Papers

We extend Gårding's theorem to homogeneous posynomials: if a finite positive sum of monomials with arbitrary nonnegative real exponents is zero-free on a product of right half-planes, then its degree-normalized root is concave. Consequently, zero-freeness in a sector of aperture $\alpha\pi$ implies $\alpha$-fractional log-concavity. This sharpens generic mixing and domain-sparsification guarantees for fixed-size matchings and nonsymmetric determinantal point processes. The result was developed in an AI-assisted interaction initiated and checked by the author; Codex also assisted with assembling and typesetting the manuscript.

Schedule-Free methods have attracted growing interest for alleviating the burden of designing and tuning a learning rate scheduler, while matching and sometimes even outperforming optimizers with tuned schedulers. Despite their strong empirical results, their convergence theory in nonconvex optimization, where modern machine learning objectives typically arise, has remained largely unexplored. In this paper, we provide worst-case analyses of Schedule-Free gradient descent and Schedule-Free stochastic gradient descent, in their standard form and without auxiliary modifications or restrictive conditions, for smooth but possibly nonconvex objectives. Based on a Lyapunov analysis derived from the continuous-time limiting ordinary differential equation associated with these methods, we show that Schedule-Free gradient descent and Schedule-Free stochastic gradient descent achieve the optimal worst-case convergence rates attainable among first-order methods. We further formulate Schedule-Free gradient descent as a nonautonomous dynamical system and prove strict-saddle avoidance under an arbitrarily small one-time perturbation. These theoretical results provide a better understanding of the strong performance that Schedule-Free methods demonstrate.

Let $p_t(N)$ denote the number of $t$-multipartitions of a positive integer $N$. In this article, we obtain an asymptotic formula for $p_t(N)$, when $t \ll N^{1 - \epsilon}$, for any $\epsilon > 0$.

In this paper, we introduce multiple Schur series, which are defined by Schur-type sums over semi-standard Young tableaux and generalize both Schur multiple zeta values and multiple Eisenstein series. To study their algebraic structure, we construct a connected, commutative, graded Hopf algebra of Young tableaux and identify its linearized quotient with the quasi-shuffle algebra. Within this Hopf algebra and its quotient, we establish several relations, including a hook formula and the Jacobi–Trudi formula. Furthermore, we relate this Hopf algebra to the ring of symmetric functions, which yields polynomial reduction formulas for tableaux with constant entries. As applications, we recover Schur multiple zeta values, introduce Schur multiple Eisenstein series together with a $q$-analogue of Schur multiple zeta values, and discuss their (quasi)modularity.

Let $A$ be a bounded accretive operator on a Hilbert space and $f(t)$ an operator monotone function on $(0, \infty)$ with $f(0)>-\infty$. Then, for $\epsilon >0$, analytic function $f (A+\epsilon I) $ is defined by Riesz-Dunford integral. We define $f(A)$ as the norm limit of it and show $$ f(A) = f(0)I + b A + \int_0^{\infty} (\frac{1}{\lambda} I - (\lambda I + A)^{-1}) d\mu(\lambda).$$ This is a generalization of fractional powers $$A^r = \frac{\sin r \pi}{\pi} \int_0^{\infty} (\frac{1}{\lambda} I - (\lambda I + A)^{-1}) \lambda ^{r} d\lambda \quad (0<r<1).$$ Let $A$ and $B$ be strictly accretive matrices, namely those real parts are positive definite. The geometric mean $A\# B$ has been introduced in Drury[6] and subsequently general matrix mean $A\sigma_f B$ in Bedrani-Kittaneh-Sababheh [3]. We extend these means to accretive, not necessarily strictly accretive, operators $A$ and $B$, and verify that $$A\# B= A^{1/2} B^{1/2}$$ if $A$ and $B$ are normal and commutative. Let $A$ be a strictly accretive operator. Then we show that $$0 \leqq \frac{1}{2} (A + A^*) \leqq A \# A^* \leqq 2(A^{-1} + (A^*)^{-1})^{-1},$$ and that $A \# A^* = | A |$ if and only if $A$ is normal. For a normal and strictly accretive operator $A$ we get \begin{align*} &|A|= \frac{1}{\pi}\int_0^{\infty}A (\lambda A + A^*)^{-1} A^* \lambda^{-1/2} d \lambda, \\ &A + A^* \leqq A^{1-r} A^{*r} + A^r A^{*(1-r)} \quad (0\leqq r \leqq 1). \end{align*}

