Papers

We study the problem of fair online resource allocation, motivated by applications such as refugee resettlement and airline scheduling, where agents arrive sequentially and must be assigned to facilities with limited capacities. We introduce a model that maximizes the overall welfare subject to resource constraints and a Lipschitz fairness requirement, which ensures that similar agents arriving in the same batch receive similar expected outcomes. We first analyze the offline problem, proving that the value of the optimal fair allocation is at least an $\Omega(1/\gamma)$ fraction of the optimal unfair allocation, where $\gamma$ is the fairness coefficient, thereby bounding the price of fairness. For the online setting, we propose an algorithm based on dual mirror descent that enforces fairness constraints within batches while estimating optimal dual variables. We prove that this algorithm achieves sublinear regret relative to the optimal offline fluid benchmark. Finally, we validate our theoretical results using real-world data from the Refugee Economies Programme, demonstrating the algorithm's performance and examining the trade-offs between welfare maximization and fairness enforcement.

This paper studies the numerical approximation of a singularly perturbed convection-diffusion problem over a bounded polygonal domain in $\mathbb{R}^d$ ($d=2,3$), where the velocity field is modeled by a log-uniform random field, a setting typical in uncertainty quantification. We introduce a novel numerical framework for computing the expected value of the linear functionals of the solution. The approach combines a finite element discretization of the problem, a truncated Karhunen–Loève expansion to represent the stochastic velocity field, and a lattice-based quasi-Monte Carlo (QMC) method to estimate expectations over the parameter space. We provide a rigorous error analysis of the proposed scheme, establishing bounds on the mean squared error and demonstrating that the QMC method achieves a nearly linear optimal convergence rate, with a constant independent of the integration dimension. Furthermore, the convergence rate is shown to be independent of the singular perturbation parameter.

Let $\Gamma$ be a geometrically finite thin subgroup of an arithmetic lattice $\Gamma_0 < G := \operatorname{SO}(n, 1)$ and consider the congruence covers of $\Gamma \backslash G$. In the breakthrough work of Bourgain-Gamburd-Sarnak, the expansion machinery was used to establish a uniform spectral gap in the setting $(G, \Gamma_0) = (\operatorname{SL}_2(\mathbb{R}), \operatorname{SL}_2(\mathbb{Z}))$ when the critical exponent satisfies $\delta_\Gamma > \frac{1}{2}$. The main applications are affine sieve for $\Gamma$-orbits and uniform resonance-free half-planes for the resolvent of the Laplacian. These results were generalized in subsequent works by Mohammadi-Oh, Oh-Winter, the author, and Edwards-Oh. Yet, the region $\delta_\Gamma \in \bigl(\frac{1}{2}, n - 2\bigr]$ for $n \geq 3$ remains to be treated when there are cusps. The purpose of this paper is to fill in this gap in the literature. The difficulty lies in working with a countably infinite coding due to the presence of cusps. In particular, we incorporate new tools to prove the Zariski density and full trace field properties of the return trajectory subgroups.

We study covert block-activity information transmission over a thermal-loss bosonic channel, where messages are encoded in weight-constrained activity patterns that Bob must recover through block-local decisions by prescribed deadlines. Shared circular Gaussian displacement modulation makes Willie's averaged lost-light state exactly thermal, with a strictly convex relative-entropy cost in signal energy, whereas a fixed Gaussian receiver at Bob yields a Gaussian mean-shift divergence linear in energy. This asymmetry produces an exact finite-block energy–information frontier and a detector-independent latency converse, matched in order by a block-reset cumulative-sum detector. For block covertness budget $\delta_b$ and error target $\epsilon_b$, the required active length scales as $\delta_b^{-1}\log^2(1/\epsilon_b)$ up to an explicit channel–receiver factor. Lifting this local law to communication yields the matching transmission limit $\log M=\Theta(\sqrt n/\log n)$ for the symmetric coordinatewise architecture under a fixed total covertness budget, maximal-message covertness, vanishing maximal error, and local deadlines. A relaxed full-horizon reference with the same modulation family and fixed measurement supports $\Theta(\sqrt n)$, showing that covertness and the selected measurement alone do not impose the extra logarithmic factor. Finally, public quadrature phase-shift keying codebooks selected by $O(\sqrt n)$ secret bits remove ideal continuous shared randomness without changing the payload order.

