We conjecture that a class of Artinian Gorenstein Hilbert algebras called full Perazzo algebras always have minimal Hilbert function, fixing codimension and length. We prove the conjecture in length four and five, in low codimension. We also prove the conjecture for a particular subclass of algebras that occurs in every length and certain codimensions. As a consequence of our...
In this paper we show that if X is a $$T_1$$ -space with a $$\pi $$ -base whose elements have compact closure, then $$d(X)\le c(X)\cdot 2^{\psi (X)}$$ and therefore, for such spaces we have $$d(X)^{\psi (X)} = c(X)^{\psi (X)}$$ . This result allows us to restate several known upper bounds of the cardinality of a Hausdorff space X by replacing in them d(X) with c(X). In addition...
In this paper we introduce the concept of infinite pointwise dense lineability (spaceability), and provide a criterion to obtain density from mere lineability. As an application, we study the linear and topological structures within the set of infinite differentiable and integrable functions, for any order $$p \ge 1$$ , on $$\mathbb R^N$$ which are unbounded in a pre-fixed set.
Let $${\mathcal {C}}$$ be a class of topological semigroups. A semigroup X is called absolutely $${\mathcal {C}}$$ -closed if for any homomorphism $$h:X\rightarrow Y$$ to a topological semigroup $$Y\in {\mathcal {C}}$$ , the image h[X] is closed in Y. Let $$\textsf {T}_{\!\textsf {1}}\textsf {S}$$ , $$\textsf {T}_{\!\textsf {2}}\textsf {S}$$ , and $$\textsf {T}_{\!\textsf {z...
The complex orbifold structure of the moduli space of Riemann surfaces of genus g ( $$g\ge 2$$ ) produces a stratification into complex subvarieties named equisymmetric strata. Each equisymmetric stratum is formed by the surfaces where the group of automorphisms acts in a topologically equivalent way. The Riemann surfaces in the equisymmetric strata of dimension one are of two...
Orlicz-type modules are module analogues of classical Orlicz spaces. We study duality and stable compactness in Orlicz-type modules. We characterize the conditional Köthe dual of an Orlicz-type module as the space of all $$\sigma $$ -order continuous module homomorphisms. We find an order continuity criterion for stable compactness in Orlicz-type modules. As an application, we...
Appell sequences of polynomials can be extended to the Dunkl context replacing the ordinary derivative by the Dunkl operator on the real line, and the exponential function by the Dunkl kernel. In a similar way, discrete Appell sequences can be extended to the Dunkl context; here, the role of the ordinary translation is played by the Dunkl translation, which is a much more...
In this paper, we prove two recently conjectured supercongruences (modulo $$p^3$$ , where p is any prime greater than 3) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.
In this paper, we deal with seven types of Ky Fan type relations between bivariate, symmetric and homogeneous means. For each relation we determine necessary and sufficient conditions for means to be in this relation. Additionally, we investigate the dependencies between these relations.
For singular n-manifolds in $${{\mathbb {R}}}^{n+k}$$ with a corank 1 singular point at $$p\in M^n_{{\text {sing}}}$$ we define up to $$l(n-1)$$ different axial curvatures at p, where $$l=\min \{n,k+1\}.$$ These curvatures are obtained using the curvature locus (the image by the second fundamental form of the unitary tangent vectors) and are therefore second order invariants. In...
Generalized analytic functions are naturally defined in manifolds with boundary and are built from sums of convergent real power series with non-negative real exponents. In this paper we deal with the problem of reduction of singularities of these functions. Namely, we prove that a germ of generalized analytic function can be transformed by a finite sequence of blowing-ups into a...
We consider the Hilbert-type operator defined by $$\begin{aligned} H_{\omega }(f)(z)=\int _0^1 f(t)\left( \frac{1}{z}\int _0^z B^{\omega }_t(u)\,du\right) \,\omega (t)dt, \end{aligned}$$ where $$\{B^{\omega }_\zeta \}_{\zeta \in \mathbb {D}}$$ are the reproducing kernels of the Bergman space $$A^2_\omega $$ induced by a radial weight $$\omega $$ in the unit disc $$\mathbb {D...
