Papers

A two-level analytical framework is presented for modeling random walk transport of messenger ribonucleic acid (mRNA) molecules along neuronal microtubules from soma to synapses. Motivated by empirical observations of mRNA cargo motion, the transport within a dendrite is modeled by a persistent telegraph process with pauses. Theoretical expressions for the probability of traversing the dendrite and the mean time for such travel are derived for different and equal probabilities of persistence. These results are used for the construction of a semi-Markov model of motion of mRNA cargo within the whole neuron. The semi-Markov model provides the probabilities of absorption at a given synapse and corresponding mean first-passage times (MFPTs) from the soma, where mRNA is transcribed. The theoretical expressions, together with experimentally obtained parameter values, are used to calculate MFPTs for neurons with empirically reconstructed morphology. The model predicts that when retrograde persistence is stronger, the MFPT to each synapse is effectively the same. Otherwise, when the persistence is more pronounced in the anterograde direction, the transport in the neuron resembles the motion along a single dendrite – nearly linear dependence of MFPT on the distance between soma and synapse. These findings are theoretically justified when the lengths of dendrites are considerably longer than the distance traversed during a typical run.

An $n$-dimensional subspace of $\ell^p$ has Plücker coordinates indexed by the $n$-element subsets of $\N$. We show these coordinates lie in $\ell^p\In{n}$ — the exponent is preserved — with multilinear norm exactly $1$ for $0<p\le 2$; for $p>2$ the sharp constant exceeds $1$ and its determination contains the Hadamard maximal determinant problem. A reconstruction lemma shows every nonzero solution of the quadratic Plücker relations in $\ell^p\In{n}$ is decomposable with frame in $\ell^p$; consequently $\Gr_n(\ell^p)$ is a closed Banach-analytic submanifold of $\mathbb{P}\big(\ell^p\In{n}\big)$ cut out by the Plücker relations alone, with no auxiliary summability condition and no polarization. For mixed sums $\bigoplus \ell^{p_i}$ the exterior power is graded by compositions of $n$; the support of the grading is the lattice-point set of a generalized permutohedron determined by the intersection pattern of the subspace with partial sums, this stratification is canonical for the isometry group though not for $\GL$, and each stratum admits a tubular neighborhood whose normal coordinates are precisely the Plücker blocks vanishing on it. We record what is proved and what is conjectured; the finitary and single-space core of the theory, including full proofs of Cauchy–Binet and Hadamard's inequality, has been formally verified in Lean~4.

In this paper, we study GL-equivariant modules over the infinite-variable divided power algebra $D = \text{Div}(k^{\infty})$ with $k$ an algebraically closed field of characteristic $p > 0$. Unlike previously analyzed GL-algebras, the divided power algebra is not noetherian or even finitely generated. We show that $D$ is GL-coherent and prove a “shift theorem” for finitely presented $D$-modules. Using this, we obtain a (semi-infinite) semi-orthogonal decomposition of its bounded derived category with one piece corresponding to each Frobenius twist $D^{(r)}$ of $D$. Crucial to our approach is the fact that $D$ is a flat colimit of subalgebras which are GL-noetherian.

We study the three-dimensional incompressible Boussinesq equations on a bounded domain with smooth boundary and Navier boundary conditions. We construct global weak solutions by a Galerkin approximation and establish the associated Leray–Hopf energy inequalities. For nonnegative boundary friction, the total energy decays exponentially when the friction coefficient is positive on a boundary subset of positive surface measure. In the frictionless case, the scalar field and the velocity component orthogonal to the rigid-motion kernel decay exponentially; when the kernel is trivial, this is exponential decay of the total energy. When the scalar initial datum vanishes, this also proves exponential decay for the Navier–Stokes system on solids of revolution for every friction coefficient $\alpha\in L^\infty(\partial\Omega)$ such that $\alpha\ge0$ almost everywhere and $\alpha\not\equiv0$, resolving the corresponding case left open in \cite{Kelliher2025}. The proof uses a weighted Korn–Poincaré inequality and a two-time Gronwall-type inequality with an exponentially decaying forcing term.

