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In this paper, we prove that for any bounded set of finite perimeter \Omega \subset \mathbb {R}^n, we can choose smooth sets E_k \Subset \Omega such that E_k \rightarrow \Omega in L^1 and \begin{align*} \limsup _{i \rightarrow \infty } P(E_i) \le P(\Omega)+C_1(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In the above \Omega ^1 is the measure-theoretic interior of \Omega, P(\cdot) denotes the perimeter functional on sets, and C_1(n) is a dimensional constant. Conversely, we prove that for any sets E_k \Subset \Omega satisfying E_k \rightarrow \Omega in L^1, there exists a dimensional constant C_2(n) such that the following inequality holds: \begin{align*} \liminf _{k \rightarrow \infty } P(E_k) \ge P(\Omega)+ C_2(n) \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1). \end{align*} In particular, these results imply that for a bounded set \Omega of finite perimeter, \begin{align*} \mathscr {H}^{n-1}(\partial \Omega \cap \Omega ^1)=0 \end{align*} holds if and only if there exists a sequence of smooth sets E_k such that E_k \Subset \Omega, E_k \rightarrow \Omega in L^1 and P(E_k) \rightarrow P(\Omega). [ABSTRACT FROM AUTHOR]

In this paper the authors introduce an analogue of the nilpotent cone, N, for a classical Lie superalgebra, g, that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g0 ⊕ g1 with Lie G0 = g0, it is shown that there are finitely many G0-orbits on N. Later the authors prove that the Duflo-Serganova commuting variety, X, is contained in N for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper. [ABSTRACT FROM AUTHOR]

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real (resp. complex) analytic spaces to \mathbb {R}^m (resp. \mathbb {C}^m). It is shown that the flatness of analytic maps from germs of real (resp. complex) analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection (resp. Cohen-Macaulay) analytic germs can be approximated arbitrarily well by polynomial (resp. Nash) maps in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney's tangent cone. [ABSTRACT FROM AUTHOR]

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras. [ABSTRACT FROM AUTHOR]

In this paper we study the motion of an internal water wave and an internal wave in a porous medium. For these problems we establish that, if the free boundary and, in the case of the Euler equations, also the tangential velocity at the interface are sufficiently smooth, the depth cannot vanish in finite time. This results holds regardless of gravity and surface tension effects or, if applicable, the stratification in multiphase flows. [ABSTRACT FROM AUTHOR]

In 1974, Askey and Steinig proved that for all n ≥ 0 and x ∈ (0,2π) the trigonometric sums sin (x/4)/1 + sin (5x/4)/2 ⋅⋅⋅ + sin ((4n+1)x/4)/n+1 and cos (x/4)/1 + cos (5x/4)/2 + ⋅⋅⋅ + cos ((4n+1)x/4)/n+1 are positive. Recently, the Askey-Steinig inequalities were improved by the present authors. In this paper, we further improve these inequalities and provide new sharp upper and lower bounds for the two sums given above. [ABSTRACT FROM AUTHOR]

The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we introduce the notion of 'finite splitting height' which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-1 subgroup H in G, we show that the Kropholler-Roller conjecture holds when H has finite splitting height in G. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include relatively quasiconvex subgroups of relatively hyperbolic groups with virtually polycyclic peripheral subgroups. In particular, our results extend Stallings' theorem and generalise a theorem of Sageev on decomposing a hyperbolic group by quasiconvex subgroups. [ABSTRACT FROM AUTHOR]

In this paper we show that evolution algebras over any given field \Bbbk are universally finite. In other words, given any finite group G, there exist infinitely many regular evolution algebras X such that Aut(X)\cong G. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme \mathbf {G} over \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra. [ABSTRACT FROM AUTHOR]

In this paper we describe the relationship between the finite free resolutions of perfect ideals in split format (for Dynkin formats) and certain intersections of opposite Schubert varieties with the big cell for homogeneous spaces G/P, where P is a maximal parabolic subgroup. [ABSTRACT FROM AUTHOR]

We consider the elliptic-elliptic Davey-Stewartson system in the three-dimensional Euclidean space, and we give sufficient conditions for the existence of finite time blow-up solutions in non-isotropic spaces. The proof is based on some general results on distributions defined via homogeneous symbols, in conjunction with a convexity argument. [ABSTRACT FROM AUTHOR]

