Papers

We study the ambient-domain Hardy–Littlewood maximal operator \[ \mathcal M_\Omega f(x) := \sup_{B\ni x}\frac1{|B|}\int_{B\cap\Omega}|f(y)|\,dy, \qquad x\in\Omega \] on weighted Musielak–Orlicz spaces over a general open set \(\Omega\subset\mathbb R^n\), where the supremum is taken over all Euclidean balls \(B\subset\mathbb R^n\). For a Musielak–Orlicz function \(\varphi\), we use the pointwise lower Matuszewska–Orlicz index \(p_\varphi(\cdot)\) and the lower-index normalization \[ \psi_\varphi(x,t)=\varphi(x,t)^{1/p_\varphi(x)}. \] This factorizes the modular as a weighted variable-exponent modular applied to \(\psi_\varphi(x,|f|)\). Under endpoint lower growth, normalized weighted generalized Orlicz \((A0)\)–\((A2)\) assumptions and an admissible whole-space extension hypothesis for the weight at the lower-index exponent, we prove the boundedness of \[ \mathcal M_\Omega:L^{\varphi(\cdot)}_\omega(\Omega)\to L^{\varphi(\cdot)}_\omega(\Omega). \] For the converse direction we use the natural Köthe-associate ambient ball condition \(A_\varphi(\Omega)\). Under the local characteristic-function hypothesis, boundedness of \(\mathcal M_\Omega\) implies \(\omega\in A_\varphi(\Omega)\). On the whole space \(\mathbb R^n\), this framework provides a weighted characterization conditional on an associate-to-lower-index product reduction. In the present paper this reduction is verified for uniformly lower-index-power-equivalent models; it remains open for genuinely two-phase growth such as \(t^p+a(x)t^q\). As an application, we prove density of \(C_c^\infty(\mathbb R^n)\) in weighted Musielak–Orlicz–Sobolev spaces.

We study the separation of geometric effects from quantum interference in quantum optimization algorithms. Constrained optimization problems such as routing, assignment, and scheduling are often encoded as product spaces of local variables, together with global feasibility penalties. The central algorithmic question we address is how a constraint-preserving mixing operator transports quantum amplitude across an exponential search space in the presence of local and global constraints. We develop a framework that separates three effects that are usually intermixed: amplitude transport, coherent interference among transported amplitudes, and problem-dependent classical postprocessing. We show that the mixing operator alone does not have a target-seeking ability. Concretely, the normalized distribution induced by its amplitude transport moves toward the distance profile of a uniformly random configuration. Thus, quantum sampling advantage may only arise when the phases of the many computational paths reaching a target configuration are sufficiently aligned for their amplitudes to reinforce. We show that, when the cost phases are engineered so that these paths add coherently, a number of circuit alternations growing only logarithmically with problem size suffices to convert the sum of their absolute contributions into a lower bound on the target amplitude, yielding a certified success probability independent of the ambient Hilbert-space dimension, the search-space size, or the feasible-set cardinality. We develop applications to problem-specific transpilation diagnostics, scalable hardware probes, constraint-induced classical maps of quantum-generated samples, the attribution of solution quality between the quantum distribution and classical post-processing in hybrid quantum-classical workflows and connections to distance-partitioned product spaces from classical coding theory.

We study the existence and asymptotic behaviour of normalized solutions to the following pseudo-relativistic Schrödinger equation with logarithmic nonlinearity \[ (\sqrt{-\Delta+m^2 }-m )u+i(v\cdot \nabla )u+\lambda u = u\log|u|^2+|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N, \] under the mass constraint \[ \|u\|_2^2=a, \] where $m,a>0$, $2<p\le\frac{2N}{N-1}$ with $N\ge 2$, $v\in \mathbb{R}^N$ is the travelling velocity with $|v|<1$, and $\lambda\in\mathbb{R}$ appears as Lagrange multiplier, as minima of the corresponding energy on the constraint. By applying variational method, we first provide a complete classification of the existence and nonexistence of such minima. In particular, for the mass-critical case $p=2+\frac{2}{N}$, we show that there exists a constant $a^\ast_v$ which is a threshold for the existence. Based on this, we analyse the blow-up behaviour of such minimizers as $a$ approaches $a^\ast_v$ from below. Finally, we investigate the limiting profiles of minimizers to problem when $\lim\limits_{n\to\infty}a_n=a_0\in(0,+\infty)$ with $\{a_n\}\subset(0,+\infty)$ in the mass-subcritical case $2<p<2+\frac{2}{N}$ and $\lim\limits_{n\to\infty}a_n= a_0\in(0,a^\ast_v)$ with $\{a_n\}\subset(0,a^\ast_v)$ in the mass-critical case $p=2+\frac{2}{N}$, respectively.