Let $\Bbbk$ be an algebraically closed field of characteristic zero, let $V=\Bbbk^\ell$ and $W=\Bbbk^m$, and set \[ \mathcal R_{\ell,m}=\operatorname{Sym}(\operatorname{Sym}^2V\otimes W)^{U_V}. \] We construct an explicit skew-symmetrizable seed $\Sigma_{\ell,m}$ by restricting and folding the determinantal seed for the flagged $m$-arrow Kronecker quiver. For every $\ell,m\ge2$, we have \[ \mathcal R_{\ell,m}=\mathcal U(\Sigma_{\ell,m}), \] with polynomial frozen coefficients, and $\Sigma_{\ell,m}$ admits a reddening sequence. The theta basis extends across the frozen boundary exactly for parameters in a rational polyhedral cone $\mathscr C_{\ell,m}$. Its weight fibers count the multigraded highest-weight multiplicities of $\mathcal R_{\ell,m}$, and the Jacobi–Trudi identity expresses symmetric-square plethysm coefficients as finite alternating sums of these counts. Optimized frozens give an explicit finite system of inequalities for $\mathscr C_{\ell,m}$.

The enthalpy method was introduced in the late 1970s to simulate phase-change problems on fixed grids without explicitly tracking the moving phase-change front. It remains one of the most widely used approaches in academic and commercial software for the simulation of industrial melting and solidification processes. For pure phase-change materials (PCMs) that melt or solidify at a single temperature, the enthalpy method introduces an artificial mushy region bounded by the solidus temperature, $T^{\rm sol}$, and the liquidus temperature, $T^{\rm liq}$. As the numerical parameter $\Delta T=T^{\rm liq}- T^{\rm sol}$ approaches zero, the solution obtained with the enthalpy method is generally assumed to converge to that of the classical Stefan problem, in which the phase-change front is infinitesimally thin. This assumption is largely based on benchmark studies performed under the simplifying assumption of equal solid and liquid densities. In this work, we systematically investigate the accuracy and spatio-temporal convergence properties of the enthalpy method for both low- and high-density-ratio phase-change problems. Because the limiting behavior $\Delta T\rightarrow0$ is difficult to realize numerically, owing to the diminishing thickness of the mushy region, we formulate and analyze the finite-$\Delta T$ mushy Stefan problem solved by the enthalpy method. We show that this problem possesses a self-similar structure that reduces the governing equations to a boundary-value problem involving two unknown parameters. The theoretical analysis also enables improvements to our previously developed low-Mach enthalpy method, enhancing its stability and accuracy as the density ratio between the two phases increases from $\mathcal{O}(1)$ to $\mathcal{O}(3)$.

A graph G is said to be maximal non-Hamiltonian if G is non-Hamiltonian, but $G+e$ is Hamiltonian for every nonedge $e$ of $G.$ In 1994, Vu Dinh Hoa conjectured that if $C$ is a longest cycle of a maximal non-Hamiltonian graph $G,$ then $G-V(C)$ is a complete graph. We disprove this conjecture by constructing a counterexample of every order $n\ge 56.$ We also pose several related open problems.

We characterize shadowing for continuous endomorphisms of compact Hausdorff groups. First, we prove that an endomorphism has shadowing if and only if its restriction to the identity component does, reducing the problem to compact connected groups. Passing to the stable image then reduces the analysis to surjective endomorphisms. For a compact connected abelian group $A$, let $T_{\mathbb Q}$ be the rational linear endomorphism induced by the dual map on $\widehat A\otimes_{\mathbb Z}\mathbb Q$. We show that the endomorphism of $A$ has shadowing if and only if every finite-dimensional $T_{\mathbb Q}$-invariant subspace is hyperbolic. For a general compact connected group, let $A$ and $S$ denote respectively the connected central and semisimple parts of its stable image. The induced endomorphism on $S/Z(S)$ determines an injective map on the set of simple factors. We prove that shadowing holds exactly when the rational dual map on $A$ is hyperbolic on every finite-dimensional invariant subspace and the induced map on the simple factors has no periodic point.

