Papers

We study exact representability by bias-free shallow polynomial neural networks using algebraic geometry. Over $\mathbb{C}$, a width-$r$ network with activation $z\mapsto z^d$ computes a sum of $r$ $d$-th powers of linear forms, whose Zariski closure is a Veronese secant variety. Ideal elimination therefore yields polynomial certificates of nonrepresentability. We implement this construction as a generic architecture-to-certificate pipeline. For quadratics, we recover the exact symmetric determinantal description and explain its dimension through orthogonal symmetry. In higher degree, the implementation recovers classical catalecticant and secant equations and maps the practical reach of direct elimination across a finite architecture sweep. We also derive the exact population loss floor for a rank-two quadratic network on the sphere, illustrating how an algebraic obstruction induces irreducible approximation error.

In the present paper, we study the existence of isomorphic embeddings from Lorentz function spaces $L_{p_1,q_1}$ into Lorentz ideals $\mathcal{C}_{p_2,q_2}$ in $B(H).$ In particular, we show that, for $p,q\in (0,\infty),$ the quasi-Banach function space $L_{p,q}(0,1)$ isomorphically embeds into the ideal $\mathcal{C}_{p,q}$ if and only if $(p,q)=(2,2).$ This extends several existing results in the literature and answers a question due to Astashkin et. al.

The conformal geometric algebra Cl(4,2) ~ R(8) carries a natural Z_2-grading defined by the dilatation bivector D=\e_4e_0, which satisfies $D^2=1$ and splits the eight-dimensional spinor module into two four-dimensional eigenspaces $S=S_+\oplus S_-$. We prove that the ad_D-eigenvectors of Cl(4,2), those elements with eigenvalue~$\pm1$ under the Lie bracket $[D,\cdot]$, are exhausted by 16 doublets of the form $T^\pm_J=n_\bullet e_J$, where $n_\bullet \in {n_\infty,n_o}$ is a null vector and $e_J$ is a product of Lorentz-sector basis vectors indexed by $J\subseteq\{1,2,3,5\}$. These doublets satisfy universal algebraic identities: $(T^\pm_J)^2=0$ (nilpotency) and $T^+_J\,T^-_J=(-1)^{k(k+1)/2+1}\,\eta_J 2\Pi_+$ (single-projector proportionality), where $k=|J|$, $\eta_J=\prod_{j\in J}\eta_{jj}$ is the spectator metric, and $\Pi_+=\tfrac12(1+D)$. The contracted sums $\sum_{|J|=k}T^+_J T^-_J$ are shown to be controlled by the elementary symmetric polynomials $S_k$ of the Lorentz signature values $(+1,+1,+1,-1)$, with the vanishing of $S_2=0$ reflecting the $(3,1)$ signature of physical spacetime. The even-grade doublets (belonging to Cl^+(4,2) ~ Cl(4,1)) contribute $8\Pi_+$ to the unweighted grand total, while the odd-grade doublets cancel exactly. The 32 doublet eigenvectors generate, under the commutator bracket, the 63-dimensional Lie algebra sl(8,R) in which the conformal algebra so(4,2) embeds as a 15-dimensional subalgebra. All identities, including a closed form for the commutators across the Levi interface, are proved in every conformal extension Cl(p+1,q+1); there the commutator span misses precisely the center of the algebra, a parity dichotomy invisible at the Lorentz signature. Two applications are developed: the top rung of the tower computes the Dirac versus Majorana dichotomy of rank-five pseudoscalars ... [shortened]

Gross, Mansour, and Tucker~[European Journal of Combinatorics, 95 (2021): 103329] introduced the partial-twuality polynomials of ribbon graphs. Recently, Deng, Jin, and Yan generalized the partial-twuality polynomials to the framework of matrix algebra and investigated several of their basic properties. They asked whether there exist matrix operations, called partial duality $\delta$ and partial Petrie duality $\tau$, on pairs $(M,A)$, where $M$ is a square matrix whose rows and columns are indexed by a finite set $V$ and $A\subseteq V$, such that $\delta^2=\tau^2=(\delta\tau)^3=id$ and the exponent of the partial-twuality polynomials coincides with some parameter of the matrix obtained by applying \(\bullet\) to \((M, A)\). In this paper, we introduce a paired-matrix framework for partial-twuality polynomials over the binary field $\mathbb{GF}(2)$. We prove that there exist two local operations \(\delta\) and \(\tau\) on \(\bigl((M,I_{|V|}),A\bigr)\) satisfying $\delta^2=\tau^2=(\delta\tau)^3=id$ and $P_{\langle \bullet \rangle}((M,I_{|V|}),z)=P_{\langle \bullet \rangle}(M,z)$ for $\bullet \in \{\delta, \tau, \delta\tau, \tau\delta, \delta\tau\delta\}$, thereby answering their question affirmatively. Finally, we establish a recurrence relation for the partial \(\langle\delta\tau\delta\rangle\)-polynomial with respect to an edge. This recurrence enables the computation of the partial \(\langle\delta\tau\delta\rangle\)-polynomial for certain bouquets, simple graphs, and simple signed graphs.

