Papers

We study a nonlinear size-structured transport model with distributed harvesting and prescribed recruitment flux, where environmental feedback is determined by a scalar population functional. After establishing global well-posedness on $L^1$, we reduce stationary equilibria to a scalar closure equation. This reduction reveals that loss of equilibrium uniqueness occurs through a generic fold, mathematically characterizing critical population transitions. On uniformly nonresonant equilibrium branches, we prove the existence of optimal stationary harvesting policies via the direct method of the calculus of variations. We then derive a boundary-corrected adjoint equation and establish an identity equating equilibrium sensitivity with the adjoint loop gain. This relation yields explicit criteria for the persistence and creation of optimal harvesting thresholds. Collectively, these results provide a unified analytical framework connecting environmental feedback, equilibrium structure, and optimal harvesting.

Let $\pi=(<_D,<_A)$ be a pair of distinct orders on a $k$-letter alphabet $A. $ The periodic trajectories $v_i^\infty $ of a discrete $k$-interval exchange transformations $T$ with permutation $\pi$ are characterized by the following order symmetry : $v_i^\omega <_D v_j^\omega$ (lexicographically) if and only if $v_i^{-\omega}<_Av_j^{-\omega}$ (reverse lexicographically). For general words $u$ and $v$ over $A$, the orders need not agree in which case either $u^\omega<_A v^\omega<_D u^\omega$ (Type 1) or $v^\omega<_A u^\omega<_D v^\omega$ (Type 2). We partition all such order crossings amongst the set of conjugates of two words $u$ and $v$ into disjoint families $T_1(u,v)$ and $T_2(u,v)$ and define the index $i(u,v)$ by $|T_1(u,v)|+|T_2(u,v)|.$ Remarkably the difference, $|T_2(u,v)|-|T_1(u,v)|,$ depends only on the Parikh vectors $\lambda(u)$ and $\lambda(v).$ We show that $|T_2(u,v)|-|T_1(u,v)|=\lambda(u)^T \Omega \lambda(v)$ where $\Omega$ is a $k\times k$ skew symmetric matrix depending only on $\pi.$ It follows that the Parikh vectors of the trajectories of a discrete interval exchange are orthogonal with respect to $\Omega.$ Applied to dimension $3,$ we obtain an arithmetic formula for the number of orbits in a discrete $3$-interval exchange and hence a characterization of minimality. For general $k,$ the orthogonality of the trajectories gives an upper bound $\lfloor \frac{k+d}2 \rfloor$ on the number of distinct trajectories where $d=\dim \ker (\Omega).$ If $T$ is symmetric, then the number of distinct trajectories is at most$\lfloor \frac{k+1}2 \rfloor.$ An alternate interpretation of this result is that on an ordered $k$-letter alphabet, there are at most $\lfloor \frac{k+1}2 \rfloor$ primitive, pairwise non conjugate perfectly clustering words which perfectly cluster collectively in a single array in which all their conjugates are arranged in increasing order.

We develop the analytical foundations for integrals over configuration spaces used to relate chain-level $S^1$-equivariant string topology to perturbative Chern-Simons theory. As an application, we prove that the composition of Chen's iterated integral with homotopy transfer intertwines the involutive Lie bialgebra structures on homology.

In this paper, we describe “Archeolab: Computed Tomography and Virtual Unwrapping of Scrolls”, an interdisciplinary undergraduate course guiding students through the virtual reading of scrolls. By studying the theoretical foundations of X-ray Computed Tomography, genetic algorithms, and conducting hands-on experiments and programming sessions, we aim to encourage students to study applied mathematics and promote active participation in lectures.

