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This paper considers a specific autonomous Schwarzian differential equation given by Sf=c(f−σ1)(f−σ2)/[(f−τ1)(f−τ2)]$$\begin{equation*} \hspace*{35pt} S_f=c(f-\sigma _1)(f-\sigma _2)/[(f-\tau _1)(f-\tau _2)] \end{equation*}$$and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the p -Laplacian for any p ∈ (1 , + ∞) (this paper covers the case p = 1 whereas the case p = + ∞ was already known). [ABSTRACT FROM AUTHOR]

This paper addresses the issue of the import of convergence arguments in theory assessment. A first part is devoted to making the point of the different types of strategies based on convergence, providing new distinctions with respect to the existing literature. Specific attention is devoted to robustness vs consilience arguments and one representative example for each category is then discussed in some detail. These are: (a) Perrin's famous robustness argument on behalf of the atomic hypothesis on the grounds of the concordance of thirteen different procedures to the same result for the Avogadro number; (b) the consilience argument motivating the trust in the viability of the extra-dimension conjecture in the context of early string theory. These two cases are expressly chosen in order to highlight possible differences, also including whether the convergence obtains in terms of empirical or theoretical procedures. Notwithstanding these various differences, in both cases the evaluation of the assessment strategy similarly depends, in a significant way, on how the convergence argument is interpreted, as shown in the final part of the paper. [ABSTRACT FROM AUTHOR]

The walk matrix of an n-vertex graph G with adjacency matrix A, denoted by $ W(G) $ W (G) , is $ [e,Ae,\ldots,A^{n-1}e] $ [ e , Ae , ... , A n − 1 e ] , where e is the all-ones vector. Let $ G\circ P_m $ G ∘ P m be the rooted product of G and a rooted path $ P_m $ P m (taking an endvertex as the root), i.e. $ G\circ P_m $ G ∘ P m is a graph obtained from G and n copies of $ P_m $ P m by identifying each vertex of G with an endvertex of a copy of $ P_m $ P m . Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112–127.] and Mao and Wang [Generalized spectral characterization of rooted product graphs. Linear Multilinear Algebra. 2022. DOI:10.1080/03081087.2022.2098226.] proved that, for m = 2 and $ m\in \{3,4\} $ m ∈ { 3 , 4 } , respectively \[ \det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m, \] det W (G ∘ P m) = ± a 0 ⌊ m 2 ⌋ (det W (G)) m , where $ a_0 $ a 0 is the constant term of the characteristic polynomial of G. Furthermore, in the same paper, Mao and Wang conjectured that the formula holds for any $ m\ge 2 $ m ≥ 2. In this paper, we verify this conjecture using the technique of Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$ -extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ , where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$ -extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$ -extension, then it holds for almost all $\mathbb{Z}_{p}$ -extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p -adic Lie extension. [ABSTRACT FROM AUTHOR]

This paper shows that there exist Reed-Solomon (RS) codes, over large finite fields, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving list-decoding capacity. In particular, we show that for any ε E (0,1] there exist RS codes with rate Ω(ε(log(1/∈)+1) that are list-decodable from radius of 1-ε. We generalize this result to list-recovery, showing that there exist (1-ε,l,O(l/ε)) -list-recoverable RS codes with rate Ω(εl√(log(1/ε)+1)) . Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. We also state a new conjecture that generalizes the tree-packing theorem to hypergraphs, and show that if this conjecture holds, then there would exist RS codes that are optimally (non-asymptotically) list-decodable. [ABSTRACT FROM AUTHOR]

