Papers

This paper presents a first-order multimodal logic programming system called MMLP. The system accepts arbitrary formulas as both programs and queries, without restricting either side to Horn clauses or a separate goal grammar. Its declarative semantics is given independently by a Hilbert system for selected D, T, I, B, 4, and 5 modal principles. Execution uses a nested proof calculus with finite grammar certificates for modal propagation. Certificate reachability is equivalent to the associated Horn closure, certificate existence is decidable, and the calculus is cut-free complete. For quantified answer computation, we give a unification algorithm based on permission sets that controls eigenparameter scope. The algorithm always terminates and fails exactly when no admissible solution exists. A successful run returns a unifier that is itself admissible and through which all admissible solutions factor. Computed answers are declaratively correct, and each declaratively correct answer is an ordinary output instance of a computed answer. In proof search, MMLP admits syntactic focalization and a fair and complete enumeration of focused answers. It represents exactly the same answer substitutions as Nguyen's pure logical KDI4s5-MPROLOG on their common fragment, while admitting a strictly larger program and query language.

We consider a processing network model with $m$ job classes or buffers, exogenous input flows into some classes, $n$ processing activities, and $p$ servers. Each activity is either a specified server processing jobs of a specified class, or a fictional input server delivering jobs of a specified class; jobs change class in Markovian fashion after completing service. A standard multiclass queueing network, with its one-to-one correspondence between job classes and activities, is a special case, but our general model allows two or more ways to process a given class, and some or all input flows may be turned away at the system manager's discretion. Costs are linear: a holding cost per time unit for each class $i$ job in the system, and a rejection penalty for each class $i$ arrival denied access $(i=1,\ldots,m)$. The system manager makes input control, job routing, and order-of-service decisions to minimize expected discounted costs over an infinite horizon. We formulate an approximating Brownian control problem (BCP) whose state space is the $m$-dimensional nonnegative orthant; control is a drift vector chosen from a bounded polyhedral set, based on dynamic state observations. Using recently developed computational methods, the BCP can be solved numerically in dimensions up to at least $m=50$, and we explain how the numerical solution is translated into an implementable control policy for the queueing system of original interest. Previous work on heavy traffic diffusion approximations suggests that this policy is nearly optimal in the heavy traffic parameter regime, and numerical examples support that conjecture. We also discuss its advantage over an alternative approach, featured in our previous work, where the BCP is replaced by a lower-dimensional "equivalent workload formulation" that is computationally efficient but difficult to interpret in the network of original interest.

This article studies dominant local rings, a notion introduced by Takahashi. We prove that, for a flat local homomorphism R–>S between commutative noetherian local rings, dominance descends from S to R, thereby answering Takahashi's descent question affirmatively. The proof makes use of Krause's realization of the idempotent completion of the singularity category as the subcategory of compact objects in the homotopy category of acyclic complexes of injective modules.

Modern online retailers leverage networks of distributed warehouses to rapidly fulfill customer orders. We study a problem of jointly deciding (i) how to allocate inventories across warehouses in the network subject to warehouse capacity and product supply constraints, (ii) how to dynamically select personalized product assortments based on customer preferences and location, as well as real-time stock levels, and (iii) which warehouse to use to fulfill the product chosen by the customer to maximize expected profit. In our model, the firm chooses an inventory placement across warehouses at the start of the selling horizon. Customers of different types then arrive at discrete time periods over a finite time horizon according to a known distribution. When offered an assortment, each customer type chooses at most one product according to a discrete choice model. The firm then chooses a feasible warehouse from which to fulfill the chosen product, depleting its stock and earning a profit that depends on the product, customer type, and warehouse choice. We develop approximation algorithms and policies with provable theoretical guarantees and strong empirical performance. Under general probabilistic choice models, we establish asymptotically optimal approximation algorithms as warehouse capacities and product supplies scale. Under the multinomial logit (MNL) choice model, we obtain constant-factor approximation algorithms, with approximation factors ranging from 0.080 in the most general setting to 0.199 under stationarity and unlimited product supplies, which means that only warehouse capacity constraints are present. To the best of our knowledge, these results provide the first provably efficient algorithms for the joint inventory placement, assortment personalization, and order fulfillment problem.