In this paper, we study $(-1)$-homogeneous steady solutions to the Euler equations on $\mathbb{R}^n \setminus \{0\}$. In low dimensions $n=2,3$, such flows are known to be essentially trivial. In contrast, we show that in higher dimensions $n \ge 4$, every $(-1)$-homogeneous Euler flow is a geodesible vector field with constant Bernoulli function. Moreover, any $(-1)$-homogeneous geodesible field is induced by a geodesible field on the sphere $\mathbb{S}^{n-1}$. In particular, in the case $n=4$, every $(-1)$-homogeneous Euler flow is obtained as an extension of a Beltrami field on $\mathbb{S}^{3}$.

We study infinite exchangeable sequences with Gaussian one-dimensional marginals. We formulate the conjecture that joint Gaussianity of a single pair of coordinates forces the entire sequence to be a Gaussian process. Although this conjecture remains open, we prove that joint Gaussianity of the first four coordinates is sufficient. We also establish the corresponding two-point criterion under the additional assumption that the directing measure is almost surely infinitely divisible.

Motivated by the Kodaira-Spencer local stability theorem for Kahler structures and by C. Li's study of Kahler structures on holomorphic submersions between compact complex manifolds, we establish an equivalent characterization of the Kahlerness of the total space near a Kahler fiber, in an optimal manner. The proof combines a (1,1)-type lifting argument for flat sections of a local system, observations on torsion-freeness and a Kahler neighborhood criterion for compact Kahler submanifolds.

The analytical determination of critical scaling in three-dimensional strongly coupled field theories is inherently restricted by the Borel non-summability of perturbative expansions. In this paper, we establish a symplectic isomorphism that maps the infrared dynamics of scalar quantum field theories onto the finite-strain bifurcation mechanics of hyperelastic continua. By reformulating the renormalization group flow as an effective spatial Hamiltonian evolution within a symplectic phase space, we identify the critical fixed point with the spectral degeneration of a continuous Riccati-Lyapunov dynamical system. The algebraic integrity of this geometric correspondence is rigorously verified against exactly solvable limits, recovering both the Gaussian free-field scaling and the localized Jackiw-Rebbi zero-mode of the continuum Su-Schrieffer-Heeger model. Applying this framework to the 3D Ising universality class, we construct a self-consistent algebraic scaling Ansatz for the anomalous dimension $\eta$. This derivation dispenses with arbitrary perturbative truncations, demonstrating instead that interaction-induced non-perturbative screening emerges as a strict consequence of the nonlinear self-consistency demanded by the continuous Algebraic Riccati Equation. The resulting rational scaling law functions as a geometric Padé approximant that asymptotically satisfies the unitarity bound limits, providing a mathematically transparent, first-principles macroscopic impedance analogue for the dynamic suppression of strong-coupling divergences.

