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Abstract: The paper contains the study of the weighted \(L^{p_1}\times L^{p_2}\times \ldots \times L^{p_m}\rightarrow L^p\) estimates for the multilinear maximal operator, in the context of abstract probability spaces equipped with a tree-like structure. Using the Bellman function method, we identify the associated optimal constants in the symmetric case \(p_1=p_2=\ldots =p_m\) , and a tight constant for remaining choices of exponents. PubDate: 2024-05-01

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Abstract: Abstract We study stable trace ideals in one dimensional local Cohen–Macaulay rings and give numerous applications. PubDate: 2024-05-01

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Abstract: Abstract This paper is devoted to a deep analysis of the process known as Cheeger deformation, applied to manifolds with isometric group actions. Here, we provide new curvature estimates near singular orbits and present several applications. As the main result, we answer a question raised by a seminal result of Searle–Wilhelm about lifting positive Ricci curvature from the quotient of an isometric action. To answer this question, we develop techniques that can be used to provide a substantially streamlined version of a classical result of Lawson and Yau, generalize a curvature condition of Chavéz, Derdzinski, and Rigas, as well as, give an alternative proof of a result of Grove and Ziller. PubDate: 2024-05-01

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Abstract: Abstract Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to k-planes with the restriction that their intersection has a prescribed dimension. We study dimensions of restricted secant of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the BDdG conjecture [Baur et al. in Exp Math 16(2):239–250, 2007, Conjecture 4.1] on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay 2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory. PubDate: 2024-05-01

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Abstract: Abstract Due to \(\mathbb R_+=(0,\,\infty )\) not being a group under addition, \(L^2(\mathbb R_+)\) admits no traditional wavelet or Gabor frames. This paper addresses a class of modulation-dilation frames ( \(\mathcal MD\) -frames) for \(L^2(\mathbb R_+)\) . We obtain a \(\Theta \) -transform matrix-based expression of adding generators to generate \(\mathcal MD\) -tight frames from a \(\mathcal{M}\mathcal{D}\) -Bessel sequences in \(L^2({\mathbb R}_+)\) ; and present criteria on \(\Phi \) with \(\mathcal{M}\mathcal{D}(\Psi \cup \Phi ,\,a,\,b)\) being a Parseval frame (an orthonormal basis) for an arbitrary Parseval frame sequence (an orthonormal sequence) \(\mathcal{M}\mathcal{D}(\Psi ,\,a,\,b)\) in \(L^2(\mathbb R_+)\) . Some examples are also presented. PubDate: 2024-05-01

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Abstract: Abstract We characterise those Banach spaces X which satisfy that L(Y, X) is octahedral for every non-zero Banach space Y. They are those satisfying that, for every finite dimensional subspace Z, \(\ell _\infty \) can be finitely-representable in a part of X kind of \(\ell _1\) -orthogonal to Z. We also prove that L(Y, X) is octahedral for every Y if, and only if, \(L(\ell _p^n,X)\) is octahedral for every \(n\in {\mathbb {N}}\) and \(1<p<\infty \) . Finally, we find examples of Banach spaces satisfying the above conditions like \({\textrm{Lip}}_0(M)\) spaces with octahedral norms or \(L_1\) -preduals with the Daugavet property. PubDate: 2024-05-01

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Abstract: Abstract The set of smooth cubic hypersurfaces in \({{\mathbb {P}}}^n\) is an open subset of a projective space. A compactification of the latter which allows to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points is termed a 1–complete variety of cubic hypersurfaces, in analogy with the space of complete quadrics. Imitating the work of Aluffi for plane cubic curves, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. In the end, we derive the desired numbers in the case of cubic surfaces. PubDate: 2024-05-01

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Abstract: Abstract The depth of squarefree powers of a squarefree monomial ideal is introduced. Let I be a squarefree monomial ideal of the polynomial ring \(S=K[x_1,\ldots ,x_n]\) . The k-th squarefree power \(I^{[k]}\) of I is the ideal of S generated by those squarefree monomials \(u_1\cdots u_k\) with each \(u_i\in G(I)\) , where G(I) is the unique minimal system of monomial generators of I. Let \(d_k\) denote the minimum degree of monomials belonging to \(G(I^{[k]})\) . One has \({\text {depth}}(S/I^{[k]}) \ge d_k -1\) . Setting \(g_I(k) = {\text {depth}}(S/I^{[k]}) - (d_k - 1)\) , one calls \(g_I(k)\) the normalized depth function of I. The computational experience strongly invites us to propose the conjecture that the normalized depth function is nonincreasing. In the present paper, especially the normalized depth function of the edge ideal of a finite simple graph is deeply studied. PubDate: 2024-05-01