We develop an abstract Banach principle in vector lattices and apply it to obtain lattice-theoretic versions of the individual and maximal ergodic theorems, without recourse to any measure representation. Considering a sequence of bounded operators with values in a Dedekind $\sigma$-complete vector lattice endowed with a locally solid topology satisfying the $\sigma$-Lebesgue property, we prove that the set of points at which the sequence is order convergent is a closed subspace and coincides with the whole space whenever convergence holds on a dense subset. Investigating positive, power-bounded, mean ergodic operators on order continuous Banach lattices, we construct a topology on the universal completion induced by a strictly positive order continuous functional and prove that the Cesàro means converge in order in the universal completion and, in particular, uo-converge in the original lattice. Moreover, we introduce the notion of a superinvariant pair and derive a lattice-theoretic Hopf inequality together with a weak type estimate via band projections, which yields an abstract maximal ergodic theorem. A spectral-theoretic version of the theorem follows from classical Perron–Frobenius theory. Finally, we specialise the abstract framework to the model space $L^0(\Omega)$ and revisit the classical Banach principle, the Hopf–Dunford–Schwartz theorem and Doob's martingale convergence theorem.

There are two possible answers to "$(a+b)^2 = a^2 + b^2$". Either "this is completely wrong" or "I assume, you are in characteristic $2$". If we go for the first answer, we could ask ourselves: "How wrong is it actually?" In this expository article, we introduce and use resolution of singularities and the real log canonical threshold (RLCT) as a measure for such errors. The RLCT is an invariant with connections to the minimal model program and, as recently discovered and investigated, to Bayesian statistics and machine learning. Inspired by typical mistakes from high school mathematics, we compute this invariant for several classes of examples, among them the error arising from the "freshman's dream" above and from a flawed computation of means in elementary statistics. In some examples, we highlight how a computer algebra software can be used in order to solve this problem.

We present SymQuPS, a SymPy-based algebra system in Python mainly aimed toward the phase space representation of quantum mechanics within the Cahill-Glauber formalism (including the Glauber-Sudarshan $P$, Wigner, and Husimi $Q$ representations). By extension, the package serves as an algebraic venue for canonical quantization. A key feature is the phase space representation of an arbitrary Lindblad master equation, which gives the phase space equation of motion of the quantum system. We describe the core functionalities of the package, consisting of $s$-ordered operators, the star products, and the phase space representation. Some examples of use are given to illustrate the application of the package, and the package's performance in typical use cases is discussed.

For reaction-diffusion with blow-up nonlinearities, we consider the question whether the sup norm of any positive blow-up solution must be eventually monotone nondecreasing in time. While some sufficient conditions are known, especially for radial solutions, this natural and basic question for the blow-up theory does not seem to have been addressed so far in full generality. We construct surprising (nonradial) counter-examples of blow-up solutions with oscillatory $L^\infty$ norm, for any Sobolev supercritical power nonlinearity, which show that this property may fail. In addition, this provides examples of type II blow-up for any supercritical power, which considerably increases the known range of powers for which type II blow-up may occur. Moreover, whereas all the type II blow-up rates known so far were at most polynomial, the blow-up in our counter-examples can be arbitrarily singular. As a related question, we clarify the gradient estimates obtained and used in previous works. In particular we show that these estimates hold only at times when the $L^\infty$ norm is maximal with respect to the past.

Let $R$ be a commutative noetherian ring. Denote by $\operatorname{mod} R$ the category of finitely generated $R$-modules. Let $\Delta$ be a subset of $\operatorname{Spec} R$, and let $\operatorname{Ass}^{-1}\Delta$ stand for the full subcategory of $\operatorname{mod} R$ consisting of finitely generated $R$-modules whose associated prime ideals belong to $\Delta$. In this paper, we consider classifying torsion pairs in $\operatorname{Ass}^{-1}\Delta$ and some other full subcategories of $\operatorname{mod} R$.

