A hybrid mean value involving a new Gauss sums and Dedekind sums

Document Type : Research Paper


1 School of Mathematics‎, ‎Northwest University‎, ‎Xi'an‎, ‎Shaanxi‎, ‎710127‎, ‎P‎.R‎. ‎China.

2 School of Mathematics‎, ‎Northwest University‎, ‎Xi'an‎, ‎Shaanxi‎, ‎710127‎, ‎P‎.‎R‎. ‎China.


‎In this paper‎, ‎we introduce a new sum‎ ‎analogous to Gauss sum‎, ‎then we use the properties of the‎ ‎classical Gauss sums and analytic method to study the hybrid mean‎ ‎value problem involving this new sums and Dedekind sums‎, ‎and‎ ‎give an interesting identity for it.


Main Subjects

B. Ahmad, On some sets and mappings in topological spaces, J. Natur. Sci. Math. 2 (1986) 37--46.
O.T. Alas, L.R. Junqueira and R.G. Wilson, Countability and star covering properties, Topology Appl. 158 (2011) 620--626.
D.R. Anderson and J.A. Jensen, Semi-continuity on topological spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 42 (1967), no. 8, 782--783.
M. Caldas, S. Jafari and R.K Saraf, semi-Ө-open sets and new classes of maps, Bull. Iranian Math. Soc. 31 (2005), no. 2, 37-52.
S.G. Crossley, A note on semitopological properties, Proc. Amer. Math. Soc. 72 (1978), no. 2, 409--412.
S.G. Crossley and S.K. Hildebrand, Semi-closure, Texas J. Sci. 22 (1971) 99--112.
S.G. Crossley and S.K. Hildebrand, Semi-topological properties, Fund. Math. 74 (1972) 233--254.
G. Di Maio and T. Noiri, Weak and strong forms of irresolute functions, Suppl. Rend. Circ. Mat. Palermo, Ser. II 18 (1988), no. 3, 255--273.
K. Dlaska, N. Ergun and M. Ganster, On the topology generated by semi-regular sets, Indian J. Pure Appl. Math. 25 (1994), no. 11, 1163-1170.
C. Dorsett, Semi compactness, semi separation axioms, and product spaces, Bull. Malays. Math. Sci. Soc. 4 (1981), no. 1, 21--28.
C. Dorsett, Semi-regular spaces, Soochow J. Math. 8 (1982) 45--53.
R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
M. Ganster, D.S. Janković and I.L. Reilly, On compactenss with respect to semi-open sets, Comment. Math. Univ. Carolin. 31 (1990), no. 1, 37--39.
D. Kocev, Almost Menger and related spaces, Mat. Vesnik 61 (2009) 173--180.
L.D.R. Kočinac, Star-Menger and related spaces, Publ. Math. Debrecen 55 (1999) 421-- 431.
L.D.R. Kočinac, Star-Menger and related spaces II, Filomat 13 (1999) 129--140.
L.D.R. Kočinac, Selected results on selection principles, in: Proc. 3rd Seminar on Geometry & Topology, pp. 71--104, Azarb. Univ. Tarbiat Moallem, Tabriz, 2004.
L.D.R. Kočinac, Star selection principles: A survey, Khayyam J. Math. 1 (2015), no. 1, 82--106.
N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), no.1, 36--41.
M.V. Matveev, On the extent of SSM spaces, preprint.
M.V. Matveev, A survey on star covering properties, Topology Atlas, Preprint No. 330, 1998.
A. Sabah, M. ud Din Khan and L.D.R. Kočinac, Covering properties defined by semiopen sets, J. Nonlinear Sci. Appl. 9 (2016) 4388--4398.
M. Sakai, Star covering versions of the Menger property, Topology Appl. 176 (2014) 22--34.
M. Sakai and M. Scheepers, Combinatorics of open covers, in: K.P. Hart, J. van Mill, P. Simon (eds.), Recent Progress in General Topology III, pp. 751--799, Atlantis Press, Paris, 2014.
M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology Appl. 69 (1996) 31--62.
W. Sierpiński, Sur un ensemble nondénombrable, donc toute image continue est de mesure nulle, Fund. Math. 11 (1928) 301-304.
Y.-K. Song, Remarks on strongly star-Menger spaces, Comment. Math. Univ. Carolin. 54 (2013) 97--104.
Y.-K. Song, A note on star-Rothberger spaces, Houston J. Math. 40 (2014) 645--653.
Y.-K. Song, Remarks on star-Menger spaces, Houston J. Math. 40 (2014) 917--925.
B. Tsaban, Combinatorial aspects of selective star covering properties in ѱ-spaces, Topology Appl. 192 (2015) 198--207.
E.K. van Douwen, G.M. Reed, A.W. Roscoe and I.J. Tree, Star covering properties, Topology Appl. 39 (1991) 71--103.
Volume 43, Issue 6
November and December 2017
Pages 1957-1968
  • Receive Date: 05 July 2016
  • Revise Date: 07 December 2016
  • Accept Date: 07 December 2016