In this paper, we study the semilinear wave equations $$ u_{tt}-\Delta u+b(t)u_t+m^2(t)u=|u|^p, \quad t \geq 0, \quad x\in\mathbb{R}^n $$ with effective time-dependent damping and a time-dependent mass dominated by the damping. D'Abbicco, Girardi and Reissig established global small-data existence in supercritical ranges and identified the scale $$ p_{\beta,\eta}(n)=1+\frac{2\eta}{n+2\eta\beta} $$ for initial data in $(L^\eta(\mathbb R^n)\cap H^1(\mathbb R^n))\times(L^\eta(\mathbb R^n)\cap L^2(\mathbb R^n))$ with $1\leq\eta<2$, where $\beta$ is the lower mass index associated with the damping-mass pair. To support the expected sharpness of this scale, they also established an analogous subcritical nonexistence result for the corresponding diffusion equation with nonnegative initial data in $L^\eta(\mathbb{R}^n)$, leaving the wave-equation counterpart with effective damping and time-dependent mass open. We address this problem for $\eta=1$ under an intrinsic accumulated-mass balance and a Liouville nonoscillation condition. By constructing a positive slow adjoint mode, we prove nonexistence of global weak solutions for $$ 1<p<p_{\beta,1}(n)=1+\frac{2}{n+2\beta}, $$ and also treat the critical case $p=p_{\beta,1}(n)$ under an Osgood divergence condition. Conditional lifespan upper bounds and explicit admissible coefficient families are also given.

In this paper, we systematically incorporate the Polyak step-size into the randomized iterative method to improve its efficiency for solving generalized absolute value equations. In particular, we adopt the Polyak step-size within a stochastic iterative setting where the objective function updates dynamically at every step, unlike the classical Polyak step-size designed for deterministic optimization with fixed objective functions. Consequently, this novel implementation differs from the conventional Polyak scheme and demands a dedicated convergence analysis. We rigorously analyze the convergence properties of the proposed method and establish its linear convergence in expectation. Numerical experiments demonstrate that the incorporation of the Polyak step-size substantially improves the computational performance of randomized iterative methods with constant step-sizes.

We consider the defocusing energy-supercritical nonlinear Schrödinger equation $i\partial_{t}u+\Delta u=|u|^p u$ in dimensions $d\ge5$. Killip-Visan [Comm. Partial Differential Equations, 2010] and Li-Li [Siam J. Math. Anal., 2022] proved that for $s_c:=\frac{d}{2}-\frac{2}{p}>1$, any solution that remains bounded in the critical Sobolev space $\dot H_x^{s_c}(\mathbb{R} ^d)$ must be global and scatter. In dimensions \(d \ge 8\), their results required either that \(p\) be even or that \(s_c < \frac{d+2-\sqrt{(d-2)^2-16}}{4}\). In this paper, we improve the upper bound on \(s_c\) to \(s_c<1+p\) by establishing some new nonlinear estimates. This allows us to cover all cases in which \(p\) lies in the local existence range.

Let \((M^m,g)\) be a closed Riemannian manifold with \(\sec_g\leq-1\). We prove that the bottom spectrum of its universal cover attains McKean's lower bound if and only if the universal cover is hyperbolic space of constant sectional curvature \(-1\). More generally, for every \(1<p<\infty\), the variational \(p\)-fundamental tone satisfies \[ \lambda_{1,p}(\wti M) \geq\left(\frac{m-1}{p}\right)^p, \] and equality for some \(p\in(1,\infty)\) holds if and only if \(\wti M\cong\bH^m(-1)\). In that case, equality holds for every \(p\in(1,\infty)\). The proof converts the two McKean defects of a minimizing sequence into a stationary probability measure on the compact horospherical suspension; heat-kernel positivity then forces its zero-defect support to contain a complete leaf.

