Papers

In this paper, we introduce a broad class of flows, including the prescribed Hermitian-Yang-Mills flow: $$\frac{\partial h}{\partial t}=-\Lambda_{\omega_g}\left(\sqrt{-1} R^h\right)+P$$ where $P\in\Gamma(M,E^*\otimes\bar{E}^*)$ is a prescribed Hermitian tensor associated with a holomorphic vector bundle $E$ over a Kähler (or Hermitian) manifold $(M,\omega_g)$. We establish the long-time convergence of the flow to a limiting metric $h_{\infty}$ and use it to solve the prescribed Hermitian-Yang-Mills tensor equation $$\Lambda_{\omega_g}\left(\sqrt{-1} R^{h_\infty}\right)=P, $$ for a general class of prescribed Hermitian tensors $P$. The crucial uniform $C^0$-estimate of $\{h(t)\}$ along the flow is obtained via a parabolic comparison principle.

We study the multigraded regularity of the complete flag variety under the Plücker embedding. In particular, we prove inductive relationships about the regularity regions, and we provide some inner and outer bounds on the regions.

In this paper we study the logarithm of order $m$ of the Schrödinger operator $\mathcal L_V$ in $\mathbb R^d$, for certain nonnegative potentials $V$. First, the operator $\log^m\mathcal L_V$, $m\in \mathbb N$, is defined by using the spectral measure associated with the self-adjoint operator $\mathcal L_V$ on a suitable subspace of $L^2(\mathbb R^d)$. Then, the semigroup of operators $\{T_t^V\}_{t>0}$ generated by $\mathcal L_V$ allows us to extend the definition of $\log^m\mathcal L_V$ to a wider class of Lipschitz functions. By using logarithmic operators $\log^m\mathcal L_V$, $m\in \mathbb N$, we prove Taylor expansions for the fractional powers $\mathcal L_V^s$ and $\mathcal L_V^{-s}$ with respect to the order $s\in (0,1)$, where the convergence is understood in $L^p(\mathbb R^d)$, $1<p<\infty$.

Every Hölder continuous path $X$ admits a geometric rough path lift $\boldsymbol{X}$ by the Lyons–Victoir extension theorem. A natural question that emerges when lifting more than one path segment at once is that of locality, namely whether the lift $\boldsymbol{X}_{s,t}$ only depends on the increments $X_{s,u}$, $u \in [s,t]$. We investigate the locality of rough path lifts in deterministic and stochastic settings. On the deterministic side, we show that no local, homogeneous rough path lift can be defined on $\gamma$-Hölder paths for all $\gamma \leq 1/2$. More strongly, we show that no Lévy area can be defined which is at the same time bounded, with no further regularity assumptions, and either local and homogeneous or time translation–invariant. We moreover show that the boundedness requirement is sharp: an unbounded, local, time translation–invariant, and bilinear Lévy area can be defined on all continuous paths. On the stochastic side, we classify all local, square-integrable rough path lifts of $d$-dimensional fractional Brownian motion with Hurst parameter $H \in (0,1/2]$. For $H \leq 1/4$, we show that no such lifts exist when $d \geq 2$, while for $H>1/4$, we show that all such lifts are stochastic translations of the canonical rough path. We further refine the classification by requiring invariance in law under time translation, scaling, and coordinate permutation, and show that only the canonical lift satisfies these constraints except at \(H=1/3\), for which there is a one-parameter family of lifts. Finally, we present an argument showing that scale-invariance forces a local $\eta$-Hölder rough path lift of fractional Brownian motion to be square-integrable, allowing scale-invariance to replace square-integrability in our classification and non-existence results.

