Conjecture Research Articles

Proof of the magnetic gap-labelling conjecture for principal solenoidal tori

In this note, we prove the magnetic spectral gap-labelling conjecture as stated in [8], in all dimensions, for principal solenoidal tori.

An Improved Bound in Vizing’s Conjecture

A well-known conjecture of Vizing (Vycisl Sistemy 9:30–43, 1963) is that $$\gamma (G \square H) \ge \gamma (G)\gamma (H)$$ for any pair of graphs G, H, where $$\gamma $$ is the domination number and $$G \square H$$ is the Cartesian product of G and H. Suen and Tarr [7], improving a result of Clark and Suen [5], showed $$\gamma (G \square H) \ge \frac{1}{2}\gamma (G)\gamma (H) + \frac{1}{2}\min (\gamma (G),\gamma (H))$$. We further improve their result by showing $$\gamma (G \square H) \ge \frac{1}{2}\gamma (G)\gamma (H) + \frac{1}{2}\max (\gamma (G),\gamma (H))$$.

Near-Optimal Finite-Length Scaling for Polar Codes Over Large Alphabets

For any prime power $q$ , Mori and Tanaka introduced a family of $q$ -ary polar codes based on the $q\,\,\times \,\,q$ Reed–Solomon polarization kernels. For transmission over a $q$ -ary erasure channel, they also derived a closed-form recursion for the erasure probability of each effective channel. In this paper, we use that expression to analyze the finite-length scaling of these codes on the $q$ -ary erasure channel with erasure probability $\epsilon \in (0,1)$ . Our primary result is that for any $\gamma >0$ and $\delta >0$ , there is a $q_{0}$ , such that for all $q\geq q_{0}$ , the fraction of effective channels with erasure rate at most $N^{-\gamma }$ is at least $1-\epsilon -O(N^{-1/2+\delta })$ , where $N=q^{n}$ is the blocklength. Since this fraction cannot be larger than $1-\epsilon -O(N^{-1/2})$ , this establishes near-optimal finite-length scaling for this family of codes. Our approach can be seen as an extension of a similar analysis for binary polar codes by Hassani, Alishahi, and Urbanke. A similar analysis is also considered for $q$ -ary polar codes with $m \times m$ polarizing matrices. This separates the effect of the alphabet size from the effect of the matrix size. If the polarizing matrix at each stage is drawn independently and uniformly from the set of invertible $m \times m$ matrices, then the linear operator associated with the Lyapunov function analysis can be written in the closed form. To prove near-optimal scaling for polar codes with fixed $q$ as $m$ increases, however, two technical obstacles remain. Thus, we conclude by stating two concrete mathematical conjectures that, if proven, would imply near-optimal scaling for fixed $q$ .

Quantum Contract Signing with Entangled Pairs

We present a quantum scheme for signing contracts between two clients (Alice and Bob) using entangled states and the services of a third trusted party (Trent). The trusted party is only contacted for the initialization of the protocol, and possibly at the end, to verify clients’ honesty and deliver signed certificates. The protocol is fair, i.e., the probability that a client, say Bob, can obtain a signed copy of the contract, while Alice cannot, can be made arbitrarily small, and scales as , where is the total number of rounds (communications between the two clients) of the protocol. Thus, the protocol is optimistic, as cheating is not successful, and the clients rarely have to contact Trent to confirm their honesty by delivering the actual signed certificates of the contract. Unlike the previous protocol (Paunković et al., Phys. Rev. A 84, 062331 (2011)), in the present proposal, a single client can obtain the signed contract alone, without the need for the other client’s presence. When first contacting Trent, the clients do not have to agree upon a definitive contract. Moreover, even upon terminating the protocol, the clients do not reveal the actual contract to Trent. Finally, the protocol is based on the laws of physics, rather than on mathematical conjectures and the exchange of a large number of signed authenticated messages during the actual contract signing process. Therefore, it is abuse-free, as Alice and Bob cannot prove they are involved in the contract signing process.

New Refinements of the Erdös–Mordell Inequality and Barrow’s Inequality

In this paper, two new refinements of the Erdös–Mordell inequality and three new refinements of Barrow’s inequality are established. Some related interesting conjectures are put forward.

Breaking Boundaries: Discovering the Impossible Counterproof of Beal’s Conjecture

This paper will attempt to logically differentiate between two types of fractions and discuss the idea of Zero as a neutral integer. This logic can then be followed to create a counterexample and a proof for Beal’s conjecture.

A -ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE

We give a $q$-analogue of the following congruence: for any odd prime $p$, $$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p),\end{eqnarray}$$ which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author [‘A $q$-analogue of the (L.2) supercongruence of Van Hamme’, J. Math. Anal. Appl. 466 (2018), 749–761].

On Frank's conjecture on k-connected orientations

We disprove a conjecture of Frank [3] stating that each weakly 2k-connected graph has a k-vertex-connected orientation. For k≥3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.

Has Carmichael's Totient Conjecture Been Proven? No, No, it has not.

Has Carmichael's Totient Conjecture Been Proven? No, No, it has not.