Finding the sharp estimate of max | z | = 1 | p ′ (z) | in terms of max | z | = 1 | p (z) | for the class of polynomials p(z) satisfying p (z) ≡ z n p (1 / z) has been a well-known open problem for a long time and many papers in this direction have appeared. The earliest result is due to Govil, Jain and Labelle [9] who proved that for polynomials p(z) satisfying p (z) ≡ z n p (1 / z) and having all the zeros either in left half or right half-plane, the inequality max | z | = 1 | p ′ (z) | ≤ n 2 max | z | = 1 | p (z) | holds. A question was posed whether this inequality is sharp. In this paper, we answer this question in the negative by obtaining a bound sharper than n 2 . We also conjecture that for such polynomials max | z | = 1 | p ′ (z) | ≤ ( n 2 - 2 - 1 4 (n - 2)) max | z | = 1 | p (z) | and provide evidence in support of this conjecture. [ABSTRACT FROM AUTHOR]

The biclique B of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graph K B (G) of a graph G is the graph with the bicliques of G as its vertices where two vertices of K B (G) are adjacent if and only if they have a common vertex in G. In this paper, we disprove a conjecture of Groshaus and Montero concerning Helly property of biclique graphs (Groshaus and Montero, 2021) and two conjectures of Montero concerning vertex removal in biclique graphs (Montero, 2022). In addition, we confirm a conjecture of Montero about the structural characterization of biclique graphs in the same paper. [ABSTRACT FROM AUTHOR]

El Sahili conjectured that every oriented n -path, where n ≥ 8 , is contained in every n -chromatic digraph. This conjecture was confirmed by Gallai–Hasse–Roy–Vitaver's theorem for directed paths and by Addario-Berry et al. for oriented paths with two blocks. A block of an oriented path is a maximal directed subpath. El Joubbeh showed that every (4 n − 4) -chromatic digraph contains every oriented n -path with three blocks. For oriented paths with more than three blocks, Addario-Berry et al. established that every oriented n -tree, and hence every oriented n -path with any number of blocks, is contained in every ( n 2 2 − n 2 + 1) -chromatic digraph. This paper proposes an improvement to the bound on the chromatic number for digraphs that must include all paths consisting of four blocks. In fact, we extend El Joubbeh's bound on the chromatic number from (4 n − 4) for digraphs that should contain every n -path with three blocks to (4 n − 2) for digraphs that should be contain every n -path with four blocks. This bound surpasses the one provided by Addario-Berry et al. Furthermore, we show that every oriented n -path with t blocks, t ≥ 4 , is contained in every (4 r (t) n + q (t)) -chromatic digraph where r (t) = log 2 (t − 1) and q (t) = − 20 7 8 r (t) − 1 + 6 7 . Finally, the paper highlights an improvement in the bound of Addario-Berry et al. for n -paths with t blocks, where t ≤ n − 1 2 + 1. [ABSTRACT FROM AUTHOR]

This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem. [ABSTRACT FROM AUTHOR]

The classical hypercontractive inequality for the noise operator on the discrete cube plays a crucial role in many of the fundamental results in the Analysis of Boolean functions, such as the Kahn-Kalai-Linial theorem, Friedgut's junta theorem and the invariance principle of Mossel, O'Donnell and Oleszkiewicz. In these results the cube is equipped with the uniform (1/2-biased) measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general p-biased measures. However, simple examples show that when p is small there is no hypercontractive inequality that is strong enough for such applications. In this paper, we establish an effective hypercontractivity inequality for general p that applies to 'global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, making progress on two conjectures of Kahn and Kalai (both these conjectures were open when we arXived this paper in 2019; one of them was solved in 2022; the other is still open), and proving a p-biased analogue of the seminal invariance principle of Mossel, O'Donnell, and Oleszkiewicz. In this 2023 version of our paper we will also survey many further applications of our results that have been obtained by various authors since we arXived the first version in 2019. [ABSTRACT FROM AUTHOR]

In an edge-colored graph G , let d m o n (v) denote the maximum number of edges with the same color incident with a vertex v in G , called the monochromatic-degree of v. The maximum value of d m o n (v) over all vertices v ∈ V (G) is called the maximum monochromatic-degree of G , denoted by Δ m o n (G). Li et al. in 2019 conjectured that every edge-colored complete graph G of order n with Δ m o n (G) ≤ n − 3 k + 1 contains k vertex-disjoint properly colored (PC for short) cycles of length at most 4, and they showed that the conjecture holds for k = 2. Han et al. showed that every edge-colored complete graph G of order n with Δ m o n (G) ≤ n − 2 k contains k PC cycles of different lengths. They further got the condition Δ m o n (G) ≤ n − 6 for the existence of two vertex-disjoint PC cycles of different lengths. In this paper, we consider the problems of the existence of edge-disjoint PC cycles of length at most 4 (different lengths) in an edge-colored complete graph G of order n. [ABSTRACT FROM AUTHOR]

