Papers

We construct and classify all global solutions to the singular equation $$-\Delta u=f(X)\cdot u^{-\gamma}$$ supported in a general Lipschitz epigraphical cone, where $f(X)$ is a locally Dini continuous function with $0<\lambda\leq f(X)\leq\Lambda$. The existence and non-existence of a global solution is solely determined by the exponent $\gamma$ of the equation and the “frequency" of the cone. Moreover, in order to classify all global solutions, we introduce several new methods. First, we use the local data to estimate the global growth rate, which in turn establishes the boundedness of the “asymptotic slope" of the global solution. Second, by establishing a nonlinear variant of Kemper's boundary Harnack principle, we classify all global solutions through an “oscillation reduction" argument on the “asymptotic slope".

Plane hypermaps are plane maps endowed with a proper coloration of their inner faces in black or white. We consider the problem of enumerating plane hypermaps with prescribed face degrees and a $k$-alternating boundary condition: by this we mean the colors of inner faces incident to the outer face alternates at most $2k$ times when turning around the hypermap. The present paper deals with the cases $k=1,2$, the general case being left to the forthcoming part II. Our approach relies on the so-called slice decomposition and uses crucially the notion of accessibility, which exploits the canonical orientation of hypermaps and the marking variable $t$ associated with vertices, to enumerate pointed hypermaps by decomposing them according to the set of all vertices that can access to the marked vertex. This process enables us to express the generating functions of hypermaps with mixed boundaries in terms of the generating functions of hypermap slices and to recover, in a purely combinatorial way, some formulas previously obtained through algebraic methods.

In two dimensions, we show that dissipation in one spatial direction is sufficient to enforce the energy equality for every weak solution at the natural energy level. In particular, neither anomalous energy loss nor creation can occur. The main difficulty is that the missing directional regularity prevents the usual self-testing argument. We overcome this obstruction through two observations: The pressure is square-integrable by a directional Riesz-transform estimate, and the less regular component is still a renormalized solution. As an application of energy rigidity, we derive a weak-strong uniqueness principle.

In this paper, dissipative singular $q$-Dirac operators are examined in Hilbert space $\mathcal{L}_{q}^{2}(q^{{\mathbb{Z}}};\mathbb{C}^{2})$, where they arise as extensions of a minimal symmetric operator in the limit-point case. We develop a selfadjoint dilation and get its incoming and outgoing spectral representations, enabling the explicit computation of the scattering matrix associated with the dilation. Furthermore, we construct a functional model for the dissipative operator and express its characteristic function in terms of the Titchmarsh-Weyl function of the corresponding selfadjoint operator. Finally, we establish results concerning the completeness of the system of eigenvectors and associated vectors for these dissipative Dirac operators.

For fixed graphs $H$ and $F$, let $\ex(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. In this note, we prove the generalized rational exponents conjecture, posed by Gerbner and Palmer, showing that for every rational number $\alpha\ge1$, there exist fixed graphs $H_\alpha$ and $F_\alpha$ such that \[ \ex(n,H_\alpha,F_\alpha)=\Theta(n^\alpha). \] Furthermore, the counting graph $H_\alpha$ can always be chosen connected with diameter at most $3$. Our argument hinges on a localization–compression–shift framework, which transforms the Bukh–Conlon finite family construction for edges into a generalized Turán problem setting with a single forbidden graph.

