Papers

We develop the theory of first- and higher-order codifferential calculi over coalgebras $C$ over fields $k$ with characteristic $\mathrm{char}(k)\neq 2$. For a given first-order codifferential calculus, we introduce its maximal prolongation by means of an explicit construction that associates to it a differential graded coalgebra, satisfying a universal property. For module coalgebras over a Hopf algebra $U$, we introduce the notion of an equivariant codifferential calculus. If $C$ is of the form $U\otimes_H k$ for a Hopf algebra $U$ and a right coideal subalgebra $H$ such that $U$ is faithfully flat as a left- and right $H$-module, we show that equivariant first-order codifferential calculi correspond to certain right coideals $T\subseteq \ker(\varepsilon\colon C\rightarrow k)$ called quantum tangent spaces. If $H$ is a sub bialgebra and the right $C$-coaction on $T$ is trivial, then the maximal prolongation is described in terms of a quadratic coalgebra. We further relate codifferential calculi to differential calculi and Cartan pairs over the dual algebra $C^\ast$, or more generally subalgebras thereof. We explicitly compute codifferential calculi on the coalgebra pre duals of the Podleś sphere and the quantized projective spaces. As an application, we give a new proof that the antiholomorphic Heckenberger–Kolb calculi on quantized projective spaces have classical dimension.

Numeric format proliferation in machine learning hardware – FP8 (E4M3 and E5M2), BF16, MXFP4, microscaling block formats, and dozens of research variants – has outpaced the availability of vendor-neutral, bit-exact reference material. Engineers porting models across accelerators encounter silent divergences that are difficult to diagnose without a shared ruler. This paper describes a catalog of 109 numeric formats spanning 12 clusters (83 at v2; the count is a catalog invariant, not a fixed number), a suite of six bit-exact conformance packs covering GF16, MXFP4 element, BF16, FP8 E4M3, FP8 E5M2, and E8M0 block scale, and an IEEE P3109 v3.2.0 cross-walk that maps each pack to its corresponding standards-track configured format. Each pack is a self-contained JSON document with a SHA-256 fingerprint, a shared row schema, and an anchor vector that encodes 3.0 – the identity phi^2 + 1/phi^2 = 3 – as a cross-pack sanity check. Packs are cross-validated against ml_dtypes 0.5.4 (Google/JAX); any divergence is documented explicitly and interpreted as a spec-permitted interpretation gap rather than hidden. The work is framed as registry filling: it does not propose new formats, make model-accuracy claims, or assert superiority over any vendor's implementation. All artifacts are publicly available at https://github.com/gHashTag/t27 under an open license.

To any essential right ideal $I$ in a bounded HNP ring $R$ we may assign a divisor $\partial I$, the image of the finite length module $R/I$ in the Grothendieck group $K_0(\text{fl. mod-$R$})$. We show that there is a composition of divisors $\circ$ for which $\partial I J = \partial I \circ \partial J$. Additionally, we describe this composition, and show that $\partial$ is faithful.

For a contractive multiplier $\varphi$ in the multiplier algebra $M(k)$ of a kernel $k$, the associated de Branges–Rovnyak kernel is given by $k^\varphi(x,y) = (1-\varphi(x)\overline{\varphi(y)})k(x,y)$. Motivated by recent developments clarifying the structural and geometric features of reproducing kernel Hilbert spaces associated with kernels admitting a complete Pick factor, we investigate the precise conditions for a general de Branges-Rovnyak kernel to admit a complete Pick factor, thereby extending the framework introduced by Ahmed, Das and Panja (J. Geom. Anal., 2025). In this paper, we characterize the existence of a complete Pick factor for $k^\varphi$ across a broad class of base kernels encompassing both complete Pick and non-complete Pick architectures (such as the Szegő kernel on the polydisk). Our first characterization is formulated in terms of operator-valued holomorphic functions satisfying an interpolation condition. We also show that $k^\varphi$ admits a complete Pick factor if and only if $(\widetilde k)^\varphi$ is itself a complete Pick kernel, where $\widetilde k$ is an auxiliary kernel constructed from the given data. Notably, our main result is completely new even when specialized to the classical Szegö kernel of the unit disk. As an application of our framework, we obtain a structural insight into a classical theorem of Chu (J. Funct. Anal., 2020) and provide an alternative proof of a recent result by Luo and Zhu (Canad. J. Math., 2024). The results are illustrated by concrete examples.