We study stable equivalences of Brauer graph algebras. We prove that a basic symmetric algebra without semisimple summands which is stably equivalent to a Brauer graph algebra is again a Brauer graph algebra. In particular, Brauer graph algebras are closed up to semisimple summands under stable equivalence of Morita type. As consequences, we reprove the closure under derived equivalence established by Antipov and Zvonareva and obtain a solution of the reconstruction problem posed by Rickard and Rouquier for algebras stably equivalent of Morita type to Brauer graph algebras.

In analogy with the notion of Krull dimension, for every commutative ring $R$, we introduce the notion of pure dimension $\pdim(R)$ by considering chains of purely-prime ideals. Then we see that the inequality $\pdim(R)\leqslant\dim(R)$ holds for a very broad class of rings $R$, including pseudo-Noetherian rings (e.g., Noetherian rings), Gelfand rings (e.g., local rings), mp-rings (e.g., integral domains), and rings with finitely many maximal ideals or finitely many minimal primes. In fact, for all of these rings $R$, we always have $\pdim(R)=0$, i.e., every purely-prime ideal of $R$ is purely-maximal. \\ However, we prove a classification theorem which states for each ordinal number $\Omega$ the proper pure ideals of the commutative ring $$R_{\Omega}=k[X_{\alpha} : \alpha <\Omega]/\big(X_{\alpha}(1-X_{\beta}) : \alpha<\beta<\Omega\big)$$ are exactly of the form $I_{\lambda}=(x_{\alpha}: \alpha<\lambda)$ with $\lambda=0$ or $\lambda\leqslant\Omega$ a limit ordinal (we write $x_{\alpha}$ for the image of $X_{\alpha}$ in $R_{\Omega}$). In addition, all of them are purely-prime ideals and form the following unique strictly ascending chain: $$I_{0}=0\subset I_{\omega}\subset I_{\omega\cdot2}\subset I_{\omega\cdot 3}\subset\ldots\subset I_{\omega\cdot\alpha}$$ where $\alpha$ is a unique ordinal with $\Omega=\omega\cdot\alpha+n$ and $n\in\omega$. To prove this theorem, we use the full strength of ...

This paper defines Lyra almost Ricci-Bourguignon solitons on twisted warped product manifolds and investigates the interaction between the twisting function, the Lyra scale, and the soliton potential field. We derive the horizontal, vertical, and mixed components of the soliton equation and obtain trace, gradient, and factor-inheritance characterizations. The mixed equation yields new rigidity phenomena: a base-dependent Lyra scale preserves the ordinary separability obstruction, whereas a fiber-dependent scale admits a weighted compensation law that allows genuinely nonseparable twisting functions. We also establish Einstein reductions for conformal, Killing, homothetic, concurrent, and gradient potential fields. Exact flat and non-flat examples, together with local and compact nonexistence results, demonstrate the geometric significance of the scale?twisting interaction.

In 2019 La Bretèche, Stoll and Tennenbaum showed that the ratio of the sum of digits function $s_{q_1}(n)/s_{q_2}(n)$ of two multiplicatively independent bases $q_1$ and $q_2$ is dense in $\mathbb{Q}^+$. Recently Spiegelhofer proved that in the special case $s_2(n)/s_3(n)=1$ we have infinitely many solutions. Spiegelhofer extended this jointly with Drmota to show that the pair $(s_2(n),s_3(n))$ attains almost every value of $\mathbb{N}^2$ and hence, in particular, that every rational ratio is attained infinitely many times.\\ In this paper we show that, indeed, for any pair of multiplicatively independent bases $p$ and $q$, that the ratio attains every rational number infinitely many times. We also study this problem in the multiplicatively dependent case, hence giving a complete characterisation in the case of 2 bases.