In 2013, while studying a relevant class of polyominoes that tile the plane by translation, i.e., double square polyominoes, Blondin Massé et al. found that their boundary words, encoded by the Freeman chain coding on a four letters alphabet, have specific interesting properties that involve notions of combinatorics on words such as palindromicity, periodicity and symmetry. Furthermore, they defined a notion of reducibility on double squares using homologous morphisms, so leading to a set of irreducible tile elements called prime double squares. The authors, by inspecting the boundary words of the smallest prime double squares, conjectured the strong property that no runs of two (or more) consecutive equal letters are present there. In this paper, we prove such a conjecture using combinatorics on words' tools, and setting the path to the definition of a fast generation algorithm and to the possibility of enumerating the elements of this class w.r.t. standard parameters, as perimeter and area. [ABSTRACT FROM AUTHOR]

In this work, we analyze the problem of catastrophicity of encoders of convolutional codes over the Laurent series with coefficients in ℤ p r , ℤ p r ((d)). Kuijper and Pinto proved in [M. Kuijper and R. Pinto, On minimality of convolutional ring encoders, IEEE Trans. Autom. Control55(11) (2009) 4890–4897] that, contrary to what happens for codes over ((d)) , where is a field, when dealing with ℤ p r ((d)) there are convolutional codes that do not admit non-catastrophic encoders. Nevertheless it was conjectured that any catastrophic convolutional code admits another type of non-catastrophic encoder called p -encoder. In this paper we solve this conjecture for a class of (2 , 1) convolutional codes over ℤ p 2 and show that, in fact, these codes always admit a non-catastrophic p-encoder. We also describe a constructive procedure that allows us to obtain a non-catastrophic p -encoder. [ABSTRACT FROM AUTHOR]

In this paper, we study certain variational aspects of the volume product functional restricted to the space of small projective deformations of a fixed convex body. In doing so, we provide a short proof of a theorem by Klartag: a strong version of the slicing conjecture implies the non-symmetric Mahler conjecture. We also exhibit an interesting family of critical convex bodies in dimension |$2$| containing saddle points for this functional. [ABSTRACT FROM AUTHOR]

In this paper we further develop the theory of equivariant Seiberg–Witten–Floer cohomology of the two authors, with an emphasis on Brieskorn homology spheres. We obtain a number of applications. First, we show that the knot concordance invariants θ(c)$\theta ^{(c)}$ defined by the first author satisfy θ(c)(Ta,b)=(a−1)(b−1)/2$\theta ^{(c)}(T_{a,b}) = (a-1)(b-1)/2$ for torus knots, whenever c$c$ is a prime not dividing ab$ab$. Since θ(c)$\theta ^{(c)}$ is a lower bound for the slice genus, this gives a new proof of the Milnor conjecture. Second, we prove that a free cyclic group action on a Brieskorn homology 3‐sphere Y=Σ(a1,⋯,ar)$Y = \Sigma (a_1, \dots, a_r)$ does not extend smoothly to any homology 4‐ball bounding Y$Y$. In the case of a non‐free cyclic group action of prime order, we prove that if the rank of HFred+(Y)$HF_{red}^+(Y)$ is greater than p$p$ times the rank of HFred+(Y/Zp)$HF_{red}^+(Y/\mathbb {Z}_p)$, then the Zp$\mathbb {Z}_p$‐action on Y$Y$ does not extend smoothly to any homology 4‐ball bounding Y$Y$. Third, we prove that for all but finitely many primes a similar non‐extension result holds in the case that the bounding 4‐manifold has positive‐definite intersection form. Finally, we also prove non‐extension results for equivariant connected sums of Brieskorn homology spheres. [ABSTRACT FROM AUTHOR]

