Papers

Pairwise comparisons are fundamental in the analytic hierarchy process. Various consistency indices have been proposed to assess inconsistencies in these comparisons. Since Saaty first proposed his consistency index, the assessment of the degree of consistency in pairwise comparison matrices has remained an open and hot topic in the study of the analytic hierarchy process. The consistency indices CI and CR proposed by Saaty are defined using the principal eigenvalue of the pairwise comparison matrix. In our previous study, we introduced an alternative index derived from the relationship between the coefficient of the characteristic polynomial and the consistency of comparisons. Saaty proposed a fixed threshold of 0.1 for CI or CR as a guideline for an acceptable level of consistency, regardless of the matrix size. However, whether this threshold represents an equivalent level of consistency across different matrix sizes, that is, across different numbers of evaluation items, remains unclear. This study analysed the relationship between consistency and matrix size by examining pairwise comparison matrices constructed from subsets of evaluation items. Based on this analysis, we propose the fundamental property to be satisfied by a size-independent consistency index. Furthermore, we refine our previously proposed index to ensure that it satisfies this property, demonstrating that it coincides with the existing consistency index. Finally, we visualise the relationship between the matrix size and consistency index values using randomly generated pairwise comparison matrices, thereby providing insights into the impact of matrix size on consistency evaluation.

We prove that for a Manneville–Pomeau type interval map, the set of points whose orbit closures miss a prescribed countable set is absolute winning in the sense of McMullen. The proof has three parts. First we directly prove that the exceptional set for the distinguished endpoint of the induced first-return map is absolute winning. Then we use the finite-branch winning theorem of Hu–Li–Yu, together with the one-dimensional implication from $1/2$-strong winning to absolute winning, to obtain absolute winning for all countable induced targets. Finally, a quasisymmetric pullback argument transfers these induced results back to the original map.

For irreducible lattices in semisimple Lie groups of real rank at least $2$, we prove a cohomological vanishing result implying that any action on a CAT(0) cube complex fixes a vertex whenever every hyperplane stabilizer is solvable. As an application, we prove regularizability for actions of all higher-rank lattices by birational transformations on projective surfaces. We first use superrigidity for actions on infinite-dimensional real hyperbolic spaces to reduce to the de Jonquières group, and then apply our fixed-point theorem to the Jonquières complex. Our proof bypasses the direct use of property FW.

We investigate the minimum size of finite abelian Cayley graphs that admit an isometric embedding of a finite connected graph. While every connected graph on n vertices embeds isometrically into a binary Cayley graph of dimension at most n-1, the smallest possible abelian host has remained largely unexplored. We establish fundamental lower bounds showing that every binary host has dimension at least max(diam(G), floor(log2 n)), whereas every finite abelian host has order at least max(n, 2^diam(G)). Moreover, we prove that the minimum host order equals n if and only if G is itself an abelian Cayley graph. Exact binary dimensions are obtained for several important graph families. Hypercubes, complete graphs of order 2^k, and even cycles attain the lower bound. For stars we prove k_min(K1,q)=floor(log2 q)+1 using maximum sum-free sets, yielding an exponential improvement over the naive and isometric dimensions. For odd cycles we prove k_min(Cm)=m-1 for all m<17 and reduce the general case to a cyclic-interval lemma, showing that the universal upper bound is tight. Our computational contribution is a certified exhaustive census of all 995 connected graphs with 2<=n<=7 vertices under general abelian compactifications. The data reveal an "abelian dividend": 569 graphs (57 percent) admit a strictly smaller abelian host than the best binary host, 707 (71 percent) admit an optimal host containing a cyclic factor Zm with m>2, and only 17 graphs attain the theoretical order floor max(n,2^diam(G)). These results demonstrate that compact non-binary abelian hosts are typical rather than exceptional, while binary hosts remain the universal worst-case construction. 2020 MSC:05C12, 05C25, 05C30, 11B75, 20K01

We give the first pair of non-diffeomorphic exotic $\mathbb{R}^4$'s made by attaching diffeomorphic Casson handles onto diffeomorphic disk complements. Our examples are obtained using the RBG link construction to find slice knots with diffeomorphic slice disk complements, but whose Whitehead doubled disk complements are not diffeomorphic. We distinguish the exotic $\mathbb{R}^4$'s using end Floer homology.

