Odd Parts Research Articles

Almost partition identities

An almost partition identity is an identity for partition numbers that is true asymptotically [Formula: see text] of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.

PAPR Reduction in DCO-OFDM Visible Light Communication Systems Using Optimized Odd and Even Sequences Combination

This paper proposes a novel peak-to-average power ratio (PAPR) reduction scheme, which has several advantages, such as low-complexity candidate construction and no side information (SI) transmission in direct current-biased optical orthogonal frequency division multiplexing (DCO-OFDM) visible light communication (VLC) systems. In the proposed scheme, the candidates can be generated by linearly combining time-domain even and odd sequences with cyclic shift and phase rotation. Even and odd sequences are obtained by adopting an inverse fast Fourier transform on the even and odd parts of a Hermitian symmetry encoded signal, respectively. Furthermore, a deterministic and efficient selection method of cyclic shift and phase rotation is derived to improve PAPR reduction performance and ensure that each candidate is unique, so that the receiver can recover the transmitted candidate signal without the use of SI. Simulation results show that the proposed scheme can achieve comparable PAPR reduction performance with the conventional selected mapping (CSLM) scheme with random and chaotic sequences, but at a considerably low computational complexity. Moreover, the proposed blind SI detection scheme provides almost the same bit error rate (BER) performance as that of the CSLM scheme with a perfect SI detection and symmetric selected mapping for a DCO-OFDM VLC system with 1024 subcarriers and 16-QAM. In contrast to other related time-domain sequences that combine methods used for DCO-OFDM systems, the proposed scheme improves BER performance with a slight increase in computational complexity and PAPR.

Berry curvature and Hall viscosities in an anisotropic Dirac semimetal

We investigate parity-odd non-dissipative transport in an anisotropic Dirac semi-metal in two spatial dimensions. The analysis is relevant for interacting electronic systems with merging Dirac points at charge neutrality. For such systems the dispersion relation is relativistic in one direction and non-relativistic in the other. We give a proposal how to calculate the Berry curvature for this system and use it to derive more than one odd viscosities, in contrast to rotationally invariant systems. We observe that in such a model the odd part of stress tensor is parameterised by two independent transport coefficients and one that is identically zero.

Overpartition $M2$-rank differences, class number relations, and vector-valued mock Eisenstein series

We prove that the generating function of overpartition $M2$-rank differences is, up to coefficient signs, a component of the vector-valued mock Eisenstein series attached to a certain quadratic form. We use this to compute analogs of the class number relations for $M2$-rank differences. As applications we split the Kronecker-Hurwitz relation into its even and odd parts and calculate sums over Hurwitz class numbers of the form $\sum_{r \in \mathbb{Z}} H(n - 2r^2)$.

Magnetic field induced local structural transformations in the optically excited states 5F2 and 5F3 of HoFe3(BO3)4 single crystal

Abstract We measured absorption spectra of transitions 5I8 → 5F2, 5F3 in Ho3+ ion in the HoFe3(BO3)4 single crystal at temperature 2 K as a function of magnetic field 0–65 kOe directed along a and c axes. The spectra were decomposed to the Lorentz shape components and magnetic field dependences of their parameters: position, intensity and line width, were obtained. Substantial difference between dependences of the energy and intensity of the transitions on magnetic field was revealed. So, the abrupt changes of the electron transition energies were mainly observed at the reorientation magnetic phase transitions in the crystal. A number of singularities in the magnetic field dependences of the electron transition intensities were revealed both at the reorientation transitions and outside of them. The singularities in the average intensities of transitions into multiplets at the reorientation magnetic transitions are due to the changes of the odd part of the crystal field, responsible for the allowance of f-f transitions, and also are due to the changes of the magnetic state. The singularities in behavior of intensity of each separate transition are due to the local even distortions in the excited states, which mix the electron states.

On the Generalized Fractional Brownian Motion

The generalized fractional Brownion motion (gfBm) is a new extension of both fractional and sub-fractional Brownian motions, introduced very recently. We show that this process could serve to obtain new models, better than those constructed from fractional and sub-fractional Brownian motions, permitting to take the level of correlation between the increments of the studied phenomenon into account. We also expand explicitly this process, we study the rate of convergence of the obtained expansion and, we apply our result to get a computer generation of some gfBm sample paths. In particular we present some sample paths of the even and odd parts of the fractional Brownian motion.

