Papers

We develop an operator-algebraic framework for boundary conditions and bulk-boundary correspondence in one-dimensional gapped phases with categorical symmetry. Working directly in the thermodynamic limit, we construct half-infinite fusion spin chains and commuting-projector boundary Hamiltonians from a unitary fusion category $\mathcal{C}$, an indecomposable semisimple right $\mathcal{C}$-module category $\mathcal{M}$, a Q-system $Q\in\mathcal{C}$ specifying the bulk phase, and a right $Q$-module $K\in\mathcal{M}_{Q}$, regarded as an object of $\mathcal{M}_{Q}^{\mathrm{op}}$, specifying the boundary. We prove that these Hamiltonians have unique ground states and that the resulting realization functor $\mathcal{M}_{Q}^{\mathrm{op}}\to\mathrm{BCond}$ is an equivalence, so simple boundary conditions are classified by simple objects of $\mathcal{M}_{Q}$ and general boundary conditions by their finite direct sums. We also give a microscopic formulation of the boundary symmetry topological field theory using DHR bimodules of the boundary quasi-local algebra. For a half-infinite fusion spin chain, the boundary DHR category is monoidally equivalent to $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$, and the canonical action of the bulk DHR category on it agrees with the categorical action of $Z_1(\mathcal{C}^{\mathrm{rev}})$. Finally, we identify the action of the boundary DHR category on boundary conditions with the categorical action of $(\mathcal{C}_{\mathcal{M}}^{\vee})^{\mathrm{rev}}$ on $\mathcal{M}_{Q}^{\mathrm{op}}$. This yields a one-dimensional bulk-boundary correspondence: the enriched monoidal category describing the bulk is the enriched center of the enriched category describing the boundary.

It has been well established that information improves decisions, pushing the population forward as more information becomes available. Nevertheless, a wide range of empirical evidence shows that humans avoid complexity, delegate judgement, and prefer simplified social worlds. This tension raises an evolutionary puzzle: if knowledge is economically valuable and therefore evolutionarily beneficial, why do populations not converge towards universally informed and complex agents? In this study, we propose a theory of cognitive economy in which information has positive utility but costly acquisition, processing, and coordination. In complex environments, selection can favour heterogeneous populations: most individuals use low-cost heuristics and simplified choice architectures, whereas a minority of agents or institutions specialize in information processing. This cognitive division of labour reduces decision costs while preserving much of the value created by knowledge. We formalize this trade-off by comparing societies of uniformly complex agents with societies containing simpler agents and a specialized decision-making centre. The latter can dominate when the costs saved by distributed simplicity exceed the utility lost through reduced individual autonomy and imperfect delegation. Crucially, the specialized decision-maker need not face a volunteer's dilemma, because its private payoff can exceed that available under universal complexity through rents, status, control or superior information. The framework links bounded rationality, rational inattention, hierarchy, markets, and cultural evolution, and suggests that simplicity is not a failure of adaptation but a precondition for scalable social organization.

Let $\lambda(n)$ be the exponent of the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^{\times}$, and set $L(x,y) = \#\{n\le x: \lambda(n)\le y\}$. We prove an upper bound for $\log \frac{L(x,y)}{x}$ valid for $\exp((\log_2{x})^{1+\epsilon}) \le y\le x/\exp((\log_2{x})^{1+\epsilon})$. Our bound is asymptotically sharp under a plausible hypothesis on powersmooth shifted primes. As an application, we obtain a new upper bound on the count of odd $n\le x$ for which the order of $2$ modulo $n$ is appreciably smaller than $x^{1/2}$.

The $p$-rotor walk on $\mathbb{Z}$ is a self-interacting walk that interpolates between the simple random walk and the deterministic rotor walk. While the weak convergence of this model to a perturbed Brownian motion is known, its almost sure asymptotic boundaries have not been characterized. In this paper, we establish the exact Law of the Iterated Logarithm (LIL) for the $p$-rotor walk. Utilizing the decomposition of the walk into a martingale perturbed by its running extrema, we obtain first a functional Law of the Iterated Logarithm for the linearly interpolated paths of the $p$-walk. We then obtain the classical LIL constants by solving a calculus of variations problem over the perturbed Strassen set.