For every irreducible depth-zero representation of the general linear group over a non-Archimedean local field, parameterized by multisets of segments in the sense of Zelevinsky, we determine the complete decomposition of the space of vectors fixed by the pro-unipotent radical of a hyperspecial maximal compact subgroup. The decomposition is described explicitly in terms of the Zelevinsky parameters: the irreducible constituents are governed by Kostka numbers and Zelevinsky decomposition numbers, and they are naturally bounded between two partitions arising from the Zelevinsky segments of the representation and of its Aubert dual. In the generic case the result is given entirely by Kostka numbers. For general depth-zero Bernstein blocks, the multiplicities are products of the corresponding single-block multiplicities. We first prove, for an arbitrary unramified connected reductive group, that the functor of taking pro-unipotent fixed vectors intertwines Zelevinsky–Aubert duality on the p-adic side with Alvis–Curtis duality on the finite side. The proof for general linear groups combines Bushnell–Kutzko types, finite Harish–Chandra series, Iwahori–Hecke algebras, and the Zelevinsky classification to provide a complete and effective hyperspecial branching law for all irreducible depth-zero representations.

Building on the family of geometric constants introduced by Amini-Harandi and Rahimi, we define a skew-parameterized family on real normed spaces by replacing the classical symmetric pair with a rotated coefficient pair. This modification reveals new extremal behavior, particularly for asymmetric homogeneous weight functions. We establish reduction formulas, comparison inequalities, and parameter-stability estimates. Under an appropriate differential balance condition, we determine the exact values of these constants on Hilbert spaces and obtain a converse characterization in dimensions at least three. We further derive sufficient conditions for uniform non-squareness and normal structure, together with explicit computations in classical normed spaces. These results provide a unified framework for detecting Hilbertian and fixed-point-related geometric properties of Banach spaces.

Torquato and Stillinger used a pair-correlation-function optimization framework to conjecture an exponential improvement of Minkowski's lower bound on the maximal density of sphere packings in $\mathbb{R}^d$, with rate $2^{-(0.7786524795\ldots+o(1))d}$. Conditional on their realizability conjecture, we show that a family of hyperuniform pair correlation functions yields polynomial improvements of the form $\phi_{\max}\gtrsim d^\beta 2^{-d}$ for every fixed $\beta>1$ as $d\to\infty$. As the polynomial exponent increases with dimension, this family approaches the conjectured exponential improvement. From the Cohn–Elkies dual linear-programming upper bound, we independently derive the Torquato–Stillinger rate and show that its radial test functions cannot asymptotically exclude packings with this density scaling. For the near-contact families considered, any fixed number of shells, gaps, or radial bands can improve subexponential factors but not the leading exponential rate. To surpass this rate, we introduce an explicit hyperuniform construction based on Gauss–Radau quadrature. It contains $\lfloor(d-1)/4\rfloor$ positive delta-function shells and has rate $2^{-(0.622556248918\ldots+o(1))d}$, showing how growing radial complexity improves upon the Torquato–Stillinger rate. Finally, we prove strong duality between the unrestricted pair-correlation program and its Cohn–Elkies dual: their optimal values coincide, with no duality gap. By approximation with ordinary finite-band functions, we show that the unrestricted pair-correlation program attains the optimal Cohn–Elkies rate $2^{-(0.6044005\ldots+o(1))d}$. This result is optimal but supplies no comparable closed-form family. Together, these results support the possibility of exceptionally dense disordered sphere packings in high dimensions and strengthen the case for the Torquato–Stillinger realizability conjecture.