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Abstract: Abstract Let \(f:X\rightarrow Y\) be a semistable fibration between smooth complex varieties of dimension n and m. This paper contains an analysis of the local systems of de Rham closed relative one forms and top forms on the fibers. In particular the latter recovers the local system of the second Fujita decomposition of \(f_*\omega _{X/Y}\) over higher dimensional base. The so called theory of Massey products allows, under natural Castelnuovo-type hypothesis, to study the finiteness of the associated monodromy representations. Motivated by this result, we also make precise the close relation between Massey products and Castelnuovo-de Franchis type theorems. PubDate: 2024-05-01

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Abstract: Abstract We construct examples of twice differentiable functions in \({\mathbb {R}}^n\) with continuous Laplacian and unbounded Hessian. The same construction is also applicable to higher order differentiability. PubDate: 2024-05-01

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Abstract: Abstract In this paper, we generalize some halfspace type theorems for self-shrinkers of codimension 1 to the case of arbitrary codimension. PubDate: 2024-05-01

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Abstract: Abstract In this paper we obtain new estimates for bilinear pseudodifferential operators with symbol in the class \(BS_{1,1}^m\) , when both arguments belong to Triebel-Lizorkin spaces of the type \(F_{p,q}^{n/p}({\mathbb {R}}^n)\) . The inequalities are obtained as a consequence of a refinement of the classical Sobolev embedding \(F^{n/p}_{p,q}({\mathbb {R}}^n)\hookrightarrow \textrm{bmo}({\mathbb {R}}^n)\) , where we replace \(\textrm{bmo}({\mathbb {R}}^n)\) by an appropriate subspace which contains \(L^\infty ({\mathbb {R}}^n)\) . As an application, we study the product of functions on \(F_{p,q}^{n/p}({\mathbb {R}}^n)\) when \(1<p<\infty \) , where those spaces fail to be multiplicative algebras. PubDate: 2024-05-01

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Abstract: Abstract Assume that G is a graph with edge ideal I(G). We provide sharp lower bounds for the depth of \(I(G)^2\) in terms of the star packing number of G. PubDate: 2024-05-01

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Abstract: Abstract A marked Prym curve is a triple \((C,\alpha ,T_d)\) where C is a smooth algebraic curve, \(\alpha \) is a \(2-\) torsion line bundle on C, and \(T_d\) is a divisor of degree d. We give obstructions—in terms of Gaussian maps—for a marked Prym curve \((C,\alpha ,T_d)\) to admit a singular model lying on an Enriques surface with only one ordinary singular point of multiplicity d, such that \(T_d\) is the pull-back of the singular point by the normalization map. More precisely, let (S, H) be a polarized Enriques surface and let (C, f) be a smooth curve together with a morphism \(f:C \rightarrow S\) birational onto its image and such that \(f(C) \in H \) , f(C) has exactly one ordinary singular point of multiplicity d. Let \(\alpha =f^*\omega _S\) and \(T_d\) be the divisor over the singular point of f(C). We show that if H is sufficiently positive then certain natural Gaussian maps on C, associated with \(\omega _C\) , \(\alpha \) , and \(T_d\) are not surjective. On the contrary, we show that for the general triple in the moduli space of marked Prym curves \((C,\alpha ,T_d)\) , the same Gaussian maps are surjective. PubDate: 2024-04-30