We study the structure of the initial layer arising in the relaxation-time limit of the collisional quantum hydrodynamic (QHD) system. When the initial data are not well prepared, a fast transient regime appears near the initial time, which prevents the uniform-in-time convergence of the momentum density to its limiting value. Using the method of matched asymptotic expansions, we derive a systematic asymptotic description of the solution with respect to the relaxation-time parameter $\tau$. In particular, we identify the fast time scale $t/\tau^{2}$ governing the initial layer and explicitly construct the corresponding inner expansion for the momentum density together with the outer expansion describing the slow dynamics. The leading-order outer dynamics are shown to coincide with the quantum drift-diffusion equation. The asymptotic expansion is rigorously justified by establishing uniform in $\tau$ energy estimates for the remainder terms under suitable regularity assumptions on the solutions. As a consequence, we prove the strong convergence of the momentum density in $L^\infty$ in time after subtracting the leading initial-layer correction. The analysis further shows that the convergence rate of order $\tau$ is optimal for general initial data and explains the improved rate in the well-prepared case.

We propose level-set physics-informed neural networks (PINNs) for domain inverse problems of gravimetry. The domain inverse problem establishes a correctness class for ill-posed inverse gravimetry, which we solve within the PINNs framework. Directly representing the domain inverse problem via neural networks is problematic due to the discontinuous nature of interfaces. We consider a level-set formulation where the neural network represents a continuous level-set function, and its zero level-set depicts sharp interfaces. To overcome the challenges of exploding and vanishing gradients caused by sharp interfaces during training, we propose an interface-aware backpropagation strategy. By redefining the derivative associated with interface evolution, this strategy enables a broader support region to drive the evolution process. Detailed analysis is provided to justify the efficacy of this strategy. Additionally, we introduce a simple procedure for adaptive refinement of collocation points near interfaces. The selection of network architecture is investigated by studying the solution spaces and the approximation properties of neural networks. Finally, extensive 2D and 3D numerical examples demonstrate the effectiveness of the proposed method.

Weighted estimates for the fractional integral operator $I_\alpha$ are established and subsequently applied to derive corresponding Hardy–Sobolev inequalities. The weights are constructed from distance functions to bounded median porous sets and possess mixed homogeneity, which enables us to extend earlier results obtained for porous sets to a significantly broader class of geometries.

We study the connection between mixed Hodge modules on quiver moduli spaces and the K-theoretic Hall algebra of preprojective algebras by taking associated graded modules. Furthermore, we show that the functor taking associated graded modules commutes with reflection functors.

The hyperinvariant subspace problem for Hilbert space operators $T$ containing a unilateral shift is addressed. The discussion is based on a similarity model of $T$, which is an operator-matrix $\widehat T= [T_{i,j}]_3$ where $T_{1,1}$ is the simple unilateral shift $S$ and $T_{3,3}$ is a cyclic diagonal operator $D$. The existence of $D$ is established by the technique resulting almost invariant half-spaces in \cite{APTT}; see also \cite{Tc} and \cite{HP}. For any operator $Q=[Q_{i,j}]_3$ in the commutant of $\widehat T$, the entry $Q_{3,1}$ intertwines $S$ with $D$ up to a transformation of rank at most 1. These entries form a linear manifold ${\cal L}_{3,1}$. We focus on 3-dimensional cross-sections of ${\cal L}_{3,1}$. These are subspaces of $3\times 3$ complex matrices, transformed into singular matrices by a canonical mapping. If such a subspace ${\cal L}$ is not transitive, then $T$ has a nontrivial hyperinvariant subspace. A throrough study reveals that ${\cal L}$ can be transitive only if it has a very specific basis. Consequences of the existence of nontrivial hyperinvariant subspaces in the presence of shift-type invariant subspaces are also discussed.

In the functional-integral formulation of Euclidean field theory, exact renormalization group (ERG) transformations are realized by functional-integral kernels. Unlike the ERG flow equations that describe infinitesimal ERG transformations, these ERG kernels explicitly depend on the global topology of the configuration space. This paper explores this topology dependence for multiply connected configuration spaces. We show that the ERG kernel is in general given by a weighted sum of kernels on its universal covering space, where the weight factors are determined by a one-dimensional representation of the fundamental group. These weight factors are shown never to be renormalized under the ERG. We also show that these factors can be interpreted as Aharonov-Bohm phases with respect to a background magnetic flux penetrating the infinite-dimensional configuration space. From this viewpoint, a normalization condition for ERG transformations corresponds to a flux-quantization condition, which is equivalent to the level-quantization condition for Wess-Zumino-Witten terms in nonlinear sigma models. Finally, we present an alternative gauge-equivalent form of the ERG flow equation that incorporates this topological information locally.

