Papers

Understanding the fundamental limits of recovering planted subgraphs in random graphs is a central challenge in high-dimensional statistics and theoretical computer science. While existing work has largely focused on special subgraph families such as cliques, bicliques, or dense blocks, the exact recovery of a general planted subgraph in Erdős–Rényi random graphs remains poorly understood. In this paper, we study the exact recovery of an arbitrary planted subgraph $\Gamma = \Gamma_n$ embedded in a dense Erdős–Rényi random graph $\mathcal{G}(n,q_n)$, where edges within $\Gamma$ are present independently with probability $p_n > q_n$. Our main results identify sharp conditions under which exact recovery is possible with high probability, and we establish matching lower bounds showing the necessity of these conditions. The resulting statistical threshold is characterized by a new graph-theoretic quantity, which we term the minimal maximum subgraph density. This quantity is defined as the maximum subgraph density of the smallest induced balanced subgraph of $\Gamma$. We then turn to the problem of recovery under polynomial-time constraints. We propose a computationally efficient recovery algorithm that applies to arbitrary planted subgraphs and analyze its performance in terms of certain spectral properties of the adjacency matrix. In addition, we derive computational lower bounds for recovery using the low-degree polynomial framework, establishing regimes where recovery is statistically possible but computationally hard. Finally, we consider several extensions of our setting, including recovery in semi-random models and weaker notions of recovery.

Model order reduction techniques have become an attractive approach for obtaining fast approximations of multidimensional problems. Besides computational efficiency, ensuring the reliability of the resulting approximations is of primary importance. This work focuses on the certification of PGD-based reduced-order models based on the separation of spatial variables, which are particularly well suited to plate and shell geometries. Considering diffusion problems defined in plate-like domains, we introduce a guaranteed global error estimate associated with the PGD approximation. To this end, the error bounds are derived from the Constitutive Relation Error (CRE) method. The main difficulty of this approach lies in the construction of equilibrated fluxes, for which a dedicated procedure is proposed. Based on the resulting estimator, an adaptive strategy is developed to control both the discretization error and the number of PGD modes. This certification procedure is further extended to the error control in quantities of interest. We provide several numerical examples illustrating the reliability and efficiency of our procedure.

We study the three-dimensional incompressible free-boundary ideal magnetohydrodynamic (MHD) equations with surface tension and a closed free surface. Our first result establishes $H^3$ a priori estimates in general bounded domains, without graph structure, periodicity, or simple connectedness; in particular, for surface-tension ideal MHD in general domains this lowers the previously available threshold from $H^6$. Compared with the free-boundary problem for incompressible Euler equations, the feature is that the Lorentz force enters the elliptic pressure estimates, and the frozen-in magnetic field must preserve the tangential boundary constraint. Using these estimates, we prove a refined finite-time blow-up criterion for $H^3$ solutions that separates topological self-intersection, loss of boundary regularity, blow-up of the normal velocity, and interior MHD blow-up. The interior condition has an intrinsic magnetic-field asymmetry: besides $\|\nabla u\|_{L^\infty}$ and $\|\nabla h\|_{L^\infty}$, with $u$ and $h$ denoting the velocity and magnetic field, respectively, it requires the additional control of $\|\nabla^2 h\|_{L^2}$, a quantity arising from the Lorentz-force contribution to the pressure estimates and having no velocity analogue. Finally, we construct regular initial data whose solutions develop finite-time boundary self-intersection while the Sobolev regularity and curvature remain controlled up to the contact time. Thus, neither surface tension nor the ideal magnetic coupling precludes topological self-intersection of the free boundary.

This paper considers the unconstrained minimization of a lower semicontinuous function. Exploiting first and second subderivatives, directional limiting subdifferentials, and directional proximal subdifferentials, necessary and sufficient first- and second-order optimality conditions are derived that build upon the recently introduced notion of directional local minimality. These results then also yield optimality conditions for conventional nondirectional local minimality which are stated in terms of so-called critical directions and variational objects depending on them. Illustrative examples show that the derived conditions allow for a finer analysis than classical nondirectional optimality conditions.