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Abstract: Abstract Let \(I\subset S\) be a graded ideal of a standard graded polynomial ring S with coefficients in a field K, and let \({\text {v}}(I)\) be the \({\text {v}}\) -number of I. In previous work, we showed that for any graded ideal \(I\subset S\) , then \({\text {v}}(I^k)=\alpha (I)k+b\) , for all \(k\gg 0\) , where \(\alpha (I)\) is the initial degree of I and b is a suitable integer. In the present paper, using polarization, we extend Simon conjecture to any monomial ideal. As a consequence, if Simon conjecture holds, I is a monomial ideal generated in a single degree and all powers of I have linear quotients, then \(b\in \{-1,0\}\) . This fact suggests that if I is an equigenerated monomial ideal with linear powers, then \({\text {v}}(I^k)=\alpha (I)k-1\) , for all \(k\ge 1\) . We verify this conjecture for monomial ideals with linear powers having \({\text {depth}}\,S/I=0\) , edge ideals with linear resolution, polymatroidal ideals, and Hibi ideals. PubDate: 2024-04-23

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Abstract: Abstract Let I be a perfect ideal of height two in \(R=k[x_1, \ldots , x_d]\) and let \(\varphi \) denote its Hilbert–Burch matrix. When \(\varphi \) has linear entries, the algebraic structure of the Rees algebra \({\mathcal {R}}(I)\) is well-understood under the additional assumption that the minimal number of generators of I is bounded locally up to codimension \(d-1\) . In the first part of this article, we determine the defining ideal of \({\mathcal {R}}(I)\) under the weaker assumption that such condition holds only up to codimension \(d-2\) , generalizing previous work of P. H. L. Nguyen. In the second part, we use generic Bourbaki ideals to extend our findings to Rees algebras of linearly presented modules of projective dimension one. PubDate: 2024-04-04

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Abstract: Abstract Let \((\mathcal {G},\otimes )\) be any closed symmetric monoidal Grothendieck category. We show that K-flat covers exist universally in the category of chain complexes and that the Verdier quotient of \(K(\mathcal {G})\) by the K-flat complexes is always a well generated triangulated category. Under the further assumption that \(\mathcal {G}\) has a set of \(\otimes\) -flat generators we can show more: (i) The category is in recollement with the \(\otimes\) -pure derived category and the usual derived category, and (ii) The usual derived category is the homotopy category of a cofibrantly generated and monoidal model structure whose cofibrant objects are precisely the K-flat complexes. We also give a condition guaranteeing that the right orthogonal to K-flat is precisely the acyclic complexes of \(\otimes\) -pure injectives. We show this condition holds for quasi-coherent sheaves over a quasi-compact and semiseparated scheme. PubDate: 2024-04-03

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Abstract: Abstract We prove the Kato–Ponce inequality (fractional normed Leibniz rule) for multiple factors in the setting of multiple weights ( \(A_{\vec P}\) weights). This improves existing results to the product of m factors and extends the class of known weights for which the inequality holds. PubDate: 2024-03-25

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Abstract: Abstract In this paper we introduce the atomic Hardy space \(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\) associated with the non-doubling probability measure \(d\gamma _\alpha (x)=\frac{2x^{2\alpha +1}}{\Gamma (\alpha +1)}e^{-x^2}dx\) on \((0,\infty )\) , for \({\alpha >-\frac{1}{2}}\) . We obtain characterizations of \(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\) by using two local maximal functions. We also prove that the truncated maximal function defined through the heat semigroup generated by the Laguerre differential operator is bounded from \(\mathcal {H}^1((0,\infty ),\gamma _\alpha )\) into \(L^1((0,\infty ),\gamma _\alpha )\) . PubDate: 2024-03-19

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Abstract: Abstract In this article, we introduce a relation including ideals of an evolution algebra A and hereditary subsets of vertices of its associated graph and establish some properties among them. This relation allows us to determine maximal ideals and ideals having the absorption property of an evolution algebra in terms of its associated graph. In particular, the maximal ideals can be determined through maximal hereditary subsets of vertices except for those containing \(A^2\) . We also define a couple of order-preserving maps, one from the sets of ideals of an evolution algebra to that of hereditary subsets of the corresponding graph, and the other in the reverse direction. Conveniently restricted to the set of absorption ideals and to the set of hereditary saturated subsets, this is a monotone Galois connection. According to the graph, we characterize arbitrary dimensional finitely-generated (as algebras) evolution algebras under certain restrictions of its graph. Furthermore, the simplicity of finitely-generated perfect evolution algebras is described on the basis of the simplicity of the graph. PubDate: 2024-03-14