Double-categorical database schemas are enriched with universal quantification in the form of right adjoints to substitution. This allows phrasing of the important query, relational division. It is shown that such right adjoints together with suitable tabulators interpret modal operators, provide cartesian closed structure, and, when combined with global cocartesian structure, interpret first-order predicate logic and thus description logic. These structures are applied throughout to querying and optimization. It is seen, for example, that Frobenius reciprocity and Beck-Chevalley hold in any suitably structured double database schema and that these provide pushdown optimization rules. Such optimization rules are thus provably error-free as a property of the schema. Likewise, negation queries are derived from cocartesian and local implication structure.

The goal of this article is to derive observability properties for wave and plate equations with rough coefficients in the principal part, including the case of interfaces, through a regularization process. Specifically, we demonstrate that if a singular coefficient can be approximated by a sequence of smooth coefficients corresponding to uniformly observable systems, then the observability inequalities can be passed to the limit. This allows us to derive observability properties for the limit system. While this strategy is natural, we show that it can be used to establish new observability results in several settings, particularly in the presence of interfaces meeting the boundary, or multiple interfaces intersecting at a point, under a suitable multiplier-type condition that imposes sign constraints on the jumps of the coefficients at the interfaces. A key advantage of this approach is that it avoids the need for a detailed analysis of the regularity of solutions near the sets where the coefficients are singular.

In this short note, we present a proof of a relatively expected result in a convex-optimization problem. In fact, we will show that a matrix at which the maximum of the SWAP fidelity is achieved has determinant zero.

Soon after the proof of the Flat Cover Conjecture around the year 2000, two related concepts were introduced for classes in Grothendieck categories: Deconstructible classes and the strictly weaker Kaplansky classes. All commonly-studied Kaplansky classes, such as the class $\mathcal{FM}$ of Flat Mittag-Leffler modules in $R$-Mod and the class $\mathcal{D}$ of Drinfeld vector bundles in Qcoh($X$), satisfy a stronger property we introduce here: they are Uniformly Stationary Kaplansky classes. While such classes generally lack the key feature (existence of precovers) that make deconstructible classes so central to modern relative homological algebra, they often suffice for model-theoretic stability. This is true even in the absence of the Amalgamation Property, with various restricted classes of morphisms, and in some non-additive settings. For example, $\mathcal{FM}$ with pure embeddings, and $\mathcal{D}$ with (either categorical or geometric) pure embeddings, are stable in all sufficiently closed cardinals.

Lalou, Mbarek, Skender and Togni (arXiv:2607.25917) proved that the $n$-dimensional dual-cube $F_n$ admits two completely independent spanning trees for every $n\ge 5$, observed that none exist for $n\le 3$, and identified $F_4$ as the first unresolved case, reporting more than 700 hours of inconclusive computation. We settle this case affirmatively by an explicit construction, completing the classification: $F_n$ admits two completely independent spanning trees if and only if $n\ge 4$. The internal-vertex sets of the two trees are the level sets of a single ten-term cubic polynomial over $\mathbb{F}_2$ in the seven vertex bits, and correctness reduces to finite connectivity checks that are machine-verified by a solver-free program distributed with the certificate. In $F_4$ the two trees necessarily use 254 of the 256 edges. We also report exact infeasibility results for simpler rules of the same shape: within the search model, no affine or quadratic rule works, and ten terms is the fewest possible for a cubic rule.

Fixed-reference entropy-regularized transport yields smooth distributionally robust optimization (DRO) objectives, but its value as a function of the adversarial marginal is generally minimized at a kernel-smoothed law rather than at the nominal distribution. Consequently, the smallest raw ambiguity set need not recover nominal risk. Building on prior information-geometric plan-KL orientations, we study the forward tangent-centered discrepancy obtained by subtracting the value and first variation at the nominal law. This is the unique affine recentering, up to the simplex gauge; for symmetric costs, its plan-KL representation coincides with the forward candidate noted by Amari et al. (2018). The representation shows that the discrepancy dominates marginal KL, is uniquely zero at the nominal law, and is strongly convex in total variation. The resulting DRO problem retains a scalar log-sum-exp dual and closed-form conditional adversarial weights. On finite supports with symmetric zero-diagonal costs, the discrepancy converges uniformly to the unregularized transport cost at low temperature; for any finite cost, its rescaled value converges uniformly to marginal KL at high temperature. We derive its exact local Hessian, an explicit finite-sample containment radius, and a generalized chi-square calibration for multinomial data. Controlled simulations verify nominal centering, both limiting regimes, and practical plug-in coverage.