In this paper, for d ≥ 1 and s ∈ (0 , d 2) , we study the Bianchi–Egnell quotient Q (f) = inf f ∈ H ˙ s (R d) \ B ‖ (- Δ) s / 2 f ‖ L 2 (R d) 2 - S d , s ‖ f ‖ L 2 d d - 2 s (R d) 2 dist H ˙ s (R d) (f , B) 2 , f ∈ H ˙ s (R d) \ B , where S d , s is the best Sobolev constant and B is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when d = 1 , there is a neighborhood of B on which the quotient Q (f) is larger than the lowest value attainable by sequences converging to B . This behavior is surprising because it is contrary to the situation in dimension d ≥ 2 described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for d = 1 , Q (f) has no minimizer on H ˙ s (R d) \ B , which again would be contrary to the situation in d ≥ 2 . As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every d ≥ 1 . For d ≥ 2 , this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For d = 1 we make some numerical observations which support the conjecture stated above. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups. [ABSTRACT FROM AUTHOR]

In this paper we give a complete and explicit description of the Howe correspondence of unipotent characters for a finite reductive dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization. That is, the conjecture by Aubert-Michel-Rouquier is confirmed. [ABSTRACT FROM AUTHOR]

We show that any n-absorbing ideal must be strongly n-absorbing, which is the first of Anderson and Badawi's three interconnected conjectures on absorbing ideals. We prove this by introducing and studying objects called maximal and semimaximal collections, which are tools for analyzing the multiplicative ideal structure of commutative rings. Furthermore, we discuss and make heavy use of previous work on absorbing ideals done by Anderson and Badawi, Choi and Walker, Laradji, and Donadze, among others. At the end of this paper, we pose three conjectures, each of which implies the last unsolved conjecture made by Anderson and Badawi. [ABSTRACT FROM AUTHOR]

We develop new techniques to compute the weighted L 2 -cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of L 2 -harmonic forms obtained in a companion paper, this allows us to compute the reduced L 2 -cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of n points on C 2 , for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3. [ABSTRACT FROM AUTHOR]

It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the -primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa's result on the p -primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties. [ABSTRACT FROM AUTHOR]

Let N k (m , n) denote the number of partitions of n with Garvan k -rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k -rank for k = 1 and k = 2 , respectively. In this paper, we prove that the sequences (N k (m , n)) | m | ≤ n − k − 71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently. [ABSTRACT FROM AUTHOR]

We prove Conjecture 4.16 of the paper [ 2 ] of Elagin and Lunts; namely, that a smooth projective curve of genus at least |$1$| over a field has diagonal dimension |$2$|⁠. [ABSTRACT FROM AUTHOR]

Alishahi and Hajiabolhassan found that for some classes of graphs, the wonderful equality χ G , r K 2 = | E (G) | − ex G , r K 2 holds as an amazing relationship between chromatic number and generalized Turán number. This powerful equality enabled them to determine chromatic numbers of some interesting and important classes of graphs. They conjectured that the aforementioned equality holds for all connected graphs G. Iradmusa, by a nice elegant use of a class of cubic graphs, called snarks, made counterexamples to this conjecture for which χ G , r K 2 = 1 and | E (G) | − ex G , r K 2 = 3. In this paper, for any arbitrary positive integer Θ , we explicitly construct a sequence of trees T r r = 1 ∞ for which lim r → ∞ (| E T r | − ex T r , r K 2 ) = + ∞ while χ T r , r K 2 = Θ for all r ≥ 3. [ABSTRACT FROM AUTHOR]

