In this paper, we study the problem of the existence of left invariant Ricci flat metrics on nilpotent Lie groups. We mainly prove that any nilpotent Lie algebra obtained by a double extension of an Abelian Lie algebra admits at least one left invariant Ricci flat metric. As an application, we obtain certain new nilpotent Lie algebras which admit left invariant Ricci flat metrics.

If A, B are bounded linear operators on a complex Hilbert space, then we prove that $$\begin{aligned} w(A)\le & {} \frac{1}{2}\left( \Vert A\Vert +\sqrt{r\left( |A||A^*|\right) }\right) ,\\ w(AB \pm BA)\le & {} 2\sqrt{2}\Vert B\Vert \sqrt{ w^2(A)-\frac{c^2(\mathfrak {R}(A))+c^2(\mathfrak {I}(A))}{2} }, \end{aligned}$$ where \(w(\cdot ),\left\| \cdot \right\| \), and \(r(\cdot )\) are the numerical

Let \((\pi ,\,H_\pi )\) be a unitary representation of a locally compact group G. The characterization of the minimality of the topological center induced by \(\pi \) is already accessed. Also, some conditions which guarantee that the center is maximal are known. Here, we set our goal to characterizing the maximality of the center by concentrating on weakly almost G-periodic operators. In fact, we

We provide a useful extension of the Schönemann–Eisenstein irreducibility criterion. In the end, we illustrate our result through examples.

We compute the determinant of a matrix containing Ramanujan sums associated to the divisors of an integer n, and use this computation to prove a weak version of So’s conjecture on circulant graphs with integral spectrum.

In this paper, we treat an open problem related to the number of periodic orbits of Hamiltonian diffeomorphisms on closed symplectic manifolds, the so-called (generic) Conley conjecture. The generic Conley conjecture states that generically Hamiltonian diffeomorphisms have infinitely many simple contractible periodic orbits. We prove the generic Conley conjecture for very wide classes of symplectic

A Correction to this paper has been published: 10.1007/s00013-020-01457-0

Tartar gave an alternative proof of the Riesz–Thorin interpolation theorem for operators of strong types (1, 1) and \((\infty ,\infty )\). His method characterizes the \(L^{p}\) norm in terms of the Lebesgue spaces \(L^{1}\) and \(L^{\infty }\), and works not only for complex Lebesgue spaces but also for real Lebesgue spaces. The aim of this paper is to extend the proof for operators of strong types

There are many results that give information on the structure of a finite group in terms of properties that refer to its character degrees (or class sizes or element orders) and at most two primes. In this note, we try to extend some of these results considering three primes.

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation \(i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u\) in \(H^1\). The well-posedness theory in \(H^1\) has been intensively studied in recent years, but the currently known approaches do not work for the critical case \(\beta =(4-2\alpha )/(n-2)\). It is still an open problem.

Let \({{\mathcal {A}}}\) be a Banach algebra and let \(\varphi \) be a non-zero character on \({{\mathcal {A}}}\). Suppose that \({{\mathcal {A}}}_M\) is the closure of the faithful Banach algebra \({{\mathcal {A}}}\) in the multiplier norm. In this paper, topologically left invariant \(\varphi \)-means on \({{\mathcal {A}}}_M^*\) are defined and studied. Under some conditions on \({{\mathcal {A}}}\)

Let \((r_n)_{n=1}^\infty \) be a non-decreasing sequence of radii in \((0, \infty )\), and let \((\theta _n)_{n=1}^\infty \) be a sequence of independent random arguments uniformly distributed in \([0, 2\pi )\). In this paper, we establish a new sufficient condition on the sequence \((r_n)_{n=1}^\infty \) under which \((r_ne^{i\theta _n})_{n=1}^\infty \) is almost surely a zero set for Fock spaces

We give another proof of the recent result of Ringel, which asserts equality between the finitistic dimension and delooping level of Nakayama algebras. The main tool is the syzygy filtration method introduced in our earlier work.

In this paper, we extend compactness theorems of Cheeger, Gromov, Taylor, and Sprouse to the Bakry–Émery Ricci tensor and generalized quasi-Einstein tensors. Our results generalize previous results obtained by Yun and Wan.

Let G be a finite non-abelian p-group and let \(W(G)/Z(G)=\Omega _1(Z_2(G)/Z(G))\). A longstanding conjecture asserts that G admits a non-inner automorphism of order p. We confirm the conjecture in case Z(G) is not cyclic and W(G) is non-abelian.

