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Feit and Tits (1978) proved that a nontrivial projective representation of minimal dimension of a finite extension of a finite nonabelian simple group G factors through a projective representation of G, except for some groups of Lie type in characteristic 2; the exact exceptions for G were determined by Kleidman and Liebeck (1989). We generalise this result in two ways. First we consider all low-dimensional projective representations, not just those of minimal dimension. Second we consider all characteristically simple groups, not just simple groups.

The continued fraction mapping maps a number in the interval [0, 1) to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space (mathbb {R}), the continued fraction mapping is a homeomorphism onto the product space (mathbb {N}^{mathbb {N}}), where (mathbb {N}) is a discrete space. In this short note, we examine the continuity of the continued fraction mapping, addressing both irrational and rational points of the unit interval.

In this paper, we establish two new anisotropic Liouville type theorems for the stationary 3D Navier–Stokes equations. Under certain anisotropic integrability conditions on the components of the velocity, we show that the solution is trivial. Our results extend and improve the results of Chae (Appl Math Lett 142:108655, 2023) and Luo and Yin (Arch Ration Mech Anal 224:209–231, 2017).

Let (T_{a}) be a pseudo-differential operator with symbol a. When (ain S^m_{rho ,1},m=n(rho -1)), it is well known that (T_{a}) is not always bounded on ({L^1}({mathbb {R}^n})). However, under extra assumptions on a, we prove that (T_{a}) is bounded on ({L^p}({mathbb {R}^n})) for (1 le p le infty ) when (a in {L^infty }S_rho ^{n(rho - 1)}(omega )).

Let p be an odd prime and P a Sylow p-subgroup of a finite group G. If P is either metacyclic or each of its elements of order p lies in the center, then (N_G(P)) controls strong G-fusion in P, as established in Martino and Priddy (Math. Z. 225(2):277–288, 1997, Theorems 2.7 and 4.1). First, we provide alternative proofs for these results without relying on the Alperin fusion theorem, thereby simplifying the theoretical framework. Second, we establish an equivalence for the control of fusion in terms of a permutation character. Specifically, we define the permutation character induced by the action of G on (Syl_p(G)) as the Sylow p-character of G. Now let (Pin Syl_p(G)), and (N_G(P)le N le G ). Set (chi ,psi ) to be the Sylow p-characters of G and N, respectively. Then we prove that N controls G-fusion in P if and only if (frac{chi (g)}{psi (g)}=frac{|C_G(g)|}{|C_N(g)|} text { for all } gin P.) In the case that N is a p-local subgroup, further results are obtained.

Klarner and Rivest showed that the growth of the number of polyominoes, also known as Klarner’s constant, is at most (2+2sqrt{2}<4.83) by viewing polyominoes as a sequence of twigs with appropriate weights given to each twig and studying the corresponding multivariate generating function. In this short note, we give a simpler proof by a recurrence on an upper bound. In particular, we show that the number of polyominoes with n cells is at most G(n) with (G(0)=G(1)=1) and for (nge 2),

It should be noted that G(n) has multiple combinatorial interpretations in the literature.

We investigate ((2+1))-dimensional oscillatory integral operators characterized by a certain class of polynomial phase functions. By employing Stein’s complex interpolation, we derive sharp (L^2rightarrow L^p) decay estimates for these operators.

The non-inner automorphism conjecture (NIAC) and the divisibility problem (DP) are two famous problems in the study of finite p-groups. We observe that the verification of NIAC can be reduced to purely non-abelian finite p-groups. In connecting NIAC with DP, as a consequence of our results obtained on NIAC, we provide a short and cohomology-free proof of a theorem of Yadav, which states that if G is a finite p-group such that (G, Z(G)) is a Camina pair, then |G| divides (|{{,mathrm{!Aut},}}(G)|).

The work is concerned with the concept of stochastic maximal (L^p)-regularity. In fact, we proved that such a regularity is stable under admissible observation operators.

The aim of this work is to describe new classes of disjoint hypercyclic Toeplitz operators on the Hardy space (H^2({mathbb {D}})) in the unit disc ({mathbb {D}}). We examine the disjoint hypercyclicity of the coanalytic Toeplitz operators, the Toeplitz operators with the symbols (a{bar{z}}+b+cz), where (a,b,cin {mathbb {C}}), and the Toeplitz operators with the symbols (p(bar{z})+varphi (z)), where p is a polynomial and (varphi in H^infty (mathbb {D})). The hypercyclicity of these classes of Toeplitz operators has been characterized by G. Godefroy and J. Shapiro (J. Funct. Anal., 98, 1991), S. Shkarin (arXiv:1210.3191v1, 2012), and A. Baranov and L. Lishanskii (Results Math., 70, 2016), respectively. Based on their results, we first provide a criterion for the bounded linear operators to be disjoint hypercyclic. Using this criterion, we then establish certain conditions under which the aforementioned classes of Toeplitz operators are disjoint hypercyclic in terms of their symbols.