A set \(\mathcal{G}\) of integers is a \(g\)-Golomb ruler if each positive difference appears at most \(g\) times between any 2 elements of the set, and \(G(g,n)\) denotes the minimum diameter of such a ruler with \(n\) marks. We prove a general lemma for passing from certain modular constructions to ordinary \(g\)-Golomb rulers. The key point is that, in a modular \(g\)-Golomb ruler, no cyclic gap length can occur more than \(g\) times. This gives a larger guaranteed cut than the previous average gap argument. We apply this lemma to cyclic relative difference sets, Singer sets, Ruzsa–Spence rulers, and Paley quadratic residues to provide many competing constructions for \(g\)-Golomb Rulers. A computation on the grid \(1\le g\le500\), \(n=g+b\), \(2\le b\le500\), compares the four resulting construction families.

We study generalized global symmetries of the bosonic M2-brane in eleven-dimensional backgrounds with non-trivial four-form flux. Focusing on compactifications of the form $AdS_5\times S^2_{1} \times S^2_{2}\times T^2$, we show that the monopole and winding symmetry sectors of the membrane do not organize as an ordinary direct product once the Wess–Zumino coupling is included. Instead, they combine into a non-trivial 2-group global symmetry. We identify the corresponding mixed background-gauge structure and show that the quantized target space flux determines the associated Postnikov class. This provides a concrete realization of higher-group symmetry in the worldvolume theory of the M2-brane induces a flux quantization on the worldvolume and relates its global symmetry structure to the introduction of a quantized flux in M-theory.

A factorization exact for the continuous Laplace–rotation flow need not preserve time symmetry when implemented stage by stage on a Fourier grid. We analyze two explicit exact integrators (EEIs) for rotating Bose–Einstein condensates. For the seven-factor EEI, a constant-state Fourier coefficient proves that its fixed-grid logarithm has a nonzero quadratic defect on every centered even grid for any nonzero rotation vector. For a distinct five-operator EEI, a nonconstant-mode certificate proves the same obstruction on nonsingular coefficient branches whenever at least two rotation components are nonzero. In real-coefficient compact splitting, merging adjacent linear stages with fractions $c_j$ multiplies either defect by the strictly positive factor $\kappa=\sum_jc_j^2$; nominal high order therefore cannot be guaranteed uniformly at fixed grid. For the seven-factor implementation, the accumulated-response formula predicts $\kappa_4/\kappa_6\approx0.7606$ for nominal fourth- and sixth-order compact schemes (CS4/CS6). Scalar dipolar and spin-1 tests of that implementation verify the predicted amplitude ratio and reveal a crossover toward first-order compact behavior under time refinement. Pairing a forward linear half-step with its method adjoint around the exact nonlinear flow restores symmetry of the complete step; classical compositions then recover fourth and sixth order at fixed grid. The analysis provides explicit structural diagnostics and a direct structure-preserving repair.

We study when, and how compactly, a finite connected graph (G) embeds isometrically into a Cayley graph of a finite abelian group. The classical theory of partial cubes answers this for isometric subgraphs of hypercubes through the Djokovic-Winkler relation (\theta); we extend the question to the full family of abelian Cayley graphs, whose hosts may carry composite generators and cyclic factors of any order. We introduce an involutive edge relation (\varphi), defined by two simultaneous distance equalities, which coincides with (\theta) exactly on partial cubes and remains informative beyond them, together with an oriented relation (\Phi) for non-involutive hosts, where generator classes are constrained to be partial permutations rather than matchings.The central result is a quotient labeling theorem: for any partition of the edge set into candidate generator classes, the most generic consistent vertex labeling is the quotient of the free module on the classes by the lattice of signed cycle-class incidences, computed by the Smith normal form; the binary case is its reduction modulo two. We prove that the finest partition always yields an isometric labeling, that compactifying the resulting universal group is itself an instance of the same quotient construction, and that the whole construction is algorithmic and certifiable. Worked examples include the triangle, the Petersen graph (embedding into the Clebsch graph of order 16), the Pappus graph (a 1024-fold compaction), and the diamond (a non-diagonal fold). Sharp dimension bounds and an exhaustive census of small graphs are developed in a companion paper. 2020 MSC: 05C12, 05C25, 20K01, 05C50