New necessary and sufficient conditions for two-weight estimates for fractional maximal and integral operators are established by means of Harnack-type estimates for Muckenhoupt weights. Applications include weighted Hardy inequalities with weights based on the Hardy-Littlewood maximal operator.

Let $g$ be a continuous metric on $\mathbb R^3$ which is asymptotically flat in the sense that $\vert g_{ij}(x) - \delta_{ij}\vert = O(\vert x\vert^{-\tau})$ for some $\tau > \frac{1}{2}$. Further assume that $g$ can be uniformly approximated on compact sets by smooth metrics with almost non-negative scalar curvature. For such a metric $g$, we define a synthetic ADM mass $m(g)$ using harmonic functions. The harmonic mass $m(g)$ coincides with the usual ADM mass whenever $g$ is smooth and decays rapidly enough that the latter is defined. The harmonic mass can also be computed as a limit of the $C^0$ local mass introduced by Burkhardt-Guim. Our main result is a positive mass theorem: the harmonic mass satisfies $m(g)\geq 0$ and if $m(g) = 0$ then $g$ is flat.

We introduce and study finite free perpetuities, defined as monic polynomial solutions of degree $n$ to the affine fixed-point equation \[ p(z) = \mathbb{E}\!\left[ A^{n}\,p\!\left(\frac{z-B}{A}\right)\mathbf{1}_{\{A\neq0\}} \right] + \mathbb{E}\!\left[ (z-B)^n\mathbf{1}_{\{A=0\}} \right], \] where $A$ and $B$ are complex-valued random variables with finite moments up to order $n$. Equivalently, if $p(z)=\mathbb{E}[(z-X)^n]$, then $p$ encodes a truncated moment version of the classical perpetuity equation $X\stackrel{d}{=}AX+B$ with $X$ and $(A,B)$ independent. This places finite free perpetuities between classical perpetuities and free-probabilistic fixed-point laws. We prove existence and uniqueness under weak conditions, and we identify a broad class of admissible pairs $(A,B)$ for which the resulting polynomial has only real, nonnegative zeros. Our approach uses finite free additive and multiplicative convolutions together with a probabilistic representation via the $U$-transform. As a motivating example, we exhibit an explicit family of finite free perpetuities expressed in terms of Jacobi polynomials and show that their empirical root distributions converge to a free-beta-prime law. More generally, for admissible sequences of parameters, we prove weak convergence of the empirical root distributions of finite free perpetuities to the law of a free perpetuity characterized by the corresponding free fixed-point equation. This yields a finite-degree polynomial model approximating free perpetuities and clarifies the connection between classical affine recursions, finite free convolutions, and free probability.

We construct a sequence of new universal quantum knot invariants that are lifts of both the semi-simple and non semi-simple $U_q(sl_2)$ quantum knot invariants. More specifically, for any level $\mathscr N$ we define a “level $\mathscr N$ universal invariant” $ \widetilde{\Omega}_{\mathscr N}(L)$ arising from quantum traces on finite quotients of the generic Verma module over certain quotient rings. We show that for $\mathscr N$ prime, this is the maximal invariant that can arise from the $\mathscr N$-part of the Verma module, and it is a specific interpolation between the $\mathscr N^{th}$ coloured Jones and $\mathscr N^{th}$ ADO polynomials. For $\mathscr N$ non prime $ \widetilde{\Omega}_{\mathscr N}(L)(q,s)$ has a richer structure, it recovers the $\mathscr N^{th}$ coloured Jones and $\mathscr N^{th}$ ADO polynomials, but it could contain more information which is not seen in the sequence of coloured Jones and ADO invariants.

