London Math Research Articles

A new lower bound for the number of conjugacy classes

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G G is a finite group, p p is a prime dividing the group order, and k ( G ) k(G) is the number of conjugacy classes of G G , then k ( G ) ≥ 2 p − 1 k(G)\geq 2\sqrt {p-1} , and they proved this conjecture for solvable G G and showed that it is sharp for those primes p p for which p − 1 \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p p , and we prove it for solvable groups, and when p p is large, also for arbitrary groups.

Shrinking target problems in d-decaying Gauss-like iterated function systems

Let Φ={ϕn}n≥1 be an infinite iterated function system (IFS) on [0,1] satisfying the open set condition with the open set (0,1) and let J be its attractor. For any x∈J, there is a unique integer sequence {ωn(x)}n≥1, called the digit sequence of x, such thatx=limn→∞⁡ϕω1(x)∘⋯∘ϕωn(x)(1) except at countably many points. In this paper, we investigate the shrinking target problem in the infinite iterated function system with a polynomial decay of the derivative. More precisely, we consider the size of the setW(T,x0)={x∈J:Tn(x)∈Itn(x0) for infinitely many n∈N}, where T:J⟼J is the expanding map induced by the left shift and Itn(x0) denotes the tn-th cylinder of x0∈J. As a continuation of one result of Li et al. (2014) [11], we obtain the exact Hausdorff dimension of the set W(T,x0) in some general infinite function systems.

Common best proximity point theorems in Hausdorff topological spaces

In the present paper, we have obtained common best proximity point theorems of nonself maps in Hausdorff topological space. Further, our results extend the results due to Gerald F. Jungck, thereby proving a generalized version of Kirk’s theorem (J. London Math. 1(1):107–111, 1969).

Discrete spectrum of the magnetic Laplacian on almost flat magnetic barriers

The magnetic Laplacian with a step magnetic field has been intensively studied during the last years. We adapt the construction introduced by Bonnaillie-Noël et al. [Bull. London Math. Soc. 56, 2529 (2024)] to prove the existence of bound states of a new effective operator involving a magnetic step field on a domain with an almost flat magnetic barrier. This result emphasizes the fact that even a small non-smoothness of the discontinuity region can cause the appearance of eigenvalues below the essential spectrum. We also give an example where this effective operator arises.

A note on ideal C$$^*$$-completions and amenability

AbstractFor a discrete group G, we consider certain ideals $$\mathcal {I}\subset c_0(G)$$ I ⊂ c 0 ( G ) of sequences with prescribed rate of convergence to zero. We show that the equality between the full group C$$^*$$ ∗ -algebra of G and the C$$^*$$ ∗ -completion $$\textrm{C}^*_{\mathcal {I}}(G)$$ C I ∗ ( G ) in the sense of Brown and Guentner (Bull. London Math. Soc. 45:1181–1193, 2013) implies that G is amenable.

On the space-time analyticity of the inhomogeneous heat equation on the half space with Neumann boundary conditions

We consider the inhomogeneous heat equation on the half-space R + d with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol [A direct approach to Gevrey regularity on the half-space. In: Partial differential equations in fluid mechanics. Cambridge: Cambridge Univ. Press; 2018. p. 268–288. (London math. soc. lecture note ser.; vol. 452).].

Extremal functions for the critical Trudinger–Moser inequality with logarithmic Kernels

In this paper we study Moser–Trudinger type inequalities for some nonlocal energy functionals in presence of a logarithmic convolution potential, when the domain is a ball. In particular, we perform a blow-up analysis to prove existence of extremal functions in the borderline case of critical growth. Using this, we sharpen the results in [S. Cingolani and T. Weth J. London Math. Soc. 105 (2022) 1897–1935] under critical growth assumptions and give answers to some questions left open in [S. Cingolani and T. Weth J. London Math. Soc. 105 (2022) 1897–1935].