Dynamic optimal transport unifies optimal transport, fluid mechanics, and gradient-flow theory within a continuous dynamical framework, offering a geometry-aware language for applications across physics, biology, and machine learning. However, conventional formulations cast it as a constrained optimization problem that must explicitly satisfy the continuity equation, hindering the reconstruction of the underlying dynamics directly from data. We propose the energetic variational method for dynamic optimal transport (EVMDOT), which reformulates it within an energetic variational framework by combining the flow map, the least action principle, and the maximum dissipation principle. The flow map recasts the constrained problem as an unconstrained one by automatically enforcing the continuity equation, while the balance between the conservative and dissipative forces determines the velocity field. Applied to the Fokker–Planck equation, the EVMDOT reconstructs both the energy landscape and the Waddington landscape directly from time-series density data. Through numerical experiments, we reveal that the EVMDOT achieves an intrinsic balance between data quantity and data quality: a sufficient data quantity compensates for limited data quality, making the reconstruction robust to the choice of the observation window. We further apply the EVMDOT to the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset to infer the potential landscape of amyloid-$\beta$ and tau, revealing two wells corresponding to the cognitively normal and Alzheimer's disease stages and the transition pathway between them.

We use linear programming to prove a separable version of Ky Fan's majorization relation for a sum of two operators that are each a tensor product of $n$ positive semidefinite operators. We give an example showing that such a relation does not hold in general for sums of three or more tensor products of three or more positive semidefinite operators.

We show unconditionally that the cubic and quartic large sieves are not perfectly orthogonal. The main obstruction to perfect orthogonality comes from the bias exhibited by Gauss sums. Our proof requires two main inputs: a Lindelöf-on-average upper bound for the second moment of Kubota's Dirichlet series, and a tight average lower bound for the Fourier coefficients of a certain Rankin-Selberg convolution of metaplectic theta functions. The latter input is particularly important in the quartic case, where much less is known about Fourier coefficients of metaplectic theta functions. To establish both of these inputs, we adapt a Rankin-Selberg regularization method due to Zagier (1981). In addition to the cubic and quartic cases considered in this paper, we expect that the family of Hecke characters of each fixed order $n \geq 3$ over a number field $K \supset \mathbb{Q}(\zeta_n)$ is not perfectly orthogonal. We provide a precise conjecture for the operator norm of these ensembles for each $n$.

The framed Beltrami–Vekua equation $\Phi(w_{\bar z} - \mu w_z) + \Psi(\overline{w_z} - \mu\overline{w_{\bar z}}) + \mathfrak{a}w + \mathfrak{b}\bar w = \mathfrak{f}$, with $|\mu|<1$ and $|\Phi|>|\Psi|$, carries a numerator field $N = \Phi\mathfrak{b} - \Psi\mathfrak{a} - W_L(\Phi,\Psi)$ whose weighted modulus integrates to the pseudo-analytic mass. This paper extracts the integer carried by the same field. When the zero set of $N$ is compactly contained in a bounded simply connected domain, the winding number of $N$ along any enclosing curve – the pseudo-analytic charge $n \in \mathbb{Z}$ – is invariant under every recombination $w = \varphi w' + \psi\bar w'$ of the unknown, every scaling of the equation, and every orientation-preserving $C^1$ change of variables: recombinations multiply $N$ by the positive factor $|\varphi|^2 - |\psi|^2$, so their invariance is exact, while on multiply connected domains the other two actions fix the component charges only in $\mathbb{Z}/2\mathbb{Z}$ and the total charge exactly. The charge is a Brouwer degree: it localizes at the zeros of $N$, vortices which no action of the class creates or destroys; an isolated vortex persists under perturbation of the data precisely when its local charge is non-zero. It involves the Beltrami coefficient only through the $L$-Wronskian of the frame, and is $\mu$-independent wherever $W_\partial(\Phi,\Psi) \equiv 0$ – in particular at the trivial frame, where $N = \mathcal{B}$ and the charge is the gauge-invariant winding of the coefficient of the Beltrami–Vekua equation. Mass and charge are independent: every pair in $(0,\infty)\times\mathbb{Z}$ is realized.

We present a technique for computing a finite set of generators and relations for the ring $\mathrm{h}^*(G)$ in terms of formal Demazure operators, where $\mathrm{h}^*$ is an oriented cohomology theory satisfying the localization axiom and $G$ is a semisimple algebraic group. Using this technique, we give minimal presentations for the oriented cohomology rings of the adjoint and simply-connected groups of types $A_1$, $A_2$, and $B_2$.

