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Potentialism is the view that objects are successively generated in an incompletable process. A strict version of the view adds that truths are successively determined. Strict potentialism can be analyzed using two modalities: one for the generation of objects, another for truths becoming determined. The result is a classical bimodal logic. We obtain simpler and more user-friendly theories by invoking so-called mirroring theorems to ``switch off'' one or both modalities, in return for a less classical logic. When the modality of object generation is switched off, we obtain a restricted plural logic. When the modality of truth determination is switched off, the logic becomes intuitionistic. Finally, the value of this general approach to strict potentialism is illustrated by applications to a Weyl-inspired predicative set theory, Cantor's domain principle, and strict potentialism about Cantorian sets.

We utilize multi-virtual knot theory where there are a multiplicity of virtual crossings to study strict virtual linkoids. In strict virtual linkoid theory, local moves define all virtual moves and Reidemeister moves. In the strict equivalence, no moves, classical or virtual, can transfer an arc across a linkoid endpoint. By taking closures of strict virtual linkoids that are multi-virtual knots and links, we obtain new invariants for strict virtual linkoids. Generalized bracket polynomial invariants and generalized loop bracket polynomial invariants (for planar strict virtual linkoids) are studied in this context. The paper defines virtual polar links where there are degree two nodes in virtual link diagrams across which isotopies are forbidden. The paper shows how multi-virtual theory and its concepts can be applied to obtain invariants for polar virtual links.

The construction of Arf rings and strictly closed rings has been studied widely; however, there has been no clear description of the structure of the strict closure R^* when the integral closure of R is not a finitely generated R-module. In this paper, we investigate the construction and finite generation of the strict closure of rings. We determine its structure when R is a Cohen-Macaulay semi-local ring of dimension one, with dim R_M=1 for every Maximal ideal M in R. Using this, a characterization of the finite generation of the strict closure is given.

This paper investigates the finite generation of the strict closure of rings in arbitrary dimension. For a Noetherian local ring $(R, \mathfrak{m})$, we provide a sufficient condition under which the strict closure $R^*$ is finitely generated as an $R$-module. Using this result, we characterize the finite generation of the strict closure over excellent rings.

We define strict C(n) small-cancellation complexes, intermediate to C(n) and C(n+1), and we prove groups acting properly cocompactly on a simply-connected strict C(6) complex are hyperbolic relative to a collection of maximal virtually free abelian subgroups of rank 2. We study geometric walls in a simply-connected strict C(6) complex, and we use them to prove a convex cocompact (cosparse) core theorem for (relatively) quasiconvex subgroups of strict C(6) groups. We provide an examples showing the convex cocompact core theorem is false without the strict C(6) assumption.

In characteristic zero, the category of strict polynomial functors is well-known to be the tensor abelian category freely generated by one object. We show that this property fails in positive characteristic, but that it can be repaired by restricting the class of tensor abelian categories considered. The new universal property recovers several known constructions and shows that Ext-algebras of strict polynomial functors act on cohomological computations in many other contexts.

Given commuting $d$-tuples $\mathbb{S}_i$ and $\mathbb{T}_i$, $1\leq i\leq 2$, Banach space operators such that the tensor products pair $(\mathbb{S}_1\otimes\mathbb{S}_2,\mathbb{T}_1\otimes\mathbb{T}_2)$ is strict $m$-isometric (resp., $\mathbb{S}_1$, $\mathbb{S}_2$ are invertible and $(\mathbb{S}_1 \otimes \mathbb{S}_2, \mathbb{T}_1 \otimes\mathbb{T}_2)$ is strict $m$-symmetric), there exist integers $m_i >0$, and a non-zero scalar $c$, such that $m=m_1+m_2-1$, $(\mathbb{S}_1, {\frac{1}{c}}\mathbb{T}_1)$ is strict $m_1$-isometric and $(\mathbb{S}_2, c\mathbb{T}_2)$ is strict $m_2$-isometric (resp., there exist integers $m_i >0$, and a non-zero scalar $c$, such that $m=m_1+m_2-1$, $(\mathbb{S}_1,{\frac{1}{c}}\mathbb{T}_1)$ is strict $m_1$-symmetric and $(\mathbb{S}_2, c\mathbb{T}_2)$ is strict $m_2$-symmetric. However, $(\mathbb{S}_i,\mathbb{T}_i)$ is strict $m_i$-isometric (resp., strict $m_i$-symmetric) for $1\leq i\leq 2$ implies only that $(\mathbb{S}_1\otimes \mathbb{S}_2, \mathbb{T}_1\otimes \mathbb{T}_2)$ is $m$-isometric (resp., $(\mathbb{S}_1 \otimes \mathbb{S}_2, \mathbb{T}_1\otimes\mathbb{T}_2)$ is $m$-symmetric).

Varkonyi and Domokos (2006) proved that convex homogeneous bodies with exactly one stable and one unstable equilibrium point exist. Sloan (2023) gave the first analytical parameterization, with radial function $R(θ,φ)$ having exactly two critical points on $S^2$. This is the v2 amendment-of-record of arXiv:2604.17120. v1 claimed Sloan's parameterization does not produce mono-monostatic bodies and reported a 13-member catalog of Fourier/radial extensions certified at ECS=1 via mesh-vertex drainage-basin analysis. Following correspondence with P. L. Varkonyi (BME), an analytical verification suite was built around the Varkonyi-Gauss identity. Finding 1: Sloan's parameterization does produce mono-monostatic bodies in a strictly-convex sub-regime ($β\lesssim 0.036$), where $K_{\min} > 0$ and the identity certifies ECS=1. v1 missed this because its mesh-vertex oracle over-counted on shallow COM-height landscapes. At Sloan's published $β=0.05$, strict convexity is lost ($K_{\min}=-0.569$; $K<0$ over 4.01% of surface); the identity's precondition fails. v1's "global surface information" mechanism is replaced by the strict-convexity precondition. Finding 2: Of v1's 13 catalog instanc

We study the strict stability of calibrated cones with an isolated singularity. For special Lagrangian cones and coassociative cones, we prove the strict stability. In the complex case, we give non-strictly stable examples.