This paper concerns the inverse elastic scattering problem of determining an unknown penetrable obstacle from far-field data. Using the Helmholtz decomposition, the coupled boundary value problem is reformulated as a coupled scalar Helmholtz system. We prove the uniqueness of the associated coupled Helmholtz boundary value problem and of the corresponding boundary integral equation system, which is then discretized by a Nyström method for efficient numerical computation. Leveraging the relation between the compressional and shear far-field patterns of the Navier system and those of the coupled Helmholtz system, we employ three indicators to reconstruct the location and shape of the obstacle. Furthermore, we analyze the decay properties of these indicators and establish corresponding stability estimates. Numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method, even for limited-aperture data.

This paper is devoted to the study of solutions for a class of operator equations governed by strongly monotone operators on real Riesz spaces continuously embedded into Banach lattices. By exploiting the intrinsic lattice structure of Banach lattices, we establish refined growth assumptions on nonlinear terms to guarantee that suitable neighbourhoods of positive and negative cones are invariant under the descending flow. Combining the descending flow invariant set technique with the theory of strongly monotone operators, we derive abstract existence theorems: the operator equation possesses at least one positive solution, one negative solution and one sign-changing solution. These abstract results are further applied to \((p,q)\)-Laplacian boundary value problems, yielding corresponding multiplicity conclusions on positive, negative and sign-changing solutions.

This paper investigates the existence of critical points for functionals defined on a Hilbert space $X$ which is continuously embedded into a Banach lattice $E$. A lattice decomposition of $E$ is constructed, which possesses both order disjointness and inner-product orthogonality. Accordingly, a corresponding decomposition of the Hilbert space $X$ is obtained. Under this decomposition, the associated functional satisfies the energy collapse condition and order-preserving property on certain subspaces, while exhibiting coerciveness on others. By combining the descending flow invariant set method with Morse theory, we establish the existence of multiple critical points for the abstract indefinite problems. Finally, applications to elliptic boundary value problems are provided.

The equal-charge branch of supersymmetric rotating AdS$_5$ black holes has $Q_1=Q_2=Q_3$. The corresponding microcanonical sector of the $\mathcal{N}=4$ superconformal index is obtained by projecting to equal charges, or equivalently by extracting the constant term in the two charge-difference fugacities. We prove that for the large-$N$ multigraviton sector the projected index factorizes exactly as \[ \mathcal{I}^{\rm eqQ}_\infty(x,p) =\prod_{k\ge1}(1-p^kx^{3k})(1-p^{-k}x^{3k}) \sum_{n\ge0}\mathsf{p}(n)^3x^{6n}, \] where $\mathsf{p}(n)$ is the partition function. This factorization gives, for every spin sector, an explicit onset energy below which the large-$N$ coefficient is zero. Exact $U(3)$ computations show that finite-rank coefficients can nevertheless appear at energies where the large-$N$ coefficient vanishes, including beyond the classical $U(3)$ black-hole bound. We also determine the full line $j=6J_R+6$. In particular, with $j^*(J_R)$ denoting this large-$N$ onset energy, \[ d_3^{\rm eqQ}(87,\tfrac{27}{2})=1, \qquad j^*(\tfrac{27}{2})-87=1554, \] and the first giant-graviton sector already contributes one unit at this point. All coefficients are coefficients of the $(-1)^F$-graded index, not positive degeneracies. The main conclusion is that the high-spin tail survives the exact equal-charge projection.

We show that the submonoids of the tied-symmetric monoid containing the symmetric group form a distributive lattice. Furthermore, we determine a minimal generating set for any such submonoid.