Forecast value depends not only on accuracy but also on the information structure available to the operating model. We study a diagnostic current-endpoint-only forecast for offshore-wind thermal firming: at hour t, the controller observes the current net-demand state z_t and one perfect future target z_{t+h}, but not the intermediate path or earlier endpoint messages. Unlike rolling path forecasts, these endpoint information sets are not nested in h. We embed the signal in a cyclostationary MDP using quantile Fourier regression states estimated from ISO New England load and offshore wind data, and solve annual state-action-frequency LPs for h = 1, . . . , 6. A one-hour endpoint forecast reduces annual firming cost by 8.07%, while a six-hour endpoint reduces it by 2.28%. The decreasing profile shows that a single farther endpoint is less actionable for a one-step ramping decision, without implying that longer rolling forecasts are less valuable.

In this short note we prove that every system of proportionally modular numerical semigroups is representable by a canonical equivariant resolution of a weighted homogeneous surface singularity with rational homology sphere link. The construction starts from the quotient descriptions of proportionally modular numerical semigroups by two-generator numerical semigroups, realizes each quotient by a two-legged canonical equivariant resolution graph, and then glues these graphs with suitable multiplicities.

In this article, we introduce a weak Galerkin finite element method (WG-FEM) for a class of parabolic singularly perturbed boundary turning point problems (SPBTPPs). The proposed numerical scheme employs an implicit $\theta$-scheme for temporal discretization over a uniform mesh and applies WG-FEM spatial discretization on a layer-adapted Shishkin mesh. Rigorous stability estimates are established for both the semi-discrete and fully-discrete formulations. Furthermore, we derive error estimates in the energy norm and prove that the convergence of the scheme is uniform with respect to the perturbation parameter. Numerical tests are conducted to verify the theoretical findings and illustrate the efficiency of the proposed method. In addition, the theoretical framework developed in this work lays the foundation for future extensions to higher-dimensional problems using ADI-type operator splitting WG-FEM schemes.

Automatic scientific discovery has long been a goal of computational scholars - a machine that can discover nature's secrets on its own, moving computational systems beyond data-fitting tools toward the generation and refinement of mechanistic models of the universe. Recent advances in symbolic regression (SR) and large-language-model (LLM)-based agents suggest that such systems can recover equations from data, incorporate domain priors, and automate parts of the research workflow. However, most existing approaches either focus on narrow equation-discovery benchmarks or broad end-to-end automation pipelines, while biological systems remain comparatively underexplored. Here, we introduce the MEDA system, an LLM- and SR-powered agentic framework for discovering ordinary-differential-equation (ODE) models of biological and biologically inspired dynamical systems. MEDA retrieves background knowledge, defines admissible variables, generates mechanistic constraints, proposes candidate ODEs, and fits and evaluates them. We evaluate it across canonical model retrieval, reasoning-based extrapolation to unseen variants, and open-ended discovery, with and without experimental data. Across these settings, MEDA recovered the correct state variables, achieved strong structural recovery in retrieval and extrapolation tasks, and produced biologically plausible discovery-oriented models. Ablation and robustness analyses show that knowledge-guided formalization and mechanistic constraints are load-bearing components, whereas numerical fitting alone can preserve trajectory-compatible but biologically incorrect equations.

In this paper, we derive three generalized continued fractions of any even order $k$ with the aid of a general continued fraction identity of Ramanujan and we establish general theta function identities for these continued fractions. As an application of continued fraction of order seventy-six, we obtain partition theoretic identities and some vanishing coefficient results.

