Papers

Robotics increasingly relies on GPUs for parallel simulation, large-scale learning, and neural-network inference. For model predictive control (MPC) to scale with this paradigm, solvers must run efficiently on this hardware while remaining fast, differentiable, and compatible with expressive MPC formulations used in robotics. We present TurboMPC, a differentiable MPC solver that runs entirely on the GPU and supports state and control inequality constraints, implicit integrators, cross-time-coupled costs, and slack variables. TurboMPC combines sequential quadratic programming (SQP), an alternating direction method of multipliers (ADMM) inner solver, implicit differentiation, and a co-designed JAX-CUDA implementation for efficiency and ease of use. In simulation, we validate TurboMPC on constrained planning, humanoid imitation learning, and reinforcement learning with neural-network cost function tasks, achieving up to $15\times$ and $58\times$ speedups over state-of-the-art CPU and GPU differentiable solvers, respectively. We deploy TurboMPC on a full-scale car for minimum-time racing and find that batched, GPU-accelerated tuning of MPC parameters via Bayesian optimization yields significantly faster driving than a hand-tuned baseline. TurboMPC also scales to planning horizons of over $8000$ knot points while maintaining control of the vehicle. We open-source TurboMPC at: https://github.com/ToyotaResearchInstitute/turbompc

This note describes an integration of the sim-to-real performance estimate with betting (from Chen et al.) and the safe anytime-valid inference (from Ramdas et al.). Using the scaled simulators. The method produces efficient, reliable certificates for the mean estimate, an approach that is especially valuable in robot performance testing. This note gives a primary, self-contained account of the construction; preliminaries of the respective methods are kept at a minimum, and one shall refer to the original works for full detail. Some synthetic examples demonstrating the proposed algorithm can be found at https://github.com/ISUSAIL/Bet4Sim2Real-EProcess.

We extend the study of discrete analytic (DA) Schur functions to rhombic lattices, utilizing suitably defined shift operators. There is a number of important differences with the classical case, including eigenvalues of the backward shift operator. As an application we solve a basic interpolation problem in a weighted Hardy space of DA functions, introducing a discrete counterpart of the Blaschke factor.

Hermann Weyl's The Classical Groups is a landmark work connecting classical invariant theory with modern representation theory. It shows how polynomial invariants of the general linear, orthogonal, and symplectic groups can be systematically understood through linear representations and tensor methods. The current note is primarily based on a personal reading of the book of Weyl and of the more accessible books Classical Invariant Theory by Kraft and Procesi and by Olver. It is neither exhaustive, nor original, nor state of the art. We focus on a few selected aspects, aiming for an elementary and concrete approach. We work over the field of reals R with the classical groups GL(n), SL(n), O(n), and SO(n). Most of our efforts have been devoted to carefully worked examples, introducing just enough of the general theory to handle them effectively.

Telecommunications is a high-leverage domain for large language model (LLM)-based reasoning because routine engineering workflows require joint grounding in normative specifications, operational telemetry, vendor-specific fault evidence, and exact RF/network calculations. However, current LLM integration in telecom remains bottlenecked by a two-sided capability gap: generic reasoners often lack telecom-specific grounding, while domain-specific telecom LLMs remain limited in structured, multi-step reasoning. To bridge this gap, we release TelecomGPT-R1-9B, a unified open-source telecom reasoner that ranks top-performing on the GSMA open telco leaderboard. Specifically, we curate a 67,427-example supervised fine-tuning (SFT) corpus organized around four complementary reasoning axes: protocol, knowledge, modeling, and fault. The corpus is built from axis-matched public web sources and enhanced through axis-specific chain-of-thought (CoT) generation and prefix-continuation self-validation. Starting from Qwen3.5-9B, we further develop a two-stage post-training recipe. First, multi-teacher low-rank adaptation (LoRA)-based SFT injects telecom knowledge and induces axis-specific reasoning formats. Second, group relative policy optimization (GRPO), stabilized by decoupled clip and dynamic sampling policy optimization (DAPO), optimizes the policy using four axis-aligned binary verifier rewards. Across seven public telecom benchmarks, TelecomGPT-R1-9B ranks first among open-source telecom LLMs and achieves a seven-axis mean comparable to state-of-the-art closed-source frontier reasoners.

