Motivated by the foundational result that a monomial complete intersection has the strong Lefschetz property (SLP) in characteristic zero, it is natural to ask when monomial almost complete intersections have the SLP. In this paper, using the Hilbert series as a central tool, we investigate the strong Lefschetz property for certain monomial almost complete intersections: those with the non-pure-power generator having support in two variables, and those with symmetric Hilbert series. In the former case, we give a complete classification for when the SLP holds, and in the latter case, we prove that such algebras always have the SLP.
In this article, we address a question raised in Dao et al. (Trans Am Math Soc Ser B 10:355–380, 2023) regarding the preservation of reflexivity under taking trace. We answer this question negatively. We also study a few cases where the question has a positive answer in a one dimensional analytically unramified local Cohen-Macaulay ring.
This paper establishes new rigidity theorems for complete spacelike xi-translator immersed in the pseudo-Euclidean space R-p(n+p) . By imposing suitable upper bounds on the norm parallel to xi parallel to or geometric constraints on the norm of the second fundamental form parallel to A parallel to, we apply generalized maximum principles due to Da Silva, Lima Jr. and de Lima (Arch Math 118:663-673, 2022), Chen and Qiu (Adv Math 294:517-531, 2016), and Alias, Caminha and Nascimento (J Math Anal Appl 474: 242-247, 2019). These conditions force the vanishing of the second fundamental form, implying that any such xi-translator is necessarily a spacelike hyperplane of Rn+p (p) .
Consider a bounded strictly pseudoconvex domain Ω with smooth boundary ∂Ω . In this paper, we establish the higher-order Taylor expansion in terms of the horizontal vector fields via the Folland–Stein approximation of ∂Ω by the Heisenberg group in terms of the vector fields, and then provide an explicit construction of the Alpert wavelet with general higher-order cancellation. These results are of independent interest. As a direct application of this Alpert wavelet, we obtain the characterization of the endpoint Schatten estimate for the commutator of the Cauchy–Szegö projection.
Let α∈ (0,n) . In this article, the authors study the Lorentz properties of the fractional commutator [b,I_α ] which is generated by the Riesz potential and a symbol b. Moreover, the equivalent characterizations of the boundedness of [b, I_α] on Lorentz spaces in terms of b ∈BMO_β _1(ℝ^n) , and the Lorentz compactness characterizations via b ∈CMO_β _2(ℝ^n) are also established, where β _1∈ [0,1] , β _2∈ [0,1) , and α +β _i∈ (0,n) for i = 1,2 .
We construct affine charts of a smooth projective toric variety which contain its nonnegative points, and which admit a closed embedding into the total coordinate space of Cox's quotient construction. We show that such positive charts arise from smooth subcones of the nef cone. To each positive chart we associate an algebraic moment map, the fibers of which are the critical points of a monomial function in Cox coordinates. This work provides a toric framework for the theory of u-equations in positive geometry.
We obtain weighted estimates in terms of the multilinear Fujii-Wilson constant introduced by Nieraeth (PhD thesis, Delft University of Technology (2020); arXiv preprint arXiv:2401.15725 (2024)) for multilinear fractional maximal function and multilinear fractional integral operators. Our Theorem 1.1 provides a positive answer to the conjecture posed in Nieraeth (Conjecture 3.3.6, PhD thesis, Delft University of Technology (2020)). Furthermore, the weighted estimates yield classical A_P⃗,q estimates. In Lerner and Moen (Theorem 1.6, Studia Math. 219(3), 247–267 (2013)) and Cruz-Uribe and Moen (Theorem 2.5, Integral Equations Operator Theory 76(3), 421–446 (2013)), by means of extrapolation, the weighted estimates for multilinear Calderón-Zygmund operators and multilinear fractional integral operators were obtained. Without extrapolation we also provide the weighted estimates for multilinear fractional integral operators.
Let X_n be the projective plane blown up at n ⩾ 10 general points. In this paper we give several consequences of the Segre–Harbourne–Gimigliano–Hirschowitz Conjecture, that pertain to complete linear systems on X_n . We begin by classifying such systems |C| with general irreducible member of genus g ⩾ 2 (up to Cremona equivalence), in terms of invariants of the adjoint systems |C+mK| . We then use this to prove that, for fixed n ⩾ 10 and g⩾ 2 , up to the action of the Cremona group, there exist finitely many complete linear systems on X_n whose general member is irreducible of genus g. Further, there is a function g↦ n(g) such that every such (effective) system is Cremona equivalent to a system in X_n(g) . The latter result is based on the explicit computation of the minimum possible self-intersection of an irreducible linear system with given n and (|C|) . We classify those systems which achieve the minimal self-intersection. We also classify the systems with C^2 ⩽ 5 , whether or not they have minimal C^2 for the given n and dimension. We finish by proving several statements concerning systems that are base-point-free, and systems that give birational maps to their image.
We study the constant 𝒞_d,p defined as the smallest constant C such that ‖ P‖ _∞ ^p ≤ C‖ P‖ _p^p holds for every polynomial P of degree d, where we consider the L^p norm on the unit circle. We conjecture that 𝒞_d,p≤ dp/2+1 for all p ≥ 2 and all degrees d. We show that the conjecture holds for all p ≥ 2 when d ≤ 4 and for all d when p ≥ 6.8.
