International Journal of Mathematics and Mathematical Sciences

Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. The fundamental theorem of algebra tells us that every nonconstant polynomial with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to if we restrict the coefficients to be real. Let and be the vector space of all polynomials of degree or less with real coefficients. In this article, we give explicit forms of polynomials in such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on which preserve real roots of polynomials in a certain subset of .

For parabolic Shilov equations with continuous coefficients, the problem of finding classical solutions that satisfy a modified initial condition with generalized data such as the Gelfand and Shilov distributions is considered. This condition arises in the approximate solution of parabolic problems inverse in time. It linearly combines the meaning of the solution at the initial and some intermediate points in time. The conditions for the correct solvability of this problem are clarified and the formula for its solution is found. Using the results obtained, the corresponding problems with impulse action were solved.

For a Gaussian prime and a nonzero Gaussian integer with and , it was proved that if where , , belong to a complete residue system modulo , and the digits and satisfy certain restrictions, then the polynomial is irreducible in . For any quadratic field , it is well known that there are explicit representations for a complete residue system in , but those of the case are inapplicable to this work. In this article, we establish a new complete residue system for such a case and then generalize the result mentioned above for the ring of integers of any imaginary quadratic field.

The concept of frames in Hilbert spaces continues to play a very interesting role in many kinds of applications. In this paper, we study the notion of dual continuous -g-frames in Hilbert spaces. Also, we establish some new properties.

Any continuous function with values in a Hausdorff topological space has a closed graph and satisfies the property of intermediate value. However, the reverse implications are false, in general. In this article, we treat additional conditions on the function, and its graph for the reverse to be true.

Combinatorial sums and binomial identities have appeared in many branches of mathematics, physics, and engineering. They can be established by many techniques, from generating functions to special series. Here, using a simple mathematical induction principle, we obtain a new combinatorial sum that involves ordinary powers, falling powers, and binomial coefficient at once. This way, and without the use of any complicated analytic technique, we obtain a result that already exists and a generalization of an identity derived from Sterling numbers of the second kind. Our formula is new, genuine, and several identities can be derived from it. The findings of this study can help for better understanding of the relation between ordinary and falling powers, which both play a very important role in discrete mathematics.