Let $A$ be a ring, let ${\bf T}$ be a finite-term big tilting complex over $A$, and choose a bounded projective representative $\cpx{P}$ of ${\bf T}$. Put $B=\End_{\D{A}}(\cpx{P})$. We prove that the kernel of the associated derived tensor functor ${\bf R}:\D{B}\to\D{A}$ is equivalent, through the derived inclusion, to $\D{\mathscr E}$, where $\mathscr E=\Ker({\bf R})\cap B\Modcat$ carries the inherited exact structure. Consequently, $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof combines a dg double-centralizer argument with a functorial realisation of the kernel on unbounded complexes. The category $\mathscr E$ is definable and $d$-symmetric whenever $d$ is at least the projective amplitude of the underlying perfect right $B$-complex. We also characterise when the kernel is induced by a homological ring epimorphism: this holds precisely when $\mathscr E$ is an abelian subcategory, or equivalently when the standard $t$-structure restricts to the kernel. In amplitude at most one, this recovers the universal-localisation description for two-term big tilting complexes. Over semiperfect endomorphism rings, the amplitude is detected by reduction modulo the Jacobson radical. Finally, we construct non-compact examples of arbitrarily large minimal projective amplitude. One family yields algebraic Calkin quotients; another, over connected rings constructed from singular local hypersurfaces, has nonzero, non-abelian exact kernels and cannot be obtained from modules by derived equivalences between ordinary rings.

Phylogenetic networks are graphs that represent the evolutionary history of species. Recently, the class of orchard phylogenetic networks, which can be reduced by so-called cherry-picking sequences, has gained attention for its computational and biological aspects. In this paper, we study a fundamental question on orchards and their cherry-picking sequences by considering the CoveringNumber problem: given an orchard network $N$, how many cherry-picking sequences are needed to reduce all subnetworks of $N$? We initiate this study by considering the problem for trees. We then show that the covering number can be computed for binary trees recursively using a similar but more fine-grained notion of survival covering number. We also give a recursive formula for the survival covering number of non-binary trees. However, computing the covering number for non-binary trees appears to be considerably more challenging. For this case, we show that the covering number of star trees (whose root is adjacent to all leaves) is equivalent to the so-called SubsetConnectivity problem, which we introduce in this paper. Finally, we show that if there is no restriction on the sequence length, a single sequence of minimum length $\binom{n}{2}$ suffices to reduce all subtrees of a tree on $n$ leaves.

We address the practical computation of feedforward steering inputs for a car with n trailers via differential flatness. The standard route (iterated symbolic differentiation of the flat output) scales prohibitively with n: even with computer-algebra software the expression sizes grow super-exponentially and become intractable beyond n approx. 5. We propose an algorithm that combines four ingredients (angular intermediate variables, factorisation of the recursion in submaps, rescaled higher-order product/quotient rules, and a composition rule based on truncated formal power series instead of Faa di Bruno's formula). We benchmark the proposed method against (i) direct symbolic differentiation in SymPy and (ii) Faa di Bruno's formula evaluated via Bell polynomials. Both reference methods hit a clear computational wall: SymPy reaches a 20 s timeout at derivative order r = 14, while the Bell-polynomial recursion exceeds a 60 s timeout beyond r = 24. The proposed approach has O(r3) arithmetic complexity in the derivative order and remains under one millisecond up to r = 40, with numerical agreement to machine precision in the regime where the reference methods succeed. As an end-to-end illustration, a 20-trailer parking manoeuvre is computed and animated.

This paper is devoted to the study of the numerical range of linear relations in Banach spaces. We present a new definition adapted to the multivalued nature of linear relations and analyze its main properties. We establish spectral inclusion results showing that the spectrum is contained in the union of the closure of the numerical range of a linear relation and the numerical range of its Banach adjoint, together with resolvent estimates related to the distance to the numerical range. As an application, we derive spectral enclosures for operator pencils by associating them with suitable linear relations and introduce corresponding numerical ranges for operator pencils in Banach spaces. We show that this approach may provide sharper information than classical numerical ranges of operator pencils. Furthermore, we use the numerical abscissas to show that the Banach space numerical range can yield strict exponential decay for semigroups that cannot be obtained from the corresponding Hilbert space numerical range.