Standing waves and jets on a sessile, incompressible bubble

We show numerically that large amplitude, shape deformations imposed on a spherical-cap, incompressible, sessile gas bubble pinned on a rigid wall can produce a sharp, wall-directed jet. For such a bubble filled with a permanent gas, the temporal spectrum for surface-tension-driven, linearized perturbations has been studied recently in D. Ding and J. Bostwick [“Oscillations of a partially wetting bubble,” J. Fluid Mech. 945, A24 (2022)]. in the potential flow limit. We reformulate this as an initial-value problem analogous in spirit to classical derivations in the inviscid limit by Kelvin [“Oscillations of a liquid sphere,” Math. Phys. Papers 3, 384–386 (1890)], Rayleigh [“On the instability of jets,” Proc. London Math. Soc. s1-10, 4–13 (1878)] or by Prosperetti [“Viscous effects on small-amplitude surface waves,” Phys. Fluids 19, 195–203 (1976)] and Prosperetti [“Motion of two superposed viscous fluids,” Phys. Fluids 24, 1217–1223 (1981)] for the viscous case. The first test of linear theory is reported here by distorting the shape of the pinned, spherical cap employing eigenmodes obtained from linearized theory. These are employed as the initial shape distortion of the bubble in numerical simulations. It is seen that linearized predictions show good agreement with nonlinear simulations at small distortion amplitude producing standing waves, which oscillate at the predicted frequency. Beyond the linear regime as the shape distortions are made sufficiently large, we observe the formation of a dimple followed by a slender, wall-directed jet, analogous to similar jets observed in other geometries from collapsing wave troughs [Farsoiya et al., “Axisymmetric viscous interfacial oscillations–theory and simulations,” J. Fluid Mech. 826, 797–818 (2017) and Kayal et al., “Dimples, jets and self-similarity in nonlinear capillary waves,” J. Fluid Mech. 951, A26 (2022).] This jet can eject with an instantaneous velocity exceeding nearly 20 times that predicted by linear theory. By projecting the shape of the bubble surface around the time instant of jet ejection, into the eigenspectrum we show that the jet ejection coincides with the nonlinear spreading of energy into a large number of eigenmodes. We further demonstrate that the velocity-field associated with the dimple plays a crucial role in evolving it into a jet and without which, the jet does not form. It is also shown that evolving the bubble shape containing a dimple but zero initial velocity-field everywhere, via linear theory, does not produce the jet. These conclusions accompanied by first principles analysis provide insight into the experimental observations of Prabowo and Ohl [“Surface oscillation and jetting from surface attached acoustic driven bubbles,” Ultrason. Sonochem. 18, 431–435 (2011)], where similar jets were reported earlier, albeit via acoustic forcing. Our inferences also complement well-known results of Naude and Ellis [“On the mechanism of cavitation damage by nonhemispherical cavities collapsing in contact with a solid boundary,” J. Fluids Eng. 83, 648–656 (1961)] and Plesset and Chapman [“Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary,” J. Fluid Mech. 47, 283–290 (1971)] demonstrating that wall-directed jets can be generated from volume preserving, shape deformations of a pinned bubble.

Gravity–capillary wave-making resistance on deep water

For supercritical cases (forcing speed > the minimum phase speed, 0.23 m/s), the problem of two-dimensional linear, inviscid gravity–capillary waves generated by a moving delta-function type pressure source is well known. Using harmonic functions or Fourier transform, Lamb [Hydrodynamics, 6th ed. (Cambridge University Press, 1993)] and Rayleigh [Proc. London Math. Soc. s1-15(1), 69–78 (1883)] detailed the steady-state full-space wave-profile solution using an artificial viscosity. Whitham [Linear and Nonlinear Waves (Wiley-Interscience, 1974)] presented the same solution for the region that is far-from-the-forcing using a slowly varying exponential function. For the same problem, but, considering not only supercritical but also subcritical cases, and, without using the artificial viscosity, the present work provides a detailed solution procedure to find full-space wave-profile solutions based on Fourier transform where complex integration is needed; different analytical expressions of the same wave profile will be provided depending on different paths. The associated wave-making resistance is shown to be equal to the integral of the product of a moving pressure source, and the resultant wave slope and is calculated in two ways. One is a direct calculation in the physical domain, which requires the wave-profile solution, and the other is an indirect calculation in the wavenumber domain, which does not require the wave-profile solution. For supercritical cases, short and long sinusoidal waves are calculated ahead of and behind the pressure source. The associated wave-making resistance decreases toward a certain constant as the forcing speed increases and the associated required power features a minimum at the forcing speed of 0.3027 m/s. For subcritical cases, a simple symmetric dimple is calculated and the wave-making resistance becomes zero due to its symmetry.