We study some semi-linear equations for the (m,p)-Laplacian operator on locally finite weighted graphs. We prove existence of weak solutions for all m\in \mathbb {N} and p\in (1,+\infty) via a variational method already known in the literature by exploiting the continuity properties of the energy functionals involved. When m=1, we also establish a uniqueness result in the spirit of the Brezis–Strauss Theorem. We finally provide some applications of our main results by dealing with some Yamabe-type and Kazdan–Warner-type equations on locally finite weighted graphs. [ABSTRACT FROM AUTHOR]

Spaces with positive weights are those whose rational homotopy type admits a large family of "rescaling" automorphisms. We show that finite complexes with positive weights have many genuine self-maps. We also fix the proofs of some previous related results. [ABSTRACT FROM AUTHOR]

In this paper, we give a new proof of the Bessonov theorem [Proc. Amer. Math. Soc. 142 (2014), pp. 1301-1313] on finite rank truncated Toeplitz operators on model spaces via the product of Hankel operators on Hardy space. [ABSTRACT FROM AUTHOR]

A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré's upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting. [ABSTRACT FROM AUTHOR]

We study a family of Brascamp-Lieb forms acting on families of weighted Lp-spaces and Lorentz spaces where the weight is a power of the distance to the origin. We establish a set of necessary conditions and a set of sufficient conditions for the finiteness of these forms in Lorentz spaces. The conditions are close to optimal. [ABSTRACT FROM AUTHOR]

We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact \Bbbk \text {-}linear monoidal functor can be lifted to a tensor equivalence under some certain conditions. [ABSTRACT FROM AUTHOR]

Let S_1 and S_2 be disjoint finite unions of parallelepipeds. We describe necessary and sufficient conditions on the sets S_1,S_2 and exponents p such that the canonical projection P from PW_{S_1\cup S_2}^p to PW_{S_1}^p is a contraction. [ABSTRACT FROM AUTHOR]

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set A of integers with \min (A)=0 and \gcd (A)=1 there exist two sets, the "head" and the "tail", such that if m\ge \max (A)-|A|+2, then the m-fold sumset mA consists of the union of the head, the appropriately shifted tail, and a long block of consecutive integers separating them. We give sharp estimates for the length of the block, and find all those sets A for which the bound \max (A)-|A|+2 cannot be substantially improved. [ABSTRACT FROM AUTHOR]

We propose a notion of Frobenius-Perron dimension for certain free \mathbb {Z}-modules of infinite rank and compute it for the \mathbb {Z}-modules of finite dimensional complex representations of unitary groups with nonnegative dominant weights. The definition of Frobenius-Perron dimension that we are introducing naturally generalizes the well-known Frobenius-Perron dimension on the category of finite dimensional complex representations of a finite group. [ABSTRACT FROM AUTHOR]

Given a quadruple of finite index subfactors we explicitly compute the Pimsner-Popa probabilistic constant for the pair of intermediate subfactors and relate it with the corresponding Connes-Størmer relative entropy between them. This generalizes an old result of Pimsner and Popa [Ann. Sci. École Norm. Sup. (4) 19 (1986), pp. 57–106]. [ABSTRACT FROM AUTHOR]

Let G be a discrete group, VN(G) the von Neumann algebra generated by the left translation operators and A(G) the Fourier algebra of G. Then VN(G) can be considered as a subspace of \ell ^2(G) and A(G) = \ell ^2(G)*\ell ^2(G). The Banach space dual of A(G) is VN(G). The duality between \varphi \in VN(G) and u \in A(G) is \langle \varphi,u \rangle = \sum _{x \in G}{\varphi (x)u(x)}, if \varphi \in \ell ^{1}(G) or u \in \ell ^2(G); in these two cases, \varphi u \in \ell ^{1}(G). We will show that if G is infinite then there exist \varphi \in VN(G) and u \in A(G) such that \varphi u \notin \ell ^{1}(G). When G is countably infinite, this implies that there exist \varphi \in VN(G), u \in A(G) and an increasing sequence of finite subsets F_n, G = \bigcup F_n such that the sequence \sum _{x \in F_n}{\varphi (x)u(x)} is divergent and therefore \|{\chi _{F}}_n \|_{MA(G)} is unbounded. This leads us to give a preliminary study of the growth of \|\chi _{F}\|_{MA(G)} as |F|, the size of the finite subset F of G, is increasing. Recall that when G = \mathbb {Z}, the additive group of integers, \|\chi _{F}\|_{MA(\mathbb {Z})} = \frac {1}{2\pi }\int _0^{2\pi } \big |\sum _{k \in F} {e^{ikx}}\big | \, d{x}. [ABSTRACT FROM AUTHOR]

