Papers

Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is $1$-Lipschitz. Motivated by this theorem, Milman [Mil18] formulated several conjectures for the round sphere and for weighted manifolds satisfying the curvature-dimension condition $\operatorname{CD}(\rho,\infty)$. A contracting transport map as in these conjectures implies a corresponding spectral comparison. In the spherical setting, this comparison was also conjectured by Colding and Minicozzi [CM98] for compact manifolds with Ricci curvature lower bounds. We construct counterexamples to these conjectured spectral comparisons on spheres in dimensions $d\geq4$ and on smooth complete weighted manifolds diffeomorphic to $\mathbb{R}^d$ satisfying the $\operatorname{CD}(1,\infty)$ condition in dimensions $d\geq4$, thereby obtaining obstructions to contracting transport maps. The spherical counterexamples can be chosen arbitrarily close to the unit round metric in $C^\infty$, while retaining $\operatorname{Ric_g}\geq(d-1)g$. The weighted counterexamples can be chosen with non-negative sectional curvature $\operatorname{Sec}_g\geq0$ when $d\geq4$. For $d\geq5$, we also construct counterexamples satisfying $\operatorname{Ric}_g\geq0$ and $\nabla_g^2V\geq g$, where $\mu=Z^{-1}e^{-V}\operatorname{dvol}_g$.

In this paper, we determine the asymptotic behavior of the singular values of the commutator $\mathscr{C}_u := [M_u, \mathbb{P}_\alpha]$ acting on $L^2(\mathbb D, dA_\alpha)$, where $M_u$ is the operator of multiplication by a subharmonic function $u$ on $\mathbb D(R)$ and harmonic outside the origin, with $R>1$, and $\mathbb{P}_\alpha$ is the orthogonal projection onto the space $\mathscr{H}_\alpha^2(\mathbb D)$ of harmonic functions on $\mathbb D$ that are square-integrable with respect to the weighted measure $dA_\alpha$. We prove that if $u(z) = U(z) + \overline{U(z)} + \nu_u \log|z|^2$, where $U$ is a holomorphic function on $\mathbb D(R)$ and $2\nu_u$ is the Lelong number of $u$ at $0$, then $$s_n(\mathscr{C}_u) \underset{n\to+\infty}{\sim} \frac{\sqrt{\alpha+1}}{2\pi n} \int_{\partial \mathbb D} \sqrt{\nu_u^2 + |U'(z)|^2} \; |dz|.$$ In particular, the operator $\mathscr{C}_u$ belongs to the Von Neumann-Schatten class $\mathcal C_p$ for any $p>1$.

Motivated by Kontsevich's graph complexes, this paper gives a systematic study of matroid complexes. We construct deletion and contraction bicomplexes on the vector space spanned by matroid classes equipped with ground-set orientations, organizing the several naturally arising variants into a single unified framework. We show that direct sum and restriction-contraction make this space into a connected graded Hopf algebra extending Schmitt's matroid Hopf algebra, and use the resulting dg-algebra structure to prove broad acyclicity results. We compute the total, simple, loopless, regular, binary, and ternary matroid complexes through ground-set size $9$, and the connected quotient of the simple loopless regular complex through ground-set size $15$. These computations detect nontrivial homology and lead to a conjectural description in terms of odd-wheel matroids.

We discuss the role of the Feynman-Hellmann theorem for abstract one-parameter families of Hamiltonians in sum rules and trace identities of Harrell and the author and its application to spectral theory. In particular, we derive a sum rule for the second derivative of eigenvalues of a one-parameter family of Hamiltonians extending thereby concepts of second order perturbation theory. We present applications to semiclassical eigenvalue bounds of Schrodinger operators as Lieb-Thirring inequalities, zeros of Bessel functions, eigenvalue inequalities for sums of matrices and trace inequalities.

