Cycle Lengths Modulo k in Large 3-connected Cubic Graphs, Advances in Combinatorics

The existence of cycles with a given length is classical topic in graph theory with a plethora of open problems. Examples related to the main result of this paper include a conjecture of Burr and Erdős from 1976 asked whether for every integer $m$ and a positive odd integer $k$, there exists $d$ such that every graph with average degree at least $d$ contains a cycle of length $m$ modulo $k$; this conjecture was proven by Bollobás in [Bull. London Math. Soc. 9 (1977), 97-98]( https://doi.org/10.1112/blms/9.1.97). Another example is a problem of Erdős from the 1990s asking whether there exists $A\subseteq\mathbb{N}$ with zero density and constants $n_0$ and $d_0$ such that every graph with at least $n_0$ vertices and the average degree at least $d_0$ contains a cycle with length in the set $A$, which was resolved by Verstraete in [J. Graph Theory 49 (2005), 151-167]( https://doi.org/10.1002/jgt.20072). In 1983, Thomassen conjectured that for all integers $m$ and $k$, every graph with minimum degree $k+1$ contains a cycle of length $2m$ modulo $k$. Note that the parity condition in the first and the third conjectures is necessary because of bipartite graphs. The current paper contributes to this long line of research by proving that for every integer $m$ and a positive odd integer $k$, every sufficiently large $3$-connected cubic graph contains a cycle of length $m$ modulo $k$. The result is the best possible in the sense that the same conclusion is not true for $2$-connected cubic graphs or $3$-connected graphs with minimum degree three.

Simple Stochastic Games with Almost-Sure Energy-Parity Objectives are in NP and coNP

We study stochastic games with energy-parity objectives, which combine quantitative rewards with a qualitative $$\omega $$ ω -regular condition: The maximizer aims to avoid running out of energy while simultaneously satisfying a parity condition. We show that the corresponding almost-sure problem, i.e., checking whether there exists a maximizer strategy that achieves the energy-parity objective with probability 1 when starting at a given energy level k, is decidable and in $$\mathsf {NP}\cap \mathsf {coNP}$$ NP ∩ coNP . The same holds for checking if such a k exists and if a given k is minimal.

Risk Return Trade-Off in Relaxed Risk Parity Portfolio Optimization

This paper formulates a relaxed risk parity optimization model to control the balance of risk parity violation against the total portfolio performance. Risk parity has been criticized as being overly conservative and it is improved by re-introducing the asset expected returns into the model and permitting the portfolio to violate the risk parity condition. This paper proposes the incorporation of an explicit target return goal with an intuitive target return approach into a second-order-cone model of a risk parity optimization. When the target return is greater than risk parity return, a violation to risk parity allocations occurs that is controlled using a computational construct to obtain near-risk parity portfolios to retain as much risk parity-like traits as possible. This model is used to demonstrate empirically that higher returns can be achieved than risk parity without the risk contributions deviating dramatically from the risk parity allocations. Furthermore, this study reveals that the relaxed risk parity model exhibits advantageous traits of robustness to expected returns, which should not deter the use of expected returns in risk parity model.

Reducibility of Schrödinger Equation on the Sphere

In this article we prove a reducibility result for the linear Schrödinger equation on the sphere $\mathbb{S}^n$ with quasi-periodic in time perturbation. Our result includes the case of unbounded perturbation that we assume to be of order strictly less than $1/2$ and satisfying some parity condition. As far as we know, this is one of the few reducibility results for an equation in more than one dimension with unbounded perturbations. Letus note that, surprisingly, our result does not require the use of the pseudo-differential calculus although the perturbation is unbounded.

The number of equivalence classes arising from partition involutions

Involutions have played important roles in many research areas including the theory of partitions. In this paper, for various sets of partitions, we give relations between the number of equivalence classes in the set of partitions arising from an involution and the number of partitions satisfying a certain parity condition. We examine the number of equivalence classes arising from the conjugations on ordinary partitions, overpartitions, and partitions with distinct odd parts. We also consider other types of involutions on partitions into distinct parts, unimodal sequences with a unique marked peak, and partitions with distinct even parts.

A Version of the Berglund–Hübsch–Henningson Duality With Non-Abelian Groups

A. Takahashi suggested a conjectural method to find mirror symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. Here we generalize the Saito duality between Burnside rings to a case of non-abelian groups and prove a “non-abelian” generalization of the statement about the equivariant Saito duality property for invertible polynomials. It turns out that the statement holds only under a special condition on the action of the subgroup of the permutation group called here PC (“parity condition”). An inspection of data on Calabi–Yau three-folds obtained from quotients by non-abelian groups shows that the pairs found on the basis of the method of Takahashi have symmetric pairs of Hodge numbers if and only if they satisfy PC.

Testing the interest-parity condition with Irving Fisher's example of Indian rupee and sterling bonds in the London financial market, 1869–1906

Following the pioneering work of Irving Fisher, this article assesses the uncovered interest-parity (UIP) condition by comparing Indian interest and exchange rates during the 1869 to 1906 period. The Indian case provides a good example of the UIP condition, since Indian rupee and sterling bonds were simultaneously traded in the London financial market and subject to negligible default risks. Large deviations from the UIP condition arose when India suffered from pervasive levels of uncertainty about the future of its silver-based currency system. Otherwise, a relatively close correlation arises between sterling-to-rupee interest-rate differences and exchange-rate changes.

On the Strength of Unambiguous Tree Automata

This work is a study of the class of non-deterministic automata on infinite trees that are unambiguous i.e. have at most one accepting run on every tree. The motivating question asks if the fact that an automaton is unambiguous implies some drop in the descriptive complexity of the language recognised by the automaton. As it turns out, such a drop occurs for the parity index and does not occur for the weak parity index.More precisely, given an unambiguous parity automaton [Formula: see text] of index [Formula: see text], we show how to construct an alternating automaton [Formula: see text] which accepts the same language, but is simpler in terms of the acceptance condition. In particular, if [Formula: see text] is a Büchi automaton ([Formula: see text]) then [Formula: see text] is a weak alternating automaton. In general, [Formula: see text] belongs to the class [Formula: see text], what implies that it is simultaneously of alternating index [Formula: see text] and of the dual index [Formula: see text]. The transformation algorithm is based on a separation procedure of Arnold and Santocanale (2005).In the case of non-deterministic automata with the weak parity condition, we provide a separation procedure analogous to the one used above. However, as illustrated by examples, this separation procedure cannot be used to prove a complexity drop in the weak case, as there is no such drop.

Chowla’s theorem over function fields

Sarvadaman Chowla proved that if [Formula: see text] is an odd prime, then [Formula: see text] ([Formula: see text]) are linearly independent over the field of rational numbers. We establish an analog of this result over function fields. As an application, we prove an analog of the Baker–Birch–Wirsing theorem about the non-vanishing of Dirichlet series with periodic coefficients at [Formula: see text] in the function field setup with a parity condition.