We consider a mean-field model of competitive firms operating under an AK production technology, where output is proportional to capital and production generates emissions. In this setting, we introduce a regulator whose objective is the reduction of cumulative emissions, in the spirit of Emission Trading Systems (ETS), where firms must hold and trade permits to cover their emissions in a regulated market. The regulator acts as a central planner and controls the supply of permits, balancing emission reduction and aggregate output. Permits are allocated through a dynamic auction mechanism that adjusts supply in real time to achieve both market efficiency and the regulator's long-term goals. The regulator and the firms interact through the endogenous permit price, which is determined by the regulator's policy and affects firms' optimal strategies via the market clearing condition. The regulator's optimal policy is derived within a Mean-Field Control (MFC) framework. Exploiting the linear-quadratic structure and the presence of common noise, we characterise the equilibrium via a system of coupled FBSDEs and associated Riccati equations. Our results provide a tractable characterisation of optimal permit allocation policies in large economies and offer insights into the design of efficient emission trading mechanisms.

Deep learning has shown strong potential for massive multiple-input multiple-output (Massive MIMO) physical-layer tasks, including channel state information (CSI) feedback and channel estimation. However, environmental heterogeneity can severely degrade CSI models in unseen scenarios, while conventional adaptation requires target-domain data and substantial computation. This paper proposes Channel Conditional Parameter Generation (CCPG), an end-to-end pipeline for rapid deployment of CSI models in dynamic wireless environments. CCPG identifies scene-sensitive adaptation bottlenecks through component-freezing experiments and generates only lightweight LoRA weights instead of full model parameters. It compresses high-dimensional channel features into compact latent conditions using cascaded SVD and a Perceiver Resampler. An energy-based canonicalization mechanism mitigates permutation and sign ambiguities in LoRA weights, while a diffusion-based generator incorporates structural information and an asymmetric size-aware loss for topology-aware parameter generation. Experiments on DeepMIMO and WAIR-D for CSI feedback and channel estimation show that CCPG adapts to new scenarios in about 3 seconds with a single forward pass, without target-scenario training or fine-tuning, and achieves cross-domain recovery performance comparable to costly online adaptation. These results demonstrate that CCPG enables efficient deployment of CSI models in large-scale dynamic wireless scenarios for intelligent 6G communications.

In this paper, we study dual orthogonal tripotent matrices, defined by $\hat{A}^3=\hat{A} = \hat{A}^*$, and examine their fundamental algebraic properties. Additionally, we establish several characterizations of this class of matrices using matrix averages involving $\hat{A}$, $\hat{A}_e$, $\hat{A}^*$, $\hat{A}\sp{\scriptscriptstyle N}$, as well as integer powers of products such as $\hat{A}\hat{A}^*$ and $\hat{A}^*\hat{A}$. These results enrich the theory of dual generalized matrix classes and reveal new perspectives on related dual quaternion matrices.

This paper presents a detailed application of the Unified Transform Method (Fokas method) to the one-dimensional heat equation on $[0,1]$ with Dirichlet boundary conditions. The analysis formulates the Initial-Boundary Value Problem and derives an integral representation of the solution via a generalised spatial Fourier transform with complex spectral parameter $\lambda \in \mathbb{C}$, yielding the Global Relation – an algebraic identity coupling the initial datum, prescribed boundary values, and unknown Neumann data. The unknowns are eliminated by exploiting the symmetry $\lambda \mapsto -\lambda$, reducing the solution to a contour integral over $\partial D^+$. An explicit evaluation is carried out for exponential initial datum $u_0(x)=e^{-x}$ and Dirichlet conditions $g_0(t)=\cos(t)$, $h_0(t)=e^{-1}\cos(t)$. The integral representation is analysed in the complex plane, with emphasis on exponential decay and analyticity, providing rigorous justification for contour deformation via Cauchy's Theorem and Jordan's Lemma. Numerical implementation in Maple uses a trapezoidal contour parametrisation ensuring exponential decay along each segment; the solution over $x\in[0,1]$, $t\in[0,2\pi]$ matches prescribed data to machine precision. The results confirm the analytical and numerical efficacy of the Unified Transform for classical parabolic problems and illustrate how rigorous contour analysis yields stable, accurate solutions.