The numerical simulation of high-dimensional, multi-rate Delay Differential Equations (DDEs) is fundamentally bottlenecked by synchronous time-stepping and the dynamic memory allocation required for continuous history tracking. In this paper, we introduce the Asynchronous Adaptive Taylor Solver (AATS), an event-driven integration framework designed to overcome these high-performance computing limitations. By assigning independent local clocks to individual coordinates and advancing them using high-order Taylor polynomials generated via compile-time Automatic Differentiation, AATS restricts computational work to actively evolving sub-graphs. To eliminate the severe memory overhead endemic to traditional DDE solvers, AATS utilizes statically allocated circular buffers to store polynomial segments, achieving interpolation-free continuous dense-output evaluation with a verified zero-allocation runtime memory footprint. Alongside this software architecture, we establish a novel continuous proof of convergence for asynchronous Taylor expansions and formally prove that the framework's algorithmic complexity scales linearly (O(N)). Extensive benchmarks against state-of-the-art synchronous solvers (Julia SciML) validate these theoretical bounds. On large-scale benchmarks (upto $N = 10000$ coordinates) AATS fundamentally minimizes the constant factor of algorithmic work by avoiding redundant evaluations, delivering empirically consistent with O(N) execution scaling and significant wall-clock speedups.

In this paper, we provide sufficient conditions for the absolute continuity of one-dimensional push-forwards of dependent random vectors. Suppose that $X$ has an absolutely continuous distribution and that the conditional distribution of an $\mathbb{R}^d$-valued random vector $Y$ given $X=x$ is nondecreasing in $x\in \mathbb{R}$ in the usual stochastic order. For Borel maps $g\colon \mathbb{R}\times\mathbb{R}^d\to\mathbb{R}$ satisfying a coordinatewise monotonicity condition in $Y$ and a uniform lower-increment condition in $X$, we prove that $g(X,Y)$ has an absolutely continuous distribution. The result requires neither independence nor a joint density, and allows the marginal law of $Y$ to be completely arbitrary. Moreover, the result remains valid if $\mathbb{R}^d$ is replaced by an arbitrary measurable space endowed with a reflexive binary relation. We discuss consequences for monotone risk aggregation and extensions of the familiar regularization by convolution beyond independent random variables.

In 2022 Aliniaeifard, Li, and van Willigenburg defined Schur functions in the algebra of symmetric functions in noncommuting variables (NCSym), answering an open question posed by Rosas and Sagan in 2004. These Schur functions are not monomial positive, since they are defined via a noncommutative analogue of the Jacobi-Trudi determinant. We introduce Hall-Littlewood functions ${\bf P}_{\pi}({\bf x};t)$ indexed by set partitions $\pi$ in noncommuting variables ${\bf x}=({\bf x}_1,{\bf x}_2,\ldots)$, and define Schur functions in noncommuting variables to be ${\bf s}_{\pi}({\bf x})={\bf P}_{\pi}({\bf x};0)$. We prove that the set of Hall-Littlewood functions $\{{\bf P}_{\pi}({\bf x};t)\}$ for all set partitions $\pi$ of $[n]$ forms a $\mathbb{Q}[t]$-basis of NCSym of homogeneous degree $n$, and that this basis is invariant under any permutation acting on set partitions. These Hall-Littlewood functions in NCSym map to classical Hall-Littlewood functions under commutation, up to a scalar factor. We also show that the Hall-Littlewood functions ${\bf P}_{\pi}({\bf x};t)$ naturally refine the lifted Hall-Littlewood functions in NCSym. Specifically, the Schur functions ${\bf s}_{\pi}({\bf x})$ are monomial positive and refine the lifted Schur function introduced by Rosas and Sagan. Moreover, we introduce a star product of two polynomials in NCSym and develop the star-multiplication rule for a lifted and a non-lifted Hall-Littlewood functions in NCSym. This rule is a noncommutative analogue of the product rule for two Hall-Littlewood functions and, in particular, of the Littlewood-Richardson rule. Finally, our approach extends to the algebra of quasisymmetric functions in noncommuting variables (NCQSym) indexed by set compositions.