We prove the optimality of the Gagliardo–Nirenberg inequality: $$\begin{aligned} \Vert \nabla u\Vert _{X}\lesssim \Vert \nabla ^2 u\Vert _Y^{1/2}\Vert u\Vert _Z^{1/2}, \end{aligned}$$ where Y and Z are rearrangement invariant Banach function spaces, and $$X = Y^{1/2}Z^{1/2}$$ is the Calderón–Lozanovskii space. By optimality, we mean that for a certain pair of spaces on the right...
The purpose of the present article is to present a simplified proof of Serre’s modularity conjecture using the strong modularity lifting results currently available.
The linear space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology (induced from the product space $$\mathbb {R}^X$$ ) is denoted by $$C_p(X).$$ In this paper we continue the systematic study of sequences spaces $$c_{0}$$ and $$\ell _{q}$$ (for $$0
Recently, Andrews and Merca have given a new combinatorial interpretation of the total number of even parts in all partitions of n into distinct parts. We generalise this result and consider many more variations of their work. We also highlight some connections with the work of Fu and Tang.
We introduce the Jordan-strict topology on the multiplier algebra of a JB $$^*$$ -algebra, a notion which was missing despite the forty years passed after the first studies on Jordan multipliers. In case that a C $$^*$$ -algebra A is regarded as a JB $$^*$$ -algebra, the J-strict topology of M(A) is precisely the well-studied C $$^*$$ -strict topology. We prove that every JB...
The conditional probability formula is supposed to reflect the correct updating of probability assignments when new information is incorporated. Starting from a non-atomic probability measure, it is proved that the conditional probability formula provides the only transformed probability measure satisfying a “minimum requirement” relational assumption. This result applies to the...
Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is continuous with respect to changes in the filtration function. However, due to the discrete setting of simplicial complexes, the simplices terminating matched bars cannot change continuously for arbitrary perturbations of filtration...
A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field $$\xi ^{S}$$ on their sphere bundles is Killing with respect to some invariant Riemannian metric. The set of all these metrics is determined, as well as the set of all those invariant contact metric...
The generalized Cesàro operators $$C_t$$ , for $$t\in [0,1]$$ , were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in $${{\mathbb {C}}}^{{{\mathbb {N}}}_0}$$ , such as $$\ell ^p$$ , $$c_0$$ , c, $$bv_0$$ , bv and, as recently shown in Curbera et al. (J Math Anal Appl 507:31, 2022) [26], also in the discrete Cesàro spaces...
We establish several bounds on the cardinality of a topological space involving the Hausdorff pseudocharacter $$H\psi (X)$$ . This invariant has the property $$\psi _c(X)\le H\psi (X)\le \chi (X)$$ for a Hausdorff space X. We show the cardinality of a Hausdorff space X is bounded by $$2^{pwL_c(X)H\psi (X)}$$ , where $$pwL_c(X)\le L(X)$$ and $$pwL_c(X)\le c(X)$$ . This generalizes...
Let $${\overline{M}}^{2n}$$ , $$n>1$$ , be a complete, noncompact Kählerian manifold, endowed with a nontrivial closed conformal vector field $$\xi $$ having at least one singular point. Under a reasonable set of conditions, we show that $$\xi $$ has just one singular point p and that $${\overline{M}}{\setminus }\{p\}$$ is isometric to a one dimensional cone over a simply...
Given a polynomial map $$F:\mathbb C^n\longrightarrow \mathbb C^p$$ with finite zero set, $$p\geqslant n$$ , we introduce the notion of global multiplicity $${\text {m}}(F)$$ associated to F, which is analogous to the multiplicity of ideals in Noetherian local rings. This notion allows to characterize numerically the Newton non-degeneracy at infinity of F. This fact motivates us...
We investigate Orlicz–Lorentz function spaces equipped with the Orlicz norm generated by any Orlicz function and any non-increasing weight function. As far as we know, this is the first time such a general research is conducted. First we show some basic properties of the Orlicz norm, including its equality to the Amemiya norm, the problem of attainability of infimum in the...