Among the many types of acceleration techniques designed to improve the performance of Benders decomposition, unified cut generation schemes have recently attracted a keen interest. Unified cuts aim for a better balance between the generation of optimality and feasibility cuts, while also providing a way to compare the strengths of different feasibility cuts. Our goal is to assess the experimental performance of a broad selection of unified and distinct Benders cuts in the context of the multi-commodity fixed-charge network design problem (MCFNDP). We express under a common mathematical structure and notation the construction of each unified or distinct Benders cut considered. We also explain how the generic formulations of the Benders cuts can be specialized to conform to the specifications of the MCFNDP. In addition, we suggest bespoke methods for comparing the performance of several solution methods when the benchmark is made up of heterogeneous problem instances. We report the results of a systematic empirical analysis comparing the performances of 50 Benders methods involving unified or distinct cuts in applications to a common testing bench made up of standardized MCFNDP instances. The analysis identifies a small number of leading Benders methods, namely those featuring the static Brandenberg-Stursberg cuts and the Hosseini-Turner l1-deepest cuts. In addition, we also report results obtained by using both Gurobi and CPLEX as the supporting solver to the SMS++ computation library.

Let $X\subset\mathbb{P}^3$ be the Schur quartic \[ x_0^4-x_0x_1^3-x_2^4+x_2x_3^3=0. \] We exhibit a connected arrangement of $24$ lines on $X$, defined over $\mathbb{Q}(\sqrt{-3})$, whose singular locus consists of $32$ ordinary triple points and no other intersections. Each line contains four triple points. The resulting reduced divisor $D$ satisfies $D\sim6H$, where $H$ is the hyperplane class. If $\pi:Y\to X$ blows up the triple points and $B=(\pi^{-1}D)_{\mathrm{red}}$, then \[ \overline{c}_{1}^{2}(Y,B)=112,\qquad \overline{c}_{2}(Y,B)=40, \qquad \frac{\overline{c}_{1}^{2}(Y,B)}{\overline{c}_{2}(Y,B)}=\frac{14}{5}. \] This gives a negative answer to the K3-surface specialization of the proposed $8/3$ bound for transversal arrangements of rational curves. The configuration is one half of the $48$ lines of the second kind on $X$; an explicit projective automorphism exchanges the two halves. We deliver the line parametrizations and all $32$ triple-point coordinates. Ancillary exact-arithmetic data record the $120$ line-containment coefficients and all $276$ pair-incidence determinants. A finite-field mixed-integer search is described only as the discovery procedure and is not used in the proof.

We study approximation and structural simplification of optimal control policies for controlled regime-switching diffusion processes for discounted, ergodic, finite-horizon, and exit-time criteria. We first establish continuity of the cost functionals over classes of Markov and stationary Markov policies by exploiting elliptic and parabolic regularity of the corresponding Hamilton–Jacobi–Bellman and Poisson equations. Using density results of policies with finite-action, piecewise-constant, and Lipschitz continuous, we show that each control problem admits $\varepsilon$-optimal policies within these structured subclasses. We then construct an Euler–Maruyama approximation of the controlled regime-switching diffusion under piecewise-constant controls. We prove strong convergence of the controlled state process and establish convergence of the associated finite-horizon value functions with rate $O(h^{\gamma/2})$. Building on this discretization, we develop a finite-state approximation of the induced discrete-time Markov chain via state-space quantization. We show that the value functions of the finite models converge uniformly on compact sets to the value function of the original problem, and that optimal policies of the approximating models are asymptotically optimal. These results provide a systematic framework for approximating regime-switching diffusion control problems and justify the use of structured policies and finite-state models for numerical implementation.

The logistic-normal integral appears in problems of statistical estimation for logistic models with Gaussian random effects, and generalized linear mixed models. We study the numerical evaluation of this integral and of its derivatives, and give closed form evaluations at certain points and series expansions. There is a continuum of possible series expansions, and we single out one series expansion which is optimal for numerical evaluation. We propose an algorithm for a precise numerical evaluation, based on the optimal series, with good approximation error control in the tails. As an application we give explicit results for the first four moments of a logistic-normal random variable.

In this note, we construct algebraic correspondences as matings of Fuchsian $(p,q,\infty)$-triangle groups with Blaschke products. Combined with the results of [MM25], this proves mateability of all cusped triangle groups with suitable Blaschke products. The proof of the main result involves associating two piecewise analytic circle maps to the $(p,q,\infty)-$triangle group, mating these maps with appropriate Blaschke products to produce two commensurable conformal matings, and finally constructing the desired algebraic correspondence as a common lift of the two conformal matings.