In this paper, Cimmino's classical reflection algorithm for solving the $n\times n$ nonsingular linear system $A\bx=\bb$ is analysed through the lens of spectral theory. Reformulating the weighted iteration as $\e^{(\nu+1)}=M_w\,\e^{(\nu)}$, where $M_w = I - A^\top D_w A$, the error is shown to contract by the spectral radius $\sprad(M_w)$ at every step, with a sharp, asymptotically tight bound. For $n=2$, a closed-form expression for the contraction factor is derived, \[ \sprad(M_w) \;=\; |1-\mu| + \tfrac{1}{2}\sqrt{(w_1-w_2)^2 + 4w_1w_2\cos^2\!\theta}, \] where $\mu=(w_1+w_2)/2$ and $\theta$ denotes the angle between the hyperplane normals. A central result of this paper is that the standard unit weights $w_1^*=w_2^*=1$ are globally optimal over all positive weight pairs, uniquely achieving the minimum contraction factor $\sprad^*=|\cos\theta|$ – a quantity determined solely by the geometry of the hyperplane normals. The inter-normal angle $\theta$ thus emerges as the single diagnostic parameter governing both convergence speed and weight selection. Extensions to a single-step convergence criterion at $\theta=\pi/2$ and to an exact spectral rate for general~$n$ are also established.

We introduce the Morse ensemble polynomial $\ME_K(z_0,\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\ME_K = \sum_M \prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$, where $c_i(M)$ counts critical $i$-simplices. This polynomial records the complete critical-vector distribution over all acyclic matchings, equivalently over all discrete gradient vector fields arising from discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes. In dimension one, this invariant recovers the Chari–Joswig graph formula for the $f$-vector of the discrete Morse complex in a two-variable Morse-vector normalization: $\ME_G=z_1^{m-n}\det(z_0z_1\,I_n+L_G)$ for a connected graph $G$. The main new contributions are higher-dimensional and structural. First, we prove a Top-Face Recursion for adding a top-dimensional simplex, with a non-liftable correction term $F(K,\sigma,\tau)$. The vanishing and leading obstruction of this correction term are controlled by the top incidence graph: an incidence-separation criterion detects exactly when $F=0$, a leading obstruction layer is governed by shortest obstruction paths, and a tree-like (incidence-forest) regime of the top incidence structure gives a correction-free higher-dimensional recursion, including stacked balls as a concrete class. Second, we introduce the independence ME polynomial $\Phi(G):=\ME_{\mathrm{Ind}(G)}$, a graph invariant which strictly refines the graph-level Morse ensemble, separates examples not distinguished by $T_G$ and $I(G;t)$, and records collapse-level information of $\mathrm{Ind}(G)$ through coefficients such as $[z_0]\Phi(G)$.

We prove that linear collisional kinetic equations in the whole space without confinement mechanism display a long-time self-similar behaviour. This drastically improves the recently known results (decay estimates) about the solutions in such a context, providing the first result regarding this self-similar behaviour. As a consequence, we also establish a uniform-in-time convergence of the suitably rescaled solutions to their diffusion limit, which is also new. The class of equations considered includes some BGK type equations, some kinetic nonlocal Fokker–Planck-type equations and some kinetic (possibly fractional) Fokker–Planck equations, for which we are able to write explicitly solutions through a Wild sum (or Dyson series) or we can manage some accurate computations on the Fourier side.

In this article, we present a method to find a solution to a one-dimensional nonlocal conservation law that respects a space-dependent mapping, referred to as the obstacle. This is achieved by generalizing existing results for the local conservation law: We consider a relaxation of the velocity, that explicitly depends on the obstacle. We prove existence of solutions to the relaxed problem and show that, as the relaxation mapping converges to a Heaviside-type function, the corresponding solutions converge to a weak solution of a discontinuous nonlocal conservation law. Moreover, we can characterize the limiting flux in several cases.\\ The paper concludes with a numerical study that illustrates the aforementioned convergence.

We consider immigration processes with binomial catastrophes and random survival parameters. Two sources of randomness are analyzed. In the first model, the survival parameter is independently resampled at each catastrophe. In the second model, individuals are assigned independent survival parameters at birth, which remain fixed over time. We show that the first model exhibits almost sure extinction, as in the classical case with a fixed survival parameter. In contrast, the second model exhibits a phase transition, admitting survival with positive probability, depending on the distribution of individual survival parameters. We provide explicit formulas for both the survival probability and the expected time to extinction. Finally, our proofs establish a novel methodological bridge with the Firework process, unifying population dynamics with spatial models of information spreading.

