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In this paper we introduce some conjectures analogous to the well-known Collatz conjecture. [ABSTRACT FROM AUTHOR]

Let G be a classical group defined over a local field F of characteristic zero. For any irreducible admissible representation \pi of G(F), which is of Casselman-Wallach type if F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F. In particular, if \pi has a generic local L-parameter, we introduce the spectral first occurrence index {\mathfrak {f}}_{\mathfrak {s}}(\pi) and the arithmetic first occurrence index {\mathfrak {f}}_{{\mathfrak {a}}}(\pi) of \pi and prove in this paper that {\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{{\mathfrak {a}}}(\pi). Based on the theory of consecutive descents of enhanced L-parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L-packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality. [ABSTRACT FROM AUTHOR]

In recent years, a variety of multiple zeta values (MZVs) variants have been defined and studied. One way to produce these variants is to restrict the indices in the definition of MZVs to some fixed parity pattern, which include Hoffman's multiple t-values, Kaneko and Tsumura's multiple T-values, and Xu and this paper's author's multiple S-values. Xu and this paper's author have also considered the so-called multiple mixed values by allowing all possible parity patterns and have studied a few important relations among these values. In this paper, we turn to the finite analogs and the symmetric forms of the multiple mixed values, motivated by a deep conjecture of Kaneko and Zagier, which relates the finite MZVs and symmetric MZVs, and a generalized version of this conjecture by the author to the Euler sum (i.e., level two) setting. We present a few important relations among these values such as the stuffle, reversal, and linear shuffle relations. We also compute explicitly the (conjecturally smallest) generating set in weight one and two cases. In the appendix, we tabulate some dimension computations for various subspaces of the finite multiple mixed values and propose a conjecture. [ABSTRACT FROM AUTHOR]

Mutual exclusion algorithms are at the heart of concurrent/parallel and distributed systems. It is well known that such algorithms are very difficult to analyze, and in the literature, different conjectures about starvation freedom and the number of by-passes (also called the overtaking factor) exist. The overtaking factor affects the (hopefully) bounded waiting time that a process competing for entering the critical section has to suffer before accessing the shared resource. This paper proposes a novel modeling approach based on Timed Automata and the Uppaal toolset, which proves effective for studying all the properties of a mutual exclusion algorithm for N ≥ 2 processes, by exhaustive model checking. Although the approach, as already confirmed by similar experiments reported in the literature, is not scalable due to state explosion problems and can be practically applied until N ≤ 5 , it is of great value for revealing the true properties of analyzed algorithms. For dimensions N > 5 , the Statistical Model Checker of Uppaal can be used, which, although based on simulations, can confirm properties by estimations and probabilities. This paper describes the proposed modeling and verification method and applies it to several mutual exclusion algorithms, thus retrieving known properties but also showing new results about properties often studied by informal reasoning. [ABSTRACT FROM AUTHOR]

This paper is devoted to finding a necessary and sufficient condition for the solvability of the overdetermined problem for Poisson's equation with both the Dirichlet and Neumann conditions on the entire boundary. The proof is based on the boundary condition formula for the Newton potential. The obtained results are also extended to general second-order linear elliptic equations. As a byproduct, we present a characterization of the Schiffer property. It gives a definitive answer to the Schiffer problem. [ABSTRACT FROM AUTHOR]

This paper explores the Apéry-like series and demonstrates the derivation of closed-form expressions using fractional calculus. We consider a variety of Apéry-like functions, which were categorized by their functional forms and coefficients by applying the Riemann–Liouville fractional integral and derivative to examine their properties across various domains. The study focuses on establishing rigorous mathematical frameworks that unveil new insights into the behaviors of these series, contributing to a deeper understanding of number theory and mathematical analysis. Key results include proofs of convergence and divergence within specified intervals and the derivation of closed-form solutions through fractional integration and differentiation. This paper also introduces a method aimed at conjecturing mathematical constants through continued fractions as an application of our results. Finally, we provide the proof of validation for three unproven conjectures of continued fractions obtained from the Ramanujan Machine. [ABSTRACT FROM AUTHOR]

In this paper, we consider the general dual fractional parabolic problem ∂ t α u (x , t) + L u (x , t) = f (t , u (x , t)) in R n × R. We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons' conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L and the Marchaud time derivative ∂ t α , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons' conjecture. It is worth noting that our results are novel even for a special case of L , the fractional Laplacian (−Δ)s, and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators. [ABSTRACT FROM AUTHOR]