In this article, we investigate the asymptotic Bayes optimality under sparsity (ABOS) of the Gavrilov-Benjamini-Sarkar (GBS) step-down multiple testing procedure (Gavrilov et al., 2009) in the sparse Gaussian sequence model. While the asymptotic optimality properties of the Benjamini-Hochberg procedure have been extensively studied, corresponding results for the GBS procedure remain unavailable despite its favorable finite-sample performance in several settings. Within the spike-and-slab Bayesian formulation and the asymptotic decision-theoretic framework of Bogdan et al. (2011), we establish that the GBS procedure is ABOS over a broad class of sparse asymptotic regimes. Existing analyses of the Benjamini-Hochberg procedure rely on approximating the random rejection threshold by a suitable deterministic surrogate. In contrast, our approach proceeds through a direct analysis of the Bayes risk by separately controlling the false discovery and false nondiscovery components. The analysis combines two key ingredients: a new finite-sample inequality for shifted order statistics associated with the GBS critical constants and a signal-crossing argument for ordered alternative $p$-values that exploits the sequential step-down structure of the procedure. For diverging expected signal counts, these tools yield ABOS without threshold localization. For bounded positive expected signal counts, a direct argument based on the first GBS critical constant establishes ABOS as well. In particular, our results cover polynomial sparsity $p_m\asymp m^{-\beta}$ for every fixed $0<\beta\le1$. To the best of our knowledge, this is the first asymptotic decision-theoretic analysis of the GBS procedure and the first proof of its asymptotic Bayes optimality under sparsity.

For a graph $G,$ let $\alpha(G)$ denote its second smallest Laplacian eigenvalue. The Laplacian Spread Conjecture is that $\alpha(G)+\alpha(\overline{G}) \geq 1,$ where $\overline{G}$ is the complement of $G.$ In this article, we provide a new proof of the Laplacian spread conjecture by means of linear algebra, which is more concise.

The objective of this paper is to review recent developments in Edge Multiscale Finite Element Methods (EMsFEM) for partial differential equations with heterogeneous coefficients or highly oscillatory solutions. Using elliptic equations with heterogeneous coefficients as an illustrative example, we present the key ideas of the method. We also analyze the approach while accounting for the discrete error in the multiscale basis functions. Extensive numerical tests are provided to validate the performance of the method.

Large-scale Capacitated Vehicle Routing Problems (CVRPs) are commonly solved by partitioning customers into smaller routing problems that can be optimized independently. While this substantially reduces computational complexity, independently constructed routing solutions may leave some customer demand unserved even when sufficient resources exist elsewhere in the fleet. We present Collaborative Routing Constructors (CoRC), a routing framework that enables independently solved subproblems to exchange customers and vehicles during optimization rather than relying solely on a fixed partition or a subsequent global re-optimization stage. Computational experiments on AGS benchmark instances and synthetic instances containing up to 200,000 customers compare CoRC against independent routing, post-routing global re-optimization, and state-of-the-art, end-to-end routing frameworks. Across all evaluated partitioning strategies, CoRC consistently constructs feasible routing solutions where competing partition-based methods do not. Furthermore, it remains effective on problem instances for which the evaluated end-to-end routing frameworks did not produce solutions under the same computational budget. These results demonstrate that collaboration between routing subproblems provides a robust and scalable approach for feasible large-scale route construction.

Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=\Lambda(x)U(x)$. The diagonal factor $\Lambda(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.

We investigate the collective dynamics of networks composed of diffusively coupled Gumowski-Mira maps and analyze how modifications in the intrinsic dynamics of the local oscillator reorganize the emergent phase structure of the network. The coupling strength and the local control parameter are treated as bifurcation parameters, and the resulting collective states are quantified using the largest Lyapunov exponent, a synchronization error measure, cluster-count statistics, and collective phase-classification diagrams. Two representative regimes of the local dynamics are examined. In the first regime, the network exhibits a smooth and highly organized collective parameter space, featuring a synchronization wedge embedded within an extended region of periodic cluster states. In the second regime, the same coupling architecture yields a fragmented phase organization, comprising disconnected synchronization islands, incoherent domains, and enhanced chaotic-cluster states. These results indicate that variations in the intrinsic dynamics of the individual Gumowski-Mira oscillator do not simply shift synchronization thresholds but can fundamentally restructure the topology of the collective phase space. Our findings thus establish a direct relationship between local nonlinear dynamics and the emergent organization of collective phases in coupled discrete-time networks.