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues, respectively. Akbari, Kumar, Mohar, Pragada, and Zhang conjectured that, when $k$ is odd, the value of $k$ modulo $4$ determines which of $s^+(G)$ and $s^-(G)$ is greater than $n$. More precisely, if $k\equiv 3\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\equiv 1\pmod 4$, then $s^+(G)<n<s^-(G)$. We confirm this conjecture.

Selecting a finite dictionary of observables whose span is Koopman-invariant is a central challenge in data-driven Koopman operator approximation. We address this problem by exploiting zero-block structure in Extended Dynamic Mode Decomposition (EDMD) matrices. We show that any sub-dictionary whose span is Koopman-invariant induces an exact zero block in the EDMD matrix, even for finite data. We then show that such blocks can be detected by applying PageRank to a row-normalized EDMD matrix constructed from a large initial dictionary. The theory extends to approximately invariant subspaces and yields stronger guarantees for personalized PageRank (PPR) when the seed observables lie inside the target block and reach all observables in that block. Combining EDMD concentration bounds with PageRank perturbation theory gives end-to-end detection guarantees with $O(1/\sqrt{M})$ finite-sample scaling and explicit constants. More generally, without assuming an invariant subspace exists, high PPR mass on a sub-dictionary controls discounted multi-step leakage from the seed observables. Numerical experiments on the Duffing oscillator, Van der Pol oscillator, Lorenz system, and a three-well Ramachandran potential suggest that the method identifies compact, interpretable dictionaries with accurate predictions.

We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $\varphi$-finite element method ($\varphi$-FEM). $\varphi$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $\varphi$. To impose the boundary conditions, Dirichlet problems adopt the $\varphi$-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with $\varphi$-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. When labeled reference data are available, an optional data-augmented variant (WINO+data) can further combine this physics-informed loss with a supervised term. After training, WINO outputs can seed the nonlinear $\varphi$-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show substantial accuracy of WINO together with total training times of about 15%-70% of those of supervised $\varphi$-FEM-FNO across all cases, without requiring reference-solution generation.

This paper studies the optimal control problems of stochastic evolution equations with infinite delay of general functional type. By introducing a non-anticipative path derivative and its infinite-window dual operator, we derive the infinitely anticipated backward stochastic evolution equation (IABSEE) as the adjoint equation, and establish both necessary and sufficient maximum principles. As an application, we solve an infinite-delay linear-quadratic (LQ) control problem and obtain the explicit optimal control.

We study a regulation problem for stochastic systems subject to both continuous fluctuations and rare but significant shocks, modeled as a jump-diffusion with uncertainty in both the drift and the jump intensity. Such settings arise in applications including inventory control, cash management, and capacity planning. We formulate the problem as a robust ergodic singular control problem in which a decision maker applies upward and downward interventions while accounting for model ambiguity through entropy-penalized distortions. The resulting max-min problem involves a long-run average performance criterion. We show that the associated Hamilton–Jacobi–Bellman equation reduces to a nonlinear integro-differential free-boundary problem with a tractable structure. The worst-case model exhibits a bang-bang form, and the optimal policy is characterized by reflecting barriers. Under exponentially distributed jumps, the problem further reduces to a system of ordinary differential equations, enabling efficient numerical computation.

Quadratic regularization has emerged as a potential alternative to the popular entropic regularization in computational optimal transport, offering the theoretical advantage of producing sparse couplings through its hinge density structure. Despite recent progress in one-dimensional settings and general upper bounds, fundamental questions about the localization rate of QOT optimizers around the Monge coupling have remained open. In this work, we establish a general lower bound showing that the support of the QOT optimizer cannot concentrate around the Monge graph faster than order $\varepsilon^{\frac{1}{d+2}}$ in the directed Hausdorff distance, matching the conjectured optimal exponent under standard regularity assumptions in \citet{wiesel2025sparsity}. We also show that the QOT value gap controls the mean-squared deviation $\mathbb E_{\pi_\varepsilon}\|y-T(x)\|^2$ by the scale of $\varepsilon^{\frac{2}{d+2}}$. As a corollary, in the affine Brenier regime, which includes Gaussian-to-Gaussian transport, we derive a sharp pointwise tube bound of order $\varepsilon^{\frac{1}{d+2}}$ by reducing the problem to self-transport and applying recent self-transport sparsity results. Finally, we validate our theoretical bound with a synthetic experiment in high-dimensional settings.

