Narrow Results

In this paper we give the cost, in terms of states, of some basic operations (union, intersection, concatenation, and Kleene star) on regular languages in the unary case (where the alphabet contains only one symbol). These costs are given by explicitly determining the number of states in the noncyclic and cyclic parts of the resulting automata. Furthermore, we prove that our bounds are optimal. We also present an interesting connection to Jacobsthal's function from number theory.

If x is a rational number, 0<x≤1, then A(x)^c is a context-free language, where A(x) is the set of factors of the infinite Sturmian words with asymptotic density of 1’s smaller than or equal to x. We also prove a “gap” theorem i.e. A(x) can never be an unambiguous co-context-free language. The “gap” theorem is established by proving that the counting generating function of A(x) is transcendental. We show some links between Sturmian words, combinatorics and number theory.

We review the Batyrev approach to Calabi–Yau spaces based on reflexive weight vectors. The Universal CY algebra gives a possibility to construct the corresponding reflexive numbers in a recursive way. A physical interpretation of the Batyrev expression for the Calabi–Yau manifolds is presented. Important classes of these manifolds are related to the simple-laced and quasi-simple-laced numbers. We discuss the classification and recurrence relations for them in the framework of quantum field theory methods. A relation between the reflexive numbers and the so-called Berger graphs is studied. In this correspondence the role played by the generalized Coxeter labels is highlighted. Sets of positive roots are investigated in order to connect them to possible new algebraic structures stemming from the Berger matrices.

Motivated by the Goldbach conjecture in number theory and the Abelian bosonization mechanism on a cylindrical two-dimensional space–time, we study the reconstruction of a real scalar field as a product of two real fermion (so-called prime) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators — labeled by prime numbers p — acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allows us to prove that the theory is not renormalizable. We also comment on the potential consequences of this result concerning the validity or breakdown of the Goldbach conjecture for large integer numbers.

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re(s) = 1/2. Hilbert and Pólya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory. Using the construction of the so-called super-zeta functions or secondary zeta functions built over the Riemann nontrivial zeros and the regularity property of one of this function at the origin, we show that it is possible to extend the Hilbert–Pólya conjecture to systems with countably infinite number of degrees of freedom. The sequence of the nontrivial zeros of the Riemann zeta function can be interpreted as the spectrum of a self-adjoint operator of some hypothetical system described by the functional approach to quantum field theory. However, if one considers the same situation with numerical sequences whose asymptotic distributions are not "far away" from the asymptotic distribution of prime numbers, the associated functional integral cannot be constructed. Finally, we discuss possible relations between the asymptotic behavior of a sequence and the analytic domain of the associated zeta function.

The sequence of nontrivial zeros of the Riemann zeta function is zeta regularizable. Therefore, systems with countably infinite number of degrees of freedom described by self-adjoint operators whose spectra is given by this sequence admit a functional integral formulation. We discuss the consequences of the existence of such self-adjoint operators in field theory framework. We assume that they act on a massive scalar field coupled to a background field in a (d+1)-dimensional flat space–time where the scalar field is confined to the interval [0, a] in one of its dimensions and there are no restrictions in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even-dimensional space–time, the series that defines the regularized vacuum energy is finite. For the odd-dimensional case, to obtain a finite vacuum energy per unit area, we are forced to introduce mass counterterms. A Riemann mass appears, which is the correction to the mass of the field generated by the nontrivial zeros of the Riemann zeta function.

We present a remarkable connection between the asymptotic behavior of the Riemann zeros and one-loop effective action in Euclidean scalar field theory. We show that in a two-dimensional space, the asymptotic behavior of the Fourier transform of two-point correlation functions fits the asymptotic distribution of the nontrivial zeros of the Riemann zeta function. We work out an explicit example, namely the nonlinear sigma model in the leading order in 1/N expansion.

We construct a supersymmetric quantum mechanical model in which the energy eigenvalues of the Hamiltonians are the products of Riemann zeta functions. We show that the trivial and nontrivial zeros of the Riemann zeta function naturally correspond to the vanishing ground state energies in this model. The model provides a natural form of supersymmetry.

In this paper, we explore some recent advancements in enumerative algebraic geometry, focusing particularly on the role of quantum K-theory of quiver varieties as viewed through the lens of integrable systems. We highlight a number of conjectures and open questions.

