Odd Parts Research Articles

A sufficient condition for a rational differential operator to generate an integrable system

For a rational differential operator \({L=AB^{-1}}\), the Lenard–Magri scheme of integrability is a sequence of functions \({F_n, n \geq 0}\), such that (1) \({B(F_{n+1})=A(F_n)}\) for all \({n \geq 0}\) and (2) the functions \({B(F_n)}\) pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of \({B(F_n)}\) is unbounded, property (2) holds if and only if L belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator L is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence (Fn) starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.

Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions

We study the stationary Wigner equation on a bounded, one-dimensional spatial domain with inflow boundary conditions by using the parity decomposition of L. Barletti and P. F. Zweifel [Transport Theory Statist. Phys., 2001, 30(4-6): 507–520]. The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd L 2-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.

Overpartitions related to the mock theta function $\omega (q)$

It was recently shown that $q\omega(q)$, where $\omega(q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega}(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less than twice the smallest part. In this paper, we study the overpartition analogue of $p_{\omega}(n)$, and express its generating function in terms of a ${}_3\phi_{2}$ basic hypergeometric series and an infinite series involving little $q$-Jacobi polynomials. This is accomplished by obtaining a new seven parameter $q$-series identity which generalizes a deep identity due to the first author as well as its generalization by R.P.~Agarwal. We also derive two interesting congruences satisfied by the overpartition analogue, and some congruences satisfied by the associated smallest parts function.

On 3-regular partitions with odd parts distinct

We consider \(\text {pod}_3(n)\), the number of 3-regular partitions with odd parts distinct, whose generating function is $$\begin{aligned} \sum _{n\ge 0}\text {pod}_3(n)q^n=\frac{(-q;q^2)_\infty (q^6;q^6)_\infty }{(q^2;q^2)_\infty (-q^3;q^3)_\infty }=\frac{\psi (-q^3)}{\psi (-q)}, \end{aligned}$$ where $$\begin{aligned} \psi (q)=\sum _{n\ge 0}q^{(n^2+n)/2}=\sum _{-\infty }^\infty q^{2n^2+n}. \end{aligned}$$ For each \(\alpha >0\), we obtain the generating function for $$\begin{aligned} \sum _{n\ge 0}\text {pod}_3\left( 3^{\alpha }n+\delta _\alpha \right) q^n, \end{aligned}$$ where \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha }}\) if \(\alpha \) is even, \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha +1}}\) if \(\alpha \) is odd.

Simple right alternative superalgebras of Abelian type whose even part is a field

We study central simple unital right alternative superalgebras of Abelian type of arbitrary dimension whose even part is a field. We prove that every such superalgebra , except for the superalgebra , is a double, that is, the odd part can be represented in the form for a suitable . If the generating element commutes with the even part , then is isomorphic to a twisted superalgebra of vector type introduced by Shestakov [1], [2]. But if commutes with the odd part , then is isomorphic to a superalgebra introduced in [3] and called an -double. We prove that if the ground field is algebraically closed, then is isomorphic to one of the superalgebras , , .

Energetics of synchronization in coupled oscillators rotating on circular trajectories.

We derive a concise and general expression of the energy dissipation rate for coupled oscillators rotating on circular trajectories by unifying the nonequilibrium aspects with the nonlinear dynamics via stochastic thermodynamics. In the framework of phase oscillator models, it is known that the even and odd parts of the coupling function express the effect on collective and relative dynamics, respectively. We reveal that the odd part always decreases the dissipation upon synchronization, while the even part yields a characteristic square-root change of the dissipation near the bifurcation point whose sign depends on the specific system parameters. We apply our theory to hydrodynamically coupled Stokes spheres rotating on circular trajectories that can be interpreted as a simple model of synchronization of coupled oscillators in a biophysical system. We show that the coupled Stokes spheres gain the ability to do more work on the surrounding fluid as the degree of phase synchronization increases.