We show that the barrier function in Root's solution to the Skorokhod embedding problem is continuous and finite at every point where the target measure has no atom and its absolutely continuous part is locally bounded away from zero. [ABSTRACT FROM AUTHOR]

The Kronecker polynomial K(z) is a finite product of cyclotomic polynomials C_j(z). Any Kronecker polynomial K(z) of degree N+1 with simple roots on the unit circle generates a finite set \Phi _0=1, \Phi _1(z), ..., \Phi _N(z) of polynomials (para) orthogonal on the unit circle (POPUC). This set is determined uniquely by the condition \Phi _N(z) = (N+1)^{-1} K'(z). Such set can be called the set of Sturmian Kronecker POPUC. We present several new explicit examples of such POPUC. In particular, we define and analyze properties of the Sturmian cyclotomic POPUC generated by the cyclotomic polynomials C_M(z). Expressions of these polynomials strongly depend on the decomposition of M into prime factors. [ABSTRACT FROM AUTHOR]

We study the finite generation of homotopy groups of closed manifolds and finite CW-complexes by relating it to the cohomology of their fundamental groups. Our main theorems are as follows: when X is a finite CW-complex of dimension n and \pi _1(X) is virtually a Poincaré duality group of dimension \geq n-1, then \pi _i(X) is not finitely generated for some i unless X is homotopy equivalent to the Eilenberg–MacLane space K(\pi _1(X),1); when M is an n-dimensional closed manifold and \pi _1(M) is virtually a Poincaré duality group of dimension \ge n-1, then for some i\leq [n/2], \pi _i(M) is not finitely generated, unless M itself is an aspherical manifold. These generalize theorems of M. Damian [Trans. Amer. Math. Soc. 361 (2009), pp. 1791–1809] from polycyclic groups to any virtually Poincaré duality groups. [ABSTRACT FROM AUTHOR]

We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products, and affine Hopf algebras of low Gelfand-Kirillov dimension. [ABSTRACT FROM AUTHOR]

Let X be a finite connected poset and K a field. We give a full description of the Lie automorphisms of the incidence algebra I(X,K). In particular, we show that they are in general not proper. [ABSTRACT FROM AUTHOR]

A group \Gamma is said to be uniformly HS-stable if any map \varphi : \Gamma \to U(n) that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We present a complete classification of uniformly HS-stable groups among finitely generated residually finite ones. Necessity of the residual finiteness assumption is discussed. A similar result is shown to hold assuming only amenability. [ABSTRACT FROM AUTHOR]

Given an infinite countable discrete amenable group \Gamma, we construct explicitly sharply weak mixing nonsingular Poisson \Gamma-actions of each Krieger's type: III_\lambda, for \lambda \in [0,1], and II_\infty. The result is new even for \Gamma =\mathbb {Z}. As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli \Gamma-actions and infinite direct product of finite type systems of each possible Krieger's type. [ABSTRACT FROM AUTHOR]

A nilpotent Cantor action is a minimal equicontinuous action Φ : Γ × X → X on a Cantor space X, where Γ contains a finitely-generated nilpotent subgroup Γ0 ⊂ Γ of finite index. In this note, we show that these actions are distinguished among general Cantor actions: any effective action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application of this result, we obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence. [ABSTRACT FROM AUTHOR]

We classify principal blocks of finite groups with semidihedral defect groups up to splendid Morita equivalence. This completes the classification of all principal 2-blocks of tame representation type up to splendid Morita equivalence and shows that Puig's Finiteness Conjecture holds for such blocks. [ABSTRACT FROM AUTHOR]

We introduce the forcing property "almost strong properness" which sits between properness and strong properness. As an application, we introduce a simple forcing with finite conditions to force Mapping Reflection Principle. [ABSTRACT FROM AUTHOR]