Boundary layers provide a demanding test for numerical solvers because the solution may remain almost constant over most of the domain while changing rapidly in a narrow region near the boundary. This paper studies a singularly perturbed one-dimensional transport boundary-value problem with increasing Peclet number $(\mathrm{Pe})$. A local Legendre wavelet collocation method (LWM) is compared with a standard soft-boundary physics-informed neural network (PINN) for this benchmark. The wavelet approximation uses locally supported Legendre polynomial basis functions and converts the problem into a square algebraic collocation system with residual, boundary, and interface-continuity equations. Numerical experiments are performed for $\mathrm{Pe}=1,10,100,$ and $1000$. The LWM captures all four cases, with the largest error remaining below $5\times 10^{-3}$. The standard soft-boundary PINN performs well for the mild cases but fails to resolve the sharp boundary layer for the larger Peclet numbers. The results show that local wavelet collocation is more reliable than the standard soft-boundary PINN for this benchmark, while dense near-boundary evaluation helps reveal errors that may be missed on coarse grids.

We present Trellis: an autoformalization system that leverages LLM agents in a deterministically constrained workflow to enforce incremental progress in Lean autoformalization tasks through iterative refinement of natural language proofs. Our approach is motivated by the common mathematician's notion of what it means to have a rigorous proof in the first place: namely, that it would be routine to elaborate any part of the proof in further detail. The result is a system which aims to achieve reliable autoformalization on a modest budget and with generalist agents, with specialization to autoformalization coming not from any task-specific agent training but instead from a meaning-of-rigor inspired workflow enforced by process semantics. We link to an end-to-end Lean formalization of a recent Ramsey theory breakthrough produced by the process.

We introduce the notion of a uniformly recurrent subalgebra (URA) for a trace-preserving action of a countable discrete group $\Gamma$ on a finite von Neumann algebra $M$, providing an operator-algebraic counterpart to the theory of uniformly recurrent subgroups (URS). We also show that the Effros-Maréchal space $\text{Sub}(M)$ is compact if and only if $M$ lacks a diffuse direct summand. Leveraging this, we show that URAs can exhibit arbitrary topological complexity and construct exotic URAs homeomorphic to any prescribed minimal Polish space. In the context of crossed products $M \rtimes \Gamma$ with amenable coefficients, we utilize URAs to formulate a new characterization of C*-simplicity, proving that $\Gamma$ is C*-simple if and only if the only amenable URA of the crossed product containing $M$ is $\{M\}$. Finally, to bypass the failure of compactness in $\text{Sub}(M)$, we develop a generalized state-space machinery using Baire-category methods on the weak-* compact space of trace-extending states. This construction captures compact, discrete, and exotic URAs, while recovering the classical URS framework as a special case.

Purpose: Up to 4% of adults will develop strabismus in their lifetime. The most common surgical intervention involves adjusting the length of one or more extraocular muscles to correct the angular deviation. This correction depends on surgical expertise and statistical reference tables, which often fail to yield optimal results for patients with atypical eye morphology. Our work proposes a physics-based modeling approach to personalized surgical planning, accounting for patient-specific eye anatomy. Methods: We built a physics-based simulator of the eye and its muscles, incorporating patient-specific geometry and Hill-type muscle biomechanics. We solve an optimization problem to find the surgical dosage that minimizes angular deviation. The model is implemented as a fully differentiable simulation, enabling efficient optimization. We validated the framework by comparing its predictions with standard surgical tables for emmetropic eyes before applying it to anatomically atypical virtual patients. Results: Our model's predictions for emmetropic eyes were first validated, demonstrating a strong fit with standard surgical tables. More importantly, for high-myopia models, the framework computed a clinically significant increase in the required surgical dosage compared to standard eyes. This computed recession difference is highly relevant as surgical plans are adjusted in 0.5 mm increments. Conclusion: Our results show that our model provides a calibrated surgical plan that, unlike standard tables, also accounts for pathologies involving atypical eye shapes. This patient-specific model represents a step toward personalized surgical planning, with the potential to improve dosage accuracy and surgical outcomes for atypical cases.