We study weak shock reflection from a rigid wall in the joint limit of weak shock strength and nearly glancing incidence. In the distinguished scaling $\Mach=1+\lambda\alpha^2$, the inner reflection region is governed by the unsteady transonic small-disturbance (UTSD equation and is controlled, to leading order, by the single parameter $a_0=1/(2\sqrt{\lambda})$, independent of the ratio of specific heats $\gamma$. Thus the known UTSD detachment value $a_d=\sqrt2$ corresponds in this scaling to $\lambda_d=1/8$, with Guderley–Mach reflection for $\lambda>1/8$. The physical trajectory angle is obtained by multiplying the canonical UTSD trajectory function $g(a)$ by the Mach-number strength scale $\delta=\sqrt{2(\Mach^2-1)}$, so that $\chi_{\rm phys}=\delta g(a)+O(\delta^2)=2\sqrt{\lambda}\,\alpha g(a_0)+O(\alpha^3)$. We rederive the self-similar UTSD reduction, sonic parabola, and shock polar in order to make the convention and the detachment map self-contained. We also record a formal adjoint solvability expression for the first correction $H(a;\gamma)$, while specifying the free-boundary data required to evaluate it. Finally, a time-marching solver for the full leading-order canonical UTSD system is benchmarked at $a_0=0.5$: retaining the transverse compression $u>1$ gives a $u=0.5$ contour location consistent with the Hunter–Tesdall triple-point benchmark. This computation is used only as a leading-order benchmark, not as a substitute for an adaptive self-similar Guderley free-boundary solver.

This work proposes a Dual Time Phase Field Inversion (DTPFI) method for recovering unknown potential functions in phase field models. The reconstruction is formulated as an optimization problem that minimizes the mismatch between model predictions and observed fields at the final measurement time. We prove the differentiability of the measured field with respect to the unknown potential and establish the local convexity of the regularized objective function, thereby ensuring the existence of a local optimal solution. For numerical implementation, automatic differentiation is employed to compute gradients, avoiding the expensive evaluation of analytic gradients. Extensive numerical experiments demonstrate that DTPFI accurately reconstructs both polynomial and logarithmic potentials and remains robust under measurement noise. The framework is further extended to inverse problems involving field dependent mobility and joint parameter identification in coupled Cahn-Hilliard-Allen-Cahn systems.

We investigate the global well-posedness of a class of generalized Hillen–Painter systems – specifically, supercritical volume-filling chemotaxis models – in $\mathbb{R}^2$ under the influence of Couette flow. It is well established that, in the absence of fluid flow, solutions to this system may develop finite-time singularities (blow-up) for arbitrary initial cell mass. It is proved that the introduction of a Couette flow with sufficiently large amplitude guarantees the global existence of solutions for all initial masses. By employing a novel frequency decomposition technique, we successfully remove the mass threshold limitation presented in previous studies on the domain $\mathbb{T}\times\mathbb{R}$ (Wang et al., Commun. Contemp. Math.), thereby establishing global regularity in the whole space without any smallness assumptions.

This is the second of a series of three papers, in which we prove a formula expressing the modular height of a unitary Shimura variety over a CM number field in terms of the logarithmic derivative of the Hecke L-function associated with the CM extension. The main idea of our proof is to compare the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series and the arithmetic degree of a generating series of divisors on unitary Shimura varieties. In this paper, we define the arithmetic generating series of divisors on unitary Shimura varieties, compute the corresponding arithmetic intersection numbers, and derive the modular height formula for unitary Shimura curves as well as the height formula for a CM point on them.

We present a new classification of the class of all the hypergeometric orthogonal polynomial sequences. The classification uses properties of the coefficients $\alpha_n$ and $\beta_n$ in the three-term recurrence relation satisfied by the orthogonal polynomial sequences. Such coefficients are rational functions of $n$ and are determined by a set of six parameters. The partial fraction decomposition of $\alpha_n$ is a linear combination of five linearly independent functions of $n$ with coefficients $d_j$, for $0 \le j \le 4$, that are polynomials in our six parameters. For each value of a parameter $r$, associated with the eigenvalues, we classify the $\alpha_n$ according to the set of coefficients $d_j$ that are nonzero. There are certain particular values of $r$ that must be considered separately. Our classification yields a collection of 53 disjoint classes and it is different from the Askey scheme in several aspects. It is similar to the classifications proposed recently by Koornwinder.

We give a complete recipe for constructing extremal van der Geer–van der Vlugt curves over finite fields of odd characteristic. The input data consist of a nonzero element of the base field together with a linear subspace satisfying a certain trace condition. This construction may be viewed as an odd-characteristic analogue of the one previously obtained by the authors in characteristic two.