We prove asymptotic formulae for $(p,q)$-Rogers–Szegő polynomials with positive real parameters when the leading quadratic term in the logarithmic expansion about the middle coefficient vanishes. For coefficient indices whose distance from $n/2$ is of order $n^{3/4}$, the coefficient ratios have a quadratic–quartic exponential limit. A uniform bound valid for all indices allows the ratios to be summed and gives locally uniform convergence of the centred and normalised generating polynomials to a Fourier–Laplace integral. We also obtain an asymptotic formula for the sum of the coefficients, convergence of all rescaled moments, weak convergence of the normalised coefficient measures, and convergence with multiplicity of the zeros near $w=1$ for which $n^{3/4}\log w$ remains bounded.

Seismic surface waves, particularly low-frequency Rayleigh waves, are notoriously destructive and remain difficult to control using conventional engineering methods. Recent advances in elastic metamaterials have opened new avenues for manipulating and guiding seismic waves at subwavelength scales. In this work, we present a rigorous mathematical study of subwavelength resonances and seismic-wave localization in arbitrarily shaped Matryoshka-type elastic metamaterials, which are composed of nested high-contrast resonators made of materials much stiffer than the background medium. By employing the displacement-to-traction map and a variational formulation, we derive necessary and sufficient conditions that characterize subwavelength resonances. Using the Gohberg–Sigal theory and Puiseux series of multivalued algebraic functions, we establish the existence of subwavelength resonances and obtain their asymptotic characterization in terms of the eigensystem of the generalized stiffness tensor, which serves as the elastic analogue of the capacitance matrix in the celebrated Minnaert acoustic-cavitation systems. Furthermore, in the concentric spherical case, the stiffness tensor exhibits a block-diagonal structure separating translational and rotational components, with each block being tridiagonal and possessing positive, simple eigenvalues. Based on this matrix formulation,we rigorously demonstrate that, for structures with a sufficiently large number of layers, ring defects in layered subwavelength resonators can act as effective radially laminated seismic waveguides, supporting both wave localization and guided propagation along the defects. In particular, appropriately placed ring defects can induce eigenmodes that are exponentially localized at multiple defect sites simultaneously,with precisely quantified amplitude ratios.

Let $p$ be an odd prime, and let $G$ be a finite group whose order is divisible by $p$. Suppose that every proper subgroup of $G$ whose order is divisible by $p$ is supersolvable, while $G$ itself is not. In this paper, we investigate the arithmetic and structural properties of such groups, addressing both the solvable and nonsolvable cases.

We study the optimal transport of optimally controlled agents from a compactly supported absolutely continuous source to a discrete target measure. The ground cost for the transport is induced by the optimal cost of the agents' motion. When this ground cost satisfies the twist condition, the optimal transport map is given almost everywhere in terms of a Laguerre tessellation of the state space. We refer to this control-theoretic generalization of Laguerre tessellation as Control Laguerre Tessellation (CLT), and illustrate it for two ground costs induced by linear controlled agents with minimum energy and minimum time objectives.

In this note, we study distortion risk measures of step-weighted distribution.

We study conditions under which stability of the origin of stochastic differential equations is robust to small perturbations. We express robustness in two ways, firstly in the sense that stochastic stability is maintained under small parametric perturbations not exceeding a state-dependent bound vanishing at the origin but positive elsewhere, and secondly via stochastic input-to-state stability (ISS) which allows non-zero perturbations everywhere. We prove the former property assuming the existence of a Lyapunov function certifying stochastic stability of the nominal system. Under the same assumption, stochastic ISS holds under a suitable state-dependent perturbation scaling. Stochastic exponential stability is maintained under proportionally bounded perturbations and implies exponential ISS even without perturbation scaling. Finally, we propose a novel approach to stochastic integrator backstepping in pure-feedback form that uses the tools from our robustness analysis.

