Papers

Thurston constructed codimension one foliations on $S^3$ thereby proved that the homomorphism $gv: \pi_3(B\overline{\Gamma}^\infty_1)\rightarrow \mathbb{R}$ induced by the Godbillon-Vey invariant is surjective. By another real analytic construction, he proved that the homomorphism $gv: H_3(B\overline{\Gamma}^\omega_1)\rightarrow \mathbb{R}$ is also surjective where $B\overline{\Gamma}^\omega_1$ is a $K(\pi,1)$ space by Haefliger. Tsuboi proved that the former surjection splits so that $\pi_3(B\overline{\Gamma}^\infty_1)= \mathbb{R}\oplus \mathrm{Ker}\,gv$. He further showed that the subgroup of $H_3(B\overline{\Gamma}^\infty_1;\mathbb{Z})$ generated by all the Thurston's constructions coincides with his direct summand $\mathbb{R}$. In this paper, we prove that Thurston's second surjection splits and also that the subgroup of $H_3(B\overline{\Gamma}^\omega_1;\mathbb{Z})$ generated by all the Thurston's cycles is equal to our direct summand $\mathbb{R}$ which is a lift of Tsuboi's one. To show this, we modify the arguments of Thurston and Tsuboi by replacing Reeb components with a real analytic construction. We prove certain uniqueness of them by showing acyclicity of the affine group in the Haefliger group $\pi_1(B\overline{\Gamma}^\omega_1)$. We also prove the existence of a new kind of characteristic class of foliations in $H^4(B\overline{\Gamma}^\omega_1;\mathbb{Z})$.

This paper studies conditional independence testing under the Gaussian additive noise model (GANM), where two variables are modeled as nonlinear functions of covariates with independent bivariate Gaussian regression errors. Under this framework, conditional independence can be characterized by the correlation coefficient of the regression errors, which motivates a test based on the Pearson correlation coefficient computed from the fitted residuals. Despite its simple form, the asymptotic behavior and statistical efficiency of the resulting test have not been well understood. In this paper, we develop the semiparametric efficiency theory under GANM and show, surprisingly, that the efficient estimator coincides exactly with the ordinary residual Pearson correlation estimator. We further establish the asymptotic properties of the proposed test and develop the corresponding inference procedure. Simulation studies demonstrate that the proposed method achieves near-oracle efficiency and competitive empirical power while maintaining valid Type I error control. We further apply the proposed test to conditional dependence analysis of U.S. stock returns.

Engression is a recently proposed and effective framework for conditional distribution learning. Its multi-step Reverse Markov extension further improves generative flexibility by decomposing complex conditional sampling into sequential reverse transitions. Despite their strong empirical performance, rigorous finite-sample statistical guarantees for these methods remain unavailable. In this paper, under deep neural network parameterizations, we establish nonasymptotic convergence bounds for Engression by directly controlling the Energy Distance between the learned and target conditional distributions. For the Reverse Markov framework, we further develop an Energy-Distance-based chain rule that enables a rigorous analysis of error propagation across reverse steps. Our analysis yields corresponding excess-risk bounds that are near-optimal up to logarithmic factors relative to the classical minimax rate over a general Hölder class.

The equilibrium shapes of fluid lipid membranes are governed by nonlinear differential equations derived from the Helfrich bending energy. In the tensionless, pressure-free, and zero-spontaneous- curvature limit, the governing equation reduces to the Willmore equation. We show that the combination of the Zheng–Liu and Langer–Singer first integrals reduces the third-order axisymmetric Willmore equation to a first-order ordinary differential equation. This formulation recovers the sphere, axisymmetric minimal surfaces, and the Clifford torus as special cases, while organizing the local solution space according to two integration constants. For nonzero spontaneous curvature, the axisymmetric Helfrich functional is reformulated as the energy of an inhomogeneous hyperbolic elastica with a preferred curvature proportional to the distance from the axis. This formulation identifies a constant-mean-curvature branch and a logarithmic branch associated with a biconcave disk profile. The resulting exact relations provide analytical constraints and benchmark solutions for configurations of axisymmetric vesicles and biomembranes.