This paper explores the effectiveness of various novel deterministic and traditional stochastic data assimilation (DA) and parameter recovery (PR) algorithms given noisy data from chaotic systems. We use semi-analytic methods to numerically construct synthetic data from the Lorenz '63 and multiscale Lorenz '96 chaotic dynamical systems, adding white noise. Our findings show that, for different noise levels, deterministic PR algorithms paired with deterministic DA algorithms are shown computationally to be overall more accurate and stable than stochastic PR algorithms. Additionally, deterministic PR methods have demonstrated greater speed and efficiency, requiring less computational power than stochastic PR methods. This suggests that future work should consider exploring the full potential of deterministic PR algorithms in the presence of noise.

We introduce a closure operator for valuated matroids and prove that it yields a cryptomorphic definition of valuated matroids. As an application, we introduce a class of modules over the tropical semifield (geometric modules) and prove that there is one-to-one correspondence between projective equivalence classes of simple valuated matroids and isomorphism classes of finitely generated geometric modules. We further illustrate how this correspondence can be lifted to simple infinite valuated matroids and geometric modules.

The scalar auxiliary variable (SAV) method constructs linear, unconditionally energy-stable time discretizations of gradient flows. Eliminating the auxiliary variable in a first-order SAV step shows that the state equation is a semi-implicit update augmented by a rank-one positive semidefinite correction from the previous nonlinear force. The multiple-SAV (MSAV) method produces this correction componentwise, with rank up to the number of energy components. This separates two mechanisms usually coupled in MSAV: the number of scalar variables tracking the nonlinear energy and the rank of the correction. We introduce a pullback-corrected SAV (PB-SAV) family that keeps a single scalar auxiliary variable but replaces the rank-one SAV correction by the pullback correction induced by an admissible component decomposition. The correction remains positive semidefinite, has rank at most the number of components, and may change from step to step without changing the scalar auxiliary variable. We prove modified-energy dissipation laws for fixed and step-dependent decompositions, establish first-order convergence after a fixed spatial discretization under standard smoothness and positivity assumptions, derive a refinement identity whose gain is an explicit weighted variance, and give a Sherman–Morrison–Woodbury implementation of the low-rank perturbation of the standard semi-implicit solve. We also show, in finite dimensions, that the pullback correction is the Gauss–Newton matrix of a least-squares representation of the nonlinear energy. Numerical experiments on finite-dimensional gradient flows, Allen–Cahn dynamics, a finite-rank nonlocal gradient flow, and a linear nonlocal Cahn–Hilliard model show regimes in which PB-SAV and SAV differ only in the first-order error constant and regimes in which PB-SAV reduces the trajectory error by a large factor.

Yun, Sra, and Jadbabaie (COLT 2021, open question) conjectured the SS–RS–GD inequalities: for well-conditioned symmetric matrices $A_1,\dots,A_n$, the operators $W_{ss}$, $W_{rs}$, and $W_{gd}$ that encode the expected iterate of single-shuffle SGD, random-reshuffle SGD, and gradient descent on a quadratic finite sum should satisfy \[ \|W_{ss}\|\le \| W_{rs}\|\le \|W_{gd}\|. \] The conjecture is resolved, $\bullet$ SS-RS inequality fails. Already for $n=3$, $K=2$, and $d=4$, we exhibit explicit PSD matrices whose condition number is arbitrarily close to $1$, yet $\|W_{ss}\|>\|W_{rs}\|$. $\bullet$ RS-GD inequality holds. For every symmetric $A_i$ with $\bigl(1-\frac1{4n^2+1}\bigr)I\preceq A_i\preceq I$, one has $\|W_{rs}\|\le\|W_{gd}\|$. The proof was found via GPT-5.5 Pro extended prompted by the author.