The main focus of this paper is on the algebraic structure of two-dimensional $(\lambda,\mu)$-constacyclic codes of length $\ell\mathrm{m}$ over finite chain rings with residue field $\mathbb{F}_q$, where $q \equiv 1 \pmod{r\mathrm{m}}$ and $r$ denotes the multiplicative order of $\bar{\mu}$. In this paper, the structure of two-dimensional $(\lambda,\mu)$-constacyclic codes is obtained. Our approach relies on analysing primitive idempotents within the finite chain ring to determine the generators of these codes. We also find the condition under which two-dimensional constacyclic codes are maximum Hamming distance with respect to rank (MHDR) over finite chain rings.

Recalling that the so-called Elliott-Halberstam conjecture twisted by the Möbius function $\mu(n)$ claims that \[ \sum_{q\leq N^{\theta}}\max_{y\leq N}\max_{(a,q)=1}\left|\sum_{\underset{{\scriptstyle n\equiv a\,\mod\,q}}{n\leq y}}\Lambda(n)\mu\left(N-n\right)-\frac{1}{\varphi\left(q\right)}\sum_{n\leq y}\Lambda(n)\mu\left(N-n\right)\right|\ll\frac{N}{\log\left(N\right)^{A}} \] for every $A>0$, where $0<\theta<1$ is fixed, and also recalling that the validity of this conjecture, in combination with the validity of the classical Elliott-Halberstam for suitable $\theta$, proves the binary Goldbach conjecture, in this paper we study weighted average variants of this problem. We will show that, under Generalized Riemann Hypothesis, a weak version of the Gonek-Hejhal conjecture and working with weights belonging to the Sobolev space $W^{2,1}$ or in the Hölder-Zygmund spaces $\mathcal{C}^{\delta}$ for suitable range of $\delta$, the bound of the average is consistent with the bound of the “diagonal versions” of this conjecture (that is, taking $y=N$ and taking $n\equiv N\mod q)$. In particular, in the case of weights in Sobolev space, the consistent upper bound holds for the whole $0<\theta<1$ and, in the case of weights in the Hölder-Zygmund class $\mathcal{C}^{\delta}$, for $\theta$ that depends on the choice of $\delta$ but still not below the $1/2-2\varepsilon$ threshold.

We compute the magnetic curvatures (in the sense of Assenza) of left-invariant magnetic systems on Lie groups and explore relations between curvature properties and algebraic properties, extending to the magnetic setting a number of results from Milnor's classic 1976 paper on the geometry of left-invariant metrics. We also discuss the existence of non-trivial bi-invariant magnetic systems and exhibit the resulting curvature formulas in some concrete examples, including the Heisenberg group.

The logit dynamic, a nonlinear dynamical system, subject to a const constraint with heterogeneous agents is formulated, and its well-posedness is studied from both agent-based (stochastic differential equation: SDE) and probabilistic (Fokker-Planck equation: FPE) viewpoints. The state space of agent actions is a compact domain in a finite-dimensional Euclidean space. The SDE is of the McKean-Vlasov type and is driven by jumps with finite variations, whereas the FPE is a nonlinear integro-differential equation. A key to our mathematical analysis of the FPE is adding a regularization factor into the cost constraint to mitigate the blow-up of the logit function. This property carries over to the McKean-Vlasov SDE. We also present a numerical method based on a naïve finite difference discretization for computing the FPE along with demonstrative computational examples, showing that the regularization method does not critically affect numerical solutions while preventing the breakdown of numerical computation. Finally, we conduct another demonstrative application study in which environmental, energy, and fishery resources intersect.

We solve the equation $\sum\limits_{j=1}^{k}jU_{j}(x,y)^{p}=U_{n}(x,y)^{q}$ positive integers $x,p,q,k,n$, with $y=\pm1$ and $\max\{p,q\}\leq11$, where $U_{m}(x,y)=\frac{\alpha^{m}-\beta^{m}}{\alpha-\beta}$ for $\alpha$ and $\beta$ roots of the polynomial $t^2-xt+y$. This generalizes existing results on similar equations, wherein the sequence was fixed as either the Fibonacci or Pell numbers. In addition, we find all solutions with $k=2$ and $y=\pm1$.