Galois actions of finitely generated groups rarely have model companions

AbstractWe show that if is a finitely generated group such that its profinite completion is “far from being projective” (i.e., the kernel of the universal Frattini cover of is not a small profinite group), then the class of existentially closed ‐actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is “far from being projective,” the main result of this paper corrects an error in our paper, Beyarslan and Kowalski (Proc. London Math. Soc., (2) 118 (2019), 221–256), by showing the negation of Theorem 3.26 in that paper.

Identifying 1-rectifiable measures in Carnot groups

Abstract We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of M. Badger and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of analyst’s traveling salesman theorem, which characterizes the subsets of rectifiable curves in R 2 {{\mathbb{R}}}^{2} (P. W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1–15), in R n {{\mathbb{R}}}^{n} (K. Okikiolu, Characterization of subsets of rectifiable curves in R n {{\bf{R}}}^{n} , J. London Math. Soc. (2) 46 (1992), no. 2, 336–348), or in an arbitrary Carnot group (S. Li) in terms of local geometric least-squares data called Jones’ β \beta -numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R n {{\mathbb{R}}}^{n} that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space.

A class of t-weight codes and its applications

In this paper, we constructed a class of [Formula: see text]-weight linear codes over [Formula: see text] under the homogeneous weight metric by their generator matrices, where [Formula: see text] and [Formula: see text] The Gray images of some class of these codes over [Formula: see text] are [Formula: see text]-ary nonlinear codes, which have the same weight distributions as that of the two-weight [Formula: see text]-ary linear codes of type SU1 in the sense of [R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18(2) (1986) 97–122]. Also, we obtained the minimum distance of the dual codes of the constructed codes. Further, we discussed some optimal linear codes over [Formula: see text] with respect to Plotkin-type bound from the constructed codes when [Formula: see text] Furthermore, we investigated the applications in strongly regular graphs and secret sharing schemes.

Distribution of recursive matrix pseudorandom number generator modulo prime powers

Given a matrix A ∈ G L d ( Z ) A\in \mathrm {GL}_d(\mathbb {Z}) . We study the pseudorandomness of vectors u n \mathbf {u}_n generated by a linear recurrence relation of the form u n + 1 ≡ A u n ( mod p t ) , n = 0 , 1 , … , \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*} modulo p t p^t with a fixed prime p p and sufficiently large integer t ⩾ 1 t \geqslant 1 . We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of u n \mathbf {u}_n and control the p p -adic orders of their coefficients in polynomial representations.

Positivity of Sylow branching coefficients for symmetric groups

In this article we investigate the positivity of Sylow branching coefficients for symmetric groups when p=3. In particular, we complete the discussion begun by Giannelli and the author in [4, J. Algebra 506 (2018), 409–428] and developed in [5, J. London Math. Soc. (2) 103 (2021), 697–728] concerning the case of odd primes.

When are Baer modules extending?

The well-known notion of an extending module is closely linked to that of a Baer module. A right [Formula: see text]-module [Formula: see text] is called extending if every submodule of [Formula: see text] is essential in a direct summand. On the other hand, a right [Formula: see text]-module [Formula: see text] is called Baer if for all [Formula: see text], [Formula: see text] where [Formula: see text]. In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc. 21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: “a module[Formula: see text] is[Formula: see text]-nonsingular extending if and only if[Formula: see text] is[Formula: see text]-cononsingular Baer”. [Formula: see text] is called [Formula: see text]-nonsingular if [Formula: see text] such that [Formula: see text], [Formula: see text]. Moreover, [Formula: see text] is called [Formula: see text]-cononsingular if for any [Formula: see text] with [Formula: see text] for all [Formula: see text], implies [Formula: see text]. In view of this result, every Baer module which happens to be [Formula: see text]-cononsingular will automatically become an extending module. In this paper, our main focus is the study of [Formula: see text]-cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of [Formula: see text]-cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of [Formula: see text]-cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman’s result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided.

Corrigendum: On a problem of Lang for matrix polynomials

AbstractThis note is a corrigendum to the author's paper “On a Problem of Lang for Matrix Polynomials, Bull. London Math. Soc. 54 (2022), 1552–1567”.