The depth structure of motivic multiple zeta values

In this paper, we construct some maps related to the motivic Galois action on depth-graded motivic multiple zeta values. From these maps we give some short exact sequences about depth-graded motivic multiple zeta values in depth two and three. In higher depth we conjecture that there are exact sequences of the same type. We will show from three conjectures about depth-graded motivic Lie algebra we can nearly deduce the exact sequences conjectures in higher depth. At last we give a new proof of the result that the modulo $$\zeta ^{\mathfrak {m}}(2)$$ version motivic double zeta values are generated by the totally odd part. We reduce the well-known conjecture that the modulo $$\zeta ^{\mathfrak {m}}(2)$$ version motivic triple zeta values are generated by the totally odd part to an isomorphism conjecture in linear algebra.

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck’s conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher’s bijection, we give the combinatorial proofs of these three conjectures directly or by proving more generalized results.

Nonperturbative one-loop effective action for QED with Yukawa couplings

We derive the one-loop effective action for scalar, pseudoscalar, and electromagnetic fields coupled to a Dirac fermion in an extension of QED with Yukawa couplings. Using the Schwinger proper-time formalism and zeta-function regularization, we calculate the full nonperturbative effective action to one loop in the constant background field approximation. Our result is nonperturbative in the external fields, and goes beyond existing results in the literature which treat only the first nontrivial order involving the pseudoscalar. The result has an even and odd part, which are related to the modulus and phase of the fermion functional determinant. The even contribution to the effective action involves the modulus of the effective Yukawa couplings and is invariant under global chiral transformations while the odd contribution is proportional to the angle between the scalar and pseudoscalar couplings. In different limits the effective action reduces either to the Euler–Heisenberg effective action or the Coleman–Weinberg potential. We also comment on the relationship between the odd part of the effective action and the chiral anomaly in QED.

On 5-regular bipartitions with even parts distinct

In 2010, Andrews, Michael D. Hirschhorn and James A. Sellers considered the function ped(n), the number of partition of an integer n with even parts distinct (the odd parts are unrestricted). They obtained infinite families of congruences in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). Let b(n) denote the number of 5-regular bipartitions of a positive integer n with even parts distinct (odd parts are unrestricted). In this paper, we establish many infinite families of congruences modulo powers of 2 for b(n). For example, $$\begin{aligned} \sum _{n=0}^{\infty } b\left(16\cdot 3^{2\alpha }\cdot 5^{2\beta }n+14\cdot 3^{2\alpha }\cdot 5^{2\beta }+1\right) q^n \equiv 8f_2^3f_5^3 \pmod {16} , \end{aligned}$$where $$\alpha , \beta \ge 0$$.

On overpartition pairs into odd parts modulo powers of 2

In 2012, Lin (Electron. J. Combin. 19(2) (2012) #P17) investigated the 2 and 3-divisibility properties for pp¯o(n), the number of overpartition pairs into odd parts. Using modular forms, he proved that for a fixed positive integer k, pp¯o(n) is almost always divisible by 2k. In this paper, we prove several congruences for pp¯o(n) modulo higher powers of 2 in an elementary way.

Minimal dark matter in SU(2)L × U(1)Y × U(1)B−L

The minimal gauge group extension to the standard model (SM) by the local U(1)B−L (MBLSM) is well known as the minimal model to understand neutrino mass origins via the seesaw mechanism, following the gauge principle. This “small” symmetry also has deep implication to another big thing, dark matter (DM) stability. We demonstrate it in the framework of minimal dark matter (MDM), which aims at addressing two basic questions on DM, stability and the nature of interactions. However, stability and perturbativity may only allow the fermionic quintuplet. The situation is very different in the MBLSM, which leaves the subgroup of U(1)B−L, the matter parity (−1)3(B−L), unbroken; it is able to stabilize all of the MDM candidates after assigning a proper U(1)B−L charge. For the candidates with nonzero hypercharge, the phenomenological challenge comes from realizing the inelastic DM scenario thus evading the very strict DM direct detention bounds. We present two approaches that can slightly split the CP-even and -odd parts of the neutral components: 1) using the dimension 5 operators, which works for the U(1)B−L spontaneously breaking at very high scale; 2) mixing with other fields having zero hypercharge, which instead works for a low U(1)B−L breaking scale.