The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation.

We construct a van Est map for strict Lie 2-groups from the Bott-Shulman-Stasheff double complex of the strict Lie 2-group to the Weil algebra of its associated strict Lie 2-algebra. We show that, under appropriate connectedness assumptions, this map induces isomorphisms in cohomology. As an application, we differentiate the Segal 2-form on the loop group.

It is an old idea to use gradient flows or time-discretized variants thereof as methods for solving minimization problems. In some applications, for example in machine learning contexts, it is important to know that for generic initial data, gradient flow trajectories do not get stuck at saddle points. There are classical results concerned with the non-degenerate situation. But if the Hessian of the objective function has a non-trivial kernel at the critical point, then these results are inconclusive in general. In this paper, we show how relevant information can be extracted by ``blowing up'' the objective function around the non-strict saddle point, i.e., by a suitable non-linear rescaling that makes the higher order geometry visible.

We prove that $i)$ if $\mathcal{A}$ is $λ$-accessible and it is axiomatizable in (finitary) coherent logic then $λ$-pure maps are strict monomorphisms and $ii)$ if there is a proper class of strongly compact cardinals and $\mathcal{A}$ is $λ$-accessible then for some $μ\vartriangleright λ$ every $μ$-pure map is a strict monomorphism.

Currently, the simplex method and the interior point method are indisputably the most popular algorithms for solving linear programs, LPs. Unlike general conic programs, LPs with a finite optimal value do not require strict feasibility in order to establish strong duality. Hence strict feasibility is seldom a concern, even though strict feasibility is equivalent to stability and a compact dual optimal set. This lack of concern is also true for other types of degeneracy of basic feasible solutions in LP. In this paper we discuss that the specific degeneracy that arises from lack of strict feasibility necessarily causes difficulties in both simplex and interior point methods. In particular, we show that the lack of strict feasibility implies that every basic feasible solution, BFS, is degenerate; thus conversely, the existence of a nondegenerate BFS implies that strict feasibility (regularity) holds. We prove the results using facial reduction and simple linear algebra. In particular, the facially reduced system reveals the implicit non-surjectivity of the linear map of the equality constraint system. As a consequence, we emphasize that facial reduction involves two steps where, the

A strict bramble of a graph $G$ is a collection of pairwise-intersecting connected subgraphs of $G.$ The order of a strict bramble ${\cal B}$ is the minimum size of a set of vertices intersecting all sets of ${\cal B}.$ The strict bramble number of $G,$ denoted by ${\sf sbn}(G),$ is the maximum order of a strict bramble in $G.$ The strict bramble number of $G$ can be seen as a way to extend the notion of acyclicity, departing from the fact that (non-empty) acyclic graphs are exactly the graphs where every strict bramble has order one. We initiate the study of this graph parameter by providing three alternative definitions, each revealing different structural characteristics. The first is a min-max theorem asserting that ${\sf sbn}(G)$ is equal to the minimum $k$ for which $G$ is a minor of the lexicographic product of a tree and a clique on $k$ vertices (also known as the lexicographic tree product number). The second characterization is in terms of a new variant of a tree decomposition called lenient tree decomposition. We prove that ${\sf sbn}(G)$ is equal to the minimum $k$ for which there exists a lenient tree decomposition of $G$ of width at most $k.$ The third characterizatio

Recently, Armstrong, Guzmán, and Sing Long (2021), presented an optimal $O(n^2)$ time algorithm for strict circular seriation (called also the recognition of strict quasi-circular Robinson spaces). In this paper, we give a very simple $O(n\log n)$ time algorithm for computing a compatible circular order for strict circular seriation. When the input space is not known to be strict quasi-circular Robinson, our algorithm is complemented by an $O(n^2)$ time verification of compatibility of the returned order. This algorithm also works for recognition of other types of strict circular Robinson spaces known in the literature. We also prove that the circular Robinson dissimilarities (which are defined by the existence of compatible orders on one of the two arcs between each pair of points) are exactly the pre-circular Robinson dissimilarities (which are defined by a four-point condition).

We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to be strictifiable. Our counterexample comes from higher category theory: the strict algebras are strict 3-categories, and the pseudoalgebras are a type of semi-strict 3-category lying in between Gray-categories and tricategories. Thus, the result follows from the fact that not every Gray-category is equivalent to a strict 3-category, connecting 2-categorical and higher-categorical coherence theory. In particular, any nontrivially braided monoidal category gives an example of a pseudoalgebra that is not equivalent to a strict one.

We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary and sufficient conditions under which a convex combination and composition of strict pseudocontractions as well as demicontractions that share a common fixed point is again a strict pseudocontraction or a demicontraction, respectively. Moreover, we introduce a generalized relaxation of composition of demicontraction and give its properties. We apply these properties to prove the weak convergence of a class of algorithms that is wider than the Douglas-Rachford algorithm and projected Landweber algorithms. We have also presented two numerical examples, where we compare the behavior of the presented methods with the Douglas-Rachford method.

We attack the problem of getting a strict ranking (i.e. a ranking without equally ranked items) of $n$ items from a pairwise comparisons matrix. Basic structures are described, a first heuristical approach based on a condition, the $\mathcal{R}-$condition, is proposed. Analyzing the limits of this ranking procedure, we finish with a minimization problem which can be applied to a wider class of pairwise comparisons matrices. If solved, it produces consistent pairwise comparisons that produce a strict ranking.