We consider the inhomogeneous doubly nonlinear parabolic systems of the form \begin{equation*}\partial_t (|u|^{q-1}u)-\operatorname{div}(|Du|^{p-2}Du)=\operatorname{div}(|F|^{p-2}F)\end{equation*} in a bounded space-time cylinder $\Omega_T=\Omega\times(0,T)\subset \mathbb{R}^{N+1}$. We study the local regularity properties for weak solutions in the subcritical range $p\leq\frac{N(q+1)}{N+q+1}$ and $0<q<\frac{N+2}{N-2}$. Under an extra integrability assumption $|u|\in L_{\loc}^{\rr}(\Omega_T)$, we establish a quantitative bound for $|u|$. Here, the exponent $\rr$ satisfies $\mathrm{\lambda}_{\rr}=N(p-q-1)+p\rr>0$. In addition, we prove local higher integrability of $|Du|$ in the range $p=\frac{N(q+1)}{N+q+1}$ and $\frac{N+p}{N-p}<q<\frac{N+2}{N-2}$, provided that $|u|\in L_{\loc}^{\rr}(\Omega_T)$ holds.

A learning architecture induces a prior over its expressible models. Recent work links architectural success to a broad complexity spectrum: model probabilities spanning many orders of magnitude. Understanding how depth shapes these priors and prediction may help explain the success of neural networks and transformers. We study this connection in large-alphabet prediction through the *layered simplex architecture* (LSA), which multiplies independent uniform simplex vectors coordinate-wise and normalizes. At depth zero, the prior is concentrated on the uniform distribution; at depth one, the Bayesian predictor is Laplace's rule. Each additional layer shifts prior mass toward more concentrated distributions. We derive exact Bayesian marginal likelihood and regret expressions using reusable depth-specific kernels. Mixing over depths up to a cutoff logarithmic in alphabet size yields an explicit predictor with rigorous guarantees. For targets uniform on a specified sparse support, the mixture's prior complexity is asymptotic to the support-description cost at every fixed positive KL resolution, establishing a broad complexity spectrum. We also prove a regret lower bound for every label-invariant predictor: the expected description length of the discovered symbol set minus the source's uncertainty about its identity given its size. For Zipf sources in a growing-alphabet regime, the depth mixture attains this bound to leading order. Thus, the symbol-discovery rate determines the regret exponent. Although introduced to study depth, the LSA predictor competes with established large-alphabet methods, including Good–Turing, Ristad's law and an adaptive Dirichlet mixture, while substantially outperforming classical add-constant rules on concentrated sources.

Let $G=N\rtimes H$ be a finite Frobenius group with $|N|=q$ and $|H|=q-1$. We classify all Ramanujan Cayley graphs of $G$ whose connection sets are normal, in the sense of being unions of conjugacy classes. The group-theoretic input is a simple blow-up phenomenon: every such Cayley graph is either $Y[\overline{K_q}]$ or $Y[K_q]$ for a connected regular Cayley graph $Y$ on the complement $H$. We first prove a graph-theoretic result classifying all Ramanujan graphs of these two forms when $Y$ is an arbitrary connected regular graph on $q-1$ vertices. The proof combines the classical characterization of regular graphs with least eigenvalue greater than $-2$ with a second-moment identity in the bipartite case. Translating the resulting five graph types back to $G$ yields a complete classification for all ratio-one Frobenius groups, and in particular for $\operatorname{AGL}(1,q)$ over every finite field.

Let $\mathsf{P}$ be a finite graded poset and $\Delta_a^{\mathsf{P}}$ the height-$a$ thickening of its diagonal, with projections $q_1,q_2$ to $\mathsf{P}$. We study the window convolution $C_a=\operatorname{Lan}_{q_1}\circ q_2^\ast$ on $\mathrm{Shv}(\mathsf{P};k)$, a discrete analogue of convolution against a thickening kernel. An interleaving distance needs the homotopy window convolution $\mathbb{C}_a$ to compose as a flow, $\mathbb{C}_a\mathbb{C}_b\simeq\mathbb{C}_{a+b}$; where the meet assignment $\Phi$ is total it is a functor and carries a comparison map. Finality is sufficient, and necessary at every minimal apex and wherever the finality defect of $\Phi$ is essential; where $\Phi$ is total at a minimal apex with unit windows, it is the failure of a length-two interval above the apex to have a single interior element. The flow fails at every branching length-two interval with minimal bottom element, and with it on the face poset of every finite regular cell complex of dimension $\ge2$. It survives on tame posets, where $\mathrm{id}\Rightarrow\mathbb{C}_a$ gives a canonical extended interleaving pseudometric on $\operatorname{D^{b}}(\mathrm{Shv}(\mathsf{P};k))$; in the saturation cases computed here it takes no finite value above the length of $\mathsf{P}$, and is finite if and only if the derived colimits agree.

