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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]

One of the most interesting open problems of a digraph is Seymour's Second Neighborhood Conjecture (SSNC), which asserts that every digraph D has a vertex v satisfying d + + (v) ≥ d + (v) , we called such vertex v a Seymour vertex. Sullivan stated two compromise conjectures on SSNC. Let v ∈ V (D) if d + + (v) ≥ d − (v) , then v is called a Sullivan-1 vertex. If d + + (v) + d + (v) ≥ 2 d − (v) , then v is called a Sullivan-2 vertex. In this paper, we give a condition that the digraph has n Seymour vertices. And we also prove that every random tournament T n has n Sullivan-1 vertices and at least n 2 − n log n Sullivan-2 vertices with high probability. [ABSTRACT FROM AUTHOR]

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at ⌊ n 2 ⌋. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter. [ABSTRACT FROM AUTHOR]

The domination game is played on a graph G by two players, Dominator and Staller, who alternate in selecting vertices until each vertex in the graph G is contained in the closed neighbourhood of the set of selected vertices. Dominator's aim is to reach this state in as few moves as possible, whereas Staller wants the game to last as long as possible. In this paper, we prove that if G has n vertices and minimum degree at least 2, then Dominator has a strategy to finish the domination game on G within 10 n / 17 + 1 / 17 moves, thus making progress towards a conjecture by Bujtás, Iršič and Klavžar. [ABSTRACT FROM AUTHOR]

For a vertex v of a graph, the local complementation at v is an operation that replaces the neighborhood of v by its complement graph. Two graphs are locally equivalent if one is obtained from the other by a sequence of local complementations. A graph H is a vertex-minor of a graph G if H is an induced subgraph of a graph locally equivalent to G. Although this concept was introduced in the 1980s, it was not widely known and except for the survey paper of Bouchet published in 1990, there is no comprehensive survey listing all the new developments. We survey classic and recent theorems and conjectures on vertex-minors and related concepts such as circle graphs, cut-rank functions, rank-width, and interlace polynomials. [ABSTRACT FROM AUTHOR]

Sarnak's conjecture in quantum computing concerns when the groups PU2$\operatorname{PU}_{2}$ and PSU2$\operatorname{PSU}_{2}$ over cyclotomic rings Z[ζn,1/2]${\mathbb {Z}}[\zeta _{n}, 1/2]$ with ζn=e2πi/n$\zeta _n=e^{2\pi i/n}$, 4|n$4|n$, are generated by the Clifford‐cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group G$G$ has corankG>0$\operatorname{corank}G>0$ only if G$G$ is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary groups in the families n=2s$n=2^s$ and n=3·2s$n={3\cdot 2^s}$, n⩾8$n\geqslant 8$, by letting them act on Bruhat–Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for the families n=2s$n=2^s$ and n=3·2s$n=3\cdot 2^s$, the corank grows doubly exponentially in s$s$ as s→∞$s\rightarrow \infty$; it is 0 precisely when n=8,12,16,24$n= 8,12, 16, 24$, and indeed, the cyclotomic unitary groups are generated by torsion elements (in fact by Clifford‐cyclotomic gates) for these n$n$. We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over Fn=Q(ζn)+$F_n={\mathbf {Q}}(\zeta _n)^+$ via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section, we execute a program of Sarnak to show that our results for the n=2s$n=2^s$ and n=3·2s$n={3\cdot 2^s}$ families are sufficient to give a second proof of Sarnak's conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we study a multiplier operator which is induced by the Schwarzian derivative of univalent functions with a quasiconformal extension to the extended complex plane. As applications, we show that the Brennan conjecture is satisfied for a large class of quasidisks. We also establish a new characterization of asymptotically conformal curves and of the Weil–Petersson curves in terms of the multiplier operator. [ABSTRACT FROM AUTHOR]

For a simple graph G on n vertices the Seidel Estrada index of G, denoted by $ SEE(G) $ SEE (G) , is defined as $ SEE(G)=\sum _{i=1}^ne^{\theta _i} $ SEE (G) = ∑ i = 1 n e θ i , where $ \theta _1,\ldots,\theta _n $ θ 1 , ... , θ n are the Seidel eigenvalues (the eigenvalues of the Seidel matrix) of G. In this paper, we find the maximum and minimum values of the Seidel Estrada index among all graphs with the fixed number of vertices. Our results confirm some conjectures on Seidel Estrada index of graphs that have been posed in [M. Hakimi-Nezhaad, M. Ghorbani, On the Estrada Index of Seidel Matrix, Mathematics Interdisciplinary Research5 (2020) 43–54]. [ABSTRACT FROM AUTHOR]

In this paper, we report on a computation demonstrating that if A and B are nonempty subsets of a torsion-free group such that A B has no unique product, then | A | + | B | ≥ 1 6. Moreover, this bound is sharp, as there are examples where | A | = | B | = 8 , and in fact A = B for at least two such examples. More generally, when | A | is small, we find lower bounds on | B |. One consequence of this work is that any counterexample to Kaplansky's zero-divisor conjecture must be quite complicated, if it exists at all. Another advance is that we give new examples of torsion-free groups that are not unique product groups. [ABSTRACT FROM AUTHOR]