We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property $TMCP^+(K,N)$. The cost for this problem is the time separation function $\ell(x,y)$ (the Lorentz distance), which represents the maximum amount a particle can age when traveling from $x$ to $y$. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we show that the Lorentzian timelike measure contraction property $TMCP^+(K,N)$ is equivalent to the $TMCP^+_{rLip}(K,N)$ condition, where the curvature dimension conditions are defined on gradient flow curves of reverse 1-Lipschitz functions.

Starting from a Hilbert plane and removing the Side-Angle-Side (SAS) congruence axiom, we investigate to what extent SAS can be recovered synthetically from the remaining classical triangle congruence criteria. We show that the Angle-Side-Angle criterion, together with a ray correspondence principle corresponding to Theorem 13 of Hilbert's Grundlagen der Geometrie, suffices to reconstruct SAS. We further show that both the Side-Side-Side and the Side-Angle-Angle criteria also suffice, once combined with the ray correspondence principle and suitable auxiliary principles – the existence of midpoints and a hypotenuse-angle criterion for right triangles in the first case, and the existence of angle bisectors, the congruence of supplements of congruent angles, and the Pons Asinorum in the second. Although the two routes rely on auxiliary principles of different character, we show that they converge on a single final argument once a common hypotenuse-angle criterion is established. A metamathematical analysis, based on an explicit model adapted from Hilbert's own independence construction, complements these reconstructions: it shows that the ray correspondence principle alone cannot reconstruct any of the classical criteria, and that the Pons Asinorum and the hypotenuse-angle criterion are each independent of the remaining auxiliary principles used in their respective reconstructions. The resulting picture is not a formal hierarchy of the congruence criteria, but it does show that the Angle-Side-Angle reconstruction rests on a provably more economical basis than those obtained from Side-Side-Side or Side-Angle-Angle.

We study the algebra of integro-differential operators with smooth coefficients and kernels on a subspace of $C^\infty[a,b]$. We find a normal form for elements of this algebra and determine its unit group. The formulation of inverses gives explicit solutions of inhomogeneous linear Volterra integro-differential equations and Volterra integral equations of first kind with smooth kernels.

We determine the sharp total-support uncertainty for the standard complete set of mutually unbiased bases in every prime dimension at least five, proving that the minimum is \(T_s(p)=p^2-p+2\). This resolves all prime dimensions left open by Fiorentino and Weigert, beginning with dimension eleven, and gives a uniform analytic determination of the optimal lower bound in dimension seven, where the earlier determination was partly numerical. The main new ingredient is a sharp zero-count theorem for arbitrary complex Fourier data supported on the finite parabola. A cyclotomic reduction and truncated local expansion produce a first nonzero finite-field jet that is affine in the transverse variable, reducing the planar zero count to a one-dimensional multiplicity problem. We then classify all extremizing projective states and derive an exact enumeration formula. In dimension seven, this gives analytic four-point extremizers and exactly \(12348\) such rays, completing the analytic description and count left open in the earlier work.

Network analysis for multivariate time series is popular in many fields, from neuroscience to seismology. The inverse spectral density is a common choice for time series network analysis due to its representation of the frequency domain correlation between two variables after removing the best linear predictor of all other variables. In many applications, the goal is to study how these networks change across different conditions. For example, in neuroscience, one might be interested in how the brain connectivity network changes before and after stimulation. Towards this goal, we develop an inference framework based on a direct estimate of the difference in two high-dimensional inverse spectral densities. We develop a new Gaussian approximation error bound for any de-biased D-trace estimation procedure which is then leveraged to both inform optimal window sizes of Welch's estimators of the spectral density and establish asymptotic normality of our de-biased D-trace estimator. Moreover, we develop an efficient algorithm based on a generalized D-trace estimation procedure to overcome the computational complexity of high-dimensional inference. The method is illustrated on synthetic data experiments and on experiments with electroencephalography data.