The notion of a transposed Poisson n -Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson n -Lie algebra with a derivation gives rise to a transposed Poisson (n + 1) -Lie algebra. In this paper, we focus on transposed Poisson n -Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of (n + 1) -Lie algebras from transposed Poisson n -Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases. The notion of a transposed Poisson -Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson -Lie algebra with a derivation gives rise to a transposed Poisson -Lie algebra. In this paper, we focus on transposed Poisson -Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of -Lie algebras from transposed Poisson -Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases. [ABSTRACT FROM AUTHOR]

The Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees T2,...,n-1, Tn Ti has i vertices pack into Kn. The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that k largest trees pack. The latter is true if none tree is a star, but in general, it is known only for k = 5. In this paper we prove, among other results, that seven largest trees pack. [ABSTRACT FROM AUTHOR]

A distance antimagic labeling of graph G = (V, E) of order n is a bijection f: V(G) → {1, 2, . . ., n} with the property that any two distinct vertices x and y satisfy ω(x), ω(y), where ω(x) denotes the open neighborhood sum PΣaN(x)f(a) of a vertex x. In 2013, Kamatchi and Arumugam conjectured that a graph admits a distance antimagic labeling if and only if it contains no two vertices with the same open neighborhood. A circulant graph C(n; S) is a Cayley graph with order n and generating set S, whose adjacency matrix is circulant. This paper provides partial evidence for the conjecture above by presenting distance antimagic labeling for some circulant graphs. In particular, we completely characterized distance antimagic circulant graphs with one generator and distance antimagic circulant graphs C(n; {1, k}) with odd n. [ABSTRACT FROM AUTHOR]

The Asymptotic Weak Gravity Conjecture (WGC) has been proposed as a special case of the Tower WGC that probes infinite distances in the moduli space corresponding to weakly coupled gauge regimes. The conjecture has been studied in M-theory on a Calabi–Yau threefold (CY3) with finite volume inducing a 5D effective quantum field theory. In this paper, we extend the scope of the previous study to encompass lower dimensions, particularly we generalize the obtained 5D Asymptotic WGC to the effective field theory (EFT |$_{3D}$|⁠) coupled to 3D gravity that descends from M-theory compactified on a Calabi–Yau fourfold with an emphasis on |$K3\times K3$|⁠. We find that the CY4 has three fibration structures labeled as line Type- |$\mathbb {T}^{2}$|⁠ , surface Type- |$\mathbb {S}$|⁠ , and bulk Type- |$\mathbb {V}$|⁠. The emergent EFT |$_{3D}$| is shown to have 2+2 towers of particle states termed as the BPS |$\mathcal {T}_{M_{\mathrm{k}}\rightarrow 0}^{\rm{{\small BPS}}}$| and |$\mathcal {T}_{M_{\mathrm{k}}\rightarrow \infty }^{\rm{{\small BPS}}}$| as well as the non-BPS |$\mathcal {T}_{M_{\mathrm{k}}\rightarrow 0}^{\rm{{\small N-BPS}}}$| and |$\mathcal {T}_{M_{\mathrm{k}}\rightarrow \infty }^{\rm{{\small N-BPS}}}$|⁠. To ensure the viability of the 3D Asymptotic WGC, we give explicit calculations to thoroughly test the Swampland constraint for both the weakly and strongly gauge coupled regimes. Additional aspects, including the gauge symmetry breaking and duality symmetry, are also investigated. [ABSTRACT FROM AUTHOR]

We study in this paper the global attractivity for a higher order rational difference equation. As application, our results not only include and generalize many known ones, but also formulate some new results for several conjectures presented by Camouzis and Ladas, et al. [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]

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]

This paper presents some generalizations of a Thébault conjecture, provides an analog of the Thébault conjecture for the -simplex, and also solves a conjecture in a 2022 paper by the authors by using linear algebra. [ABSTRACT FROM AUTHOR]