Let $F_1=F_2=1$ and $F_{t+2}=F_{t+1}+F_t$ for $t\geq1$. For every $n\geq2$, there are unique integers $a,b,t$ such that $n=aF_t+bF_{t-1}$ with $t\geq2$ and $1\leq a\leq b\leq F_t$. The Fibonacci walk with initial pair $(b,a)$ reaches $n$ as late as possible, and the term following $n$ in this walk is $\lfloor\phi n\rfloor$ when $t$ is even and $\lceil\phi n\rceil$ when $t$ is odd, where $\phi=(1+\sqrt5)/2$. Let $D=\{d_1<d_2<\cdots\}$ and $U=\{u_1<u_2<\cdots\}$ be the sets corresponding to even and odd $t$, respectively. For $\ell,m\geq1$, define $D_\ell=\{d_{k+\ell}-d_k:k\geq1\}$, $U_\ell=\{u_{k+\ell}-u_k:k\geq1\}$, $D_\ell(m)=\{d_k:d_{k+\ell}-d_k=m\}$ and $U_\ell(m)=\{u_k:u_{k+\ell}-u_k=m\}$. Chung, Graham and Spiro conjectured that $D_\ell=U_\ell$ for all $\ell$, and asked for the densities of $D_\ell(m)$ and $U_\ell(m)$, especially when $\ell=1$. In this paper, we determine the third and fourth order gap spectra, and show that the conjecture holds for $\ell=3$ but fails for $\ell=4$. We also answer their density question by characterizing when $D_\ell(m)$ and $U_\ell(m)$ have natural densities and proving that their logarithmic densities always exist and are equal. For $\ell=1$, we give the exact logarithmic densities.

A combinatorial game is a two-player game without hidden information or chance elements. The main object of combinatorial game theory is to determine the outcome (i.e., which player has a winning strategy) of a given position in combinatorial games. Nim is a well-known and fundamental ruleset in combinatorial game theory. This paper proposes a novel partizan variant of Nim called Partizan-Serial-Nim, defined as follows: there are $n$ piles of stones indexed by $1, 2, \ldots, n$; the two players have permutations $\mathbf{\sigma}^L$ and $\mathbf{\sigma}^R$ of $(1, 2, \ldots, n)$, respectively; a move is to remove any positive number of stones from the non-empty pile with the minimum value in the player's permutation; the player who cannot make a move loses. This ruleset is a generalization of Serial-Nim and Partizan-End-Nim. We give an algorithm to compute the outcome of a given position in Partizan-Serial-Nim in $O(n^2)$ time, provided that each arithmetic and comparison operation is performed in $O(1)$ time. Also, for the case where all non-empty piles have the same number $m$ of stones, we prove that the outcome does not depend on $m$ for $m \geq 2$ and present an algorithm to compute the outcome in $O(n)$ time. Further, we prove that the atomic weight of every position in Partizan-Serial-Nim is an integer.