For all prime powers $q\geq5$, we construct lattices $\mathcal{L}_q\subseteq\mathbb{Z}^q$ with kissing numbers \[ \tau(\mathcal{L}_q)\geq \left(\frac{1}{2\pi e^2}+o(1)\right)\sqrt{q}\,e^{2\sqrt{q}}. \] The same asymptotic bound holds on a set of integer dimensions of natural density $1$, and in every sufficiently large integer dimension $n$ with an additional factor $e^{-\tfrac{1}{2}n^{1/40}}$. The construction is an extension of a previous construction by Bennett-Peikert based on Reed-Solomon codes.

The classical theory of compact translation surfaces is organized through the stratification of moduli spaces according to the orders of the zeros of the associated Abelian differentials. In the setting of tame translation surfaces of infinite topological type, no analogous notion of strata has been systematically developed. In this article, we introduce a stratification theory for tame translation surfaces based on their singularity data. To each non-compact tame translation surface, we associate a sequence recording the cardinalities of finite-angle cone singularities of each possible cone angle together with the cardinality of the set of infinite-angle singularities. We call this sequence the singularity data of the surface. We study the realization problem for singularity data on the Loch Ness monster. Our first main result shows that there is no tame translation structure on the Loch Ness monster having only a finite number of finite cone angle singularities. We then prove that this is the only obstruction: every singularity data not excluded by the previous result is realized by a tame translation structure on the Loch Ness monster. As a consequence, we obtain a complete characterization of the strata of tame translation structures on this surface in terms of their singularity data. These results provide the first realization theorem for strata of non-compact tame translation surfaces and reveal a strong interaction between singularity data and the ends space.

In the pioneer work of vectorial free boundary problems \cite[Adv. Math. 280 (2015)]{ASUW15}, Andersson, Shahgholian, Uraltseva, and Weiss investigated the regularity theory of solutions and the free boundary of a singular cooperative system \begin{equation*} \Delta\mathbf{u} = \frac{\mathbf{u}}{|\mathbf{u}|} \chi_{\{|\mathbf{u}|>0\}} \quad\text{in }\Omega\subset\mathbb{R}^n, \qquad \mathbf{u}:\Omega\longrightarrow \mathbb{R}^m, \qquad n,m\ge 2. \end{equation*} At the level of solutions, the boundedness of the right-hand side directly yields $W^{2,p}_{\mathrm{loc}}$ regularity for every $1<p<\infty$ by standard elliptic estimates, while the corresponding $W^{2,\infty}_{\mathrm{loc}}$ regularity question was explicitly left as an open problem on page~753 of that paper. In this paper, we give a positive answer by establishing optimal $W^{2,\infty}_{\mathrm{loc}}$ regularity through a scale-invariant interior Hessian estimate for every weak solution. Unlike the scalar case, the classical Alt–Caffarelli–Friedman monotonicity formula cannot be applied directly to the present system. Our proof instead combines a localized Newtonian potential decomposition with an explicit projection onto quadratic harmonic polynomials on the unit sphere, which isolates the symmetric trace-free quadratic part of the Hessian. A key new observation is that the resulting matrix coefficients satisfy an exact linear ordinary differential equations with constant coefficients on the logarithmic scale. The main novelty of this work lies in a so-called two-regime argument based on the competition between the affine and quadratic parts of the rescaled solution, together with a continuity argument that connects the two regimes and yields a uniform bound for the coefficient. The resulting a priori estimate provides an analytic basis for further study of the fine structure of the free boundary.

This paper provides methods to compute degrees of Fano quiver moduli and gives explicit degree formulas for two classes of them. One such class is a series of toric Fano varieties that parametrize isomorphism classes of representations of the bipartite quiver $K_{(2,q)}^m$ with dimension vector $\underline{1}$, and the other class is the moduli space of q-point configurations on $\mathbb{P}^1$. In the former case, we obtain a formula comprised of sums of binomial coefficients in $q$ and $m$. Its special case $m = 1$ turns out to be the same as the central MacMahon numbers (OEIS entry A177043). In the latter case, we obtained a formula over $\mathbb{Q}$. These formulas are much more efficient than computing directly in Chow rings.