We study stochastic fixed-point equations $\mathbf{T}(\mathbf{x}) = \mathbf{x}$ over normed spaces $(\mathcal{E}, \|\cdot\|)$, where the operator $\mathbf{T}$ is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given $\epsilon > 0, \delta \in (0, 1)$, the goal is to output $\mathbf{x} \in \mathcal{E}$ such that $\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq \epsilon$ with probability at least $1-\delta$. We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping $\tau(\mathbf{x}; \xi)$ itself, we clip stochastic differences at the Lipschitz scale $\gamma\|\mathbf{x} - \mathbf{y}\|$. This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound: on a single event of probability at least $1 - \delta$, the residual decreases nearly geometrically across epochs, up to lower-order logarithmic factors. Under only bounded variance, displaying only the dependence on the target error $\epsilon$ and Lipschitz constant $\gamma \in (0, 1]$ of $\mathbf{T}$, the resulting oracle complexity is $\min\{\epsilon^{-5}, (1-\gamma)^{-3}\epsilon^{-2}\}$. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding $\epsilon^{-3}$ nonexpansive rate (i.e., for $\gamma = 1$), and under samplewise nonexpansiveness to $\epsilon^{-2}$.

As fusion energy moves from theoretical feasibility toward commercialization, design of new reactor concepts, autonomous tokamak control, and high-performance scenario optimization are becoming increasingly important. Traditionally, such optimization tasks have relied on costly trial-and-error or brute-force parameter searches, based on black-box experiments or simulations. Recently, advances in differentiable programming are changing the paradigm of numerical simulation. Unlike conventional simulations, which are typically executed as locally connected step-by-step procedures, differentiable simulation represents the entire simulation pipeline as a connected computational graph. In such a framework, machine parameters, actuator waveforms, and plasma responses are linked through differentiable operations, allowing Jacobians to propagate across the full simulation. This enables direct gradient-based control and optimization using the internal sensitivities of the simulator, rather than treating the simulator as a black box. Here, we present TokaGrad, an end-to-end differentiable tokamak transport simulator for full-scenario modeling, including ramp-up, L-mode operation, and H-mode access. TokaGrad self-consistently integrates differentiable models for plasma equilibrium, transport, heating, L-H transition, and pedestal formation. To our knowledge, this is the first differentiable tokamak simulator capable of self-consistently modeling dynamic full-discharge scenarios where actuators and plasma evolve together with equilibrium, pedestal, and confinement-regime transitions. We demonstrate that, when coupled to gradient-based optimizers, TokaGrad enables reactor-design optimization, actuator control, and full-scenario waveform optimization. This framework provides a pathway toward automated, differentiable optimization of burning-plasma scenarios and reactor concepts.

This paper studies graph matching under the correlated $\text{Erd\H{o}s-R\'{e}nyi}$ (ER) graph pair model. This model first samples an $\mathrm{ER}(n,\frac{\lambda}{ns})$ base graph, whose edges are then independently subsampled twice with probability $s$ to produce two correlated $\mathrm{ER}(n,\frac{\lambda}{n})$ graphs. We propose a graph matching algorithm that has $n^{2+o(1)}$ time complexity and achieves almost exact recovery with high probability under the assumptions $\lambda=(\log n)^{\alpha+o(1)}$ for some $\alpha\in(0,1)$ and $s\in(\sqrt{C_{\mathrm{Otter}}},1]$, where $C_{\mathrm{Otter}}\approx 0.338$ is Otter's tree-counting constant. This is the first algorithm with almost quadratic time complexity in this regime of $\lambda$, while the best known result in this regime is the chandelier-counting algorithm with time complexity $O(n^{c(s)})$, where $c(s)\rightarrow \infty$ as $s$ approaches $\sqrt{C_\mathrm{Otter}}$ from above. The proposed algorithm is based on local tree correlation tests. It uses a rank-based algorithm to match the vertex pairs instead of threshold-based rules in the literature. This avoids the need of computing an explicit threshold, which is computationally difficult to obtain. To prove the almost exact recovery result, we establish a new analysis of tree correlation tests in the diverging-degree regime, where both the mean degree and the tree depth grow with $n$. Based on this new result, we establish the existence of a threshold for a threshold-based graph matching algorithm via local tree correlation tests. Finally, we couple the performance of the rank-based algorithm with the threshold-based algorithm to show almost exact recovery.