A folklore theorem states that for any pair of distinct knots in $S^3$, performing $p/q$-Dehn surgery on each knot yields orientation-preservingly homeomorphic manifolds for at most finitely many slopes $p/q$. In this paper, we provide a proof based on the JSJ decomposition of knot exteriors. In particular, for any given pair of distinct knots, it provides an effective bound on the maximal number of shared surgeries between the knots.

We prove the antipode formula for the $q$-deformations of quasi-symmetric functions. We also define a fundamental basis for non-commutative quasi-symmetric functions, and establish a partial antipode formula there. Our method also reduce to a new proof for the known antipode formula on the usual quasi-symmetric functions.

We give an exact solution of the ferromagnetic Ising model on a random regular graph ensemble via analytic combinatorics. Expressing the partition function as the generating function of labeled edge-bicolored graphs, we obtain the free energy in the thermodynamic limit from the asymptotic enumeration of these graphs. A simple analysis of the resulting formula reveals a second-order phase transition with critical exponents of the mean-field universality class.

Given a graph $G$ and an integer $\ell>\chi(G)$, the reconfiguration graph of the $\ell$-colorings of $G$ has as its vertices as the proper $\ell$-colorings of $G$, with an edge between two colorings whenever they differ on exactly one vertex. We say that $G$ is recolorable if this reconfiguration graph is connected for every $\ell>\chi(G)$. Belavadi and Cameron determined which $(F_1,F_2)$-free graphs are recolorable whenever $F_1$ and $F_2$ are graphs on at most four vertices, with the single exception of $(2K_2,K_4)$-free graphs. Gaspers and Huang showed such graphs are $4$-colorable. The $3$-colorable case within this class has also been resolved, leaving the open question of whether every $(2K_2,K_4)$-free graph with chromatic number $4$ is recolorable. In this paper, we provide evidence toward an affirmative answer by establishing recolorability for three subclasses: $(2K_2,K_4,C_5)$-free graphs, $(2K_2,K_4,H_a,H_b)$-free graphs for any distinct $a,b\in \{2,3,4\}$, and $(2K_2,K_4,H_4)$-free graphs containing an induced $W_5$, where $H_i$ denotes the unique $2K_2$-free graph obtained from a $W_5$ by keeping exactly $i$ edges from the universal vertex to the cycle.

Let $X$ be an affine smooth real algebraic variety (in the sense of Bochnak, Coste, and Roy) and let $V$ be a topological vector bundle over $X(\mathbb{R})$. We investigate the problem of deciding whether $V$ is topologically isomorphic to an algebraic vector bundle using motivic homotopy theory. We prove that if $\dim X\leq 3$, then the algebraicity of Stiefel-Whitney classes is a necessary and sufficient condition for $V$ to be algebraizable. Next, we show that when $\dim X=4$ and $X(\mathbb{R})$ is compact, even if the characteristic classes of $V$ are algebraic, there is still an obstruction to algebraizing $V$ related to the Pontryagin class $p_1$ and the Stiefel-Whitney class $w_4$. Then we give some applications of this result. Namely, we give an example where this obstruction is nontrivial, and we investigate the group $\mathrm{K}_0(X)$.

We give mathematically self-contained formulations, in the complex-time (kime) representation, of three open problems from the foundations of classical mechanics: (I) the extension of the classical entropic uncertainty principle to non-canonical variables and to multiple degrees of freedom; (II) the characterization of coordinate-invariant measures and entropies, i.e., the question of why continuous physical quantities must be paired for an invariant entropy to exist; and (III) the construction of a classical relativistic directional degree of freedom (a classical analogue of a spin-1/2 system). Throughout, the kime phase is interpreted {statistically as a latent circular random variable whose law \Phi models the intrinsic trial-to-trial variability of repeated, identically controlled experiments indexed by the kime magnitude. The mathematical bridge is an exact symplectic identification of the kime cone with the action-angle chart of a one-degree-of-freedom phase space, under which the kime measure is the Liouville measure and the phase law becomes the angular conditional of a Liouville density. Specifically, we (i) prove a sharp entropic uncertainty relation on the kime cylinder whose extremal family is von Mises x Gaussian, together with a sharp circular Fisher-information inequality saturated exactly by von Mises laws; (ii) prove an exact non-canonical uncertainty relation in which the correction term is the geometric mean of the Poisson bracket, clarifying the conjectured role of the expected bracket; (iii) prove aggregate multi-degree-of-freedom bounds via the Williamson normal form and Fischer's inequality, and isolate the per-degree-of-freedom refinement as a precise open problem of symplectic Schur-Horn type; (iv) prove that diffusion of the kime phase produces monotone entropy growth with the equipartitioned (Haar-uniform) phase law.