Spline functions, particularly B-splines, play a fundamental role in approximation theory, numerical analysis, and spectral representations. Although generating function techniques are widely used in combinatorics and special function theory, their systematic use in spline constructions remains relatively limited. Motivated by this observation, we introduce a generating function framework for the construction and analysis of generalized exponential B-spline families associated with Apostol-Bernoulli polynomials. Using backward shift operators and operator-based representations of uniform B-splines, we construct new Apostol-Bernoulli-type B-spline functions and derive explicit generating functions, recurrence relations, and structural identities. The present approach also yields generalized exponential spline families depending on the parameters $(\lambda,\alpha)$, which interpolate between classical polynomial splines, exponential splines, and Apostol-type extensions. Furthermore, by combining de Boor recurrence relations with differential operator techniques, we establish analytic recurrence formulas involving parameter derivatives of the associated spline sequences. We also investigate the Fourier transform of the generating functions and obtain explicit rational-type representations in the frequency domain, revealing a direct connection between discrete difference operators, exponential spline structures, and spectral analytic behavior. These results provide a unified analytic approach for constructing generalized spline families and suggest further applications in approximation theory, operator-based spline analysis, and generalized spectral methods.

In this paper, we introduce and study the notion of fuzzy normed BCK-algebras and fuzzy normed BCI-algebras as a natural extension of clas sical normed algebraic structures into the fuzzy setting. A fuzzy norm on a BCK/BCI-algebra is defined as a mapping from the algebra and a positive real parameter into the unit interval satisfying suitable axioms analogous to those of fuzzy normed linear spaces. Several examples are presented to illustrate the validity of the axioms. Fundamental properties of fuzzy normed BCK/BCI algebras are established, including monotonicity, chained triangle inequalities, and order-related behaviors. It is shown that every BCK/BCI-algebra admits a fuzzy norm, and the behavior of fuzzy norms under algebra homomorphisms is investigated. Necessary and sufficient conditions are obtained for the transfer of fuzzy norms via injective, surjective, and bijective homomorphisms. A charac terization theorem is proved showing that the main fuzzy norm inequality can be reduced to a simpler condition. These results generalize known concepts in fuzzy algebra and provide a new analytical framework for studying BCK/BCI algebras.

In this paper, we propose a matrix-based polyalphabetic data encoding and decoding scheme using Fibonacci, Leonardo, Jacobsthal, and Lucas sequences. The method employs three sequence-based alphabets for character substitution and a Lucas-based auxiliary alphabet for word separators. A position-dependent selector, \[ \sigma=\bigl(v^2+(i-1)+(j-1)\bigr)\pmod 3, \] distributes repeated plaintext symbols among different numerical alphabets, thereby reducing frequency concentration. The resulting numerical matrix is divided into $3\times 3$ blocks and transformed using powers of the Leonardo $Q$-matrix with block-dependent keys generated from pre-shared parameters $(s,p)$. A collision-free public prime $P$ is used to keep ciphertext entries bounded while preserving unique decoding. A worked example and preliminary statistical, entropy, avalanche, and timing results indicate that the proposed modular construction is computationally efficient and provides improved distributional behavior compared with standard monoalphabetic substitution.

In this paper, we introduce a matrix-based binary tree that forms the foundation of both the Stern–Brocot and Calkin–Wilf trees. We then explore its connection to continued fractions. Building on this relationship, we provide a path-based representation of Brillhart's proof of the sum of two squares.

We compare spanning-tree counts at a fixed number of vertices for Cartesian products of paths $P_r$, $r\ge1$, or cycles $C_s$, $s\ge3$. With the number $d$ of factors fixed, an equal-volume tuple that is more balanced in the logarithmic majorization order has at least as many spanning trees, with equality only for permutations. This comparison persists after adjoining any fixed finite connected Cartesian factor; equal-sided blocks therefore uniquely maximize the count at admissible block volume $n^d$. We prove the comparison through strict log-convexity of one-dimensional heat traces in logarithmic length. The Hartman–Watson representation expresses the cycle interpolation as a positive mixture of theta functions, and the path case follows from it. The same heat-trace order decreases every positive inverse-power sum of the nonzero Laplacian eigenvalues, and hence the total effective resistance. Cylinders behave differently: square balancing can fail. For $P_r\square C_s$ of area $N=rs$, extending the product formula to real height and optimizing over integer circumferences $3\le s\le N$ yields an optimum on the scale $N^{1/3}$. Along areas having a divisor asymptotic to that optimum, quantitative estimates place every graph-maximizing circumference on the same scale. For two-dimensional tori, balancing remains optimal, but the gain in the logarithmic spanning-tree count admits no positive uniform lower bound proportional to the decrease in the side sum.