We study a function-on-function linear regression model in which the response at time $t$ depends on both the past trajectory of a predictor and its concurrent value. The model combines an $L^2$-historical effect with a point-evaluation effect, and these two coefficient functions are not automatically identifiable. We characterize the resulting non-identifiability and show that the concurrent and historical effects are separately identifiable whenever the covariance eigenfunctions of the covariance operator of the predictor are not pointwise square-summable. This mild and novel condition prevents the concurrent point evaluation from being represented by an $L^2$-historical effect. Building on an orthogonalized representation of the predictor process, we propose a smoothing-spline estimator for both coefficient functions and establish consistency rates. The rates reveal an interesting trade-off between path regularity and eigenvalue decay: smoother predictor trajectories lead to faster convergence of the historical-effect estimator, whereas rougher trajectories lead to faster convergence of the concurrent-effect estimator. Simulation studies and two real-data applications demonstrate the practical importance of disentangling concurrent and historical effects.

Let K_q(n,R) denote the minimum cardinality of a q-ary code of length n with covering radius R. We improve the known bounds on K_q(n,R) in 83 cases (82 distinct cells). On the upper-bound side we give 25 improved bounds for 5<=q<=15 – twenty-four found by search and one propagated by monotonicity – using two complementary methods: an engineered focused local search seeded with structural constructions, and a large-neighbourhood search driven by exact full-space coverage transforms that evaluates every candidate codeword position simultaneously. These are, to our knowledge, the first improvements to any upper bound on K_q(n,R) with q >= 5 since the 2011 revision of Keri's tables; several bounds decrease by more than 20%, e.g. K_6(8,4)<=166 (previously 216) and K_8(10,5)<=1883 (previously 2461). On the lower-bound side we give 58 improved bounds for 6<=q<=21, obtained from the semidefinite programming hierarchy of Gijswijt and Polak, whose published results cover q<=5, by combining an exact-arithmetic reimplementation of the reduced program with a multiprecision solution pipeline. Every new lower bound is certified by a rational dual solution validated by a standalone exact-arithmetic checker; no floating-point computation is part of the trusted base. The same pipeline also gives strong numerical evidence of limits: on a dozen further cells the certified value of the relaxation, which the solver reports as optimal to within its working precision, lies below the best known bound, indicating that no improvement is available there at this level of the hierarchy. One cell is improved from both sides: 441<=K_6(10,4)<=2751, previously 417–2952. All codes and certificates are provided in machine-readable form together with standalone verifiers.

We give a short, conceptual account of Dwyer–Kan's hammock localization in the setting of $\infty$-categories. Starting from Mazel-Gee's formula for localization via the relative Rezk nerve, we show that its Segalification can be described explicitly as a Segal anima of zig-zags. We also compute its mapping animae and recover and generalize Dwyer–Kan's hammock formula. As an application, we show that, when a relative $\infty$-category supports fractions, the mapping animae of localization admit a simple description. This gives a unifying treatment for several formulas of this type in the existing literature.

Marton's inner bound, the best-known achievable region for a general discrete memoryless broadcast channel, was proposed by Katalin Marton in 1979, and whether it always achieves the capacity region has remained open since then. In this paper, we establish its strict sub-optimality: we show that the capacity region of some discrete memoryless broadcast channels can be strictly larger than Marton's inner bound.

We introduce a model with control vacations instead of standard queueing control systems with permanent control. The researched model is an M/M/1 queue with temporary periods of service rate control with two available service rates. After a control period of exponentially distributed length, a control vacation is initiated during which a fixed service rate $\mu$ is used. The start of the next control period needs to be scheduled directly at a certain cost. We will use the Markov Decision Process from Kanavetas et al. arXiv:2605.31573 to find a sufficient condition that ensures that the average expected cost can be reduced w.r.t. the model that only uses the fixed service rate. Under this condition we will use properties of this related process to construct a cost reducing policy. The process with control vacations under specific policies induces a renewal reward process. We use a Tauberian theorem to relate the average expected cost of this renewal reward process to a vanishing discount method and analytically determine a lower bound of the average expected cost reduction. Finally, we study the actual attained average cost reduction for these policies through simulation.