Validity and entailment in modal and propositional dependence logics

The computational properties of modal and propositional dependence logics have been extensively studied over the past few years, starting from a result by Sevenster showing NEXPTIME-completeness of the satisfiability problem for modal dependence logic. Thus far, however, the validity and entailment properties of these logics have remained mostly unaddressed. This paper provides a comprehensive classification of the complexity of validity and entailment in various modal and propositional dependence logics. The logics examined are obtained by extending the standard modal and propositional logics with notions of dependence, independence, and inclusion in the team semantics context. In particular, we address the question of the complexity of validity in modal dependence logic. By showing that it is NEXPTIME-complete we refute an earlier conjecture proposing a higher complexity for the problem.

A number of conjectures are likely to setup

A number of conjectures are likely to setup

What is Falconer's Conjecture?

What is Falconer's Conjecture?

The first screenings of Lumière films in China: Conjectures and new findings

The first screenings of Lumière films in China: Conjectures and new findings

Young Scientists Discovering Food Web: An IBSE (Inquiry Based Science Education) Activity at Zoo Delle Maitine (Benevento, Italy)

The IBSE approach in the study of science allows pupils to formulate themselves the questions to which they need to find an answer. Is a highly stimulating approach that was used in an Educational Activity about food web at the Zoo. Starting from the observation of living creatures (from insects to reptiles and small mammals as well) and then posing themselves questions, kids were stimulated to find answers and to hypothesize conjectures. The success, calculated by questionnaires, both between kids and teachers, and checking every five minutes the attention of students, was great and the lesson about food web in nature was, for sure, learned. A total of 480 students participated in the activities; 69 teachers were present and questioned at the end of activities: 25 had the IBSE lab, 44 the non IBSE. Opinions about both activities, IBSE and NON IBSE, were generally good but the activity conducted with the IBSE method scored a higher value so the IBSE method was a great success.

A Propose of modal education for Korean learners - around ‘-(eu)lkka ha-/bo-’ used to mean conjecture and intention.

A Propose of modal education for Korean learners - around ‘-(eu)lkka ha-/bo-’ used to mean conjecture and intention.

Sticky matroids and Kantor’s Conjecture

We prove the equivalence of Kantor’s Conjecture and the Sticky Matroid Conjecture due to Poljak und Turzik.

A q-analogue of a curious supercongruence of Guillera and Zudilin

ABSTRACTThe following curious supercongruence of Guillera and Zudilin: plays an important part in their proof of a ‘divergent’ Ramanujan-type supercongruence. In this paper, we give a q-analogue of this supercongruence by using the q-WZ method twice. Meanwhile, we confirm a q-analogue of the mentioned ‘divergent’ Ramanujan-type supercongruence, which was early conjectured by the author.

A q-analogue of the (I.2) supercongruence of Van Hamme

We give a [Formula: see text]-analogue of the supercongruence (I.2) of Van Hamme. As a conclusion, we confirm a recent conjecture of Swisher. We also give a [Formula: see text]-analogue of the corresponding [Formula: see text] series (I.1) along with some similar results.

Corrigendum The Generalized Artin Primitive Root Conjecture (EUROPEAN J. OF PURE AND APPLIED MATH. Vol. 11, No. 1, 2018, 23-34)

The subset of integers $\mathcal{N}_2= \{ n\in \mathbb{N}:\text{ord}_n(2)=\lambda(n) \}$ in page 24, \cite{CN18}, should be \\ $$\label{eq2-40} \mathcal{N}_u =\left\{ n\in \mathbb{N}:\ord_n(u)=\lambda(n) \text{ and } p \mid n \Rightarrow \ord_p(u)=p-1, \ord_{p^2}(u)=p(p-1) \right\} $$ where $u \ne \pm 1, v^2$.In addition, Lemma 3.4 was corrected. These changes do not affect the main result. The proof of Theorem 1.1 remains the same. The new version of the paper is available at arXiv:1504.00843v9.

Composantes isotypiques de pro-$p$-extensions de corps de nombres et $p$-rationalite

Soit p un nombre premier et soit K/k une extension galoisienne finie de corps de nombres de groupe de Galois ∆ d’ordre etranger a p. Soit S un ensemble fini de places finies de k contenant l’ensemble S p des places p-adiques et soit KS la pro-p-extension maximale de K non-ramifiee en dehors de S. Soi tG:“GS{Hun quotient de GS:“Gal KS/K) sur lequel ∆ agit trivialement. Posons X:“H{rH, Hs. Dans ce travail,nous etudions la φ-composante X φ de X pour les caracteres Qp-irreductibles φ de ∆ et nous montrons entre autres, sous la conjecture de Leopoldt et pour tout caractere non trivial φ, que X φ est ZpvGw-libre si et seulement si la φ-composante de la Zp-torsion de Gs /{Gs, Gs est triviale. Nous faisons egalement une etude numerique sur la liberte de Xφ dans des extensions cycliques K{Qde degre 3 et de degre 4 (`a partir de familles de polynomes donnees par Balady, Lecacheux et plus recemment par Balady et Washington),et ainsi que dans des extensions diedrales de degre 6 de Q. Les resultats de nos calculs renforcent une recente conjecture de Gras