Given a metric space (X,d), a set S ⊆ X is called a k-metric generator for X if any pair of different points of X is distinguished by at least k elements of S. A k-metric basis is a k-metric generator of the minimum cardinality in X. We prove that ultrametric spaces do not have finite k-metric bases for k >2. We also characterize when the metric and 2-metric bases of an ultrametric space are finite and, when they are finite, we characterize them. Finally, we prove that an ultrametric space can be easily recovered knowing only the metric basis and the coordinates of the points in it. [ABSTRACT FROM AUTHOR]

We give examples of pseudoconvex domains of finite type in C2 where the Kohn algorithm for subelliptic estimates fails to yield an effective lower bound for the order of subellipticity in terms of the type. We show how to modify the algorithm to obtain an effective procedure to prove subellipticity on domains of finite type in C2 with real analytic boundary satisfying a condition slightly stronger than pseudoconvexity. We close with a generalization to higher dimensions. [ABSTRACT FROM AUTHOR]

We consider a variant of the Hegselmann-Krause model of consensus formation where information between agents propagates with a finite speed c > 0. This leads to a system of ordinary differential equations (ODE) with state-dependent delay. Observing that the classical well-posedness theory for ODE systems does not apply, we provide a proof of global existence and uniqueness of solutions of the model. We prove that asymptotic consensus is always reached in the spatially one-dimensional setting of the model, as long as agents travel slower than c. We also provide sufficient conditions for asymptotic consensus in the spatially multidimensional setting. [ABSTRACT FROM AUTHOR]

Given a countable amenable group G, a Følner sequence (FN) ⊆ G, and a set E ⊆G with d(FN) (E) = lim supN→∞ |E ∩ FN|/|FN| > 0, Furstenberg's correspondence principle associates with the pair (E,(FN)) a measure preserving system X = (X,B,μ,(Tg)g∈G) and a set A ∈ B with μ (A) = d(FN)(E), in such a way that for all r ∈ N and all g1,⋅⋅⋅, gr ∈ G one has d(FN)(g1−1 E ∩ ⋅⋅⋅ ∩ gr−1 E) ≥ μ ((Tg1)−1 A ∩ ⋅⋅⋅ ∩ (Tgr)−1 A). We show that under some natural assumptions, the system X is unique up to a measurable isomorphism. We also establish variants of this uniqueness result for non-countable discrete amenable semigroups as well as for a generalized correspondence principle which deals with a finite family of bounded functions ƒ1, ⋅⋅⋅ , ƒ&#8467 : G → C. [ABSTRACT FROM AUTHOR]

We study the behaviour of the smallest possible constants dn and cn in Hardy's inequalities ∑k=1n 1/k ∑j=1k aj)2 ≤ dn ∑k=1n ak 2, (a1, . . . ,an) ∈ Rn and ∈t0∞ 1/x∈t0x ƒ(t)dt)2 dx ≤ cn ∈t0∞ ƒ2(x) dx, ƒ∈ Hn, for the finite dimensional spaces Rn and Hn := ƒ : ∈t0x ƒ(t) dt = e-x/2p(x) : p∈ Pn, p(0)= 0, where Pn is the set of real-valued algebraic polynomials of degree not exceeding n. The constants dn and cn are identified to be expressed in terms of the smallest zeros of the so-called continuous dual Hahn polynomials and the two-sided estimates for dn and cn of the form 4 - c/ln n< dn, cn < 4 - c/ln2 n, c > 0 are established. [ABSTRACT FROM AUTHOR]

We prove that pushouts A*CB of residually finite-dimensional (RFD) C*-algebras over central subalgebras are always residually finite-dimensional provided the fibers Ap and Bp, p ∈ spec C are RFD, recovering and generalizing results by Korchagin and Courtney-Shulman. This then allows us to prove that certain central pushouts of amenable groups have RFD group C*-algebras. Along the way, we discuss the problem of when, given a central group embedding H ≤ G, the resulting C*-algebra morphism is a continuous field: this is always the case for amenable G but not in general. [ABSTRACT FROM AUTHOR]

In this short note we answer a query of Brodzki, Niblo, Špakula, Willett and Wright [J. Noncommut. Geom. 7 (2013), pp. 583-603] by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser [Trans. Amer. Math. Soc. 372 (2019), pp. 2855-2874] proved that if Γ is a finitely generated group and {Hi}∞i=1 is a Farber sequence of finite index subgroups, then the associated Schreier graph sequence is of Property A if and only if the group is amenable. We show however, that there exist a non-amenable group and a nested sequence of finite index subgroups {Hi}∞i=1 such that ∩Hi = eΓ, and the associated Schreier graph sequence is of Property A. [ABSTRACT FROM AUTHOR]