Prescribed positive bidiagonal factorizations of semi-infinite banded matrices reveal spectral, approximation, and integrable structures for arbitrary lower and upper bandwidths. Every cyclic permutation of the factors admits a normalized lower–upper positive factorization. Under suitable degree conditions, factor transfers induce matrix Christoffel transformations, linking the cyclic Darboux orbit to mixed-type Favard theory. Christoffel words determine minimum-height positive refactorizations and the cyclic products that reach them. The associated matrices have two block displacements, reducing to two diagonals for coprime bandwidths. Their finite nonzero spectra lie on a star, and planar networks yield positive radial Stieltjes moment sequences. In the compact radial case star-supported representing measures are characterized by conditions at the origin and negative fractional moments. For coprime bandwidths, a radial moment deformation gives determinant solutions of a sparse Lax hierarchy and synchronizes the cyclic Toda flows. Retaining the prescribed factor order yields matrix continued fractions with explicit Padé-type contact, denominator factorizations, backward evaluation, and error bounds. For bounded nonnegative factors, the convergents are monotone and converge to the least nonnegative solution in the admissible domain. Under the Favard identification, they approximate the mixed-type Weyl matrix. Unbounded factorizations retain the formal approximation; analytic convergence requires a closed realization and stability hypotheses, with an additional identification for measure-defined Weyl matrices. Piñeiro and Jacobi-like systems give applications, including global positive coefficientwise integrable solutions under the stated positivity and AT hypotheses.

For a graph \(G\), let \(\operatorname{cp}(G)\) and \(\operatorname{cc}(G)\) be the minimum numbers of cliques in an edge partition and a clique cover of \(G\), respectively. Set $\sigma_n = \max_{\lvert V(G)\rvert=n} \bigl(\operatorname{cp}(G)-\operatorname{cc}(G)\bigr), d_n=\left\lfloor\frac{n^2}{4}\right\rfloor-\sigma_n.$ In 1983, Erdős, Faudree, and Ordman asked whether \(d_n=O(n)\). Caccetta, Erdős, Ordman, and Pullman previously constructed graphs showing \(d_n=O(n^{3/2})\). We prove that $d_n=\Theta(n^{4/3}),$ thereby determining the correct order of the deficit and answering their question in the negative.

For a set $V$ of $n-1$ distinct positive integers let $\mathrm{LR}(V)=\max_{t}\min_{v\in V}\|vt\|$, where $\|x\|$ is the distance from $x$ to the nearest integer. The Lonely Runner Conjecture asserts $\mathrm{LR}(V)\ge 1/n$, and $V$ is called tight if equality holds. Goddyn and Wong determined which of the sets obtained from $[n-1]=\{1,\dots,n-1\}$ by multiplying a single entry by an integer are tight. We complete the classification of arbitrary one-entry modifications $([n-1]\setminus\{r\})\cup\{w\}$, $w\ge n$: for $n\ge4$ such a set is tight if and only if it is of Goddyn–Wong type or is one of the two sporadic sets $\{1,3,4,7\}$ and $\{1,3,4,5,9\}$ found by Wills. The proof is elementary and rests on an exact description of the region left uncovered by the retained speeds. We then show that the largest speed $M(n)$ occurring in a tight one-entry modification satisfies $M(n)<0.60\,n\log n+52\,n$ and $\limsup_{n\to\infty}M(n)/(n\log n)=1/2$; in particular, tight sets are not confined to speeds $O(n)$. Along the way we give a short proof of Liu's closed formula for $\mathrm{LR}([n-1]\setminus\{2\})$ and show that $\mathrm{LR}([n-1]\setminus\{r\})>1/(n-1)$ for every $r\le n-2-2\sqrt n$, extending the range $r\le(n-1)/2$ covered by Liu's theorem. For two-entry modifications we show that no tight instance removes a speed $q$ with $2\le q\le n/10$ once $n\ge\max(40,10q)$, whatever the second removed speed and the inserted speeds are; an exact-arithmetic census for $8\le n\le140$ with inserted speeds at most $8n$ finds only two tight two-entry modifications, both already listed by Goddyn and Wong.