By incorporating leading $c\,$-dependent corrections to the Carroll transformations, we introduce the “post-Carroll transformations”. We demonstrate that these transformations are consistent with post-Carrollian mechanics \cite{Najafizadeh:2025ksm}; furthermore, they give rise to the so-called “post-Carroll algebra”. We show that, unlike the Carroll algebra, this new structure allows for a central charge in higher dimensions; we refer to it as the “Carroll-Bargmann algebra”. To construct conformal extensions, we first build the conformal extension of the post-Carroll algebra and study field theories invariant under this symmetry. We then construct the conformal extension of the Carroll-Bargmann algebra, referred to as the “Carroll-Schrödinger algebra”, and demonstrate that it precisely matches the symmetry algebra of the higher-dimensional Carroll-Schrödinger theory \cite{Najafizadeh:2024imn}. Finally, we derive the general form of two-point functions in a post-Carrollian CFT, which in $1+1$ dimensions exhibits both electric and magnetic sectors, while in higher dimensions only the magnetic sector survives.

We propose a class of neural ODEs that universally approximates locally exponentially stable dynamics and the region of attraction from trajectory data. The model dynamics are constrained by the gradient field of a jointly learned maximal Lyapunov function. Under this constraint, we show that exponentially stable dynamics can be approximated arbitrarily well within the region of attraction. Furthermore, the region of attraction of the constrained model is exactly characterized by the 1-sublevel set of the jointly learned Lyapunov function, and we derive conditions under which it approximates the true region of attraction arbitrarily well. We validate the approach experimentally on nonlinear systems with nonconvex regions of attraction.

Terwilliger algebras are subalgebras of a matrix algebra constructed from an association scheme. Rie Tanaka defined what it means for a Terwilliger algebra to be almost commutative and gave five equivalent conditions in the case where the association scheme is commutative. A sixth condition for a Terwilliger algebra coming from a commutative Schur ring to be almost commutative has since been discovered. In this paper we first provide a classification of orbit Schur rings that produce an almost commutative Terwilliger algebra for a finite cyclic group. In particular, we show that the subgroup of automorphisms used to form the orbit Schur ring is either trivial, or the whole automorphism group when the cyclic group has prime power order. If the cyclic group has order $2^n$, these are the only options. If the cyclic group has order $p^n$, for an odd prime $p$, then the automorphism subgroup of order $p^{n-1}$ also works. If the group has a non-prime power order then the only orbit Schur ring that produces an almost commutative Terwilliger algebra comes from the trivial subgroup of the automorphism group. We then give a condition for when a wedge product of Schur rings produces an almost commutative Terwilliger algebra. This allows us to determine exactly when a Schur ring over a cyclic group produces an almost commutative Terwilliger algebra.

We provide a statistical theory explaining why empirical score matching can achieve minimax optimal performance for diffusion generative models when the data distribution is supported on a low-dimensional manifold. More precisely, we establish convergence rates for empirical risk minimization over suitably constrained Sobolev classes and show that, under appropriate regularity conditions, the resulting score estimators lead to minimax optimal Wasserstein-1 rates for the generated distribution, up to logarithmic factors. Our results provide sufficient conditions for score-based generative modeling to generalize from the data instead of overfitting, while the resulting statistical rates depend on the intrinsic dimension of the data manifold rather than its ambient dimension.

Sphere packings in circular cylinders have attracted substantial research interest, among which the discovery of chiral helical structures is the most iconic. However, recent experimental results on zebrafish do not match the known packing structures in circular cylinders. To account for the inherent imperfections of biological tubes, we take elliptic cylinders as the canonical deformation of circular cylinders and investigate the densest packings of hard spheres in them using simulation, theory, and experiments. Starting from the chiral structures in circular cylinders, we demonstrate that even a weak cross-sectional deformation can trigger entirely new phases, including ones that either eliminate global chirality or significantly complicate the chiral structures. This reveals the significant effect of cylindrical anisotropy. The new helical phases under anisotropic confinement remain chiral and develop hierarchical periodic structures, which are difficult to obtain by simulations but are predicted by our newly developed theory for helical phases in elliptic cylinders. The theory also predicts double oscillated-chain phases without chirality, which perfectly match the simulations. Our work offers fresh insights into understanding packings in anisotropic cylinders, which will help researchers to design new materials and to understand many living systems.

