We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analog of the Plücker ring studied classically in invariant theory. We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gröbner basis for the grove ideal.

Building on work of Furstenberg and Tzkoni, we introduce $\mathbf {r}$-flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermassintegrals as averages on flag manifolds (where the Grassmannian can be considered as a special case). We establish affine and linear invariance properties and extend fundamental results to this new setting. In particular, we prove several affine isoperimetric inequalities from convex geometry and their approximate reverse forms. We also introduce functional forms of these quantities and establish corresponding inequalities.

The main theme of this paper is to study $\tau $-tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $-cotorsion torsion triples and investigate a bijection between the collection of $\tau $-cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $-tilting subcategories of $\mathscr {A}$, generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$. General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$, where R is a unitary associative ring, we characterize all support $\tau $-tilting (resp. all support $\tau ^-$-tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support $\tau $-tilting (resp. support $\tau ^{-}$-tilting) subcategory of $\mathrm{Mod}\mbox {-}R$. We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$, where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $-tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $-tilting subcategories of $\mathscr {A}$.

Hearts of cotorsion pairs on extriangulated categories are abelian categories. On the other hand, hearts of twin cotorsion pairs are not always abelian. They were shown to be semi-abelian by Liu and Nakaoka. Moreover, Hassoun and Shah proved that they are quasi-abelian under certain conditions. In this article, we first show that the heart of any twin cotorsion pair has a largest exact category structure and is always quasi-abelian. We also provide a sufficient and necessary condition for the heart of a twin cotorsion pair being abelian. Then by using the results we have got, we investigate the almost split sequences in the hearts of twin cotorsion pairs. Finally, as an application, we show that a Krull–Schmidt, Hom-finite triangulated category has a Serre functor whenever it has a cluster tilting object.

The main objective of the present paper is to present a version of the Tannaka–Krein type reconstruction theorems: if $F:{\mathcal B}\to {\mathcal C}$ is an exact faithful monoidal functor of tensor categories, one would like to realize ${\mathcal B}$ as category of representations of a braided Hopf algebra $H(F)$ in ${\mathcal C}$. We prove that this is the case iff ${\mathcal B}$ has the additional structure of a monoidal ${\mathcal C}$-module category compatible with F, which equivalently means that F admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fiber functors, and we give some applications. One particular motivation was the logarithmic Kazhdan–Lusztig conjecture.

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, k-point configuration sets given by

$$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$

$E\subset \mathbb {R}^d$

$k\ge 3$

$\Delta ^{\mathrm{diag}}(E)$

(0.1)

$$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$

$C^\omega $

Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.

The local invariants of a meromorphic quadratic differential on a compact Riemann surface are the orders of zeros and poles, and the residues at the poles of even orders. The main result of this paper is that with few exceptions, every pattern of local invariants can be obtained by a quadratic differential on some Riemann surface. The exceptions are completely classified and only occur in genera zero and one. Moreover, in the case of a nonconnected stratum, we show that, with three exceptions in genus one, each configuration of invariants can be realized in each non-hyperelliptic connected component of the stratum. In the hyperelliptic components with two poles the residues at both poles coincide. These results are obtained using the flat metric induced by the differentials. We give an application by bounding the number of disjoint cylinders on a primitive quadratic differential.

The meridional rank conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres.

We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$-polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the h-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.

In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the continuous setting. The results present interactions between discrete geometry, geometric measure theory, and graph theory.

Our goal of the paper is to investigate the Waring problem for upper triangular matrix algebras, which gives a complete solution of a conjecture proposed by Panja and Prasad in 2023.

We prove some conditions for higher-dimensional algebraic fibering of pro-p group extensions, and we establish corollaries about incoherence of pro-p groups. In particular, if $1 \to K \to G \to \Gamma \to 1$ is a short exact sequence of pro-p groups, such that $\Gamma $ contains a finitely generated, non-abelian, free pro-p subgroup, K a finitely presented pro-p group with N a normal pro-p subgroup of K such that $K/ N \simeq \mathbb {Z}_p$ and N not finitely generated as a pro-p group, then G is incoherent (in the category of pro-p groups). Furthermore, we show that if K is a finitely generated, free pro-p group with $d(K) \geq 2$, then either $\mathrm{Aut}_0(K)$ is incoherent (in the category of pro-p groups) or there is a finitely presented pro-p group, without non-procyclic free pro-p subgroups, that has a metabelian pro-p quotient that is not finitely presented, i.e., a pro-p version of a result of Bieri–Strebel does not hold.

Given an affine Coxeter group W, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan–Lusztig cells for W. Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W. Low elements in W were introduced to study the word problem of the corresponding Artin–Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in W. In this article, we show, in the case of an affine Coxeter group, that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a by-product of our proof, we show that the descent walls – the walls that separate a region from the fundamental alcove – of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element.

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann–Shannon entropy, as the noise parameter $\varepsilon $ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon \log (1/\varepsilon )+O(\varepsilon )$ for some explicit dimensional constants C depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or locally semiconcave costs for a finer estimate, and lower bounds for $\mathscr {C}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for nondegenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

Finite Cartesian products of operators play a central role in monotone operator theory and its applications. Extending such products to arbitrary families of operators acting on different Hilbert spaces is an open problem, which we address by introducing the Hilbert direct integral of a family of monotone operators. The properties of this construct are studied, and conditions under which the direct integral inherits the properties of the factor operators are provided. The question of determining whether the Hilbert direct integral of a family of subdifferentials of convex functions is itself a subdifferential leads us to introducing the Hilbert direct integral of a family of functions. We establish explicit expressions for evaluating the Legendre conjugate, subdifferential, recession function, Moreau envelope, and proximity operator of such integrals. Next, we propose a duality framework for monotone inclusion problems involving integrals of linearly composed monotone operators and show its pertinence toward the development of numerical solution methods. Applications to inclusion and variational problems are discussed.

We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.