Vizing conjectured that the list edge chromatic number of any graph with maximum degree $\Delta$ is at most $\Delta + 1$. This conjecture has been confirmed for several important classes of graphs, in particular, Lang proved that it holds for all graphs of treewidth $3$. In this paper, we introduce the list edge coloring of signed graphs, a framework that generalizes both classical list edge coloring and the signed edge coloring introduced by Behr. We extend Lang's result by proving the signed analogue of Vizing's conjecture for all signed graphs of treewidth $3$, as well as for signed graphs of treewidth $4$ with maximum degree $\Delta \ge 10$.

We study a shape optimization problem among convex bodies in $\mathbb{R}^n$ that minimize or maximize weighted perimeters of the form $\int_{\partial\Omega} \phi(|x|) \, \mathrm{d} H^{n-1}(x)$ under a standard perimeter constraint. We prove the existence of extremals for general weight functions in any dimension. In dimension two, we prove that the degenerate needle configuration $(-a,a)\times \{0\} \subset \mathbb{R}^2$ is the optimizer for a wide family of weights, including $|x|^p$ for $p \in (0,2]$ and $|x|^{-\alpha}$ for $\alpha \in (0,1)$, among convex curves satisfying a symmetry assumption.

Resilience has emerged as a fundamental capability for future wireless systems operating in dynamic and uncertain environments. Although resilience has attracted growing attention across academia, industry, and standardization, its conceptual scope, enabling mechanisms, and realization techniques remain fragmented. This paper presents a systematic framework of resilience in wireless systems from a trustworthiness perspective. We first formalize the concept of resilience and distinguish it from related uncertainty-aware terminologies, including reliability, robustness, adaptability, survivability, and recoverability. We then establish a hierarchical framework that organizes resilience into capability dimensions and enabling mechanisms, and that quantifies resilience through different technical aspects. We further use physical links and unmanned aerial vehicle networks as representative wireless scenarios to demonstrate how resilience can be systematically realized through the joint design of architectures, operations, and algorithms. Finally, we examine the fundamental trade-offs in resilience engineering: Improving resilience generally incurs costs in resource efficiency, nominal performance, information acquisition, implementation complexity, and latency.

Augmented Lagrangian methods are effective for nonlinear equality-constrained optimization, but solving their nonlinear primal subproblems can be expensive. For smooth nonconvex problems with deterministic or stochastic objectives, we propose a nonlinear-residual linearized augmented Lagrangian method (NR-LALM) that replaces this subproblem by a regularized Gauss-Newton-type step while retaining the classical multiplier update based on the nonlinear constraint residual. The resulting step is computed from one symmetric positive-definite linear system, but the mismatch between the linearized primal model and the nonlinear-residual update produces a quadratic constraint-linearization error in the multiplier identity. We show that this error can be controlled under local regularity; multiplier boundedness and trajectory localization are derived rather than assumed. With fixed, accuracy-independent parameters, deterministic NR-LALM finds an $\varepsilon$-approximate Karush-Kuhn-Tucker (KKT) pair in $O(\varepsilon^{-2})$ iterations and first-order oracle evaluations. For stochastic objectives, a projected stochastic path-integrated differential estimator with safeguarded restarts requires, in expectation, $O(\varepsilon^{-3})$ stochastic-gradient evaluations and $O(\varepsilon^{-2})$ constraint and Jacobian evaluations. Compactness and a Kurdyka-Lojasiewicz condition further yield finite-length convergence of the deterministic primal-dual sequence. An optional minimum-norm second-order correction reduces the constraint-linearization error from second to fourth order without changing the complexity orders. All theoretical results are formalized in Lean 4. Numerical experiments confirm the predicted error orders and show favorable performance on high-dimensional deterministic and stochastic problems.

For $n\geq 4$, let $c(n)$ be the minimum number of distinct circles containing at least three points of an $n$-point set in the Euclidean plane, where the set is neither collinear nor concyclic. Put[F(n)=1+\binom{n-1}{2}-\left\lfloor\frac{n-1}{2}\right\rfloor.]We determine $c(n)$ for every $n\geq 4$: it equals $F(n)$ apart from three exceptional orders. We also solve the variant in which no three points are collinear; that variant has a single exceptional order. The proofs and exact finite verifications were developed through a collaboration between human researchers and artificial-intelligence systems.

