The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with

Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes, and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration, the associated polytope of the toric variety coincides with the matching field polytope. We study combinatorial mutations, which are analogues

In this paper, we use variational approaches to establish the existence of weak solutions for a class of Kirchhoff-type equations with fractional \(p_{1}(x)\) & \(p_{2}(x)\)-Laplacian operator, for \(1\le p_{1}(x,y)

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least

Given a finite order ideal \({\mathcal {O}}\) in the polynomial ring \(K[x_1,\ldots , x_n]\) over a field K, let \(\partial {\mathcal {O}}\) be the border of \({\mathcal {O}}\) and \({\mathcal {P}}_{\mathcal {O}}\) the Pommaret basis of the ideal generated by the terms outside \({\mathcal {O}}\). In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations

In this paper we are concerned with a long-standing conjecture of Huneke and Wiegand. We introduce a new class of ideals and prove that each ideal from such class satisfies the conclusion of the conjecture in question. We also study the relation between the class of Burch ideals and that of the ideals we define, and construct several examples that corroborate our results.

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let S be a graded (\(\pm 1\))-skew polynomial algebra in n variables of degree 1 and \(f =x_1^2 + \cdots +x_n^2 \in S\). We prove that the stable category \(\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))\) of graded maximal Cohen–Macaulay module over S/(f) can be completely computed

Let A be a regular ring containing a field of characteristic zero and let \(R = A[X_1,\ldots , X_m]\). Consider R as standard graded with \(\deg A = 0\) and \(\deg X_i = 1\) for all i. In this paper we present a comprehensive study of graded components of local cohomology s \(H^i_I(R)\) where I is an arbitrary homogeneous ideal in R. Our study seems to be the first in this regard.

An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate

Let R be a Noetherian local k-algebra whose derivation module \({\mathrm{Der}}_k(R)\) is finitely generated. Our main goal in this paper is to investigate the impact of assuming that \({\mathrm{Der}}_k(R)\) has finite projective dimension (or finite Gorenstein dimension), mainly in connection with freeness, under a suitable hypothesis concerning the vanishing of (co)homology or the depth of a certain

Let \(S=K[x_1,\ldots ,x_n]\), where K is a field, and \(t_i\) denotes the maximal shift in the minimal graded free S-resolution of the graded algebra S/I at degree i. In this paper, we prove: If I is a monomial ideal of S and \(a\ge b-1\ge 0\) are integers such that \(a+b\le \mathrm {proj\,dim}(S/I)\), then $$\begin{aligned} t_{a+b}\le t_a+t_1+t_2+\cdots +t_b-\frac{b(b-1)}{2}. \end{aligned}$$ If \(I=I_{\Delta

We show that Szczarba’s twisting cochain for a twisted Cartesian product is essentially the same as the one constructed by Shih. More precisely, Szczarba’s twisting cochain can be obtained via the basic perturbation lemma if one uses a ‘reversed’ version of the classical Eilenberg–Mac Lane homotopy for the Eilenberg–Zilber contraction. Along the way we prove several new identities involving these homotopies

On the basis of fractional calculus, we introduce an integral of controlled paths with respect to Hölder rough paths of order \(\beta \in (1/4,1/3]\). Our definition of the integral is given explicitly in terms of Lebesgue integrals for fractional derivatives, without using any arguments from discrete approximation. We demonstrate that for suitable classes of \(\beta\)-Hölder rough paths and controlled

We define a discrete version of the bilinear spherical maximal function, and show bilinear \(l^{p}(\mathbb {Z}^d)\times l^{q}(\mathbb {Z}^d) \rightarrow l^{r}(\mathbb {Z}^d)\) bounds for \(d \ge 3\), \(\frac{1}{p} + \frac{1}{q} \ge \frac{1}{r}\), \(r>\frac{d}{d-2}\) and \(p,q\ge 1\). Due to interpolation, the key estimate is an \(l^{p}(\mathbb {Z}^d)\times l^{\infty }(\mathbb {Z}^d) \rightarrow l^{p}(\mathbb

Let \(x=(x_1,x_2)\) with \(x_1,x_2 \in \mathbb {R}^n\) and let \(K(x)={\Omega \big ({x}/{|x|}\big )}{\big |x\big |^{-2n}}\), where \(\Omega \in L^{\infty }(\mathbb {S}^{2n-1})\) and satisfies \(\int _{\mathbb {S}^{2n-1}}\Omega =0\). We show that the maximal truncated bilinear singular integrals with rough kernel \(K(x_1,x_2)\) satisfy a sparse bound by (p, p, p)-averages for all \(p>1\). As consequences

