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In the paper we study the existence of balanced metrics of Hodge-Riemann type on non-K\"ahler complex manifolds. We first find some general obstructions, for instance that a Hodge-Riemann balanced manifold of complex dimension $n$ has to be $(n - 2)$-K\"ahler. Then, we focus on the case of compact quotients of Lie groups by lattices, endowed with an invariant complex structure. In particular, we prove non existence results on non-K\"ahler complex parallelizable manifolds and some classes of solvmanifolds, and we show that the only nilmanifolds admitting invariant structures of this type are tori. Finally, we construct the first non-K\"ahler example of a Hodge-Riemann balanced structure, on a non-compact complex manifold obtained as the product of the Iwasawa manifold by $\C$., Comment: Comments are welcome!

In the present paper we study SKT and generalized K\"ahler structures on solvable Lie algebras with (not necessarily abelian) codimension two nilradical. We treat separately the case of $J$-invariant nilradical and non $J$-invariant nilradical. A classification of such SKT Lie algebras in dimension six is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized K\"ahler compact solvmanifolds., Comment: 27 pages

In recent years, power analysis has become widely used in applied sciences, with the increasing importance of the replicability issue. When distribution-free methods, such as Partial Least Squares (PLS)-based approaches, are considered, formulating power analysis turns out to be challenging. In this study, we introduce the methodological framework of a new procedure for performing power analysis when PLS-based methods are used. Data are simulated by the Monte Carlo method, assuming the null hypothesis of no effect is false and exploiting the latent structure estimated by PLS in the pilot data. In this way, the complex correlation data structure is explicitly considered in power analysis and sample size estimation. The paper offers insights into selecting statistical tests for the power analysis procedure, comparing accuracy-based tests and those based on continuous parameters estimated by PLS. Simulated and real datasets are investigated to show how the method works in practice.

The project CAESAR (Comprehensive spAce wEather Studies for the ASPIS prototype Realization) is aimed to tackle all the relevant aspects of Space Weather (SWE) and realize the prototype of the scientific data centre for Space Weather of the Italian Space Agency (ASI) called ASPIS (ASI SPace Weather InfraStructure). This contribution is meant to bring attention upon the first steps in the development of the CAESAR prototype for ASPIS and will focus on the activities of the Node 2000 of CAESAR, the set of Work Packages dedicated to the technical design and implementation of the CAESAR ASPIS archive prototype. The product specifications of the intended resources that will form the archive, functional and system requirements gathered as first steps to seed the design of the prototype infrastructure, and evaluation of existing frameworks, tools and standards, will be presented as well as the status of the project in its initial stage., Comment: 4 pages, 2 figures, ADASS XXXII (2022) Proceedings

A $p$-K\"ahler structure on a complex manifold of complex dimension $n$ is given by a $d$-closed transverse real $(p,p)$-form. In the paper we study the existence of $p$-K\"ahler structures on compact quotients of simply connected Lie groups by discrete subgroups endowed with an invariant complex structure. In particular, we discuss the existence of $p$-K\"ahler structures on nilmanifolds, with a focus on the case $p =2$ and complex dimension $n = 4$. Moreover, we prove that a $(n-2)$-K\"ahler almost abelian solvmanifold of complex dimension $n\geq3$ has to be K\"ahler., Comment: Final version. To appear in Annali di Matematica Pura ed Applicata

In this note we study $T^2$-invariant pluriclosed metrics on the Kodaira-Thurston surface. We obtain a characterization of $T^2$-invariant Vaisman metrics, and notice that the Kodaira-Thurston surface admits Vaisman metrics with non-constant scalar curvature. Then we study the behaviour of the Vaisman condition in relation to the pluriclosed flow. As a consequence, we show that if the initial metric on the Kodaira-Thurston surface is a $T^2$-invariant Vaisman metric, then the pluriclosed flow preserves the Vaisman condition, extending to the non-constant scalar curvature case the previous results in [6].

