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- ItemMonotonicity character of convex combinations of two sequences generated by the square of the cube root function(2025-10-15) Stevic, Stevo; Iričanin, Bratislav D.; Kosmala, Witold; Šmarda, ZdeněkThere is a natural connection between the finite sums of the reciprocals of the squares of the cube roots of positive natural numbers and two definite integrals. By using the differences between the sums and the integrals are formed in a natural way two sequences of real numbers converging to the same finite limit. We completely determine the monotonicity character of the sequences which are obtained by convex combinations of the two sequences, in an elegant way. We also present an interesting result about monotonicity character of a real function on the interval , which is used in the proof of the main result. The results are important for solving difference equations.
- ItemSolutions with prescribed mass to Kirchhoff equations: generic double-behaviour nonlinearities(2025-10-30) Cai, Li; Radulescu, VicentiuIn this paper, we study the Kirchhoff equation {-(a+b integral(3)(R)|del u|(2)dx)Delta u+lambda u=f(u), x is an element of R-3, integral(3)(R)|u|(2)dx=c(2), x is an element of R-3, (lambda,u)is an element of RxH(1)(R-3), where a,b,c>0, lambda is an element of R is unknown as a Lagrange multiplier. We provide generic assumptions about the nonlinearity f(u), which correspond to the L-2-subcritical, L-2-critical, L-2-supercritical, and Sobolev critical cases. Making use of the minimization of the energy functional over a linear combination of the Nehari and Pohozaev constraints intersected with the product of the closed balls in L-2(R-3) of radii c, we prove the existence of normalized solutions to the Kirchhoff equation.
- ItemInquiry-Based Linear Algebra Teaching and Learning in a Flipped Classroom Framework: A Case Study(Taylor & Francis, 2024-07-17) Fredriksen, Helge; Rebenda, Josef; Rensaa, Ragnhild Johanne; Pettersen, PetterFlipped Classroom (FC) approaches, which utilize video distribution via modern internet platforms, have recently gained interest as a pedagogical framework. Inquiry Based Mathematics Education (IBME) has proven to be a valid form of task design to motivate active learning and enhance classroom interactivity. This article presents a practical combination of introductory videos and inquiry-based class activities adoptable in a basic linear algebra course for stimulating students’ exploration of the underlying mathematics. Teachers’ and students’ work addressed in the article was realized in two case studies in engineering programs in Norway and the Czech Republic. The learning objective was to connect different interpretations of the matrix equation Ax=b, which is often perceived as challenging for engineering students. Feedback from classroom sessions, interviews, and questionnaires encourage further research and inspired us as teachers to closely examine the mathematics behind the task design.
- ItemOn a product-type system of difference equations of second order solvable in closed form(BioMed Central, 2015-10-12) Stevič, Stevo; Iričanin, Bratislav; Šmarda, ZdeněkSufficient conditions for solving a system of difference equations of second order in closed form are given. Moreover, sufficient conditions for existence of periodic solutions and asymptotic stability of solutions are given as well.
- ItemSolving certain classes of Lane-Emden type equations using differential transformation method(Springer Nature, 2012-10-05) Khan, Yasir; Svoboda, Zdeněk; Šmarda, ZdeněkIn this paper, differential transformation method (DTM) is applied to solve singular initial problems represented by certain classes of Lane-Emden type equations. Some new differential tranformation formulas for certain exponential and logarithmic nonlinearities are derived.