In this short note, we show that given a cost function c, any coupling \(\pi \) of two probability measures where the second is a discrete measure can be associated to a certain bipartite graph containing a perfect matching, based on the value of the infinity transport cost \(\Vert c \Vert _{L^\infty (\pi )}\). This correspondence between couplings and bipartite graphs is explicitly constructed. We

It is shown that the Ornstein–Uhlenbeck operator perturbed by a multipolar inverse square potential $$\begin{aligned} A_{\Phi ,G}+V=\Delta -\nabla \Phi \cdot \nabla +G\cdot \nabla +\sum \limits _{i=1}^{n}\frac{c}{|x-a_{i}|^{2}} \end{aligned}$$ with suitable domain generates a quasi-contractive and positive analytic \(C_{0}\)-semigroup on the weighted space \(L^{2}(\mathbb {R}^{N},d\mu )\), where \(d\mu

We consider the blowup X(a, b, c) of a weighted projective space \({\mathbb {P}}(a,b,c)\) at a general nonsingular point. We give a sufficient condition for a curve to be a negative curve on X(a, b, c) in terms of \(\chi ({\mathcal {O}}_X(C))\). This can be applied to find the effective cone of X(a, b, c) and can serve as a starting point to prove the Mori dreamness of blowups of many weighted projective

In this paper, we obtain a version of the strong maximum principle for the spectral Dirichlet Laplacian. Specifically, let \(d \in \{1,2,3,\ldots \}\), \(s \in (\frac{1}{2},1)\), and \(\Omega \subset \mathbb {R}^d\) be open, bounded, connected with Lipschitz boundary. Suppose \(u \in L^1(\Omega )\) satisfies \(u \ge 0\) a.e. in \(\Omega \) and \((-\Delta )^s u\) is a Radon measure on \(\Omega \). Then

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. Every skew projection on that Hilbert space is contained in such an algebra and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm.

We prove a more general vector-valued variant of the following statement: If \(X_1\) and \(X_2\) are Choquet simplices such that the spaces of affine continuous functions \({\mathfrak {A}}(X_1, {\mathbb {R}})\) and \({\mathfrak {A}}(X_2, {\mathbb {R}})\) are isomorphic, then the cardinality of \({\text {ext}}X_1\) is equal to the cardinality of \({\text {ext}}X_2\).

We obtain a new bound for trilinear exponential sums with Kloosterman fractions which in some ranges of parameters improves that of S. Bettin and V. Chandee (2018). We also obtain a similar result for more general sums.

For quadratic forms in 4 variables defined over the rational function field in one variable over \(\mathbb C(\!(t)\!)\), the validity of the local-global principle for isotropy with respect to different sets of discrete valuations is examined.

In this paper, we show a quantitative version of the theorem stating that relatively weakly compact sets in \(\ell _1\) coincide with those having the Banach-Saks property. Namely, we prove that the measure of the weak noncompactness based on the Eberlein double limit criterion is equal to the measure of the non-Banach-Saks property defined by the arithmetic separation of sequences.

In this paper, the arithmetic-geometric mean inequalities of indefinite type are discussed. We show that for a J-selfadjoint matrix A satisfying \(I \ge ^J A\) and \({\mathrm{sp}}(A) \subseteq [1, \infty ),\) the inequality $$\begin{aligned} \frac{I + A}{2} \le ^J \sqrt{A} \end{aligned}$$ holds, and the reverse does for A with \(I \ge ^J A\) and \({\mathrm{sp}}(A) \subseteq [0, 1]\). We also prove

We give some regularity criterion (of a weak\(-L^{p}\) Serrin type) of a weak solution to the 3D Navier–Stokes equations in a bounded domain \(\Omega \subset \mathbb {R}^3\) with a smooth boundary. In particular, in case of the half space, we give a regularity condition of a weak solution with respect to a tangential component of the velocity flow vector.

Let K be a finitely generated extension of a field k of characteristic \(p\not =0\). In 1947, Dieudonné initiated the study of maximal separable intermediate fields. He gave in particular the form of an important subclass of maximal separable intermediate fields D characterized by the property \(K\subseteq k({D}^{p^{-\infty }})\), and which are called the distinguished subfields of K/k. In 1970, Kraft

We establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short survey of the recent field of maximal Fourier restriction theory.

We prove a general stability theorem for p-class groups of number fields along relative cyclic extensions of degree \(p^2\), which is a generalization of a finite-extension version of Fukuda’s theorem by Li, Ouyang, Xu, and Zhang. As an application, we give an example of a pseudo-null Iwasawa module over a certain 2-adic Lie extension.

