Odd Parts Research Articles

A proof of Subbarao's conjecture

Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) for which p(N) is even, and infinitely many integers M ≡ r (mod t) for which p(M) is odd. In the even case the conjecture was settled by Ken Ono. In this paper we prove the odd part of the conjecture which together with Ono's result implies the full conjecture. We also prove that for every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) such that p(N) ≢ 0 (mod 3), which settles an open problem posed by Scott Ahlgren and Ken Ono.

CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE pod FUNCTION

In this paper, we prove arithmetic properties modulo 5 and 7 satisfied by the function pod (n) which denotes the number of partitions of n wherein odd parts must be distinct (and even parts are unrestricted). In particular, we prove the following: for all n ≥ 0, [Formula: see text] and [Formula: see text]

Numerical Estimation of the Current Large Deviation Function in the Asymmetric Simple Exclusion Process with Open Boundary Conditions

We numerically study the large deviation function of the total current, which is the sum of local currents over all bonds, for the symmetric and asymmetric simple exclusion processes with open boundary conditions. We estimate the generating function by calculating the largest eigenvalue of the modified transition matrix and by population Monte Carlo simulation. As a result, we find a number of interesting behaviors not observed in the exactly solvable cases studied previously as follows. The even and odd parts of the generating function show different system-size dependences. Different definitions of the current lead to the same generating function in small systems. The use of the total current is important in the Monte Carlo estimation. Moreover, a cusp appears in the large deviation function for the asymmetric simple exclusion process. We also discuss the convergence property of the population Monte Carlo simulation and find that in a certain parameter region, the convergence is very slow and the gap between the largest and second largest eigenvalues of the modified transition matrix rapidly tends to vanish with the system size.

Stickelberger elements and Kolyvagin systems

AbstractIn this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is r > 1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

Symmetry of the CMB sky as a new test of its statistical isotropy. Non cosmological octupole?

In this article we propose a novel test for statistical anisotropy of the CMB ΔT(n̂ = (θ,ϕ)). The test is based on the fact, that the Galactic foregrounds have a remarkably strong symmetry with respect to their antipodal points S1: = n̂↔−n̂,n̂ = (θ,ϕ) and with respect to the Galactic plane, while the cosmological signal should not be symmetric or asymmetric under these transitions.We have applied the test for the octupole component of the WMAP ILC 7 map, by looking at a3,1 and a3,3, and their ratio to a3,2 both for real and imaginary values. We find abnormal symmetry of the octupole component at the level of 0.58%, compared to Monte Carlo simulations. By using the analysis of the phases of the octupole we found remarkably strong cross-correlations between the phases of the kinematic dipole and the ILC 7 octupole, in full agreement with previous results. We further test the multipole range 2 < l < 100, by investigating the ratio between the l+m = even and l+m = odd parts of power spectra. We compare the results to simulations of a Gaussian random sky, and find significant departure from the statistically isotropic and homogeneous case, for a very broad range of multipoles. We found that for the most prominent peaks of our estimator, the phases of the corresponding harmonics are coherent with phases of the octupole. We believe, our test would be very useful for detections of various types of residuals of the foreground and systematic effects at a very broad range of multipoles 2 ⩽ l ⩽ 1500−3000 for the forthcoming PLANCK CMB map, before any conclusions about primordial non-Gaussianity and statistical anisotropy of the CMB.

Stickelberger elements and Kolyvagin systems

AbstractIn this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank isr &amp;gt;1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets

In this paper we derive the most general first-order symmetry operator commuting with the Dirac operator in all dimensions and signatures. Such an operator splits into Clifford even and Clifford odd parts which are given in terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous forms respectively. We study commutators of these symmetry operators and give necessary and sufficient conditions under which they remain of the first-order. In this specific setting we can introduce a Killing-Yano bracket, a bilinear operation acting on odd Killing-Yano and even closed conformal Killing-Yano forms, and demonstrate that it is closely related to the Schouten-Nijenhuis bracket. An important non-trivial example of vanishing Killing-Yano brackets is given by Dirac symmetry operators generated from the principal conformal Killing-Yano tensor [hep-th/0612029]. We show that among these operators one can find a complete subset of mutually commuting operators. These operators underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all dimensions [arXiv:0711.0078].

ℓ-adic properties of smallest parts functions

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on ℓ-adic properties of certain modular forms and mock modular forms of weight 3/2 with respect to the Hecke operators T ( ℓ 2 m ) .