Simple Jordan superalgebras with associative nil-semisimple even part

Under study are the simple infinite-dimensional abelian Jordan superalgebras not isomorphic to the superalgebra of a bilinear form. We prove that the even part of such superalgebra is a differentially simple associative commutative algebra, and the odd part is a finitely generated projective module of rank 1. We describe unital simple Jordan superalgebras with associative nil-semisimple even part possessing two even elements which induce a nonzero derivation.

Proof of a limited version of Mao’s partition rank inequality using a theta function identity

Ramanujan’s congruence \(p(5k+4) \equiv 0 \pmod 5\) led Dyson (Eureka 8:10–15, 1944) to define a measure “rank”, and then conjectured that \(p(5k+4)\) partitions of \(5k+4\) could be divided into subclasses with equal cardinality to give a direct proof of Ramanujan’s congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer (Some properties of partitions 4:84–106, 1954), who proved Dyson’s conjecture. More recently, Mao (Number Theory 133:3678–3702, 2013) proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 and for \(M_2\)-rank differences for partitions with no repeated odd parts modulo 6 and 10 (Mao in Ramanujan J 37:391–419, 2015). Alwaise et al. (Ramanujan J. doi:10.1007/s11139-016-9789-x, 2016) proved four of Mao’s conjectured inequalities, while leaving three open. Here, we prove a limited version of one of the inequalities conjectured by Mao.

A proof of some conjectures of Mao on partition rank inequalities

Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo 10, and additional results modulo 6 and 10 for the \(M_2\) rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao’s rank inequality conjectures for both the rank and the \(M_2\) rank modulo 10 using elementary methods.

Linear Super-Commuting Maps and Super-Biderivations on the Super-Virasoro Algebras

Let SVir be the well-known super-Virasoro algebras. In this paper, we first prove that any super-skewsymmetric super-biderivation of SVir is inner. Based on this, we show that every linear super-commuting map ψ on SVir is of the form ψ(x) = f(x)c, where f is a linear function from SVir to ℂ mapping the odd part of SVir to zero, and c is the central charge of SVir.

Pulses in binary wave guide arrays and long wave PDE approximations

Binary waveguide arrays are linear arrays of optical waveguides with binary alternation of parameters, and have been of recent interest. They can be modeled by systems of nonlinear ODEs with forms related to the discrete nonlinear Schrödinger equation. Such equations can also arise in semi-classical molecular models of polymers with excitable states in each monomer, and coupling between these.An important class of solutions arises from an initially highly localized signal, such as input to a single element of the array. Simulations show that for a wide array of parameter values and of such initial data, a pulse is generated that travels approximately as a traveling wave. After a suitable phase shift in the variables, this pulse quickly develops a slow spatial variation, leading to a long-wave approximation by a system of coupled third order PDEs; one each for nodes of even and odd indices.This system of PDEs is presented, and verified to quite accurately reproduce the pulse propagation seen in the ODE system; further there is often a strong tendency for the behavior of the two PDE components to converge, with a corresponding convergence of the even and odd index parts of the ODE system solution. The PDE model gives some indication of why this occurs.

Core partitions into distinct parts and an analog of Euler’s theorem

A special case of an elegant result due to Anderson proves that the number of (s,s+1)-core partitions is finite and is given by the Catalan number Cs. Amdeberhan recently conjectured that the number of (s,s+1)-core partitions into distinct parts equals the Fibonacci number Fs+1. We prove this conjecture by enumerating, more generally, (s,ds−1)-core partitions into distinct parts (the special case d=1 was independently also proved by Xiong). We do this by relating them to certain tuples of nested twin-free sets.As a by-product of our results, we obtain a bijection between partitions into distinct parts and partitions into odd parts, which preserves the perimeter (that is, the largest part plus the number of parts minus 1). This simple but curious analog of Euler’s theorem appears to be missing from the literature on partitions.

On partitions with fixed number of even-indexed and odd-indexed odd parts

This article is an extensive study of partitions with fixed number of odd and even-indexed odd parts. We use these partitions to generalize recent results of C. Savage and A. Sills. Moreover, we derive explicit formulas for generating functions for partitions with bounds on the largest part, the number of parts and with a fixed value of BG-rank or with a fixed value of alternating sum of parts. We extend the work of C. Boulet, and as a result, obtain a four-variable generalization of Gaussian binomial coefficients. In addition, we provide combinatorial interpretation of the Berkovich–Warnaar identity for Rogers–Szegő polynomials.