A new class of analog-to-digital (A/D) and digital-to-analog (D/A) converters that uses a flaky quantizer, known as a β-encoder, has been shown to have exponential bit rate accuracy and a self-correcting property for fluctuations of the amplifier factor β and quantizer threshold ν. The probabilistic behavior of this flaky quantizer is explained by the deterministic dynamics of a multivalued Rényi–Parry map on the middle interval, as defined here. This map is eventually locally onto map of [ν - 1, ν], which is topologically conjugate to Parry's (β, α)-map with α = (β - 1)(ν - 1). This viewpoint allows us to obtain a decoded sample, which is equal to the midpoint of the subinterval, and its associated characteristic equation for recovering β, which improves the quantization error by more than 3 dB when β > 1.5. The invariant subinterval under the Rényi–Parry map shows that ν should be set to around the midpoint of its associated greedy and lazy values. Furthermore, a new A/D converter referred to as the negative β-encoder is introduced, and shown to further improve the quantization error of the β-encoder. Then, a switched-capacitor (SC) electronic circuit technique is proposed for implementing A/D converter circuits based on several types of β-encoders. Electric circuit experiments were used to verify the validity of these circuits against deviations and mismatches of the circuit parameters. Finally, we demonstrate that chaotic attractors can be observed experimentally from these β-encoder circuits.

In this paper we present an efficient algorithm for enumerating all the grid points in a convex polygon, where the enumeration is done by scanning the grid points with a set of parallel grid lines. Given a convex n-gon, enumeration can be done in time, where K is the number of the grid points reported, l is the diameter of the polygon and w is the length of the shorter sides of the rectangle enclosing the polygon with longer sides parallel to the line segment connecting the diametral pair of the polygon. We also show that the ratio of the number of the scanned grid lines to the minimal is less than some constant.

The design of fiducials for precise image registration is of major practical importance in computer vision, especially in automatic inspection applications. We analyze the sub-pixel registration accuracy that can, and cannot, be achieved by some rotation-invariant fiducials, and present and analyze efficient algorithms for the registration procedure. We rely on some old and new results from lattice geometry and number theory and efficient computational-geometric methods.

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems.

In this paper, we propose an arc recognition method based on the number theoretic properties of isothetic covers. A definition of digital circles is given based on the dilation of Euclidean circles with unit squares. We show that a variant of the digital circle is equivalent to the grid centers between the isothetic covers of disks. Number theoretic properties of the isothetic covers of disks are explored and show that the distribution of square numbers can be used to find the run lengths of the isothetic covers. Arc recognition algorithms are developed based on the number theoretic properties.

In this note we apply the general multifractal analysis for growth rates derived in [10], and show that this leads to some new results in ergodic theory and the theory of multifractals of numbers. Namely, we consider Stern–Brocot growth rates and introduce the Stern–Brocot pressure P. We then obtain the results that P is differentiable everywhere and that its Legendre transformation governs the multifractal spectra arising from level sets of Stern–Brocot rates.

Generalized splines on a graph $G$ with edge labels in a commutative ring $R$ are vertex labelings such that if two vertices share an edge in $G$, the difference between the vertex labels lies in the ideal generated by the edge label. When $R$ is an integral domain, the set of all such splines is a finitely generated $R$-module $R_G$ of rank $n$, the number of vertices of $G$. We find determinantal conditions on subsets of $R_G$ that determine whether $R_G$ is a free module, and if so, whether a so-called “flow-up class basis” exists.

Let $n$ be a positive integer and let ${𝔽}_q$ be a finite field with $q$ elements, where $q$ is a prime power and $gcd(n,q)=1$. In this paper, we give the explicit factorization of $x^n-1$ over ${𝔽}_q$ and count the number of its irreducible factors for the following conditions: $n,q$ are odd and $\text{rad}(n)|(q^2+1)$. First, we present a method to obtain the set of all representatives of $q$-cyclotomic cosets modulo $m$, where $m=gcd(n,q^2+1)$. This set of representatives is then used to find the irreducible factors of $x^n-1$ and the cyclotomic polynomial ${\mathrm{\Phi }}_n(x)$ over ${𝔽}_q$. The form of irreducible factors of $x^n-1$ is characterized such that the coefficients of these irreducible factors are followed by second-order linear recurring sequences.

Kohnen showed that the zeros of the Eisenstein series E_k in the standard fundamental domain other than i and ρ are transcendental. In this paper, we obtain similar results for a more general class of modular forms, using the earlier works of Kanou, Kohnen and the recent work of Getz.

Recently, Duke and Jenkins have studied a certain family of modular forms f_k,m, which form a natural basis of the space of weakly holomorphic modular forms of weight k on SL₂(ℤ). In particular, they prove that for all but at most values of m, the zeros of these functions all lie on the unit circle. Using a method of Kohnen, we observe that other than i, ρ, these zeros are transcendental. In addition we observe that in the remaining cases there are at most finitely many algebraic zeros and examine the values of these algebraic zeros in a special case.

Inspired by recent congruences by Andersen with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author for the function spt(n). We show that a normalized form of the generating function of spt(n) is an eigenform modulo 32 for the Hecke operators T(ℓ²) for primes ℓ ≥ 5 with ℓ ≡ 1, 11, 17, 19 (mod 24), and an eigenform modulo 16 for ℓ ≡ 13, 23 (mod 24).