In a continuous-time stochastic Stackelberg differential game, a leader steers strategic followers through the information structure and a transfer mechanism, not the dynamics. We pose two problems, neither formulated as a control problem with strategic followers: information control, where the leader commits only to a disclosure policy, and information-mechanism control, which adjoins transfers. The first is equilibrium-constrained and admits no dynamic programming principle; the second is tractable, and the transfer is why: alignment makes the lower level a potential game, collapsing the bilevel problem to one stochastic control problem with an exact first-order condition. Disclosure precision becomes a control input and the belief a controlled state obeying a Riccati equation. The latent environment is a jump-diffusion whose belief filter is exactly finite-dimensional under publicly observed epochs. A marginal-contribution transfer makes truthful reporting dominant and efficient action the Nash response. Equilibrium feedback is saturated on strictly convex components and bang-bang on linear ones, hence discontinuous. The master value is the unique viscosity solution of a partial integro-differential Hamilton-Jacobi-Bellman equation whose nonlocal term carries the epochs; verification holds without smoothness, and semiconcavity makes the switching set null, giving a well-posed Filippov closed loop. Instantiated on multi-area power systems, the levers are complements under a one-factor common shock. Calibrated to 2021 Winter Storm Uri, coupling removes 7.4% of social cost relative to autarky, rising to 35% at a 10-gigawatt tie, and disclosure is worth 8.7%; under a European renewable-drought calibration it is worth 37% under autarky and 48% under coupling.

We study public-good provision when a planner observes agents' preferences only through a fixed local-privacy channel that randomizes each report before it reaches the planner. We characterize the optimal reduced-form allocation: the project is implemented when an aggregate posterior score is positive, where each agent's score combines the posterior expected valuation and posterior virtual value. Privacy enters through these posterior objects, muting the responsiveness of provision to private preferences and, under weak monotone likelihood ratios, potentially generating pooling. We then distinguish the optimal reduced-form allocation from its implementation through signal-measurable transfers: the required transfers solve a Fredholm integral equation whose solution is unique under completeness when it exists, while existence requires a separate range condition. Maximum reduced-form revenue exhibits three population regimes: it is asymptotically linear, of square-root order, or exponentially small according as the lower endpoint of the valuation distribution is positive, zero, or negative. Finally, welfare comparisons depend on the privacy calibration. At a common noise scale, Laplace Blackwell-dominates logistic noise, while under a common tight $\mu$-GDP calibration the ordering reverses for the maximally separated binary endpoint experiment. Thus the preferred privacy channel depends on the standard used to hold privacy fixed.

For a compact 3-manifold W. Thurston introduced a norm on the first cohomology group of the manifold. The unit ball $B$ of this norm is a polyhedron and the set of cohomology classes that are representable by fibrations over a circle is a union of cones on some of the open faces of $B$. In the present paper we study the fibred faces of the Thurston polyhedra of exteriors of links in $S^3$. Our approach is based on the non-abelian Novikov homology associated with the universal covering of the exterior of the link. We prove in particular that for a 2-component 2-bridge link $L$ a cohomology class $\xi\in H^1(E(L))$ can be represented by a fibration over a circle if and only if its 2-variable Alexander polynomial is $\xi$-monic. We compute the Morse-Novikov numbers for a majority of 2-component prime links with number of crossings $\leq 8$.