The weak cosmic censorship conjecture is a hypothesis about the nature of event horizons and singularities during the formation of black holes. It posits that singularities are always enveloped by the event horizons of black holes, thereby preventing naked singularities from affecting the causal structure of spacetime. In this paper, we study the effect of rotating short haired black holes on the weak cosmic censorship conjecture. Discussion of whether the event horizons of a rotating short haired black hole is disrupted by studying incident neutral test particles and scalar fields. In the context of short haired black holes, when incident neutral test particles are considered for extreme and near extreme cases, our research results indicate that neutral test particles can destroy the event horizons of short haired black holes, violating the weak cosmic censorship conjecture. In the case of scalar field incidence in short haired black holes, for extreme situations, when the incident wave modes fall within the range of 1 2 κ M + β M κ < ω m < 1 2 M κ + β M κ - 1 2 κ , the results indicate that the event horizons of rotating short haired black holes is disrupted. For near extreme cases, the presence of hair allows for the disruption of the event horizons of rotating short haired black holes, as indicated by our results. Therefore, these conclusions are intriguing and will provide new insights for us to further understand the weak cosmic censorship conjecture and explore the properties of short haired black holes. [ABSTRACT FROM AUTHOR]

In this paper, many important formulas of the subtrigonometric, subhyperbolic, pretrigonometric, prehyperbolic, supertrigonometric, and superhyperbolic functions sin Wiman class are developed for the first time. The subsine, subcosine, subhyperbolic sine, and subhyperbolic cosine associated with Kohlrausch–Williams–Watts (KWW) function and their scaling-law ODEs are proposed. The supersine, supercosine, superhyperbolic sine, and superhyperbolic cosine functions associated with Mittag-Leffler (ML) function and their fractional ODEs are obtained. The conjectures for the supercosine functions containing ML function are presented in detail. [ABSTRACT FROM AUTHOR]

In this paper, we address the entire Fourier sine and cosine integrals related to the Mittag-Leffler function. We guess that the entire functions have the real zeros in the entire complex plane. They can be connected with the well-known conjectures in analytic number theory. They are considered as the special solutions for the time-fractional diffusion equation within the Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

In this paper, we prove several conjectures raised by Zhi-Wei Sun on determinants and permanents by the eigenvectors-eigenvalues identity recently highlighted by Denton, Parke, Tao and Zhang. [ABSTRACT FROM AUTHOR]

For a given sequence A of nonnegative integers, let P (A) be the set of all finite subsequence sums of A. A is called complete if P (A) contains all sufficiently large integers. A real number α > 0 is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of α. Hegyvári conjectured that A α , β is complete if α or β is i.d.f. and α / β ≠ 2 l (l ∈ ℤ) , where A α , β = { [ α ] , [ β ] , ... , [ 2 n α ] , [ 2 n β ] , ... } is a sequence of integers. In this paper, we give a partial result of Hegyvári's conjecture. [ABSTRACT FROM AUTHOR]

Let I(G; x) denote the independence polynomial of a graph G. Alavi, Malde, Schwenk and Erdős conjectured for any tree T that I(T; x) is unimodal. In this paper, we derive that rooted products of some trees preserve real-rootedness of independence polynomials. In consequence, we obtain more and more trees whose independence polynomials have only real zeros. This in particular implies that their independence polynomials are unimodal. [ABSTRACT FROM AUTHOR]

Let r 1 , ... , r t be positive integers and let R 2 (r 1 , ... , r t) be the set of positive odd integers that can be represented as p + 2 k 1 r 1 + ⋯ + 2 k t r t , where p is a prime and k 1 , ... , k t are positive integers. It is easy to see that if r 1 − 1 + ⋯ + r t − 1 < 1 , then the set R 2 (r 1 , ... , r t) has asymptotic density zero. In this paper, we prove that if r 1 − 1 + ⋯ + r t − 1 ≥ 1 , then the set R 2 (r 1 , ... , r t) has a positive lower asymptotic density. Several conjectures and open problems are posed for further research. [ABSTRACT FROM AUTHOR]