In the sense of Baire categories, we prove that the set of pairs of bivariate copulas that are comparable – in either direction – under the increasing convex order is nowhere dense in the space of all pairs of bivariate copulas equipped with the uniform metric. As a consequence, a topologically generic pair of bivariate copulas is not comparable under this order. We further extend the Baire-category programme to two additional stochastic orders on the space of bivariate copulas: the bivariate convex order and the stop-loss order on the sum of the components. For each of these orders, we establish that the set of comparable pairs is closed and nowhere dense, and we show that a topologically generic pair of bivariate copulas is simultaneously incomparable in all three orders. These results complement those obtained in [F. Durante, J. Fernández-Sánchez, C. Ignazzi (2022). Baire category results for stochastic orders. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, article 188] for the lower orthant order on copulas.

We determine the complete Benjamin-Feir spectrum near the origin for small-amplitude hydroelastic Stokes waves of the two-dimensional finite-depth irrotational Euler equations with surface tension and elastic bending. For the non-resonant Stokes branch and away from an intrinsic characteristic-collision surface $\mathfrak D$, we resolve all four Bloch eigenvalues bifurcating from the origin in the long-wave Floquet regime. Exploiting the Hamiltonian and reversible structure of the problem, we reduce the linearized Bloch operator to the four-dimensional spectral subspace bifurcating from the generalized kernel at the origin and conjugate the resulting matrix to the direct sum of a Benjamin-Feir block and a long-wave block. The long-wave pair remains purely imaginary, whereas the Benjamin-Feir pair is governed by an explicit closed-form instability index $\operatorname{Ind}(\mathtt{h},\kappa,b)$: a positive index produces a local figure-eight spectral curve with nonzero real part, while a negative index implies that all four small eigenvalues remain purely imaginary. Together with the Wilton-type resonance loci and the characteristic-collision surface $\mathfrak D$, this index yields a three-parameter spectral-stability diagram in the depth $\mathtt{h}$, surface tension $\kappa$, and bending rigidity $b$. The diagram recovers the classical pure-gravity critical-depth limit and, on the zero-bending boundary, the gravity–capillary stability diagram. It also reveals a genuinely hydroelastic phenomenon: all Wilton-type resonances disappear whenever $b\geq 1/14$ or $\kappa\geq 1/2$. This provides the first complete rigorous characterization of the local Benjamin-Feir spectrum for a hydroelastic free-boundary problem.

The purpose of this note is to address the gap in the stable rank one/purely infinite dichotomy of selfless $C^*$-probability spaces. In particular, we show that nonfaithful selfless $C^*$-probability spaces are purely infinite, simple. This completes the dichotomy: Every selfless $C^*$-algebra either has stable rank one or is purely infinite. Notably, this shows that every selfless $C^*$-algebra is pure. To accomplish this, we show that infinite reduced free products of $C^*$-probability spaces with nonfaithful states inducing faithful GNS representations are often purely infinite, simple. Having resolved the dichotomy, we improve existing permanence properties of selfless $C^*$-probability spaces, make progress on a conjecture of Choda and Dykema, and produce several new isomorphisms arising from reduced free products.

Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena. As a promising alternative to traditional first principles, data-driven differential equation discovery has attracted increasing attention for its ability to infer governing laws directly from experimental or simulated data, especially when the underlying physics is unclear. However, the field has expanded rapidly along diverse methodological directions, particularly with the emergence of AI-based approaches, and still lacks a clear organizing perspective. In this Review, we propose a problem-oriented perspective on data-driven differential equation discovery. We first introduce a two-dimensional phase diagram of equation discoverability, where discovery problems are organized according to structural complexity and coefficient complexity. This phase diagram shows how the field has moved from the discovery of sparse equations with simple coefficients toward more complex governing laws with richer structures and more flexible parameterizations. It also clarifies why different methodological families succeed or fail in different problem settings. We then present the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process. By identifying the core problems of equation discovery that persist across algorithmic variations, REO shifts the discussion from individual algorithms to the fundamental principles that determine discoverability. We connect these perspectives to applications across physics and adjacent sciences, and argue that the next challenge is not merely recovering equations, but using them to revise existing theories, distil mechanisms and form new scientific concepts.

Modulated circulant matrices form a special class of N-parametric circulant matrices, recently introduced in the literature, with a structured spectral decomposition based on a Vandermonde type basis. Motivated by this definition, in this work we define the Modulated Quantum Fourier Transform (MQFT), a quantum primitive tailored to this matrix family.