Lagrangian formulations for (ir)reducible integer higher-spin massless and massive Poincare group representations subject to Young tableau with $k$ columns $Y[\hat{s}_1,\hat{s}_2,...,\hat{s}_k]$ in $d$-dimensional Minkowski space-time are firstly presented. The particles are described in a metric-like formulation by tensor fields with $k$ groups of antisymmetric Lorentz indices $\Phi_{\mu^1[{\hat{s}_1}],\mu^2[{\hat{s}_2}],..., \mu^k[{\hat{s}_k}]}$ by means of the BRST procedure with complete, $Q$, and incomplete, $Q_c$, BRST operators. Starting from a description of bosonic mixed-antisymmetric higher-spin fields in terms of an auxiliary Fock space associated with a special Poincare module, we realize a conversion of the initial operator constraint system into a system of first-class operator constraints. To this aim, we find, in first time, by means of Verma module the auxiliary representations of the constraint subalgebra, to be isomorphic due to Howe duality to $so(k,k)$ algebra, and containing the subsystem of second-class operators in terms of new oscillator variables forming the Fock module. An unconstrained (with $Q$) and constrained (with $Q_c$ and BRST invariant algebraic constraints) gauge Lagrangian formulations with equivalent dynamics, but different configuration spaces are found. Concept of consistent cubic interactions are suggested within $Q_c$-approach together with evaluation the structure of cubic vertices for $p\geq 3$ copies of interacting massless mixed-antisymmetric fields.

A positive integer is Zeckendorf-Niven (respectively, Lucas-Niven) if it is divisible by the number of summands in its Zeckendorf decomposition (respectively, Lucas decomposition). We show that there exist infinitely many Zeckendorf-Niven numbers and Lucas-Niven numbers in every arithmetic progression. Furthermore, we provide bounds on the maximum number of consecutive Zeckendorf-Niven terms in certain arithmetic progressions.

In this paper we construct compact manifolds without Kähler structures that admit both a symplectic form satisfying the Hard Lefschetz Condition (HLC) and another symplectic form that does not. Our construction builds upon the orbifold introduced by Fernández and Muñoz and its symplectic resolution studied by Cavalcanti, Fernández, and Muñoz. By considering a one-parameter family of symplectic forms on the orbifold, we show that the corresponding resolved manifolds fail to satisfy the HLC for all parameters. However, after performing a suitable symplectic blowup along a union of tori, we obtain a family of symplectic manifolds for which the HLC holds for all non-zero parameters but fails at the central parameter. As a consequence, we exhibit a smooth manifold with no Kähler structure whose space of symplectic forms contains both HLC and non-HLC structures in the same connected component. This provides new examples of the subtle interplay between symplectic topology and the Hard Lefschetz property.

In terms of the existence of a single clopen set and two related nets, we characterize when the full Boolean algebra, ${\mathfrak B}(G)$, of clopen subsets of a topological group $G$ is left introverted. We employ this characterization to show that when $G$ is a first countable, $\sigma$-compact, totally disconnected locally compact group, ${\mathfrak B}(G)$ is left introverted if and only if $G$ is compact or discrete, thus providing a strong positive answer to a question posed in Stephens and Stokke (Q J Math 2023). Examples of clopen sets and nets witnessing our non-introversion theorem are presented. Some hereditary properties of left introversion of ${\mathfrak B}(G)$ are proved and then employed to extend our main result to other classes of topological groups.

A microwave linear analog computer (MiLAC) is a tunable microwave network that performs linear operations directly on radio-frequency signals through wave propagation. Used as an antenna-array front end, it can map many antenna signals to a small number of active RF chains. While lossless reciprocal MiLACs have been shown to provide flexible or capacity-achieving beamforming for wireless communications, their sensing performance remains largely unexplored. We analyze direction-of-arrival estimation for $K$ far-field targets using a tunable receive-side lossless reciprocal MiLAC combiner. We show that the Fisher information matrix depends on the combiner only through the orthogonal projector onto its row space and never exceeds that of a fully digital receiver. Equality holds when the row space contains the $2K$-dimensional joint steering–derivative subspace, establishing a zero-gap threshold of two RF chains per target. A dimension-counting argument lower-bounds the number of tunable components required to achieve the digital Cramér–Rao bound for every target configuration. The stem-connected MiLAC attains this bound asymptotically, up to an antenna-count-independent additive overhead, while scaling linearly with the antenna and target counts. Unlike a phase-shifter front end with the same number of RF chains, MiLAC can exactly attain the fully digital bound. Numerical results validate the analysis.