Modern service systems, ranging from cloud data centers and ride-hailing platforms to healthcare facilities, operate at massive scales where it is important to handle congestion. Queueing theory is used to understand the delay and queue length behavior in these systems. Except in simple queues, it is not possible to obtain a closed form solution for the quantities of interest, and so, one studies the system in certain asymptotic regimes such as the heavy traffic. The transform method, presented in this tutorial, is a framework to understand the steady-state behavior of Stochastic Processing and Matching Networks (SPNs/SMNs). By exploiting the zero-drift property of exponential test functions, the method derives explicit functional equations (acting as a proxy for global balance equations) for the transforms (such as moment-generating functions) of queue-length distributions. These functional equations can be used to either characterize the exact behavior of the system in an asymptotic regime or to obtain non-asymptotic performance bounds on the mean, higher order moments, or tail bounds on the queue lengths. Since its introduction for load-balancing in data center networks, the transform method, as a framework, has been extended to handle various features that arise in different systems, including customer abandonment, state-dependent arrivals, Markov-modulated arrivals, large-system scale, and multi-dimensional networks with multiple bottlenecks. This tutorial presents an overview of the transform method starting with the simplest setting viz., a single server queue. The transform method is introduced as a three step procedure. We then illustrate how the method can be adapted within this three-step framework to handle the aforementioned features.

We introduce the Generalized Neural Distributional Regression (GNDR) framework, which seamlessly embeds deep neural networks into the parameter space of classical probability distributions. To reconcile the inherent non-identifiability of deep architectures with maximum likelihood theory, we propose a two-step semi-parametric estimation procedure. By isolating the terminal prediction heads and treating the upstream network as a fixed, non-linear basis expansion, GNDR enables the extraction of analytical Fisher Information matrices. This facilitates rigorous uncertainty quantification, generating observation-specific confidence bands and tolerance intervals via the multivariate Delta method. We demonstrate the framework's versatility and superior distributional calibration across diverse data modalities, including overdispersed clinical counts, right-censored transcriptomic survival profiles under a mixture cure framework, and zero-truncated age distributions derived directly from unstructured facial images. The methodology is natively implemented in the open-source Python package thetaflow.

We propose a mathematical framework for redistricting compactness grounded in the classical soap bubble problem: the partition of a planar region into N subregions of prescribed population that minimizes total shared boundary length. This formulation, which we call the soap bubble model, yields a physically and mathematically natural optimality criterion that generalizes all existing per-district compactness measures. We carefully distinguish two formulations: minimizing physical boundary length in geographic coordinates, and minimizing a density-weighted length in the coordinates produced by the Monge–Ampere density-flattening map. In the transformed coordinates, the first-order conditions for a minimizer are precisely the classical two-dimensional Plateau's laws – circular arc boundaries, 120-degree triple junctions, and perpendicular boundary contact. In geographic coordinates, the Euler–Lagrange equation shows arc curvature varies proportionally with local population density, so boundaries curve more tightly through dense areas and gently through sparse ones. Remarkably, the 120-degree junction condition and 90-degree boundary condition are unchanged by the density, as they follow from a force-balance argument that does not involve density. We adopt the physical-length objective as the natural redistricting criterion – it minimizes actual miles of district boundary and requires no coordinate transformation – and derive the corresponding density-modified volume-preserving mean curvature flow as its gradient descent. The N-state Potts model on the census block adjacency graph with population-weighted pressure updates is the natural discrete implementation of this flow. We define a multiplicative gerrymandering metric decomposed into tactical and systemic components and discuss practical computation, including grid-pinning effects in the discrete setting.

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs). For Singh contractions, we allow $T^p$ to satisfy the condition, providing greater flexibility. Examples illustrate the applicability of all results. The work highlights the utility of generalized contractions in differential equations, particularly when the Banach contraction principle fails.