The rapid expansion of AI globally has led to the proliferation of energy-intensive hyperscale data centres (DCs), making them as a structurally challenging component in power system planning and operation. Using a spatially explicit optimisation model of Europe across 21 AI growth scenarios, we systematically quantify additional demand, capacity requirements, emissions, and operational impacts of DCs. Results indicate that AI could drive 73-723 TWh of extra demand by 2050, risking cumulative emissions overshoots of 67-181 MtCO2 between 2030 and 2050. Our analysis indicates that after 2030, the geography of AI infrastructure will be shaped more by firm power and system flexibility than by the mere abundance of clean energy. In moderate scenarios, AI requires an additional of 200 hours of firm generation, which increases LCOE by 35 EUR/MWh in key hubs. We show that even under the pessimistic scenarios, existing infrastructure would require 70 GW additional capacity, while under managed growth pathways, this expansion could reach 226 GW. We further find DCs workload dynamics strongly shape energy dispatch, system flexibility, and emissions, while improved efficiency significantly reduces capacity needs, and system peaks. While our findings suggest that net-zero targets for 2050 may be achieved, critical emission risks may appear in the intermediate years, and the EU may compromise its carbon-neutral goals unless policies adapt to this accelerating digital transformation.

In this paper, we resolve the question of whether the Neumann problem for the parabolic PDE $-\partial_tu + \mathrm{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic with bounded and measurable coefficients that satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for any $1<p<\infty$ the Neumann problem is solvable under the assumption that both the Carleson norm of coefficients and the Lipschitz constant of the domain are sufficiently small (with dependence on $p$). The question of what happens in the "large Carleson norm/large Lipschitz constant" regime remains open, and even for elliptic PDEs this question has only been resolved in two dimensions. This paper complements results from our recent manuscript (by the same authors) in which the parabolic regularity problem has been fully resolved in both the small and large Carleson norm regime. Previously, the Dirichlet problem had been resolved under the same conditions by various authors.

This paper investigates Polynomial Differential Variational Inequalities (PDVIs), a class of non-smooth dynamical systems in which an ordinary differential equation is coupled with a time-dependent variational inequality (VI) defined by polynomials. As a foundation for polynomial differential variational inequalities, we investigate the parametric polynomial variational inequality and establish that, under mild assumptions, its parameter space admits a stratification into finitely many disjoint semi-algebraic regions, each corresponding to a qualitatively distinct solution structure. Furthermore, we establish the existence of maximal regular solutions to the PDVI for almost every initial condition in the parametric feasible set, where each solution has finitely many switches and evolves according to continuous semi-algebraic selection laws between switching times. Leveraging the structured partition of the parameter space, in which region boundaries correspond to discontinuities in the solution map of the parametric VI, we propose the structure-aware SOS-PDVIRK4 algorithm to handle solution multiplicity and structural transitions. This method integrates a fourth-order Runge–Kutta integrator with the Moment–SOS hierarchy and features a novel hybrid strategy that is explicitly guided by the underlying parameter space geometry. The efficacy of our structure-aware framework is demonstrated through numerical experiments on a constrained cyber-physical control system. Our numerical experiments demonstrate that, from identical initial states but different initial controls, trajectories can diverge significantly, revealing the discontinuous structure of the solution map of the parametric polynomial VI. The experiments also include numerical convergence studies and highlight the potential of our algorithm to explore the long-term behavior of PDVI systems.

Covering codes in finite Hamming spaces ask for small sets of words whose Hamming balls cover the whole space. This paper presents a Lean 4 formalization of the elementary theory of q-ary covering codes, centered on certificate predicates for upper bounds, lower bounds, and exact covering numbers $K_q(n,r)$. The formalization proves the q-ary Hamming-ball volume formula, the sphere-covering lower bound, elementary exact cases, product and relation rules, and selected small exact certificates. It also demonstrates an end-to-end workflow for checking explicit upper bounds transcribed from van Laarhoven et al. (1989). The accompanying database is proof-carrying: stored bounds have traces that replay to Lean proofs of the corresponding upper- or lower-bound predicates. The contribution is not new record bounds or a reproduction of known tables, but a reusable, auditable foundation for machine-checked covering-code certificates.

