In the original publication of this article [Monatsh. Math. 209 (2026), 185–213.], oversight occurred in the typographical notation of the extremal functions, where the presence of the absolute value in both the upper limit of integration and the differential inadvertently compromised their analyticity. Additionally, the parameter α was omitted from its absolute value notation in several subsequent upper bounds and norm estimates throughout Section 3. This corrigendum rectifies these expressions to their rigorous analytic forms and ensures the mathematical precision of the sharpness computations. These corrections do not alter the validity of the main results or the underlying analytical proofs.
The polynomial chaos expansion method has recently gained prominence for solving stochastic differential equations. Representing solutions as stochastic processes through orthogonal series expansions raises two key questions: how to compute fundamental probabilistic properties – such as expectation, variance, covariance, and finite-dimensional distributions (via cumulative distribution functions or probability density functions) – and how to model input parameters (e.g., coefficients, driving forces, and initial conditions) with prescribed probability distributions within the chaos expansion framework. This paper addresses both issues. Building on the Wiener–Itô chaos expansion with Hermite polynomials, we derive explicit formulas for computing the cumulative distribution function and probability density function from chaos expansion coefficients. Because moments and moment determinacy play a critical role in these expressions, we also provide a formula for evaluating nth-order moments from chaos expansion coefficients. Conversely, we propose an algorithm for determining the Wiener–Itô chaos expansion of a random variable with a known distribution, using a generalized Legendre polynomial expansion. Both approaches are illustrated with simple yet instructive examples that demonstrate the applicability of our results.
We utilize a proof-theoretically tame approach to the dual of an abstract Banach space in systems amenable to methods from proof mining, as recently introduced by the author, to provide similar such systems with accompanying logical metatheorems on the extraction of uniform quantitative information from proofs pertaining to the theory of monotone set-valued operators on Banach spaces as introduced by Browder. With that, we finally extend proof mining methods to this important class of objects which are at the heart of many seminal results from nonlinear functional analysis, and the metatheorems presented here in particular provide the first logical basis for a range of recent applications of proof mining methods to this branch of mathematics. Further, we provide a characterization of the extensionality principle for these operators using the analytical notion of maximality, extending previous analogous results for accretive operators on Banach spaces and monotone operators on Hilbert spaces, and with that further illustrate the central importance and special position of extensionality issues in proof mining applications dealing with set-valued operators.
In this paper, we investigate the existence and uniqueness of monotone bounded solutions for a differential equation arising from stratified arctic gyres. We show that the existence of the solution can be established for the equations with more weak hypotheses. Using the Dominated Convergence Theorem we received a uniqueness result with some generalization of Osgood’s conditions and it seems that this generalization can play a key role in the proofs of uniqueness theorems.
We prove composition results for the Saphar p-summing multilinear operators. All these results have no linear analogue. For some concrete operators we apply the general results to find the necessary and sufficient conditions for these to be Saphar p-summing.
A polynomial D∈ℤ[x] is called Pellian over ℤ if the polynomial Pell equation P^2-DQ^2=1 has a non-trivial solution in ℤ[x] . It is an open problem to determine all the polynomials D∈ℤ[x] that are Pellian over ℤ . In the literature, the study of the Pellian polynomials over ℤ has been mostly restricted to monic polynomials as there are many key difficulties involved in the non-monic case. In this article, we overcome those difficulties and characterise all the quadratic polynomials in ℤ[x] that are Pellian over ℤ by providing a necessary and sufficient condition for Pellianity over ℤ . This seems to be the first instance of a comprehensive study on the Pellianity over ℤ of non-monic quadratic polynomials in ℤ[x] . A key difficulty is to find solutions in ℤ[x] , when the necessary condition is satisfied. Our proof exhibits a constructive method to do so.
Let E_1, … , E_s be s, not necessary distinct, elliptic curves over ℚ . We give upper bounds on the frequency of s-tuples of points in E_1(ℚ)×…× E_s(ℚ) whose denominators or x-coordinates are multiplicatively dependent. More precisely, we give such bounds in two scenarios: one in which we fix s non-torsion ℚ -rational points P_i ∈ E_i(ℚ) and arbitrary ℚ -rational points Q_i ∈ E_i(ℚ) , i =1, … , s , and we count s-tuples (n_1P_1+Q_1,… , n_sP_s+Q_s) ∈ E_1(ℚ) ×…× E_s(ℚ) with n_1, … , n_s in an arbitrary interval of length N, and the second in which we count points (P_1,… ,P_s) ∈ E_1(ℚ) ×…× E_s(ℚ) of bounded canonical height.
We show that there exists a unique non-trivial separable Hilbert space of analytic functions on ℂ^2n for which certain irreducible unitary representations of the twisted Heisenberg group act in a uniformly bounded manner. Moreover, we prove Sarason’s product problem for twisted Fock spaces, which are non-radial weighted spaces distinct from conventional Fock and Fock-Sobolev spaces.