A digraph G $G$ is k $k$‐geodetic if for any (not necessarily distinct) vertices u,v $u,v$ there is at most one directed walk from u $u$ to v $v$ with length not exceeding k $k$. The order of a k $k$‐geodetic digraph with minimum out‐degree d $d$ is bounded below by the directed Moore bound M(d,k)=1+d+d2+⋯+dk $M(d,k)=1+d+{d}^{2}+\cdots +{d}^{k}$. The Moore bound can be met only in the trivial cases d=1 $d=1$ and k=1 $k=1$, so it is of interest to look for k $k$‐geodetic digraphs with out‐degree d $d$ and smallest possible order M(d,k)+ϵ $M(d,k)+{\epsilon }$, where ϵ ${\epsilon }$ is the excess of the digraph. Miller, Miret and Sillasen recently ruled out the existence of digraphs with excess one for k=3,4 $k=3,4$ and d≥2 $d\ge 2$ and for k=2 $k=2$ and d≥8 $d\ge 8$. We conjecture that there are no digraphs with excess one for d,k≥2 $d,k\ge 2$ and in this paper we investigate the structure of minimal counterexamples to this conjecture. We severely constrain the possible structures of the outlier function and prove the nonexistence of certain digraphs with degree three and excess one, as well closing the open cases k=2 $k=2$ and d=3,4,5,6,7 $d=3,4,5,6,7$ left by the analysis of Miller et al. We further show that there are no involutary digraphs with excess one, that is, the outlier function of any such digraph must contain a cycle of length ≥3 $\ge 3$. [ABSTRACT FROM AUTHOR]

Given a planar graph family F ${\rm{ {\mathcal F} }}$, let exP(n,F) $e{x}_{{\mathscr{P}}}(n,{\mathscr{F}})$ and spexP(n,F) $spe{x}_{{\mathscr{P}}}(n,{\mathscr{F}})$ be the maximum size and maximum spectral radius over all n $n$‐vertex F ${\rm{ {\mathcal F} }}$‐free planar graphs, respectively. Let tCℓ $t{C}_{\ell }$ be the disjoint union of t $t$ copies of ℓ $\ell $‐cycles, and tC $t{\mathscr{C}}$ be the family of t $t$ vertex‐disjoint cycles without length restriction. Tait and Tobin determined that K2+Pn−2 ${K}_{2}+{P}_{n-2}$ is the extremal spectral graph among all planar graphs with sufficiently large order n $n$, which implies the extremal graphs of both spexP(n,tCℓ) $spe{x}_{{\mathscr{P}}}(n,t{C}_{\ell })$ and spexP(n,tC) $spe{x}_{{\mathscr{P}}}(n,t{\mathscr{C}})$ for t≥3 $t\ge 3$ are K2+Pn−2 ${K}_{2}+{P}_{n-2}$. In this paper, we first determine spexP(n,tCℓ) $spe{x}_{{\mathscr{P}}}(n,t{C}_{\ell })$ and spexP(n,tC) $spe{x}_{{\mathscr{P}}}(n,t{\mathscr{C}})$ and characterize the unique extremal graph for 1≤t≤2 $1\le t\le 2$, ℓ≥3 $\ell \ge 3$ and sufficiently large n $n$. Second, we obtain the exact values of exP(n,2C4) $e{x}_{{\mathscr{P}}}(n,2{C}_{4})$ and exP(n,2C) $e{x}_{{\mathscr{P}}}(n,2{\mathscr{C}})$, which solve a conjecture of Li for n≥2661 $n\ge 2661$. [ABSTRACT FROM AUTHOR]

In this paper, we study the Wiener index of the orientation of trees and theta‐graphs. An orientation of a tree is called no‐zig‐zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree T $T$ achieving the maximum Wiener index is no‐zig‐zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta‐graph Θa,b,c ${{\rm{\Theta }}}_{a,b,c}$ with a≥b≥c $a\ge b\ge c$ and b>1 $b\gt 1$, the maximum Wiener index is achieved by the one in which the union of the paths between u1 ${u}_{1}$ and u2 ${u}_{2}$ forms a directed cycle of length a+b+2 $a+b+2$, where u1 ${u}_{1}$ and u2 ${u}_{2}$ are the vertex of degree 3. We confirm the validity of the conjecture. [ABSTRACT FROM AUTHOR]

In our previous work with Grifo and Hà, we showed the stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of sufficiently many general points in P N. In this paper, we establish the conjectures for all remaining cases, and hence, give the affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture for the general points in P N for all N. Our new technique is to develop the Cremona reduction process that provides effective lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective spaces. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Ruzsa number R m is bounded by 1 9 2 for any positive integer m , which improves the prior bound R m ≤ 2 8 8 given by Chen in 2008. [ABSTRACT FROM AUTHOR]

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the |$3$| -diagonal map acting on affine equivalence classes of centrally symmetric octagons. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail. [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]

Let a , b , c , and n be positive integers such that a + b = c 2 , 2 ∤ c and n > 1 . In this paper, we prove that if gcd (c , n) = 1 and n ≥ 117.14 c , then the equation (a n 2 + 1) x + (b n 2 − 1) y = (c n) z has only the positive integer solution (x , y , z) = (1 , 1 , 2) under the assumption gcd (a n 2 + 1 , b n 2 − 1) = 1 . Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer n , if (a , b , c) = (12 , 13 , 5) , (18 , 7 , 5) , or (44 , 5 , 7) , then this equation has only the solution (x , y , z) = (1 , 1 , 2) . This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently. [ABSTRACT FROM AUTHOR]

In this paper, we consider generalized Möbius functions associated with two types of L -functions: Rankin–Selberg L -functions of symmetric powers of distinct holomorphic cusp forms and L -functions derived from Maass cusp forms. We show that these generalized Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak's conjecture holds for these two types of Möbius functions. [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 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]

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]

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]

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, 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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]