Compact differences of two weighted composition operators acting from the weighted Bergman space \(A^p_{\omega }\) to another weighted Bergman space \(A^q_{\nu }\), where \(0

We introduce and study a family of spaces of entire functions in one variable that generalise the classical Paley–Wiener and Bernstein spaces. Namely, we consider entire functions of exponential type a whose restriction to the real line belongs to the homogeneous Sobolev space \(\dot{W}^{s,p}\) and we call these spaces fractional Paley–Wiener if \(p=2\) and fractional Bernstein spaces if \(p\in (1

We investigate the resurgence and asymptotic resurgence numbers of fiber products of projective schemes. Particularly, we show that while the asymptotic resurgence number of the k-fold fiber product of a projective scheme remains unchanged, its resurgence number could strictly increase.

We obtain a sharp estimate to the norm of the traceless second fundamental form of complete hypersurfaces with constant scalar curvature immersed into a locally symmetric Riemannian manifold obeying standard curvature constraints (which includes, in particular, the Riemannian space forms with constant sectional curvature). When the equality holds, we prove that these hypersurfaces must be isoparametric

The symplectic Brill–Noether locus \({{{\mathcal {S}}}}_{2n, K}^k\) associated to a curve C parametrises stable rank 2n bundles over C with at least k sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions

Notions of convergence and continuity specifically adapted to Riesz ideals \(\mathscr {I}\) of the space of continuous real-valued functions on a Lindelöf locally compact Hausdorff space are given, and used to prove Stone–Weierstrass-type theorems for \(\mathscr {I}\). As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on \(\mathscr {I}\)

Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules \(H^i_I(M)\) are I-cofinite, for all finitely generated R-modules M and all \(i\in {\mathbb {N}}_0\), if and only if the R-modules \(H^i_I(R)\) are I-cofinite with dimension not exceeding 1, for all integers \(i\ge 2\); in addition, under these equivalent conditions it is shown that, for each minimal prime ideal \({{\mathfrak

We use techniques from time-frequency analysis to show that the space \({\mathcal{S}}_\omega\) of rapidly decreasing \(\omega\)-ultradifferentiable functions is nuclear for every weight function \(\omega (t)=o(t)\) as t tends to infinity. Moreover, we prove that, for a sequence \((M_p)_p\) satisfying the classical condition (M1) of Komatsu, the space of Beurling type \({\mathcal{S}}_{(M_p)}\) when

The Orlicz spaces \(X^{\varPhi }\) associated to a quasi-Banach function space X are defined by replacing the role of the space \(L^1\) by X in the classical construction of Orlicz spaces. Given a vector measure m, we can apply this construction to the spaces \(L^1_w(m),\)\(L^1(m)\) and \(L^1(\Vert m\Vert )\) of integrable functions (in the weak, strong and Choquet sense, respectively) in order to

A gauge\(\gamma\) in a vector space X is a distance function given by the Minkowski functional associated to a convex body K containing the origin in its interior. Thus, the outcoming concept of gauge spaces\((X, \gamma )\) extends that of finite dimensional real Banach spaces by simply neglecting the symmetry axiom (a viewpoint that Minkowski already had in mind). If the dimension of X is even, then

We give an explicit formula for the Hilbert–Poincaré series of the parity binomial edge ideal of a complete graph \(K_{n}\) or equivalently for the ideal generated by all \(2\times 2\)-permanents of a \(2\times n\)-matrix. It follows that the depth and Castelnuovo–Mumford regularity of these ideals are independent of n.