Semigroups of holomorphic self-maps of the unit disc with an interior fixed point are, by the classical Berkson–Porta representation, entirely determined by a single holomorphic function constrained only by a positivity condition on its real part. This paper uses that representation to determine exactly when the associated flow contracts the Kobayashi metric of the disc at its best possible rate — the rate dictated by linearization at the fixed point — rather than at some smaller, conservative rate of the kind ordinarily obtained through auxiliary metric constructions. The question is reduced to a single pointwise inequality on the representing function, and this inequality is resolved completely for a natural one-parameter family of nonlinearities, yielding an exact threshold rather than a sufficient condition of undetermined tightness. Beyond this family, an explicit representing function is exhibited for which the inequality fails almost everywhere on the disc, and the Herglotz integral representation underlying the associated Carathéodory class is used to trace this failure to concentration of the representing measure, explaining rather than merely documenting why no threshold-free general theorem is available. The results are illustrated by direct numerical verification of the sharp threshold and of the explicit obstruction, and the paper closes by identifying the precise class of representing measures — point masses and their neighborhoods — that any future general sufficient condition would need to exclude.

This paper studies incremental stability of holomorphic dynamical systems through the infinitesimal Kobayashi metric, an intrinsic pseudometric on complex manifolds invariant under holomorphic transformations and free of the coordinate dependence inherent in auxiliary Riemannian or Hermitian formulations. Contraction is formalized as an upper Dini-derivative inequality on the Kobayashi metric along trajectories; the passage from this differential condition to exponential distance contraction follows the classical Finsler-metric contraction mechanism of Forni and Sepulchre, instantiated here for the case in which the Finsler structure is the Kobayashi metric itself. Since the intrinsic condition is difficult to verify directly, a practical criterion is developed through a smooth Hermitian metric, the complex-Hermitian analogue of the classical real matrix contraction inequality: contraction with respect to such a metric implies intrinsic contraction on forward-invariant compact subsets, the two notions related through explicit local equivalence constants. Building on this, a Nagumo-type invariance result is established for Laplacian-coupled holomorphic networks, giving verifiable conditions for forward invariance in a class of systems not previously treated this way, and the framework extends to feedback-controlled holomorphic systems, with consequences for equilibria and periodic orbits following directly from intrinsic contraction. Numerical experiments on a network of coupled holomorphic oscillators verify the Hermitian condition analytically on a proven invariant set, and reveal that the observed synchronization rate substantially exceeds this guaranteed rate; the gap matches, to three decimal places, a closed-form combination of the node-wise rate and the network graph-Laplacian spectral gap, identified here as a target for a network-aware extension rather than resolved in full.

The purpose of the present paper is to advertise and illuminate the applied research of Mikhail Alekseyevich Lavrentiev, a distinguished scientist who generated new ideas and obtained many outstanding results in mechanics and mathematics. He also initiated and maintained numerous flourishing activities in mechanics and applied mathematics in the former Soviet Union. He was notably an outstanding researcher in pure and applied mathematics as well as in engineering. On the applied side, his research spanned fluid dynamics, explosion physics, mechanics, technology, and several branches of engineering. It was always aimed at the practical challenges of his time and achieved impressive successes. It has been fascinating to uncover all of this. Much of his biographical material and a review of his activities in pure mathematics have been published, but no review of his applied achievements is available. This paper aims to give Lavrentiev's applied research the recognition it deserves in the communities of fluid dynamics, applied mathematics, engineering, and the history of science.