This paper studies the uniqueness of bound states for the problem \Delta u + u\log u ^2=0, \quad u\in H^1(\RN), \quad n\geq 2, which arises from the logarithmic Schrödinger equation. We prove that for every integer $k\geq 1$, there exists a unique radial solution $u(r)=u(|x|)$ that has exactly $k$ simple zeros for $r>0$. This resolves an open problem posed by Troy [{Arch. Ration. Mech. Anal.} 222 (2016), 1581–1600] and confirms the Berestycki-Lions conjecture for the logarithmic nonlinearity. The proof combines the shooting method with suitable auxiliary functions introduced by Tang [{Invent. math.} 243 (2026), 245–291]. A major difficulty arises from the singular behavior of the nonlinearity $f(u)=u \log u^2$ at origin. We overcome it by establishing asymptotic convergence and sharp decay rates at infinity for any ground state or bound state. More precisely, every such solution satisfies \lim_{r\to\infty}\frac{ u'(r)}{u(r) \sqrt{\abs{ \log u^2(r)}}}=\lim_{r\to\infty}\frac{u'(r)}{ru(r)} = -1, \quad \limsup_{r\to\infty}|u(r)|e^{(\frac12-\epsilon)r^2}<\infty, ~~\forall \epsilon \in ( 0,\frac{1}{2}). These asymptotic behaviors are of independent interest and may be useful for other problems involving logarithmic nonlinearities.

We study random Schrödinger operators on closed Riemannian manifolds with Anderson-type potentials. We prove high-probability spectral inclusion bounds showing that eigenvalues remain close to those of the Laplacian, with deviations controlled by a norm of the potential coefficients. Compared with deterministic bounds, this yields a square-root cancellation gain. The proof is based on a general principle showing that randomisation improves operator norm bounds for multiplier-type operators, which we formulate in both discrete and continuous settings.

In this paper, we consider the quasilinear Schrödinger-Poisson system with concave and convex nonlinearities \begin{align*} \begin{cases} -\Delta_{p} u+\lambda V(x)|u|^{p-2}u + \mu \phi |u|^{p-2}u= a(x)|u|^{m-2}u + b(x)|u|^{q-2}u & \ \ \ \mathrm{in}\ \mathbb{R}^{3}, -\Delta \phi=|u|^{p} &\ \ \ \mathrm{in}\ \mathbb{R}^{3}, \end{cases} \end{align*} where $\lambda >0, ~\mu >0$, $\frac{3}{2}<p<3$, $1< q<p < m < 2p$ and $\Delta_{p} u= \hbox{div}(|\nabla u|^{p-2}\nabla u)$. We assume that $V(x) \in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well, while $a(x)$ and $b(x)$ are allowed to be sign-changing and satisfy some suitable assumptions in $\mathbb{R}^3$. By using the Ekeland's variational principle and combining the constraint approach, we prove that the system admits two positive solutions.

Floer persistence barcodes provide a quantitative way to encode action-filtered Floer homology. Inspired by the Shannon entropy of persistence barcodes in topological data analysis, we introduce a Floer-theoretic entropy invariant, called persistent entropy, which measures the asymptotic linear growth rate, under iteration, of the Shannon entropy determined by the distribution of finite bar lengths. This is complementary to the barcode entropy of Çineli–Ginzburg–Gürel, which records the exponential growth rate of the number of not-too-short bars. We prove that, for Hamiltonian diffeomorphisms, the relative and absolute persistent entropies coincide with the corresponding barcode entropies. For Liouville domains, we prove general comparison inequalities and a subexponential length-growth criterion which gives equality beyond the case of vanishing symplectic homology. We also compute the persistent entropy of cotangent disk bundles of negatively curved manifolds and relate it to the topological entropy of the geodesic flow. In addition, we prove Hofer-stability estimates for finite-level Shannon entropy and derive flexibility and rigidity-type questions for barcode and persistent entropies of Reeb flows.

