Periodic Modulo Research Articles

Circular code in introns

A massive statistical analysis based on the autocorrelation function of the circular code X observed in genes is performed on the (eukaryotic) introns. Surprisingly, a circular code periodicity 0 modulo 3 is identified in 5 groups of introns: birds, ascomycetes, basidiomycetes, green algae and land plants. This circular code periodicity, which is a property of retrieving the reading frame in (protein coding) genes, may suggest that these introns have a coding property. In a well-known way, a periodicity 1 modulo 2 is observed in 6 groups of introns: amphibians, fishes, mammals, other animals, reptiles and apicomplexans. A mixed periodicity modulo 2 and 3 is found in the introns of insects. Astonishing, a subperiodicity 3 modulo 6 is a common statistical property in these 3 classes of introns. When the particular trinucleotides N1N2N1 of the circular code X are not considered, the circular code periodicity 0 modulo 3, hidden by the periodicity 1 modulo 2, is now retrieved in 5 groups of introns: amphibians, fishes, other animals, reptiles and insects. Thus, 10 groups of introns, taxonomically different, out of 12 have a coding property related to the reading frame retrieval. The trinucleotides N1N2N1 are analysed in the 216 maximal C3 self-complementary trinucleotide circular codes. A hexanucleotide code (words of 6 letters) is proposed to explain the periodicity 3 modulo 6. It could be a trace of more general circular codes at the origin of the circular code X.

Mylar Balloon and Associated Geometro-Mechanical Moments

Starting with identifications of the very fundamental geometric characteristics of a Mylar balloon such as the profile curve, height, volume, arclength, surface area, crimping factor, etc., using the geometrical moments In(x) and In, we present explicit formulas for them and those of the mechanical moments of both solid and hollow balloons of arbitrary order. This is achieved by relying on the recursive relationships among elliptic integrals and the final results are expressed via the fundamental mathematical constants such as π, lemniscate constant ω˜, and Gauss’s constant G. An interesting periodicity modulo 4 was detected and accounted for in the final formulas for the moments. The principal results are illustrated by two tables, a few graphics, and some direct relationships with other fundamental areas in mathematics, physics and geometry are pointed out.

Sharp operator‐norm asymptotics for thin elastic plates with rapidly oscillating periodic properties

We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order of smallness. Assuming that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response, such as the case in e.g. acoustic wave propagation, we derive a class of "hybrid", homogenisation dimension-reduction, norm-resolvent estimates for the plate, under different energy scalings with respect to the plate thickness.

On expansions of (Z,+,0)

Call a (strictly increasing) sequence (rn) of natural numbers regular if it satisfies the following condition: rn+1/rn→θ∈R>1∪{∞} and, if θ is algebraic, then (rn) satisfies a linear recurrence relation whose characteristic polynomial is the minimal polynomial of θ. Our main result states that (Z,+,0,R) is superstable whenever R is enumerated by a regular sequence. We give two proofs of this result. One relies on a result of E. Casanovas and M. Ziegler and the other on a quantifier elimination result. We also show that (Z,+,0,<,R) is NIP whenever R is enumerated by a regular sequence that is ultimately periodic modulo m for all m>1.

Romik's conjecture for the Jacobi theta function

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point i. He then conjectured that the Taylor coefficients d(n) either vanish or are periodic modulo any prime p; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, with the latter trio considering arbitrary half integral weight modular forms. We refine previous work for p≡1(mod4) by displaying a concise algebraic relation between d(n+p−12) and d(n) related to the p-adic factorial, from which we can deduce periodicity with an effective period.

Axion scales and couplings with St\xfcckelberg mixing

We study the axion field range and low energy couplings in models with Stückelberg mixing between axions and U(1) gauge bosons. It is noted that the gauge- invariant axion combination ξ in the model is periodic modulo an appropriate shift of gauge-variant axions eaten by the massive U(1) gauge bosons, which in some cases makes the connection between the field range and the low energy couplings less transparent. We derive the field range of ξ for generic forms of the axion kinetic metric and U(1) charges, and identify the field basis for which all non-derivative couplings of ξ are quantized in a manner manifestly consistent with the periodicity of ξ. Generically Stückelberg mixing reduces the axion field range. In particular, the mixings between N axions and (N − 1) U(1) gauge bosons typically result in an exponentially reduced field range {M}_{xi }=mathcal{O}left({k}^{-left(N-1right)}f/sqrt{N!}right) for the residual gauge-invariant axion ξ in the limit N ≫ 1, where f and k denote the typical decay constant and the root mean square of the U(1) gauge charges of the original N axions. Using simple examples, we study also the reparameterization-invariant physical quantities such as the axion effective potential and 1PI couplings to gauge bosons, which are determined by the reparameterization-dependent axion couplings in the model.