The Koopman operator provides a linear framework for analyzing nonlinear dynamical systems through spectral properties. Extended Dynamic Mode Decomposition (EDMD) approximates this operator from data, but non-invariant dictionaries can introduce spurious eigenvalues. We introduce the Projected Koopman Operator Approximation framework for constructing Filtered EDMD operators. The framework projects the Koopman action onto admissible dictionary subspaces that need not be invariant, while exactly preserving every represented Koopman eigenpair with nonzero eigenvalue. A forward–intersection chain provides a canonical hierarchy of compatible subspaces, connecting the full dictionary to its maximal invariant core while retaining useful intermediate models. We analyze two projection geometries: a coordinate-orthogonal projector, which requires no function-space Gram-matrix estimate but is basis-dependent, and a function-space orthogonal projector, which recovers population EDMD at the unfiltered level. We characterize their relationship to EDMD and existing subspace-selection methods. We also develop SVD-based algorithms for constructing sampled forward–intersection spaces and implementing the coordinate projector. Under independent noiseless sampling and exact-rank identifiability, the resulting empirical operators converge almost surely to their population counterparts. Experiments on a Kronecker flow, a polynomial system, and the Van~der~Pol oscillator demonstrate reduced spectral pollution. For Van~der~Pol, the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, whereas the $L^2(\mu)$ projector approximates the limit-cycle spectrum on the same certified subspace.

The numerical approximation of a class of quasilinear hyperbolic equations arising in dynamic strain-limiting elasticity-whose nonlinear constitutive law relates stress and strain nonlinearly-is investigated. Ellipticity may be lost by such problems in regions of large strain, leading to significant challenges in the design of stable and accurate numerical methods. To address these difficulties, a fully discrete continuous Galerkin finite element framework is developed, combining continuous linear finite elements for spatial discretization with the Hilber-Hughes-Taylor (HHT-alpha) time integration scheme. A lifting technique is incorporated within the proposed formulation to treat non-homogeneous Dirichlet boundary conditions, and a consistent Newton iteration is employed for the efficient solution of the resulting nonlinear systems. Robustness is enhanced by the algorithmic dissipation introduced through the HHT-alpha method by suppressing nonphysical high-frequency oscillations near degenerate regions, while consistency and accuracy are preserved. Under suitable structural assumptions on the nonlinear constitutive coefficient, the weak formulation is established in appropriate energy spaces. Theoretical second-order convergence in the L^2-norm and first-order convergence in the H^1-norm, rapid nonlinear convergence with nearly mesh-independent Newton iterations, physically consistent wave-speed evolution, and stable energy dissipation are demonstrated through comprehensive numerical experiments. Furthermore, it is confirmed by the results that an accurate, stable, and computationally efficient framework for simulating nonlinear strain-limiting wave propagation is provided, offering a solid foundation for future extensions to multidimensional nonlinear elastodynamics, adaptive finite element methods, and fracture and damage mechanics.

Inference-behaviour Semantics (I-bS) is a sequent calculus-based substructural approach to proof-theoretic semantics (P-tS), focussed on modelling the relationships of connective meanings across different logics. This paper provides a case study of how I-bS can be extended to any finitely multi-dimensional sequent calculus. To this end, we give I-bS for the 3-dimensional sequent calculi K3 (Strong Kleene) and LP (Logic of Paradox). We find that the K3 and LP connectives have the same meaning and that their meaning conservatively extends the meaning of the corresponding classical LK connectives. We situate this result within the wider project of integrating I-bS with MUltlog. The goal is to automate the generation of I-bS for arbitrary multi-valued logics using a MUltlog-style system, thus contributing towards the computational automation of P-tS approaches.