In the paper, an absolutely convergent Dirichlet series whose shifts approximate a wide class of analytic functions is constructed. This series is close in the mean to the Riemann zeta-function.

Let (M, F) be a closed \(C^\infty \) Finsler manifold and \(\varphi \) its geodesic flow. In the case that \(\varphi \) is Anosov, we extend to the Finsler setting a Riemannian vanishing result of M. Gromov about the \(L^\infty \)-cohomology.

This work investigates the spectrality of a self-affine measure \(\mu _{M,D}\) and the related digit set D in the case when \(|\mathrm{det}(M)|=p^{\alpha }\) is a prime power and \(|D|=p\) is a prime, where \(\alpha \in {\mathbb {N}}\), and \(\mu _{M,D}\) is generated by an expanding matrix \(M\in M_{n}({\mathbb {Z}})\) and a digit set \(D\subset {\mathbb {Z}}^{n}\) of cardinality |D|. We obtain that

Let A be a group acting on a solvable group G and let N be an A-invariant normal subgroup of G such that \([G,A]\nsubseteq N\). We prove the inequality \(h([G,A])\le h([G,A]N/N)+h([N,A])\) where h(G) denotes the Fitting height of G. As an application of this result, we obtain several Fitting height inequalities. A new concept “fixed point free separability” and a new characteristic subgroup Y(G) is

Recently, the different types of unbounded convergences \((uo, un, uaw, ua w^*)\) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of linear functionals for these convergences. Then we define the corresponding unbounded dual spaces and get their exact form. Based on these results, we discuss

We strengthen a sufficient condition, due to R.S. Vieira, for a (reciprocal) polynomial \(R(x)=(x-\alpha _{1})(x-\alpha _{1}^{-1})\cdots (x-\alpha _{s})(x-\alpha _{s}^{-1})\in \mathbb {Z} [x],\) with degree at least 4, to have \((2s-2)\) zeros on the unit circle. Also, we show that R is a Salem polynomial if and only if there is a natural number n such that the polynomial \((x-(\alpha _{1}^{n}+\alpha

Let G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank \({\mathrm{rk}}(G)\le 3+\sqrt{n+1}\). Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by Biran and Cieliebak on subcritical polarisations of symplectic manifolds. Our proof is based on a simple homological argument using ideas of Kulkarni–Wood.

We consider groups of the form \({G} = {AB}\) with two locally cyclic subgroups A and B. The structure of these groups is determined in the cases when A and B are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case where A and B are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic subgroups

We show that for a quadratic polynomial \(f(x)=x^2-c\), where \(c=(8k+2)(8k+3)\) or \(c=(4k+1)(4k+2)+1\) with \(k\in {\mathbb {N}}\cup \{0\}\), the Galois group of the splitting field of each iterate \(f^n\) of f is isomorphic to the automorphism group of a complete binary rooted tree of height n.

In this paper, we establish a sharp integral inequality for n-dimensional closed spacelike submanifolds with constant scalar curvature immersed with parallel normalized mean curvature vector field in the de Sitter space \(\mathbb S_p^{n+p}\) of index p, and we use it to characterize totally umbilical round spheres \(\mathbb S^n(r)\), with \(r>1\), of \(\mathbb S_1^{n+1}\hookrightarrow \mathbb S_p^{n+p}\)

In this article, we study hypersurfaces \(\Sigma \subset {\mathbb {R}}^{n+1}\) with constant weighted mean curvature, also known as \(\lambda \)-hypersurfaces. Recently, Wei-Peng proved a rigidity theorem for \(\lambda \)-hypersurfaces that generalizes Le–Sesum’s classification theorem for self-shrinkers. More specifically, they showed that a complete \(\lambda \)-hypersurface with polynomial volume

We prove a solvability theorem for the Stieltjes problem on \(\mathbb {R}^d\) which is based on the multivariate Stieltjes condition \(\sum _{n=1}^\infty L(x_j^{n})^{-1/(2n)} =+\infty \), \(j=1,\dots ,d.\) This result is applied to derive a new solvability theorem for the moment problem on unbounded semi-algebraic subsets of \(\mathbb {R}^d\).