We prove the local mixing theorem for geodesic flows on abelian covers of finite volume hyperbolic surfaces with cusps, which is a continuation of the work [Hee Oh and Wenyu Pan, Int. Math. Res. Not. 19 (2019), pp. 6036-6088]. We also describe applications to counting problems and the prime geodesic theorem. [ABSTRACT FROM AUTHOR]

In this note we give an alternative proof of existence of finite energy weak solutions to the quantum hydrodynamics (QHD) system. The main novelty in our approach is that no regularization procedure or approximation is needed, as it is only based on the integral formulation of NLS equation and the a priori bounds given by the Strichartz estimates. The main advantage of this proof is that it can be applied to a wider class of QHD systems. [ABSTRACT FROM AUTHOR]

We prove that an iterated function system of similarities on R that satisfies the weak separation condition and has an interval as its self-similar set is of generalized finite type. It is unknown if the assumption that the self-similar set is an interval is necessary. [ABSTRACT FROM AUTHOR]

Let A be a type 1 subdiagonal algebra in a σ-finite von Neumann algebra M with respect to a faithful normal conditional expectation Φ. We give necessary and sufficient conditions for which A is maximal among the σ-weakly closed subalgebras of M. In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of Arveson's finiteness problem in 1967 in type 1 case. [ABSTRACT FROM AUTHOR]

A group G is said to be an involutive Yang-Baxter group, or simply an IYB-group, if it is isomorphic to the permutation group of an involutive, nondegenerate set-theoretic solution of the Yang-Baxter equation. We give new sufficient conditions for a group that can be factorised as a product of two IYB-groups to be an IYB-group. Some earlier results are direct consequences of our main theorem. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Joshi and Zhang on eigenform product identities for Hilbert modular forms of full level. Namely, we prove that for a fixed positive integer n, there are finitely many eigenform product identities over all totally real number fields of degree n and all Hilbert eigenforms of weight 2 or greater and of full level. [ABSTRACT FROM AUTHOR]

We establish a lower bound for the Donaldson-Futaki invariant of optimal degenerations produced by the Kähler-Ricci flow in terms of the greatest Ricci lower bound on arbitrary Fano manifolds. As an application, we can generalize the finiteness of the Futaki invariants on Kähler-Ricci solitons obtained by Guo-Phong-Song-Sturm to the space of all Fano manifolds. Also, we discuss the relation to Hisamoto's inequality for the infimum of the H-functional. [ABSTRACT FROM AUTHOR]

In this note, we prove that if Λ is an Artin algebra with a simple module S of finite projective dimension, then the finiteness of the finitistic dimension of Λ implies that of (1−e)Λ (1−e) where e is the primitive idempotent supporting S. We derive some consequences of this. In particular, we recover a result of Green-Solberg-Psaroudakis: if Λ is the quotient of a path algebra by an admissible ideal I whose defining relations do not involve a certain arrow α, then the finitistic dimension of Λ is finite if and only if the finitistic dimension of Λ/Λ α Λ is finite. [ABSTRACT FROM AUTHOR]

The Frobenius test exponent Fte(R) of a local ring (R,m) of prime characteristic p > 0 is the smallest e0 ∈ N such that for every ideal q generated by a (full) system of parameters, the Frobenius closure qF has (qF )[pe0 ] = q[pe0 ]. We establish a sufficient condition for Fte(R) < ∞ and use it to show that if R is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e., Hjm(R)/0FHjm(R)is finite length for 0 ≤ j < dim(R), then Fte(R) < ∞. [ABSTRACT FROM AUTHOR]

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core, and Alexander quandles) is established. Further, residual finiteness of automorphism groups of some residually finite quandles is also discussed. [ABSTRACT FROM AUTHOR]

The author establishes the finite generation of the cohomology ring of a classical Lie superalgebra relative to an even subsuperalgebra. A spectral sequence is constructed to provide conditions for when this relative cohomology ring is Cohen-Macaulay. With finite generation established, support varieties for modules are defined via the relative cohomology, which generalize those of Boe, Kujawa, and Nakano [Trans. Amer. Math. Soc. 262 (2010), no. 12, 6551-6590]. [ABSTRACT FROM AUTHOR]