We study skew product measure-preserving systems driven by an irrational rotation and the step function with values $1$ and $-1$ on each half of the circle. These systems can be seen as a deterministic version of Kalikow's $[T,T^{-1}]$ system, and are particular instances of Rokhlin cocycle extensions. Under the assumption of ergodicity, we show that these systems obey an interesting rigidity property for isomorphisms. That is, we show that two skew products are isomorphic if and only if the rotations in the base are isomorphic, and the fiber transformations are flip isomorphic. Under mild extra assumptions, we are able to characterize all isomorphisms. We prove that by choosing the rotation angle suitably, these systems can realize arbitrarily low measure-theoretic complexity in the sense of lower slow entropy. We apply this result to obtain new examples of systems that fail the variational principle for slow entropy. These examples can have a set of invariant measures as rich as desired (any metrizable Choquet simplex, up to affine homeomorphism).

Control allocation for multirotors with bidirectional propellers is commonly formulated in signed-thrust variables, where the wrench map is linear. The signed-quadratic map from physical rotor speed to thrust, however, is not a local diffeomorphism at zero speed. This work derives two distinct consequences. Under nonredundant full actuation, a single-propeller reversal removes one instantaneous task direction; hence, no global continuously differentiable exact static allocator exists over the complete task space. Under redundant actuation, the physical task map may remain regular, yet a transverse pseudoinverse zero crossing requires an unbounded rotor-speed derivative. We define differential realizability as regularity of the physical lift of an actuator-output section, derive exact and first-order validity conditions, and distinguish structural rank loss from an allocator- induced rate singularity. A local nullspace deformation repairs isolated pseudoinverse reversals, while a global fixed-orthant construction establishes existence of regular sections at the cost of persistent task-preserving internal actuation.

The Monge-Kantorovich problem with the Euclidean distance cost is degenerate, typically admitting infinitely many optimal plans. A unique optimal plan is selected by perturbing the distance to a strictly convex or increasing strictly concave cost and passing to the limit: the convex side gives the plan induced by the map which is monotone on each transport ray, while the concave side was understood only on the real line, where Juillet proved that power-cost optimizers converge to the so-called excursion coupling, which is induced by a map when the source is atomless. This paper extends the concave selection to $\mathbb{R}^n$ for mutually singular finite positive Borel measures $\mu$ and $\nu$ with equal total mass and finite first moments, assuming that $\mu$ is absolutely continuous with respect to the Lebesgue measure: for a broad class of increasing strictly concave perturbations of the distance with a well-defined first-order profile, the corresponding optimal maps converge in $\mu$-measure to the same intrinsic limit $t_\#$, independent of the perturbation family and its profile. The map $t_\#$ is obtained by disintegrating the transport along its maximal rays and applying the one-dimensional excursion coupling on each ray. The same limit is established for the power costs $\|x-y\|^{1-\varepsilon}$ under a finite $\|x\|\ln(1+\|x\|)$ moment condition, establishing a conjecture of Juillet.

Associated with the classical Kloosterman sums, we consider the quadratic twisted symmetric power moments. We investigate the cohomological properties and construct the motives over the field of the rationals underlying such moments. We determine the associated modular forms in the lower degree cases.

Let $\Gamma^s(2,q)$ be the graph induced in the Grassmann graph by the $q$-ary simplex codes of dimension 2. For $q=4$, this graph is known to have diameter 3. We prove that the same diameter occurs for every prime power $q\geq5$. Together with the elementary cases $q=2,3$, this gives diam$\Gamma^s(2,q)=0,2,3$ for $q=2$, $q=3$, and $q\geq4$, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For $q\geq5$ an explicit diagonal pair of simplex lines gives the matching lower bound; we give two proofs, one using moments and one using products. For $q\geq7 we also retain an independent counting proof. The case $q=4$ is handled separately.

We prove the sharp $\ell^p$ bound $p^\star-1$ for certain second-order discrete Riesz transforms on products of integers, in all dimensions. The family includes the imaginary part of the discrete Beurling-Ahlfors operator in dimension two as a special case. While these operators admit a martingale representation via compound Poisson jump processes and heat extensions, the martingale attached to a function and the martingale attached to its transform do not enjoy strong differential subordination, so the operator-norm estimate is not accessible through prior work.

We study the mixing behavior of three q-deformed card shuffles on $\mathcal{H}_q(S_n)$: the short systematic scan introduced in \cite{DiaconisRam2000}, the $q$–deformed random–to–random shuffle introduced in \cite{AxelrodFreedBraunerChiangComminsLang2024}, and the $q$–deformed $k$–star transposition shuffle, whose $q=1$ counterpart was studied in \cite{ArfaeeNestoridi}. The first two chains were diagonalized in \cite{DiaconisRam2000} and \cite{AxelrodFreedBraunerChiangComminsLang2024}, respectively. Here, we diagonalize the $q$–deformed $k$–star transposition shuffle and use the spectra of the three chains to study their mixing behavior. For fixed $q>1$, we prove that all three exhibit total variation and $\ell^2$ cutoff. For the short systematic scan, these improve both the upper and lower bounds of Diaconis and Ram \cite{DiaconisRam2000}.

