Institute of Mathematics AS CR, v. v. i.
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Exponential and Logarithmic Functions – Discovery and Argumentation
No full textsummary:V článku uvádzame príklady úloh a prezentácie učiva o exponenciálnej a logaritmickej funkcii, ktoré obsahujú podnety pre žiacke objavovanie a argumentáciu. Pozornosť venujeme zvlášť pojmu exponenciálneho rastu a logaritmickej škály, viaceré úlohy využívajú vzťahy medzi grafom logaritmickej a exponenciálnej funkcie, resp. medzi grafmi exponenciálnych, resp. logaritmických funkcií s rôznym základom. V článku uvádíme příklady úloh a prezentace učiva o exponenciální a logaritmické funkci, které obsahují podněty pro žákovské objevování a argumentaci. Pozornost věnujeme zvláště pojmu exponenciálního růstu a logaritmické stupnice, více úloh využívá vztahy mezi grafem logaritmické a exponenciální funkce, resp. mezi grafy exponenciálních (logaritmických) funkcí při různých základech.summary:The article presents examples of tasks and presentations of the curriculum on exponential and logarithmic functions, which contain stimuli for student discovery and argumentation. We pay particular attention to the concept of exponential growth and logarithmic scale, several tasks use the relationships between the graph of the logarithmic and exponential function, or between the graphs of exponential (logarithmic) functions with different bases
A novel study of properties, functional equations and families of fuzzy implications through strict monotonicity
Full text linksummary:It is well known that monotonicity has been an important defining criterion for fuzzy logic connectives, such as fuzzy negations, t-norms, t-conorms and fuzzy implications. Also, a stronger version of monotonicity, namely strict monotonicity, establishes some significant representation theorems of continuous fuzzy negations, continuous t-norms and continuous t-conorms. In this work, we propose the strict monotonicity for fuzzy implications and investigate some necessary conditions on fuzzy implications to fulfill the same. Also, the relationship between the basic properties, functional equations of fuzzy implications and the strict monotonicity will be investigated. Further, we examine the strict monotonicity for fuzzy implications that do come from different families of fuzzy implications and show that the strict monotonicity is a necessary condition for fuzzy polynomial implications, fuzzy rational implications and some subclasses of and -generated fuzzy implications
Non-injective inductions and restrictions of modules over finite groups
No full textsummary:We extend the inductions and restrictions of modules over finite groups to non-injective group homomorphisms, establishing transitivity, Frobenius reciprocity, Mackey's formula, etc
Maximal and Riesz potential operators on Musielak-Orlicz spaces over unbounded metric measure spaces
No full textsummary:We are concerned with the boundedness of modified Hardy-Littlewood maximal operator and Sobolev inequalities for the variable Riesz potentials on Musielak-Orlicz spaces over unbounded metric measure spaces, as an improvement of our recent paper, see T. Ohno, T. Shimomura (2025a). As an application, we give the boundedness of and Sobolev inequalities for for the multi-phase functionals with variable exponents where , , and are log-Hölder continuous, for , and , and are nonnegative, bounded, and Hölder continuous
-prime ideals of commutative rings
No full textsummary:Let be a commutative ring with identity and , be positive integers. We introduce the class of -prime ideals which lies properly between the classes of prime and -closed ideals. A proper ideal of is called -prime if for , implies either or Several characterizations of this new class with many examples are given. Analogous to primary decomposition, we define the \hbox {-decomposition} of ideals and show that every ideal in an -Noetherian ring has an -decomposition. Furthermore, the -prime avoidance theorem is proved
The risk probability optimal problem for infinite discounted semi-Markov decision processes
Full text linksummary:This paper investigates the risk probability minimization problem for infinite horizon semi-Markov decision processes (SMDPs) with varying discount factors. First, we establish the standard regularity condition to guarantee the state process is non-explosive. Furthermore, based only on the non-explosion of the state process, we use value iteration technique to establish the optimality equation satisfied by the value function, and prove the uniqueness of the solution and the existence of the risk probability optimal policy. Our condition is weaker than the first arrival condition commonly used in existing literature. Finally, we develop a value iteration algorithm to compute the value function and optimal policy, and illustrate the feasibility and effectiveness of the algorithm through a numerical example
On the sequences of -generalized Fibonacci numbers
Full text linksummary:We consider a new family of recurrence sequences, the -generalized Fibonacci numbers. These sequences naturally extend the well-known sequences of -generalized Fibonacci numbers and generalized -order Pell numbers. Further, we obtain the Binet formula and study the asymptotic behavior of the dominant root of the characteristic equation. The proof methods exploit pairs of characteristic polynomials which allow several auxiliary results