We study an obliquely reflected optimal stopping problem in the nonnegative quadrant with nonsmooth max-type payoff \(G(x)=x_1\vee\alpha x_2\), and we develop a measure-valued potential-theoretic formulation of the associated obstacle problem. The kink of \(G\) on the diagonal \(x_1=\alpha x_2\) produces a singular surface measure in the distributional generator, while the oblique reflection directions generate boundary local-time contributions on the coordinate faces. Together with the absolutely continuous stopping gain, these terms define a total signed stopping measure \(\Gtot\). We derive the corresponding reflected Itô–Tanaka identity, prove a killed-resolvent representation of the value function in the continuation region, and show that the unrestricted reflected resolvent is generally incorrect because the process is not absorbed on the stopping set. The free boundary is formulated through a continuation-side trace condition for the killed potential. Under a vertical monotonicity hypothesis on \(V-G\), the stopping set is shown to have an epigraph form. We finally prove a verification theorem: any admissible epigraph candidate satisfying contact, strict continuation, reflected Neumann compatibility, growth, the trace condition, and measure-superharmonicity coincides with the value function, and its first entry time is optimal.

We study the vorticity formulation of the 3D Navier-Stokes equation driven by transport noise in a periodic channel with Navier-slip boundary conditions. We consider both non-degenerate transport noise and degenerate tangential transport noise. For any prescribed $T>0$ and $\epsilon>0$, we prove that, by choosing the noise intensity sufficiently large and concentrating the noise on sufficiently high modes, the solution exists up to $T$ with probability at least $1-\epsilon$. A main contribution of this work is to identify and analyze the interaction between enhanced dissipation induced by transport noise and physical boundary effects. The no-flux condition breaks the isotropy of the noise and changes the scaling limit of the Itô-Stratonovich corrector. In the non-degenerate case, a boundary feedback term appears in the limiting effective operator; in the degenerate case, the limiting operator is a nonlocal anisotropic tangential dissipation. The proof is based on a combination of a boundary correction operator, a Meyers-type estimate, a scaling-limit analysis of the Itô-Stratonovich corrector, and resolvent estimates for the deterministic limiting equations.

We present GPU algorithms for Ewald summation methods for accelerating N-body Stokes flow problems in periodic domains. Like most N-body codes, Ewald sums use a near-field/far-field decomposition. The near field involves particle-to-particle (P2P) interactions. The far field primarily involves particle-to-grid (P2G) and grid-to-particle (G2P) interactions, as well as Fast Fourier Transforms. For each interaction, we investigate several algorithmic variants. Our implementation uses PyKokkos, a Python interface for the Kokkos C++ parallel programming framework, which supports portability to AMD/NVIDIA GPU and ARM/x86 CPU architectures. Double and single-precision numerical results, alongside analytical performance models, confirm the efficiency of our algorithms on AMD and NVIDIA GPU and on ARM and AMD CPU architectures. The P2P interaction achieves around 73% compute efficiency on NVIDIA H200, 84% on NVIDIA A100, 60% on AMD MI300, 52% on Grace CPU, and 68% on AMD Epyc CPU. A straightforward implementation of the P2G kernel can become a computational bottleneck. We introduce a novel P2G algorithm that achieves up to 16$\times$ speedup compared to a baseline GPU implementation. The overall Ewald sum code processes approximately 8 million particles per second on a H200 GPU, and about a half-million particles per second on a Grace CPU, for nine digits of accuracy. We also perform a multi-GPU weak scaling test on up to 256 million particles (64 GPUs) that shows bounded communication cost for all stages except the all-to-all particle sorting, which can be reduced to neighbor communication in the relevant time-stepping regime.