A 3‐graph F ${\rm{ {\mathcal F} }}$ is U(s,2s+1) $U(s,2s+1)$ if for any s $s$ edges e1,...,es∈E(F) ${e}_{1},\ldots ,{e}_{s}\in E({\rm{ {\mathcal F} }})$, ∣e1∪⋯∪es∣≤2s+1 $| {e}_{1}\cup \cdots \cup {e}_{s}| \le 2s+1$. Frankl and Kupavskii proposed the following conjecture: For any 3‐graph F ${\rm{ {\mathcal F} }}$ with n $n$ vertices, if F ${\rm{ {\mathcal F} }}$ is U(s,2s+1) $U(s,2s+1)$, then e(F)≤maxn−12,(n−s−1)s+12+s+13,2s+13. $\begin{array}{c}e({\mathscr{F}})\le \max \left\{\left(\genfrac{}{}{0ex}{}{n-1}{2}\right),(n-s-1)\left(\genfrac{}{}{0ex}{}{s+1}{2}\right)\right.\\ \,\left.+\left(\genfrac{}{}{0ex}{}{s+1}{3}\right),\left(\genfrac{}{}{0ex}{}{2s+1}{3}\right)\right\}.\end{array}$ In this paper, we confirm Frankl and Kupavskii's conjecture. [ABSTRACT FROM AUTHOR]

Let G $G$ be a simple graph. Let Δ(G) ${\rm{\Delta }}(G)$ and χ′(G) $\chi ^{\prime} (G)$ be the maximum degree and the chromatic index of G $G$, respectively. We call G $G$overfull if ∣E(G)∣∕⌊∣V(G)∣∕2⌋>Δ(G) $| E(G)| \unicode{x02215}\lfloor | V(G)| \unicode{x02215}2\rfloor \gt {\rm{\Delta }}(G)$, and critical if χ′(H)<χ′(G) $\chi ^{\prime} (H)\lt \chi ^{\prime} (G)$ for every proper subgraph H $H$ of G $G$. Clearly, if G $G$ is overfull then χ′(G)=Δ(G)+1 $\chi ^{\prime} (G)={\rm{\Delta }}(G)+1$. The core of G $G$, denoted by GΔ ${G}_{{\rm{\Delta }}}$, is the subgraph of G $G$ induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attack the overfull conjecture. Along this direction, we in this paper show that for any integer k≥2 $k\ge 2$, if G $G$ is critical with Δ(G)≥23n+3k2 ${\rm{\Delta }}(G)\ge \frac{2}{3}n+\frac{3k}{2}$ and δ(GΔ)≤k $\delta ({G}_{{\rm{\Delta }}})\le k$, then G $G$ is overfull. [ABSTRACT FROM AUTHOR]

Gálvez-Carrillo et al. [Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals, Adv. Math. 334 (2018) 544–584] constructed a decomposition space U of all Möbius intervals, as a recipient of Lawvere's interval construction for Möbius categories, and conjectured that U enjoys a certain universal property: for every Möbius decomposition space X , the space of culf functors from X to U is contractible. In this paper, we work at the level of homotopy 1-types to prove the first case of the conjecture, namely for locally discrete decomposition spaces. This provides also the first substantial evidence for the general conjecture. This case is general enough to cover all locally finite posets, Cartier–Foata monoids, Möbius categories and strict (directed) restriction species. The proof is 2-categorical. First, we construct a local strict model of U , which is then used to show by hand that the Lawvere interval construction, considered as a natural transformation, does not admit other self-modifications than the identity. [ABSTRACT FROM AUTHOR]

In 2018, Luca and Patel conjectured that the largest perfect power representable as the sum of two Fibonacci numbers is 38642 = F36 + F12. In other words, they conjectured that the equation has no solutions with a ≥ 2 and ya > 38642 While this is still an open problem, there exist several partial results. For example, recently Kebli, Kihel, Larone and Luca proved an explicit upper bound for ya, which depends on the size of y. In this paper, we find an explicit upper bound for ya, which only depends on the Hamming weight of y with respect to the Zeckendorf representation. More specifically we prove the following: If y = Fn1 + · · · + Fnk and equation (*) is satisfied by y and some non-negative integers n, m and a ≥ 2, then Here, ϵ > 0 can be chosen arbitrarily and C(ϵ) is an effectively computable constant. [ABSTRACT FROM AUTHOR]