In this paper, we construct a wide class of new counterexamples to the generalized Zariski cancellation problem. The cylinders over these counterexamples have infinitely many non-conjugate maximal tori in their regular automorphism groups. We provide an example of a variety with maximal tori of different dimensions in its automorphism group. Additionally, we prove that a cylinder over a non-rigid trinomial variety without a line factor is generically flexible, and hence, has a trivial Makar-Limanov invariant.

In this paper, we provide a characterization of pairwise orthogonal frames with generalized translation-invariant (GTI) structures, based on the unconditional convergence property (UCP). These GTI systems are generated by translating functions over a countable family of closed, co-compact subgroups of a locally compact abelian (LCA) group $G$, where the families of subgroups associated with each system may differ. As an application of this characterization, we establish necessary and sufficient criteria for the orthogonality of various structured systems, including Gabor, wavelet, and shearlet systems on LCA groups. Furthermore, we derive a characterization of GTI Parseval (tight) frames and present explicit constructions of pairs of GTI systems using filters. Each constructed system satisfies the $\infty$-UCP and admits a Calderón sum equal to one. As a consequence of these results, the constructed systems form Parseval frames and are pairwise orthogonal. The proposed construction improves upon the technique in \cite{RGS} by relaxing the stationary assumption on the families of subgroups. Finally, we illustrate the results with examples using $B$-splines as generating functions and discuss applications of pairwise orthogonal frames in sampling theory.

A single-track Gray code is a cyclic Gray code with codewords of length $n$, over an alphabet of size $m$, such that all the $n$ tracks that correspond to the $n$ distinct coordinates of the codewords are cyclic shifts of the first track. Such codes have advantages over the conventional Gray codes in certain quantization and coding applications. Unless $n=2$, there are no such binary codes that contain all the $2^n$ codewords of length $n$. In this paper, we prove that such codes of length $p^t$, $n \geq 2$, with $p^{p^t}$ codewords, over $\F_p$, $p$ prime, exist, for $p=3$ and $p=5$. For larger prime $p$ an appropriate code for $t=2$, implies the existence of such a code for any $t >2$. If the alphabet size $m$ is not a prime there are also indications that such codes exist.

Fast and accurate classical simulation of quantum systems is a central challenge in the design and control of quantum computers, but the highly oscillatory dynamics of these systems severely limit the efficiency of standard numerical methods. To address this, we adapt Filon quadrature for oscillatory integrals into two numerical methods, called Filon and Controlled Filon, for solving linear systems of ODEs with highly oscillatory solutions. We tailor both methods for efficient implementation in controlled quantum systems, and the Controlled Filon method additionally accounts for the oscillatory structure of the control pulses. We show by numerical experiments that these methods significantly reduce the computational cost of accurately simulating systems of superconducting transmon qubits by decreasing the number of timesteps needed to reach a given level of precision, with only a modest increase in the cost per timestep. For a realistic simulation of the dynamics of a CNOT gate, the Controlled Filon method is the most efficient method tested at every target accuracy, outperforming the best Hermite method by up to 6x and the Hermite method of the same order by up to 500x.

We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Neumann data. We propose a weighted-residual a posteriori error estimator that is an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be parabolically scaled prismatic, i.e., their elements are tensor-products $J\times K$ of elements in time $J$ and space $K$ with $|J| \eqsim \text{diam}(K)^2$. In the considered numerical experiments on two-dimensional domains in space, an adaptive algorithm steered by the derived estimator yields significantly faster convergence compared to uniform refinement, achieving near-optimal rates even in the presence of strong singularities.

An eigenvalue of a graph is called main if its eigenspace is not orthogonal to the all-ones vector. Introduced by Cvetković in the early 1970s and systematically studied by Rowlinson and others, graphs with exactly one or two main eigenvalues are now well understood. However, the classification of graphs with precisely three main eigenvalues remains a challenging open problem in spectral graph theory. This paper provides a complete classification of all trees of diameter 5 with exactly three main eigenvalues. Using equitable partitions, the spectral condition reduces to the unique solvability of linear systems over the rationals, leading to Diophantine equations involving branch lengths and pendant counts. We prove that every such tree is isomorphic either to a symmetric tree $T_r(a)$ or to a member of a parametric family $\mathcal{T}$ determined by arithmetic divisibility conditions. We also construct an infinite family of such trees with unbounded diameter.