Hierarchical edge-cloud computing-aided Internet of Things (IoT) networks offer low-latency and cost-efficient services to a growing number of data-intensive IoT devices. However, optimizing service placement, which involves determining the most suitable locations within a network to deploy various services, is critical to balancing workloads dynamically and ensuring efficient resource utilization. In this paper, we jointly optimize service placement, edge/cloud cooperation, task offloading, and bandwidth allocation to enhance processing efficiency and response times. The main objective is to minimize both the overall end-to-end latency and the system cost, including service deployment and operational costs. The formulated problem belongs to the class of non-convex mixed-integer nonlinear programming, where finding a feasible solution is already challenging. Towards a stable system, we first transform the original problem into a more tractable form and then decompose it into sub-problems which are solved at different timescales. Combining tools from relaxation and the successive convex approximation method, we develop iterative algorithms to solve these problems efficiently. With an appropriate penalty parameter, the proposed algorithms guarantee convergence to at least a local optimum. We produce extensive numerical results to demonstrate the superior performance of the proposed algorithms over benchmark schemes as well as emphasize the significance of the joint service placement and resource allocation in enhancing system performance and efficiency.

For a commutative ring $R$ with identity, the weakly zero-divisor graph $W\Gamma(R)$ has vertex set $Z(R)^{\ast}$, with distinct vertices $x$ and $y$ adjacent whenever there exist nonzero $r\in{\rm Ann}(x)$ and $s\in{\rm Ann}(y)$ with $rs=0$. The Laplacian spectrum of $W\Gamma(Z_n)$ has been determined by Shariq, Mathil, and Kumar, who also established that $W\Gamma(Z_n)$ is Laplacian integral. Building on the structural description of $W\Gamma(R)$ due to Nikmehr, Azadi, and Nikandish, we extend the Laplacian spectrum and integrality results from $Z_n$ to every finite commutative ring $R$: we restate $W\Gamma(R)$ in unified form as a complete multipartite graph whose parts are made explicit by the local-ring decomposition of $R$, compute the full Laplacian spectrum in closed form, prove Laplacian integrality of $W\Gamma(R)$, and give a sharp bound on the number of distinct Laplacian eigenvalues. As consequences we obtain explicit formulas for the algebraic connectivity and number of spanning trees of $W\Gamma(R)$, and recover the Laplacian spectrum of $W\Gamma(Z_n)$ in compact form.

Using the Chern-Gauss-Bonnet theorem, we establish a sharp inequality for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds $M^n$ with nullity index at least $n-3$. Consequently, the Euclidean isoperimetric inequality extends to $M^n$, which proves the Cartan-Hadamard conjecture for these spaces.

In this short note, we prove a general nilpotence theorem for a rational rigid 2-ring all of whose objects satisfy a certain “moderate growth condition” inspired from the theory of tensor categories. This applies in particular to the category of modules over a rational $E_{\infty}$-ring, to the derived category of any super-Tannakian category in characteristic zero, and conjecturally to Voevodsky's rational category of mixed motives over a field $DM_{\mathbb{Q}}$. In fact, we further prove that any such category has enough tt-fields, which can be chosen to be of the form Perf(L) for an even 2-periodic field L.

This paper started as a review of seven results pertaining to a family of bilinear models with rank one NGM introduced by Fall, Iggidr, Sallet and Bonzi, which utilize the explicit eigenvectors of the NGM to compute the unique endemic equilibrium (EE), and Lyapunov functions at both the disease free equilibrium (DFE) and EE, and of results of Shuai and Van den Driessche (2013), which essentially deal with the same "DFE -EE stability exchange" in the non-rank one case, when the eigenvectors are not explicit. Recently, these results were complemented by Earn and McCluskey (2025), who proved as well a “strong threshold theorem", namely that when the DFE is unstable, a second equilibrium which is globally asymptotically stable must exist. Below, we obtain in Theorem \ref{thm:TK_DFE} some results that extend beyond rank one. For example, a nontrivial positive equilibrium exists if and only if the spectral equation \(\rho(\widetilde K(S))=1\), admits a strictly positive solution. Also, we showed that Bonzi-Iggidr-Sallet bilinear models with rank one NGM may be classified in two classes, with slight variations in the eigenvector formulas, and that extensions in the presence of feedback from infectious to susceptible are possible. Another takeout from the previous works, which we clarify in a revisit of the seven results in the rank one case, is that Lyapunov functions for both the DFE and the EE may be constructed using as weights the left Perron eigenvector $\pi(S)$ of the \NGM\ (NGM) $K=F(S) V^{-1}$, where $S$ denote all the non-infectious variables, and when a positive ODE leaves a siphon face, it does so along the right Perron eigenvector $w(S)$ of $\T K= V^{-1}F(S)$. The question of whether this continues to be true beyond rank one is explored in ongoing work.

