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

The purpose of this note is to present a construction of sequences which do not have metric Poissonian pair correlations (MPPC) and whose additive energies grow at rates that come arbitrarily close to a threshold below which it is believed that all sequences have MPPC. A similar result appears in work of Lachmann and Technau and is proved using a totally different strategy. The main novelty here is

A subgroup H of a topological group G is called cocompact (or uniform) if the quotient space \(G/{\overline{H}}\) is compact, where \({\overline{H}}\) denotes the closure of H in G. The aim of this note is to determine all topological groups with the property that every non-trivial discrete subgroup is cocompact.

Let G be a non-abelian finite simple group. In addition, let \(\Delta _G\) be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with distinct subgroups joined by an edge if and only if they intersect non-trivially. We prove that the diameter of \(\Delta _G\) has a tight upper bound of 5, thereby resolving a question posed by Shen (Czechoslov Math J 60(4):945–950

We show that all elementary lattices that are free \(\mathbb {Z}_p C_p\)-modules admit an orthogonal decomposition into a sum of a free unimodular and a p-modular \(\mathbb {Z}_p C_p\)-lattice.

We obtain a result regarding the existence of regular orbits for self-dual modules, thus generalizing a theorem of A. Espuelas for symplectic modules.

This short note is concerned with a quasilinear diffusion equation under initial and Neumann boundary value condition. To be more precise, the authors establish a gradient maximum principle of classical solutions via the maximum principle and Hopf’s lemma. The result generalizes a recent work obtained by Kim (Proc Amer Math Soc 145:1203–1208, 2017).

Let G be a group. The orbits of the natural action of \({{\,\mathrm{Aut}\,}}(G)\) on G are called automorphism orbits of G, and the number of automorphism orbits of G is denoted by \(\omega (G)\). Let G be a virtually nilpotent group such that \(\omega (G)< \infty \). We prove that \(G = K \rtimes H\) where H is a torsion subgroup and K is a torsion-free nilpotent radicable characteristic subgroup

In this note, we shall study semistable aCM bundles on an abelian surface of Picard number 1.

We obtain a necessary and sufficient condition for the occurrence of the Bohr phenomenon for Banach space valued analytic functions defined on the open unit disk \({{\mathbb {D}}}\). We also discuss its interesting consequences, which include the connection of the Bohr phenomenon with the strong maximum modulus principle, as well as some more specific criteria regarding the existence of nonzero Bohr

This paper deals with the asymptotic behavior of solutions of the half-linear differential equation $$\begin{aligned} (p(t)|x'|^{\alpha }\mathrm {sgn}\,x')' + q(t)|x|^{\alpha }\mathrm {sgn}\,x = 0, \quad t \ge t_{0}. \end{aligned}$$ It will be shown that, for any solution x(t) of this equation, there are only two types of asymptotic behavior as \(t\rightarrow \infty \).

We show that on an Abelian variety over an algebraically closed field of positive characteristic, the obstruction to lifting an automorphism to a field of characteristic zero as a morphism vanishes if and only if it vanishes for lifting it as a derived autoequivalence. We also compare the deformation space of these two types of deformations.

A profinite group is index-stable if any two isomorphic open subgroups have the same index. Let p be a prime, and let G be a compact p-adic analytic group with associated \(\mathbb {Q}_p\)-Lie algebra \(\mathcal {L}(G)\). We prove that G is index-stable whenever \(\mathcal {L}(G)\) is semisimple. In particular, a just-infinite compact p-adic analytic group is index-stable if and only if it is not virtually

Let G be a finite group of even order that has no 2-rank 1. We will prove, using only elementary methods, that there is an involution \(t \in G\) such that \(|G| < |C_{G}(t)|^{6}\).

Let A be a PI-algebra. If A is an associative algebra, the sequence of codimensions \(c_n(A), n=1,2,\ldots ,\) of A is asymptotically non-decreasing. For the non-associative case, there are examples of PI-algebras whose sequence of codimensions is not eventually non-decreasing. For a associative PI-algebra A with involution \(*: A\rightarrow A\), it was recently shown that its sequence of \(*\)-codimensions

We study a geometric flow on curves, immersed in \({\mathbb {R}}^3\), that have strictly positive torsion. The evolution equation is given by $$\begin{aligned} X_{t}=\frac{1}{\sqrt{\tau }} \mathbf{B} \end{aligned}$$ where \(\tau \) is the torsion and \(\mathbf{B} \) is the unit binormal vector. In the case of constant curvature, we find all of the stationary solutions and linearize the PDE for the

We study \((\sigma ,\tau )\)-derivations of a group ring RG where G is a group with center having finite index in G and R is a semiprime ring with 1 such that either R has no torsion elements or that if R has p-torsion elements, then p does not divide the order of G and let \(\sigma ,\tau \) be R-linear endomorphisms of RG fixing the center of RG pointwise. We generalize Main Theorem 1.1 of Chaudhuri

We investigate the behavior of the second fundamental form of an isometric immersion of a space form with negative curvature into a space form so that the extrinsic curvature is negative. If the immersion has flat normal bundle, we prove that its second fundamental form grows exponentially.

A closed Riemann surface S (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map \(\beta :S \rightarrow E\) with at most one branch value, where E is a genus one Riemann surface. In this case, \((S,\beta )\) is called an origami pair and \(\mathrm{Aut}(S,\beta )\) is the group of conformal automorphisms \(\phi \) of S such that \(\beta =\beta \circ \phi \).

Let \(\mathcal {S}\) be a nonempty semigroup endowed with a binary associative operation \(*\). An element e of \(\mathcal {S}\) is said to be idempotent if \(e*e=e\). Originated by one question of P. Erdős to D.A. Burgess: If \(\mathcal {S}\) is a finite semigroup of order n, does any \(\mathcal {S}\)-valued sequence T of length n contain a nonempty subsequence the product of whose terms, in some

We show that every solvable group is a subgroup of some monomial real group. This extends a result of Dade, who proved that every solvable group is a subgroup of a monomial group.

In this paper, we first prove the local existence of strong solutions to the 3D Boussinesq equations in a bounded domain with Navier boundary conditions. Then we show the global stability of strong large solutions under a suitable integral condition.

In a commutative ring R with unity, a unit u is called exceptional if \(u-1\) is also a unit. For \(R = {\mathbb {Z}}/n{\mathbb {Z}}\) and for any \(f(X) \in {\mathbb {Z}}[X]\), an element \({\overline{u}} \in {\mathbb {Z}}/n{\mathbb {Z}}\) is called an “f-exunit” if \(gcd(f(u),n) = 1\). Recently, we obtained the number of representations of a non-zero element of \({\mathbb {Z}}/n{\mathbb {Z}}\) as

We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in Bonheure et al. (Trans Amer Math Soc 370(10):7081–7127, 2018). We apply it to generalized p-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions. Based on a generalized hidden convexity result, we show that uniqueness holds among strongly positive solutions

In this article, we prove an isoperimetric inequality for the harmonic mean of the first \((n-1)\) nonzero Steklov eigenvalues on bounded domains in n-dimensional hyperbolic space. Our approach to prove this result also gives a similar inequality for the first n nonzero Steklov eigenvalues on bounded domains in n-dimensional Euclidean space.

The Liouville–von Neumann equation describes the change in the density matrix with time. Interestingly, this equation was recently regarded as a wave equation for wave functions but not as an equation for density functions. This setting leads to an extended form of the Schrödinger wave equation governing the motion of a quantum particle. In this paper, we obtain the integrability of the wave propagator