Topological string free energy has a rich collection of non-perturbative contributions which are labeled by D-brane charge vectors, and the associated Stokes constants are conjectured to coincide with BPS or DT invariants, i.e. D-brane multiplicities. In this paper, we provide additional evidence to this conjecture by studying modular properties of non-perturbative contributions. We argue using resurgence theory that non-perturbative contributions form orbits of local monodromy group induced by singular points inside a stability chamber, and that the associated Stokes constants must be the same across the orbits. In some examples, this allows generation of infinitely many Stokes constants, which reproduce the entire BPS spectrum. In addition, following [DK26], we also show that generators of Stokes transformations of non-holomorphic partition function satisfy Lie brackets of the Kontsevich-Soibelman Lie algebra, making it possible to identify the global Stokes transformation with the Kontsevich-Soibelman wall-crossing invariant.

Traditional variational Kalman filtering with unknown noise statistics suffers from inconsistent process covariance estimation and slow convergence speed, limiting its practical utility. To address these issues, we introduce a surrogate variable representing the process-noise-free state, which enables explicit modeling and inference of process noise statistics. In addition, we reformulate the conventional coordinate ascent variation inference (CAVI) as a marginalized maximum a posteriori problem, followed by a single-step hyperparameter fitting. This reformulation obviates the need for multiple inner iterations inherent to CAVI and decouples the design of the covariance tracking filters. Consequently, this architecture permits the deployment of higher-order filters for covariance tracking and enables sliding-window hyperparameter estimation. Notably, when this window encompasses all historical data, the covariance tracking estimator intrinsically operates as a zero-phase filter. Numerical simulations validate the theoretical framework, demonstrating the enhanced convergence speed and superior estimation accuracy compared with existing methods.

W}e study the equation $P(i\nabla)u=0$ on $\mathbb{R}^d$ for a class of admissible symbols $P$ whose zero set is the unit sphere $S^{d-1}$ and which vanish there to some finite order. Working in the framework of Lizorkin distributions, and hence without any boundedness or decay hypothesis on $u$, we give a complete classification of the solutions: $u$ solves $P(i\nabla)u=0$ if and only if $\hat{u}$ is a multi-layer distribution on $S^{d-1}$ of order at most $N$. Alternatively, $u$ solves $P(i\nabla)u=0$ if and only if $(1+\Delta)^{N+1}u=0$ if $P$ satisfies a flatness condition. The proof recasts the equation as a division problem and combines the order of vanishing of $P$ with the structure theorem for distributions. This unifies and extends known Helmholtz-type rigidity results, which correspond to a simple zero on the sphere, to symbols with zeros of arbitrary finite order.

We consider a Bose gas on the three-dimensional torus in the Gross–Pitaevskii regime and explain how to remove the assumptions of radiality and compact support on the interaction potential in the proof of Bose–Einstein condensation and convergence of the excitation spectrum to Bogoliubov's prediction. In particular, we sketch the proof of [55].

We investigate the class of measures of finite energy integrals and the behavior of potentials and co-potentials associated with non-symmetric closed forms. In particular, we compare these objects with their symmetric counterparts from three viewpoints: a non-symmetric version of Stollmann–Voigt's inequality, non-symmetric perturbations of symmetric forms, and closed forms associated with non-symmetric jump-type forms. Our results indicate that measures of finite energy integrals, potentials, and co-potentials behave differently in the non-symmetric setting, requiring more delicate analysis than in the symmetric case.

We consider the problem of designing input signals for an unknown linear time-invariant system in such a way that the resulting data, within a finite horizon, is suitable for identification with a desired accuracy. We consider both noise-free and noisy settings with $\ell_\infty$–bounded noise models. We will take into account general prior knowledge of the system parameters. Central in our study is the concept of universal inputs. An input is called universal for identification if, when applied to any system complying with the prior knowledge, it yields data suitable for accurate identification. We provide new methods for designing such universal inputs. Our results generalize the experiment design approach based on Willems et al.'s fundamental lemma that relies on persistently exciting inputs, and that is limited to prior knowledge on controllability. It turns out that for other types of prior knowledge, there exist universal inputs that outperform the persistently exciting ones, e.g., in terms of sample efficiency. Moreover, we investigate types of prior knowledge that enable experiment design for exact identification in the presence of noise.

Schwarz type domain decomposition methods generally require a partition of unity to combine solutions on subdomains. However, in mesh-based methods it is common to organize subdomains with minimal overlap, if any, which is facilitated by the availability of a mesh. This study analyzes how the continuity of the partition of unity affects the algebraic Schwarz method for Poisson and Stokes equations from a meshless point of view, whereby the underlying differential operators are discretized using the radial basis function finite difference (RBF-FD) method. We demonstrate numerically that, in this setting, small overlaps improve the performance of the domain decomposition, leading to smaller iteration counts, and therefore no disjoint partitioning technique is required.