This ArXiv preprint of my doctoral dissertation, which, at this stage, has not yet been accepted by the doctoral thesis committee, is intended both to lay the groundwork for a series of papers, and to confirm the existence of a first proposed solution to the "Dual Amplituhedron" problem posed in 2014 by Princeton physicists N. Arkani-Hamed and J. Trnka, a mathematical object which encapsulate the calculation of scattering amplitudes in high-energy particle colliders. The first part of this thesis deals with a rewriting of the positive real Grassmannian and performing the singular gluing of its positroid varieties in a new way via Spectral Algebraic Geometry of J. Lurie on "structured" spaces and a categorification of his "Tannaka Duality for Quasi-coherent Stacks", finding in this formal moduli problem, a compact yet holistic formulation via perverse intersection complexes of P. Deligne. This algebraico-geometric perspective paired with a synthetic differential-geometric perspective, namely the Differential Cohesion of B. Lawvere and U. Schreiber, subsumes infinitesimal thickenings, crystalline cohomology of De Rham stacks, the "Modalities of Structured Geometries" and the unification of their cohomologies of the underlying concrete topological etale algebraic space. The second part uses this rewriting of positive Schubert geometry and "The Cohomology of Brauer-Grothendieck Spaces" of B. Toën and B. Antieau, to show that, instead of the Grassmannian spectral Deligne-Mumford stack which is autodual in the infinity-category of prestable infinity-categories of modules on E\infty-ring spectra, the Amplituhedron, categorified in a functor between infinity topoï, possesses a concrete non-trivial dual, and has a De Rham volume, facts of interest for the expected Duality between the Standard Model of particles and String Theory. This new construction yields the Dual Amplituhedron.

Minimax problems arise in many applications, including robust learning and Stackelberg models. Most existing methods for minimax problems address unconstrained or projection-friendly settings, while functional constrained minimax problems remain far less explored. We study a class of convex-(strongly-)concave minimax problems with functional constraints. By exploiting strong duality, we incorporate the inner-maximization functional constraints into the objective. This allows us to efficiently obtain inexact gradients of the primal function of the reformulation and to design a proximal augmented Lagrangian method (PALM). Each PALM subproblem is solved by an inexact accelerated proximal gradient scheme to handle inexact gradients arising from approximately solving an auxiliary maximization subproblem. We show that the proposed method returns an $\varepsilon$-KKT point and a primal $\varepsilon$-optimal solution, with $\tilde{\mathcal{O}}(\varepsilon^{-1})$ first-order oracle and iteration complexity in the convex-strongly-concave case. For the convex-concave case, the complexity remains the same for the primal gradient evaluation but increases to $\tilde{\mathcal{O}}(\varepsilon^{-\frac{3}{2}})$ for the dual part.

This paper proposes and analyzes a fractional-order SVIR epidemic model based on a hybrid Mittag-Leffler-Caputo-Fabrizio (MLCF) fractional operator with a nonsingular kernel. This model captures short- and long-term memory effects in epidemic transmission dynamics. The positivity and boundedness of the solutions are proven through an integrated formulation of the MLCF operator and a fractional Gronwall inequality. The basic reproduction number $\mathcal{R}_0$, equilibrium points, and their local and global stability properties are rigorously investigated through Jacobian analysis, logarithmic Lyapunov functionals, and a fractional LaSalle invariance principle. To approximate the model, a $\theta$-weighted nonstandard finite difference (NSFD) method is developed. This method preserves the continuous system's key qualitative properties, including positivity and boundedness, and is unconditionally stable in the fully implicit case. Consistency and first-order convergence are also proven. Numerical experiments, together with sensitivity and bifurcation analyses, illustrate the impact of fractional memory parameters on epidemic evolution and demonstrate the effectiveness of the proposed approach.

