This confirms the three-elements digit set conjecture on the non-spectrality of self-affine measures in the plane.
In this paper we offer a new redundant decimal digit set [−8, 9] and a fully redundant addition/subtraction scheme.
We study spectral properties of the self-affine measure μM,D generated by an expanding integer matrix M∈Mn(Z) and a consecutive collinear digit set D= {0,1,…,q−1}v where v∈Zn∖{0} and q≥2 is an integer.
In most primates the foot is adapted for grasping (i.e., is prehensile), with the first digit set at an angle from the others.
In the proposed approach, only the odd values (rather than all the 2k values) of the quotient digit set are used to generate the multiples of divisor.
A set D = { d 1, d 2, …, d N } of coset representatives of the quotient Z n / L Z Z n ) is called a digit set.
In this paper, we consider the non-spectral problem for the planar self-affine measures μM,D generated by an expanding integer matrix M∈M2(Z) and a finite digit set D⊂Z2.
In the proposed fully redundant adder (VS semi-redundant ones such as decimal carry-save adders) both operands and sum are redundant decimal numbers with overloaded decimal digit set [0, 15].
The proposed digit set, faithfully encoded as a mix of posibits, negabits, and unibits, is shown to obviate the need for any compare-to-9 operations and leads to minimal penalty subtraction using the addition circuitry.
In the present paper we show that for an expanding integer matrix M∈M2(Z) and the three-elements digit set D given byM="[abdc]andD="{ 00),(10),(01)}, if ac−bd∉3Z, then there exist at most 3 mutually orthogonal exponentials in L2 μM,D), and the number 3 is the best.
It is assumed that 0 ∈ D. By standard algebra results, for D to be a digit set it is necessary that | D | = | det L |. Consider an affine IFS F : = F ( L, D ) = { R n ; f 1, f 2, …, f N }, where f i ( x ) = L − 1 ( x − d i ).