Let $M_n$ denote the largest interpoint distance among independent random points $X_1,\dots,X_n$ uniformly distributed in a compact set in $\mathbb{R}^d$. Weak limit laws for $M_n$ are known in several geometric settings, in particular for ellipsoids with a unique major axis. In this paper we treat the simplest nontrivial case in which the largest semi-axis is not unique, namely the rotational ellipsoid $\{(x_1,x_2,x_3)\in\mathbb{R}^3: (x_1^2+x_2^2)/h^2 + x_3^2/a^2 \le 1\}$, where $0<a<h$. The diameter of this ellipsoid is attained by all antipodal pairs on the equatorial circle, so the extremal points are not isolated. We prove that $n^{4/7}(2h-M_n)$ converges in distribution to a Weibull-type limit law with explicit parameter. The proof combines geometric localization arguments with a Chen–Stein Poisson approximation for rare nearly diametral pairs.

We sharpen the constants in two degree inequalities for circle-valued Sobolev maps in degenerate regimes, as $p \to 1^+$ or $\delta \to 0^+$. The two proofs use the same power trick together with elementary estimates. The results answer two open problems posed by Brezis. The proofs are obtained by generative AI and are verified by the authors.

Motivated by the notion of integrability introduced by Bogoyavlenskij for vector fields, we propose a definition of smooth integrability for general diffeomorphisms. In brief, we say that a diffeomorphism is integrable if it commutes with the flows of {\color{black}commutative} vector fields and shares their first integrals. We establish a Liouville-Arnold type theorem and prove that on connected invariant level sets of the first integrals, a smoothly integrable diffeomorphism is conjugate to a skew translation on a toroidal cylinder, and in particular, when the level sets are compact, the induced dynamics is quasiperiodic, as in the classical Hamiltonian case. We further show that linear diffeomorphisms on real or complex Euclidean spaces are integrable in our sense, as is the case for linear vector fields in the sense of Bogoyavlenskij, and symplectic diffeomorphisms that are integrable in their standard Liouville-type definition are integrable in our framework. We also modify the cotangent lift construction, which is useful in the treatment of integrability of vector fields, for diffeomorphisms, and show that integrability is preserved under this correspondence. Several examples are provided to illustrate the scope of the theory.

We present a theoretical and numerical analysis of Monte Carlo methods for the estimation of statistical moments of random variables $X:\Omega\rightarrow E$ taking values in a Banach space $E$. For practical computation, we consider finite-dimensional approximation subspaces ${(E_\ell)_{\ell\in\mathbb{N}}\subset E}$ of increasing dimension. We develop a refined error analysis that explicitly accounts for a dependence of the Rademacher type constants on the dimension of $E_\ell$, leading to novel complexity results for single- and multilevel Monte Carlo methods to estimate the mean and injective moments of arbitrary order, which are, in certain cases, sharper than those derived in [Kirchner, Schwab, J. Funct. Anal, 2024]. Moreover, we show that, in favorable cases, the resulting error-vs.-work bounds are independent of the Rademacher type of $E$. We then focus on $L^p(S)$-valued random variables for a $\sigma$-finite measure space satisfying certain approximation properties, and prove that for a random variable $X\in L^q(\Omega;L^p(S))\cap L^p(S;L^q(\Omega))$, with $q\in (1,\infty)$ and $p\in [1,\infty)$, the $L^q$-convergence rate of a Monte Carlo estimator is determined exclusively by the integrability parameter $\min\{q,2\}$, with no dependence on the Rademacher type $\min\{p,2\}$ of $L^p(S)$. We further investigate the impact of measuring the (multilevel) Monte Carlo error in the $L^q(\Omega;L^p(S))$-norm while $X$ possesses additional regularity, $X\in L^{\tilde{q}}(\Omega;L^p(S))\cap L^p(S;L^{\tilde{q}}(\Omega))$ with $\tilde{q}\in [q,\infty)$. This analysis reveals an interplay between the sampling error and the strong approximation error, and leads to optimized error-vs.-work bounds for both single- and multilevel Monte Carlo methods. Numerical experiments confirm the sharpness of the analyses presented.