A simple graph G with maximum degree Δ is overfull if | E (G) | > Δ ⌊ | V (G) | / 2 ⌋. The core of G , denoted G Δ , is the subgraph of G induced by its vertices of degree Δ. Clearly, the chromatic index of G equals Δ + 1 if G is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if G is a simple connected graph with Δ ≥ 3 and Δ (G Δ) ≤ 2 , then χ ′ (G) = Δ + 1 implies that G is overfull or G = P ⁎ , where P ⁎ is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case Δ = 3 in 2003, and Cranston and Rabern proved the next case, Δ = 4 , in 2019. In this paper, we give a proof of this conjecture for all Δ ≥ 4. [ABSTRACT FROM AUTHOR]

The classical Harer conjecture states the stable homology triviality of the canonical embedding ϕ:B2g+2↪Γg$\phi: B_{2g+2} \hookrightarrow \Gamma _{g}$, which was proved by Song and Tillmann. The main part of the proof is to show that Bϕ+:BB∞+→BΓ∞+$\operatorname{B}\phi ^{+}: \operatorname{B}B_{\infty }^{+} \rightarrow \operatorname{B}\Gamma _{\infty }^{+}$, induced from ϕ$\phi$ is a double‐loop space map. In this paper, we give a proof of the generalized Harer conjecture concerning the homology triviality for every regular embedding ϕ:Bn↪Γg,k$\phi: B_{n} \hookrightarrow \Gamma _{g,k}$. The main strategy of the proof is to remove all the interchangeable subsurfaces from Sg,k$S_{g,k}$ and collapse the new boundary components. Then, we obtain (the union of) covering spaces over a disk with n$n$ marked points that we can analyze. The final goal is to show that the map Φ:C→S$\Phi: \mathcal {C} \rightarrow \mathcal {S}$ induced by Bϕ:Confn(D)→Mg,k$\operatorname{B}\phi:\operatorname{Conf}_n(D)\rightarrow \mathcal {M}_{g,k}$ preserves the actions of the framed little 2‐disks operad. [ABSTRACT FROM AUTHOR]

For degree ± 1 harmonic maps from R 2 (or S 2 ) to S 2 , Bernand-Mantel et al. (Arch Ration Mech Anal 239(1):219–299, 2021) recently establish a uniformly quantitative stability estimate. Namely, for any map u : R 2 → S 2 with degree ± 1 , the discrepancy of its Dirichlet energy and 4 π can linearly control the H ˙ 1 -difference of u from the set of degree ± 1 harmonic maps. Whether a similar estimate holds for harmonic maps with a higher degree is unknown. In this paper, we prove that a similar quantitative stability result for a higher degree is true only in a local sense. Namely, given a harmonic map, a similar estimate holds if u is already sufficiently near to it (modulo Möbius transforms) and the bound in general depends on the given harmonic map. More importantly, we thoroughly investigate an example of the degree 2 case, which shows that it fails to have a uniformly quantitative estimate like the degree ± 1 case. This phenomenon shows the striking difference between degree ± 1 ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation. [ABSTRACT FROM AUTHOR]

Kim gave a new proof of Siegel's Theorem that there are only finitely many S-integral points on \mathbb {P}^1_\mathbb {Z}\setminus \{0,1,\infty \}. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of S increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where S has size 2 which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim. [ABSTRACT FROM AUTHOR]

Let |${\textsf k}$| be a |$p$| -adic field, and let |$\textbf {G}({\textsf k})$| be the |${\textsf k}$| -points of a connected reductive group, inner to split. The set of Aubert–Zelevinsky duals of the constituents of a tempered L-packet forms an Arthur packet for |$\textbf {G}({\textsf k})$|⁠. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang–Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets, which we call weak Arthur packets and describe their constituents in terms of their Langlands parameters. [ABSTRACT FROM AUTHOR]