This paper proposes a new technique for computer simulation of linear filters. It allows simulating continuous-time linear filters based on the spectral method for analyzing linear control systems. Applying the spectral method implies that the input and output signals are represented by ordered sets of expansion coefficients in a chosen basis, and the filter itself is specified by a two-dimensional nonstationary transfer function. The described technique is tested on Bessel, Papoulis, and Legendre filters of various orders. For each filter, the corresponding two-dimensional nonstationary transfer function, i.e., the matrix of the linear transformation relating expansion coefficients of the input and output signals, is obtained.

The dimension of a polynomial spline space of bi-degree $(d_1,d_2)$ over a T-mesh $\mathscr{T}$ with the highest order of smoothness $(d_1-1,d_2-1)$ depends on both mesh topology and geometric configurations. Under the assumption that the T-connected components of the T-mesh $\mathscr{T}$ contain no vanishable T $l$-edges, we develop explicit upper and lower bounds of the dimension of the polynomial spline space. By introducing a decoupling technique within the completely non-diagonalizable component (CNDC) of the T-mesh $\mathscr{T}$, we separate tightly coupled multi-vertex constraints and transform global conformality conditions into localized linear equations along each interior large edge. Based on the decoupling technique, a new dimension formula of the polynomial spline space is then presented, and from which sharp upper and lower bounds of the dimension are obtained. The bounds are sharp in the sense that different geometric realizations of T-meshes with the same topology can attain the lower and upper bounds for the dimension of the polynomial spline space. We further prove that the new formula is consistent with Mourrain's homological dimension formula, and a sharper lower bound is obtained by our method.

We study a symmetric system of transport equations with solenoidal coefficients. This system reduces to an evolutionary equation with a skew-symmetric spatial operator in the real Hilbert space of square-integrable vector-functions, and by general results we claim that there always exists a generalized solution of the Cauchy problem. Uniqueness of this solution is equivalent to skew-adjointness of the spatial operator. We demonstrate that in the case of locally Lipschitz coefficients satisfying a linear growth condition the spatial transport operator is indeed skew-adjoint. For scalar transport equation this result remains true under the weaker DiPerna-Lions conditions.

Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[ \textsf{P}\left\{ \left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert >t \right\} \le C\exp\left[-c_K\mathcal I_{n,d}\left( \frac{t}{L n^{(d-1)/2}} \right)\right], \] where \[ \mathcal I_{n,d}(s)= \min\left\{ \frac{s^2}{d^2}, \frac{s^2}{d\log(e+nd/s^2)} \right\},\qquad s>0, \qquad \mathcal I_{n,d}(0)=0. \] The first rate is forced by changes in $\|X\|$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears.

Causal Bayesian networks (CBNs) and structural causal models (SCMs) are the dominant frameworks for graphical causal reasoning, but they cannot adequately represent all real-world causal systems. In particular, systems at equilibrium—where feedback mechanisms create cyclic causal dependencies—can exhibit causal semantics that are fundamentally incompatible with these frameworks: different interventions that enforce the same variable value may have different effects, rendering the standard “perfect intervention” do($X = x$) ambiguous. We propose bipartite graphical causal models (BGCMs), in which the structure of a system of equations is encoded by a bipartite graph with variable and equation nodes. In this framework, a hard intervention do($f_j : X_v = \xi_v$) specifies which equation is replaced, which variable is targeted, and at what value—resolving the ambiguity of the standard notion. We demonstrate, through a detailed case study of a physical system, that this representation naturally corresponds to distinct real-world interventions. We formulate a Markov property in terms of a new graphical separation criterion (B-separation) that exploits the functional determinism inherent in the equations, and we extend it to settings with non-random inputs. We show how this gives rise to a do-calculus for reasoning about domain invariances. BGCMs strictly generalize CBNs and SCMs while retaining the ability to perform graphical causal reasoning.