First-Principle Derivation of Entropy Production in Transport Phenomena

The linear response framework was established by Kubo a half century ago, but no clear explanation of irreversibility namely entropy production has been given in this scheme. This has been now solved. The serious puzzle up to now is the following. Even using the linear response density matrix ρlr = ρ0 + ρ1(t), it has been difficult to derive the entropy production.Surprisingly, the correct entropy production is given by the second-order term ρ2(t) as . It is shown to agree with the ordinary expression J·E/T = σE2/T in the case of electric conduction for a static electric field E, where σ denotes the electric conductivity expressed by the famous canonical current-current time correlation functions in equilibrium.The present article gives a review of the derivation of entropy production (M.S., Physica A 390(2011)1904-1916) based on the first-principle of using the projected density matrix ρ2(t) or more generally ρeven(t), while the previous standard argument is due to the thermodynamic energy balance. This new derivation clarifies conceptually the physics of irreversibility in transport phenomena. In general, the transport phenomena are described by the odd part ρodd(t) of the density matrix and the entropy production (namely irreversibility) is described by the even part ρeven(t). These are related to each other through the coupled equations as , where "q" denotes "even" (symmetric) or "odd" (antisymmetric), Pq is the projection operator to the "q" part of ρ(t), and ℋ1(t) denotes the partial Hamiltonian due to the external force such as in electric conduction.The concept of a stationary temperature Tst in steady states with current (say electric current) is also proposed by using the projected and symmetry-separated von Newmann equation introduced by the present author. The entropy production of the relevant steady state depends on this stationary temperature. A mechanical formulation of thermal conduction is given by introducing a thermal field ET and its conjugate "heat" operator for a local internal energy hj of the thermal particle j.

Congruences for bipartitions with odd parts distinct

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider the number of bipartitions with odd parts distinct. Let this number be denoted by pod−2(n). We obtain two Ramanujan-type identities for pod−2(n), which imply that pod−2(2n+1) is even and pod−2(3n+2) is divisible by 3. Furthermore, we show that for any α≥1 and n≥0, \(\mathit{pod}_{-2}(3^{2\alpha+1}n+\frac{23\times 3^{2\alpha}-7}{8}) \) is a multiple of 3 and \(\mathit{pod}_{-2}(5^{\alpha+1}n+\frac{11\times5^{\alpha}+1}{4})\) is divisible by 5. We also find combinatorial interpretations for the two congruences modulo 2 and 3.

EASY MONEY OR HARD TIMES? HEALTH AND 401(k) LOANS

1. INTRODUCTION Medical debt has become increasingly commonplace. Frequently, households rely on debt to finance health care; as health insurance coverage has declined, out-of-pocket expenditures have increased and health care costs have risen. Loans from defined contribution (DC) retirement savings plans, such as 401 (k) plans, are among the most easily accessed resources because savers act as lenders to themselves. As DC plans have become more prevalent, households have increased access to this type of loan. Not only has the prevalence of DC plans increased, account balances have grown and more employers offer the option to borrow from DC plans to their employees. All of these factors increase a household's ability to borrow from itself. The financing of unexpected emergencies, such as medical care, is one reason for allowing loans from tax-advantaged retirement accounts. The logic is that households will accrue savings in their retirement accounts faster if they know that they can access those funds in an emergency. On the other hand, rising medical debt is one of the most consistent predictors of personal bankruptcy. The personal bankruptcy rate has risen over the decades along with growing personal debt, especially medical debt. The ease with which DC loans can be accessed, though, may mean that these loans are used for a variety of consumption purposes, including all kinds of goods and services. It is possible that DC loans are systematically related to a number of factors other than health status, for instance, homeownership or educational levels. Understanding the reasons why people borrow from their retirement plans has clear policy implications. DC loans may pose a dilemma for policymakers if these loans are related to health status and health insurance coverage. The money in such accounts is intended to supplement other retirement income. A number of federal tax incentives support this goal. People may borrow from their DC plans to pay for their health care. However, borrowing is likely to result in a reduction in their retirement savings and may increase their chance of future bankruptcy. More DC loans may consequently reduce immediate hardships for households, but increase future financial insecurity. In this article, we focus on the relationship between health status, access to health insurance, and the likelihood of having a DC loan and the loan amount. We find that households with a household member in poor health are more likely to borrow, and borrow larger amounts, from their DC plans than do healthy households. Health status is, in fact, as important as or even more important than other potential reasons for DC loans--such as education and home-ownership. The link between health status and DC loans remains after we control for health insurance coverage. Even those households with health insurance tend to systematically increase their probability of borrowing from DC plans when a family member is in bad health. The rest of the article is organized as follows. We provide an overview of the relevant literature in Section II, followed by a discussion of some summary data in Section III. Our multivariate analyses of the links between health status, health insurance coverage, and DC loans follow in Section IV. Section V contains our concluding remarks. II. LITERATURE REVIEW A. Effects of DC Loans DC account loans are an odd sort of loan. The odd part of these loans is that credit approvals and the ability to pay back the loan are not subject to outside review. With this type of loan, the borrower is also the banker, albeit within some limits (GAO 1997). People with a DC plan may borrow $50,000 or one-half of the vested balance from the account, whichever is lower. Loans must be repaid within 5 years, except for the loans used for the first-time purchase of a home, which can be repaid over 15 years. The interest rates on these loans are generally favorable. …

On the number of partitions with distinct even parts

Let p e d ( n ) be the number of partitions of n wherein even parts are distinct (and odd parts are unrestricted). We obtain many congruences for p e d ( n ) mod 2 and mod 4 by the theory of Hecke eigenforms.