Parity of the partition function p(n,k)

Let [Formula: see text] denote the number of partitions of [Formula: see text] into parts less than or equal to [Formula: see text]. We show several properties of this function modulo 2. First, we prove that for fixed positive integers [Formula: see text] and [Formula: see text], [Formula: see text] is periodic modulo [Formula: see text]. Using this, we are able to find lower and upper bounds for the number of odd values of the function for a fixed [Formula: see text].

Goldie Rings Graded by a Group with Periodic Quotient Group Modulo the Center

In this paper, we study gr-prime and gr-semiprime Goldie rings graded by a group with periodic quotient group modulo the center. We enhance the theorem of Goodearl and Stafford (2000) about gr-prime rings graded by Abelian groups; we extend the Abelian group class to the class of groups with periodic quotient group modulo the center. We also decompose the orthogonal graded completion Ogr(R) of a gr-semiprime Goldie ring R (graded by a group satisfying the same condition) into a direct sum of gr-prime Goldie rings R1, . . . , Rn and prove that the maximal graded quotient ring Qgr(R) equals the direct sum of classical graded quotients rings of R1, . . . , Rn.

An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices.

In this work, we revisit a criterion, originally proposed in Friesecke & Pego (Friesecke & Pego 2004 Nonlinearity17, 207-227. (doi:10.1088/0951715/17/1/013)), for the stability of solitary travelling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-travelling frame, as well as at that of the Floquet problem arising when considering the travelling wave as a periodic orbit modulo shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the travelling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

Diletter circular codes over finite alphabets

The graph approach of circular codes recently developed (Fimmel et al., 2016) allows here a detailed study of diletter circular codes over finite alphabets. A new class of circular codes is identified, strong comma-free codes. New theorems are proved with the diletter circular codes of maximal length in relation to (i) a characterisation of their graphs as acyclic tournaments; (ii) their explicit description; and (iii) the non-existence of other maximal diletter circular codes. The maximal lengths of paths in the graphs of the comma-free and strong comma-free codes are determined. Furthermore, for the first time, diletter circular codes are enumerated over finite alphabets. Biological consequences of dinucleotide circular codes are analysed with respect to their embedding in the trinucleotide circular code X identified in genes and to the periodicity modulo 2 observed in introns. An evolutionary hypothesis of circular codes is also proposed according to their combinatorial properties.

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability.

In this work, we provide two complementary perspectives for the (spectral) stability of solitary traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One is as an eigenvalue problem for a stationary solution in a cotraveling frame, while the other is as a periodic orbit modulo shifts. We connect the eigenvalues of the former with the Floquet multipliers of the latter and using this formulation derive an energy-based spectral stability criterion. It states that a sufficient (but not necessary) condition for a change in the wave stability occurs when the functional dependence of the energy (Hamiltonian) H of the model on the wave velocity c changes its monotonicity. Moreover, near the critical velocity where the change of stability occurs, we provide an explicit leading-order computation of the unstable eigenvalues, based on the second derivative of the Hamiltonian H^{''}(c_{0}) evaluated at the critical velocity c_{0}. We corroborate this conclusion with a series of analytically and numerically tractable examples and discuss its parallels with a recent energy-based criterion for the stability of discrete breathers.

Free subgroup numbers modulo prime powers: The non-periodic case

In [J. Algebra 452 (2016) 372–389], we characterise when the sequence of free subgroup numbers of a finitely generated virtually free group Γ is ultimately periodic modulo a given prime power. Here, we show that, in the remaining cases, in which the sequence of free subgroup numbers is not ultimately periodic modulo a given prime power, the number of free subgroups of index λ in Γ is — essentially — congruent to a binomial coefficient times a rational function in λ modulo a power of a prime that divides a certain invariant of the group Γ, respectively to a binomial sum involving such numbers. These results, apart from their intrinsic interest, in particular allow for a much more efficient computation of congruences for free subgroup numbers in these cases compared to the direct recursive computation of these numbers implied by the generating function results in [J. London Math. Soc. (2) 44 (1991) 75–94].