In this paper, we demonstrate the existence, uniqueness and non-existence of the general Orlicz Minkowski problem involving p=0 for a class of Orlicz functions. The results are based on variational arguments.
Let γ _s(G) and Z_s(G) denote the s-th terms of the lower and upper central series of a group G, respectively. A classical theorem by R. Baer states that if Z_s(G) has finite index in G, then γ _s+1(G) is also finite. In this paper, we prove that if G is a generalized soluble group such that γ _s(G)/(γ _s(G) ∩ Z_t(G)) has finite rank r for some s, t, then the rank of γ _s+t(G) is finite and (r, s, t)-bounded. Moreover, a corresponding result replacing the finite-rank assumption by the condition that γ _s(G)/(γ _s(G) ∩ Z_t(G)) is a Chernikov group of bounded size is also obtained. These results extend recent generalizations of the classical Baer’s theorem.
This paper studies the convolution operator on weighted Hahn sequence spaces. The boundedness and compactness of these operators, together with the multiplier algebras of the weighted Hahn space and its dual, are investigated. A complete characterization of the spectrum and fine spectrum is obtained, with illustrative examples. The introduction of the weighted framework leads to the emergence of new multiplier and spectral properties.
Starting from the general equations (in a rotating frame) for a compressible, viscous fluid, coupled to an equation of state and the first law of thermodynamics, we present a derivation of a general set of governing equations based solely on the thin-shell approximation. This uses a single small parameter (ε), measuring the thinness of the shell, keeping all other parameters fixed as ε→ 0 . The resulting equations retain the essentials of the spherical geometry, but are inviscid (at leading order) and allow an arbitrary variation of density with height. This system is rewritten, producing a form that is suitable for seeking solutions which can be accessed (in principle) without further approximation. Arguments are presented which show that a solution exists which recovers the structure and properties of the hexagon that surrounds Saturn’s northern pole. The elucidation of some of the details requires the use of a numerical approach, because the resulting system of equations does not possess a suitable solution expressible in closed form. The hexagon structure is obtained, as well as a representation of the random-looking flows that sit outside the hexagon. The associated properties of the jet stream within the hexagon, and its temperature field, are described, including some requirements that the internal heat source must satisfy in order to maintain the hexagonal structure. All these results are compared with the available data, demonstrating an encouraging level of agreement. A critique of this work, and its shortcomings, are discussed, together with suggestions for future study.
We study the modulational instability of smooth, small-amplitude periodic traveling wave solutions to the generalized Fornberg-Whitham equation. Our approach is based on applying spectral perturbation theory to the associated linearization. We derive a modulational instability index that depends on the nonlinear parameter and the wave number of the underlying wave, and prove that the sufficiently small periodic traveling waves of the generalized Fornberg-Whitham equation are spectrally unstable to long-wavelength perturbations when the modulational instability index is negative, this confirms the well-known Benjamin-Feir instability for the generalized Fornberg-Whitham equation.
In this paper, we derive the multifractal spectra for the recurrence rate of the first return time in expanding Markov maps, including the cases returning to the ball and cylinder. The results can be applied to some dynamical systems on fractal sets, such as cookie-cutter systems.
We provide some exact solutions for the governing equations of geophysical fluid dynamics representing azimuthal equatorial flows with discontinuous density and surface tension. The fluid domain is bounded above by the free surface-which is an unknown of the problem and below by the ocean bottom. The fluid consists of two layers, each having a continuous density and are separated by an interface along which the densities do not match. After deriving formulas for the velocity and for the pressure the existence of functions describing the free surface and the interface is shown by a functional analytic approach.
We provide a complete description of the automorphism group Aut (W) of a Coxeter group W admitting a star-shaped finite Coxeter diagram. We prove that each automorphism decomposes as a product of inner and diagram automorphisms, along with three additional types: transvections and two families of partial conjugations. Furthermore, we investigate the splitting criteria for the natural short exact sequence 1 →Inn (W) →Aut (W) →Out (W) → 1 . Using Moussong’s criteria for hyperbolicity, we show that these groups possess the R_∞ -property. Finally, we establish rigidity properties for these groups using known techniques and provide a solution to the isomorphism problem within the class of star-shaped Coxeter systems. This generalizes the framework established for odd Coxeter groups, investigated in Naik and Singh [32], to a significantly broader class that incorporates even edge labels.
A Pollard-like exact solution for nonlinear trapped lee waves in the f-plane approximation is derived. The solution is described in the Lagrangian framework. Moreover, we analyze the physical properties of the solution, including the dispersion relation, the vorticity distribution, the density and pressure gradients, which are examined in detail in Section 3.