We consider the compressible 3D magnetohydrodynamic (MHD) equations under general pressure laws. For all hypo-viscosities $(-\Delta)^{\alpha_1}$ and hypo-resistivity $(-\Delta)^{\alpha_2}$ with $\alpha_1,\alpha_2\in(0,1)$, we prove the non-uniqueness of weak solutions to 3D MHD equations which reveals that there exist infinitely many weak solutions with the same initial data. Also, for the weak solutions in $C^{\widetilde{\beta}}_{t,x}$ to the compressible ideal MHD, where $\widetilde{\beta}>0$, we prove that they are the strong vanishing viscosity and resistivity limit of the weak solutions to the hypo-dissipative compressible MHD.

We prove that, whenever the monodromy maps are fully-punctured, the finite quotients of a fibered hyperbolic $3$-manifold group detect the taut polynomials of the fibered faces of the Thurston norm ball, as well as the Teichmüller polynomials of suitable Dehn fillings. Toward this, we develop a general framework for the profinite invariance of twisted multivariable Alexander polynomials, which is of independent interest; we also prove that being fully-punctured and preserving transverse orientations are profinite properties of the monodromy maps. As an application, we identify specific one-cusped hyperbolic $3$-manifolds \blue{whose fundamental groups} are profinitely rigid among $3$-manifold groups, by a strategy using normalized dilatations and the veering census; notably, the taut polynomials distinguish a pair of mapping tori sharing the fiber and the dilatation.

Multiparametric programming offers a powerful solution to the computational burden of solving optimal control problems repeatedly online. By solving the problem once offline, it yields the optimal control laws as explicit closed-form functions of the initial system state, reducing online execution to a direct evaluation with no iterations required. Most existing works build this idea on a discrete-time foundation, slicing the time horizon into intervals and applying KKT conditions to the resulting algebraic system. This discretization forces a tradeoff: too few intervals and the model fails to capture the true system dynamics, while too many cause the problem size, the number of decision variables, and the number of critical regions to grow rapidly, making both offline preparation and online lookup increasingly expensive. This work develops a multiparametric framework that works directly with the continuous-time problem. Pontryagin's Maximum Principle (PMP) is applied without any model discretization, and the optimal control is recovered as an explicit function of the initial state. Compared to the discrete-time formulation, the continuous-time approach produces substantially fewer critical regions, and this number remains fixed regardless of accuracy requirements, since it reflects the structure of the problem itself rather than a discretization grid. The framework also yields the switching times as explicit functions of the initial state, directly exposing when and how the optimal control structure changes over the horizon. Knowing these switching times in advance allows the real-time controller to skip unnecessary computations between them, further reducing the online execution cost. Results from a PAROC framework case study demonstrate that the continuous-time multiparametric approach is a rigorous alternative to the conventional discrete-time formulations.

We show that the Samuelson condition, an area condition, is not satisfied by the Lagrangian 2-web formed by the tangent lines to a non-degenerate real quadratic curve.

This paper investigates physical-layer security for uplink low Earth orbit (LEO) satellite communications in the presence of multiple non-colluding satellite eavesdroppers. Secure beamforming design in such systems is challenging due to time-varying orbital geometry and probabilistic fading-induced outage constraints. To address this, we first derive tractable closed-form expressions for both connection and secrecy outage probabilities under Nakagami-m fading, and develop differentiable upper-bound cost functions that are amenable to optimization. Next, to exploit the predictable orbital dynamics and temporal correlation of satellite mobility, we reformulate the non-convex secrecy rate maximization problem as a constrained Markov decision process. We then develop a primal-dual soft actor-critic algorithm with a multi-head cost critic that jointly optimizes beamforming while enforcing average outage constraints via Lagrangian relaxation. Numerical results show that the proposed framework improves the ergodic secrecy rate over maximum ratio transmission across all eavesdropper configurations, and outperforms zero-forcing in dense eavesdropping regimes. It achieves within 7 percent of an offline successive convex approximation benchmark while requiring only a single forward pass, enabling low-complexity real-time operation. These results indicate that the proposed approach is applicable to secure beamforming in dynamic LEO satellite environments.