The paper contains a study of weighted exponential inequalities for differentially subordinate martingales, under the assumption that the underlying weight satisfies Muckenhoupt’s condition \(A_{\infty }\). The proof exploits certain functions enjoying appropriate size conditions and concavity. The martingales are adapted, uniformly integrable, and càdlàg - we do not assume any path-continuity restrictions

In this paper, we study vanishing and splitting results on a complete smooth metric measure space \((M^n,g,\mathrm {e}^{-f}\mathrm {d}v)\) with various negative m-Bakry-Émery Ricci curvature lower bounds in terms of the first eigenvalue \(\lambda _1(\Delta _f)\) of the weighted Laplacian \(\Delta _f\), i.e., \(\mathrm {Ric}_{m,n}\ge -a\lambda _1(\Delta _f)-b\) for \(0

In previous work (Dawsey et al. in J Algebra 533:339–343, 2019), the authors confirmed the speculation of J.G. Thompson that certain multiquadratic fields are generated by specified character values of sufficiently large alternating groups \(A_n\). Here we address the natural generalization of this speculation to the finite general linear groups \(\mathrm {GL}_m\left( {{\mathbb {F}}}_q\right) \) and

Unfortunately, the corresponding author was published incorrectly in the original article. The correct corresponding author is Dr. Bidisha Roy.

Estimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L of higher order \(2m\ge 4\) have been shown so far only when the principal part of L is the polyharmonic operator \((-\Delta )^m\). In the present note, it is shown that such kind of result still holds when

In this note, we obtain a simpler expression for the constant number given by Costin–Maz’ya on the sharp Hardy–Leray inequality for a class of solenoidal (namely divergence-free) vector fields, with respect to any radial power-weighted measure. The dependence of the constant on the weight exponent will be clear.

We construct whole-space extensions of functions in a fractional Sobolev space of order \(s\in (0,1)\) and integrability \(p\in (0,\infty )\) on an open set O which vanish in a suitable sense on a portion D of the boundary \({{\,\mathrm{\partial \!}\,}}O\) of O. The set O is supposed to satisfy the so-called interior thickness condition in \({{\,\mathrm{\partial \!}\,}}O {\setminus } D\), which is

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound \(\mu (I^2)\ge 9\) for the number of minimal generators of \(I^2\) with \(\mu (I)\ge 6\). Recently, Gasanova constructed monomial ideals such that \(\mu (I)>\mu (I^n)\) for any positive integer n. In reference

We estimate the number of primes represented by a general quadratic polynomial with discriminant \(\Delta \), assuming that the corresponding real character is exceptional.

We study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth \(\phi \), under the action of the centraliser \(C(\phi )\). As a corollary, we get that the same holds for the action of \( <\phi>\) on \(Min(\phi )\). Finally, we prove that the minimally displaced

A definition of the Laplacian on Cartesian products with mixed boundary conditions using quadratic forms is proposed. Its consistency with the standard definition for homogeneous and certain mixed boundary conditions is proved and, as a consequence, tensor representations of the corresponding Sobolev spaces of first order are derived. Moreover, a criterion for the domain to belong to the Sobolev space

Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset \(\Delta \) of a finite group G is called a p-base (where p is a prime) if \(\langle \Delta \rangle \) is a p-group and \(\mathrm {C}_G(\Delta )\) is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups

We investigate tensor products of random matrices, and show that independence of entries leads asymptotically to \(\varepsilon \)-free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular \(\varepsilon \) arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary

We study the validity of the distributivity equation $$\begin{aligned} ({\mathcal {A}}\otimes {\mathcal {F}})\cap ({\mathcal {A}}\otimes {\mathcal {G}}) ={\mathcal {A}}\otimes \left( {\mathcal {F}}\cap {\mathcal {G}}\right) , \end{aligned}$$ where \({\mathcal {A}}\) is a \(\sigma \)-algebra on a set X, and \({\mathcal {F}}, {\mathcal {G}}\) are \(\sigma \)-algebras on a set U. We present a counterexample

A simple graph defines an associated Bernstein superalgebra. Costa and Grishkov showed that graphs with isomorphic algebras are isomorphic using algebraic group techniques. We prove combinatorially that the superalgebra determines the graph for any field of characteristic not 2, and we describe the automorphisms of the superalgebra.

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra

This paper is devoted to studying the Tate–Hochschild cohomology for periodic algebras. We will prove that the Tate–Hochschild cohomology ring of a periodic algebra can be written as the localization of the non-negative part of the Tate–Hochschild cohomology ring.

We show that if all proper subgroups of a locally graded group G are finite-by-abelian-by-finite, then G contains a finite normal subgroup N such that all proper subgroups of G/N are abelian-by-finite. Then we apply this result to the study of groups which are minimal-non-\({\mathcal {P}}\) also for related group properties \({\mathcal {P}}\). Finally we see how for locally (soluble-by-finite) groups