We introduce several very weak first-order theories of unary concatenation of dyadic strings and investigate their relationships to other previously studied veey weak first-order theories, namely the theory WT of binary trees of Kristiansen and Murwanashyaka, the theory WD of binary concatenation of Murwanashyaka, and Robinson's very weak arithmetic R. We prove that all these theories are mutually formally interpretable with the theories of unary concatenation studied in the paper, thus establishing essential undecidability of the latter. In the process we show that binary concatenation is first-order definable from unary concatenation modulo the presence of the initial segment relation plus either the end segment relation or the inverse operation on words, thus giving a positive solution to a problem posed by Karlov.

We establish a Fefferman-Szegő characterization of local CR sphericity for smoothly bounded strongly pseudoconvex domains in complex dimension three. We derive the boundary expansion of the normalized determinant of the Fefferman-Szegő metric and prove that its second-order coefficient is a universal multiple of the squared Chern-Moser curvature. Hence, vanishing of the second-order deviation from the ball model is equivalent to local sphericity. A logarithmic stability theorem for the associated Monge-Ampère determinant controls the remainder and completes the dimension-three case.

We investigate kink antikink collisions in higher order scalar field theories described by the higher order models and their shifted periodic extensions. Both classes of models possess three degenerate vacuum states and support topological kink solutions with asymmetric profiles and algebraically decaying tails. By extending conventional polynomial potentials across multiple spatial sectors, we construct shifted periodic high order field theories and examine how this modification affects the scattering dynamics of topological defects. The primary objective of this study is to provide a comparative numerical analysis of kink collisions in the standard and shifted periodic versions of these higher order models. Using direct numerical simulations, we determine the critical velocities that separate capture from escape regimes and identify resonance structures associated with energy exchange between translational and internal vibrational degrees of freedom. Particular attention is devoted to the emergence of escape windows, quasi-fractal patterns, and the role of algebraic tails in shaping the collision outcomes. Our results demonstrate that, although the conventional and shifted periodic models exhibit similar kink antikink configurations, important quantitative differences arise in their critical velocities, resonance structures, and scattering characteristics. The findings further confirm that both classes of models support resonant energy transfer mechanisms analogous to those observed in lower order theories, while simultaneously exhibiting novel features associated with higher-order interactions and long range effects. These results contribute to the growing understanding of nonlinear excitations in scalar field theories and provide new insights into the dynamics of topological solitons in shifted periodic systems

A Markov Logic Network (MLN) is a probabilistic relational model used in Statistical Relational Artificial Intelligence for defining a probability distribution on the set of possible worlds with domain $D$ for an arbitrary finite domain $D$. An MLN consists of soft constraints with associated weights which are nonnegative real numbers. In this study we consider a language speaking about a property $P(x)$ and a relation $R(x, y)$. We consider an MLN for which every Boolean combination of $P(x)$ and $R(x, y)$ is a soft constraint (with associated weight). Let $n$ denote the size (cardinality) of the domain. We show that, for every choice of weights, if the weights are scaled by $1/n$ then, for every first-order sentence $\varphi$, the probability that $\varphi$ holds tends to either 0 or 1 as $n \to \infty$; that is, a 0-1 law for first-order logic holds. Morover, the limit probability does not depend on the weights. If we instead use the standard semantics of MLNs, in the case of which the weights are not scaled, then the limit behaviour is more complicated and depends on the weights. With unscaled weights we get 7 qualitatively different cases which depend on the weights. In some cases we have a 0-1 law for first-order logic, in some cases not, but we may still have a convergence law. The influence of the weights on the asymptotic probability of a first-order sentence may be in the form of a sudden “phase transition” from one of the 7 cases to another. The presence of a convergence law has positive implications for inference on large domains.