We prove lossless inhomogeneous Strichartz estimates for solutions to the Schrödinger equation on compact manifolds with nonpositive curvature over frequency-dependent time intervals of length $\log \lambda \cdot \lambda^{-1} $. As applications, we improve upon the Sobolev norm growth bounds for the cubic NLS established by Planchon, Tzvetkov and Visciglia on 3-dimensional compact manifolds when the manifold also has nonpositive curvature and extend the lossless homogeneous Strichartz estimates on logarithmic time intervals established by the first author and Sogge to Schrödinger operators with critically singular potentials.

We show that one of the main cases of the stable forking conjecture, stability of the forking relation over a base in a finite-rank supersimple theory, is determined by finitely many pregeometries in each rank. This case of the stable forking conjecture has long had an implicitly well-known pregeometric interpretation: there is a set of matroids $\mathcal{G}_{n}$ such that the forking instability in rank $n$ is equivalent to the pregeometry on some rank-one partial type (over a finite set) embedding a matroid in $\mathcal{G}_{n}$. Our contribution is to show that this set of matroids $\mathcal{G}_{n}$, determining based forking stability in rank $n$, can be chosen to be finite. The main part of our proof was already accomplished in rank $3$ by Peretz, but does not extend as stated to higher ranks (and may or may not directly extend in a weaker sense to higher ranks, by shrinking terms). However, we obtain a sufficient substitute for Peretz's work in ranks $n > 3$: we turn Peretz's original universal result into an existence theorem. The rest of our proof refines an argument from multi-experiment parameter definability, originating from work in applied model theory by Li, Meshkat, Ovchinnikov, Pillay, Pogudin and Scanlon.

Reconstructing the location and shape of an unknown impenetrable scatterer from far-field measurements is a fundamental inverse problem in elastic scattering. In this paper, we propose monotonicity-based shape characterization theorems and develop corresponding algorithms for rigid and traction-free impenetrable scatterers. By establishing the factorization of the elastic far-field operator and constructing localized wave functions, we derive a sharp monotonicity-based characterization criterion for determining the shape and position of the impenetrable scatterer. This criterion is based on the spectral properties of the monotonicity operator, defined as a specific linear combination of the far-field and Herglotz probing operators. Building on this theoretical foundation, we first present a counting-based monotonicity sampling method that evaluates the number of negative eigenvalues of the monotonicity operator. To address the inherent sensitivity of eigenvalue-counting to measurement noise, we further develop two novel monotonicity spectral sampling algorithms that exploit the magnitudes, rather than merely the signs, of the negative eigenvalues. The single-frequency monotonicity spectral sampling method provides robust stability against data perturbations, while the multi-frequency monotonicity spectral sampling method extension aggregates frequency information into a multiscale indicator that balances noise robustness with high-resolution geometric fidelity. Numerical experiments across various scatterer geometries and noise levels demonstrate sharp boundary localization and accurate reconstruction of complex concave features, confirming the effectiveness of the single-frequency and multi-frequency monotonicity spectral sampling methods.

In this paper, we prove an asymptotic formula for the moment of the first order derivative of modular $L$-functions at the center of the critical strip, weighted by generalized divisor functions formed by primitive quadratic characters. Such moments were previously studied by Munshi, which naturally arise in the study of elliptic fibration. To the best of our knowledge, such asymptotic formulae have only been proven in the setting of higher-order derivatives, or under the specialization to dihedral forms.

This work studies the asymptotic stability of solutions to the good Boussinesq equation in dispersive wave region when the reflection coefficients associated with the initial data belong to weighted Sobolev space. The Dbar-steepest descent method is applied to the Riemann-Hilbert problem and the long-time asymptotic expansion of the solution are obtained up to an optimal error of order $\mathcal{O}(t^{-3/4})$. Compared with previous results, we extend the initial data from the rapidly decaying Schwartz space to a weighted Sobolev space, and prove the asymptotic stability of the solution in dispersive wave region.