In this paper, we present a survey about the latest results in global stability concerning the discrete-time evolutionary Ricker competition model with n species, in both, autonomous and periodic models. The main purpose is to convey some arguments and new ideas concerning the techniques for showing global asymptotic stability of fixed points or periodic cycles in these kind of discrete-time models. In order to achieve this, some open problems and conjectures related to the evolutionary Ricker competition model are presented, which may be a starting point to study global stability, not only in other competition models, but in predator–prey models and Leslie–Gower-type models as well. [ABSTRACT FROM AUTHOR]

In this paper, we will study finite multiple T-values (MTVs) and their alternating versions, which are level two and level four variations of finite multiple zeta values, respectively. We will first provide some structural results for level two finite multiple zeta values (i.e., finite Euler sums) for small weights, guided by the author's previous conjecture that the finite Euler sum space of weight, w, is isomorphic to a quotient Euler sum space of weight, w. Then, by utilizing some well-known properties of the classical alternating MTVs, we will derive a few important Q -linear relations among the finite alternating MTVs, including the reversal, linear shuffle, and sum relations. We then compute the upper bound for the dimension of the Q -span of finite (alternating) MTVs for some small weights by rigorously using the newly discovered relations, numerically aided by computers. [ABSTRACT FROM AUTHOR]

A long-standing conjecture due to R. Fox states that the coefficients of the Alexander polynomial of an alternating knot exhibit a trapezoidal pattern. In other words, these coefficients increase, stabilize, then decrease in a symmetric way. A stronger version of this conjecture states that these coefficients form a log-concave sequence. This conjecture has been recently highlighted by J. Huh as one of the most interesting problems on log-concavity of sequences. In this expository paper, we shall review the various versions of the conjecture, highlight settled cases and outline some future directions. [ABSTRACT FROM AUTHOR]

A long-standing conjecture asserts that every finite nonabelian p-group has a non-inner automorphism of order p. This paper proves the conjecture for finite p-groups of coclass 4 and 5 (p ≥ 5). We also prove the conjecture for an odd order nonabelian p-group G with cyclic center satisfying CG(Gpγ3(G)) ∩ Z3(G) ≤ Z(Φ(G)). [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices and m edges. A (G) and I denote, respectively, the adjacency matrix of G and an n by n identity matrix. For a graph G, the permanent of matrix (I + A (G)) is called the permanental sum of G. In this paper, we give a relation between the Hosoya index and the permanental sum of G. This implies that the computational complexity of the permanental sum is N P -complete. Furthermore, we characterize the graphs with the minimum permanental sum among all graphs of n vertices and m edges, where n + 3 ≤ m ≤ 2 n − 3 . [ABSTRACT FROM AUTHOR]

In this paper, we conjecture a monotonicity property that we call monotonicity under coarse-graining for a class of multi-partite entanglement measures. We check these properties by computing the measures for various types of states using different methods. [ABSTRACT FROM AUTHOR]

The burning number b(G) of a graph G, introduced by Bonato, is the minimum number of steps to burn the graph, which is a model for the spread of influence in social networks. In 2016, Bonato et al. studied the burning number of paths and cycles, and based on these results, they proposed a conjecture on the upper bound for the burning number. In this paper, we determine the exact value of the burning number of Q graphs and confirm this conjecture for Q graph. Following this, we characterize the single tail and double tails Q graph in term of their burning number, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the following question. Assume ϕ is a discrete Gaussian free field (GFF) on Λ⊂ 1/n ​ Z² and that we are given eiTϕ , or equivalently ϕ(mod2πT). Can we recover the macroscopic observables of ϕ with o(1) precision? We prove that this statistical reconstruction problem undergoes the following Kosterlitz–Thouless type phase transition: If T< T - rec, one can fully recover ϕ from the knowledge of ϕ(mod2πT). In this regime our proof relies on a new type of Peierls argument which we call annealed Peierls argument and which allows us to deal with an unknown quenched ground state. If T>T+rec, it is impossible to fully recover the field ϕ from the knowledge of ϕ(mod2πT). To prove this result, we generalize the delocalization theorem by Fröhlich–Spencer to the case of integer-valued GFF in an inhomogeneous medium. This delocalization result is of independent interest and we give an application of our techniques to the random-phase sine-Gordon model in Appendix B. Also, an interesting connection with Riemann theta functions is drawn along the proof. This statistical reconstruction problem is motivated by the two-dimensional XY and Villain models. Indeed, at low temperature T, the large scale fluctuations of these continuous spin systems are conjectured to be governed by a Gaussian free field. It is then natural to ask if one can recover the underlying macroscopic GFF from the observation of the spins of the XY or Villain model. Another motivation for this work is that it provides us with an “integrable model” (the GFF) that undergoes a KT transition. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional hereditary algebra over an algebraically closed field k. In this paper, we study the tilting quiver of A from the viewpoint of τ -tilting theory. First, we prove that there exists an isomorphism between the support τ -tilting quiver Q(s τ -tilt A) of A and the tilting quiver Q(tilt A ¯ ) of the duplicated algebra A ¯ . Then, we give a new method to calculate the number of arrows in the tilting quiver Q(tilt A) when A is representation-finite. Finally, we study the conjecture given by Happel and Unger, which claims that each connected component of Q(tilt A) contains only finitely many non-saturated vertices. We provide an example to show that this conjecture does not hold for some algebras whose quivers are wild with at least four vertices. [ABSTRACT FROM AUTHOR]