The Unlimited Sampling Framework (USF) overcomes the traditional trade-off between dynamic range and digital resolution, achieving performance unattainable with standard ADCs. Its multi-channel extension (MC-USF) enables reconstruction from multiple folded measurements at critical sampling rates. Existing MC-USF methods typically rely on Chinese Remainder Theorem (CRT)-based unfolding, which requires strict channel-level sampling synchronization and is therefore vulnerable to timing mismatch, jitter, and drift. This paper introduces an asynchronous MC-USF architecture that eliminates the need for synchronization. By viewing spatial-temporal signal lifting as inducing smoothness over a graph of sensing channels, we develop a reconstruction strategy robust to temporal misalignment. Numerical experiments validate the approach, demonstrating accurate recovery and enabling more practical multi-channel USF implementations.

We establish laws of the iterated logarithm for weighted sums of linear and nonlinear functionals of long-memory stationary Gaussian sequences. The leading Wiener chaos determines the scale of the almost-sure fluctuations; all higher chaoses are negligible on it and the exponent of the iterated-logarithm factor is half the Hermite rank. In the linear case, we obtain the exact LIL constant, and it is the classical one. In the nonlinear case, we identify a functional cluster set and give a variational formula for the LIL constant. We also present explicit upper and lower bounds for this constant. The lower bound is expressed through beta functions and depends on the memory parameter, the weight exponent, and the Hermite rank; numerically it matches the constant to within a fraction of one percent, and can therefore be used in its place. Thus, the weights affect the normalization, the geometry of the limiting cluster set, and the nonlinear LIL constant itself.

We formulate a source-adapted Chern–Connes description of M5-brane magnetic charge in physically motivated noncommutative and quotient linking sectors. The starting point is the local codimension-five geometry of an M5-brane source: the magnetic charge is measured on a four-dimensional link. In the smooth local model the link is the ordinary four-sphere, and the pairing reduces to the standard flux integral of the M-theory four-form field strength. The construction is motivated by the published even-codimensional Noncommutative Foliation Invariant (NCFI for short), where Connes' transverse fundamental cyclic cocycle is paired with an even transverse-geometric \(K\)-class. For an M5-brane the relevant codimension is odd, so the required odd \(K\)-class is supplied instead by the boundary map of the source/linking extension. Noncommutative or groupoid algebras are used only when they arise from physical mechanisms: Matrix/M-theory compactification on noncommutative tori, reduction to D-branes in background \(B\)-fields, large \(C\)-field M5 limits, or quotient/orbifold source links. The result is a topological charge formula, not a derivation of the full M5-brane dynamics or of the complete M-theory action.

Sabok asked whether the compact convex set \(S'(X)\) attached to a separable metric space of diameter at most one is always a simplex, and whether \(S'(\mathbb U_1)\) is the Poulsen simplex. We give negative answers. For finite \(X=\{x_1,\ldots,x_m\}\), \(S'(X)\) is affinely homeomorphic to the convex hull of the rows \(r_i=(d(x_i,x_1),\ldots,d(x_i,x_m))\) of the distance matrix; it is a simplex exactly when these rows are affinely independent. The diameter-one four-cycle gives the minimal finite obstruction. For the Urysohn sphere, using the rational Urysohn sphere \(D\) as coordinates, we identify the coordinate model \(S'_D(\mathbb U_1)\) with the Katétov compactum \(K(D)\). Four explicit extreme points \(f_A,g_A,\mathbf 1,\mathbf h\) satisfy \(f_A+g_A=\mathbf 1+\mathbf h\), giving two distinct representing measures for \((3/4)\mathbf 1\). Hence \(S'(\mathbb U_1)\) is not a Choquet simplex.