In this paper we study the Stokes phenomenon of the quantum confluent hypergeometric supersystem, certain meromorphic linear system of ordinary differential equation with a second order pole, associated to the Lie superalgebra $\mathfrak{gl}_{m|n}$. We prove that its Stokes supermatrices satisfy the Yang-Baxter equation, and thus give rise to the quantum supergroup $U_q(\mathfrak{gl}(m|n))$.

We formulate a program to prove the categorical local Langlands conjecture (CLLC) of Fargues-Scholze, for all quasisplit $p$-adic groups where the Fargues-Scholze $L$-parameters agree with the semisimplification of a known "automorphic" local Langlands parametrization. A key working hypothesis - which we expect to prove elsewhere jointly with Hamann - is the compatibility of the enhanced Whittaker coefficient functor $c_\psi$ with Eisenstein series. For $\mathrm{GL}_n$, we show that this hypothesis alone implies the full CLLC. For more general groups $G$, we prove an induction principle which reduces CLLC for $G$ to CLLC for all proper Levi subgroups together with a very small amount of information about $G$. This principle applies unconditionally to many classical groups with current technology. Along the way, we establish many foundational results. In particular: - We prove a very strong finiteness theorem for spectral constant term functors. - We prove a spectral analogue of Bernstein's finite global dimension theorem for $p$-adic Hecke algebras. - We introduce and develop the theory of admissible ind-coherent sheaves and admissible duality on derived stacks. - We prove a duality theorem for the spectral action. Using all of these results, we unconditionally define a new and explicit functor $t_{\psi}$ from the spectral side to the automorphic side, which is defined on enough ind-coherent sheaves to control the entire conjecture.

This paper studies two problems that are motivated by combining two novel approaches, namely DNA composite and rank modulation. The recent approach of composite DNA takes advantage of the DNA synthesis property which generates a huge number of copies for every synthesized strand. Under this paradigm, every composite symbols does not store a single nucleotide but a mixture of the four DNA nucleotides. Instead of considering all the possible composite symbols we are interested only in the rank of the motifs in the symbol. The first problem in this paper addresses the capacity of a channel that uses such symbols, while in the second, bounds and construction of such codes are studied.

An edge-colored graph is called rainbow if all its edges have distinct colors. For a fixed graph $H$, the rainbow Turán number $\exstar(n, H)$ is the maximum number of edges in a properly edge-colored graph with $n$ vertices that does not contain a rainbow subgraph isomorphic to $H$. A $t$-fan $\Ft$ ($t \geq 2$) is a graph formed by $t$ triangles sharing a common vertex. A wheel graph $\Wn$ is constructed by connecting a new vertex $x$ to all vertices of a cycle $\Cn$ with $n$ vertices. Keevash, Mubayi, Sudakov, Verstraëte~({\small Combin. Probab. Comput., 2007}) showed that \[ \exstar(n, F_t) \geq \floor{\frac{n^2}{4}} + (t-1)\floor{\frac{n}{2}} \] by constructing extremal graphs for $n\not\equiv 2\pmod 4$. In this paper, we propose such a method: for a graph $H$, the upper bound of $\exstar(n, H)$ can be analyzed using the rainbow Turán number of $H'$, where $H'$ is obtained by deleting one vertex from $H$. By applying this method, we determine the exact values of the rainbow Turán numbers for two families of graphs when the number of vertices $n$ is sufficiently large. Specifically, we obtain: (1) when $n > 230t$, \[ \exstar(n, \Ft) = \floor{\frac{n^2}{4}} + (t-1)\floor{\frac{n}{2}} - \Dn, \] where $\Dn = 1$ if $n \equiv 2 \pmod{4}$, and $\Dn = 0$ otherwise; (2) for graphs $H$ satisfying $W_{2t} \subset H \subset \Kts$ ($t \leq s$), when $n$ is sufficiently large, \[ \exstar(n, H) = \begin{cases} \floor{\frac{n^2}{4}} + \floor{\frac{(t-1)n}{2}} - \Dn & \text{if } t \text{ is odd}, \\ \floor{\frac{n^2}{4}} + \floor{\frac{(t-1)n}{2}} & \text{if } t \text{ is even}. \end{cases} \]

