arXiv subjects

Terence R. Smith

Publications and source records attributed to Terence R. Smith.

5 recordsLinked to original sources

arXiv2026

Wave Numbers: Discrete Sequence Algebras, Sieve Projectors, and Dynamical Geometry

We establish a comprehensive algebraic, geometric, and physical classification of the wave closure space generated from primitive plane wave sequences on the discrete spatial lattice $\mathbb{Z}$. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group $\mathcal{G}$ under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to $(\mathbb{Q}/\mathbb{Z}) \times (\mathbb{Q}/\mathbb{Z})$, with canonical decomposition into Prüfer $p$-groups and maximal cyclotomic value field $\mathbb{Q}^{\mathrm{ab}}$. Extending to coordinate powers yields the polynomial phase group $\mathcal{G}_{\mathrm{poly}}$, classified by integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. Adjoining addition yields the group algebra $\mathbb{C}[\mathcal{G}_{\mathrm{poly}}]$, for which we establish the Permutation-Symmetric Phasor Superposition Theorem, factoring superpositions into collective barycentric carriers and closed Born probability envelopes. Using algebraic sieves over roots of unity, we construct idempotent projectors and complementary Not-sieve notch filters, achieving exact algebraic synthesis of both momentum and localized Kronecker position bases. Generalizing to non-abelian quaternions $\mathcal{V}_\mathbb{H}$, we prove a polar decomposition into scalar envelopes and $\mathrm{SU}(2)$ spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime $s^2 = c^2 t^2 - \|\mathbf{x}\|^2$ and the Lorentz group $\mathrm{SO}^+(1,3)$. We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.

math.NT↗

arXiv2026

Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices

We formalize the algebraic structure of rational polynomial phase sequences (discrete chirps) over the integer lattice $\mathbb{Z}$. By constructing a convolution algebra generated by polynomial phases with coefficients in $\mathbb{Q}$, we establish a rigorous operational calculus for discrete difference equations. We prove an exact orthogonality relation for these sequences over finite periodic windows, yielding a generalized discrete Fourier isometry. Furthermore, we demonstrate that the continuum limit of this discrete framework can be rigorously defined via the inductive limit of finite-dimensional $C^*$-algebras (an AF-algebra). This framework provides a rigorous algebraic substrate for finite-mode lattice approximations in quantum mechanics and the construction of mutually unbiased bases in finite-dimensional quantum information theory.

math.GM↗

arXiv2025

Sums of Exponential Terms, Conserved Quantities, and the Real Wave Numbers

There is consensus that sums $S_n={ {Σ}_{k=1}^n R_{0k} e^{i θ_k}}$ of complex exponential terms, despite their mathematical significance, only possess closed-form representations for specific values of n and special values of their parameters and that there are no generally-accepted recursive formulae for their computation. This note is focused on recursive formulae that: (1) provide closed-form analytic representations of $S_n$ for any finite n; (2) include generalizations of the usual formula for the sum of two exponentials; and (3) are representable in the form $S_n= A_n exp({ iΣ_{k=1}^n θ_k})$. The goal of the paper is to show that one may interpret the exponential term $exp(i Σ_{k=1}^n θ_k)$ of $S_n$ as representing the projection, from a field of numbers that generalizes the complex numbers onto the complex plane, of a term representing quantities that are conserved under the addition and multiplication of numbers in the extended space. In particular, it is shown that the general form of a number in the extended field generalizes the form of a sum of complex exponentials.

math.NT↗

arXiv2025

Rational Wave Numbers and the Algebraic Structure of the Cyclic Groups of the Roots of Unity

Rational wave numbers are periodic sequences ${\mathbf ω}={\bf A}{\bf w}(f,g)$ in which amplitude ${\bf A}$ a product of powers of trigonometric sequences and ${\bf w}(f,g)=\exp({\bf {i2}π( f {\mathbf ξ} \oplus g{\bf 1})})$ is a sequence with $ ξε\mathbb{ Z}$ and $f,g$ rational. They generalize the cyclic groups of the $n$th roots of unity and are generated from two unitary sequences. The multiplicative group ${\bf W}_M$ with ${\bf A}={\bf 1}$ is their closure wrt product, root, and reflection operators. The commutative ring ${\bf W}_A$ has additional closure wrt summation. The field ${\bf W}_I$ of invertible wave numbers has further closure wrt an inverse. Sums and differences of its elements are ${\mathbf ω}_1 \oplus {\mathbf ω}_2= {\bf 2} \cos\big( {\bf i} ln\big(\frac{{\mathbf ω}_2}{{\mathbf ω}_1}\big)^{1/2}\big) \big({\mathbf ω}_1 {\mathbf ω}_2 \big)^{1/2}$ and ${\mathbf ω}_1 \ominus {\mathbf ω}_2= {\bf {2 i}} \sin\big( {\bf i} ln\big(\frac{{\mathbf ω}_2}{{\mathbf ω}_1}\big)^{1/2}\big) \big({\mathbf ω}_1 {\mathbf ω}_2 \big)^{1/2}$. Its amplitudes form a multiplicative subgroup over which ${\bf W}_M \cup \big\{{\bf 0}\big\}$ is a vector space. Wave numbers of period $n\ ε \mathbb{N}$ possess $n$ phases, multiplicative norms, representations wrt orthonormal bases, and prime representations. Rational wave numbers may be completed with respect to Cauchy sequences of parameters $({f,g})$. Equations in invertible wave numbers have solutions corresponding to zeros of their trigonometric factors. Orthonormal bases are employed in constructing the integral wave numbers and allow definitions of the particulate and prime wave numbers in terms of permissible phase-values.

math.NT↗

arXiv2022

Natural Wave Numbers, Natural Wave Co-numbers, and the Computation of the Primes

The paper exploits an isomorphism between the natural numbers N and a space U of periodic sequences of the roots of unity in constructing a recursive procedure for representing and computing the prime numbers. The nth wave number ${\bf u}_n$ is the countable sequence of the nth roots of unity having frequencies k/n for all integer phases k. The space U is closed under a commutative and associative binary operation ${\bf u}_m \odot{\bf u}_n={\bf u}_{mn}$, termed the circular product, and is isomorphic with N under their respective product operators. Functions are defined on U that partition wave numbers into two complementary sequences, of which the co-number $ {\overset {\bf \ast }{ \bf u}}_n$ is a function of a wave number in which zeros replace its positive roots of unity. The recursive procedure $ {\overset {\bf \ast }{ \bf U}}_{N+1}= {\overset {\bf \ast }{ \bf U}}_{N}\odot{\overset {\bf \ast }{\bf u}}_{N+1}$ represents prime numbers explicitly in terms of preceding prime numbers, starting with $p_1=2$, and is shown never to terminate. If ${p}_1, ... , { p}_{N+1}$ are the first $N+1$ prime phases, then the phases in the range $p_{N+1} \leq k < p^2_{N+1}$ that are associated with the non-zero terms of $ {\overset {\bf \ast }{\bf U}}_{N}$ are, together with $ p_1, ...,p_N$, all of the prime phases less than $p^2_{N+1}$. When applied with all of the primes identified at the previous step, the recursive procedure identifies approximately $7^{2(N-1)}/(2(N-1)ln7)$ primes at each iteration for $ N>1$. When the phases of wave numbers are represented in modular arithmetic, the prime phases are representable in terms of sums of reciprocals of the initial set of prime phases and have a relation with the zeta-function.

cs.DS↗