Let $G$ be a graph and $I(G)$ its edge ideal. The $p$-th squarefree power $I(G)^{[p]}$ is the monomial ideal generated by the squarefree monomials corresponding to the matchings of size $p$ of $G$. A central highlight of our work is the complete solution of a problem posed by Bigdeli, Herzog and Zaare-Nahandi in 2017, which asked to characterize when squarefree powers of edge ideals have the first syzygy module generated by linear forms. Moreover, for a $1$-dimensional flag simplicial complex $\Delta$ and its Stanley–Reisner ideal $I_\Delta$, which is the edge ideal of the complement graph of $\Delta$, we prove that $I_\Delta^{[p]}$ has a linear resolution if and only if it is linearly related and the $(2p-1)$-st homology of the neighbourhood complex of $\Delta$ vanishes. As a consequence, we describe the shape of the Betti table of $I_\Delta^{[p]}$, and we give a combinatorial characterization of when $I_\Delta^{[p]}$ has a linear resolution, provided that the $2p$-core of $\Delta$ has maximum degree at most $2p$. Finally, we show that no such characterization exists in general: the linearity of $I_\Delta^{[2]}$ may depend on the characteristic of the field, and it cannot be described by finitely many forbidden induced subgraphs.

Hybridizable staggered discontinuous Galerkin methods are developed for arbitrary-order polyharmonic equations $(-\Delta)^m u=f$ on shape-regular polytopal meshes in $\mathbb R^d$, for any $m\ge1$, $d\ge2$, and polynomial degree $k\ge0$. The method uses the mixed variable $\sigma=\nabla^m u$ and a staggered primal–dual mesh to impose complementary continuity on scalar and tensor unknowns, without restrictions such as $d\ge m$. Local trace and bubble enrichments stabilize low-order tensor spaces without adding global unknowns. Hybridization localizes the tensor variable and yields an equivalent stabilization-free weak Galerkin formulation. Well-posedness and optimal energy error estimates are proved, and numerical experiments on polygonal and tetrahedral meshes confirm the predicted rates.

We determine the minimal absolute value of a non-vanishing sum of $n$ fifth roots of unity chosen with repetition, and characterize the corresponding sums. As a function of $n$, the minimal absolute value is monotone non-increasing over congruence classes of $n$ modulo $5$ and its only jumps occur when $n=5F_m$, $n=L_m$, or $n=2L_m$, where $F_m$ and $L_m$ denote the $m$-th Fibonacci and Lucas numbers respectively. To prove our results we reduce the problem to a series of inequalities involving rational approximations of the golden ratio $\varphi=(1+\sqrt{5})/2$, the solutions of which can be characterized using the theory of continued fractions.

Let \[ \pi_{0}=[\underbrace{2,\ldots,2}_{2s-1}],\quad \pi_{\infty}=[\underbrace{3,\ldots,3}_{s},\underbrace{1,\ldots,1}_{s-2}],\quad \pi_{1}=[\gamma_{1},\gamma_{2},\gamma_{3}] \] be three nontrivial partitions of the integer \(d=4s-2\), where \(s\ge 3\). We establish a necessary and sufficient condition for the candidate datum \(\{\pi_{0},\pi_{\infty},\pi_{1}\}\) to be realizable by a rational map.

We introduce the lower and upper Wythoff-Fibonacci sequences, obtained from the classical Wythoff sequences by a Fibonacci correction. Specifically, if we put $$\epsilon(j)=\begin{cases}(-1)^k, & \text{if }j=F_k\text{ for some }k\\ 0, & \text{in other case}\end{cases},$$ where $F_k$ is the $k$-th Fibonacci number, then we define the general terms of the lower and upper Wythoff-Fibonacci sequences by $$LWF(n)=\begin{cases} 1, & \text{if }n=1,\\ 3, & \text{if }n=2,\\ a(n)+\epsilon(n), & \text{if }n\geq 3.\end{cases}$$ and $$UWF(n)=\begin{cases} 2, & \text{if }n=1,\\ b(n)+\epsilon(n), & \text{if }n\geq 2,\end{cases}$$ respectively. We show that these sequences partition the set of natural numbers and use them to give an explicit formula for a sequence $q^{\star}_j$, defined from a greedy construction studied by the first author and his coauthors in a previous paper, but with the additional condition that $q^{\star}_1=3$, instead of being defined by the greedy rule. This sequence is a permutation of the set of non-negative integers and has the property that every integer appears exactly once in the sequence of differences $q^{\star}_j-j$. We prove that $q^{\star}_{q^{\star}_j}=j\ \forall j\geq 5$, so that $q^{\star}_j$ is an almost-involution. We also give another greedy algorithm generating $q^{\star}_j$.