In this paper, we introduce a higher-order averaging theory and method for a wide range of nonsmooth systems that are generally characterized by the classical averaging canonical form. Utilizing tools from generalized derivatives theory, we provide a nonsmooth near-identity transformation analogous to the one in smooth averaging theory. Additionally, we exploit sharp calculus rules from lexicographic differentiation theory to provide a closed formula for nonsmooth first-order averaging, and for the first time in the literature, nonsmooth second-order averaging. In fact, our approach recovers the smooth averaging results, without needing to check, if the system under consideration is smooth. Equipped with a nonsmooth second-order averaging theory, we generalize literature results and introduce a class of control-affine extremum seeking systems that tolerate nonsmoothness in the vector fields and/or the objective function by analyzing its stability based on a closed formula analogous to first-order Lie bracket approximations available in the smooth literature. We provide numerical simulation results involving complicated nonsmooth functions to demonstrate the effectiveness of our approach.

We construct a mean curvature flow with surgery starting with any compact mean convex hypersurface in $\mathbb{R}^{n+1}$, extending previous results of Huisken-Sinestrari, Brendle-Huisken, and Haslhofer-Kleiner for 2-convex flows. In contrast with previous constructions of mean curvature flow with surgery, the topological surgeries are performed by the flow itself through nondegenerate cylindrical singularities. At the same time the flow needs to be slightly adjusted at finitely many smooth times.

We show that the conflated expression graph for an arbitrary permutation has a unique minimal element and a unique maximal element, and every reduced expression sits on a maximal chain from the source to the sink. This generalizes the work of Manin-Schechtman regarding higher Bruhat orders, and gives an independent and self-contained proof of certain results in Hothem. In addition, we give explicit algorithms for elements in the top and bottom commutation classes. Given any reduced expression $\rho$, we give an explicit method for producing a maximal chain containing $\rho$.

The square-and-multiply algorithm, also known as binary exponentiation or repeated squaring, is a standard method for fast exponentiation in modern computation. Its historical origins, however, remain uncertain. This paper examines the emergence and progressive formalization of the method through selected primary sources. Particular attention is given to Jamshid al-Kashi's fifteenth-century Miftah al-Hisab, where the procedure is presented explicitly as a general computational method and claimed by al-Kashi as his own innovation. Earlier instances of successive squaring are identified in the works of al-Uqlidisi and al-Biruni, although in these cases the technique appears in particular calculations rather than as a fully articulated general rule. The earliest known antecedent is found in Pingala's prosodic studies in ancient India (c. 200 BCE), which seem to presuppose the conceptual basis of the method in their use of binary representation. As part of the historical development, Legendre's 1798 worked example is one of the earliest documented European use of the algorithm which appears as a subordinate step within a specific number-theoretic computation. The evidence suggests not a single continuous line of transmission, but the repeated independent reappearance of related procedures in distinct contexts. By the twentieth century, square-and-multiply became a special case within the broader theory of addition chains. By exploring this intellectual progression, this paper sheds some light on the historical background of an algorithm that is prominent in modern computation.