Nonlocal operators with local boundary conditions in higher dimensions

We present novel nonlocal governing operators in 2D/3D for wave propagation and diffusion. The operators are inspired by peridynamics. They agree with the original peridynamics operator in the bulk of the domain and simultaneously enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of separable kernel functions together with even and odd parts of bivariate functions on rectangular/box domains. The operators are bounded and self-adjoint. We present all possible 36 different types of BC in 2D which include pure and mixed combinations of Neumann, Dirichlet, periodic, and antiperiodic BC. Our construction is systematic and easy to follow. We provide numerical experiments that verify our theoretical findings. We also compare the solutions of the classical wave and heat equations to their nonlocal counterparts.

Exact generating functions for the number of partitions into distinct parts

Let [Formula: see text] denote the number of partitions of a non-negative integer into distinct (or, odd) parts. We find exact generating functions for [Formula: see text], [Formula: see text] and [Formula: see text]. We deduce some congruences modulo 5 and 25. We employ Ramanujan’s theta function identities and some identities for the Rogers–Ramanujan continued fraction.

The quadratic type of the 2-principal indecomposable modules of the double covers of alternating groups

The principal indecomposable modules of the double cover 2.An of the alternating group over a field of characteristic 2 are enumerated using the partitions of n into distinct parts. We determine which of these modules afford a non-degenerate 2.An-invariant quadratic form. Our criterion depends on the alternating sum and the number of odd parts of the corresponding partition.

Integer Partitions with Even Parts Below Odd Parts and the Mock Theta Functions

The paper begins with a study of a couple of classes of partitions in which each even part is smaller than each odd. In one class, a Dyson-type crank exists to explain a mod 5 congruence. The second part of the paper treats the arithmetic and combinatorial properties of the third order mock theta function $${\nu(q)}$$ and relates the even part of $${\nu(q)}$$ to the partitions initially considered. We also consider a surprisingly simple combinatorial relationship between the cranks and the ranks of the partition of n.

The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts

We interpret the second difference of the sequence of the number of strict partitions, for n≥5, as the sequence of the strict partitions of n with at least three parts, the three largest parts consecutive, and the smallest part at least two. The name butterfly describes both a sequence's interpretation and a bijection between subsets of strict partitions. Using cyclotomic polynomials, we compute generating function identities of the butterfly sequence and related sequences both as infinite products and as series filtered by the number of parts. We offer a merging and splitting construction of the butterfly sequence as a sequence of partitions with odd parts larger or equal to 3, and we interpret the butterfly sequence as a sequence of generalized pentagonal, pentagonal with domino, and non-pentagonal butterfly partitions. Euler's Pentagonal Number Theorem and a similar specialization of the Jacobi Triple Product lead to recursive algorithms to compute the butterfly sequence and related sequences.

Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts

Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts

Arithmetical Fourier and limit values of elliptic modular functions

Here, we shall use the first periodic Bernoulli polynomial $$\bar{B}_1(x)=x-[x]-\frac{1}{2}$$ to resurrect the instinctive direction of B Riemann in his posthumous fragment II on the limit values of elliptic modular functions a la C G J Jacobi, Fundamenta Nova §40 (1829). In the spirit of Riemann who considered the odd part, we use a general Dirichlet–Abel theorem to condense Arias–de-Reyna’s theorems 8–15 into ‘a bigger theorem’ in Sect. 2 by choosing a suitable R-function in taking the radial limits. We supplement Wang (Ramanujan J. 24 (2011) 129–145). Furthermore, the same method is applied to obtain in Sect. 3 a correct representation for the ‘trigonometric series’, i.e., we prove that for every rational number x the trigonometric series (3.5) is represented by $$\sum _{n=1}^{\infty }(-1)^n\frac{{\bar{B}}_1(nx)}{n}$$ as Dedekind suggested but not by $$\sum _{n=1}^{\infty }\frac{{\bar{B}}_1(nx)}{n}$$ as Riemann stated.

The d-primary Brauer–Manin obstruction for curves

For a curve over a global field we consider for which integers d the d-primary part of the Brauer group can obstruct the existence of rational points. We give examples showing it is possible that there is a d-primary obstruction for infinitely many pairwise coprime d, and also an example where the odd part of the Brauer group does not obstruct although there is a Brauer–Manin obstruction. These examples demonstrate that a slightly stronger form of a conjecture of Poonen is false, despite being supported by the same heuristic.