This paper presents a revealed preference approach for rationalizing collective consumption behavior. We introduce the Constructive Rationalization Method (CRM), which approximates the real market via a surrogate market of artificial consumers, called androids, with easy-to-compute demand functions. CRM uses observed aggregate demand and adds artificial consumers on the fly, while redistributing wealth under an empirical risk minimization principle. Unlike classical revealed preference approaches, CRM provides guarantees on the generalization risk for learning the aggregate demand function, while respecting the privacy of the underlying consumers in the real market. As an application, CRM can be used to provide reliable predictions for collective consumption behavior. Specifically, we show how to apply CRM to approximate allocations that are proportionally fair without requiring the knowledge of individual utilities.

For any upper semilattice ${\cal D}$ of locally precompact topologies on a locally compact group $G$, we define an associated generalized spine subalgebra $A^*_{\cal D}(G)$ of the Fourier-Stieltjes algebra $B(G)$. We show that $A^*_{\cal D}(G)$ is a semilattice-graded $\ell^1$-direct sum of maximal copies of Fourier algebras and we identify its spectrum as a semilattice of groups. We build a collection of examples of generalized spine algebras over whose spectra we exhibit fine control. We define notions of compatible fusions of homomorphisms and affine maps, and use these definitions to characterize all completely positive, completely contractive and, when $G$ is amenable, all completely bounded homomorphisms from a generalized spine algebra $A^*_{\cal D}(G)$ to a Fourier-Stieltjes algebra $B(H)$. These results are new, even when $A^*_{\cal D}(G)$ is the full spine algebra $A^*(G)$ and even when $G$ and $H$ are abelian. We provide examples illustrating the scope of our theorems.

The Monge-Ampère-type equation for forms of positive degree was introduced by Dinew and Popovici to prove the qualitative part of an analytic version of Demailly's transcendental Morse inequality for higher cohomology classes, conditional on the solvability of this nonlinear PDE. In this paper, we show that this can be proven unconditionally. We first show that the originally proposed Laplacian trace condition, $\Lambda^{m-2}\Delta u = 0$, is analytically too rigid to permit solutions and hence introduce a general gauge-fixing that leads to the $(a,b)$ Monge-Ampère-type equation. By deriving a priori estimates, we establish the solvability of this class of equations for different parametric values $(a,b)$. As a geometric application, for $b=0$ this framework reduces to the classical complex Monge-Ampère equation, which yields the qualitative Demailly's inequality for higher-degree forms. Further, we prove that the uniqueness of these solutions can be significantly strengthened under the kernel condition $\bar{\partial}^* u = 0$.

The Fokker-Planck equation is fundamental to statistical mechanics, yet in settings with multiple state variables, anisotropic (cross-) diffusion, and jumps, conventional discretizations frequently produce non-physical negative probability densities. Building on the operator approach of "A. Itkin, Pricing derivatives under Levy models. Modern finite difference and pseudo-differential operators approach, Springer, 2017, ISBN 978-1-4939-6792-6", we introduce a family of "Diagonal Frog" discretizations whose spatial operators are eventually M-matrices (EM-matrices). Although these operators lack a local M-matrix structure, positivity of the directional sub-operators emerges in the spirit of Zeno's paradox: the matrix exponential, assembled as the limit of infinitely many ever-smaller substeps, is provably nonnegative after a short transient even though no single substep is. For the mixed-derivative block, whose generator is not eventually nonnegative, positivity instead rests on a factorized resolvent solver and holds conditionally, on an explicit step-size window; discrete mass is conserved exactly by the splitting for every step size. The resulting schemes are second-order accurate in time and space and require O(m 2 N + m 3) operations per time step, where m is the dimension of the Krylov subspace used to apply the exponential. As stress tests, we solve a two-dimensional anisotropic Fokker-Planck equation in the strong cross-diffusion regime against an exact Gaussian reference, a Kramers escape problem in a double-well potential, and an advection-dominated problem, and observe that the schemes remain stable, nonnegative, and mass-conservative for a wide range of Pécklet numbers (so, don't need any flux limiter). Finally, we extend the construction to multidimensional processes and to the backward Kolmogorov equation with jumps.