This is the second of a pair of papers where we construct and investigate a closed monoidal structure on the category of generalized algebraic theories (in the sense of Cartmell). Having presented the tensor product of theories in a syntactic way, we now study the same structure from the perspective of contextual categories. We define the exponential $\mathcal A^\mathcal B$ between two contextual categories $\mathcal A$, $\mathcal B$, and show how this yields, as a particular case, a cotensor $\mathcal A^B$ by a small category $B$. We also introduce a concept of multimorphism $(\mathcal A_1, ..., \mathcal A_n) \rightarrow \mathcal B$ for contextual categories $\mathcal A_i$, $\mathcal B$, and describe a bijective correspondence between bimorphisms $(\mathcal A, \mathcal B) \rightarrow \mathcal C$ and morphisms $\mathcal A \rightarrow \mathcal C^\mathcal B$. We give an abstract proof that there exists a contextual category $\mathcal A \otimes \mathcal B$ such that bimorphisms $(\mathcal A, \mathcal B) \rightarrow \mathcal C$ are in natural bijection with morphisms $\mathcal A \otimes \mathcal B \rightarrow \mathcal C$. We extend $\otimes:\text{Cont} \times \text{Cont} \rightarrow \text{Cont}$ into a closed symmetric monoidal structure and give a description of certain pushout-tensor maps that, in particular, allows us to prove that the tensor product of theories from part I is functorial and presents the one constructed here.

In this survey, we extend the conformal invariance of the Ising model and of the percolation from the hexagonal lattice to the 3-12 lattice.

Neural operators provide fast surrogate models for PDE simulations, but standard architectures often treat geometry and discretization as secondary to field data. Physical states are usually represented as grid-channel stacks, even when different quantities naturally belong on vertices, edges, faces, cells, boundaries, or interfaces and must satisfy compatibility constraints. We propose Cellular Sheaf Neural Operators, a discretization-aware framework for structure-preserving neural PDE surrogates. The method represents PDE states on oriented cell complexes, couples local feature spaces through learned restriction maps, and uses incidence/Hodge-informed message passing to follow computational geometry. Learned update heads pass through coboundary or flux maps, allowing selected constraints to arise from cell-complex structure rather than only from loss penalties. For magnetohydrodynamics, this yields face-based magnetic-flux updates driven by edge electromotive fields and finite-volume-style fluid updates driven by learned face fluxes and cell sources. On turbulent MHD and fusion-equilibrium surrogate tasks, the method improves structure-sensitive diagnostics, including rollout behavior, divergence control, spectral error, and equilibrium-regression accuracy. These results indicate that cellular-sheaf structure is a useful inductive bias for neural PDE surrogates in constrained multiphysics systems.

The classical identity relating squared distances to inner products determines every off-diagonal entry of a symmetric matrix associated with a set of pairwise dissimilarities, while leaving its diagonal free. This paper studies the resulting affine family of symmetric matrices and asks which representative is selected by minimising a matrix norm. We prove that the same representative, namely the double-centred matrix used in classical multidimensional scaling and principal coordinate analysis, minimises every unitarily invariant norm on the family. The result includes the Frobenius, spectral, nuclear, Schatten, and Ky Fan norms, and requires no assumption that the dissimilarities are Euclidean distances. The proof rests on a compression-reduction principle: the centring projection maps the entire compatible set to a single element of the family, and orthogonal compression is contractive for unitarily invariant norms; the conclusion follows without solving a separate minimisation problem for each norm. We then determine how many minimisers there are: the minimiser is unique for the Frobenius norm, for the nuclear norm and for every strictly convex unitarily invariant norm, while for the spectral norm we describe the whole minimising set, a compact convex set that reduces to a single point precisely when the double-centred matrix vanishes.

We present a complete framework for the explicit factorization of $X^n-1$ over integer residue rings $\mathbb{Z}_{p^e}$ for arbitrary $n$ with $\gcd(n, p)=1$. Classical approaches face fundamental bottlenecks: polynomial Hensel lifting requires updating global cofactors (scaling with $n$), while direct multivariate Newton–Hensel iteration on the factor coefficients requires Jacobian inversion (scaling exponentially as $O(p^{(m-1)^2})$ per layer due to zero-divisors, where $m$ is the coset dimension). Our framework eliminates both bottlenecks through three contributions: (1)~the Ideal Derivation Modulo Principle, which characterizes all factor coefficients as roots of a multivariate Dickson polynomial ideal derived via modular remainder extraction; (2)~a cofactor-free Hensel lift that elevates a single seed factor from $\mathbb{F}_p$ to $\mathbb{Z}_{p^e}$ using a cached polynomial inverse computed once over $\mathbb{F}_p$; and (3)~a dual-track coefficient reconstruction mechanism that recovers all remaining factors from the lifted seed's trace array via MED-based coset dispatch, with Newton–Girard inversion as the primary path and quotient-ring Gaussian elimination as an unconditional fallback when $p \leq m$. Empirical evaluation confirms the theoretical grand total algebraic complexity of $O(n + m^3 \log p + e \cdot m^2)$ for explicitly factoring $X^n-1$ over $\mathbb{Z}_{p^e}$, validating the near-constant per-layer lifting cost $O(m^2)$ to depths exceeding $e = 1000$. The framework yields speedups of $445\times$ (including runtime auto-seeding overhead) over SageMath's C-backed FLINT/Pari engine and $33.5\times$ over the V1 scalar lift.