The chord-power integrals $I_k$ are classical integral-geometric functionals of a planar convex body, obtained by integrating powers of the chord length against the kinematic measure on the space of lines meeting the body. We establish a single-expression closed form for $I_2$ on an arbitrary triangle, involving logarithms symmetric in the sides, and derive two analytic consequences: a power-sum series representation, and a sharp isoperimetric-type inequality with explicit constant involving $\ln 3$, attained uniquely by the equilateral triangle. The set $\{I_0, I_1, I_2\}$ identifies a triangle up to congruence, complementing J. Gates's algebraic recognition via $\{I_0, I_1, I_5\}$ with the minimal index set $\{0, 1, 2\}$.

We prove that the Strong Maximal Rank Conjecture holds for quadrics in $\mathbb{P}^3$ and we prove the existence of a component of the expected dimension in $\mathbb{P}^4$, as well as in a wide range of parameters $(g,d)$ in $\mathbb{P}^r$ with $r\ge 5$. We propose the Stratified Strong Maximal Rank Conjecture which also takes into account the rank $k$ of quadrics and prove it works in most of the cases when $k=3$ and $k=4$. We also prove a partial result that concerns the unrepresentability of the canonical bundle of a general curve as a sum of $3$ pencils.

We present a method to build free immersions in critical dimension on $m$-tori for $m = 2, 3, 4, 5$ by using a factorization trick inspired by tori immersions in critical dimension. As an application, we show that the set of smooth free maps from a closed surface $M$ to $\mathbb R^5$ is nonempty. In particular, every closed surface embeds freely in $\mathbb R^5$. Finally, we find the homotopy invariants of free maps in our environment and study which invariants are achieved by our maps.

In this paper, we study ideal quotients of triangulated categories by higher cluster tilting subcategories. Koenig and Zhu proved that the ideal quotient by a $2$-cluster tilting subcategory is an abelian category; moreover, by Morita's theorem, it is equivalent to the module category over the $2$-cluster tilting subcategory. We generalize this result to higher cluster tilting subcategories. More precisely, we show that the natural DG-enhancement of the ideal quotient of a triangulated category by a $(d+1)$-cluster tilting subcategory is an abelian $d$-truncated DG-category. In the appendix, we prove a Morita-type theorem for abelian $d$-truncated DG-categories, which asserts that an abelian $d$-truncated DG-category with enough projectives is equivalent to a $d$-extended module category over a $d$-truncated DG-category. As an application, we show that the ideal quotient of a triangulated category by a $(d+1)$-cluster tilting subcategory is equivalent to a $d$-extended module category over a $d$-truncated DG-category.

In this paper, we introduce and study the notion of S-filters in bounded distributive lattices.

We propose a general robust prediction framework, termed conformity-based projective prediction (CPP), that integrates Bayesian predictive modeling with ideas from conformity-based conformal prediction. Rather than assessing conformity through residual-based scores, the CPP criterion defines conformity distributionally: a candidate value for a future response is considered conforming to the extent that its inclusion in the data leaves the leave-one-out predictive distributions of the observed responses undisturbed. The framework requires only that the leave-one-out and swapped predictive distributions are available in closed form and that the swapped predictive mean is differentiable in the candidate value. Under these conditions, we establish a general bounded-influence proposition and a general local convexity lemma, and prove that CPP dominates any plug-in predictor with unbounded influence in asymptotic variance under $\epsilon$-contamination models. When the posterior mean is linear in the observations – as in Gaussian linear models, basis-expansion regression, and Gaussian process regression – the swapped predictive mean is affine in the candidate value, yielding closed-form or one-dimensional optimization solutions and an efficient rank-two computational update; all general theoretical results specialize to explicit corollaries in this setting. Simulation experiments and two data analyses under the Gaussian linear model illustrate the finite-sample advantages of the proposed method, confirming the theoretical predictions across contamination levels, sample sizes, and predictor dimensions.