In a recent preprint \cite{Kameko2026} , Kameko presented a substantial completion of the 2-local classification of $G_2$-gauge groups over $S^4$, extending earlier work by Kishimoto, Theriault, and Tsutaya. The central strategy relies on reducing the 2-local classification to the order of the Samelson product $\langle i_3,1\rangle$ and separating specific gauge group homotopy types. The purpose of this note is to provide necessary mathematical refinements and localization clarifications to several key steps in the proof of the classification theorem. Specifically, we refine the integral isomorphism claim for gauge group homotopy to its correct 2-local form, resolve an EHP sequence extension regarding the injectivity of the Hopf invariant, and make explicit the Postnikov layer conventions required for the mod 2 Leray–Serre spectral sequence calculations. We confirm that with these adjustments, the main 2-local classification theorem in \cite{Kameko2026} also holds as claimed.

Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value–derivative structure of such layers. We propose a Layer-Resolving XNet Physics-Informed Neural Network (LRX-PINN) based on integrated Cauchy activations. The proposed basis is transition-type at the solution level, while its derivative recovers a localized Cauchy kernel. We show that this structure matches the scaling of convection-dominated layers, inherits the Cauchy approximation mechanism at the derivative-profile level, and identifies \(d/\|w\|\) as the effective physical width of a ridge neuron. For analytic layer profiles, this yields derivative-stable exponential approximation in the stretched coordinate and a layer-scaled estimate for the strong residual of the singularly perturbed operator. Numerical experiments on several convection-dominated benchmarks show that LRX-PINN achieves higher accuracy than PIKAN and Fourier-feature PINNs while using less than \(30\%\) of their trainable parameters. On more challenging benchmarks, embedding the proposed representation into hp-VPINN-based frameworks further improves the best results obtained by existing hp-VPINN-based baselines without changing their original loss functionals or stabilization strategies. These results show that neural representations aligned with layer structure provide a compact and effective approach for convection-dominated problems.

We show that Holographic Renormalisation (HR) in eight dimensions is encoded in the unique conformal gravity theory that admits an Einstein sector with constant negative curvature. We explicitly relate HR to Topological Regularisation (TR), where the latter prescribes to add the Euler term to the Einstein-Hilbert Lagrangian density, with a precise coefficient so as to ensure that the resulting density is polynomial in the anti de Sitter (AdS) curvature. The polynomial is still asymptotically divergent. We find that the aforementioned, unique conformal gravity action in 8D, reproduces the polynomial, together with extra boundary terms that cancel the divergent terms in the action. We conjecture that, in arbitrary even dimension, HR is equivalent to conformally completing the Einstein-Hilbert action with negative cosmological constant, plus the Euler term with fixed coupling prescribed by TR.

Bacterial chemotactic movement and collective aggregation have long attracted substantial interest in mathematical biology and applied modeling. Classical Keller–Segel-type systems, however, are typically formulated under idealized laboratory assumptions, such as smooth agar substrates, and thus cannot adequately capture the gliding dynamics of myxobacteria in naturally rough environments like soil. In this paper, we propose a unified framework that integrates stochastic modeling, rigorous analysis, and deep neural network-based simulation of chemotactic gliding–diffusion and aggregation processes on bounded domains. Starting from a lattice-based discrete agent description and a subordinated Langevin equation driven by an inverse stable subordinator at the microscopic level, we characterize anomalous gliding dynamics on rough surfaces and derive a macroscopic time-nonlocal Keller–Segel-type chemotaxis model with logarithmic sensitivity. We then establish a comprehensive solution theory for the resulting model, covering mass conservation, novel regularity results, local well-posedness in any spatial dimension, and global well-posedness in two and three. The analysis relies on several newly developed ingredients, including a fractional Lyapunov functional, a variational inequality adapted to the time-nonlocal structure, logarithmic Sobolev-type estimates, Bregman distance techniques, and a weighted bootstrap mechanism adapted to the singular sensitivity and time-nonlocal memory. Finally, we design a mesh-free, positivity-preserving, multi-objective, time-marching physics-informed neural network method with separate architectures and tailored variable transformations. Numerical experiments on complex geometries, including a butterfly-shaped domain, demonstrate the robustness, accuracy, and flexibility of the proposed computational framework across a range of Keller–Segel-type systems.