Arithmetic properties of generalizations of Schenker sums

In this paper, we consider sequences defined by [Formula: see text], [Formula: see text], where [Formula: see text]. They are generalizations of the sequence of Schenker sums. We study such arithmetic properties of these sequences as: periodicity modulo [Formula: see text], where [Formula: see text], [Formula: see text]-adic valuations, asymptotics and prime divisors.

Periodicity of free subgroup numbers modulo prime powers

We characterise when the sequence of free subgroup numbers of a finitely generated virtually free group is ultimately periodic modulo a given prime power.

The Number of Labeled Connected Graphs Modulo Prime Powers

Let $c_n$ denote the number of vertex-labeled connected graphs on $n$ vertices. Using group actions and elementary number theory, we show that the infinite sequence, $c_n : n \ge 1,$ is ultimately periodic modulo every positive integer. We state and prove our results for sequences defined by a weighted generalization of $c_n$ and conjecture that these results are suggestive of similar periodic behavior of the Tutte polynomial evaluations of the complete graph $K_n$ at integer points.

On the Number of Tilings of a Square by Rectangles

We develop a recursive formula for counting the number of rectangulations of a square, i.e., the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was calculated and extend to the non-generic case. An interesting feature of the number of rectangulations is that it appears to have an 8-fold periodicity modulo 2. We verify this periodicity for small values of n, but the general result remains elusive, perhaps hinting at some unseen structure on the space of rectangulations, analogous to Reading’s discovery that generic rectangulations are in 1–1 correspondence with a certain class of permutations. Finally, we use discrete Morse theory to show that the space of tilings by ≤ n rectangles is homotopy-equivalent to a wedge of some number of (n−1)-dimensional spheres. Combining this result with the formulae for the number of tilings, the exact homotopy type is computed for n ≤ 28.

On the mass-critical generalized KdV equation

We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear Schrodinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution.

The Pentagram Map: A Discrete Integrable System

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from $${\mathbb Z}$$ into $${{\mathbb{RP}}^2}$$ that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].

Regularity of almost periodic modulo scaling solutions for mass-critical NLS and applications

In this paper, we consider the Lx solution u to mass critical NLS iut + ∆u = ±|u| 4 d u. We prove that in dimensions d ≥ 4, if the solution is spherically symmetric and is almost periodic modulo scaling, then it must lie in H x for some e > 0. Moreover, the kinetic energy of the solution is localized uniformly in time. One important application of the theorem is a simplified proof of the scattering conjecture for mass critical NLS without reducing to three enemies [17], [18]. As another important application, we establish a Liouville type result for Lx initial data with ground state mass. We prove that if a radial Lx solution to focusing mass critical problem has the ground state mass and does not scatter in both time directions, then it must be global and coincide with the solitary wave up to symmetries. Here the ground state is the unique, positive, radial solution to elliptic equation ∆Q − Q + Q 4 d = 0. This is the first rigidity type result in scale invariant space Lx.

Radiation damping and reciprocity in nuclear magnetic resonance: The replacement of the filling factor

The basic equation describing radiation damping in nuclear magnetic resonance (NMR) is rewritten by means of the reciprocity principle, to remove the dependence of the damping constant upon filling factor – a parameter which is neither uniquely defined for easily measured. The new equation uses instead the transceive efficiency, i.e. the peak amplitude of the radiofrequency B field in laboratory coordinates, divided by the square root of the resistance of the detection coil, for which a simple and direct means of measurement exists. We use the efficiency to define the intrinsic damping constant, i.e. that which obtains when both probe and preamplifier are perfectly matched to the system impedance. For imperfect matching of the preamp, it is shown that the damping constant varies with electrical distance to the probe, and equations are given and simulations performed, to predict the distance dependence, which (for lossless lines) is periodic modulo a half wavelength. Experimental measurements of the radiation-damped free induction NMR signal of protons in neat water are performed at a static B field strength of 14.1 T; and an intrinsic damping constant measured using the variable line method. For a sample of 5 mm diameter, in an inverse detection probe we measure an intrinsic damping constant of 204 s −1, corresponding to a damping linewidth of 65 Hz for small tip angles. The predicted intrinsic linewidth, based upon three separate measurements of the efficiency, is 52.3 Hz, or 80% of the measured value.