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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.

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.

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 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.

The characterization of dual Steffensen–Popoviciu measures has so far been an open problem. As the main contribution of this paper, we give a complete characterization of dual Steffensen–Popoviciu measures on compact intervals.

The aim of this study is twofold. Initially, by employing a perturbation semigroup approach and admissible observation operators, a novel variation of constants formula is presented for the mild solutions of a specific set of integrodifferential equations in Banach spaces. Subsequently, utilizing this formula, an examination of the maximal (L^p)-regularity for such equations is conducted through the application of the sum operator method established by Da Prato and Grisvard. Importantly, it is demonstrated that the maximal (L^p)-regularity of an integrodifferential equation is equivalent to that of the same equation when the integral term is omitted. Furthermore, a finding concerning the strong solution of an initial value integrodifferential equation is provided when the initial condition pertains to the trace space.

We determine the average size of the eigenvalues of the Hecke operators acting on the cuspidal modular forms space (S_k(Gamma _0(N))) in both the vertical and the horizontal perspective. The “average size” is measured via the quadratic mean.

In this article, we investigate some isoperimetric-type inequalities related to the first eigenvalue of the fractional composite membrane problem. First, we establish an analogue of the renowned Faber–Krahn inequality for the fractional composite membrane problem. Next, we investigate an isoperimetric inequality for the first eigenvalue of the fractional composite membrane problem on the intersection of two domains - a problem that was first studied by Lieb (Invent Math 74(3):441–448, 1983) for the Laplacian. Similar results in the local case were previously obtained by Cupini–Vecchi (Commun Pure Appl Anal 18(5):2679–2691, 2019) for the composite membrane problem. Our findings provide further insights into the fractional setting, offering a new perspective on these classical inequalities.

Let (C_{n}) be the n-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each (kge 3), there are infinitely many k-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of (xin mathbb {N}_{+}) such that (C_{x}C_{x+1}) is a power-full number and prove that there are infinitely many such x. Moreover we present some numerical results which motivate further problems.