Given a one form $\lambda$ in $\mathbb{R}^N$ and $f: \mathbb{S}^{n} \to \mathbb{R}^N$ with $f^\ast \lambda = 0$ we discuss the maximal Hölder regularity of extensions $F: \mathbb{B}^{n+1} \to \mathbb{R}^N$ such that $F^\ast\lambda = 0$ in distributional sense. Our analysis applies to the Heisenberg groups $\mathbb{H}_n$. It implies in particular that for all $n \geq 1$ any smooth horizontal map $f: \mathbb{S}^{n} \to \mathbb{H}_n$ can be extended to a $C^\alpha$-map $F: \mathbb{B}^{n+1} \to \mathbb{H}_n$ for some $\alpha > 1/2$. Moreover, if $n \geq 3$ we find $C^\alpha$-embeddings from $\mathbb{B}^{n+1}$ into $\mathbb{H}_n$ for some $\alpha > \frac{1}{2}$. In the appendix we discuss an (as of now unverified) approach to extend these arguments to find $C^\alpha$-embeddings from $\mathbb{B}^{2}$ into $\mathbb{H}_1$ for some $\alpha > \frac{1}{2}$, assisted by GPT-6 Astra.

The Gaussian Error Linear Unit is usually motivated as the expected output of an input-dependent Bernoulli gate. This work gives an alternative interpretation: GELU is the expected output of a hard linear gate with a Gaussian random threshold. This view provides a generative interpretation for the Bernoulli gate: the gate opens once the input clears a latent Gaussian threshold. This interpretation stems from a decomposition based on well-known results in stochastic inventory theory and leads to a threshold-transmission family that includes ReLU, GELU, SiLU/Swish, and hard swish as special cases. By considering a latent uniform threshold, we recover a hard-swish-like piecewise-polynomial gate whose nonlinear transition is confined to a finite interval, yielding fixed- and learned-width variants. Controlled experiments on compact vision and language models show that calibrated or learned uniform-threshold gates are consistently competitive with GELU, ReLU, and SiLU/Swish, display architecture-dependent learned widths, and use the finite transition region nontrivially.

Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity, and put $P_2$={$m$ $\ge$ 1:$\Omega(m)$ $\le$ 2}. We prove that the sumset $P_2$+{$a^a$: a$\ge$ 1} has positive lower density. The proof uses the Romanoff second moment method, in the spirit of Li and Pan's theorem on $P_2$+$2^{\mathcal P}$. The main new ingredient is the following average estimate for the singular factor \[ \frac{1}{K(K-1)} \sum_{\substack{1\le a,b\le K\\a\ne b}} \prod_{p\mid a^a-b^b}\left(1+\frac{\kappa}{p}\right) \le C_\kappa \] for some constant $C_\kappa>0$, which is valid for all $K \ge 2$ and any fixed $\kappa>0$. This estimate controls the average arithmetic correlation among the shifts $a^a$ and allows the Romanoff argument to be carried out.

We study the regularity of the rotation number as a function of the parameter in monotone one-parameter families of circle diffeomorphisms. We prove that for an open and dense set of $C^2$ monotone families, the optimal Hölder exponent is exactly $1/2$, showing therefore that the earlier result by J. Graczyk is sharp. For a $C^1-$ dense set of $C^{1+\alpha}$ families, we show that the rotation number cannot be more regular than $\alpha/(1+\alpha)-$Hölder. As a consequence, we get that the rotation number for a generic $C^1$ family of circle diffeomorphisms is not Hölder.

In this paper, we establish lower bounds for extreme values of the real part of the Riemann zeta function on the critical line. This work relies on the resonance method of Bondarenko and Seip, together with lower bounds for certain integrals associated with Dirichlet series with non-negative coefficients. Our results extend the work of Bondarenko and Seip (2018).