Agro-food transitions are commonly assessed using optimization models or scenario approaches that return a single feasible configuration. These approaches provide estimates of system performance but limited information on feasible nitrogen cycling configurations. In this study, we develop a network-based modelling framework to characterize alternative steady-state configurations of agro-food nitrogen systems and identify phase transitions between these configurations. We formulate a continuous-time compartmental network model in which production, allocation, recycling, and losses are represented as mass-conserving flows. The system is analyzed under steady-state conditions. A coarse-graining procedure identifies dominant recycling cycles and derives aggregate indicators describing system configuration. The model is parameterized using French reference data and applied to case studies covering fertilization, livestock, and dietary patterns. The analysis identifies distinct steady-state configurations of the nitrogen network, separated by phase transitions in dominant recycling structures. These transitions correspond to changes in system performance indicators. For the French reference system, cropland-based feeding capacity is about 8.7 people per hectare of cropland (depending on constraints), while strict dietary autonomy requires about 19.7 million hectares of agricultural land. Reducing synthetic fertilization without changes in diet or nitrogen sourcing leads to higher land requirements or greater dependence on external inputs. The framework provides a method for mapping feasible nitrogen system configurations and analyzing phase transitions in agro-food nitrogen networks under alternative parameters and constraints.

We study the convergence and acceleration of Qvalue iteration (QVI) with heavy-ball momentum. Although acceleration of value iteration has been studied extensively, there has been less work on the acceleration of heavy-ball QVI for control tasks. We analyze heavy-ball QVI from the viewpoint of switched linear system (SLS) theory and the joint spectral radius (JSR). First, we convert heavy-ball QVI into an exact SLS and interpret its convergence through the JSR. We also give JSR-based conditions under which heavy-ball QVI can be faster than standard QVI.

In this article, we introduce a group linear representation associated to a graph. We study this representation as well as the canonical decomposition of the associated module. A character study is conducted, demonstrating the relevance of this approach to graph theory. We also develop the links that exist between algebra and graphs, particularly in the language of representations.

A complete theory of null Cartan normal helices in Minkowski space-time $\mathbb{E}^4_1$ is developed. Two algebraic conditions, obtained by successive differentiation of the helix invariant along a unit $C$-constant normal field, fully characterize null Cartan helices; the quadratic condition yields two mutually orthogonal helix axes in the Lorentzian metric. Special field types are analyzed and null Cartan cubics are shown to be normal helices. On a timelike hypersurface, a Darboux frame with six curvature functions is constructed from first principles, the normal isophotic condition is shown to reduce to a linear first-order ODE, and the existence of normal silhouettes in $\mathbb{E}^4_1$ is established.

Networks are shaped by competing structural mechanisms, such as communities, geometry, or hubs. In a dynamic network the most predictive mechanism can change, and a model tied to one mechanism, or to fixed weights, cannot adapt as the dominant structure shifts. We develop dynamic Bayesian predictive synthesis for networks, in which a mechanism is an agent forecasting the next snapshot's edges and a synthesis layer combines them with time-varying weights. At each step the method returns a calibrated edge forecast and inference on the mechanism weights, with intervals valid given the fitted agents, so it also reports which mechanism is most informative. Inference of this kind requires a sparse-safe parametrization and an identification theory, under which a single graph identifies and estimates the weights. A sharp threshold separates distinguishable from indistinguishable mechanisms, a change in the active mechanism is tracked at an optimal per-switch cost, and for a single snapshot the method reduces to calibrated link prediction. On real networks, simulations, and benchmarks, the synthesis gives accurate, calibrated forecasts and recovers the leading mechanism when

A three-dimensional solid has the Rupert property if a congruent copy of the solid can pass through a hole cut through it without splitting it. We extend this idea to pairs of convex solids: two solids are called friends if each can pass through a suitable hole in the other. A solid is called lonely if it has no friends, including itself. We show that a convex solid is lonely if and only if it has constant width. We also show that every convex solid that does not have constant width has a particularly simple friend: an arbitrarily long and arbitrarily thin rectangular cuboid. Finally, we prove that all non-constant-width convex solids lie in a single connected component of the friendship graph. More precisely, any two such solids are connected by a chain of at most “three handshakes”.