We investigate the role of compressibility in the modified quasi-linear viscoelastic (MQLV) constitutive framework for soft solids at finite strain, where shear and bulk responses are governed by distinct relaxation functions. Analytical and semi-analytical results are derived for simple shear and torsion, under incompressible and slightly compressible assumptions. We show that compressibility affects the response only when volume changes occur: under isochoric deformations, the bulk contribution vanishes, while even small deviations from isochoricity significantly alter the normal response. Shear stress and torque are largely insensitive to compressibility, whereas normal stress and axial force exhibit pronounced sensitivity due to the coupling between shear and bulk relaxation. We further demonstrate that volumetric effects interact with the Poynting effect: in simple shear they oppose each other, reducing relaxation, while in torsion they reinforce each other, enhancing it. These trends agree with brain tissue experiments but reveal limitations of the slightly compressible model for highly compressible materials, such as agarose gels. Overall, the results emphasise the importance of accounting for compressibility in modelling normal stress responses and motivate the development of fully compressible formulations and numerical implementations.

We study a notion of fractional $s$-mass for codimension-two currents on closed Riemannian manifolds, defined via energy minimization with a prescribed Jacobian constraint. We prove equi-coercivity and $\Gamma$-convergence, with respect to the flat topology, of the $s$-mass on general codimension-two currents. We also prove several additional results for fixed $s$. We establish improved regularity for $s$-harmonic maps that are minimizing among competitors with vanishing Jacobian and show that their singular set has Minkowski dimension at most $n-3$. Moreover, we show that the $s$-mass defined via weak linking, as recently introduced by the authors, agrees with the prescribed Jacobian formulation used here, clarifying the extent to which the $s$-mass depends, or ultimately does not depend, on the way singularities are prescribed.

This paper is devoted to the shape analysis of the energy of a caloric family of Schauder functions defined on a bounded perforated domain $\Omega^o \setminus \overline{\Omega^i[\phi]}$ of $\mathbb{R}^n$, where the outer boundary is fixed, and the inner boundary is obtained by a $C^{1,\alpha}$-perturbation $\phi$ of the boundary of a reference cavity $\Omega^i$. Without imposing any boundary conditions, we prove that in a suitable neighborhood of the identity $\phi_0$, the domain-to-energy map is of class $C^{\infty}$. The proof is based on the construction of a global diffeomorphism, smoothly depending on $\phi$, from the reference annulus onto the perturbed one and on suitable regularity and smoothness assumptions on the pull-back family onto the reference domain. We then apply our main result to two boundary value problems: a nonlinear mixed Robin-type problem and a linear Dirichlet problem. After recalling some known existence and shape analysis results for the solutions, we prove that the corresponding domain-to-energy map is of class $C^{\infty}$. The proof is based on a decomposition of the fixed domain into near, intermediate, and far regions relative to the cavity, and on the smooth dependence of the layer heat potentials upon support perturbations.

A Minkowski normed space is the Euclidean space equipped with a (possibly asymmetric) uniformly convex and smooth norm, forming a particular class of Finsler manifolds. We construct a stochastic process with one-dimensional time marginal densities given by the fundamental solution to the nonlinear Finsler heat equation in Minkowski normed spaces. This process is constructed as a solution to a singular McKean–Vlasov stochastic differential equation and constitutes a nonlinear Markov process in the sense of McKean. We show that solutions to this stochastic differential equation are pathwise unique, and, thus, probabilistically strong solutions, though the equation has singular coefficients beyond the subcritical regime. Since our construction is a natural extension of the construction of standard Brownian motion from the standard heat kernel, we call this process Brownian motion in Minkowski normed spaces. Furthermore, we obtain a universality property, namely that whichever Minkowski norm is chosen, the radial part of this process is the standard Bessel process. To the best of our knowledge, this is the first construction of stochastic processes associated with nonlinear heat equation in Finslerian spaces.