Statistical validation of synthetic multivariate data requires assessing whether a generator preserves the joint dependence structure of the target population without merely reproducing observed records. We develop a model-agnostic framework based on full conditional distributions. For each coordinate, we normalize the conditional probability assigned to the observed value by the largest conditional probability available in the same record context; averaging this quantity yields a one-sided MAP-alignment statistic that can be estimated using a conditional model fitted on held-out real data. The mathematical contribution is twofold: under strict positivity and compatibility, the complete normalized conditional profile identifies the joint distribution, and its integrated L1 difference defines a metric on finite-state generative processes; we also establish consistency and finite-sample concentration for the corresponding empirical estimators. Because high conditional alignment alone can arise from copying or concentration on conditional modes, we pair it with nearest-real similarity as a separate record-level novelty diagnostic. We evaluate the framework on NSHAP health and aging data, influenza B genomic surveillance, and 34 General Social Survey waves. In GSS, the Large Science Model matched the original-data control in mean conditional alignment while retaining substantial novelty, indicating preservation of conditional structure without row reuse. In influenza B, a Chow-Liu generator matched the control alignment but had almost no novelty, revealing near-reproduction of observed records. The framework therefore distinguishes three statistically different failure modes: loss of dependence, record reuse, and mode concentration, and provides a principled basis for validating synthetic health, surveillance, and population data.

Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region $A$ if and only if the mutual information of the inner region and the exterior, conditioned on the part of $A$ inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.

The growth of e-commerce has intensified the need for last-mile delivery systems that can jointly manage customer choice and operational efficiency. We study the Dynamic Offering and Pricing of Mixed Delivery Options problem, in which a logistics service provider dynamically selects and prices attended home-delivery time slots and out-of-home pickup options for sequentially arriving customers. Each decision affects immediate revenue, customer acceptance, and the route-dependent fulfillment cost realized at the end of the booking horizon. We formulate DOPMDO as a finite-horizon Markov decision process and propose State-Value Anchored Pricing via Approximate Dynamic Programming. The method learns a continuation-value approximation on an aggregate state representation of the mixed-delivery system and uses accepted-versus-rejected value differences to estimate option-level opportunity costs. These opportunity costs are embedded in an anchored trust-region pricing problem around a calibrated fine-static benchmark. Computational experiments on the real-world Seattle instance show that SVAP–ADP increases mean episode profit by 7.0% relative to current practice (95% CI: 6.7–7.2%), primarily by reducing terminal fulfillment cost while maintaining a stable home–locker–opt-out mix. Assortment-control experiments show that dynamic pricing is most effective when the menu exposes operationally valuable locker alternatives, with richer candidate pools delivering substantially larger gains than restricted nearest-locker menus. These results indicate that anticipatory pricing and assortment control are complementary: pricing steers customers toward lower-cost options, but the menu determines whether high-value consolidation opportunities are available in the first place.

We develop a spectral three-term modification of the classic Hestenes–Stiefel conjugate gradient algorithm, preserving its anti-jamming characteristic and, simultaneously, taking care of the sufficient descent property. We discuss how a modified secant equation can be extracted from our modification scheme, yielding a memoryless BFGS updating formula. Then, the spectral parameter of our method is obtained by steering its direction toward the given BFGS direction within a least-squares context. Using our technical improvements, we outline the general framework of our algorithm and discuss its theoretical features, including the descent and convergence properties, without the convexity assumption. We put our algorithm to the test in comparison with the three other conjugate gradient algorithms on a set of CUTEr unconstrained optimization test models, comparing the outputs using the Dolan–More measure. Next, we provide a concise evaluation of the results, highlighting the practical advantages of our algorithm. As a real-world case study, we introduce a reduced quadratic surface SVM with the rescaled loss for robust binary classification and apply the proposed algorithm to assess its accuracy and training time against several other SVMs.