In this note, we prove the existence of homogeneous gradient solitons for the G$_2$-Laplacian flow by providing the first known example of this type. This result singles out the G$_2$-Laplacian flow as the first known geometric flow admitting homogeneous gradient solitons on spaces that are one-dimensional extensions., Comment: To appear in Proceedings of the American Mathematical Society

We discuss general properties of strong G$_2$-structures with torsion and we investigate the twisted G$_2$ equation, which represents the G$_2$-analogue of the twisted Calabi-Yau equation for SU$(n)$-structures introduced by Garcia-Fern\'andez - Rubio - Shahbazi - Tipler. In particular, we show that invariant strong G$_2$-structures with torsion do not occur on compact non-flat solvmanifolds. This implies the non-existence of non-trivial solutions to the twisted Calabi-Yau equation on compact solvmanifolds of dimensions $4$ and $6$. More generally, we prove that a compact, connected homogeneous space admitting invariant strong G$_2$-structures with torsion is diffeomorphic either to $S^3 \times T^4$ or to $S^3 \times S^3 \times S^1$, up to a covering, and that in both cases solutions to the twisted G$_2$ equation exist. Finally, we discuss the behavior of the homogeneous Laplacian coflow for strong G$_2$-structures with torsion on these spaces., Comment: Final version, to appear in Pure and Applied Mathematics Quarterly

We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine the six-dimensional almost nilpotent solvmanifolds admitting an invariant complex structure and study the existence of special types of Hermitian metrics, including SKT, balanced, locally conformally K\"ahler, and strongly Gauduchon metrics. In particular, we determine new balanced solvmanifolds and confirm a conjecture by the first author and Vezzoni regarding SKT and balanced structures in the six-dimensional strongly unimodular almost nilpotent case. Moreover, we prove some negative results regarding complex structures tamed by symplectic forms, showing in particular that in every dimension such structures cannot exist on non-K\"ahler almost abelian Lie algebras., Comment: 41 pages, v2: minor changes

Starting from the product of a $3$-torus and a compact K\"ahler (respectively, hyperK\"ahler) manifold we construct via mapping tori generalized K\"ahler manifolds of split (respectively, non-split) type. In this way we obtain new non-K\"ahler examples and we recover the known examples of generalized K\"ahler solvmanifolds. Moreover, we investigate the formality and the Dolbeault cohomology of the generalized K\"ahler mapping tori., Comment: Final Version. To appear in J. Symplectic Geom

We consider a semilinear wave equation involving a time-dependent structural damping term of the form $\displaystyle\frac{1}{{(1+t)}^{\beta}}(-\Delta)^{\sigma/2} u_t$. Our results show the influence of the parameters $\beta,\sigma$ on the nonexistence of global weak solutions under assumptions on the given system data., Comment: arXiv admin note: text overlap with arXiv:2111.01433

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups., Comment: 34 pages

A locally conformal SKT (shortly LCSKT) structure is a Hermitian structure $(J, g)$ whose Bismut torsion 3-form $H$ satisfies the condition $dH = \alpha \wedge H$, for some closed non-zero 1-form $\alpha$. This condition was introduced in [6] as a generalization of the SKT (or pluriclosed) condition $dH= 0$. In this paper, we characterize the almost abelian Lie algebras admitting a Hermitian structure $(J, g)$ such that $dH = \alpha \wedge H$, for some closed 1-form $\alpha$. As an application we classifiy LCSKT almost abelian Lie algebras in dimension $6$. Finally, we also study on almost abelian Lie algebras the compatibility between the LCSKT condition and other types of Hermitian structures., Comment: 20 pages