In this paper, we present some new results on the geometrically m$m$‐step solvable Grothendieck conjecture in anabelian geometry. Specifically, we show the (weak bianabelian and strong bianabelian) geometrically m$m$‐step solvable Grothendieck conjecture(s) for affine hyperbolic curves over fields finitely generated over the prime field. First of all, we show the conjecture over finite fields. Next, we show the geometrically m$m$‐step solvable version of the Oda–Tamagawa good reduction criterion for hyperbolic curves. Finally, by using these two results, we show the conjecture over fields finitely generated over the prime field. [ABSTRACT FROM AUTHOR]

Given a finite group G and an irreducible complex character χ of G , the codegree of χ is defined as the integer cod (χ) = | G : ker (χ) | / χ (1). It was conjectured by G. Qian in [16] that, for every element g of G , there exists an irreducible character χ of G such that cod (χ) is a multiple of the order of g ; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). In this paper, we prove that Qian's conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle. [ABSTRACT FROM AUTHOR]

In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of R N { A α u = u + 2 α ⁎ − 1 + λ u − s ¯ φ 1 , u > 0 , in Ω , u = 0 , on ∂ Ω , where A α is the spectral fractional operator, λ and s ¯ are real numbers, Ω ⊂ R N is bounded, 2 α ⁎ = 2 N N − 2 α is a critical exponent, 0 < α < 1 , φ 1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain. [ABSTRACT FROM AUTHOR]

The detour order of a graph G, denoted by τ (G), is the order of a longest path in G. If a and b are positive integers and the vertex set of G can be partitioned into two subsets A and B such that τ (hAi) = a and τ (hBi) = b, we say that (A,B) is an (a, b)-partition of G. If equality holds in both instances, we call (A,B) an exact (a, b)-partition. The Path Partition Conjecture (PPC) asserts that if G is any graph and a, b any pair of positive integers such that τ (G) = a + b, then G has an (a, b)-partition. The Strong PPC asserts that under the same circumstances G has an exact (a, b)-partition. While a substantial body of work in support of the PPC has been developed over the past three decades, no results on the Strong PPC have yet appeared in the literature. In this paper we prove that the Strong PPC holds for a = 8. [ABSTRACT FROM AUTHOR]

The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1981, Akiyama, Exoo and Harary conjectured that ⌈Δ(G)/2⌉ ≤ la(G) ≤ ⌈Δ(G)+1/2 ⌉ for any simple graph G. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we confirm the conjecture for 1-planar graphs G with Δ(G) ≥ 13. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the images of polynomial derivations and their simplicity. We prove that the images of simple Shamsuddin derivations are not Mathieu–Zhao spaces. In addition, we show that the images of some simple derivations in dimension three are not Mathieu-Zhao spaces. Thus, we conjecture that the images of simple derivations in dimension greater than one are not Mathieu–Zhao spaces. We also prove that locally nilpotent derivations are not simple in dimension greater than one. [ABSTRACT FROM AUTHOR]

This paper contributes to the conjecture of R. Scott and R. Styer which asserts that for any fixed relatively prime positive integers a, b and c all greater than 1 there is at most one solution to the equation a x + b y = c z in positive integers x, y and z, except for specific cases. The fundamental result proves the conjecture under some congruence condition modulo c on a and b. As applications the conjecture is confirmed to be true if c takes some small values including the Fermat primes found so far, and in particular this provides an analytic proof of the celebrated theorem of Scott (J Number Theory 44(2):153-165, 1993) solving the conjecture for c = 2 in a purely algebraic manner. The method can be generalized for smaller modulus cases, and it turns out that the conjecture holds true for infinitely many specific values of c not being perfect powers. The main novelty is to apply a special type of the p-adic analogue to Baker's theory on linear forms in logarithms via a certain divisibility relation arising from the existence of two hypothetical solutions to the equation. The other tools include Baker's theory in the complex case and its non-Archimedean analogue for number fields together with various elementary arguments through rational and quadratic numbers, and extensive computation. [ABSTRACT FROM AUTHOR]