We investigate quadratic bilinear systems by developing novel tree-based representations of their solutions. The proposed framework decomposes the solution into a sequence of coupled bilinear subsystems whose components admit explicit expansions indexed by full binary trees. These representations yield sufficient conditions for the existence of global solutions and lead to new output bounds in terms of reachability Gramians. Motivated by these estimates, we introduce time-limited and infinite-horizon reachability and observability Gramians, establish sufficient conditions for their existence, and characterize them through nonlinear matrix equations. The associated Gramians are employed to identify dominant state-spaces and to derive exact reduced-order models obtained by removing Gramian kernels. Building on these results, we develop a balanced truncation method for quadratic bilinear systems and prove an error bound for the reduced-order approximation. The proposed framework provides a unified connection between tree-based solution representations, nonlinear Gramian theory, and balanced truncation for quadratic bilinear systems, closing several theoretical gaps in the analysis of Gramian-based model reduction for this class of systems.

We continue the study of $D(z)$-quadruples in the ring $\mathbb{Z}[\sqrt{-2}]$. Motivated by the earlier classification due to the authors and by the subsequent partial results for the remaining families, we consider the exceptional congruence classes arising in the forms $24a+5+(12b+6)\sqrt{-2}$, $24a+2+(12b+6)\sqrt{-2}$, and $48a+44+(24b+12)\sqrt{-2}$. By combining the regular extension method with new families obtained by fixing a divisor $e\mid 3z$ and a small element $v\in \mathbb{Z}[\sqrt{-2}]$, we construct explicit $D(z)$-quadruples in each of the previously unsolved congruence classes. More precisely, we show that every exceptional class contains infinitely many values of $z$ admitting a twice semi-regular $D(z)$-quadruple, i.e., a quadruple containing two regular $D(z)$-triples. We also include remarks on the exceptional values $z\in\{-1,1\pm 2\sqrt{-2}\}$ and on a computational search in the exceptional congruence classes.

We prove two inequalities introduced in our prior study of the graphical shape of the number of bit strings with a given score under an interesting scoring system. Generating functions are used to establish the inequalities, which in turn imply two of the salient graphical features, uni-modality near the zero score and shape asymmetry for positive versus negative scores. One inequality provides a lower bound on the expected value of a discrete random variable with probabilities proportional to a product of two binomial coefficients. The other inequality states that the expected value with respect to near central binomial coefficients of other binomial coefficients lying on an oblique ray in Pascal's triangle exceeds the expected value along an adjacent parallel ray to its left.

Nonlinear convection–diffusion systems play a central role in transport phenomena, including mass transfer, heat transfer, porous-media transport, and coupled continuum processes with source, exchange, and interface effects. In such systems, the key question is often not only which governing partial differential equation is used, but whether the model preserves a consistent thermodynamic balance under the operations that arise naturally in transport analysis: restriction to subdomains, coupling across interfaces, linearization near equilibrium, and discretization for computation. This paper develops a continuum-first framework for open nonlinear convection–diffusion systems in which thermodynamic consistency is formulated as a free-energy balance with nonnegative bulk dissipation and explicit boundary and source contributions. Within this setting, nonlinear transport systems are defined as structured objects built from admissible state fields, storage functionals, constitutive flux decompositions, sources, and boundary ports. We prove that the thermodynamic balance is preserved under exact structure-preserving transformations, restriction to subdomains, local-to-global reconstruction over compatible domain decompositions, and power-conserving interconnection of open subsystems. We then derive classical linear convection–diffusion models as tangent thermodynamic descendants at equilibrium and show that the same invariant survives weak formulation, semidiscretization, and fully discrete time stepping when the numerical design respects thermodynamic structure. Nonlinear drift–diffusion and porous-medium convection–diffusion are used as explicit examples. The resulting contribution is a compositional transport framework in which the second law remains visible across continuum modeling, subsystem coupling, linear approximation, and computation.

We investigate the low degree rational cohomology groups of the moduli space of twisted Prym curves ${\overline{Pr}_{g,n}^{\hspace{0.05cm}(m_1, \ldots, m_n)}} $, where the integer twists $0\leq m_i\leq 1$ have even sum over $i$. We prove that these groups vanish in odd degree $\leq3$ and that the group in degree $2$ is algebraic. In particular, the results cover the classical moduli spaces of Prym curves and Prym curves with simple ramifications.