Computations in simplicial homology are crucial for algebraic topology and computational geometry in general. Although several mature software systems provide facilities for working with simplicial complexes, there is currently limited support for such computations within the Wolfram language ecosystem. This paper presents SimplicialHomology, an open-source Wolfram language paclet for constructing abstract simplicial complexes and computing their homology groups over the integers, rational numbers, and finite fields. Furthermore, the package supports many other functionalities such as calculating Betti numbers, automorphism groups, and other constructions such as joins, stars, links, etc. Several examples are presented to illustrate the package and its performance.

Operating teams of autonomous aircraft in dynamic, uncertain, and potentially adversarial environments requires safety protocols that are reliable yet selective, and allow agents to fly in close proximity while making progress toward mission objectives. We introduce adversarial time-to-collision (aTTC), a risk metric that quantifies, for a given agent, how quickly any surrounding agent could reach it assuming adversarial intent. We embed aTTC into the control barrier function (CBF) framework, defining the barrier directly in time rather than distance or velocity. The resulting aTTC-CBF is inherently anticipatory: agents modulate their own velocity based not on whether a peer is on a collision course, but on how quickly one could reach collision given its dynamical constraints. A differentiable neural-network surrogate makes the aTTC computable in real time within a standard CBF quadratic program. Across long time-horizon simulations of 3D independent-pursuit and formation-flight scenarios, the aTTC-CBF achieves up to twice the waypoint progress at half the collision rate of a higher-order distance-based CBF baseline.

We compute Simons type equations for parallel mean curvature submanifolds in complex space forms N^n(c), with constant holomorphic sectional curvature c, and product spaces N^n(c)xR. These formulas are then used to characterize some of these submanifolds.

We introduce matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces associated with an expansive matrix $A$. Inspired by the $\A_p$-dimensions of matrix weight of Bu et al. (2025), we study the properties of matrix weight associated with $A$. Using the nice properties of matrix weight, we obtain that these spaces are equivalent with their corresponding averaging spaces and establish the discrete $\varphi$-transform of these spaces. Finally, we introduce matrix-weighted anisotropic Triebel-Lizorkin spaces for the limiting case $p=\infty$ and obtain their $\varphi$-transform. The relation between matrix-weighted anisotropic Triebel-Lizorkin spaces and matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces is also studied.

Assuming the log resolution conjecture for all log pairs birational to $X$, we prove the cone theorem for projective log canonical, $\mathbb{Q}$-factorial fourfold pairs $(X,\Delta)$ such that $K_X +\Delta\equiv M \geq 0$ over bases of positive and mixed characteristic $p > 5$.

In this paper, we study the loss of regularity for solutions to one-dimensional degenerate wave equations. We first consider the linear equation \( u_{tt}=\bigl(c(t,x)^2u_x\bigr)_x \) with \(c(t,x)\sim x^p\), and then the quasilinear equation \( u_{tt}=(u^{2a}u_x)_x. \) The initial data are assumed to satisfy \( u_0(x)\sim x^\alpha \) and \( u_1(x)\sim x^\beta \) near the degenerate point \(x=0\). In the linear problem, \(p\) describes the strength of the degeneracy, while \(\alpha\) and \(\beta\) describe the regularity of the initial data. In the quasilinear problem, \(\alpha\) also determines the initial degeneracy through \(u_0(x)^a\sim x^{a\alpha}\). Our main concern is the actual occurrence of loss of regularity, namely, the phenomenon in which the regularity of \(u(t,\cdot)\) near \(x=0\) becomes lower than that of \(u_0\) for \(t>0\). Previously, such a loss had mainly been established for special linear equations with coefficients depending only on time. In our previous work on the quasilinear equation, we proved local well-posedness without loss of regularity when \(\beta\geq\alpha\). In the present paper, we show that this condition is also necessary. More precisely, if \(\beta<\alpha\), then \( C_1tx^\beta\leq u(t,x)\leq C_2x^\beta \) near \(x=0\) for sufficiently small positive time. Thus the order of the solution changes from that of \(u_0\) to that of \(u_1\), and an actual loss of regularity occurs for both linear equations with time-space dependent coefficients and degenerate quasilinear equations.

