• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.1065-1081
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• 2011

For any field F, a polynomial f $\in$ F[$x_1,x_2,{\ldots},x_k$] can be associated with a polytope, called its Newton polytope. If the polynomial f has integrally indecomposable Newton polytope, in the sense of Minkowski sum, then it is absolutely irreducible over F, i.e., irreducible over every algebraic extension of F. We present some results giving new integrally indecomposable classes of polygons. Consequently, we have some criteria giving many types of absolutely irreducible bivariate polynomials over arbitrary fields.

• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.133-159
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• 2024

In a previous work we studied generalized Stirling numbers of the second kind S^{(a₂,b₂,p₂)}_a1,b1 (p₁, k), where a₁, a₂, b₁, b₂ are given complex numbers, a₁, a₂ ≠ 0, and p₁, p₂ are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables B_p1,p2 (x₁, x₂), and Euler polynomials in two variables E_p1p2 (x₁, x₂). By using results for S^{(1,x₂,p₂)}_1,x1 (p₁, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case.

• Honam Mathematical Journal
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• v.42 no.2
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• pp.251-268
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• 2020

In this paper, we estimate the degree of approximation by means of the complete modulus of continuity, the partial modulus of continuity, the Lipschitz-type class and Petree's K-functional for the bivariate Szász-Kantorovich operators based on Brenke-type polynomials. Later, we construct Generalized Boolean Sum operators associated with combinations of the Szász-Kantorovich operators based on Brenke-type polynomials. In addition, we obtain the rate of convergence for the GBS operators with the help of the mixed modulus of continuity and the Lipschitz class of the Bögel continuous functions.

• The Journal of Korea Institute of Information, Electronics, and Communication Technology
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• v.7 no.3
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• pp.137-141
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• 2014

RFID system is applied to various industry such as process control, distribution management, access control, environment sensing, entity identification, etc. Since RFID system uses wireless communication, it has more weak points for security. In this paper, an authentication protocol is suggested between tags and a reader, which is basic property for security. A suggested protocol use a bivariate polynomial over a finite field and is secure against snooping, replay attack, position tracking and traffic analysis.

• The KIPS Transactions:PartC
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• v.12C no.4 s.100
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• pp.473-480
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• 2005

We can obtain useful information by deploying large scale sensor networks in various situations. Security is also a major concern in sensor networks, and we need to establish pairwise keys between sensor nodes for secure communication. In this paper, we propose new pairwise key establishment mechanism based on clustering and polynomial sharing. In the mechanism, we divide the network field into clusters, and based on the polynomial-based key distribution mechanism we create bivariate Polynomials and assign unique polynomial to each cluster. Each pair of sensor nodes located in the same cluster can compute their own pairwise keys through assigned polynomial shares from the same polynomial. Also, in our proposed scheme, sensors, which are in each other's transmission range and located in different clusters, can establish path key through their clusterheads. However, path key establishment can increase the network overhead. The number of the path keys and tine for path key establishment of our scheme depend on the number of sensors, cluster size, sensor density and sensor transmission range. The simulation result indicates that these schemes can achieve better performance if suitable conditions are met.