We study a forward-time formulation of the Black-Scholes equation with state-dependent volatility. In contrast to the classical terminal-value pricing problem, where the option payoff is prescribed at maturity and the price is computed backward in time, the present problem prescribes the current option-price profile and seeks to recover the option-price profile at the expiration date T. This formulation is ill-posed, since the equation evolves in the unstable direction of the parabolic operator and high-frequency perturbations in the initial data may be strongly amplified. To address this difficulty, we introduce a price-dimensional reduction based on shifted Legendre polynomials. The original Black-Scholes equation is projected onto a finite-dimensional Legendre basis in the asset-price variable, leading to a system of ordinary differential equations in time for the expansion coefficients. This reduction acts as a spectral cutoff and also relaxes the degeneracy caused by the factor S^2 at the zero-price boundary. The main reconstruction method is a dimension-reduced Legendre–Tikhonov method. We prove existence, uniqueness, data stability, and convergence for each fixed truncation level. We also include a reduced PINN solver as a secondary computational comparison after the Legendre reduction. Numerical experiments with smooth, butterfly-spread, and European put payoffs show that the Legendre–Tikhonov method recovers the terminal option-price profile from noisy initial data, while the reduced PINN solver provides a useful additional benchmark. Comparisons with the conventional physical-space quasi-reversibility method demonstrate the stabilizing effect of the Legendre reduction.

When a learned representation is reused for a new task, which of its structure should survive? A task is a small category, a change of task is a functor $J$, and the invariants of a trained representation form a functor $F$. The left Kan extension $\operatorname{Lan}_JF$ is the invariant that the source and the change of task force on the target: by its universal property, every comparison of the source with an observed target invariant factors through it. The transfer discrepancy is the largest distance, over the target objects, between this extension and the observed invariant. For persistent homology the extension is a cokernel over comma categories, and a functorial version of the discrepancy, which also compares the maps between target objects, is $1$-Lipschitz under interleavings. On finite samples the discrepancy is a bottleneck distance, computed without approximation. For data glued from two pieces whose exclusive parts lie farther apart than the filtration scale, a Mayer-Vietoris sequence identifies what extending homology loses, while extending chains loses nothing. For a merge of labelled point clouds, the degree-zero discrepancy vanishes if and only if some $1$-Lipschitz classifier has a margin above the filtration scale at every sample point. In experiments on sampled manifolds, trained encoders and pretrained image models, every exact prediction holds on every sample. The discrepancy of truncated invariants recovers a hidden merge of separated parts far above chance and tells the intended target of a merge from a permuted control, but across pretrained models it does not track the accuracy of a linear probe.

We present a Graph Neural Network (GNN) framework for the classification of finite groups according to their solvability. Using undirected Cayley graph representations, the proposed framework learns to distinguish solvable and non-solvable groups directly from structural graph information, without relying on explicit algebraic features. The framework is evaluated on a benchmark dataset of 200 finite groups, comprising 120 solvable and 80 non-solvable groups. The experiments investigate the extent to which GNNs can learn the algebraic property of solvability from Cayley graph representations and generalize to previously unseen finite groups. The selected GNN architecture achieved a balanced accuracy (BA) of 1.000 on the independent test set. Furthermore, repeated experiments using different random seeds and learning rates consistently produced BAs between 0.956 and 1.000, demonstrating the robustness of the proposed framework with respect to the training configuration. To further evaluate generalization, the entire PSL(2,q) family was excluded from the training and validation sets and reserved exclusively for testing. The selected model correctly classified every previously unseen group in this family, demonstrating successful generalization to an entirely unseen family of finite groups.