In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber F has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when F is F-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases. We also prove an analog of Fujino's result regarding his Fujita-type conjecture. [ABSTRACT FROM AUTHOR]

The conjecture of Beineke and Harary states that for any two vertices which can be separated by k vertices and l edges for l ≥ 1 but neither by k vertices and l - 1 edges nor k - 1 vertices and l edges there are k + l edge-disjoint paths connecting these two vertices of which k + 1 are internally disjoint.In this paper we prove this conjecture for l = 2 and every k ∈ N .We utilize this result to prove that the conjecture holds for all graphs of treewidth at most 3 and all k and l. [ABSTRACT FROM AUTHOR]

Copyright of Prospectiva (1692-8261) is the property of Universidad Autonoma del Caribe and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Introduced by Erdős in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set Z. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until recently little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most 1016 . In this paper we answer another question of Erdős, asked in 1952, on the number of minimal covering systems. More precisely, we show that the number of minimal covering systems with exactly n elements is exp ((4√τ/3+o(1)n3/2/(logn)1/2) as n→∞, where τ=∑∞ t=1(logt+1/t)².En route to this counting result, we obtain a structural description of all covering systems that are close to optimal in an appropriate sense. [ABSTRACT FROM AUTHOR]

The equivariant coarse Baum--Connes conjecture interpolates between the Baum--Connes conjecture for a discrete group and the coarse Baum--Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group Γ acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/Γ admits a coarse embedding into Hilbert space and Γ is amenable, and that the Γ-orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum--Connes conjecture holds for (X; Γ). Along the way, we prove a K-theoretic amenability statement for the Γ-space X under the same assumptions as above; namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K-theory. [ABSTRACT FROM AUTHOR]

Denote by A(p, n, k) the number of commuting p-tuples of permutations on [n] that have exactly k distinct orbits. It was conjectured in Abdesselam (Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations, http://arxiv.org/abs/2309.07358, 2023) that A(p, n, k) is log-concave with respect to k for every p ≥ 2 , n ≥ 3 , and the log-concavity was proved in " p = ∞ " case. In this paper, we prove that for k = n - α , the log-concavity for A(p, n, k) holds for every p ≥ 2 for sufficiently large n. [ABSTRACT FROM AUTHOR]

We show that counterexamples of Iyengar and Walker to the algebraic version of Gunnar Carlsson's conjecture on the rank of the homology of a free complex can be extended to examples over any finite group with many choices of the complex. [ABSTRACT FROM AUTHOR]

For a simple graph G, a vertex labeling ϕ : V (G) → {1, 2, . . ., k} is called k-labeling. The weight of an edge uv in G, written wϕ(uv), is the sum of the labels of end vertices u and v, i.e., wϕ(uv) = ϕ(u) + ϕ(v). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two distinct edges u and v, wϕ(u) ≠ wϕ(v). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, written es(G). In this paper, we study the edge irregular k-labeling for cycle-star graph CSk,n−k and determine the exact value for cycle-star graph for 3 ≤ k ≤ 7 and n − k ≥ 1. Finally, we make a conjecture for the edge irregularity strength of CSk,n−k for k ≥ 8 and n − k ≥ 1. [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]