We study the relation between positive semidefiniteness and entrywise nonnegativity for solutions of continuous-time Lyapunov equations. For a real Hurwitz matrix $A$, we show that the solution of $AP+PA^\top=-Q$ is positive semidefinite for every symmetric entrywise nonnegative $Q$ if and only if the solution of $A^\top X+XA=-R$ is entrywise nonnegative for every positive semidefinite $R$. This equivalence follows from the adjointness of the two solution operators and extends to all real unmixed matrices. We then prove that both properties hold for every real Hurwitz companion matrix, settling a conjecture previously established under the additional assumption of a real spectrum. The proof combines a controllability-Gramian normalization with a pairwise positivity theorem for the coefficients of the adjugate polynomial of a real accretive matrix. The latter is obtained from a coefficient-sign property of bivariate polynomials, an auxiliary determinant that does not vanish on the product of two open right half-planes, and a rank-one perturbation argument. Covariance and energy interpretations connect these results with dissipative realizations, damped second-order systems, and comparisons between input Gramians.

The Petty conjecture asserts that the quotient \(V_n(\Pi K)/V_n(K)^{n-1}\) is minimized among all convex bodies precisely by ellipsoids. In this paper, we prove the conjecture for all convex bodies of revolution in \(\mathbb R^n\), \(n\geq3\), including the equality cases.

A finite object is called probabilistically random if it is an algorithmically random element of a finite set whose Kolmogorov complexity is relatively small. This note studies the amount of probabilistically nonrandom objects (those that are not probabilistically random) as gauged by the universal measure and without assuming any structure on the objects. The main results imply that the universal measure of probabilistically nonrandom objects is vanishingly small and establish the dependence of their universal measure on the required degree of probabilistic randomness.

For computing the ground state of the defocusing Gross–Pitaevskii energy, we propose and analyze two efficient schemes based on the Davis–Yin three-operator splitting, which treats the potential and interaction terms explicitly, and the kinetic energy by a resolvent. One iteration costs one inversion of $I-\gamma\Delta$ for the first scheme, and of $I-\gamma\Delta+\gamma V_1$ for the second, with $V_1$ denoting the separable part of the potential, and on structured meshes both operators can be inverted by simple fast GPU solvers. For monotone discrete Laplacians, including the second-order finite difference scheme and the lumped linear finite element method on simplicial meshes with suitable angle conditions, we prove global convergence to the unique positive discrete ground state, for every positive normalized initial vector, for any constant step size below an explicit threshold. In contrast, the methods previously proven to converge globally to the ground state all invert a more difficult elliptic operator that depends on the iteration variable. In three-dimensional tests with up to $999^3$ unknowns on one GPU, a simple variable step size rule makes the proposed splitting schemes efficient in practice, comparable in wall-clock time to Riemannian conjugate gradient methods that also invert only a shifted Laplacian operator, and much more robust with respect to the choice of the initial guess.

High-order entropy stable (ES) methods have attracted much interest in the computational fluid dynamics (CFD) research community over the past decade, as we seek to simulate practical engineering applications with high accuracy and robustness. As industry delves further into supersonic flight and space travel, we need schemes capable of modelling flows with multiple chemical species that interact and mix within the fluid. As such, there is a considerable amount of effort currently being focused on developing high-order ES schemes for multi-species (MS) flow. A key ingredient for these entropy stable schemes is the two-point flux, which determines the properties of the resulting high-order discretization. As such, the numerical scheme can possess important properties such as entropy conservation and kinetic energy preservation only if the two-point flux also mirrors these properties. At present, in the literature, the two-point fluxes used for high-order ES multi-species solvers are not kinetic energy preserving (KEP). In this short communication we develop a two-point flux that is both entropy conserving (EC) and kinetic energy preserving by adapting Ranocha's single-species methodology for multi-species. The newly proposed EC and KEP flux is tested against existing two-point fluxes in the literature using the density pulse advection and Taylor-Green vortex cases. It is shown that EC and KEP flux retains both properties while the other fluxes only retain one of the desired properties. Since the two-point flux dictates the properties of the resulting scheme, the proposed flux is the ideal option for simulating multi-species compressible flow.

In this work, we complete the description of the one-dimensional partial actions of 8-dimensional Hopf algebras by computing the remaining cases: the Kac-Paljutkin algebra $\mathcal{A}$ and the unique non-semisimple non-pointed Hopf algebra $\mathcal{K}$. We prove that all these partial actions are symmetric, study their associated partial smash products and we determine all their partial coactions of dimension one. Beyond the 8-dimensional setting, we investigate which right coideal subalgebras of a Hopf algebra $H$ can be realized as partial smash products $\underline{ \Bbbk \# H}$ over the base field. In particular, we show that every right coideal subalgebra of a finite-dimensional cosemisimple Hopf algebra arises in this way.