These lecture notes are intended as an accessible introduction to some basic ideas of motive theory for readers with limited background in algebraic geometry. Mathematics often reveals unexpected connections between seemingly distant areas. A simple combinatorial identity may encode geometric structures, arithmetic information, or even sophisticated categorical phenomena. In these notes, we trace a path from elementary counting arguments to Voevodsky's theory of motives. We show how some classical combinatorial identities emerge naturally from geometry, and how motivic decompositions reveal the deeper geometric and arithmetic structures underlying them. Our guiding example is provided by the Karaji–Pascal identity and its $q$-analogue, which link combinatorics and algebraic geometry. Thus our primary aim of these lecture notes is to demonstrate that some familiar combinatorial identities can provide non-experts with an entry point to some of the basic ideas of motive theory! Moreover, while introducing the reader to the subject, we also hope to encourage the view that even an elementary mathematical formula may encode a deeper underlying geometric and arithmetic structure. Along the way, the notes offer a gentle introduction to motives through a concrete example rather than through the full technical machinery of modern theory.

We approach Faà di Bruno as a composition theorem for Taylor polynomials. For $C^k$ maps $\phi: E \to F$ and $\psi: F \to G$ between Banach spaces, let $T^k_\ast(\phi; x)$ denote the reduced Taylor polynomial of $\phi$ at $x$, obtained by removing the constant term. We show that $$T^k_\ast(\psi \circ \phi; x) = \pi_{\le k}\bigl(T^k_\ast(\psi; \phi(x)) \circ T^k_\ast(\phi; x)\bigr).$$ The proof is an elementary estimate of the Peano remainder and does not use partitions or combinatorial enumeration. Expanding this composition identity recovers the classical Faà di Bruno formulas. Polarization gives the multivariate partition formula (Lévy 2006), while coefficient extraction gives the multi-index formula (Constantine and Savits 1996). Our approach separates the functorial nature of Taylor approximation from the combinatorial bookkeeping of polarization and coefficient extraction. As an application, we give a general higher-order product rule.

In a recent paper, Chatterjee, the author and Mohan posed the problem of determining all solutions of the Diophantine equation $F_n=\Phi(m)$, where $F_n$ is the $n$-th Fibonacci number and $\Phi(m)$ counts the number of nonempty sets $A \subseteq \{1, 2, \dots, m\}$ for which $\gcd(A)$ is relatively prime to $m$. In this paper, we prove that the Diophantine equation has the only solutions $(n,m)=(1,1),(2,1),(3,2)$. The main tools used in this paper are lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethő version of the Baker-Davenport reduction method in diophantine approximation.

Given a finite sequence of integers, form its difference triangle by repeatedly taking absolute differences of consecutive entries. We call the sequence Gilbreath if the leftmost entry of every row below the top is 1. The Gilbreath conjecture, which remains open, asserts that every initial segment of the primes is a Gilbreath sequence. This paper studies the local extension problem: given a Gilbreath sequence, which integers can be appended to it while preserving the Gilbreath property? We call the set of such admissible values the valid-extension set of the sequence. A previously proposed characterization in the literature predicts that this set always fills a natural parity interval around the last term. We show that this fails in general: the valid-extension set can have interior holes, with the smallest failure occurring at length 5 for the sequence (2,3,5,9,15). The paper develops a corrected theory of valid extensions. We give an exact criterion for membership in the valid-extension set, an algorithm that computes it, and a sharp condition determining exactly when the set fills the candidate interval. This last condition is an order-sensitive analogue of the classical Brown completeness criterion for subset sums. We also establish endpoint validity and reflection symmetry, determine the exact minimum size of the valid-extension set together with its unique minimizer, exhibit a family whose valid-extension set has exponentially many components, and provide enumeration data through length 11.

