bootstrap can provide accurate and computationally fast inference. Importantly, we also suggest a practical and effective approach of selecting truncation levels, specifically designed for mean response inference problems. The proposed bootstrap in functional linear regression is further illustrated through a weather data example, and an accompanying R package BTSinFLRM provides numerical implementations.