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Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.

During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.

Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

Let k be a field of characteristic $$ p\not =0 $$. For a (purely) inseparable extension K / k the notion of modularity, defined by M.E. Sweedler in the 60s, is a very important property, very much like being Galois for a separable extension. We have defined, together with M. Chellali, a generalization of the notion of modularity, called lower quasi-modularity: K / k is lower quasi-modular (lq-modular) if for some finite extension $$k'$$ over k we have that $$K/k'$$ is modular. In subsequent papers M. Chellali and the author have studied various properties of lq-modular field extensions, including the existence of lq-modular closures in case $$[k{:}k^p]$$ is finite. In this paper we prove a similar result, without the hypothesis on k but with extra assumptions on K / k: the extension needs to be q-finite, that is, there must exist an integer M such that for every positive integer n the field $$K\cap k^{p^{-n}}$$ is generated by at most M elements on k. A number of properties of lq-modular closures are determined and examples are presented illustrating the results.

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kahler metric associated with the Kahler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.

In 1933, Lehmer exhibited the polynomial $$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$ with Mahler measure $$\mu _0>1$$ . Then he asked if $$\mu _0$$ is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form $$\chi _A$$ , that is, Coxeter polynomials of a finite dimensional algebra A (for instance $$L(z)=\chi _{\mathbb {E}_{10}}$$ ). A polynomial in $$\mathbb {Z}[T]$$ which is either cyclotomic or with Mahler measure $$\ge \mu _0$$ is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for $$\chi _A$$ to be Lehmer.

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

In this paper, we show that the number of monic integer polynomials of degree \(d \ge 1\) and height at most H which have no real roots is between \(c_1H^{d-1/2}\) and \(c_2 H^{d-1/2}\), where the constants \(c_2>c_1>0\) depend only on d. (Of course, this situation may only occur for d even.) Furthermore, for each integer s satisfying \(0 \le s < d/2\) we show that the number of monic integer polynomials of degree d and height at most H which have precisely 2s non-real roots is asymptotic to \(\lambda (d,s)H^{d}\) as \(H \rightarrow \infty \). The constants \(\lambda (d,s)\) are all positive and come from a recent paper of Bertok, Hajdu, and Pethő. They considered a similar question for general (not necessarily monic) integer polynomials and posed this as an open question.

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 $${{\mathcal {A}}}$$ , we will show that the set of topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}^*$$ and on $${{\mathcal {A}}}_M^*$$ have the same cardinality. The main applications are concerned with the quantum group algebra $$L^1({\mathbb {G}})$$ of a locally compact quantum group $${\mathbb {G}}$$ . In particular, we obtain some characterizations of compactness of $${\mathbb {G}}$$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $$L^1_M({\mathbb {G}})$$ or on its bidual in some senses.

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. The condition is in terms of the sum of two characteristics involving the counting function. The sharpness of this condition is discussed and examples are presented to illustrate it.

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

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 u has a quasi-continuous representative $${\tilde{u}}$$ . Let $$a \in L^1(\Omega )$$ be such that $$a \ge 0$$ a.e. in $$\Omega $$ . Then if $$\begin{aligned} (-\Delta )^s u + au \ge 0 \quad \text {a.e.} \text { in } \Omega \end{aligned}$$ and $${\tilde{u}} = 0$$ on a subset of positive $$H^s$$ -capacity of $$\Omega $$ , then $$u = 0$$ a.e. in $$\Omega $$ .

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 that for J-positive invertible operators A, B acting on a Hilbert space of arbitrary dimension, the inequality $$\begin{aligned} \frac{A + B}{2} \ge ^J A \sharp ^J B \end{aligned}$$ holds, where $$A \sharp ^J B:= J \bigl ( (JA) \sharp (JB) \bigr )$$ . Several examples involving Pauli matrices are provided to illustrate the main results.

Recently, the different types of unbounded convergences (uo, un, uaw, uaw*) 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 order continuity and reflexivity of Banach lattices. Some related results are obtained as well., Comment: 9 pages

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 growth, bounded norm of the second fundamental form, and that satisfies $$|A|^2H(H-\lambda )\le H^2/2$$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions, if the reverse inequality holds, then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete $$\lambda $$ -hypersurface with $$H\ge 0$$ must either be a hyperplane or a generalized cylinder.