Some progress on the existence of 1-rotational Steiner triple systems

A Steiner triple system of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point. The spectrum of values of v for which there exists a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57---66, 1981), Buratti (J Combin Des 9:215---226, 2001) and Mishima (Discrete Math 308:2617---2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-rotational STS(v) under an arbitrary group has not been completely determined yet. This paper is a considerable step forward to the solution of this problem. In fact, we leave as uncertain cases only those for which we have v = (p 3?p)n + 1 ? 1 (mod 96) with p a prime, $${n \not\equiv 0}$$ (mod 4), and the odd part of (p 3 ? p)n that is square-free and without prime factors congruent to 1 (mod 6).

A Rademacher‐Type Formula for pod(n)

A Rademacher‐type formula for the Fourier coefficients of the generating function for the partitions of n where no odd part is repeated is presented.

A review of a Riesz basis property for indefinite Sturm-Liouville problems

Consider the indefinite Sturm-Liouville problem−f ′′ = λrf, f(−1) = f(1) = 0 with an indefinite weight function r ∈ L[−1, 1] satisfying xr(x) > 0. A number of conditions for the so-called Riesz basis property are reviewed, i.e. conditions such that the eigenfunctions form a Ries basis of the Hilbert space L|r|[−1, 1]. Most of these conditions were previously known to be either necessary or sufficient (or both if r is odd). Now it is shown that all are equivalent to the Riesz basis property if r is strongly odd dominated, i.e. the even part of r is dominated by the odd part in some sense. One of the equivalent conditions is the validity of the HELP-type inequality on [0, 1] (∫ 1

A family of necessary stability inequalities via quadratic forms

Inequalities relating three coefficients of the even or odd part of a Hurwitz-stable polynomial have been established recently via the Newton-MacLaurin inequalities, and via optimization techniques for multivariate functions on the positive orthant. From the theory of quadratic forms we derive a family of strict inequalities which includes and generalizes the known inequalities. For polynomials of higher degree quantifiable improvements are obtained. Benefit of these inequalities is low-cost instability testing for polynomials with varying coefficients. Mathematics subject classification (2010): 15A63, 26C10, 93D09.

On Riemann’s posthumous fragment II on the limit values of elliptic modular functions

We shall resurrect the instinctive direction of B. Riemann on his posthumous fragment on the limit values of elliptic modular functions a la C.G.J. Jacobi, Fundamenta Nova. In the spirit of Riemann who considered the odd part, we shall realize the situation where there is no singularity occurring in taking the radial limits, thus streamlining and elucidating the recent investigation by Arias de Reyna. By the new Dirichlet–Abel theorem (which should be within reach of Riemann), we may directly sum the series in question, which allows us to condense Arias de Reyna’s paper into a few pages.

Counting Ordered Pairs

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].

Arithmetic properties of partitions with even parts distinct

In this work, we consider the function ped(n), the number of partitions of an integer n wherein the even parts are distinct (and the odd parts are unrestricted). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n): We prove a number of results for ped(n) including the following: For all n 0; ped(9n + 4) 0 (mod 4) and

The classification of Leibniz superalgebras of nilindex [formula omitted] ( [formula omitted

In this paper we investigate the description of the complex Leibniz superalgebras with nilindex n + m , where n and m ( m ≠ 0 ) are dimensions of even and odd parts, respectively. In fact, such superalgebras with characteristic sequence equal to ( n 1 , … , n k | m 1 , … , m s ) (where n 1 + ⋯ + n k = n , m 1 + ⋯ + m s = m ) for n 1 ⩾ n − 1 and ( n 1 , … , n k | m ) were classified in works by Ayupov et al. (2009) [3], Camacho et al. (2010) [4], Camacho et al. (in press) [5], Camacho et al. (in press) [6]. Here we prove that in the case of ( n 1 , … , n k | m 1 , … , m s ) , where n 1 ⩽ n − 2 and m 1 ⩽ m − 1 the Leibniz superalgebras have nilindex less than n + m . Thus, we complete the classification of Leibniz superalgebras with nilindex n + m .