Recovering affine linearity of functions from their restrictions to affine lines

Motivated by recent results of Tao–Ziegler [Discrete Anal. 2016] and Greenfeld–Tao (2022 preprint) on concatenating affine-linear functions along subgroups of an abelian group, we show three results on recovering affine linearity of functions $$f: V \rightarrow W$$ from their restrictions to affine lines, where V, W are $${\mathbb {F}}$$ -vector spaces and $$\dim V \geqslant 2$$ . First, if $$\dim V < |{\mathbb {F}}|$$ and $$f: V \rightarrow {\mathbb {F}}$$ is affine-linear when restricted to affine lines parallel to a basis and to certain “generic” lines through 0, then f is affine-linear on V. (This extends to all modules M over unital commutative rings R with large enough characteristic.) Second, we explain how a classical result attributed to von Staudt (1850 s) extends beyond bijections: If $$f: V \rightarrow W$$ preserves affine lines $$\ell $$ , and if $$f(v) \not \in f(\ell )$$ whenever $$v \not \in \ell $$ , then this also suffices to recover affine linearity on V, but up to a field automorphism. In particular, if $${\mathbb {F}}$$ is a prime field $${\mathbb {Z}}/p{\mathbb {Z}}$$ ( $$p>2$$ ) or $${\mathbb {Q}}$$ , or a completion $${\mathbb {Q}}_p$$ or $${\mathbb {R}}$$ , then f is affine-linear on V. We then quantitatively refine our first result above, via a weak multiplicative variant of the additive $$B_h$$ -sets initially explored by Singer [Trans. Amer. Math. Soc. 1938], Erdös–Turán [J. London Math. Soc. 1941], and Bose–Chowla [Comment. Math. Helv. 1962]. Weak multiplicative $$B_h$$ -sets occur inside all rings with large enough characteristic, and in all infinite or large enough finite integral domains/fields. We show that if R is among any of these classes of rings, and $$M = R^n$$ for some $$n \geqslant 3$$ , then one requires affine linearity on at least $$\left( {\begin{array}{c}n\\ \lceil n/2 \rceil \end{array}}\right) $$ -many generic lines to deduce the global affine linearity of f on $$R^n$$ . Moreover, this bound is sharp.

Cesàro exponents of mixed norm spaces

In 1934, G. H. Hardy and J. E. Littlewood calculated [Proc. London Math. Soc. (2) 36 (1934), pp. 516–531] the optimal Cesàro exponent for Hardy spaces. In this paper we calculate it for mixed norm spaces, hence including the Bergman spaces in particular. The main technical challenge lies in the analysis of the example needed for the critical case.

On the error term in the explicit formula of Riemann–von Mangoldt

We provide an explicit [Formula: see text] error term for the Riemann–von Mangoldt formula by making results of Wolke [On the explicit formula of Riemann–von Mangoldt, II, J. London Math. Soc. 2(3) (1983) 406–416] and Ramaré [Modified truncated Perron formulae, Ann. Math. Blaise Pascal 23(1) (2016) 109–128] explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.

Uniform Turán density of cycles

In the early 1980s, Erdős and Sós initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H H is the infimum over all d d for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least d d contains H H . In particular, they raise the questions of determining the uniform Turán densities of K 4 ( 3 ) − K_4^{(3)-} and K 4 ( 3 ) K_4^{(3)} . The former question was solved only recently by Glebov, Král’, and Volec [Israel J. Math. 211 (2016), pp. 349–366] and Reiher, Rödl, and Schacht [J. Eur. Math. Soc. 20 (2018), pp. 1139–1159], while the latter still remains open for almost 40 years. In addition to K 4 ( 3 ) − K_4^{(3)-} , the only 3 3 -uniform hypergraphs whose uniform Turán density is known are those with zero uniform Turán density classified by Reiher, Rödl and Schacht [J. London Math. Soc. 97 (2018), pp. 77–97] and a specific family with uniform Turán density equal to 1 / 27 1/27 . We develop new tools for embedding hypergraphs in host hypergraphs with positive uniform density and apply them to completely determine the uniform Turán density of a fundamental family of 3 3 -uniform hypergraphs, namely tight cycles C ℓ ( 3 ) C_\ell ^{(3)} . The uniform Turán density of C ℓ ( 3 ) C_\ell ^{(3)} , ℓ ≥ 5 \ell \ge 5 , is equal to 4 / 27 4/27 if ℓ \ell is not divisible by three, and is equal to zero otherwise. The case ℓ = 5 \ell =5 resolves a problem suggested by Reiher.