We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.

We study finite spin-$\tfrac12$ rings with nearest-neighbor Heisenberg exchange, Dzyaloshinsky-Moriya interaction, and an external magnetic field. We introduce a phenomenological interpolation parameter between the fully quantum Hamiltonian and a state-dependent mean-field description. Using dissipative Gisin-Schrödinger dynamics, we analyze the resulting quantum-classical crossover through local magnetization, connected spin correlations, single-site entropy, and the saturation field of the fully polarized state.

We provide a constructive existence proof for overhanging periodic gravity water waves with constant vorticity. By introducing a conformal mapping formulation, we parameterize the fluid domain to accommodate multi-valued surface heights, bypassing coordinate singularities that arise in traditional domain-flattening frameworks. We operate at fixed, macroscopic physical parameters and employ a global bifurcation framework, supplemented by a computer-assisted proof based on the Newton-Kantorovich theorem, to constructively prove the existence of the branch of exact solutions. Crucially, this allows us to provide the first rigorous existence proof of overhanging periodic gravity water waves obtained along a finite-depth global bifurcation branch from the flat state at fixed $O(1)$ physical parameters. Moreover, our analysis confirms the existence of a global solution branch that transitions to overhanging profiles and resolves the conjecture of Constantin, Strauss, and Varvaruca regarding the topological termination of this branch via physical self-intersection at the trough line.

For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems ($d\gg 10$), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, but often suffer from limited accuracy and efficiency. Recently proposed spectral-informed neural networks (SINNs) combine spectral methods with PINNs, operating directly in the spectral domain to avoid spatial derivative computations and to reduce memory consumption. In this work, we introduce Modified SINNs, which integrate coefficient decay scaling and basis embeddings motivated by harmonic analysis to enhance accuracy in high-dimensional problems and enable accurate approximation of unknown spectral coefficients. Numerical experiments on steady and time-dependent partial differential equations demonstrate that Modified SINNs outperform sparse grid spectral methods on middle-dimensional problems with incomplete spectral information and achieve superior accuracy compared to PINNs on high-dimensional problems.

This paper addresses the problem of running variable manipulation in Regression Discontinuity Designs. Leveraging the observation that manipulation often alters the density balance around the cutoff, we detect these structural imbalances using Benford's Law -a natural statistical regularity widely applied in fraud detection. Our framework serves as a vital precautionary safeguard alongside traditional McCrary-type tests. It eliminates researcher-chosen parameters that can skew outcomes, while delivering a deeper diagnostic breakdown of the density's behavior. Crucially, whereas the classic McCrary test can overlook systemic imbalances due to its rigid symmetric setup, our method separates the data into directional components. This allows researchers to pinpoint the exact origin of a deviation and spot hidden manipulation that standard frameworks fail to capture. To achieve this, we introduce an innovative method for selecting a bandwidth consistent with BL, and construct two distinct, complementary tests using threshold values adapted from Nigrini (2012) that successfully transition the law's application from digits to probabilities. Empirical applications confirm the enhanced protective value of this diagnostic framework.

Given a diffeomorphism preserving a one-dimensional expanding foliation $\mathcal F$ with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property. We then prove that for any measure of maximal $u$-entropy, its conditional measures on each leaf must be equivalent to the reference measures. When the measure of maximal $u$-entropy is a Gibbs $\mathcal F$-state (i.e., when the reference measures are equivalent to the leafwise Lebesgue measure), we prove that the log-Jacobian of $f$ must be cohomologous to a constant via a measurable function. We provide several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature.