We develop a detailed analysis of optimal traveling waves $U(t,x) = U(x - \beta t)$ for a model of invasive-species control proposed in [Bressan, Chiri, and Salehi, Math. Models Methods Appl. Sci., 2022] : the relative density $U \in [0,1]$ of the invasive species satisfies the following reaction-diffusion equation with a positive control \begin{equation} \label{Equa:PDE_abstract} U_t = U_{xx} + f(U) - \tilde \alpha(t,x) U, \quad U \in [0,1], \ \tilde \alpha \geq 0. \end{equation} The control $\tilde \alpha(t,x)$ represents the fraction of the population removed at $(t,x)$: the minimal control effort $E(\beta,f)$ required to sustain a traveling invasion front with prescribed speed $\beta$ is defined as the minimal $L^1$-norm of $\tilde \alpha$ for a traveling wave solution $U(x-\beta t)$ to the PDE. In order to study large scale dynamics $(t,x) \mapsto (\epsilon t,\epsilon x)$, a fundamental role is played by the structure of traveling waves and the convexity and regularity properties of $E$. The main results of this paper are the following: 1)In the phase plane $(U,P=U_x)$, there exists a unique optimal profile $P_\beta(U)$ minimizing the effort; 2) It satisfies explicit first-order conditions, which are both necessary and sufficient; 3) The associated control is acting on an open subset of the set $\{U : P_\beta(U) = \sqrt{U f(U)}\}$, in particular it is uniformly integrable, and it depends smoothly on $(\beta,f)$ on a dense open set; 4)The effort function $E(\beta,f)$ is only $C^1$ w.r.t. $\beta$ and Lipschitz w.r.t. $f$ in the $C^2$-topology, and is asymptocally linear for $\beta \to \infty$; 5)$\beta \mapsto E(\beta,f)$ is in general neither convex nor subadditive.

The main goal of this paper is to establish a general Lorentzian Lasry-Lions regularization theorem: let $u$ be a function defined on a globally hyperbolic spacetime. Assume that its forward Lax–Oleinik evolution $Tu$ is locally semiconcave in a neighbourhood of $(t_0,y_0)$ and has future-directed timelike superdifferentials there. Then, for $t$ close to $t_0$ and sufficiently small $s>0$, the function $\hat T_s\circ T_tu$ is of class $C_{\mathrm{loc}}^{1,1}$ in a neighbourhood of $y_0$. We give sufficient conditions ensuring the assumptions of the theorem and present an application to optimal transport: under quite general assumptions, for any two intermediate measures along a displacement interpolation, there exists a $C^{1,1}_{loc}$-regular maximizing pair in the dual formulation.

A hierarchy of regularised Wallis products is introduced by raising the reciprocal Wallis factor \[ 1-\frac1{n^2} \] to the polynomial weight $n^m$, $m=0,1,2,\ldots$. For each $m$, a minimal exponential counterterm is chosen by cancelling precisely the non-summable terms in the logarithmic expansion. This gives a convergent product $P_m$ the logarithm of which is an explicit zeta-function tail. The first non-trivial examples are \[ \prod_{n=2}^{\infty} e^{1/n} \left(1-\frac1{n^2}\right)^n = \frac{e^\gamma}{2}, \qquad \prod_{n=2}^{\infty} e\left(1-\frac1{n^2}\right)^{n^2} = \frac{\pi}{e^{3/2}}. \] The even branch has a finite closed form involving $\pi$, harmonic numbers, and odd zeta values. The odd branch reduces to finite logarithmic gamma moments, and hence to constants involving $\gamma$, logarithms, odd zeta values, and derivatives of the zeta function at positive even integers. The same subtraction rule also gives a two-factor extension involving the companion factor $1+1/n^2$. Finally, the associated $x$-dependent products factor into one-sided canonical products, giving a direct connection with Kurokawa's multiple sine functions: the even Wallis branch is obtained from odd multiple sine functions, while the odd branch appears as a symmetric companion to the even multiple sine case.

This paper proves that $L_\infty$ spaces over a dg manifold form a category of fibrant objects. Together with the first main result of the companion paper [CJ26], this implies that transitive $L_\infty$ algebroids over a dg manifold also form a category of fibrant objects.