This paper presents a brief overview of studies of magnetic filament formation in the solar convection zone. Two stages of magnetic filament formation and development are distinguished. The first stage can be described within the kinematic approximation for the MHD equation, since the average kinetic energy of convective motion exceeds the average magnetic field energy. Moreover, magnetic Reynolds number is on the order of $10^6$. Therefore, at the initial stage of magnetic filament formation, the magnetic field can be considered frozen-in. It turns out that at this stage, magnetic filaments begin to form at the boundaries of convective cells in hyperbolic regions of the flow. These regions act as attractors for the magnetic field. At the free boundary of the convective zone near the interfaces between convective cells, the magnetic field has a predominantly normal component relative to the boundary. As the magnetic field increases, the field's frozen-in nature in the filament is disrupted, and saturation occurs due to finite conductivity. It is important to note that the magnetic field growth and saturation in the filament at this stage can be determined by analyzing the behavior of the magnetic field at the free boundary, which, according to observations, can be considered flat. Moreover, the formation of magnetic filaments is independent of the internal structure of the convective cells and is determined by the behavior of the convective flow along the free surface. In the next stage, when the magnetic field in the filament is sufficiently strong, the magnetic field pressure gradient begins to impede the convective flow, leading to a shift of the hyperbolic regions toward the convective flow along the free surface. As a result, the transverse size of the filament increases and stops when the kinetic energy density and the magnetic field energy density become comparable.

A coupled conforming-nonconforming Galerkin method is proposed for Poisson's equation on two-dimensional curved domains. The method applies a weak Galerkin discretization only on a thin boundary layer of curvilinear elements near the curved boundary, while using a standard continuous Galerkin discretization in the polygonal interior. In this way, geometric flexibility is retained where it is needed most, and the number of nonconforming degrees of freedom is significantly reduced. A key ingredient is a mixed interpolation-projection operator on curvilinear weak Galerkin elements, combining $L^2$ edge projections with conforming nodal interpolation on the interface side to ensure compatibility with the continuous Galerkin trace. Based on this construction, we prove an optimal a priori error estimate of order $O(h^k)$ in the energy norm under the basic geometric assumptions of the method, and an optimal $L^2(\Omega)$ estimate of order $O(h^{k+1})$ under the additional elliptic dual regularity assumption. Numerical experiments confirm the theoretical rates and demonstrate substantial savings in degrees of freedom compared with a fully weak Galerkin discretization.

This article is the $8^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10}. In this study, we define isomorphic set, isomorphism series, isomorphism digraph or isomorphism diagram $\mathcal{D}$ and isomorphism graph $\mathcal{G}$ of circulant graphs and obtain these corresponding to $C_{16}(R)$, $C_{27}(S)$ and $C_{54}(1,3,17,19)$ and present the isomorphism digraph and the isomorphism graph of $C_{432}(T)$ which has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2,3 and of $C_{6750}(U)$ which has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 3,5 where $T$ = $\{16, 27, 48, 54, 128, 160, 189\}$ and $U$ = $\{135, 243, 250, 750, 1107, 1593, 2000, 2457, 2500, 2943\}$. We also define diameter of isomorphic set of $C_n(R)$ and isomorphic distance between any two circulant graphs. We find that the order of isomorphic sets of $C_{432}(T)$ and $C_{6750}(U)$ are 36 and 450 and their diameters are 3 and 3, respectively.

We consider the moduli space, $\mathcal{M}_d^N[\mathcal{P}]$, of degree $d$ rational self maps of $\mathbb{P}^N$ with prescribed pre-periodic structure $\mathcal{P}$ which were introduced by Doyle and Silverman. It was shown in arXiv:1305.1054 that, $\mathcal{M}_2^1[\mathcal{P}_6]$, the moduli space of degree two rational maps with a $6$-cycle is a surface of general type. Here we compute the Kodaira dimension of all the moduli spaces with up to six pre-periodic points and show that $\kappa = -\infty, 0, 1,2$ are all realized for some $\mathcal{P}$.