Daisy cubes are finite partial cubes that admit an isometric hypercube embedding whose labels form a Boolean down-set. Call a vertex a root if it can receive the all-zero label in such an embedding. Using the peripheral $\Theta$-class characterization implicit in earlier work of Taranenko and of Xie–Xu, and Vesel's description of possible zero vertices, we prove the following intrinsic refinement. If exactly $b$ classes are peripheral on both sides, then $G\cong G_0\square Q_b$, where $G_0$ has a unique root and the roots of $G$ induce the displayed convex $b$-cube. Our main results concern the daisy grids $B_{r,s}=P_3^{\square r}\square Q_s$. We classify every nonempty graph $B_{r,s}-\{x,y\}$ that is a partial cube, a daisy cube, or a minimal forbidden partial-cube minor for daisy cubes. The ambient-isometric cases admit a uniform coordinate description. A local four-cycle argument reduces every non-isometric noncorner partial-cube deletion to $r+s\leq3$, and an exact orbit analysis leaves precisely eleven sporadic orbits. None is daisy or pc-minor-minimal. Consequently, a deletion is pc-minor-minimal non-daisy exactly when the deleted vertices are opposite corners with $s\geq1$ and $2r+s\geq3$, or when a boundary $P_3$-edge is deleted from $B_{1,1}$. This recovers the known hypercube and one-$P_3$ examples and produces a new infinite family with at least two $P_3$-factors. A dependency-free program provides exact certificates for the finite orbit analysis.

We give an explicit classification as well as an explicit construction of simple $\mathfrak{sl}_2$-modules that are torsion-free of finite rank with respect to the action of the Cartan subalgebra and which admit a grading by a finite cyclic group corresponding to the rank of the module.

In this article, we consider a class of close-to-convex harmonic mappings with special analytic part in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ and obtain sharp pre-Schwarzian norm estimate. We study the theory of harmonic Bloch mapping for the considered class. Moreover, we discuss some growth and distortion theorems for analytic and co-analytic parts of such harmonic mappings.

We consider stratified steady water waves in a two dimensional channel. Our subject is branches of Stokes waves, bifurcating from laminar flows. We assume that the mass flux and the Bernoulli constant are fixed and consider the period of the wave as a parameter, which can change its value along the branch. A new class of density and Bernoulli functions is presented, for which laminar flows generate global bifurcation branches. The laminar flows are not necessary unidirectional and we show that the bifurcation branch can bifurcate from the laminar flow with arbitrary large period.

Probabilistic Boolean circuits have recently been proposed as a string-diagrammatic foundation for finite probabilistic programming. In this paper, we present a complete set of axioms for their semantics in terms of Markov kernels. Our approach is based on two intermediate results: completeness for partial Boolean circuits and completeness for probabilistic Boolean tapes, a diagrammatic language for rig categories.

In this article, we consider certain class of harmonic mappings defined in the unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}.$ Then we obtain pre-Schwarzian norm estimate of functions in the class. Next, we show that functions in the considered class are univalent and close-to-convex. Moreover, we discuss some growth and distortion theorems for associated analytic and co-analytic parts of harmonic mappings in the class. At last, we present coefficient estimate for the analytic part.

We discuss here how non self-adjoint Hamiltonians, and their related Heisenberg-like dynamics, can be used to model a political system consisting in a coalition $\C$ of different parties (forming a government) and by their (original) supporters $\Sc$. Our aim is to model how the opinion of these supporters changes depending on the efficiency, competence and coherence of the coalition $\C$, as these are perceived by $\Sc$ during their action while governing. After a rather general introduction we propose three specific models, and we describe and comment the dynamical behaviour of the { full} system, $\Sc\cup\C$. The role of the so-called balanced Hamiltonians, recently introduced by the authors in connection with integrals of motion, is discussed in details.