Hironaka’s characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition (Pol) holds, or a mild condition (*) on the singularity holds. For example, the latter is fulfilled if

In this paper we prove the conjectured upper bound for Castelnuovo–Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the number of maximal cliques of G, denoted by c(G). Moreover, we classify all chordal graphs G for which \(\mathcal {L}(G)=c(G)\), where \(\mathcal {L}(G)\) is the sum

Suppose that M is a finitely-generated graded module (generated in degree 0) of codimension \(c\ge 3\) over a polynomial ring and that the regularity of M is at most \(2a-2\) where \(a\ge 2\) is the minimal degree of a first syzygy of M. Then we show that the sum of the betti numbers of M is at least \(\beta _0(M)(2^c + 2^{c-1})\). Additionally, under the same hypothesis on the regularity, we establish

Let G be a connected and finite graph with the set of vertices V and the set of edges E. Let \(M_{G}\) be the Hardy–Littlewood maximal function defined on graph G and \(M_{\upalpha ,G}\)\((0\le \upalpha <1)\) be its fractional version. In this paper, the regularity problems related to \(M_{G}\) and \(M_{\upalpha ,G}\) will be studied. We show that \(M_{G}:\mathrm{BV}_p (G)\rightarrow \mathrm{BV}_p(G)\)

In this paper, the \(L^2 \times L^{\infty } \rightarrow L^2\) and \(L^2 \times L^2 \rightarrow L^1\) estimates of bilinear Fourier multiplier operators are discussed under weak smoothness conditions on multipliers. As an application, we prove the \(L^2 \times BMO \rightarrow L^2\) and \(L^2 \times L^2 \rightarrow H^1\) boundedness of bilinear operators with multipliers of limited smoothness satisfying

We give two-weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted \(L^p\) and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces

The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space \(({\mathcal {X}}, d, \mu )\). For such parabolic minimizers that coincide with Cauchy-Dirichlet data \(\eta \) on the parabolic boundary of a space-time-cylinder \(\varOmega \times (0, T)\) with an open subset \(\varOmega

We study the nuclearity of the Gelfand–Shilov spaces \({\mathcal {S}}^{({\mathfrak {M}})}_{({\mathscr {W}})}\) and \({\mathcal {S}}^{\{{\mathfrak {M}}\}}_{\{{\mathscr {W}}\}}\), defined via a weight (multi-)sequence system \({\mathfrak {M}}\) and a weight function system \({\mathscr {W}}\). We obtain characterizations of nuclearity for these function spaces that are counterparts of those for Köthe

We consider the dynamics of transcendental self-maps of the punctured plane, \(\mathbb {C}^*=\mathbb {C}{\setminus } \{0\}\). We prove that the escaping set \(I(f)\) is either connected, or has infinitely many components. We also show that \(I(f)\cup \{0,\infty \}\) is either connected, or has exactly two components, one containing 0 and the other \(\infty \). This gives a trichotomy regarding the

We give an algebraic and self-contained proof of the existence of the so-called Noetherian operators for primary submodules over general classes of Noetherian commutative rings. The existence of Noetherian operators accounts to provide an equivalent description of primary submodules in terms of differential operators. As a consequence, we introduce a new notion of differential powers which coincides

For a monomial ideal I, we consider the ith homological shift ideal of I, denoted by \({\text {HS}}_i(I)\), that is, the ideal generated by the ith multigraded shifts of I. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal I and any monomial prime ideal P, \({\text {HS}}_i(I(P))\subseteq {\text {HS}}_i(I)(P)\) for all i, where I(P) is the monomial localization

In this paper, by using variational approach, Mountain Pass Theorem and Krasnoselskii’s genus theory, we show the existence and multiplicity of solutions for a Schrödinger–Kirchhoff type equation involving the fractional \(p\left( .,.\right)\)-Laplacian in fractional Sobolev space with variable exponent. We also establish a Bartsch–Wang type compact embedding theorem for fractional Sobolev space with

We find a semi-algebraic description of the Minkowski sum \(\mathcal {A}_{3,n}\) of n copies of the twisted cubic segment \(\{(t,t^2,t^3)\mid -1\le t\le 1\}\) for each integer \(n\ge 3\). These descriptions provide efficient membership tests for the sets \(\mathcal {A}_{3,n}\). These membership tests in turn can be used to resolve some instances of the underdetermined matrix moment problem, which was

We make a first geometric study of three varieties in \(\mathbb {C}^m\otimes \mathbb {C}^m\otimes \mathbb {C}^m\) (for each m), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassen’s asymptotic rank conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we

In a certain Lipschitz domain \(\Omega \subset {\mathbb {R}}^n\), we establish the boundary Harnack principle for positive superharmonic functions satisfying the nonlinear differential inequality \(-\Delta u\le cu^p\), where \(c>0\) and \(1

We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge level, that is, the maximal distance between non-trivial Hodge numbers in the same row of the Hodge diamond. This allows us to classify varieties whose Hodge numbers are like that of a projective space, of a curve, or of a Calabi–Yau variety of low dimension.