Lieb concavity theorem, successfully solved the Wigner-Yanase-Dyson conjecture, which is a very important theorem, and there are many proofs of it. Generalization of the Lieb concavity theorem has been obtained by Huang, which implies that it is jointly concave for any nonnegative matrix monotone function f (x) over (Tr-k(Aqs/2 K*BspKAsq/2)1/s]1/k. In this manuscript, we obtained (Tr[hk(f(Aqs/2)K*f(Bsp)Kf(Asq/2))1/s]1/k was jointly concave for any nonnegative matrix monotone function f (x) by using Epstein's theorem, and some more general results were obtained. [ABSTRACT FROM AUTHOR]

Copyright of Baghdad Science Journal is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one. In this work, we establish the following query lower bounds: (1) sampling from strongly log-concave and log-smooth distributions in dimension \(d\ge 2\) requires \(\Omega (\log \kappa)\) queries, which is sharp in any constant dimension, and (2) sampling from Gaussians in dimension d (hence also from general log-concave and log-smooth distributions in dimension d) requires \(\widetilde{\Omega }(\min (\sqrt \kappa \log d, d))\) queries, which is nearly sharp for the class of Gaussians. Here, \(\kappa\) denotes the condition number of the target distribution. Our proofs rely upon (1) a multiscale construction inspired by work on the Kakeya conjecture in geometric measure theory, and (2) a novel reduction that demonstrates that block Krylov algorithms are optimal for this problem, as well as connections to lower bound techniques based on Wishart matrices developed in the matrix-vector query literature. [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]

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]

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]

The notion of supershift generalizes that one of superoscillation and expresses the fact that the sampling of a function in an interval allows to compute the values of the function outside the interval. In a previous paper, we discussed the case in which the sampling of the function is regular and we are considering supershift in a bounded set, while here we investigate how irregularity in the sampling may affect the answer to the question of whether there is any relation between supershift and real analyticity on the whole real line. We show that the restriction to R of any entire function displays supershift, whereas the converse is, in general, not true. We conjecture that the converse is true as long as the sampling is regular, we discuss examples in support and we prove that the conjecture is indeed true for periodic functions. [ABSTRACT FROM AUTHOR]

In this paper we construct new classes of mixed singularities that provide realizations of real algebraic links in the 3-sphere. Especially, we describe this construction in the case of semiholomorphic polynomials, which are mixed polynomials that are holomorphic in one variable. Classifications and characterizations of real algebraic links are still open. These new classes of mixed singularities may help to shed light on the Benedetti–Shiota conjecture, which states that any fibered link on the 3-sphere is a real algebraic link. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multi-species nonlinear Schrödinger systems in R N : { − Δ u j + V j (x) u j = μ j u j 3 + ∑ i = 1 ; i ≠ j d β i , j u i 2 u j in R N , u j (x) > 0 in R N , u j (x) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , d , where N = 2 , 3 , μ j > 0 are constants, β i , j = β j , i ≠ 0 are coupling parameters, d ≥ 2 and V j (x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials V j (x) and coupling parameters { β i , j } , without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52] , reveals new phenomenon in the case of N = 2 and d = 2 and is almost optimal for the coupling parameters { β i , j }. [ABSTRACT FROM AUTHOR]

An average-case variant of the k-SUM conjecture asserts that finding k numbers that sum to 0 in a list of r random numbers, each of the order rk, cannot be done in much less than \(r^{\lceil k/2 \rceil }\) time. However, in the dense regime of parameters, where the list contains more numbers and many solutions exist, the complexity of finding one of them can be significantly improved by Wagner's k-tree algorithm. Such algorithms for k-SUM in the dense regime have many applications, notably in cryptanalysis. In this article, assuming the average-case k-SUM conjecture, we prove that known algorithms are essentially optimal for k= 3,4,5. For k> 5, we prove the optimality of the k-tree algorithm for a limited range of parameters. We also prove similar results for k-XOR, where the sum is replaced with exclusive or. Our results are obtained by a self-reduction that, given an instance of k-SUM that has a few solutions, produces from it many instances in the dense regime. We solve each of these instances using the dense k-SUM oracle and hope that a solution to a dense instance also solves the original problem. We deal with potentially malicious oracles (that repeatedly output correlated useless solutions) by an obfuscation process that adds noise to the dense instances. Using discrete Fourier analysis, we show that the obfuscation eliminates correlations among the oracle's solutions, even though its inputs are highly correlated. [ABSTRACT FROM AUTHOR]