We study Ollivier–Ricci curvature on Cayley graphs of groups of polynomial growth. Our main result shows that non-negative Ollivier–Ricci curvature forces the group to be virtually abelian. As an application, we prove that connected vertex-transitive graphs of polynomial growth and non-negative Ollivier–Ricci curvature are quasi-isometric to $\mathbb{Z}^k$, for some $k\in\mathbb{N}$.

We complete the construction of tracial spectral nonassociative $\mathrm{L}^p$-spaces for general $\mathrm{JBW}^*$-algebras. More precisely, if $\mathcal{M}$ is a $\mathrm{JBW}^*$-algebra equipped with a normal finite faithful trace $\tau$ and $1 \leq p < \infty$, we prove that $\|x\|_{\mathrm{L}^p(\mathcal{M})} \overset{\mathrm{def}}{=} (\tau[(x^* \circ x)^{\frac p2}])^{\frac1p}$, where $x \in \mathcal{M}$, defines a norm on $\mathcal{M}$. This resolves the remaining exceptional case left open by the corresponding result for $\mathrm{JW}^*$-algebras. The main difficulty is the complexified Albert algebra $\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})$, which admits no embedding into an associative operator algebra. To treat this case, we establish a Jordan analogue of the joint convexity of the Kiefer map $\mathrm{M}_n \times \mathrm{H}_n^{++} \to \mathrm{H}_n^{+}$, $(a,h) \mapsto a^*h^{-1}a$, where $\mathrm{H}_n$ is the space of Hermitian matrices, and a Jordan version of a variational formula of Carlen and Lieb. This provides a complex Banach space framework for Jordan-algebraic models arising in some generalized probabilistic theories.

As it is stated in "Unsolved problems in combinatorial games" as problem A13, Nim with pass is a difficult problem. Nim was one of the first combinatorial games to be solved but when we introduce a pass the game becomes far more complex. In this article we extend the use of the split sum introduced in "Investigations of Impartial Games With a Pass" and we find cases where the split sum act the same as an classical disjunctive sum. We also introduce generalization of Grundy value for the split sum and find that games of same Grundy values can give very different properties with a split sum.

We study the minimum-volume problem for finite-volume convex hyperbolic polyhedra whose dihedral angles are all equal to a fixed number \(\alpha\). In the non-obtuse case one necessarily has \[ \frac{\pi}{3}\le \alpha\le \frac{\pi}{2}. \] We prove that throughout the full range in which the regular hyperbolic tetrahedron with dihedral angle \(\alpha\) exists, \[ \frac{\pi}{3}\le \alpha<\arccos\frac13, \] it is the unique minimum-volume equiangular hyperbolic polyhedron with prescribed angle \(\alpha\). The left endpoint is the known ideal case, while at \(\alpha=\arccos(1/3)\) the regular tetrahedron degenerates to the Euclidean one. The proof combines Andreev's theorem and the Schläfli formula with Atkinson's decomposition into atoroidal and prismatic parts, explicit volume estimates for ordinary prisms and complete orthoschemes, and a direct equiangular version of Inoue's edge surgery.