Asymptotic formulas for M2-ranks of partitions without repeated odd parts

Recently, K. Bringmann, Dousse and Mertens established asymptotic formulas for ranks and cranks of partitions which were first conjectured by Dyson. Motivated by their works, we prove asymptotic formulas for M2-ranks of partitions without repeated odd parts.

Orientifolding of the ABJ Fermi gas

The grand partition functions of ABJ theory can be factorized into even and odd parts under the reflection of fermion coordinate in the Fermi gas approach. In some cases, the even/odd part of ABJ grand partition function is equal to that of $\mathcal{N}=5$ $O(n)\times USp(n')$ theory, hence it is natural to think of the even/odd projection of grand partition function as an orientifolding of ABJ Fermi gas system. By a systematic WKB analysis, we determine the coefficients in the perturbative part of grand potential of such orientifold ABJ theory. We also find the exact form of the first few "half-instanton" corrections coming from the twisted sector of the reflection of fermion coordinate. For the Chern-Simons level $k=2,4,8$ we find closed form expressions of the grand partition functions of orientifold ABJ theory, and for $k=2,4$ we prove the functional relations among the grand partition functions conjectured in arXiv:1410.7658.

Submaximal conformal symmetry superalgebras for Lorentzian manifolds of low dimension

We consider a class of smooth oriented Lorentzian manifolds in dimensions three and four which admit a nowhere vanishing conformal Killing vector and a closed two-form that is invariant under the Lie algebra of conformal Killing vectors. The invariant two-form is constrained in a particular way by the conformal geometry of the manifold. In three dimensions, the conformal Killing vector must be everywhere causal (or null if the invariant two-form vanishes identically). In four dimensions, the conformal Killing vector must be everywhere null and the invariant two-form vanishes identically if the geometry is everywhere of Petrov type N or O. To the conformal class of any such geometry, it is possible to assign a particular Lie superalgebra structure, called a conformal symmetry superalgebra. The even part of this superalgebra contains conformal Killing vectors and constant R-symmetries while the odd part contains (charged) twistor spinors. The largest possible dimension of a conformal symmetry superalgebra is realised only for geometries that are locally conformally flat. We determine precisely which non-trivial conformal classes of metrics admit a conformal symmetry superalgebra with the next largest possible dimension, and compute all the associated submaximal conformal symmetry superalgebras. In four dimensions, we also compute symmetry superalgebras for a class of Ricci-flat Lorentzian geometries not of Petrov type N or O which admit a null Killing vector.

Padovan numbers as sums over partitions into odd parts

Recently it was shown that the Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. In this paper, we introduce a similar representation for the Padovan numbers. As a corollary, we derive an infinite family of double inequalities.

Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Partitions associated with the Ramanujan/Watson mock theta functions ω(q), ν(q)and ϕ(q)

The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function ω(q) (resp. ν(−q)). Similar results for partitions with the corresponding restriction on each even part are also obtained, one of which involves the third order mock theta function ϕ(q). Congruences for the smallest parts partition function(s) associated to such partitions are obtained. Two analogues of the partition-theoretic interpretation of Euler’s pentagonal number theorem are also obtained.

Partitions with Non-Repeating Odd Parts and Combinatorial Identities

Continuing our earlier work on partitions with non-repeating odd parts and q-hypergeometric identities, we now study these partitions combinatorially by representing them in terms of 2-modular Ferrers graphs. This yields certain weighted partition identities with free parameters. By special choices of these parameters, we connect them to the Gollnitz-Gordon partitions, and combinatorially prove a modular identity and some parity results. As a consequence, we derive a shifted partition theorem mod 32 of Andrews. Finally we discuss basis partitions in connection with the 2-modular representation of partitions with non-repeating odd parts, and deduce two new parity results involving partial theta series.