A Hermitian metric on a complex manifold $(M, I)$ of complex dimension $n$ is called Calabi-Yau with torsion (CYT) or Bismut-Ricci flat, if the restricted holonomy of the associated Bismut connection is contained in ${\rm SU}(n)$ and it is called strong K\"ahler with torsion (SKT) or pluriclosed if the associated fundamental form $F$ is $\partial \overline \partial$-closed. In the paper we study the existence of left-invariant SKT and CYT metrics on compact semi-simple Lie groups endowed with a Samelson complex structure $I$. In particular, we show that if $I$ is determined by some maximal torus $T$ and $g$ is a left-invariant Hermitian metric, which is also invariant under the right action of the torus $T$, and is both CYT and SKT, then $g$ has to be Bismut flat., Comment: Dedicated to Jean-Pierre Bourguignon for his 75th birthday

We construct new examples of non-formal simply connected compact Sasaki-Einstein 7-manifolds. We determine the minimal model of the total space of any fibre bundle over $CP^2$ with fibre $S^1\times S^2$ or $S^3/Z_p$ ($p>0$), and we apply this to conclude that the Aloff-Wallach spaces are formal. We also find examples of formal manifolds and non-formal manifolds, which are locally conformal parallel $Spin(7)$-manifolds. On the other hand, we construct associative minimal submanifolds in the Aloff-Wallach spaces and in any regular Sasaki-Einstein 7-manifold; in particular, in the space $Q(1,1,1)=(SU(2) \times SU(2) \times SU(2))/ (U(1) \times U(1))$ with the natural $S^1$-family of nearly parallel $G_2$-structures induced by the Sasaki-Einstein structure. In each of those cases, we obtain a family of non-trivial associative deformations., Comment: 35 pages, no figures; v2. added references; v3. title changed, clarifications in the Introduction and in the proof of Theorem 4.2, rewritten some parts to improve presentation

Motivated by the construction based on topological suspension of a family of compact non-K\"ahler complex manifolds with trivial canonical bundle given by L. Qin and B. Wang in [QW], we study toric suspensions of balanced manifolds by holomorphic automorphisms. In particular, we show that toric suspensions of Calabi-Yau manifolds are balanced. We also prove that suspensions associated with hyperbolic automorphisms of hyperk\"ahler manifolds do not admit any pluriclosed, astheno-K\"ahler or p-pluriclosed Hermitian metric. Moreover, we consider natural extensions for hypercomplex manifolds, providing some explicit examples of compact holomorphic symplectic and hypercomplex non-K\"ahler manifolds. We also show that a modified suspension construction provides examples with pluriclosed metrics., Comment: Minor revision. The title is changed

The main purpose of the present paper is to study the blow-up problem of the wave equation with space-dependent damping in the \textit{scale-invariant case} and time derivative nonlinearity with small initial data. Under appropriate initial data which are compactly supported, by using a test function method and taking into account the effect of the damping term ($\frac{\mu}{\sqrt{1+|x|^2}}u_t$), we provide that in higher dimensions the blow-up region is given by $p \in (1, p_G(N+\mu)]$ where $p_G(N)$ is the Glassey exponent. Furthermore, we shall establish a blow-up region, independent of $\mu$ given by $p\in (1, 1+\frac{2}{N}),$ for appropriate initial data in the energy space with noncompact support., Comment: 18 pages

We study compact almost complex manifolds admitting a Hermitian metric satisfying an integral condition involving $\overline \partial$-harmonic $(0,1)$-forms. We prove that this integral condition is automatically satisfied, if the Hermitian metric on the compact almost complex manifold is strongly Gauduchon. Under the further assumption that the almost complex structure is integrable, we show that the integral condition for a Gauduchon metric is equivalent to be strongly Gauduchon. In particular, a compact complex surface with a Gauduchon metric satisfying the integral condition is automatically K\"ahler. If we drop the integrability assumption on the complex structure, we show that there exists a compact almost complex $4$-dimensional manifold with a Hermitian metric satisfying the integral condition, but which does not admit any compatible almost-K\"ahler metric.

Some results on nonexistence of nontrivial solutions to some time and space fractional differential evolution equations with transformed space argument are obtained via the nonlinear capacity method. The analysis is then used for a $2\times 2$ system of equations with transformed space arguments.