In this paper, we consider the optimal relaxed control problem for a class of one-dimensional reflected McKean–Vlasov stochastic differential equations with Poisson jumps. Due to the presence of the jump term, the state process generally belongs to the Skorokhod space $D([0,T],\Rp)$, which makes the proof of tightness and the passage to the limit more complicated. Under Lipschitz conditions and suitable growth conditions, we establish uniform moment estimates for the state process and the reflecting process. Then, by using Aldous' tightness criterion, the continuity of the Skorokhod map, and the stability results for stochastic integrals, we prove the existence of an optimal relaxed control. Furthermore, under the Roxin convexity condition, we prove the existence of a strict optimal control. In the general case, we show that relaxed controls can be approximated by a sequence of strict controls.

Quantum nonlocality paradoxes, such as that of GHZ, provide maximally sharp logical obstructions to classical probabilistic models of quantum correlations. They are key resources in a broad variety of information-theoretic tasks that exhibit unconditional quantum advantage. For example, in nonlocal games, which are communication tasks that serve as core technical tools in recent landmark results in quantum computational complexity theory. Their role in establishing quantum advantage motivated their study by Abramsky et al. who introduced an infinite family of three-qubit paradoxes exhibiting novel conditional structure. This was later extended by the present authors into a full classification program. In this work, we completely classify all three-qubit nonlocality paradoxes established via a biconditional parity proof; this is a very large class of paradoxes that encompasses all earlier-known examples. We do this by introducing a suite of new structural and combinatorial techniques. We find that the landscape of nonlocality paradoxes is far richer than previously understood, violating regularity conditions underlying all prior constructions.

This letter studies reduced-attitude tracking for a rigid body on the 2-sphere S2 under a time-varying conic constraint. Using a kinematic model on S2, we first propose a geometric tracking law that guarantees almostglobal asymptotic and regionally exponential convergence in the unconstrained case, where the angular velocity serves as the control input. We then introduce a smooth reference-shaping mechanism that adjusts the desired direction so that the reference provided to the controller satisfies the time-varying conic constraint while preserving the smoothness required by the tracking law. The resulting approach yields smooth continuous feedback and retains the stability guarantees of the unconstrained controller, albeit at the expense of enforcing a soft version of the original constraint. Simulation results illustrate the effectiveness of the method and highlight its suitability for applications where deterministic behavior, smooth control action, and strong stability guarantees are preferred over hard constraint satisfaction.

We present recursive constructions of several families of generalized Cartan matrices associated with Kac-Moody algebras, whose sizes and coranks grow exponentially. The constructions are encoded by connected multigraphs and by block-doubling operations on their associated symmetric generalized Cartan matrices. Equivalently, the corank problem is translated into a spectral graph-theoretic problem: the corank of $2\mathrm{Id}-\operatorname{Adj}(G)$ is the multiplicity of the adjacency eigenvalue $2$. We give two explicit recursive families, compute their spectra and coranks, and emphasize the difference between absolute exponential growth and relative asymptotic density. The resulting algebras are typically indefinite and singular of corank larger than one, and therefore contain several independent central directions and several isotropic radical directions in the root lattice. We also discuss alternative constructions and possible applications to the algebraic structures appearing in gravity, supergravity, string/M-theory and related generalized symmetry problems.

This paper describes an MPI/OpenMP hybrid parallelized fast direct solver for the scattering problem of transverse electric (TE)-mode electromagnetic waves. Because TE-mode scattering can be reduced to the two-dimensional Helmholtz equation, solvers based on the hierarchically semiseparable (HSS) representation are highly attractive due to their high parallel efficiency. However, as the HSS representation applies low-rank approximations to all off-diagonal blocks, it exhibits poor compatibility with high-order discretization methods. We developed a fast direct solver with $O(h^3)$ convergence for Helmholtz transmission problems, whereas conventional HSS solvers typically yield only $O(h)$ convergence (where $h$ represents intervals between the quadrature nodes). It is based on the weakly singular Burton-Miller boundary integral equation and the Nyström method with a one-point correction. Furthermore, recognizing that matrix component calculation, rather than matrix factorization, dominates the total computational time of HSS-type boundary integral solvers, we introduced a load-balancing method to maximize parallel efficiency. Numerical results demonstrate that the direct solver achieves high-accuracy convergence and nearly ideal strong and weak scalabilities.