It is well-known that a function on an open set in ℝ^d is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance a real analytic version) to suitable closed sets, notably, sets with Hölder boundary and fat subanalytic sets satisfying a necessary topological condition. In this paper, we prove that the resulting set-theoretic identities of function spaces are bornological isomorphisms with respect to their natural locally convex topologies. Extending the results to maps with values in convenient vector spaces, we obtain corresponding exponential laws. Additionally, we show analogous results for special ultradifferentiable Braun–Meise–Taylor classes.
The v-number of a graded ideal is an invariant recently introduced in the context of coding theory, particularly in the study of Reed–Muller-type codes. In this work, we study the localized v-numbers of a binomial edge ideal J_G associated to a finite simple graph G. We introduce a new approach to compute these invariants, based on the analysis of transversals in families of subsets arising from dependencies in certain rank-two matroids. This reduces the computation of localized v-numbers to the determination of the radical of an explicit ideal and provides upper bounds for these invariants. Using this method, we explicitly compute the localized v-numbers of J_G at the associated minimal primes corresponding to minimal cuts of G. Additionally, we determine the v-number of binomial edge ideals for cycle graphs and give an almost complete answer to a conjecture from Dey et al. (Int J Algebra Comput 35(01):119–143, 2024), showing that the v-number of a cycle graph C_n is either ⌈2n/3⌉ or ⌈2n/3⌉ - 1 .
Two properties of projective hypersurfaces related to the module of Jacobian derivations, namely being tame and being plus-one generated, are discussed in this paper. Tame hypersurfaces are related to Bourbaki ideals, and free hypersurfaces are the simplest examples of tame hypersurfaces. Plus-one generated hypersurfaces are the non free hypersurfaces which are closest to the free ones, and it is an open question whether all of them are tame.
This article investigates under which conditions the symbolic powers of the extension of an ideal is the same as the extension of the symbolic powers. Our result generalizes the known scenarios. As an application, we prove formulas for the resurgence of sum of two homogeneous ideals in finitely generated k-algebra domains, where k is algebraically closed. Initially, these were known for ideals in polynomial rings.
We introduce the notion of a partial Betti splitting of a homogeneous ideal, generalizing the notion of a Betti splitting first given by Francisco, Hà, and Van Tuyl. Given a homogeneous ideal I and two ideals J and K such that I = J+K , a partial Betti splitting of I relates some of the graded Betti numbers of I with those of J, K, and J∩ K . As an application, we focus on the partial Betti splittings of binomial edge ideals. Using this new technique, we generalize results of Saeedi Madani and Kiani related to binomial edge ideals with cut edges, we describe a partial Betti splitting for all binomial edge ideals, and we compute the total second Betti number of binomial edge ideals of trees.
We show that Veronese varieties of dimension n ≥ 4 do not carry any Ulrich bundles of rank r ≤ 3. In order to prove this, we prove that a Veronese embedding of a complete intersection of dimension m ≥ 4, which if m=4 is either ℙ^4 or has degree d ≥ 2 and is very general and not of type (2), (2,2), does not carry any Ulrich bundles of rank r ≤ 3.
Consider a sequence of cadlag processes {X^n}_n , and some fixed function f. If f is continuous then under several modes of convergence X^n→ X implies corresponding convergence of f(X^n)→ f(X) , due to continuous mapping. We study conditions (on f, {X^n}_n and X) under which convergence of X^n→ X implies [ f(X^n)-f(X)] → 0 . While interesting in its own right, this also directly relates (through integration by parts and the Kunita–Watanabe inequality) to convergence of integrators in the sense ∫ _0^t Y_s-df(X^n_s)→∫ _0^t Y_s-df(X_s) . We show stability when f∈ C^1 , {X^n}_n,X are Dirichlet processes defined as in Coquet et al. (J Theor Probab 16:197, 2023) X^nX , [X^n-X]0 and {(X^n)^*_t}_n is bounded in probability. We also relax the conditions on f to being the primitive function of a cadlag function but with the additional assumption on X and that the continuous and discontinuous parts of X are independent stochastic processes (this assumption is not imposed on {X^n}_n however). For this setting we prove a new Itô decomposition that is a refinement of the one found in Coquet et al. (2023).
We classify (up to quasi-isomorphism) the free differential modules whose homology is equal to a given module M by developing a theory for deforming an arbitrary free complex into a differential module. We use an iterative approach to parameterize the deformations and obstructions in terms of certain Ext groups, giving an algorithmic realization of a result of Brown-Erman. We apply this theory to study certain rigidity properties of free resolutions and related rank conjectures.
In this paper, we study weighted Besov spaces ℬ^ω _p . By using Hardy inequalities, we derive significant weighted inequalities. Namely, we establish a Douglas-type formula for ℬ^ω _p and identify an equivalent norm that depends only on the modulus of the function.
In this paper, we investigate a class of generalized McKean–Vlasov stochastic differential equations driven by time-changed Lévy noise with two drift terms, one driven by the random change E_t and the other driven by non-random time t. Firstly, we establish some new time-changed Gronwall-like inequalities, which makes it easy to apply in practice and it can be considered as a more general tool in some situations. As applications of those inequalities, we prove the existence and uniqueness of the solution to the considered equations under some non-Lipschitz conditions by employing the Carathéodory approximation. Meanwhile, by developing some generalized Itô’s formula, some sufficient conditions are provided to guarantee the solutions to be stable in several different senses in terms of Lyapunov functions. Subsequently, by using the established new time-changed Gronwall-like inequalities, we show that the solutions of the generalized distribution dependent stochastic differential equations driven by time-changed Brownian motion can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence. Finally, we provide some examples to illustrate the practical usefulness of our theoretical results.