We investigate the equilibrium (trivial solution) stability, also known as Lyapunov stability, of a certain first-order linear complex constant coefficient dynamic equation on the time scale $\p_{\alpha,\beta}$, which has continuous intervals of length $\alpha>0$ followed by discrete gaps of length $\beta>0$. In particular, we establish results in the case of this specific time scale, for coefficient values in the complex plane, including where the exponential function alternates in sign. In our analysis, we employ the Lambert $W$ function. For increasing gap size $\beta$ relative to $\alpha$, we prove that the complex constant coefficient undergoes a bifurcation in its parameter space. We establish interesting results for both the delta dynamic equation and the nabla dynamic equation. Lastly, we connect these results to those related to Hyers–Ulam stability of the same nabla equations.

Let $r_n^{{k}}(N)$ be the number of representations of $N$ as the sum of $n$ generalized $k$-gonal numbers and $r_n^{\square}(N)$ be the number of representations of $N$ as the sum of $n$ squares. By modifying the Heath-Brown circle method, we prove a closed-form asymptotic relation between $r_n^{{k}}(N)$ and $r_n^{\square}(8(k-2)N+n(k-4)^2)$ for any $k\geq3$ and any $n\geq4$. Consequently, we determine the asymptotics of $\sum_{N\leq x}r_4^{{k}}(N)^2$ and, via a result of Bringmann, Jang, Kane, and Tse, prove a similar closed-form asymptotic relation between the number $r_{4,+}^{{k}}(N)$ of representations of $N$ as the sum of four ordinary $k$-gonal numbers and $r_4^{\square}(8(k-2)N+n(k-4)^2)$. We also show that if $4\mid k$, any strictly increasing infinite subsequence on which $r_{4,+}^{{k}}$ is bounded converges $2$-adically to $(k-4)^2/(4-2k)\in\mathbb{Z}_2$, supplementing a result of Meng and Sun, and if $4\nmid k$, there is no strictly increasing infinite subsequence on which $r_{4,+}^{{k}}$ is bounded.

We introduce the overflow semiring $S = A \uplus_{\operatorname{ord}} L$, extending a positive information algebra $A$ by a join-semilattice $L$, where elements of $L$ dominate $A$ and arithmetic in $L$ reduces to the join. This models overflow in computational systems and generalizes the transition from finite to infinite cardinal arithmetic. We characterize the idempotent elements of $S$ and $S[X]$, fully classify idempotent power series over cardinal numbers, describe the structure of prime and maximal ideals, compute the Krull dimension of $S$ ($\dim S = \dim A + |L|$ for well-ordered finite $L$), and establish Noetherian and Artinian criteria.

We study frames for Hardy spaces generated by orbits of multiplication operators. We characterize the symbols $\varphi \in H^\infty(\mathbb{T}^N)$ for which the multiplication operator $M_\varphi$ admits a frame of orbits on $H^2(\mathbb{T}^N)$. We also show that, in this setting, the existence of a frame is equivalent to the existence of a Parseval frame. Moreover, for $N=1$ we prove that finitely many orbits suffice if and only if $\varphi$ is a finite Blaschke product. For $N > 1$, no finite collection of orbits can generate a frame, regardless of the symbol. We study the analogous problem for the adjoint operator $M_\varphi^*$. Our results extend to the infinite-dimensional torus $\mathbb{T}^\infty$ and, via Bohr's transform, to the Hardy space of Dirichlet series $\mathcal{H}_2$.