The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.

Any \({\mathcal {S}} \in \mathfrak {sp}(1,{\mathbb {R}})\) induces canonically a derivation S of the Heisenberg Lie algebra \({\mathfrak {h}}\) and so, a semi-direct extension \(G_{{\mathcal {S}}}=H \rtimes \exp ({\mathbb {R}}S)\) of the Heisenberg Lie group H (Müller and Ricci in Invent Math 101: 545–582, 1990). We shall explicitly describe the connected, simply connected Lie group \(G_{{\mathcal

We establish Littlewood–Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions.

Our study is focused on the dynamics of weighted composition operators defined on a locally convex space \(E\hookrightarrow (C(X),\tau _p)\) with X being a topological Hausdorff space containing at least two different points and such that the evaluations \(\{\delta _x:\ x\in X\}\) are linearly independent in \(E'\). We prove, when X is compact and E is a Banach space containing a nowhere vanishing

We prove a sharp common generalization of endpoint multilinear Kakeya and local discrete Brascamp–Lieb inequalities.

The purpose of this paper is to exhibit infinite families of conjugate projective curves in a number field whose complement have the same abelian fundamental group, but are non-homeomorphic. In particular, for any \(d\ge 4\) we find Zariski \(\left( \left\lfloor \frac{d}{2}\right\rfloor +1\right) \)-tuples parametrized by the d-roots of unity up to complex conjugation. As a consequence, for any divisor

We characterize the weights for the Stieltjes transform and the Calderón operator to be bounded on the weighted variable Lebesgue spaces \(L_w^{p(\cdot )}(0,\infty )\), assuming that the exponent function \({p(\cdot )}\) is log-Hölder continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals

The proof of Theorem 3.2 in the paper contains an error

In this paper we study those submonoids of \(\mathbb {N}^d\) with a nontrivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the

We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen–Macaulayness and Buchsbaumness of these ideals and we describe their Castelnuovo–Mumford regularity and their Hilbert series. Conca and Varbaro (Square-free Groebner degenerations, 2018) have recently proved a conjecture of Conca, De Negri and Gorla

Let \((R,\mathfrak {m},k)\) be a local ring. We establish a totally reflexive analogue of the New Intersection Theorem, provided for every totally reflexive R-module M, there is a big Cohen–Macaulay R-module \(B_M\) such that the socle of \(B_M\otimes _RM\) is zero. When R is a quasi-specialization of a \({\text {G}}\)-regular local ring or when M has complete intersection dimension zero, we show the

We introduce and study higher order Jacobian ideals, higher order and mixed Hessians, higher order polar maps, and higher order Milnor algebras associated to a reduced projective hypersurface. We relate these higher order objects to some standard graded Artinian Gorenstein algebras, and we study the corresponding Hilbert functions and Lefschetz properties.

In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category \(\mathcal {C}\) and for arbitrary groups \(H\le G_1\times G_2\), is there an object \(\phi :A_1 \rightarrow A_2\) in \({\text {Arr}}(\mathcal {C})\) such that \({\text {Aut}}_{{\text {Arr}}(\mathcal {C})}(\phi ) = H\), \({\text {Aut}}_{\mathcal {C}}(A_1) = G_1\) and \({\text {Aut}}_{\mathcal {C}}(A_2)

Many years ago, Griego, Heath and Ruiz-Moncayo proved that it is possible to define realizations of a sequence of uniform transport processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In this paper we extend their results to the multi parameter case. We begin constructing a family of processes, starting from a set of independent standard Poisson

We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.

Grothendieck’s theorem asserts that every continuous linear operator from \(\ell _1\) to \(\ell _2\) is absolutely (1, 1)-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by \(\ell _1\) spaces is also investigated

It is proved that a module M over a Noetherian local ring R of prime characteristic and positive dimension has finite flat dimension if \({\text {Tor}}_i^R({}^{e}\!R, M)=0\) for \({\text {dim}}\,R\) consecutive positive values of i and infinitely many e. Here \({}^{e}\!R\) denotes the ring R viewed as an R-module via the eth iteration of the Frobenius endomorphism. In the case R is Cohen–Macualay,

We characterize the (regular) holonomicity of Horn systems of differential equations under a hypothesis that captures the most widely studied classical hypergeometric systems.