We investigate the asymptotic behavior of the Diophantine errors $\delta B_{n}-A_{n}$, where $\delta=-e\textrm{Ei}\left(-1\right)$ denotes the Euler-Gompertz constant and $B_{n}$ and $A_{n}$ are the $n$th Bell and Gould numbers, respectively. These errors have exponential generating function $g_{\delta}\left(z\right)=\exp\left(e^{z}-1\right)\left(\delta-{\displaystyle {\textstyle \int_{0}^{z}}\exp\left(1-e^{t}\right)dt}\right)$,$\:z\in\mathbb{C}$, and it is known that $\delta B_{n}-A_{n}=O(B_{n}\exp(-cn/\left(\ln\left(n\right)\right)^{2}))$ for some $c\in\mathbb{R}_{>0}$, implying $\lim_{n\rightarrow\infty}\left(A_{n}/B_{n}\right)=\delta$. In the present work, we prove that $\delta B_{n}-A_{n}$ oscillates without bound as $n\rightarrow\infty$; that is, both $\limsup_{n\rightarrow\infty}\left(\delta B_{n}-A_{n}\right)=\infty$ and $\liminf_{n\rightarrow\infty}\left(\delta B_{n}-A_{n}\right)=-\infty$.

We propose a rigorous and systematic mathematical framework for tracking the cognitive trajectories of a user, in the context of algorithmic sycophancy and AI-driven delusional spiraling. Using tools from dynamical systems theory and stochastic differential equations, we explore how individuals perceive, interpret, and update their beliefs as they interact with AI chatbots that possess hidden traits of sycophancy. We treat the evolving conviction as a continuous log-odds state variable, coupled into a stochastic differential equation, navigating a multi-valley potential energy landscape. Our analysis reveals several critical observations governing the stability and rigidity of belief dynamics. We demonstrate that the baseline prior perception of the individual is systematically enhanced by sycophantic feedback beyond a critical threshold. Here, the perceptual potential landscape undergoes a structural phase transition that severely deepens any incremental initial tilt present in the baseline state, transforming the landscape and giving rise to deep, highly resilient attractor basins that trap the individual in unshakeable, self-reinforcing, delusional convictions. Finally, we demonstrate that genuine external information can successfully challenge these rigid states. If this incoming evidence is strong and authentic enough to overcome the internal feedback barrier, it can correct the structural asymmetry caused by sycophancy, inducing a perception reversal that successfully restores the objective belief state.

We study the Wasserstein space $\mathcal{P}(M)$ equipped with a distance $\Wp$ constructed from the Lagrangian $L(x,v)=|v|^{p(x)}$ where $p(x)=2+\varepsilon(x)$ with $\varepsilon$ small. Building on the fundamental work of Lott and Villani on the $K$-geodesic convexity of the Boltzmann entropy in $(\mathcal{P}(M),\Wb)$, we establish a generalized inequality showing that the entropy remains $\left(K - C\|\varepsilon\|_\infty\right)$-convex along $\Wp$-geodesics. We then introduce a modified Rényi entropy that exactly compensates the logarithmic divergence that appears in the expansions of $\Wp^2$, obtaining thus a sharp equivalence that reaveals the Bakry-Émery tensor as the effective curvature in the variable exponent setting. As applications, we derive perturbed versions of the Log-Sobolev and Talagrand inequalities in variable exponent Wasserstein spaces, showing that these fundamental functional inequalities are robust under small perturbations of the transport exponent. This work generalizes the Lott-Villani theorem and its consequences (\emph{J. Lott and C. Villani, Ann. of Math. 169 (2009), 903-991}) to situations where the transport metric varies spatially.

In this paper we introduce "graph linear notation" – a complete graph invariant – which is positioned as an alternative definition for the finite graphs. This invariant is constructed using an algorithm similar to the algorithm of finding canonical forms of graphs. Storing graph linear notation instead of a regular graph allows us to greatly simplify two major problems: the construction of illustrations for graphs with regards to possible graph symmetries, and the comparison of two graphs for isomorphism. We also demonstrate the transferability to the graph linear notations such classical graph theory concepts as colourings and graph paths.