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 $$\mathbb S_1^{n+1}\hookrightarrow \mathbb S_p^{n+p}$$ . Our approach is based on a suitable lower estimate of the Cheng-Yau operator acting on the square norm of the traceless second fundamental form of such a spacelike submanifold.

The paper contains a study of weighted exponential inequalities for differentially subordinate martingales, under the assumption that the underlying weight satisfies Muckenhoupt’s condition$$A_{\infty }$$A∞. 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. Because of this generality, we need to handle jump parts of processes which forces us to construct a Bellman function satisfying a stronger condition than local concavity. As a corollary, we will establish some new weighted$$L^p$$Lpestimates for differential subordinates of bounded martingales.

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 $��(I^2)\ge 9$ for the number of minimal generators of $I^2$ with $��(I)\geq 6$. Recently, Gasanova constructed monomial ideals such that $��(I)>��(I^n)$ for any positive integer $n$. In reference to them, we construct a certain class of monomial ideals such that $��(I)>��(I^2)>\cdots >��(I^n)=(n+1)^2$ for any positive integer $n$, which provides one of the most unexpected behaviors of the function $��(I^k)$. The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of $R/I^n$ descends., 10 pages

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.

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 a sum of two f-exunits for a particular infinite family of polynomials $$f(X) \in {\mathbb {Z}}[X]$$ . In this paper, we complete this problem by proving a similar formula for any non-constant polynomial $$f(X) \in {\mathbb {Z}}[X]$$ .

We revive the concept of Lambda-freeness of Mlotkowski, which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper [SW] where the quantum symmetries underlying these mixtures of classical and free independences will be considered., Comment: We rewrote and shortened the earlier version. The third version contains mainly the results which are new compared to the paper of Mlotkowski

We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in $\mathbb{P}^3$, the initial gap interval was found by Xu (see [7] in References), and the next one in our previous paper (see [4] in References), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach., 9 pages, submitted preprint

In a recent paper, we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper, we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm–Liouville hypergroups, etc. A non-commutative hypergroup is also considered.

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 Schrodinger wave equation governing the motion of a quantum particle. In this paper, we obtain the integrability of the wave propagator arising from the Liouville–von Neumann equation in this setting.

In this paper, we mainly investigate the conjugation of the sylowizer that was introduced by Gaschutz (Math Z 122(4):319–320, 1971) and study the p-supersolvability of finite groups by analyzing the intersection between $$O^{p}(G)$$ and sylowizers of p-subgroups. As a continuation of research (Lei and Li in Arch Math (Basel) 114:367–376, 2020), we also give some characterizations on p-nilpotent groups by using the permutability of a sylowizer of a p-subgroup.

Let $$\mu _{M, D}$$ be the self-affine measure generated by an expanding integer matrix $$M\in M_{2}(\mathbb {Z})$$ and an integer three-element digit set $$D=\{(0,0)^T, (\alpha ,\beta )^T,(\gamma ,\eta )^T\}$$ . In this paper, we show that if $$3\mid \det (M)$$ and $$3\not \mid \alpha \eta -\beta \gamma $$ , then $$L^2(\mu _{M,D})$$ has an orthogonal basis of exponential functions if and only if $$M^*\varvec{u}\in 3\mathbb {Z}^2$$ , where $$\varvec{u}=(\eta -2\beta ,\; 2\alpha -\gamma )^T$$ .

In this paper, we study when a composition operator or a weighted composition operator on a Banach space of holomorphic functions is a Ritt operator or an unconditional Ritt operator. It turns out that for composition operators or weighted composition operators on a Banach space of holomorphic functions, if a composition operator or a weighted composition operator is a Ritt operator, then it is also an unconditional Ritt operator.

Two subgroups A and B of a group G are called msp-permutable if the following statements hold: AB is a subgroup of G; the subgroups P and Q are mutually permutable, where P is an arbitrary Sylow p-subgroup of A and Q is an arbitrary Sylow q-subgroup of B, $${p\ne q}$$ . In the present paper, we investigate groups that are factorized by two msp-permutable subgroups. In particular, the supersolubility of the product of two supersoluble msp-permutable subgroups is proved.

Let $$I_n(G)$$ denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$ and $$I_p(G) =I_p(S)$$ for some odd prime divisor p, then $$|G|=|S|$$ ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.