As quantum computing technology continues to mature, the US National Institute of Standards and Technology (NIST) has outlined a migration timeline for Post-Quantum Cryptography (PQC), recommending the deprecation of certain elliptic curve cryptography (ECC) by 2030. Furthermore, privacy-sensitive application scenarios, such as vehicular communications, require the use of anonymous certificates. However, existing anonymous certificate schemes are still largely based on ECC. Therefore, this study proposes a multivariate cryptography-based anonymous certificate scheme, aiming to design quantum-safe anonymous certificates suitable for privacy-sensitive application services. The proposed multivariate cryptography-based anonymous certificate scheme is supported by rigorous mathematical proofs and illustrated with computational cases.

Tensor randomized Kaczmarz (TRK) methods are efficient row-action solvers for tensor linear systems under the t-product framework. We study their behavior under a doubly noisy perturbation model. In this model, both the system tensor and the right-hand side tensor are corrupted. We first analyze standard TRK and derive an expected error recursion with two terms. One term is contractive, and the other is a persistent perturbation term. This explains the noise-limited and semi-convergent behavior that can occur when the observed tensor system is inconsistent. We then introduce a spectrally damped tensor randomized Kaczmarz method (SD-TRK). We prove an expected error recursion for SD-TRK that separates error propagation from noise injection. The bound makes explicit a speed-robustness trade-off. We also give an FFT-based implementation that applies the damped update slice-wise in the Fourier domain. This implementation allows frequency-dependent damping parameters in practice. Numerical experiments on synthetic tensor systems illustrate the stabilization behavior of SD-TRK relative to standard TRK in noisy and ill-conditioned settings. We also include a two-pass image reconstruction comparison under the same noisy reconstruction pipeline.

We study an infinite horizon stochastic optimal control problem by means of the associated Hamilton-Jacobi-Bellman equation. The problem we are studying has the particularity of having a discount factor which can take both signs, depending on the value of the state.

In this paper, we investigate Gabriel's and Frazer's problems for analytic and complex-valued harmonic weighted Bergman spaces. More precisely, we establish weighted integral inequalities of the form \[ \int_C |f(z)|^p(1-|z|^2)^{\alpha+1}\,|dz| \leq K_{p,\alpha,C} \int_{\mathbb D}|f(z)|^p(1-|z|^2)^\alpha\,dA(z), \] where $f$ is an analytic or complex-valued harmonic function on the unit disk $\mathbb D$ and $C$ is an arbitrary convex curve contained in $\mathbb{D}$. The corresponding problem was first studied by Gabriel [Proc. Lond. Math. Soc. 28 (1928), 121–127] for analytic Hardy spaces, where the inequality holds for every $0<p<\infty$. In contrast, the harmonic Hardy space analogue was recently shown to fail whenever $0<p\le1$. We prove that this phenomenon does not occur in the weighted harmonic Bergman setting by establishing Gabriel's inequality throughout the full range $0<p<\infty$. We further study Frazer's problem for circles and for the union of two intersecting diameters. As important applications of our main results, we derive Gabriel-type and Frazer-type inequalities for the analytic and harmonic Möbius invariant spaces $Q(n,p,\alpha)$ and $Q_h(n,p,\alpha)$.

Ilango (FOCS 2025) invented effectively zero-knowledge proofs, a new variant of zero-knowledge. We reformulate it in the language of logic and give simple proofs (under the same assumptions as Ilango (FOCS 2025)) of its existence and of the key property defined in Ilango (FOCS 2025) that it is "indistinguishable from true" (that property is in Ilango (FOCS 2025) a part of the definition of the prover, not its consequence). Using the theory of proof complexity generators we show that the concept can be turned it into a genuinely zero-knowledge proofs, assuming a conjecture from the theory about the existence of a hard generator and allowing the parties to share a common random string.