Machine learning potentials have emerged as a means to enhance the accuracy of biomolecular simulations. However, their application is constrained by the significant computational cost arising from the vast number of parameters compared to traditional molecular mechanics. To tackle this issue, we introduce an optimized implementation of the hybrid method (NNP/MM), which combines neural network potentials (NNP) and molecular mechanics (MM). This approach models a portion of the system, such as a small molecule, using NNP while employing MM for the remaining system to boost efficiency. By conducting molecular dynamics (MD) simulations on various protein-ligand complexes and metadynamics (MTD) simulations on a ligand, we showcase the capabilities of our implementation of NNP/MM. It has enabled us to increase the simulation speed by 5 times and achieve a combined sampling of one microsecond for each complex, marking the longest simulations ever reported for this class of simulation.

In this paper we investigate the existence of invariant SKT, balanced and generalized K\"ahler structures on compact quotients $\Gamma \backslash G$, where $G$ is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and $\Gamma$ is a lattice of $G$. We first obtain a characterization of Hermitian almost nilpotent Lie algebras $\mathfrak{g}$ whose nilradical $\mathfrak{n}$ has one-dimensional commutator and a classification result in real dimension six. Then, we study the ones admitting SKT and balanced structures and we examine the behaviour of such structures under flows. In particular, we construct new examples of compact SKT manifolds. Finally, we prove some non-existence results for generalized K\"ahler structures in real dimension six. In higher dimension we construct the first examples of non-split generalized K\"ahler structures (i.e., such that the associated complex structures do not commute) on almost abelian Lie algebras. This leads to new compact (non-K\"ahler) manifolds admitting non-split generalized K\"ahler structures., Comment: 45 pages, v3: final version to appear in Journal of Algebra

The paper investigates a class of a semilinear wave equation with time-dependent damping term ($-\frac{1}{{(1+t)}^{\beta}}\Delta u_t$) and a nonlinearity $|u|^p$. We will show the influence of the the parameter $\beta$ in the blow-up results under some hypothesis on the initial data and the exponent $p$ by using the test function method. We also study the local existence in time of mild solution in the energy space $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$.

A Hermitian metric on a complex manifold is called SKT (strong K\"ahler with torsion) if the Bismut torsion $3$-form $H$ is closed. As the conformal generalization of the SKT condition, we introduce a new type of Hermitian structure, called \emph{locally conformal SKT} (or shortly LCSKT). More precisely, a Hermitian structure $(J,g)$ is said to be LCSKT if there exists a closed non-zero $1$-form $\alpha$ such that $d H = \alpha \wedge H$. In the paper we consider non-trivial LCSKT structures, i.e. we assume that $d H \neq 0$ and we study their existence on Lie groups and their compact quotients by lattices. In particular, we classify 6-dimensional nilpotent Lie algebras admitting a LCSKT structure and we show that, in contrast to the SKT case, there exists a $6$-dimensional $3$-step nilpotent Lie algebra admitting a non-trivial LCSKT structure. Moreover, we show a characterization of even dimensional almost abelian Lie algebras admitting a non-trivial LCSKT structure, which allows us to construct explicit examples of $6$-dimensional unimodular almost abelian Lie algebras admitting a non-trivial LCSKT structure. The compatibility between the LCSKT and the balanced condition is also discussed, showing that a Hermitian structure on a 6-dimensional nilpotent or a $2n$-dimensional almost abelian Lie algebra cannot be simultaneously LCSKT and balanced, unless it is K\"{a}hler., Comment: references updated

In this paper, we consider the Cauchy problem in $\mathbb{R}^N$, $N\geq1$, for semi-linear Schr\"odinger equations with space-time fractional derivatives. We discuss the nonexistence of global $L^1$ or $L^2$ weak solutions in the subcritical and critical cases under some conditions on the initial data and the nonlinear term. Furthermore, the nonexistence of local $L^1$ or $L^2$ weak solutions in the supercritical case are studied.