In this paper, an improved bound on the chromatic number of the Pancake graph Pn,n > 9, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an equitable coloring of Pn. An equitable (n-1)-coloring based on efficient dominating sets is given and optimal equitable 4-colorings are considered for small n. It is conjectured that the chromatic number of Pn coincides with its equitable chromatic number for any n ≥ 2. [ABSTRACT FROM AUTHOR]

We study the problem of distributing a set of indivisible goods among agents with additive valuations in a fair manner. The fairness notion under consideration is envy-freeness up to any good (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this article, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture of Caragiannis et al. by showing an instance with three agents for which there is a partial EFX allocation (some goods are not allocated) with higher Nash welfare than that of any complete EFX allocation. [ABSTRACT FROM AUTHOR]

This paper proposed a resilient guidance strategy for planar pursuit evasion, in which the homing guidance problem is formulated as a Stackelberg game. As is known, the most salient challenge for pursuit evasion guidance is how to ensure an interception despite both players (i.e., missile and target) actively optimize their own interest with preferably incomplete information. Toward this, we introduce a game-theoretic guidance strategy that effectively integrates incentive feedback strategy into pursuit evasion game. In particular, the proposed resilient guidance law takes the form of Stackelberg game conjecture with missile/interceptor as the designated leader, and rigorously proves that leader's interest can be best served with a properly designed feedback gain, and an interception can be made possible in spite of incomplete knowledge on target's intentions (i.e., performance index). Simulation results verify the performance of the proposed strategy. • A resilient guidance strategy for planar pursuit evasion was proposed, formulated within a Stackelberg game framework, where the missile/interceptor is the designated leader. • A game-theoretic guidance strategy that effectively integrates incentive feedback strategy into pursuit evasion game is introduced, enhancing the capability to ensure an interception despite both players (i.e., missile and target) actively optimizing their own interests with incomplete information. • It is rigorously proved that the missile/leader's interest is best served with a properly designed feedback gain, so that an interception can be made possible in spite of incomplete knowledge of the target's intentions (i.e., performance index). • A target interception scenario where the target carries out sudden maneuvers is considered in numerical simulations. Simulation results verify the performance of the proposed strategy. [ABSTRACT FROM AUTHOR]

One of the main difficulties in general relativity is the potential conflict between the weak gravity conjecture (WGC) and weak cosmic censorship conjecture (WCCC). Cosmic censorship is a basic assumption that guarantees the coherence of the gravity theory. However, this paper examines the feasibility of harmonizing the WGC and the WCCC by studying the Kerr Newman black hole surrounded by perfect fluid dark matter (PFDM) in asymptotically flat spacetimes. These two conjectures appear to be unrelated, but a recent idea proposed that they have a surprising connection. Specifically, we present a plausible set of for the WCCC in the four-dimensional framework, considering a Kerr-Newman black hole when WGC is active. We show that by applying certain restrictions on the parameters of the metric, the WGC and the WCCC can be consistent. Moreover, we explore the characteristics of the Kerr Newman black hole in the presence of PFDM for Q > M and display some fascinating figures to verify the accuracy of the WGC and the WCCC at the same time. When PFDM is absent (λ = 0), the Kerr Newman black hole has either two event horizons if Q 2 / M 2 ≤ 1 , or none if Q 2 / M 2 > 1. The latter case leads to a naked singularity, which violates the WCCC. But when PFDM is present (λ ≠ 0), the Kerr Newman black hole has event horizons depending on Q, a, and M. This means that the singularity is always hidden, and the WGC and the WCCC are satisfied. Furthermore, we prove that there is a critical value of λ , denoted by λ e x t , that becomes the extremality Kerr Newman black hole when λ = λ e x t. In this case, the black hole has an event horizon, and the WGC and the WCCC are still satisfied. We conclude that PFDM can make the WGC and the WCCC compatible for the Kerr Newman black hole and that the WGC and the WCCC concur with each other when PFDM is present. [ABSTRACT FROM AUTHOR]

In this paper we provide an identity between the determinant and other generalized matrix functions, and give a criterion for positive semi-definite matrices to satisfy the permanental dominance conjecture. As a consequence, infinitely many classes of positive semi-definite matrices satisfying the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column. [ABSTRACT FROM AUTHOR]