Recently, Araya, Celikbas, Sadeghi, and Takahashi proved a theorem about the vanishing of self extensions of finitely generated modules over commutative Noetherian rings. The aim of this paper is to obtain extensions of their result over algebras.

Let $$(R,{\mathfrak {m}},k)$$ be a Noetherian local ring of dimension $$d\ge 4$$ . Assume that $$2\le i \le d-2$$ is an integer and $$x_1,\ldots ,x_i$$ is a part of a system of parameters for R. Let $$\Upsilon _i$$ denote the set of all prime ideals $${\mathfrak {p}}$$ of R such that $$\dim R/{\mathfrak {p}}=i+1$$ , $${\text {Supp}}H^i_{(x_1,\ldots ,x_i)R}(R/{\mathfrak {p}})=\{{\mathfrak {m}}\}$$ , and $$\dim _{k} {\text {Soc}}_R H^i_{(x_1,\ldots ,x_i)R}(R/{\mathfrak {p}})=\infty $$ . In this paper, it is shown that $$\Upsilon _i$$ is an infinite set.

Let A be a positive bounded linear operator acting on a complex Hilbert space $${\mathcal {H}}$$ . Our aim in this paper is to prove some A-numerical radius inequalities of bounded linear operators acting on $${\mathcal {H}}$$ when an additional semi-inner product structure induced by A is considered. In particular, an alternative proof of a recent result proved in Moslehian et al. (Linear Algebra Appl 591:299–321 2020) is given.

In this paper, we will prove a Liouville theorem for poly-harmonic functions on $${{\mathbb {R}}}^{n}_{+}$$ with Navier boundary conditions, that is, the nonnegative poly-harmonic functions u satisfying $$u(x)=o(|x|^{3})$$ at $$\infty $$ must assume the form $$\begin{aligned} u(x)=C x_{n} \end{aligned}$$ in $$\overline{{{\mathbb {R}}}^{n}_{+}}$$ , where $$n\ge 2$$ and C is a nonnegative constant. The assumption $$u(x)=o(|x|^{3})$$ at $$\infty $$ is optimal for us to derive the super poly-harmonic properties of u.

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, we prove the Besse conjecture for compact manifolds with pinched Weyl curvature. Moreover, we shall conclude that such a conjecture is true if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

T. Bandman and Yu. G. Zarhin have proved that for every word $$w \in F_n, w\notin F_n^2$$, the corresponding word map $${\tilde{w}}: \mathrm {PGL}_2^n(K)\rightarrow \mathrm {PGL}_2(K)$$ is surjective if K is an algebraically closed field of characteristic zero (here $$F_n$$ is a free group of rank n and $$F_n^i$$ is ith member of the derived series). For words $$w\in F_n^2, n >1,$$ which are not decomposable into a product $$w = w_1w_2$$ of two words with independent variables, there are only two examples when the corresponding word map for the algebraic group $$\mathrm {PGL}_2$$ is surjective. In this paper, we construct infinite recursive sequences of indecomposable words $$\{w_m\}_{m\in {{\mathbb {N}}}}$$ in $$F_2$$ such that the word maps $${\tilde{w}}_m$$ are surjective for the algebraic group $$\mathrm {PGL}_2$$ and, if $$w_m\in F_2^i$$, then $$w_{m+1} \in F_2^{i+1}$$. Also, we construct infinite recursive sequences of indecomposable words $$\{w^\prime _m\}_{m\in {{\mathbb {N}}}}$$ in $$F_3$$ such that word maps $${\tilde{w}}^\prime _m$$ are surjective for the algebraic group $${{\text {SL}}}_2$$ and, if $$w^\prime _m\in F_3^i$$, then $$w^\prime _{m+1} \in F_3^{i+1}$$.

In this paper, we consider a truncated version of the Timoshenko beam model and we prove an inverse inequality in order to obtain the controllability of the total system. The present system is new in the control and stabilization setting according to recent contributions due to Almeida Junior and Ramos (Z Angew Math Phys 68(145):31, 2017). The advantage here relies on mathematical results which do not depend on the classical condition between wave propagation velocities.