Bayesian inverse problems constrained by state equations are often sampled in a full parameter-state space by penalising the residual, rather than in a reduced space where the state is eliminated. We show that these formulations are not automatically equivalent as posterior measures. For finite-dimensional discretisations of equality-constrained inverse problems, assume the state equation \(c(\theta,u)=0\) has a unique solution \(u=G(\theta)\) and nonsingular state Jacobian \(\D_u c\). The reduced posterior, its graph lift, and the zero-noise residual posterior are then distinct. A local change of variables shows that an uncorrected Gaussian residual penalty converges, after marginalisation over \(u\), to the reduced density multiplied by \(\abs{\det \D_u c(\theta,G(\theta))}^{-1}\). Thus algebraically equivalent residuals can define the same feasible set but different limiting posteriors. We derive determinant corrections for unweighted, weighted, and rescaled residual penalties that have the graph-lifted reduced posterior as their hard-constraint limit. The result separates feasibility from posterior calibration: driving the residual to zero is not sufficient for exact sampling of the graph-lifted reduced posterior unless the sampling or correction step targets the corresponding corrected density.

We revisit quantum field theory in curved spacetime (QFTCS) as a semi-classical framework for quantum matter on classical geometries, emphasizing its limitations, including vacuum ambiguity and background dependence. We briefly review major approaches to quantum gravity (QG), including Loop Quantum Gravity (LQG), string theory, and asymptotic safety, highlighting their conceptual challenges. Motivated by these issues, we outline a teleparallel framework based on coframe and spin-connection variables, where gravity is encoded in torsion rather than curvature. This framework naturally incorporates local Lorentz symmetry and fermionic couplings while displaying a gauge-like structure. We argue that the coframe/spin-connection pair provides an alternative and geometrically refined description of gravitational variables, which may serve as a useful starting point for future investigations of QG. The purpose of this work is not to provide a complete quantization of teleparallel gravity but to identify the geometric and conceptual ingredients that such a formulation would require.

We study the distribution of rational points in $\mathbb{R}^n$, with denominators restricted to the interval $[Q-\Delta, Q]$, and $Q,\Delta\to\infty$ such that $\Delta/Q\to 0$. Previous results in the literature, due to Hall and others, were limited to Farey sequences, where the window size $\Delta$ is of the same order as $Q$. We prove the convergence of fine-scale statistics in a range of scaling limits and express the limit laws in terms of natural probability measures on the space of affine lattices. The key technical ingredient of our approach is an equidistribution theorem for slowly expanding horospheres, with some new exotic limit measures. Our techniques furthermore allow us to answer a recent question by Anderson, Boca, Cobeli and Zaharescu concerning the directional statistics of lattice points.

In the present paper, we propose Local Discontinuous Galerkin (LDG) approximations for a nonlinear system of $p$-Stokes type, having $(p,\delta)$-structure. On the basis of the primal formulation, we prove well-posedness and stability (a priori estimates) of the methods under truly minimal regularity assumptions. We show that the first method possesses a pressure-robust and quasi-optimal error estimate, and discuss its consequences. Moreover, we propose a second method, for which we show a pressure-robust error estimate and prove convergence and convergence rates, which are optimal for linear ansatz functions for all $p\in (1,\infty)$ and $\delta\geq 0$.

We establish an operational framework connecting the classical Kolmogorov–Arnold (KA) representation theorem to quantum information theory. By introducing and proving an algebraic, bounded-degree polynomial version of the theorem, we demonstrate that any target physical property of an unentangled multi-qubit product state can be exactly decomposed using a finite, fixed set of local “inner” observables and a shallow architecture of univariate polynomials. We further analyse the stability of this Quantum Kolmogorov–Arnold (QKA) representation under adversarial perturbations. In stark contrast to the pathological instabilities and severe reparameterisation sensitivities inherent to the classical Kolmogorov–Arnold representation theorem, our algebraic quantum framework exhibits remarkable resilience. We prove that the representation remains stable against bounded physical perturbations acting on the inner measurement operators, and show via the Heisenberg picture that it is inherently immune to adversarial quantum channel attacks acting on the input states.

We characterize every almost abelian Lie algebra endowed with an integrable complex structure by a triple, called presentation, consisting of a real number, an element in some vector space and an endomorphism of that vector space. We then classify in terms of presentations the almost abelian Lie algebras admitting $p$-Kähler or $p$-pluriclosed structures, and in particular those carrying Kähler, balanced, pluriclosed and Gauduchon metrics. Furthermore, we characterize the existence of LCK and Bismut-Ricci flat metrics in this setting.

