The notion of orthogonality for vectors in inner product spaces is simple,
interesting and fruitful. When moving to normed spaces, we have many
possibilities to extend this notion. We consider Birkhoff orthogonality and
isosceles orthogonality, which are the most used notions of orthogonality.
In 2006, Ji and Wu introduced a geometric constant D(X) to give a
quantitative characterization of the difference between these two
orthogonality types. However, this constant was considered only in the unit
sphere SX of the normed space X. In this paper, we introduce a new geometric
constant IB(X) to measure the difference between Birkhoff and isosceles
orthogonalities in the entire normed space X. To consider the difference
between these orthogonalities, we also treat constant BI(X).
In this paper, we extend normed spaces to quasi-normed spaces and prove the generalized Hyers-Ulam stability of a nonic functional equation:$$\aligned&f(x+5y) - 9f(x+4y) + 36f(x+3y) - 84f(x+2y) + 126f(x+y) - 126f(x)\\&\qquad + 84f(x-y)-36f(x-2y)+9f(x-3y)-f(x-4y) = 9 ! f(y),\endaligned$$where $9 ! = 362880$ in quasi-normed spaces.
Abstract
In this paper, we investigate the general solution and the generalized stability for the quartic, cubic and additive functional equation (briefly, QCA-functional equation)
for any k∈ℤ-{0,±1} in Menger probabilistic normed spaces.
AbstractIn this paper, we obtain Hyers-Ulam stability of the functional equationsf (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w),f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w)andf (x + y, z − w) + f (x − y, z + w) = 2f (x, z) − 2f (y, w)in 2-Banach spaces. The quadratic forms ax2 + bxy + cy2, ax2 + by2 and axy are solutions of the above functional equations, respectively.
AbstractThermoelectric modules can be used in waste heat harvesting, sensing, and cooling applications. Here, we report on the fabrication and performance of a four-leg module based on abundant silicide materials. While previously optimized Mg2Si0.3Sn0.675Bi0.025 is used as the n-type leg, we employ a fractional factorial design based on the Taguchi methods mapping out a four-dimensional parameter space among Mnx-εMoεSi1.75−δGeδ higher manganese silicide compositions for the p-type material. The module is assembled using a scalable fabrication process, using a Cu metallization layer and a Pb-based soldering paste. The maximum power output density of 53 μW cm–2 is achieved at a hot-side temperature of 250 °C and a temperature difference of 100 °C. This low thermoelectric output is related to the high contact resistance between the thermoelectric materials and the metallic contacts, underlining the importance of improved metallization schemes for thermoelectric module assembly.
The constants to measure the differences between Birkhoff and isosceles orthogonalities