We examine the modulus of continuity, in the spatial variable, of the local time process of Feller Brownian motion (FBM) on the half-line $[0,\infty)$. Briefly, a FBM is a strong Markov process on $[0,\infty)$ that moves like standard Brownian motion on $[0,\infty)$ up until it first encounters the state $0$. The process returns to $(0,\infty)$, either continuously (like reflecting Brownian motion) or by jumping to a (random) positive state chosen according to a specified measure. The present work is a continuation and application of our earlier work with Michael Marcus on the moduli of continuity for the local times of a Markov process built by piecing together (“rebirthing") the paths of another Markov process with finite lifetime. We first establish a general result on the resolvent and local times for a rebirthed process with a special holding state (the state $0$ for FBM). We show how our earlier approach using the Eisenbaum Isomorphism Theorem on an assemblage of excursions works out in this context. This knowledge is then used as an approximation device to obtain our main result on the exact uniform moduli of continuity for the local time of FBM on a spatial interval of the form $(0,1]$. Extensions are made, under certain conditions, to the more delicate situation of the spatial interval $[0,1]$. We also consider briefly the case of more general diffusions on $[0,\infty)$.

The Hessian conjecture asks whether a polynomial with nonzero constant Hessian determinant has a polynomial gradient inverse. It is known in dimensions at most three, false in dimensions at least five, and open in dimension four. We prove its four-variable quartic case. The top homogeneous part has zero Hessian determinant and, by the four-dimensional homogeneous Hesse theorem, is a cone. We divide its cone representative into three exhaustive types: a genuinely ternary quartic with nonzero ternary Hessian, a genuinely binary quartic, and a fourth power of a linear form. In the first type, the degree-seven determinant equation forces the cubic part to be affine-linear in the cone direction. In the binary type, the degree-six equation gives a constant null direction in a two-variable Hessian of the cubic part. In the unary type, the degree-five equation and a constant-direction lemma give the same conclusion. Every type therefore reduces to \[ f=P(x_1,x_2,x_3)+x_4Q(x_1,x_2,x_3)+a x_4^2, \qquad \deg Q\leq2. \] We prove, independently of the degree or top part of \(P\), that every constant-Hessian polynomial of this form has a polynomial gradient inverse. The branch \(a\ne0\) descends from the known three-dimensional Hessian conjecture after a Schur complement. When \(a=0\), an isotropic-cone rank analysis eliminates rank two, solves the rank-one exception by an explicit triangular inverse, and reduces rank zero to the two-dimensional Hessian conjecture. The coupled degree-six identity is retained throughout; no component with respect to a fixed quadratic form is separated.

Most models assess whether a newly introduced pathogen strain can grow when rare by testing it against a steady endemic background. Resident strains, however, can have more than one long-term behavior under the same parameters. They may settle to a steady state or continue through recurring outbreaks, leaving different fractions of hosts with different infection histories. We study this possibility in a stylized immune-history model. For two resident strains, we identify a parameter point where a stable endemic equilibrium and a stable outbreak cycle are both locally attracting. Numerical continuation maps a Hopf branch and a branch of folds of periodic orbits that bound a candidate bistability region, organized locally by a generalized Hopf (Bautin) point. We then embed the resident model exactly in a 20-state, three-strain system. When strain 3 infectiousness and total removal are common across four host-history classes, rare growth is determined by a weighted sum of the corresponding uninfected host fractions. The equilibrium and cycle yield different weighted sums, so one fixed strain 3 parameter set grows near one resident attractor but declines near the other. A second parameter set gives the reverse ordering, and both patterns persist over open regions of invader parameter space. Sixty full-model simulations with progressively smaller inocula reproduce these linear predictions across cycle phases and numerical solvers. Thus, knowing which resident strains are present may not suffice to predict initial invasion; the realized resident attractor can also matter. These results concern local growth from rarity, not eventual establishment, long-term coexistence, or post-invasion dynamics.