We isolate a coherent finite-separation strengthening of the Freese–Nation property and use it to characterise projective and weakly projective Boolean algebras. A Boolean algebra is projective if and only if it has a meet-closed decomposition base carrying such a coherent finite-separation map; it is weakly projective if and only if it has a meet-closed $\pi$-base carrying one. The global form of the same property is much stronger: the positive cone of a Boolean algebra carries a coherent finite-separation map if and only if the algebra is countable. Thus the projective and weakly projective characterisations are intrinsically local and cannot be strengthened by requiring the whole algebra to carry the coherent map except in the countable case.

We prove a sharp quantitative stability result for the affine \(L^p\)-Sobolev inequality, for \(p\ge2\), introduced by Lutwak–Yang–Zhang (J. Differential Geom., 62 (2002), 17–38). Moreover, the stability exponent is shown to be optimal, and equal to \(p\).

Recently, Starr used asymptotic methods to disprove a conjecture by Heim–Neuhauser and Abdesselam about the log-concavity of the D'Arcais polynomials, without giving an explicit counterexample. We refine the asymptotics, to give the necessary estimates on convolutions of $\sigma_{-1}$, and identify the first counterexample at $\lambda = 65\,214\,507\,758\,400$. We also consider the asymptotic density of such counterexamples.

We give a generating series expression for the sum of all multiple zeta values of a fixed weight, depth, and height, which end with a given string $\vec{\ell} = (\ell_1,\ldots,\ell_r)$; this builds upon the proof of the Ohno-Zagier Theorem. In particular, the sum of all multiple zeta values of fixed weight, depth and ending with $\vec{\ell}$ has bounded depth $\leq \ell_1 + \cdots + \ell_r$. We give some applications to evaluations of interpolated multiple zeta values, and to the generating series of double zeta values.

A diassociative algebra (or dialgebra), a generalization of associative algebras with two associative products, is a fundamental algebraic structure of great importance and wide-ranging applications. In this paper, we establish a bialgebra theory for dialgebras completely and systematically. Explicitly, we introduce the notion of diassociative bialgebras (bi-dialgebras) which are equivalent to double constructions of Frobenius dialgebras as well as matched pairs of dialgebras. Moreover, we show that a bi-dialgebra gives rise to a Leibniz bialgebra, thereby lifting the classical relation between dialgebras and Leibniz algebras to the bialgebra level. In the coboundary cases, we introduce the diassociative Yang-Baxter equation (DAYBE) in a dialgebra, and solutions whose skew-symmetric part is invariant give rise to the so-called quasi-triangular bi-dialgebras. Moreover, weighted $\mathcal{O}$-operators on dialgebras are defined, which afford an operator interpretation for such solutions of the DAYBE. Finally, we also develop the triangular and factorizable theories for bi-dialgebras, and introduce skew-symmetric Rota-Baxter Frobenius dialgebras of arbitrary weight. We show that skew-symmetric Rota-Baxter Frobenius dialgebras of weight zero give rise to triangular bi-dialgebras, whereas those of nonzero weight induce factorizable bi-dialgebras. As an application, we provide concrete examples of factorizable bi-dialgebras from associative algebras and dialgebras, respectively.

We construct KMS states for the Nelson model, the spinless Pauli–Fierz model, and the Pauli–Fierz model with spin by functional integral representations, and study point-source models after removal of the infrared and ultraviolet cutoffs. If the physical test-function space can distinguish the zero mode, the absence of off-diagonal long-range order, the vanishing of the zero-mode form, the vanishing of the condensate density, the order-parameter criterion, and the triviality of the BEC directions and the BEC ideal are equivalent. We also describe the infrared quotient and the BEC ideal in the resolvent algebra uniformly for the three models, and formulate an operator-algebraic sufficient condition for separately given spatially translation-invariant KMS states.

A finite group is said to be quartic-free if its order is not divisible by $p^4$ of any prime $p$. A finite group is called an $A$-group if all of its Sylow subgroups are abelian. Objective of this paper is to provide explicit structure of a quartic-free $A$-group. Further in the process of providing the explicit structure we also determine the derived length of a solvable quartic-free $A$-group.