We study directional expansion for probability-measure-preserving actions of countable groups through a representation-theoretic group property, the cyclic escape property. An infinite countable group has the cyclic escape property if every totally ergodic unitary representation has arbitrarily small fixed-vector projections along infinite cyclic subgroups. This property implies directional expansivity for all totally ergodic actions. We prove that all infinite finitely generated nilpotent groups have the cyclic escape property, and conjecture the same for all infinite finitely generated polycyclic groups. We also prove the cyclic escape property for higher-rank simple lattices whose finite-dimensional unitary representations all have finite image; in particular, for $SL_n(\mathbb Z)$, $PSL_n(\mathbb Z)$, and $PGL_n(\mathbb Z)$, $n\geq 3$. By contrast, free groups of rank at least two do not have the cyclic escape property. The proofs exhibit two independent mechanisms: central spectral structure in nilpotent groups and stationary character rigidity in higher-rank lattices.

Let $\boldsymbol{a}=(a_i)_{i=1}^\infty$ be an infinite sequence of points on a circle. The first $n$ of these points cuts the circle into $n$ pieces. For any given $r$, let $\mu^r_n(\boldsymbol{a})$ be the ratio between the maximum and minimum sizes of $r$ consecutive pieces. Addressing a question of De Bruijn and Erdős, we define a family of sequences for which the asymptotic least upper bound of this ratio, \[ \mu_r(\boldsymbol{a}) \;=\; \limsup_{n\to\infty}\mu^r_n(\boldsymbol{a}) , \] can easily be calculated. Hence, for small $r$, we present upper bounds on $\inf\mu_r(\boldsymbol{a})$.

Discrete diffusion models are widely used for learning and generating discrete distributions. As the generation process is inherently sequential, the acceleration of sampling is of significant importance. In this work, we parallelize the mainstream $\tau$-leaping algorithm for absorbing discrete diffusion in a Continuous-Time Markov Chain (CTMC) framework. By leveraging the continuous-time stochastic integral form of the $\tau$-leaping algorithm and the Picard iteration method, we achieve parallel-in-time sampling acceleration and provide a proof of exponential-factorial convergence for our algorithm. We improve the overall time complexity of $\tau$-leaping under absorbing settings from ${\mathcal{O}}(d \log S)$ to ${\mathcal{O}}(\log (d\log S)\cdot \log d)$ with respect to NFE. Empirically, our method shows consistent acceleration across synthetic and real-data settings. The new sampler achieves at most $7$–$9\times$ runtime speedup for synthetic distribution, and maintains the same quality with $50\%$ fewer NFE and $1.45$–$1.86\times$ runtime speedups in image/text tasks on a single GPU. Our research expands the potential of discrete diffusion models for efficient parallel inference, with broader implications for applications such as molecular structure and language generation.

A graph $G$ is called $k$-edge-hamiltonian if every linear forest (i.e., a disjoint union of paths) with at most $k$ edges is contained in a Hamilton cycle of $G$. In 2018, Füredi, Kostochka and Luo determined the maximum number of $t$-stars in nonhamiltonian graphs, thereby extending an earlier result of Erdős. Recently, Berikkyzy, Hogenson, Kirsch and McDonald extended this line of research by determining the maximum number of $t$-stars in graphs that are not $k$-edge-hamiltonian, as well as in graphs failing to satisfy related properties such as traceability, hamiltonian-connectedness and $k$-hamiltonicity. For sufficiently large $t$, they also characterized the extremal graphs, while for smaller values of $t$, they proposed a conjecture. In this paper, we investigate this conjecture. We show that the conjecture fails at the critical value and further establish a threshold-type result describing the behavior of the extremal graphs when $t$ is close to this critical value.

It is well known that if a diophantine equation turns out not to have a solution over the integers modulo p, for some p, then it does not have a solution over the integers per se. This is because the integers modulo p are a homomorphic image of the integers. However, the integers modulo p are of little use when faced with diophantine inequalities, as the homomorphic image of the less-than-relation is trivial. The purpose of the present paper is to introduce a way of gereralising a particular representation of the integers modulo p. The generalizations, novel to this paper, are in the form of decidable Lindenbaum-algebras, and allow for deciding whether given positive first-order formulas in the language of first-order arithmetic are solvable. Crucially if a system of diophantine inequalities turns out not to be solvable in one of the Lindenbaum-algebras, then it is not solvable over the standard integers.