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We define the order of the double hypergeometric series, investigate the properties of the new confluent Kampé de Fériet series, and build systems of partial differential equations that satisfy the new Kampé de Fériet series. We solve the Cauchy problem for a degenerate hyperbolic equation of the second kind with a spectral parameter using the high-order Kampé de Fériet series. Thanks to the properties of the introduced Kampé de Fériet series, it is possible to obtain a solution to the problem in explicit forms.

Let 𝔼*
^{𝑑}
* denote the 𝑑-dimensional Euclidean space. The 𝑟-ball body generated by a given set in 𝔼

*is the intersection of balls of radius 𝑟 centered at the points of the given set. The author [Discrete Optimization 44/1 (2022), Paper No. 100539] proved the following Blaschke–Santaló-type inequalities for 𝑟-ball bodies: for all 0 <*

^{𝑑}*𝑘*<

*𝑑*and for any set of given 𝑑-dimensional volume in 𝔼

*the 𝑘-th intrinsic volume of the 𝑟-ball body generated by the set becomes maximal if the set is a ball. In this note we give a new proof showing also the uniqueness of the maximizer. Some applications and related questions are mentioned as well.*

^{𝑑}We discuss the outline of the shapes of graphs of χ ^{2} statistics for distributions of leading digits of irrational rotations under some conditions on *m*th convergent. We give some estimates of important coefficients *L _{k}
*’s, which determine the graphical shapes of χ

^{2}statistics. This means that the denominator

*q*of

_{m}*m*th convergent and the large partial quotient

*a*

_{m}_{+1}determine the outline of shapes of graphs, when we observe values of χ

^{2}statistics with step

*q*.

_{m}In this note, we introduce the concept of semi-*-IFP, the involutive version of semi-IFP, which is a generalization of quasi-*-IFP and *-reducedness of *-rings. We study the basic structure and properties of *-rings having semi-*-IFP and give results for IFPs in rings with involution. Several results and counterexamples are stated to connect the involutive versions of IFP. We discuss the conditions for the involutive IFPs to be extended into *-subrings of the ring of upper triangular matrices. In *-rings with quasi-*-IFP, it is shown that Köthe’s conjecture has a strong affirmative solution. We investigate its related properties and the relationship between *-rings with quasi-*-IFP and *-Armendariz properties.

In the present paper, we establish the convergence rates of the single logarithm and the iterated logarithm for martingale differences which give some further results for the open question in Stoica [6].

Let *n* ∈ ℕ. An element (*x*
_{1}, … , *x _{n}
*) ∈

*E*is called a

^{n}*norming point*of

*n*-linear forms on

*E*. For

Norm(*T*) is called the *norming set* of *T*.

Let

In this paper, we classify Norm(*T*) for every

This article indicates another set-theoretic formula, solely in terms of union and intersection, for the set of the limits of any given sequence (net, in general) in an arbitrary *T*
_{1} space; this representation in particular gives a new characterization of a *T*
_{1} space.

We give all solutions of completely multiplicative functions ƒ , g, for which the equation *Ag*(*n* + 1) = *B*ƒ (*n*) + *C* holds for every *n* ∈ ℕ. We also study the equation *G*(*p* + 1) = *F*(*p* − 1) + *D* and we prove some results concerning it.

We consider a graph whose vertices are legally colored using *k* colors and ask if the graph contains a *k*-clique. As it turns out this very special type of *k*-clique problem is in an intimate connection with constructing schedules. The practicality this clique search based construction of schedules is checked by carrying out numerical experiments.

Assume that *A _{j}
*,

*j*∈ {1, … ,

*m*} are positive definite matrices of order

*n*. In this paper we prove among others that, if 0 <

*l I*≤

_{n}*A*,

_{j}*j*∈ {1, … ,

*m*} in the operator order, for some positive constant

*l*, and

*I*is the unity matrix of order

_{n}*n*, then

where *Pk* ≥ 0 for *k* ϵ {1, …, *m*} and