A homogeneous set in a graph is a clique or a stable set. The Erdős–Hajnal conjecture states that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a homogeneous set of size at least $n^c$. Nguyen, Scott and Seymour proved that for every $d>0$, graphs of VC-dimension at most $d$ have the Erdős–Hajnal property, confirming a conjecture of Fox, Pach and Suk. In particular, they showed that every such $n$-vertex graph contains a homogeneous set of size at least $n^{\eta_d}$ for some $\eta_d\ge 2^{-2^{O(d)}}$. In this paper, we give a sharper quantitative bound on the homogeneous sets in graphs of VC-dimension at most $d$, showing that one may take $ \eta_d\ge (Cd)^{-d}, $ where $C$ is an absolute constant. Equivalently, every graph $G$ of VC-dimension at most $d$ satisfies \[ \max\{\omega(G),\alpha(G)\}\ge |G|^{(Cd)^{-d}}. \] Our proof refines the iterative sparsification method of Nguyen, Scott and Seymour. The main enhancement is to apply the VC-dimension assumption directly, which gives a more efficient induction and thus improves the dependence on $d$. We also derive quantitative consequences for polynomial Rödl subgraphs, hypergraph Ramsey bounds under bounded VC-dimension, induced-free and viral formulations, tournaments, NIP and semi-algebraic graphs, Boolean combinations of relations of bounded VC-dimension, graphs whose adjacency matrices have bounded rank, graphs of bounded sign-rank, and graphs defined by dot-product threshold representations.

We derive exact analytic representations for the perimeter of a Lamé superellipse of degree $s>0$. The result is expressed in terms of two branches defined by series whose terms are Gauss hypergeometric functions: a negative branch for $0<s<1$ and a positive branch for $s>1$. For the positive branch, the convergence condition follows from the Leibniz test; the negative branch, although divergent in the ordinary sense, is shown to be Abel-summable. Consistently with the symmetry under interchange of the semi-axes, the formula is invariant under axis permutation. As $s$ varies, the family interpolates between the Lamé cross and the rectangle, while the case $s=1$ corresponds to the rhombus, which acts as the transition curve with the shortest perimeter within the family.

In thsi paper, based on the leap-frog discretization in time and the mixed virtual element discretization in space, we developed a linearized and structure-preserving numerical algorithm. The main contributions of this work lie in that we not only provide a rigorous proof of the energy dissipation property of the fully discrete numerical scheme, but also establish the unconditionally optimal convergence analysis by means of a inverse inequality. The core of the proof lies in the classified discussion of the relationship between \(\tau\) and $h$. Finally, two numerical examples are provided to validate the correctness of the theoretical analysis as well as the energy dissipation property of the proposed scheme.

To provide multiple-satellite coverage for global Internet of Things (IoT), a low Earth orbit (LEO) satellite IoT constellation usually contains multiple-layer orbits with different altitudes. However, the performance of multiple-layer LEO satellite IoT constellations under practical Rician fading satellite channels remains unknown due to complex theoretical modeling and intractable mathematical analysis. To address these challenges, this paper proposes a stochastic geometry-based modeling and analysis framework for multiple-layer LEO satellite IoT constellations, integrating Rician channel modeling and Cox point processes. Specifically, we introduce a novel channel approximation method to overcome the intractable expressions caused by the Rician fading. Building on this method, we derive exact closed-form expressions for key performance metrics, including connectivity probability, coverage probability, and transmission rate, especially in the case of IoT short-packet transmission. Extensive simulation results validate the accuracy and effectiveness of the proposed model and reveal significant design insights. The results not only provide new theoretical perspectives for modeling and analysis of LEO satellite IoT constellations but also offer practical guidance for system deployment and optimization.

We prove global and local upper bounds for the Hessian matrices of positive solutions to the heat equation on Kähler manifolds whose bisectional curvature is bounded from below. We also improve a result of Han and Zhang by weakening the curvature assumptions in their Hessian estimates on Riemannian manifolds. More precisely, we extend their global and local upper bounds, originally obtained under two-sided curvature bounds, to Riemannian manifolds with sectional curvature bounded from below.

We study pathwise well-posedness of the stochastic heat equation (SHE) with a multiplicative noise on the circle. By combining the convolution Young and rough integration theory, introduced by Gubinelli and Tindel (2010), with the random tensor estimate approach to pathwise well-posedness of stochastic dispersive PDEs with multiplicative noises, introduced by Chapouto and the second and third authors (2026), we establish pathwise well-posedness of SHE in both the Young and rough cases, improving the results in Gubinelli and Tindel (2010). In particular, in the rough case (= the white-in-time case), our result covers the case of almost space-time white noise, thus establishing an optimal result within the framework of one-parameter rough paths.