A submetric space is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space with the weak topology).
We demonstrate that on submetric spaces there exists a theory of convergence in probability, in law etc. equally effective as the Probability Theory on metric spaces. In the theory on submetric spaces the central role is played by a version of the Skorokhod almost sure representation, proved by the author some 25 years ago and in 2010 rediscovered by specialists in stochastic partial differential equations in the form of “stochastic compactness method”.

We approach the problem of integration for rough integrands and integrators, typically representing trajectories of stochastic processes possessing only some Hölder regularity of possibly low order, in the framework of para-control calculus. For this purpose, we first decompose integrand and integrator into Paley–Littlewood packages along the Haar–Schauder system. By careful estimation of the components of products of packages of the integrand and derivatives of the integrator we obtain a characterization of Young’s integral. For the most interesting case of functions with Hölder regularities that sum up to an order below 1 we have to employ the concept of para-control of integrand and integrator with respect to a reference function for which a version of antisymmetric Lévy area is known to exist. This way we obtain an interpretation of the rough path integral. Lévy areas being known for most frequently used stochastic processes such as (fractional) Brownian motion, this integral serves as a basis for pathwise stochastic calculus, as the integral in classical rough path analysis.

Let (S,+) be an abelian semigroup, let (H,+) be an abelian group which is uniquely 2-divisible, and let ϕ be an endomorphism of S. We find the solutions f,h: S→H of each of the functional equations
f(x+y) + f(x+ϕ(y)) = h(x) + f(y) + f◦ϕ(y), x,y∈S,
f(x+y) + f(x+ϕ(y)) = h(x) + 2f(y), x,y∈S,
in terms of additive and bi-additive maps. Moreover, as applications, we determine the solutions of some related functional equations.

The (fuzzy) filter of the Sheffer stroke Hilbert algebra was addressed by Oner, Katican and Borumand Saeid. The weak version of the filter was discussed by Jun and Oner. In this manuscipt, with the fuzzy version of the weak filter in mind, the notions of fuzzy weak filters and (∈.∈∨q)-fuzzy weak filters are introduced, and their properties are explored. Conditions under which t-level set, Q_t-set and t-∈∨q-set become weak filters are explored in relation to fuzzy weak filters and (∈.∈∨q)-fuzzy weak filters. The relationship and characterization of the fuzzy weak filter and the (∈.∈∨q)-fuzzy weak filter are investigated.

In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.

In this paper, we introduce one-parameter generalization of dualhyperbolic Jacobsthal numbers – dual-hyperbolic r-Jacobsthal numbers. We present some properties of them, among others the Binet formula, Catalan, Cassini, and d’Ocagne identities. Moreover, we give the generating function and summation formula for these numbers. The presented results are a generalization of the results for the dual-hyperbolic Jacobsthal numbers.

In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz-Sobolev spaces. What matters here is the behavior of the integral, not the space.

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0,∞) we consider the following integral transform
D(w,μ)(T) := ∫₀^∞w(λ)(λ+T)^-1dμ(λ),
where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.
Assume that A≥α>0, δ≥B>0 and 0<m≤B-A≤M for some constants α, δ, m, M. Then
0 ≤ -mD'(w,μ)(δ) ≤ D(w,μ)(A)-D(w,μ)(B) ≤ -MD'(w,μ)(α),
where D'(w,μ)(t) is the derivative of D(w,μ)(t) as a function of t>0.
If f:[0,∞)→ℝ is operator monotone on [0,∞) with f(0)=0, then
0 ≤ m/δ²[f(δ)-f'(δ)δ] ≤ f(A)A^-1-f(B)B^-1 ≤ M/α²[f(α)-f'(α)α].
Some examples for operator convex functions as well as for integral transforms D(·,·) related to the exponential and logarithmic functions are also provided.

In this article, we derive a great number of identities involving the ω function counting distinct prime divisors of a given number n. These identities also include Pochhammer symbols, Fibonacci and Lucas numbers and many more.

In this paper, we present the Sugeno integral of Hermite–Hadamard inequality for the case of quasi-arithmetically convex (q-ac) functions which acts as a generator for all quasi-arithmetic means in the frame work of Sugeno integral.

Report of Meeting. The Twenty-second Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), February 1–4, 2023.