In this paper, we shall investigate a semilinear elliptic boundary blow-up problem $$\Delta u=a(x)|u|^{p-1}u+h(x)$$ in $$\Omega $$ and $$u|_{\partial \Omega }=\infty $$, where $$\Omega $$ is a smooth bounded domain of $$\mathbb {R}^{N}$$. The weight a(x) and the nonhomogeneous term h(x) may be unbounded near the boundary. Furthermore, h(x) may change sign and a(x) may vanish in $$\Omega $$. The existence of a large solution for the problem under some assumptions on a(x) and h(x), and a consequent nonexistence result are established. We also prove the uniqueness of the solution.

In this paper, it is proved that the finite group G is solvable if cod $$(\chi ) \le p_{\chi }\cdot \chi (1)$$ for any nonlinear irreducible character $$\chi $$ of G, where $$p_{\chi }$$ is the largest prime divisor of $$|G:\mathrm{ker} \chi |$$ .

In a recent paper (Carlier et al. in ESAIM Control Optim Calc Var 18(3):611–620, 2012), an interpolation flow between an evolution by convexity and the geometric flow of motion by principal negative curvature was informally proposed. It is also expected that the geometric flow will eventually convexify the sub-level sets of the initial function $$u_0$$, yielding the quasiconvex envelope of $$u_0$$. In this note, we establish existence and uniqueness of the interpolation flow under appropriate conditions and provide a rigorous proof for its limit behaviour. In addition, we show by example that, contrary to intuition, the proposed geometric flow does not always convexify the sub-level sets of $$u_0$$.

In this paper, given $$f \in BMO$$, for all possible $$0< p< q

As an application of the curve shortening flow, this paper will show an inequality for the maximum curvature of a smooth simple closed curve on surfaces.

Let $\mathcal{G}$ be the class of all finite groups and consider the function $\psi'':\mathcal{G}\longrightarrow(0,1]$, given by $\psi''(G)=\frac{\psi(G)}{|G|^2}$, where $\psi(G)$ is the sum of element orders of a finite group $G$. In this paper, we show that the image of $\psi''$ is a dense set in $[0, 1]$. Also, we study the injectivity and the surjectivity of $\psi''$., Comment: 7 pages

A subgroup S of a group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. The main aim of this paper is to investigate the influence of p-sylowizers on the structure of finite groups. We obtained some new characterizations of p-nilpotent and supersolvable groups by the permutability of the p-sylowizers of some p-subgroups. In addition, we determined all p-sylowizers of arbitrary p-subgroups for the supersolvable groups.

What is the largest cardinal of a two-by-two equilateral point-set in an n-dimensional Minkowski space? It is conjectured that this cardinal is at least $$n+1$$, and several spaces are tight in this regard, such as the Euclidean space. In this paper, we prove in normed spaces X with the $$(H_1,\dots ,H_{n-1})$$-2-intersection property the existence of $$n+1$$ equilateral point-sets of large diameter inscribed to the unit ball B. This extends the construction of Makeev (J Math Sci (N Y) 140:548–550, 2007) in dimension 4.

This paper studies gradient estimates for positive solutions of the nonlinear elliptic equation $$\begin{aligned} \Delta _V(u^p)+\lambda u=0,\quad p\ge 1, \end{aligned}$$on a Riemannian manifold (M, g) with k-Bakry–Emery Ricci curvature bounded from below. We consider both the case where M is a compact manifold with or without boundary and the case where M is a complete manifold.

Given a prime p, an integer $$H\in [1,p)$$, and an arbitrary set $${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$$, where $${\mathbb {F}} _p$$ is the finite field with p elements, let $$J(H,{\mathcal {M}} )$$ denote the number of solutions to the congruence $$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$for which $$x,y\in [1,H]$$ and $$m,n\in {\mathcal {M}} $$. In this paper, we bound $$J(H,{\mathcal {M}} )$$ in terms of p, H, and the cardinality of $${\mathcal {M}} $$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).

In this paper, we compute the idempotents of the ring $$F_+(G)$$ as defined by Boltje (J Algebra 206:293–343, 1998) in the particular case when the Green biset functor F is such that for all subgroups H of a finite group G, F(H) is a torsion-free ring, finitely-generated as an Abelian group, and has only the trivial idempotents. In this case, the only idempotents in $$F_+(G)$$ are those arising from the Burnside ring B(G).