Let $\mathcal{A}$ be a Grothendieck category. In this paper we classify subcategories of $\mathcal{A}$ via a support notion defined by Krause \cite{Kr2024} based on the spectral category of $\mathcal{A}$. We show that this aligns with Nemman's support \cite{Ne1992}, and thereby extends earlier classifications of subcategories of the category of modules over a commutative noetherian ring and subcategories of its derived category. Over a locally coherent category $\mathcal{A}$, we show that this support agrees with open subsets of the Ziegler topology on $\mathcal{A}$.

In this work, we are concern with the wave equation subjected to a nonlinear feedback of logarithmic type and nonlinear polynomial source. To achieve a comprehensive understanding of this novel damping mechanism, we start with studying the well-posedness and the uniqueness of the problem. Then we establish the blow-up result of the problem under suitable initial data with negative initial energy and critical exponent of the source term. Finally, we present a brief numerical study and examples that support our result.

We identify the quadratic symmetry algebra of the two-dimensional Smorodinsky–Winternitz II system with a Laguerre-type confluent Heun algebra. The system is separable in Cartesian and parabolic coordinates. The complementary Cartesian separation operator \[ Y=\partial_y^2-\omega^2y^2+\frac{1/4-c^2}{y^2} \] is of Laguerre type, while the parabolic integral \(W=L_2\) is its algebraic Heun partner. With \(Z=[Y,W]\), the defining relations are \[ [Y,Z]=16\omega^2W-2bY,\qquad [W,Z]=6Y^2-4HY+2bW+8\omega^2(1-c^2), \] where \(H\) is central. This gives a direct superintegrable realization of the Laguerre–Heun algebra.

In this work, we study the well-posedness of a quasilinear first-order-in-time system with memory arising in nonlinear acoustics. The model features a memory kernel describing the fractional damping and covers, as special cases, first-order formulations of Kuznetsov- and Westervelt-type equations. For completely monotone convolution kernels, we prove that the Westervelt-type system admits unique local-in-time solutions in bounded domains for space dimensions $d\leq3$, under homogeneous Dirichlet boundary conditions, suitable regularity, and smallness assumptions. The analysis is based on energy estimates exploiting novel nonlinear coercivity properties of the memory terms. Under additional assumptions on the resolvent of the kernel, we show that the smallness conditions can be significantly relaxed. For finite sums of exponential kernels our results generalize to the Kuznetsov-type system. Additionally, we show that the inviscid case (absence of memory kernel) in \(\R^d\) can be treated by standard hyperbolic arguments in the case of the Kuznetsov-type system.

In the paper, we prove a rather general characterization of higher-order Sobolev spaces. We show that the $k\mathrm{th}$-order regularity, where $k \in \mathbb{N}$, is captured via inequalities involving $2^k$-tuples of points. In fact, in full generality, the obtained results characterize higher-order Sobolev spaces based on Banach function spaces. Moreover, we show an analogous characterization of higher-order Hölder spaces. Finally, we propose a way to use the obtained results to define higher-order Sobolev and Hölder spaces on metric measure spaces.

Globalization has enabled companies to leverage cost efficiencies; however, it has increased exposure to disruption risks that threaten supply stability. Among these, export bans have emerged as a systemic challenge, often arising as a secondary effect of conventional supply capacity disruptions. The pharmaceutical industry is particularly vulnerable to bans because of the criticality of keeping life-saving medications available, yet such disruptions also jeopardize patient health globally as well as challenge company operations. This paper proposes a supply chain design model that incorporates the relationship between conventional disruption risks (quality-related capacity failures and natural disasters) and export bans that may be induced when the drug is projected to be in short supply. The model is a two-stage stochastic mixed integer program, where the systemic setting yields a second-stage problem with one binary variable and continuous variables. The structure motivates the study of three tailored solution methods: the Alternating Integer L-shaped Cut, Alternating Disjunctive Cut, and Alternating Bilinear Cut methods. In the nearly-continuous second stage context, the Alternating Integer L-shaped cuts perform best. Using an oncology drug case study, we study effects of systemic, induced risk and quantify the value of incorporating disruptions into supply chain design.