Let $\emph{spex}_{\mathcal{P}}(n,F)$ and $\emph{spex}_{\mathcal{OP}}(n,F)$ be the maximum spectral radius among all $n$-vertex $F$-free planar graphs and outerplanar graphs, respectively. Define $C_{k,l}$ as a graph obtained from $C_k \cup C_l$ such that the two cycles share a common vertex, where $l \ge k \ge 3$. In the 1990s, Cvetković and Rowlinson conjectured $K_1 + P_{n-1}$ maximizes spectral radius in outerplanar graphs on $n$ vertices, while Boots and Royle (independently, Cao and Vince) conjectured $K_2 + P_{n-2} $ does so in planar graphs. Tait and Tobin [J. Combin. Theory Ser. B, 2017] determined the fundamental structure as the key to confirming these two conjectures for sufficiently large $n$. Recently, Yin and Li [Discrete Mathematics, 2026] characterized the extremal graphs for $\emph{spex}_{\mathcal{P}}(n,B_{t,l})$ and $\emph{spex}_{\mathcal{OP}}(n,B_{t,l})$ in planar and outerplanar graphs on the basis of this key idea, where $B_{t,l}$ denotes the graph obtained by $t$ edge-disjoint $l$-cycles sharing a common vertex. In this paper, we focus on planar and outerplanar graphs without $C_{k,l}$, and determine $\emph{spex}_{\mathcal{P}}(n,C_{k,l})$ and $\emph{spex}_{\mathcal{OP}}(n,C_{k,l})$ along with their unique extremal graphs for all $l \geq k \geq 3$ and large $n$.

We give an explicit formula for the order of the rational cuspidal class group of the modular curve $X_1(N)$ for an arbitrary integer $N$. The proof relies on results of Streng on the group of modular units on $X_1(N)$, and requires computing a certain determinant involving the second Bernoulli polynomial. We also define a higher weight analogue of the cuspidal class group and speculate that its order is related to a similar determinant defined using a higher degree Bernoulli polynomial.

We show that a pivoting strategy due to Stewart (based on work by Bischof) computes a strong rank-revealing factorization when applied to a matrix with orthonormal rows. When paired with the classical column selection algorithm of Golub, Klema, and Stewart (GKS) it helps achieve rank-$k$ approximation accuracy bounds and basis conditioning as good as those from applying a strong rank-revealing factorization directly to A. We then extend this framework in two directions: (1) providing analysis of GKS when only approximations of right singular vectors are available and (2) providing a randomized variant of the pivoting strategy for matrices with orthonormal rows that achieves the same theoretical guarantees but can return the desired subset two orders of magnitude faster than the deterministic variant.

For $C^{1+\mathrm{Dini}}$ diffeomorphisms, we prove a Ledrappier–Young-type characterization of SRB measures among invariant measures whose supports admit dominated splittings. When the dominating bundle has only positive Lyapunov exponents, absolute continuity of the conditional measures on the corresponding Pesin unstable manifolds is equivalent to the partial Pesin entropy formula. As an application, we obtain SRB measures for partially hyperbolic mostly expanding attractors in the $C^{1+\mathrm{Dini}}$ category. Counterexamples are provided to show that these conclusions fail in the $C^1$ category, even under uniform hyperbolicity.

This thesis develops the mathematical theory of intuitionistic dynamic logics - extensions of intuitionistic propositional logic with modalities and fixed point operators. Such systems provide formal tools for reasoning about change, such as encountered in mathematical systems evolving over time or in the knowledge state of an agent after an information update. We investigate five intuitionistic dynamic logics: intuitionistic master modality, intuitionistic common knowledge logic, intuitionistic linear temporal logic, bi-intuitionistic modal logic and bi-intuitionistic linear temporal logic. On the proof theoretic side we develop sound and complete Hilbert-style axiomatizations as well as non-wellfounded and cyclic sequent calculi. On the semantic side we study these logics over various classes of dynamic models, which are birelational Kripke models satisfying confluence and frame conditions. We establish expressivity results, the finite model property, decidability, as well as complexity bounds. The main contributions are threefold. First, we develop analytic cyclic sequent calculi for intuitionistic master modality and common knowledge logic, where completeness is obtained by a robust proof search argument. Second, we obtain the finite model property and decidability for bi-intuitionistic modal logic via an intricate combinatorial analysis of dynamic models. Third, we develop a sound and complete axiomatization for intuitionistic linear temporal logic featuring the temporal operators next, eventually and henceforth, thereby providing a positive answer to the long-standing open question concerning the existence of a finite axiomatization.