For $d \geq 2$, $p \geq 1$ and $\epsilon > 0$, let $N_p(d,\epsilon)$ be the smallest integer $N$ such that for every integer $n$ and every $A\in\mathbb{R}^{n\times d}$, there exists a matrix $\Phi\in\mathbb{R}^{N\times n}$ satisfying $(1-\epsilon)\lVert Ax\rVert_p\leq \lVert\Phi A x\rVert_p\leq (1+\epsilon)\lVert Ax\rVert_p$ for all $x\in\mathbb{R}^d$. For every constant $p\geq 1$ with $p\not\in 2\mathbb{Z}$, when $d\gtrsim_p \log(1/\epsilon)$, the bound \[ N_p(d,\epsilon) \gtrsim_{p} \frac{d}{\epsilon^2 \operatorname{polylog}(d/\epsilon)} \] is established. This improves the previous lower bound $\Omega(1/(\epsilon^2\operatorname{polylog}(1/\epsilon)))$ due to Li et al. (SICOMP 2021) and is optimal up to logarithmic factors for $1\leq p<2$. The central technical idea originated from ChatGPT 6 Pro.

Consider the action of a rational function on the Berkovich projective line over an algebraically closed field complete with respect to a nontrivial non-archimedean absolute value. We say a connected component of the fixed locus is a "non-attracting subtree" if it contains no attracting points. We define a "degree of a non-attracting subtree," in terms of the local degrees of the rational function at its points. We show that the degrees of non-attracting subtrees sum up to the degree of the rational function. This counting result is obtained by studying the potential of a discrete measure charging each non-attracting subtree with mass equal to its degree. We prove that as we iterate the rational function these measures equidistribute to the equilibrium measure. Under certain conditions, notably in the case of polynomials, this reduces to the equidistribution of repelling points. We also show that repelling type II points become negligible under iteration when the Lyapunov exponent is positive. These results give a partial and a complete answer to questions of Favre and Rivera-Letelier.

We study the structure of large $t$-intersecting families. A family of $k$-subsets of an $n$-set is $t$-intersecting if every two of its members intersect in at least $t$ elements. A $t$-intersecting family is non-trivial if no $t$-subset is contained in all its members. We prove several stability results for the seminal Hilton–Milner–Frankl theorem. First, for any fixed $\eta,\varepsilon,\theta\in(0,1)$, we prove that if $k/t\geq1+\eta$ and $n=\Omega(tk^{1+\varepsilon})$, then every non-trivial $t$-intersecting family of size greater than $(1+\theta)|\mathcal{K}|$ is a subfamily of one of the two extremal families in the theorem, where $\mathcal{K}$ is an explicit large non-trivial $t$-intersecting family. The key ingredient in the proof is a removal lemma. We also obtain a classification of all $t$-intersecting families with size bounded below by $|\mathcal{K}|$ minus an explicit lower-order term, provided that $k\geq t+4\geq6$ and $n\geq t+6\cdot\max\{(t+2)^2, k(k-t)\}$. This strengthens results of Cao–Lv–Wang (2021) and Frankl (2025) for a broad range of $k$ and $t$ (for example, when $k-t\geq2\sqrt{t}$). As an application of this classification, we determine the largest $t$-intersecting families for each prescribed lower bound on $t$-diversity not exceeding $t(n-k)$, thereby obtaining $t$-intersection versions of results of Han and Kohayakawa (2017) and Kupavskii (2025). To establish these results, we develop techniques based on the spread approximation method and the $t$-cover method, which may be useful for other intersection problems.

We prove that, for all positive integers $h$ and $t$ and every graph $X$ with $\mathrm{td}(X) \leq h$, there exists a positive integer $c(X,t)$ such that every graph $G$ with $\mathrm{tw}(G) < t$ that excludes $X$ as a topological minor is isomorphic to a subgraph of $H \boxtimes K_{c(X,t)}$ for some graph $H$ with $\mathrm{tw}(H) < 2^{h+1}-1$. This extends a result by Ding and Oporowski (Journal of Graph Theory; 1995), which states that for all positive integers $\Delta$ and $t$, here exists a positive integer $f(\Delta,t)$ such that every graph $G$ with $\mathrm{tw}(G)<t$ and $\Delta(G)\leq\Delta$ is isomorphic to a subgraph of $T \boxtimes K_{f(\Delta,t)}$ for some tree $T$.